communication in online social networks fosters cultural...

Research ArticleCommunication in Online Social Networks FostersCultural Isolation

Marijn A Keijzer Michael Maumls and Andreas Flache

ICSDepartment of Sociology University of Groningen Grote Rozenstraat 31 9712 TG Groningen Netherlands

Correspondence should be addressed to Marijn A Keijzer makeijzerrugnl

Received 17 May 2018 Revised 20 September 2018 Accepted 2 October 2018 Published 4 November 2018

Academic Editor Roberto Natella

Copyright copy 2018 Marijn A Keijzer et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Online social networks play an increasingly important role in communication between friends colleagues business partnersand family members This development sparked public and scholarly debate about how these new platforms affect dynamics ofcultural diversity Formal models of cultural dissemination are powerful tools to study dynamics of cultural diversity but they arebased on assumptions that represent traditional dyadic face-to-face communication rather than communication in online socialnetworks Unlike in models of face-to-face communication where actors update their cultural traits after being influenced by oneof their network contacts communication in online social networks is often characterized by a one-to-many structure in that usersemit messages directly to a large number of network contacts Using analytical tools and agent-based simulation we show thatthis seemingly subtle difference can have profound implications for emergent dynamics of cultural dissemination In particularwe show that within the framework of our model online communication fosters cultural diversity to a larger degree than offlinecommunication and it increases chances that individuals and subgroups become culturally isolated from their network contacts

1 Introduction

A major premise of the Internet was that it would createa public sphere that fosters democratic deliberation andconsensus formation [1ndash3] Yet there is increasing concernthat the Internet actually reinforces processes of opinionpolarization as users interact with like-minded individuals[4] a tendency that personalization algorithms installed insearch engines and online social networks further intensify[5 6] These psychological and computational homophilybiases fragment online debate into virtual echo chambers [7]Formal models of social influence in networks are powerfultools for understanding whether and under what condi-tions communication in social networks fosters processes ofconsensus formation or opinion polarization [8] Howeverexisting models have been tailored to represent offline ratherthan online communication Here we show that takinginto account that online communication is characterizedby ldquoone-to-manyrdquo communication rather than ldquoone-to-onerdquocommunication drastically changes the predictions of oneof the most prominent models [9] Specifically we show

that the one-to-many communication regime characteristicof online communication fosters the emergence of isolatedindividuals and the formation internally homogenous butmutually dissimilar subgroups

Scholars have long recognized that online communica-tion differs in important ways from its offline counterpart(see eg [10 11]) but existing research focused on differencesthat affect within-individual processes and largely ignoredthe complexity arising from the communication betweenindividuals A classical finding from social psychology forinstance is that computer-mediated communication is muchless affected by individualsrsquo physical appearance (eg agegender and ethnicity) which frees individuals from thesocial roles associated with memberships in high or lowstatus groups [12] This it is argued increases the relativeimpact that members of low-status groups have on collec-tive dynamics decreasing intergroup conflict and fosteringconsensus formation [12 13] Likewise research showed thatonline communication allows shy individuals to overcomethe communication barriers that socially isolate them inoffline settings [14]

HindawiComplexityVolume 2018 Article ID 9502872 18 pageshttpsdoiorg10115520189502872

2 Complexity

A A

B A

(a) t = 0

A A

B A

A A

B A

(＜2) t = 1one-to-many(＜1) t = 1one-to-one

Figure 1 Illustration of the intuition that one-to-many communication fosters isolation Nodes have three characteristics (color shape andletter) that are open to influence The number of traits shared by two nodes and thus the probability that a sender exerts influence on thereceiver is shown by the number of lines connecting the nodes Panel (a) shows the initial setup before the top-left agent communicated hisshape trait either under the one-to-one communication regime (Panel b1) or the one-to-many regime (Panel b2)

In contrast to the existing research we focus on thecomplexity arising from the interaction between individualsrather than on within-individual processes To this end westudy Axelrodrsquos prominent formal model of communicationthat was developed for offline social networks (and uses theone-to-one communication rule) keeping all assumptionsabout individual behavior unchanged but implementingcommunication between actors in a way that captures typicalforms of online communication (one-to-many)With analyt-ical tools and simulation we demonstrate that this changein model assumptions drastically changes model predictionsand leads to conclusions that challenge insights from researchon within-individual processes Contrary to the sketchedfinding that computer-mediated communication fosters theemergence of consensus [12 13] we find that the onlinecommunication regime fosters the emergence of mutuallydisagreeing subgroups in our simulations Likewise whilesocial-psychological research found that online communi-cation allows some individuals to overcome social isolation[14] we demonstrate that online communication increasesthe chances that individuals get socially isolated We derivethese results using an approach that is very different fromsocial-psychological research While these studies exploredhow online communication changes the way individualsbehave and respond to each other our work demonstratesthat differences between online and offline communicationarise through merely a different communication structureIn complexity terms we find that the ldquowholerdquo changes notbecause the ldquopartsrdquo have changed but because the interdepen-dencies between the ldquopartsrdquo are slightly different

Axelrodrsquos model of the dissemination of culture is one ofthe most prominent models of consensus formation and theemergence of dissimilar subgroups It is also a typical rep-resentative of models implementing offline communicationAxelrod proposed the model to address what he perceived asa fundamental puzzle in research on social influence askingldquoIf people tend to become more alike in their beliefs attitudesand behaviorwhen they interact why do not all such differenceseventually disappearrdquo [9 pp 203] Axelrod then showed withan agent-based model how assimilation at the microlevelof individual interactions can be reconciled with culturaldifferentiation at the level of society as a whole Like mostcontributions to the literature Axelrodrsquos model representsindividuals as nodes in a network that are described by a set ofcultural traits representing individualsrsquo cultural preferences

(like preferences for styles of music literature or dress)Furthermore Axelrod implemented the so-called ldquoone-to-onerdquo communication regime where in a social encounter oneagent always communicates one cultural trait to one of hernetwork contacts This one-to-one communication regimemimics the face-to-face communication present in manyoffline contexts but it differs from a form of communicationthat is ubiquitous on the Internet and that we label ldquoone-to-manyrdquo communication When Internet users blog orpost content on online social networks for instance theycommunicate content to multiple online contacts at oncerather than to just one of them

This paper wasmotivated by the following intuition aboutthe complexity arising from one-to-many communicationConsider for illustration the network of four actors depictedin Figure 1(a) All actors are described by three cultural traitsshape (circle or square) color (black or white) and letter (Aor B) The number of lines connecting the nodes representsthe number of cultural traits the respective two nodes havein common Implementing homophily [4 15 16] Axelrodassumed that trait overlap increases the likelihood that nodeswill adopt a trait from their neighbor Suppose that the top-left agent (in Figure 1(a)) communicates his shape trait underthe two different communication regimes Under the one-to-one communication regime assumed in Axelrodrsquos model thisagent communicates his trait to one of the two agents withwhom he already shares the letter and color traits Assumethat the top-right agent is selected for interaction and thisagent accepted the trait Figure 1(b) visualizes this situationshowing the increased cultural similarity between receiverand sender As a side effect the cultural overlap betweenthe top-right agent and the bottom-left agent decreased butthe overall network remains connected Figure 1(c) showsthat a different outcome arises when agents communicateunder the one-to-many regime The same agent (top-leftagent) communicates his shape trait but let us assumenow all actors with whom he has nonzero cultural overlapadopt the communicated trait This has two consequencesthat we study in this paper First a culturally homogenouscluster forms because after the communication three actorshold identical cultural traits Communication did not onlyincrease similarity between the sender and the receivers ofthe message but also preserve the similarity between thenodes who adopted the trait Under the one-to-one regimein contrast the two nodes on the right-hand side turned

Complexity 3

less similar to each other Second the bottom-left agent nolonger shares any trait with the other agents ending upculturally isolated The fact that the bottom-left agent wasnot influenced by the top-left agent did not only exclude thatthey grew more similar but also increased the dissimilaritybetween the bottom-left agent and the other two agents asthey did adopt the cultural trait communicated by the top-leftagent Counterintuitively this stylized example suggests thatcluster formation and cultural isolation are more likely underthe one-to-many communication regime even though thereare more instances of social influence than under Axelrodrsquosone-to-one regime

The intuition illustrated in Figure 1 requires a formalanalysis for two reasons First under the online one-to-manycommunication regime the sender transmitted a trait tomultiple network contacts while there was only a single actof communication under the one-to-one regime It remainsunclear whether repeated one-to-one communication couldaccount for this apparent difference between the communi-cation regimes or not Second the figure focuses on a tinypopulation with a simple network structure leaving openwhether one-to-many communication fosters isolation alsowhen larger numbers of agents communicate simultaneously

In order to test the validity of our intuition we imple-mented a one-to-many communication regime in Axelrodrsquosmodel of cultural dissemination keeping unchanged all othermodel assumptions (thus keeping our model ldquofully alignedrdquo[17])That is we included that actors simultaneously commu-nicate a trait to their whole network at once Subsequentlythe alters decide individually whether to adopt or reject thetrait according to the rules specified in the original Axelrodmodel We compared the predictions of the new model withpredictions of Axelrodrsquos original model using analytical aswell as computational tools First we compared the twomodelsrsquo predictions for very small but analytically tractablesocial networks conducting a Markov-Chain analysis andfind that indeed one-to-many communication increases thechances that individuals become isolated Second usingcomputational methods we show that our conclusions holdalso for bigger populations variations in the structure of theunderlying social network and higher cultural complexity interms of the number of cultural traits and features Moreoverwe find that medium sized clusters emerge under one-to-many communication at a low but consistent rate

2 Literature

Axelrodrsquos model of the dissemination of culture providesa prominent explanation of the emergence diffusion andstability of distinct cultural profiles In this literature anindividualrsquos culture is defined as the set of her personalcharacteristics (eg opinions beliefs and cultural behavior)that are susceptible to social influence [9 p 206-7] Culturaldynamics unfold from the conjunction of two social forcesthe selection of culturally similar communication partnersand the social influence resulting from communication Associal influence increases cultural similarity between com-munication partners it creates in conjunction with selectiona positive feedback loop that results in the emergence of

cultural clusters that grow internally increasingly similar andas a consequence mutually dissimilar Distinct cultural clus-ters remain stable when the cultural overlap between clustersdrops to zero which rules out subsequent communicationaccording to the selection principle Axelrodrsquos model sharesthis critical assumption with many alternative models suchas the prominent models of bounded confidence [18 19] assummarized in a recent literature review by Flache et al [8]

Many contributions have extended Axelrodrsquos work [20]testing the sensitivity of his predictions to adjustments inmodel assumptions about for instance the impact of massmedia [21] institutions [22] and the scale of the culturalfeatures [23ndash25] An important advancement was the intro-duction of noise in the process of communication-partnerselection and social influence [24 26ndash28] It turned outthat allowing agents to sometimes deviate from Axelrodrsquosassumptions with a small probability can cause the systemto inevitably move towards monoculture ie perfect culturalhomogeneity Model predictions are more robust howeverwhen agents are assumed to interact only with networkcontacts that share multiple cultural traits [25 29] whennetwork ties to contacts that are culturally too dissimilarare dissolved [30] or when agents are allowed to forminstitutions bottom-up that in turn influence the agentstop-down [22] Recently Battiston et al [31] conceptualizedexchange discussion networks as a multiplex system in whichdifferent topics are discussed among different peers Multipledisseminations of culture models are layered on top ofeach other creating distinct and robust clusters of culturalidentities

Another extension to the model that can explain thepersistence of cultural diversity despite random deviations isthe so-called ldquomultilateral social influencerdquo [32] a form ofsocial influence that is similar to the concept of ldquocomplexcontagionrdquo from the literature on diffusion processes in socialnetworks [33] Unlike Axelrod who modeled influence as adyadic one-to-one process where an agent adopts a culturaltrait from a network contact Flache and Macy [32] assumedthat agents always consider the cultural traits of multiplenetwork contacts when they reconsider their cultural profileand adopt the trait that dominates in their neighborhoodThis ldquomany-to-onerdquo form of cultural communication makespredictions much more robust to noise and is the reverseof the ldquoone-to-manyrdquo communication regime that we studyhereThat is while Flache and Macy assumed that an agent isalways influenced by multiple network contacts we considerthat an individual agent exerts influence onmultiple contacts

Modelers have also incorporated assumptions aboutone-to-many communication in existing models [34 35]However while there are social influence models thatimplement communication regimes similar to the one-to-many communication that we study the literature lacksan analysis of whether and under what conditions one-to-many communication generates different cultural dynamicscompared to one-to-one communication Thus unlike earliercontributions we implement one-to-many communicationin Axelrodrsquos model keeping all other model assumptionsunchanged Next we compare predictions of the new modelwith the predictions of the original approach

4 Complexity

Table 1 Basic assumptions of the dissemination of culture model with one-to-one communication and our implementation with one-to-many communication

One-to-one communication One-to-many communication

(1) Select active agent 119894 Every time step 119905 pick an agent 119894from the population1

Every time step 119905 pick an agent 119894from the population

(2) Select communication partner Pick a neighbor 119895 of agent 119894 (not needed)

(3) Select communicated trait Pick a feature 119891 on which 119894 and 119895differ (119902119894119891 = 119902119895119891)

Pick a feature 119891 on which 119894 and atleast one neighbor 119895 differ(119902119894119891 = 119902119895119891 for any 119895)

(4) Homophilous social influence

With a probability 119901119894119895 equal to theproportion of traits that 119894 and 119895share (119902119894119891 = 119902119895119891 over 119865) let 119895 adopttrait 119902119894119891 from 119894

With a probability 119901119894119895 equal to theproportion of traits that 119894 and 119895share (119902119894119891 = 119902119895119891 over 119865) let eachneighbor 119895 adopt trait 119902119894119891 from 119894

1In the original model Axelrod describes influence to go from 119895 997888rarr 119894 but in order to make the formalization of one-to-one and one-to-many communicationcomparable we change the phrasing of influence to go from 119894 997888rarr 119895 The two implementations are mathematically equivalent

3 The Model

The aim of the present analysis is to test our intuitionthat one-to-many communication generates more isolationthan one-to-one communication To this end we comparethe predictions of Axelrodrsquos prominent model of culturaldissemination which assumed one-to-one communicationwith a novel extension of the same model that captures one-to-many communication Like in the original Axelrodmodelwe generate populations of 119873 agents Every agent has acultural profile a vector 119862119894 with 119865 nominal features with 119876possible traits Features represent cultural attributes that areopen to social influence and traits refer to the distinctivecontent of a feature for a given agent Formally

119862119894 = (1199021198941 1199021198942 119902119894119865) 119902119894119909 isin 0 1 119876 minus 1 (1)

Axelrodrsquos used a very abstract representation of agentsrsquocultural characteristics Features can represent something asbasic as the personrsquos favorite song or something as complexandmultidimensional as the personrsquos music taste Likewise itcan model the personrsquos view on abortion or her much morecomplex preference for a specific political party whichmay bea function of her view on abortion and many other aspectsIn Section 426 we study how the dynamics emerging fromone-to-one and one-to-many communication are affected bycultural complexity measured in terms of the number offeatures and traits per feature

Table 1 summarizes and compares the two variants ofAxelrodrsquos model At every time step 119905 an agent 119894 is selectedat random from the population This agent is the source ofinfluence Second in the original model one of 119894rsquos neigh-bors is selected for communication with 119894 a step that isnot necessary under one-to-many communication where 119894communicates with all of her neighbors who are open toinfluence and not yet culturally identical In Step (3) oneof the cultural features on which there is not yet consensusbetween agent 119894 and her neighbors is selected In Axelrodrsquosoriginalmodel this translates into the exclusion of all featureswhere 119894 and 119895 hold the same trait In the variant with one-to-many communication however one of the features where 119894disagrees with at least one of her neighbors is picked Unlike

in Axelrodrsquos model with one-to-one communication thisimplies that 119894might transmit a trait to one of her neighbors 119895that 119895 already adopted making this dyadic communicationineffective However similar to Axelrodrsquos model also thevariant with one-to-many communication excludes that afeature is chosen in which cultural change is impossibleas there must be at least one neighbor who disagrees with119894 on the selected cultural dimension Step (4) implementssocial influence and is therefore the part where one-to-one and one-to-many communication are implemented InAxelrodrsquos original model actor 119895 adopts the selected trait witha probability equal to the overall cultural overlap between119894 and 119895 For instance when 119894 and 119895 hold the same trait onhalf of the features then the chance that 119895 will adopt the traitchosen in Step (3) is 50 percent This implements homophilythe notion that individuals tend to be influenced by like-minded communication partners Empirical research showedthat homophily is a strong force both online and offline[4 36 37] The same principle is implemented in the newversion of the model but here every neighbor of 119895 adopts theselected trait with a probability equal to the pairwise culturalsimilarity between 119894 and the respective neighbor 119895

4 Comparison of the TwoCommunication Regimes

We compared the models with two different methods Firstwe studied small populations of only four agents describedby only three dichotomous cultural features The simplicityof this setup allowed us to conduct a detailed analysis andprovide an analytical proof using a Markov-chain analysisSecond we conducted agent-based simulations in order totest whether the conclusions from the Markov-chain analysisalso hold in more complex settings with more agents highernumbers of cultural traits and features different neighbor-hood sizes and more complex network structures Using alarger population size in the second analysis also allowedus to address what Axelrod was primarily interested incultural diversity More precisely we could test in the secondanalysis how the communication regime affects the degreeof and conditions for cultural clustering the coexistence of

Complexity 5

(1 0 0

1 0 0

1 0 0

0

0

0

)(a)Symmetrical(13-13-13s)

(1 0 0

1 0 0

0 1 0

0

0

0

)(b) Semi-symmetrical(13-13-13ss)

(1 0 0

0 1 0

0 0 1

0

0

0

)(c) Non-symmetrical(13-13-13ns)

Figure 2 Example states of the 13-13-13 classes traits on features (rows) by agents (columns)

local consensus and global diversity highlighted by Axelrodrsquosoriginal analysis

41 Markov-Chain Analysis To be able to compare the twomodels with analytical tools we first analyzed a settingthat is very simple but where the intuition outlined abovenevertheless suggests that predictions of the two modelvariants differ According to the described intuition isolationin the one-to-many model might arise when an actor 119895 is notinfluenced by a network neighbor 119894 but their joint neighborsare influenced Clusters form because an actor 119895 exerts thesame influence on multiple network contacts Testing thisintuition requires a network consisting of sender 119894 receiver 119895and at least two other receivers 119896 and 119897 that is fully connectedFurthermore we set the number 119865 of cultural features to 3as this creates sufficient variation in probability that agentsinfluence each other If two agents do not share a trait on anyof the three features their communication probability119901119894119895 = 0If they share 1 trait then 119901119894119895 = 13 If they share 2 traits then119901119894119895 = 23 and if they share all traits 119901119894119895 = 1 Finally weassumed that all three features are dichotomous (119876 = 2)

The system with 119873 = 4 119865 = 3 and 119876 = 2 has a finitenumber of cultural configurations A cultural configurationis a mapping that assigns to each of the 119865 features of each ofthe 119873 agents a value from the set of possible trait values (0or 1) The total number of possible configurations is 119876119865119873 =212 = 4096

The dynamics of this system can be fully represented asa Markov chain that assigns to every ordered pair of culturalconfigurations a probability to move from one configurationto the other within one iteration of the simulation of themodel With 4096 configurations this Markov model ofthe system is prohibitively large for an exhaustive analysesHowever we can partition the set of all configurations intosubsets called classes hereafter which have the propertythat for every ordered pair of configurations 119883 and 119884 ofwhich 119883 falls into class 1198781 and 119884 falls into class 1198782 thetransition probability from119883 to 119884 is the same Analyzing thedynamics of the Markov Chain constituted by these classesand transition probabilities between them is equivalent toanalyzing the Markov Chain of all configurations As we willshow we can reduce the system to a number of classes thatis small enough to derive analytically the probabilities thatcultural isolation arises from a random start under one-to-one and one-to-many communication respectively

To arrive at a partition of the configurations into classeswe first observe that each feature 119891 is always in exactly one ofthe three different states

Consensus All agents adopt the same trait on feature 119891According to Rule 3 of both models any future communica-tion changing this feature is excluded because consensus hasbeen reached

1-3 Split One agent adopted trait 119902 while the three others have1199021015840

2-2 Split Two agents share trait 119902 while the other two agentsadopted 1199021015840

A first classification of configurations can be obtainedfrom distinguishing configurations that have a different dis-tribution of states over the three features All configurationsthat have the same number of features in the states C (con-sensus) 13 (1-3 split) or 22 (2-2 split) fall into the same semi-class The number of distinct semi-classes can be obtainedfrom computing the number of possible outcomes if for everyfeature its state is drawn randomly and independently withreplacement from the three possible values C 13 or 12 Thusfor the case where a feature can be in 119903 = 3 different states andthere are 119899 = 3 features constituting a cultural vector thisnumber is given as the number of unordered permutationsfor a set when sampling with replacement as

(119903 + 119899 minus 1)119903 (119899 minus 1)

= (3 + 3 minus 1)3 (3 minus 1)

= 10 (2)

However a semi-class can consist of several classesthus the number of classes is larger than 10 The reason isthat transition probabilities from a configuration containingfeatures with more than one 1-3 split or 2-2 split may bedifferent depending on whether the splits separate the setof agents along the same lines or are asymmetrical Wedistinguish three degrees of symmetry within the semi-classes with more than one nonconsensus feature and assignthe labels symmetrical (s) semi-symmetrical (ss) or non-symmetrical (ns)1

For example the 13-13-13 semi-class (ie all three featurescontain a ldquo1-3 splitrdquo) consists of three different classes sym-metrical semi-symmetrical and non-symmetrical Examplesare shown in Figure 2 Even though the three configurationsare part of the same semi-class they have very different prob-abilities of communication and transition into another classThe symmetrical 13-13-13 class (13-13-13s) is an absorbingstate of the dynamic and is characterized by one cluster ofthree culturally identical agents and one isolate The isolatedagent in this class is different from the same three otherson all three features and thus no further communication ispossible The configurations 13-13-13ss and 13-13-13ns shown

6 Complexity

transition probability0 1

(a) One-to-one


(b) One-to-many

Figure 3 Markov chains for the one-to-one and one-to-many communication regimes in the119873 = 4 119865 = 3 119876 = 2model Nodes representclasses and the directed edges are colored according to between-class transition probabilities Blue nodes are absorbing classes red nodesare classes from which consensus is the only reachable equilibrium and white nodes indicate that from there multiple equilibria are stillreachable

in Figure 2 instead allow for communication between someor all agents An overview of all classes and the proportion ofstates that fall into each class is included in Appendix A

For both model variants one can identify a partition intoa small number of classes of configurations and calculate forevery pair of classes the probability that the correspondingtransition occurs within one iteration Figures 3(a) and 3(b)visualize the transition probabilities for bothmodels variantsIn the figures nodes are colored according to whether theyrepresent an absorbing class (blue) a class from whichconsensus is the only reachable equilibrium (red) or whethermore equilibria are still reachable (white)2 Edge color cor-responds to the probability that the system moves from onestate to anotherwith darker edges indicating higher transitionprobability Recursive paths (self-loops) are not shown

Both model variants have three possible absorbingclasses these are classes that once selected by the dynamicswill never be left again the consensus class (C-C-C) theisolation class (13-13-13s) and the polarization class (22-22-22s) Consensus is stable since social influence will never leadto changes in agentsrsquo features Isolation and polarization aregroup split states they are stable because actors are eitherperfectly similar or perfectly dissimilar from their neighborsIn both cases communication will never lead to changes inthe cultural features As can be seen in the transition diagramas well as in the transition matrix diagonal these are theonly classes that are ldquosinksrdquo with an out-degree of zero in theMarkov graph Given that from every other class there is pathtowards at least one of the absorbing classes we know that inthe long run the system must end up in one of the absorbingclasses

Figure 3 illustrates how isolation canmore readily emergeunder one-to-many communication For example once thesystem has reached a configuration in which one agent isldquoalmostrdquo isolated but still agrees with the three others onone single feature (13-13-22s upper-right corners of Figures3(a) and 3(b)) it is under one-to-many communication three

Table 2 Stationary distributions for the119873 = 4 119865 = 3119876 = 2modelwith two communication regimes

Communication regimeClass One-to-one One-to-manyC-C-C (consensus) 801 71322-22-22s (polarization) 064 07313-13-13s (isolation) 135 215

times as likely that this agent will end up isolated after the nextinfluence than it is under one-to-one communication3 Moregenerally using the Markov chain convergence theoremone can calculate for each of the three absorbing classesthe probability to be reached under both communicationregimes given the initial distribution of configurations Thisrequires a row vector 119902 of the initial distribution of states andthe transition matrix 119879The stationary distribution 119901lowast is thengiven by

119901lowast = 119902119879infin (3)

Table 2 reports the stationary distributions for bothmodel variants given a uniform probability of initializingthe system in any of its 4096 configurations These resultssupport our intuition that isolation is a more likely outcomeof the dynamics of cultural influence under the one-to-many communication regime than under the one-to-onecommunication regime More precisely we find that theprobability of the outcome of cultural isolation is about 16times higher under one-to-many communication (215 versus135) We also observe that one-to-many communicationreduces the likelihood of consensus to emerge and thusincreases the likelihood that cultural diversity persists despitesocial influence Next we turn to exploring how one-to-many communication affects the likelihood and persistenceof isolation and cultural clustering in larger populations

Complexity 7

42 Isolation and Cultural Clustering in Bigger PopulationsThe Markov-chain analysis supported our intuition thatisolation and polarization are more prevalent in the one-to-many communication regime However with only 4 agentswe can not distinguish polarization (separation in exactly twoculturally opposed subgroups) from cultural clustering into alarger number of distinct local clusters To test whether andunder which conditions our analytical finding is robust andgeneralizes to cultural clustering also in larger networks weconducted computational experiments with bigger popula-tions always starting from a random initial assignment oftraits and 1000 independent replications per experimentalcondition All simulations were executed until dynamics hadreached an equilibrium

In the following subsections we first compare one-to-oneand one-to-many communication in three different networkconfigurations First we focused on a regular torus networkas this is the framework that Axelrod used (Section 421)Second we compared the two communication regimes inring networks with different degrees of network transitivityin order to test whether themicro-level intuition illustrated inFigure 1 is indeed responsible for the macro differences thatwe observe in bigger populations (Section 422) Using thering networks we also varied the size of the agentsrsquo neighbor-hoods to test whether or not cultural clustering and isolationpersist when individualrsquos communication networks growbigger under one-to-many communication (Section 423)Next we studied spatial random graphs as these networkshave been argued to mimic human social networks betterthan torus networks and ring networks (Section 424) Sub-sequently we describe ideal-typical simulation runs underone-to-one and one-to-many communication to illustratedifferences (Section 425) and replicate our main findings forpopulations consisting of agents with more features (119865) and ahigher number 119876 of possible traits per feature

421 Population Size Effects There are at least two reasons forincreasing the number of agents in the model First alreadyAxelrod found that monoculture (perfect cultural consensus)is virtually unavoidable once population size exceeds a criticalthreshold [9 pp 214-5] because dynamics last longer in big-ger populations This increases chances that two subgroups Aand B that have grownmaximally dissimilar at somemomentrestart communication because one agent adopted a traitfrom a third subgroup C that increased cultural similaritybetween A and BThis finding raises the question whether thedifferences between the two communication regimes persistwhen bigger populations are assumed Second the aim ofthe present analysis is to contribute to the development ofa valid representation of online communication a settingwhere huge numbers of individuals interact

Manipulating the communication regime whilst keepingall remaining characteristics of Axelrodrsquos model unchangedwe first compared the two models in the same cellular-worldstructure that Axelrod assumed in his seminal work and thatmany follow-up studies adopted (eg [9 26 32]) That iswe assumed that 119873 agents are distributed over a wrappedsquare lattice (a torus) such that every agent occupies onecell All agents are linked to their neighbors in the so-called

4 9 16 25 36 49N

64 81 100

oneminustominusoneoneminustominusmany

00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10

Share of monoculture

Figure 4 Effect of population size 119873 on the share of runs withat least one isolate (blue lines) and share of runs characterized bymonoculture (red lines) in a torus network with 119865 = 3 119876 = 2 1000replications per condition measured at equilibrium

ldquoMoore neighborhoodrdquo and can thus interact with eightother agents4 The first simulation experiment focused onpopulations characterized by a torus network and agentswith three cultural features with two possible traits To studypopulation-size effects we created populations with 1198982 = 119873agents and varied 119898 between 2 and 10 In Appendix B weshow that our findings are robust when 119898 is increased to 30which translates into populations of 900 agents

Figure 4 compares the two communication regimes interms of the share of runs that ended with at least oneisolated agent An isolate is defined as an agent that ismaximally different from all of her network contacts Theblue lines show that isolation still occurs under one-to-manycommunication even in larger populations while isolationvirtually disappears in the one-to-one regime Under theone-to-many regime isolation is most likely in very smallpopulations but once population size exceeds 36 the modelwith one-to-many communication generates a constant shareof about 12 percent of the runs that are characterized byisolation This finding was confirmed by simulations withpopulations of 900 agents (119898 = 30 see Appendix A)

Figure 4 furthermore shows that the proportion of runsthat end in monoculture (see the red lines) decreases with119873under one-to-many communication while the share of runsgenerating monoculture increases in population size underthe original model Axelrod already found that the originalmodel implies more monoculture in larger populations [9] aresult that generalizes to various extensions of themodel (eg[26 30]) Axelrod deemed this a counterintuitive findingconfronting it with contradictory empirical evidence from astudy of language diversity on islands in the South Pacificwhich found that there is more language diversity on largerislands Our results here suggest that one-to-one communi-cation plays an important role in the generation of Axelodrsquoscounterintuitive finding In his original model dynamics

8 Complexity

N = 81 N = 100

N = 36 N = 49 N = 64

N = 9 N = 16 N = 25

1 9 16 25 36 49 64 81 100 1 9 16 25 36 49 64 81 100

1 9 16 25 36 49 64 81 100

01

1

10

100

01

1

10

100

01

1

10

100

Cluster size

Shar

e of r

eplic

atio

ns


Figure 5The share of replications that contain a given cluster size by119873 and interaction regime All replications in torus network with 119865 = 3119876 = 2 1000 replications per condition

generate monoculture in big populations because whenever aculturally homogenous region begins to form the emerginglocal consensus can be disrupted by a single communicationevent of one member of the region with an outside sourceof influence In large populations these disruptions are morelikely simply because dynamics last longer than in small pop-ulations Such outside influences are also possible under one-to-many communication However the main difference isthat one-to-many communication offersmanymore possibil-ities how a ldquodeviantrdquo is reached by influences from membersinside of the emergent region to which the deviant belongsIn Axelrodrsquos original model the algorithm always randomlypicks two communication partners 119894 and 119895 (see Steps (1) and(2) in Table 1) which implies that the chance that a deviant 119895is influenced back by a neighbor who belongs to the culturalregion is only 18 in a populationwithMoore neighborhoodsWith our implementation of one-to-many communication119895 will always be targeted by 119894 as 119894 exerts influence on allneighbors This greatly increases the robustness of culturalregions Further support for this interpretation is given bysimilar findings that Flache and Macy [32] obtained with amodel assuming many-to-one communication

To test whether one-to-many communication fosters notonly isolation but also the formation of clusters Figure 5shows how population size affects the relative frequency ofclusters of different sizes Even though clusters of size oneand size 119873 are consistently the most likely outcome to begenerated by the model there is a remarkable difference

between the two communication regimes Under one-to-onecommunication the occurrence of ldquomedium sizedrdquo clusters(those larger than one and smaller than 119873) diminishesas 119873 increases whereas one-to-many communication doesgenerate clusters of all different sizes at all levels of119873 In thesimulation runs with 119873 = 100 for example we found thatwith one-to-one communication 12 of the replication runsend with at least one isolate and 16 of the runs generatemedium sized clusters Under one-to-many interaction theproportion of runs with at least one isolate rises to 77and medium sized clusters appear in equilibrium for 494of the runs Moreover these medium sized clusters seemto emerge at a similar rate Any given cluster size between2 and 99 has an average probability of exactly 100 (SD= 046) to appear in a given run (compared to 002under one-to-one interaction) This demonstrates how one-to-many communication stabilizes cultural diversity andclustering Both communication regimes typically generatecluster size distributions with peaks at both ends of thescale (at cluster size one or 119873) Independent of populationsize however one-to-one communication generates moremonoculture less isolation and less clustering than one-to-many communication

Figure 6 informs about the effect of the communicationregime on the relative size of the biggest subgroup in thepopulation (119878119898119886119909119873) a standard outcome measure in theliterature The figure shows that the few runs with biggerpopulations under the one-to-one regime that did not end

Complexity 9

oneminustominusone oneminustominusmany

9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N

Figure 6 Homogeneity after convergence in a torus network with 119865 = 3 and 119876 = 2 by population size 119873 and communication regimeBoxplots are shown in blue and averages indicated with red marks 1000 replications per condition

in monoculture were always characterized by one very bigcluster Under one-to-many communication the size of thebiggest subgroups can be much smaller in contrast

422 Effects of Network Transitivity on Cultural DiversityFigure 1 illustrates a key element in our reasoningwhy one-to-many communication fosters both isolation and clusteringAccording to our intuitive argument one-to-many commu-nication generates isolation and cluster formation when anagent is not adopting a trait from a network contact buttheir joint network contacts do adopt the trait and thereforegrow similar to each other and dissimilar to the agent whowas not influenced Such a series of events can only occurhowever when the sender and the agent that becomes isolatedhave common friends In other words a high degree oftransitivity in the sense that many network triads are closed(actor a is connected to b b is connected to c and c isconnected to a) can be expected to contribute to both culturalclustering and isolation and amplify the difference betweenthe regimes To test whether transitivity is indeed responsiblefor the differences between the two communication regimeswe compared populations characterized by different degreesof network transitivity

We replicated parts of the analyses presented in theprevious section manipulating the degree of transitivity inthe populationrsquos social network5

In this simulation experiment we focused on populationsof 100 agents (119873 = 100) holding three features (119865 = 3)that could adopt two traits (119876 = 2) To manipulate theaverage transitivity in the network we created symmetric ringnetworks where agents were connected to the four closestneighbors to the right and to the left [38] In the resultingnetwork all agents had the same degree (119896 = 8) just asin the simulations with the torus network Furthermorethe network was characterized by a very high degree oftransitivity as connected agents tend to be connected tothe same nodes (transitivity in this network is 064) Nextwe rewired network links following the algorithm proposedby Maslov and Sneppen (2002) which decreases networktransitivity while preserving the degree distribution TheMaslov-Sneppen rewiring algorithm first picks two edges

000

025

050

075

100

Tran

sitiv

ity

1eminus2 1eminus1 1eminus0 1e11eminus3MS Rewiring Share

Figure 7 Average transitivity by share ofMaslov-Sneppen rewiringObserved in Watts-Strogatz graph with 119896 = 8 for 119873 = 100 100replications per condition

119860 larrrarr 119861 and 119862 larrrarr 119863 making sure that 119860 notin 119862119863and 119861 notin 119862119863 and that 119860 999432999426999456 119863 and 119861 999432999426999456 119862 If any ofthese conditions is not met a new pair of edges is pickedOtherwise the algorithm removes the links 119860 larrrarr 119861 and119862 larrrarr 119863 and adds 119860 larrrarr 119863 and 119861 larrrarr 119862 This procedureis repeated until the algorithm has successfully rewired ashare 119877 of the total number of edges in the graph Westudied the two communication regimes for different shares119877 of Maslov-Sneppen rewiring namely 119877 = 10minus1198941010119894=minus30Figure 7 visualizes how the share of rewired links translatesinto network transitivity Transitivity is defined as the shareof closed triplets in the network or formally

Transitivity =3 times number of closed triangles

number of triplets (4)

Figure 8 depicts the association between transitivity andthe relative size of the biggest subgroup in the populationThebox plots show that under the one-to-one communicationregime network transitivity is not meaningfully related tothe outcome measure In contrast the bottom panel of

10 Complexity


01 02 03 04 05 06 01 02 03 04 05 0600

02

04

06

08

10

Transitivity

S max

N

Figure 8 Homogeneity after convergence in a Watts-Strogatz network with 119873 = 100 119865 = 3 and 119876 = 2 by degree of transitivity andcommunication regime Boxplots are shown in blue with bin width set to 0037 and a Loess curve is shown in red The graph with 100replications per experimental condition but networks with transitivity values between15 and 05 are less frequent as Figure 7 shows

the figure shows a strong association under the one-to-many communication regime This supports our conjecturethat one-to-many communication fosters cultural clusteringonly in networks characterized by a sufficient amount oftransitivity Note that the scatter plots on the very left and onthe very right of the figure represent more simulation runsas the used rewiring algorithm generates more networks withvery high and very low transitivity (see Figure 7)

423 Varying Neighborhood Size So far we have studiednetworks where all agents had a degree (119896) of eight becausethis resonates with Axelrodrsquos work However we also testedwhether one-to-many communication fosters cultural isola-tion also when agents have more than eight network contactsTo this end we studied populations of 49 agents interactingin ring networks as described in Section 422 Agents weredescribed by three features and two traits per feature To studyeffects of agentsrsquo degree we varied the number 119896 of neighborsfrom 2 to 48 in steps of (2) conducting 1000 independentreplications per condition Thus under 119896 = 2 the networkwas a perfect ring where every agent had one neighbor to theleft and one to the right Under 119896 = 4 agents were connectedto the two closest neighbors to the right and to the left andso on A degree of 119896 = 48 implemented a complete graph

Figure 9 informs about how agentsrsquo degree affected howoften we observed cultural isolation or monoculture underthe two communication regimes In line with Axelrodrsquos workthe solid lines show that under the classical one-to-one com-munication dynamics tend to generate monoculture whenagents have bigger neighborhoods The figure shows only asmall difference between the two communication regimes invery sparse networks (119896 = 2) which supports our conjecturefrom Section 422 that network transitivity is a necessaryrequirement for generating more cultural clustering underone-to-many communication A ring network with 119896 = 2 isa periodic line network with zero triplets As a consequencerejecting a trait communicated by a neighbor does not makeagents more dissimilar from their other neighbor whichimplies that the mechanism responsible for isolation underone-to-many communication (see Figure 1) is not activated


00

02

04

06

08

10


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

8 12 16 20 24 28 32 36 40 44 484Degree

Figure 9 Effect of degree 119896 on the share of runs with at least oneisolate (blue lines) and share of runs characterized by monoculture(red lines) in a Watts-Strogatz graph with 119873 = 49 119865 = 3 119876 = 21000 replications per condition

In contrast Figure 9 shows stark differences betweenthe two communication regimes when agents have biggernetwork neighborhoods Unlike Axelrodrsquos original modelthe model with one-to-many communication predicts thatmonoculture is less likely when degree is increased As 119896increases also the number of closed triads in the networkrises which sets into motion the isolation mechanism As aconsequence the proportion of runs ending in monoculturedrops to about 065 under one-to-many communication Inabout half of the simulation runs with a high degree that didnot end in monoculture there was at least one isolate

Figure 10 illustrates how degree affected the relative sizeof the biggest cultural cluster in the network Under one-to-many communication the average size of the largest clusterdecreases as degree rises from 2 to 8 However the averagecluster size rises again when degree is increased further

Complexity 11


2 8 14 20 26 32 38 44 50 2 8 14 20 26 32 38 44 5000

02

04

06

08

10

Degree

S max

N

Figure 10 Effect of degree on relative size of the biggest cultural cluster in a Watts-Strogatz graph with119873 = 49 119865 = 3 and 119876 = 2 by degreeand communication regime Boxplots are shown in blue Averages are shown with red marks 1000 replications per condition

Nevertheless even when agents have very high degreethere remains a noticeable difference between one-to-oneand one-to-many communication We believe that the non-monotone effect of degree under the one-to-many regimeresults from the interplay of two processes On the one handa higher degree increases the proportion of closed triadsin the network fostering the extent to which one-to-manycommunication can produce cultural clustering and isolatesOn the other hand a higher degree also increases the shareof the population to which an agent is directly exposed Thelarger this share the less likely it is that an agent disagreeswith all network neighbors The resulting cultural influencepushes the population towards more consensus as alreadydemonstrated byAxelrodThe combination of both processesgenerates a dynamic in which cultural clustering peaks ata degree of about 6 with lower levels of cultural clusteringobserved at both lower and higher degrees

424 Spatial Random Graphs Considering that both torusnetworks and the rewired ring networks are somewhatartificial network topologies we also studied spatial randomgraphs as these networks have been argued to mimic thestructure of human social networks [39] In particular spatialrandom graphs exhibit many features of real social networkssuch as low tie density short average geodesic distance ahigh level of transitivity a positively skewed actor-degreedistribution and a community structure [40] We conducteda third simulation experiment to test whether the differencesbetween one-to-many and one-to-one communication foundwith the torus networks also appear under these less con-trolled but more realistic conditions Like in the previoussimulation experiment we assumed that agents are describedby three features (119865 = 3) that could adopt two traits (119876 =2) We manipulated population size in the same way asin Section 421 and conducted 1000 independent runs perexperimental condition

We initialized the network in two steps First all agentswere randomly assigned two real numbers from the set [0 5]that defined their position on a 5 times 5 plane Subsequently welooped over all agents creating 119896 ties probabilistically withagents with whom they did not share a tie yet Whether a

9 16 25 36 49 64 81 100N


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10


Figure 11 Effect of population size 119873 on the share of runs withat least one isolate (blue lines) and share of runs characterized bymonoculture (red lines) in a spatial randomgraphwith119865 = 3119876 = 21000 replications per condition

tie between 119894 and 119895 was created depended on the Euclideandistance between the two agents on the plane (119889119894119895) and theparameter 119910 that controls the strength of the relationshipbetween distance and the probability to form a tie We set 119896 =8 such that each agent had a neighborhood of at least eightneighbors6 The probability that a tie was formed dependedon the value of 119891(119910 119889119894119895) proportional to the sum of thisfunction over all possible 119895rsquos where

119891 (119910 119889119894119895) = exp (minus119910 [119889119894119895]) (5)

The resulting social networks are characterized by atransitivity value of 0523 on average which is slightly moretransitive than the torus graph with Moore neighborhoods(transitivity is 0429) that we studied in Section 421 Fig-ure 11 visualizes how population size affected the share ofruns ending in monoculture (red lines) and the share ofruns where the population comprised at least one isolated

12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N

Figure 12 Homogeneity after convergence in a spatial random graph with 119865 = 3 and119876 = 2 by number of agents and communication regimeBoxplots are shown in blue and averages indicated with red marks 1000 replications per condition

agent in equilibrium (blue lines) Figure 12 shows how therelative size of the biggest cultural subgroup was affected bypopulation size and the communication regime Both figuresare markedly similar to the two corresponding figures forthe torus networks showing that our earlier findings arecorroborated also when a more realistic network structure isassumed

425 Typical Simulation Runs Figure 13 shows one typicalsimulation run for each communication regime Under Axel-rodrsquos original one-to-one regime (Figure 13(a)) one can seethat the culture that eventually dominates does not diffusefromone strong cluster In all snapshots the dominant cultureis present in all regions of the network For the lionrsquos share ofthe total body of simulation events about 13 of all attemptedcommunication events result in a change of culture by anagent In the last stage (between Snapshots 3 and 4) thisrate drops to approximately 17 as the dominant clusterassimilates the last deviants

The typical dynamics under one-to-many communi-cation differ as Figure 13(b) demonstrates Between theoutset and the second snapshot the dynamics generate threeclusters each located in a distinct region This happens at ahigh rate of about 1 cultural adjustment per simulation event7As a population with three cultural subgroups can never bestable under 119876 = 2 dynamics continue until two culturalgroups remain The rate of 119904119905 drops to 15 until convergingto a situation with a majority cluster (119873 = 78) one minoritycluster (119873 = 21) and one isolate The isolate (located atthe bottom right of the graph) has been locked inside themajority cluster from a very early stage and remains isolatedfrom communication with other clusters throughout the restof the run

426 Cultural Complexity Themain innovation of Axelrodrsquosmodel was to show how cultural diversity can emergeand persist despite relentless pressures on individuals toassimilate to cultural influence Axelrod and many follow-up studies also demonstrated how in the framework of thismodel stable cultural clustering is a feasible outcome only

in a particular ldquosweet spotrdquo in the parameter space in whichthe cultural space is not too complex meaning that neither119865 nor 119876 are too large If the cultural space consists of toomany different features (119865) this increases the chances thatneighboring agents happen to agree on at least one of themby random chance exacerbating the emergence of culturalboundaries and thus promoting monoculture If there aretoo many different traits per feature (119876) it is unlikely thattwo neighboring agents happen to have the same trait at theoutset which precludes interaction between them and entailscultural anomie [9 26]

We wanted to know whether our model can replicatethese fundamental results of Axelrodrsquos model under bothcommunication regimes to establish that besides the dif-ferences we have shown the two communication regimesgenerate consistent behavior For the region where culturalclustering is feasible according to Axelrodrsquosmodel we wantedto know whether the larger degree of cultural isolation andcultural clustering for the one-to-many regime generalizes toa broader range of parameter values for 119865 and 119876 than thosewe have used hitherto For this purpose we compared the twocommunication regimes under different assumptions aboutthe complexity of the cultural space

Figures 14 and 15 identify the region in which culturalclustering occurs both for Axelrodrsquos original model and forthe model with one-to-many communication Our resultsshow that clear differences between the communicationregimes occur throughout the region in which Axelrodrsquosoriginal model navigates in between anomie and mono-culture In this region the one-to-many regime producesmore cultural isolation and more cultural clustering thanone-to-one particularly when both the number of featuresis small (119865 = 3 or 119865 = 5) and the number of traitsis small or intermediate (depending on 119865) With high 119865or high 119876 the behavior known from Axelrodrsquos originalmodel is replicated also by the one-to-many version In thisregion the forces pushing towards monoculture or isolationlargely overwhelm the distinct effects of the communicationregime and strongly reduce the differences between themNevertheless even in those conditions we find a consistent

Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991

(a) One-to-one (b) One-to-many

Figure 13 Typical runs under the two communication regimes for spatial random graphs with 119873 = 100 119865 = 3 and 119876 = 2 Every distinctcombination of traits is visualized with its own unique color The top graphs show the initial setup and the bottom graphs show the twopopulations in equilibrium The remaining graphs visualize the distribution of cultural traits after 33 and 66 of the number of simulationevents needed to reach equilibrium 119905 is the number of simulation events and 119904 is the number of events where an agent adjusted her set ofcultural traits

but very small difference in the expected direction morecultural clustering and more isolates under the one-to-manyregime This supports our observation that one-to-manycommunication generates different influence dynamics thanone-to-one communication in those areas of the parameterspace where cultural diversity can be sustained at all underAxelrodrsquos model

5 Discussion and Directions for Future Work

Public debate about the role that online social networkspersonalization algorithms and fake news played in recentpolitical events such as Brexit and the election of DonaldTrump demonstrate that there is a need for a valid model ofinfluence dynamics in online contexts While the literature

14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q



Figure 14 Effect of the number of features (119865) and traits per features (119876) on the share of runs with at least one isolated agent in equilibrium(blue lines) and the share of runs ending with perfect monoculture (red lines) All simulations with a torus network with119873 = 49 agents and100 replications per condition

already provides a rich arsenal of formal models our analysesdemonstrated that it can bemisleading to readily adopt mod-els developed for communication dynamics in offline worldsto the analysis of online contexts In particular we comparedone-to-one communication a communication regime imple-mented in many models of offline communication with one-to-many communication which seems to be a more plausiblerepresentation of communication in online contexts such asblogs and online social networks like Twitter and FacebookWe reasoned that one-to-many communication fosters isola-tion and the emergence of cultural clusters because an agentwho happens to not be influenced by amessage received froma network contact does not only fail to grow more similarto the source of the message In addition the agent alsogrows more dissimilar to those contacts of the sender whowere influenced by the message and adopted the trait of thesource Building on Axelrodrsquos cultural-dissemination model[9] we implemented one-to-many communication where asender emits one message across his entire local networkrather than just a single network contact We started with aMarkov-Chain analysis of a simple but tractable part of theparameter space (119873 = 4 119865 = 3 119876 = 2) and found supportfor our conjecture that one-to-many communication fostersthe emergence of isolated individuals as well as polarizationNext we conducted a series of simulation experiments to

demonstrate (1) that one-to-many communication fostersthe isolation also in bigger populations (2) that networktransitivity fosters the emergence of isolated individualsand cultural clusters and (3) that these findings hold fornetwork topologies that mimic the structure of real socialnetworks

These findings add a new perspective to research on dif-ferences between online and offline communication Earlierresearch was inspired by a psychological perspective andfound that individuals are not affected by the physical appear-ance of their communication partners when communicationis mediated by a computer [12 13] As a consequence whencommunicating online individuals neglect the social rolesassociated with memberships in high or low status groupswhich decreases intergroup conflict and fosters consensusformation In contrast to this within-individual perspectivewe focused on between-individuals effects showing thatdifferences between online and offline communication maynot only arise from the fact that individuals behave differentlywhen they communicate online or offline We demonstratedthat differences between online and offline communicationcan arise from differences in communication structurebecause in many online settings individuals communicate tomultiple receivers at the same time This difference in theway communication is structured in many online settings

Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15

Figure 15 Effect of the number of features (119865) and traits perfeatures (119876) on the relative size of the biggest cultural cluster inthe population All simulations with a torus network with 119873 = 49agents Results are averaged over 100 replications per condition

turned out to foster cultural isolation and clustering ratherthan consensus formation

While our results support our conjecture that assumingone-to-one communication in models of online settings canlead to false conclusions there is reason to expect thatalso the model that we studied may still deviate in criticalways from communication in real online settings Futuretheoretical work should therefore explore further to whatextent existing models can capture important features ofonline communication and which further model develop-ments are needed for that purposeWe propose three possibledirections

First a potentially important difference between Axel-rodrsquos model and our extension on the one hand and Internetcommunication on the other hand is that online networkties are flexible On the one hand intuition and earliermodeling work suggests that making networks dynamic willfoster cultural diversity as isolated agents and subgroups willcut off ties to their dissimilar network neighbors [30] Thisshould further decrease chances that isolates are influencedby former contacts On the other hand the Internet makes iteasy to identify and connect to like-minded individuals evenwhen they are geographically very distant [41] This mightallow isolated individuals and subgroups to join clustersthat still communicate with individuals who are similar totheir former connections and thus act as a bridge overthe cultural divide Given these competing intuitions futureresearch is needed to explore the conditions under whichdynamic networks foster isolation under the one-to-manycommunication regime

Second future theoretical research should explore popu-lations that are more heterogeneous For instance empiricalresearch showed that the degree distribution of the Facebookgraph is skewed [42 pp 4] which suggests that some usersmay be more effective than others in spreading culturalattributes across the graph [43] Future research shouldtherefore study how variation in neighborhood sizes affectscultural dynamics Furthermore Internet users differ intheir online activity Research showed for instance that onFacebook politically active users emit more online contentthan users who are not politically engaged [44] It is an openquestion how these forms of heterogeneity affect isolationdynamics under the one-to-many communication regime

A third important direction for future research is thestudy of one-to-many communication with alternative mod-els of social influence Unlike Axelrodrsquos model many alterna-tive approaches represent cultural attributes on a continuousscale and not as distinct categories [8] Many politicalopinions for instance tend to vary between extremes and arethus better described by metric scales Models of continuousopinion dynamics can also capture more complex social-influence processes such as gradual opinion-adjustments[45] negative influence exerted by too dissimilar sources[46] and the reinforcement of opinions when two actorscommunicate persuasive arguments that support each othersrsquoviews [47] Future research should explore whether andunder what conditions assuming one-to-many communi-cation alters the predictions of these models We expectthat the mechanism responsible for isolation and clusteringunder the one-to-many regime is activated also in modelsassuming continuous cultural attributes If an actor refusesto be influenced by a communication partner he does notonly refuse to growmore similar to this actor In addition theactors grows more dissimilar to those joint network contactsthat were influenced and therefore were pulled closer to thesources of communication

Another important avenue of future research is toempirically test our theoretical prediction that one-to-manycommunication fosters isolation and cluster formation Wepropose a three-step design that resembles the structure ofthe theoretical analysis in this paper First we propose tostudy the minimal case that we explored with analytical toolsin a computerized laboratory environment with four humansubjects discussing their stance on three binary issues In thissetting one can manipulate whether subjects communicatein pairs (one-to-one) or emit messages to all participantsat once (one-to-many) With this experimental design onecan also test our theoretical prediction against the findingfrom the psychological literature that computer-mediatedcommunication fosters consensus formation in demograph-ically diverse groups In particular it would be interestingto test whether the integrating effects of computer-mediatedcommunication are stronger orweakerwhen communicationis implemented according to the one-to-one or to the one-to-many regime Second laboratory experiments are alsoa fruitful approach to compare the two communicationregimes in bigger populations Our theoretical analyses sug-gest that these experiments should focus on social networkscharacterized by high clustering and settings with relatively

16 Complexity

Table 3 Classes and their number of configurations for the119873 = 4119865 = 3 119876 = 2model

ClassNumber of statesincluded in the

class

Share of statesincluded in the

classC-C-C (consensus) 8 00222-C-C 72 01822-22-Cs 72 01822-22-Cns 144 03522-22-22s (polarization) 24 00622-22-22ss 144 03522-22-22ns 48 01213-C-C 96 02313-13-Cs 96 02313-13-Cns 288 07013-13-13s (isolation) 32 00813-13-13ss 288 07013-13-13ns 192 04713-22-C 576 14113-13-22s 288 07013-13-22ssa 288 07013-13-22nsb 576 14113-22-22s 288 07013-22-22ns 576 141Total 4096 1000a The outliers on the 1-3 split features are members of the same group onthe 2-2 split feature b the outliers on the 1-3 split features are members ofdifferent groups on the 2-2 split feature

small cultural complexity as the differences between the tworegimes were strongest under these conditions Third onemight try to test macro predictions in the field comparinginfluence dynamics in online communities with differentlocal network structures Contrary to intuition our resultssuggest that the chances that individuals turn culturallyisolated are higher when their local network is characterizedby high transitivity

There is strong public and scholarly interest on the effectsof communication in online worlds On the one hand ourresults illustrate that the formal analysis of abstract modelscan contribute to exploring the complexity of online commu-nication systems On the other hand our findings also showthat pundits experts scholars political decision makers andalso developers of online communication systems need tobe very careful when reasoning about the consequences ofonline communication Being based on modeling work andempirical studies focused on offline settings the currentscientific state of the art does not yet allow drawing reliableconclusions about the effects of online communication onsocietal processes of consensus formation and opinion polar-ization

Appendix

A Classes in the 119873=4 119865=3 119876=2Model

Table 3 presents the number and proportion of unique stateswithin each class

000

025

050

075

100

oneminustominusone oneminustominusmanyRegime

S max

N

Figure 16 Homogeneity after convergence in a torus network with119873 = 900 119865 = 3 and 119876 = 2 by communication regime Boxplotsare shown in blue and averages indicated with red marks 100replications per condition

0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity

Figure 17 Density and rug plot of cluster sizes after convergence ina torus network with119873 = 900 119865 = 3 and119876 = 2 by communicationregime 100 replications per condition

B Replication with Large Populations

In Section 421 we studied the effect of population sizeconducting simulations with populations of up to 100 agentsIn addition we also analyzed populations of 900 agents (30times30 grid) conducting 100 independent replications per com-munication regime Under the one-to-one interaction regimeall replications ended in complete homogeneity Under theone-to-many interaction regime only 23 replications endedin monoculture and of the remaining 77 replication runs 26generated at least one isolate

Figures 16 and 17 compare model outcomes under theone-to-one and the one-to-many communication regimesBoth figures show that the results found with smaller popula-tion are obtained also in substantially bigger networks one-to-many communication generates more cultural isolationand diversity than one-to-one communication

Complexity 17

Data Availability

No empirical data were used for this study

Disclosure

An earlier version of this paper has been presented at theXXXVIII Sunbelt conference

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to thank the Center for InformationTechnology of the University of Groningen for their supportand for providing access to the Peregrine high performancecomputing cluster

Endnotes

1 Nonconsensus features of the same type are symmetricalif the agents who agree on one feature also agree on theother If there are three nonconsensus features of thesame type and only two features are aligned we labelthis class semi-symmetrical In principal the alignmenton features of different types does not matter for stateclassification with the exception of the 13-13-22 semi-class where the two 1-3 split features are not symmetricalHere we label the class semi-symmetrical if the outlierson the 1-3 split features are members of the same groupon the 2-2 split feature and non-symmetrical if they arenot

2 Consensus is inevitable in the red classes because thereis at least one feature on which the agents have reachedconsensus As a consequence all pairs of neighbors willalways exert influence on each other with a positiveprobability This will eventually generate consensusLikewise it is not possible that a state of consensus onone or more features can be left

3 The transition probabilities for going from the 13-13-22s to the 13-13-13s class are 119901 = 33 under one-to-many communication and 119901 = 11 under one-to-onecommunication

4 Axelrod first used the smaller ldquoVon Neumannrdquo neigh-borhoods but also tested the robustness of his resultswith a ldquoMoorerdquo neighborhood identical to the one weemploy

5 Besides manipulating transitivity the implementedmethod also creates between-node heterogeneity in theirnetwork centrality and decreases the average path lengthin the graph This might in turn affect the dynamics ofour model However due to the inherit interrelatednessof network descriptive statistics there is no methodof manipulating transitivity without changing otheraspects of the network structure

6 Every 119894 formed eight ties but a 119895 that already possessedeight ties was not excluded from the set of 119894rsquo s potentialneighbors

7 It is not possible to compare these rates between commu-nication regimes without postprocessing As a senderrsquoswhole neighborhood (of 8 or more agents) can be influ-enced in a single simulation event in the one-to-manymodel a conservative approximation could be madeby dividing the number of successful communicationevents over the number of iterations times 8 Howeveronly the neighbors for whom 0 lt 119904119894119898119894119897119886119903119894119905119910119894119895 lt 1 can beinfluenced and this number varies locally as well as overtime

References

[1] J Habermas ldquoCivil society and the political public sphererdquoin Between Facts and Norms pp 329ndash387 The MIT PressCambridge UK 1998

[2] AGimmler ldquoDeliberative democracy the public sphere and theinternetrdquo Philosophy amp Social Criticism vol 27 no 4 pp 21ndash392001

[3] Y Benkler ldquoDegrees of freedom dimensions of powerrdquoDaedalus vol 145 no 1 pp 18ndash32 2016

[4] M McPherson L Smith-Lovin and J M Cook ldquoBirds ofa feather homophily in social networksrdquo Annual Review ofSociology vol 27 pp 415ndash444 2001

[5] E Pariser The Filter Bubble What the Internet Is Hiding fromYou Penguin London United Kingdom 2011

[6] E Bakshy S Messing and L A Adamic ldquoExposure to ide-ologically diverse news and opinion on Facebookrdquo AmericanAssociation for the Advancement of Science Science vol 348 no6239 pp 1130ndash1132 2015

[7] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002

[8] A FlacheMMas T Feliciani et al ldquoModels of social influenceTowards the next frontiersrdquo Journal of Artificial Societies andSocial Simulation vol 20 no 4 2017

[9] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[10] A Flache ldquoHowMay Virtual Communication Shape Coopera-tion in aWork Team A formal model based on social exchangetheoryrdquo Analyse amp Kritik vol 26 no 1 pp 258ndash278 2004

[11] S-S Wong and R M Burton ldquoVirtual teams what are theircharacteristics and impact on team performancerdquo Computa-tional amp Mathematical Organization Theory vol 6 no 4 pp339ndash360 2000

[12] T Postmes and R Spears ldquoBehavior online Does anonymouscomputer communication reduce gender inequalityrdquo Person-ality and Social Psychology Bulletin vol 28 no 8 pp 1073ndash10832002

[13] S Schumann O Klein K Douglas and M Hewstone ldquoWhenis computer-mediated intergroup contact most promisingExamining the effect of out-group membersrsquo anonymity onprejudicerdquo Computers in Human Behavior vol 77 pp 198ndash2102017

[14] L D Roberts L M Smith and C M Pollock ldquorsquoU ra lot bolderon the netrsquo Shyness and Internet userdquo in Shyness Development

18 Complexity

Consolidation and Change W R Crozier Ed pp 121ndash138Routledge New York NY USA 2000

[15] P Lazarsfeld and R Merton ldquoFrienship as a social processa substantive and methodological analysisrdquo in Freedom andControl in Modern Society M Berger T Abel and C H PageEds pp 18ndash66 Van Nostrand New York NY USA 1954

[16] AWimmer andK Lewis ldquoBeyondandbelow racial homophilyERGmodels of a friendship network documentedon facebookrdquoAmerican Journal of Sociology vol 116 no 2 pp 583ndash642 2010

[17] R Axtell RMAxelrod JM Epstein andMDCohen ldquoAlign-ing simulation models a case study and resultsrdquoComputationalamp Mathematical Organization Theory vol 1 no 2 pp 123ndash1411996

[18] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[19] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[20] X Hao-Xiang W Hui-Li and X Zhao-Guo ldquoOpinion dynam-ics a multidisciplinary review and perspective on futureresearchrdquo International Journal of Knowledge and Systems Sci-ence vol 2 pp 72ndash91 2011

[21] J C Gonzalez-Avella M G Cosenza and K Tucci ldquoNonequi-librium transition induced by mass media in a model for socialinfluencerdquo Physical Review E - Statistical Nonlinear and SoftMatter Physics vol 72 no 6 pp 1ndash4 2005

[22] R Ulloa C Kacperski and F Sancho ldquoInstitutions and culturaldiversity Effects of democratic and propaganda processes onlocal convergence and global diversityrdquo PLoS ONE vol 11 no4 2016

[23] A Stivala G Robins Y Kashima and M Kirley ldquoUltrametricdistribution of culture vectors in an extended Axelrodmodel ofcultural disseminationrdquo Scientific Reports vol 4 article 4870pp 1ndash9 2014

[24] R Huckfeldt P E Johnson and J Sprague Political Disagree-ment The Survival of Diverse Opinions within CommunicationNetworks Cambridge University Press Cambridge UK 2004

[25] A Flache M W Macy and K Takacs ldquoWhat sustains culturaldiversity and what undermines it Axelrod and beyondrdquo inAdvancing Social Simulation Proceedings of the First WorldCongress on Social Simulation S Takahashi Ed pp 9ndash16Springer Kyoto Japan 2006

[26] K Klemm V M Eguıluz R Toral and M San MiguelldquoNonequilibrium transitions in complex networks a model ofsocial interactionrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 67 no 2 Article ID 026120 6 pages2003

[27] K Klemm V M Eguıluz R Toral and M S Miguel ldquoGlobalculture a noise-induced transition in finite systemsrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 67no 4 Article ID 045101 2003

[28] MMas A Flache andDHelbing ldquoIndividualization as drivingforce of clustering phenomena in humansrdquoPLoSComputationalBiology vol 6 no 10 Article ID 1000959 2010

[29] L De Sanctis and T Galla ldquoEffects of noise and confidencethresholds in nominal and metric Axelrod dynamics of socialinfluencerdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 79 no 4 Article ID 046108 2009

[30] D Centola J C Gonzalez-Avella V M Eguıluz and M SanMiguel ldquoHomophily cultural drift and the co-evolution of

cultural groupsrdquo Journal of Conflict Resolution vol 51 no 6 pp905ndash929 2007

[31] F Battiston V Nicosia V Latora and M S Miguel ldquoLay-ered social influence promotes multiculturality in the Axelrodmodelrdquo Scientific Reports vol 7 no 1 article 1809 2017

[32] A Flache and M W Macy ldquoLocal convergence and globaldiversity From interpersonal to social influencerdquo Journal ofConflict Resolution vol 55 no 6 pp 970ndash995 2011

[33] D Centola and M Macy ldquoComplex contagions and the weak-ness of long tiesrdquo American Journal of Sociology vol 113 no 3pp 702ndash734 2007

[34] D Lim H Lee H Zo and A Ciganek ldquoOpinion Formationin the Digital Dividerdquo Journal of Artificial Societies and SocialSimulation vol 17 no 1 2014

[35] E Pulick P Korth P Grim and J Jung ldquoModeling interactioneffects in polarization Individual media influence and theimpact of town meetingsrdquo Journal of Artificial Societies andSocial Simulation vol 19 no 2 2016

[36] A Boutyline and R Willer ldquoThe social structure of politicalecho chambers variation in ideological homophily in onlinenetworksrdquo Political Psychology vol 38 no 3 pp 551ndash569 2017

[37] M D Vicario A Bessi F Zollo et al ldquoThe spreading ofmisinformation onlinerdquo Proceedings of the National Acadamy ofSciences of the United States of America vol 113 no 3 pp 554ndash559 2016

[38] D J Watts and S H Strogatz ldquoCollective dynamics of ldquosmall-worldrdquo networksrdquoNature vol 393 no 6684 pp 440ndash442 1998

[39] A Grow A Flache and R P M Wittek ldquoGlobal diversity andlocal consensus in status beliefs The role of network clusteringand resistance to belief changerdquo Sociological Science vol 4 pp611ndash640 2017

[40] L H Wong P Pattison and G Robins ldquoA spatial modelfor social networksrdquo Physica A Statistical Mechanics and itsApplications vol 360 no 1 pp 99ndash120 2006

[41] C R Sunstein Republiccom Princeton University Press NewJersey NJ USA 2002

[42] J Ugander B Karrer L Backstrom and C Marlow ldquoTheanatomy of the facebook social graphrdquo 2011 httpsarxivorgabs11114503

[43] E Bakshy W A Mason J M Hofman and D J WattsldquoEveryonersquos an influencer quantifying influence on twitterrdquo inProceedings of the 4th ACM International Conference on WebSearch and Data Mining (WSDM rsquo11) pp 65ndash74 Hong KongChina February 2011

[44] K N Hampton L S Goulet C Marlow and L Rainie ldquoWhymost Facebook users get more than they giverdquo 2012

[45] N E Friedkin and E C Johnsen Social Influence NetworkTheory A Sociological Examination of Small Group Dynamicsvol 33 of Structural Analysis in the Social Sciences CambridgeUniversity Press Cambridge UK 2011

[46] M W Macy J A Kitts A Flache and S Benard ldquoPolarizationin dynamic networks a hopfield model of emergent structurerdquoin Dynamic Social Network Modeling and Analysis pp 162ndash1732003

[47] M Mas and A Flache ldquoDifferentiation without distancingexplaining bi-polarization of opinions without negative influ-encerdquo PLoS ONE vol 8 no 11 2013

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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Decision SciencesAdvances in


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Submit your manuscripts atwwwhindawicom

2 Complexity

A A

B A

(a) t = 0

A A

B A

A A

B A

(＜2) t = 1one-to-many(＜1) t = 1one-to-one

Figure 1 Illustration of the intuition that one-to-many communication fosters isolation Nodes have three characteristics (color shape andletter) that are open to influence The number of traits shared by two nodes and thus the probability that a sender exerts influence on thereceiver is shown by the number of lines connecting the nodes Panel (a) shows the initial setup before the top-left agent communicated hisshape trait either under the one-to-one communication regime (Panel b1) or the one-to-many regime (Panel b2)

In contrast to the existing research we focus on thecomplexity arising from the interaction between individualsrather than on within-individual processes To this end westudy Axelrodrsquos prominent formal model of communicationthat was developed for offline social networks (and uses theone-to-one communication rule) keeping all assumptionsabout individual behavior unchanged but implementingcommunication between actors in a way that captures typicalforms of online communication (one-to-many)With analyt-ical tools and simulation we demonstrate that this changein model assumptions drastically changes model predictionsand leads to conclusions that challenge insights from researchon within-individual processes Contrary to the sketchedfinding that computer-mediated communication fosters theemergence of consensus [12 13] we find that the onlinecommunication regime fosters the emergence of mutuallydisagreeing subgroups in our simulations Likewise whilesocial-psychological research found that online communi-cation allows some individuals to overcome social isolation[14] we demonstrate that online communication increasesthe chances that individuals get socially isolated We derivethese results using an approach that is very different fromsocial-psychological research While these studies exploredhow online communication changes the way individualsbehave and respond to each other our work demonstratesthat differences between online and offline communicationarise through merely a different communication structureIn complexity terms we find that the ldquowholerdquo changes notbecause the ldquopartsrdquo have changed but because the interdepen-dencies between the ldquopartsrdquo are slightly different

Axelrodrsquos model of the dissemination of culture is one ofthe most prominent models of consensus formation and theemergence of dissimilar subgroups It is also a typical rep-resentative of models implementing offline communicationAxelrod proposed the model to address what he perceived asa fundamental puzzle in research on social influence askingldquoIf people tend to become more alike in their beliefs attitudesand behaviorwhen they interact why do not all such differenceseventually disappearrdquo [9 pp 203] Axelrod then showed withan agent-based model how assimilation at the microlevelof individual interactions can be reconciled with culturaldifferentiation at the level of society as a whole Like mostcontributions to the literature Axelrodrsquos model representsindividuals as nodes in a network that are described by a set ofcultural traits representing individualsrsquo cultural preferences

(like preferences for styles of music literature or dress)Furthermore Axelrod implemented the so-called ldquoone-to-onerdquo communication regime where in a social encounter oneagent always communicates one cultural trait to one of hernetwork contacts This one-to-one communication regimemimics the face-to-face communication present in manyoffline contexts but it differs from a form of communicationthat is ubiquitous on the Internet and that we label ldquoone-to-manyrdquo communication When Internet users blog orpost content on online social networks for instance theycommunicate content to multiple online contacts at oncerather than to just one of them

This paper wasmotivated by the following intuition aboutthe complexity arising from one-to-many communicationConsider for illustration the network of four actors depictedin Figure 1(a) All actors are described by three cultural traitsshape (circle or square) color (black or white) and letter (Aor B) The number of lines connecting the nodes representsthe number of cultural traits the respective two nodes havein common Implementing homophily [4 15 16] Axelrodassumed that trait overlap increases the likelihood that nodeswill adopt a trait from their neighbor Suppose that the top-left agent (in Figure 1(a)) communicates his shape trait underthe two different communication regimes Under the one-to-one communication regime assumed in Axelrodrsquos model thisagent communicates his trait to one of the two agents withwhom he already shares the letter and color traits Assumethat the top-right agent is selected for interaction and thisagent accepted the trait Figure 1(b) visualizes this situationshowing the increased cultural similarity between receiverand sender As a side effect the cultural overlap betweenthe top-right agent and the bottom-left agent decreased butthe overall network remains connected Figure 1(c) showsthat a different outcome arises when agents communicateunder the one-to-many regime The same agent (top-leftagent) communicates his shape trait but let us assumenow all actors with whom he has nonzero cultural overlapadopt the communicated trait This has two consequencesthat we study in this paper First a culturally homogenouscluster forms because after the communication three actorshold identical cultural traits Communication did not onlyincrease similarity between the sender and the receivers ofthe message but also preserve the similarity between thenodes who adopted the trait Under the one-to-one regimein contrast the two nodes on the right-hand side turned

Complexity 3




2 Literature






4 Complexity












3 The Model


119862119894 = (1199021198941 1199021198942 119902119894119865) 119902119894119909 isin 0 1 119876 minus 1 (1)






Complexity 5

(1 0 0

1 0 0

1 0 0

0

0

0


(1 0 0

1 0 0

0 1 0

0

0

0


(1 0 0

0 1 0

0 0 1

0

0

0












(119903 + 119899 minus 1)119903 (119899 minus 1)

= (3 + 3 minus 1)3 (3 minus 1)

= 10 (2)



6 Complexity


(a) One-to-one


(b) One-to-many









119901lowast = 119902119879infin (3)


Complexity 7





4 9 16 25 36 49N

64 81 100


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10






8 Complexity

N = 81 N = 100

N = 36 N = 49 N = 64

N = 9 N = 16 N = 25

1 9 16 25 36 49 64 81 100 1 9 16 25 36 49 64 81 100

1 9 16 25 36 49 64 81 100

01

1

10

100

01

1

10

100

01

1

10

100

Cluster size

Shar

e of r

eplic

atio

ns







Complexity 9


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N






000

025

050

075

100

Tran

sitiv

ity







10 Complexity


01 02 03 04 05 06 01 02 03 04 05 0600

02

04

06

08

10

Transitivity

S max

N






00

02

04

06

08

10


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

8 12 16 20 24 28 32 36 40 44 484Degree




Complexity 11


2 8 14 20 26 32 38 44 50 2 8 14 20 26 32 38 44 5000

02

04

06

08

10

Degree

S max

N





9 16 25 36 49 64 81 100N


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10




119891 (119910 119889119894119895) = exp (minus119910 [119889119894119895]) (5)


12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N









Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3




2 Literature






4 Complexity












3 The Model


119862119894 = (1199021198941 1199021198942 119902119894119865) 119902119894119909 isin 0 1 119876 minus 1 (1)






Complexity 5

(1 0 0

1 0 0

1 0 0

0

0

0


(1 0 0

1 0 0

0 1 0

0

0

0


(1 0 0

0 1 0

0 0 1

0

0

0












(119903 + 119899 minus 1)119903 (119899 minus 1)

= (3 + 3 minus 1)3 (3 minus 1)

= 10 (2)



6 Complexity


(a) One-to-one


(b) One-to-many









119901lowast = 119902119879infin (3)


Complexity 7





4 9 16 25 36 49N

64 81 100


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10






8 Complexity

N = 81 N = 100

N = 36 N = 49 N = 64

N = 9 N = 16 N = 25

1 9 16 25 36 49 64 81 100 1 9 16 25 36 49 64 81 100

1 9 16 25 36 49 64 81 100

01

1

10

100

01

1

10

100

01

1

10

100

Cluster size

Shar

e of r

eplic

atio

ns







Complexity 9


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N






000

025

050

075

100

Tran

sitiv

ity







10 Complexity


01 02 03 04 05 06 01 02 03 04 05 0600

02

04

06

08

10

Transitivity

S max

N






00

02

04

06

08

10


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

8 12 16 20 24 28 32 36 40 44 484Degree




Complexity 11


2 8 14 20 26 32 38 44 50 2 8 14 20 26 32 38 44 5000

02

04

06

08

10

Degree

S max

N





9 16 25 36 49 64 81 100N


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10




119891 (119910 119889119894119895) = exp (minus119910 [119889119894119895]) (5)


12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N









Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity












3 The Model


119862119894 = (1199021198941 1199021198942 119902119894119865) 119902119894119909 isin 0 1 119876 minus 1 (1)






Complexity 5

(1 0 0

1 0 0

1 0 0

0

0

0


(1 0 0

1 0 0

0 1 0

0

0

0


(1 0 0

0 1 0

0 0 1

0

0

0












(119903 + 119899 minus 1)119903 (119899 minus 1)

= (3 + 3 minus 1)3 (3 minus 1)

= 10 (2)



6 Complexity


(a) One-to-one


(b) One-to-many









119901lowast = 119902119879infin (3)


Complexity 7





4 9 16 25 36 49N

64 81 100


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10






8 Complexity

N = 81 N = 100

N = 36 N = 49 N = 64

N = 9 N = 16 N = 25

1 9 16 25 36 49 64 81 100 1 9 16 25 36 49 64 81 100

1 9 16 25 36 49 64 81 100

01

1

10

100

01

1

10

100

01

1

10

100

Cluster size

Shar

e of r

eplic

atio

ns







Complexity 9


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N






000

025

050

075

100

Tran

sitiv

ity







10 Complexity


01 02 03 04 05 06 01 02 03 04 05 0600

02

04

06

08

10

Transitivity

S max

N






00

02

04

06

08

10


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

8 12 16 20 24 28 32 36 40 44 484Degree




Complexity 11


2 8 14 20 26 32 38 44 50 2 8 14 20 26 32 38 44 5000

02

04

06

08

10

Degree

S max

N





9 16 25 36 49 64 81 100N


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10




119891 (119910 119889119894119895) = exp (minus119910 [119889119894119895]) (5)


12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N









Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5

(1 0 0

1 0 0

1 0 0

0

0

0


(1 0 0

1 0 0

0 1 0

0

0

0


(1 0 0

0 1 0

0 0 1

0

0

0












(119903 + 119899 minus 1)119903 (119899 minus 1)

= (3 + 3 minus 1)3 (3 minus 1)

= 10 (2)



6 Complexity


(a) One-to-one


(b) One-to-many









119901lowast = 119902119879infin (3)


Complexity 7





4 9 16 25 36 49N

64 81 100


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10






8 Complexity

N = 81 N = 100

N = 36 N = 49 N = 64

N = 9 N = 16 N = 25

1 9 16 25 36 49 64 81 100 1 9 16 25 36 49 64 81 100

1 9 16 25 36 49 64 81 100

01

1

10

100

01

1

10

100

01

1

10

100

Cluster size

Shar

e of r

eplic

atio

ns







Complexity 9


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N






000

025

050

075

100

Tran

sitiv

ity







10 Complexity


01 02 03 04 05 06 01 02 03 04 05 0600

02

04

06

08

10

Transitivity

S max

N






00

02

04

06

08

10


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

8 12 16 20 24 28 32 36 40 44 484Degree




Complexity 11


2 8 14 20 26 32 38 44 50 2 8 14 20 26 32 38 44 5000

02

04

06

08

10

Degree

S max

N





9 16 25 36 49 64 81 100N


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10




119891 (119910 119889119894119895) = exp (minus119910 [119889119894119895]) (5)


12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N









Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity


(a) One-to-one


(b) One-to-many









119901lowast = 119902119879infin (3)


Complexity 7





4 9 16 25 36 49N

64 81 100


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10






8 Complexity

N = 81 N = 100

N = 36 N = 49 N = 64

N = 9 N = 16 N = 25

1 9 16 25 36 49 64 81 100 1 9 16 25 36 49 64 81 100

1 9 16 25 36 49 64 81 100

01

1

10

100

01

1

10

100

01

1

10

100

Cluster size

Shar

e of r

eplic

atio

ns







Complexity 9


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N






000

025

050

075

100

Tran

sitiv

ity







10 Complexity


01 02 03 04 05 06 01 02 03 04 05 0600

02

04

06

08

10

Transitivity

S max

N






00

02

04

06

08

10


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

8 12 16 20 24 28 32 36 40 44 484Degree




Complexity 11


2 8 14 20 26 32 38 44 50 2 8 14 20 26 32 38 44 5000

02

04

06

08

10

Degree

S max

N





9 16 25 36 49 64 81 100N


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10




119891 (119910 119889119894119895) = exp (minus119910 [119889119894119895]) (5)


12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N









Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7





4 9 16 25 36 49N

64 81 100


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10






8 Complexity

N = 81 N = 100

N = 36 N = 49 N = 64

N = 9 N = 16 N = 25

1 9 16 25 36 49 64 81 100 1 9 16 25 36 49 64 81 100

1 9 16 25 36 49 64 81 100

01

1

10

100

01

1

10

100

01

1

10

100

Cluster size

Shar

e of r

eplic

atio

ns







Complexity 9


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N






000

025

050

075

100

Tran

sitiv

ity







10 Complexity


01 02 03 04 05 06 01 02 03 04 05 0600

02

04

06

08

10

Transitivity

S max

N






00

02

04

06

08

10


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

8 12 16 20 24 28 32 36 40 44 484Degree




Complexity 11


2 8 14 20 26 32 38 44 50 2 8 14 20 26 32 38 44 5000

02

04

06

08

10

Degree

S max

N





9 16 25 36 49 64 81 100N


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10




119891 (119910 119889119894119895) = exp (minus119910 [119889119894119895]) (5)


12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N









Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity

N = 81 N = 100

N = 36 N = 49 N = 64

N = 9 N = 16 N = 25

1 9 16 25 36 49 64 81 100 1 9 16 25 36 49 64 81 100

1 9 16 25 36 49 64 81 100

01

1

10

100

01

1

10

100

01

1

10

100

Cluster size

Shar

e of r

eplic

atio

ns







Complexity 9


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N






000

025

050

075

100

Tran

sitiv

ity







10 Complexity


01 02 03 04 05 06 01 02 03 04 05 0600

02

04

06

08

10

Transitivity

S max

N






00

02

04

06

08

10


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

8 12 16 20 24 28 32 36 40 44 484Degree




Complexity 11


2 8 14 20 26 32 38 44 50 2 8 14 20 26 32 38 44 5000

02

04

06

08

10

Degree

S max

N





9 16 25 36 49 64 81 100N


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10




119891 (119910 119889119894119895) = exp (minus119910 [119889119894119895]) (5)


12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N









Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N






000

025

050

075

100

Tran

sitiv

ity







10 Complexity


01 02 03 04 05 06 01 02 03 04 05 0600

02

04

06

08

10

Transitivity

S max

N






00

02

04

06

08

10


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

8 12 16 20 24 28 32 36 40 44 484Degree




Complexity 11


2 8 14 20 26 32 38 44 50 2 8 14 20 26 32 38 44 5000

02

04

06

08

10

Degree

S max

N





9 16 25 36 49 64 81 100N


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10




119891 (119910 119889119894119895) = exp (minus119910 [119889119894119895]) (5)


12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N









Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity


01 02 03 04 05 06 01 02 03 04 05 0600

02

04

06

08

10

Transitivity

S max

N






00

02

04

06

08

10


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

8 12 16 20 24 28 32 36 40 44 484Degree




Complexity 11


2 8 14 20 26 32 38 44 50 2 8 14 20 26 32 38 44 5000

02

04

06

08

10

Degree

S max

N





9 16 25 36 49 64 81 100N


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10




119891 (119910 119889119894119895) = exp (minus119910 [119889119894119895]) (5)


12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N









Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11


2 8 14 20 26 32 38 44 50 2 8 14 20 26 32 38 44 5000

02

04

06

08

10

Degree

S max

N





9 16 25 36 49 64 81 100N


00

02

04

06

08

10

Shar

e of r

uns w

ith is

olat

e

00

02

04

06

08

10




119891 (119910 119889119894119895) = exp (minus119910 [119889119894119895]) (5)


12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N









Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity


9 16 25 36 49 64 81 100 9 16 25 36 49 64 81 10000

02

04

06

08

10

N

S max

N









Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 13

t = 3292s = 1020

t = 0000s = 0000

t = 6585s = 2039

t = 9874s = 2416

t = 0551s = 1061

t = 0000s = 0000

t = 1102s = 0837

t = 1653s = 0991






14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








14 Complexity

F=10 F=15

F=3 F=5

21 22 23 24 25 26 27 28 29 210 21 22 23 24 25 26 27 28 29 210

00

Shar

e of r

uns w

ith is

olat

e 02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

00

02

04

06

08

10

Q







Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 15

000

025

050

075

100

21 22 23 24 25 26 27 28 29 210

Q

S max

N


F=3F=5F=10F=15








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








16 Complexity



class





Appendix



000

025

050

075

100


S max

N


0 100 200 300 400 500 600 700 800 900Cluster size


0000

0004

0008

0012

Den

sity





Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








Complexity 17

Data Availability


Disclosure




Acknowledgments


Endnotes








References















18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018








18 Complexity











































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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