Effect of Persuasion on Information Diffusionin Social Networks

Abhimanyu DasMicrosoft Research

abhidas@microsoft.com

Sreenivas GollapudiMicrosoft Research

sreenig@microsoft.com

Emre KicimanMicrosoft Research

emrek@microsoft.com

ABSTRACTOne of the key factors guiding the act of communication be-tween individuals in a social network is the desire to per-suade or influence one another. In this paper, we study theinterplay between a person writing (selecting) a message tosend to another and the effect that the message has on its re-cipient. Using large-scale online user studies, we focus ona single effect (persuading or changing a recipient’s opin-ion about a topic) and its relationship to various measurableproperties of the written message often associated with per-suasive techniques, namely the degree of emotional and log-ical content. We find that the persuasive efficacy of theseproperties varies by domain of discussion and by individualsusceptibility, and that senders appear to strategically selecttheir persuasion techniques. Based on these results, we de-velop a structural model of information diffusion and showthrough simulations that the emergent larger-scale behaviorsare consistent with current models of information cascadesand, moreover, are able to model as yet unexplained empiri-cally observed variance in the structural virality of cascades.

1. INTRODUCTIONPersuasion is the act of convincing someone to take

some action. For example, one individual might per-suade another to buy a product, vote for a candidate, orshare a message on social media. People wield a broadvariety of persuasion techniques, from logical argumentand emotional appeals to threats and bribes. Thesetechniques and their appropriate applications have beenwidely studied from many perspectives in fields includ-ing psychology, rhetoric, marketing, and neurobiology [21,13, 12]

While there is no single unified theory of persuasion,it is broadly clear from past research that the effective-ness of persuasion techniques is greatly dependent onthe target individual as well as the requested action.While some individuals may be easy to persuade, thereare many people who are challenging to convince to act.Moreover, persuasion techniques work well in some con-texts, but not in others. Certain persuasion techniquesare more likely to work for some individuals than others.In many ways, this is simply common sense. Individu-

als have different attitudes, backgrounds, assumptions,and these individual perspectives are entwined with thecontext of the action.

In this paper, we present a first investigation into therole of persuasion in information dissemination withina social network. We focus our study on the interplaybetween an individual who sends a message and the(at least one) person who receives it: can the senderpersuade the recipent to adopt the message and re-propragate it along the network?

In this setting, through constrained models and ex-periments, we study the implications of the most basicproperties of persuasion—namely, that there are manydifferent persuasion techniques, that the efficacy of aspecific technique is contextually dependent, and thatindividuals vary in their susceptibility to specific tech-niques. First, through on-line user studies, we measurethe contextual efficacy of two persuasion techniques;validate that individual susceptibility to such appealsvaries; and that in aggregate, senders appear to strate-gically select their persuasion technique.

Secondly, based on the learnings from the user stud-ies, we propose a simple structural model of persuasion’srole in information dissemination based on the micro-behavior between a sender and recipient. The modelassociates with each node a d-dimensional vector repre-sentation of preferences (or susceptibilities) to differentpersuasion techniques. A message flowing through thenetwork is also represented by a vector in the same la-tent space, and its propagation through a node is con-trolled by the distance between the message and node.Within this model, we also perform preliminary analy-sis for the problem of selecting a message for maximalpropagation.

Finally, through simulations, we study the emergentproperties of the large-scale information diffusions thatarise from the micro-behaviors specified in our model.We find that, not only are the resultant diffusion treesbroadly consistent with the previous models and em-pirical observations, they are also able to model empir-ically observe variances in structural virality that havenot been captured by past models.

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Contributions of this studyWe make the following contributions in this paper:

• We perform carefully designed user studies to es-tablish the role of persuasive properties of a mes-sage on senders and receivers of such messagesacross different topics. This lets us characterizethe persuasive impact of different message prop-erties on receivers, and the strategic behaviour ofsenders in choosing these message properties whentrying to persuade receivers.

• Based on insights from our user studies, we posit anew analytical model of information diffusion thatincorporates how the persuasive properties of amessage plays a central role in its diffusion througha social network. We also present some prelim-inary analytical results for how a sender shouldoptimize her choice of message in the context ofour model to maximize its adoption in the net-work.

• We conduct extensive simulation studies based onour above information diffusion model to charac-terize properties related to the virality rate andstructural virality [15] of information cascades inlarge scale networks. Our simulations reproduceempirically observed variances in virality of real-world cascades that could not be explained by ear-lier models.

2. RELATED WORKThere has been a huge body of work in the area of

information diffusion through networks. Several earlymodels for information diffusion were inspired from clas-sical disease propogation models in epidemiology, suchas SIR and SIS [3]. Other related diffusion models forproduct marketing included the Bass [4] model that isbased on an S-shaped adoption curve.

There has also been extensive work on modeling theadoption or spread of an idea, content or product in asocial network. Well known classes of models in this do-main include Threshold [19] and Cascade models [17],that specify how a node adopts a particular idea orproduct based on the adoption pattern prevalent in itsneighborhood. In a Threshold model, a node choosesa threshold uniformly at random, and adopts an ideaif the (weighted) number of its neighbors who haveadopted the idea is greater than the threshold. In theCascade model, a node u adopts an idea with proba-bility puv whenever a neighbor v of u also adopts theidea.

A paper that is quite relevant to our work is [15],which propose a formal measure (called structural vi-rality) of the amount of viral diffusion in a cascade.Using this measure, the authors conduct a large scale

empirical study of a billion diffusion events for news,videos, images and petitions on twitter, and observe awide range of diverse cascading structures with varyingstructural virality, and show a low correlation betweenpopularity and structural virality. The authors thenshow how a simple SIR model can capture several of theempirically-observed properties of the cascades. How-ever, they note that their model could not explain thelarge variance in structural virality that they observedempirically.

Another recent related paper [29] also looks at the ef-fect of a message’s wording on its propogation on Twit-ter. The authors take a natural language processing ap-proach to build classifiers based on textual and linguis-tic features that can predict the likelihood of propoga-tion of a message. Unlike our approach, the authors donot address the problem in the context of informationdiffusion models.

Kempe et. al [22] analyzed the Threshold and Cas-cade models in the context of influence maximization:that is, the problem of selecting a set of influential nodesto seed with an idea such that it maximizes adoption ofthe idea in the entire social network. Several other pa-pers [18, 26] studied various aspects of the phenomenonof diffusion of ideas or content in social networks suchas the problem of tracing the path used for diffusionin the network, or of modeling external out-of-networksources in the diffusion process.

Goel et al. [16] empirically observed diffusion pat-terns over seven different domains including URL cas-cades corresponding to news stories, videos, and on-line games, and observed that the vast majority of cas-cades in practice are small and terminate within a fewhops. Leskovec et al [23] studied information diffu-sion in the context of product recommendation for e-commerce, propose an algorithm for enumerating cas-cades, and observe that the distribution of cascade sizesfollow a power-law model.

Several papers [20, 1] also analyzed and characterizedinformation propogation (URLs, topics, etc) in blogsusing well known models of information diffusion. In[24], the authors study cascades in linking propogationpatterns for blogs and show that the SIS model is ableto mimic the cascades quite well.

In social psychology, there has been a large body onwork on persuasion and social influence [7, 10, 30] thattalks about various cognitive theories and psychologi-cal processes behind how people convince and persuadeeach other. The book by [27] provides a comprehen-sive overview. Guadagno and Cialdini discuss persua-sion and compliance in the context of Internet-mediatedcommunications, especially textual messages [21]. How-ever, to the best of our knowledge, not of these per-suasion processes have been modeled in the context ofinformation diffusion in social networks.

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3. PERSUASION TECHNIQUESStudies in the area of persuasion theory [27, 21, 14]

recognize several models for effecting people’s opinions.The Elaboration Likelihood Model argues that thereare two fundamental methods (or “routes”) of informa-tion processing and reasoning that occur when a subjectconsiders a persuasive attempt—a central path of rea-soning uses logic and deep reasoning; and a peripheralpath relies on heuristics. According to the elaborationlikelihood model, which path is taken depends on manyfactors, including the personal receiver’s motivation andexpertise. Learning theory views persuasion as a spe-cific form of learning, where subjects learn the best re-sponse to a persuasive message through conditioningand incentives over time. Cognitive dissonance viewspersuasion in the context of our desire to reduce dis-sonance or discomfort due to holding conflicting views.In other words, successful persuasion depends on theelimination of conflict or congnitive dissonance withinthe subject’s mental models about a given topic.

While they have many fundamental differences, sub-stantially all of these theories and models about persua-sion broadly recognize that the efficacy of persuasiontechniques varies significantly based on topical and in-dividual context. The importance and comprehensibil-ity of the topic, the subjects familiarity, expertise andattitude, all play some role in how (and how much) aperson considers and reacts to an attempt to persuadehim or her.

In this paper, we focus on a taxonomy of persuasivetechniques first proposed by Aristotle in Rhetoric, andstill in use as a general framework today [11, 28, 6].This taxonomy splits modes or strategies of persuasioninto three categories, pathos, logos, and ethos:Logos is a logical appeal or argument, attempting toconvince through facts and other detailed information.Such messages often try to simplify the issue around asingle key topic to minimize cognitive dissonance. Hereare some examples of messages making logical appeals:

I like the Surface Pro because I can install

most any program that runs on my regular PC

and it adds in stylus/touch controls.

I don’t like the Surface Pro because the

windows app store is not there yet -- some

of the apps are great, but you will find they

are higher priced than you are used to paying

in the google play or App store.}

Pathos is an emotional appeal or argument tries toconvince through feelings. Such messages may directlymention emotions, like love, hate, or injustice; or in-directly by mentioning things people care about, likefamily, celebrities, memories.

Here are some examples of messages making emo-tional appeals:

I like the book ‘‘The Valley of Amazement’’

because Oprah loved this novel.

I love coke because it reminds me of the

holidays! Always love their jingles during

the holidays.

Ethos is an ethical appeal that depends on the credibil-ity or authority of the author. For example, a leader ofa group may attempt to persuade people based on theircredibility. Previous studies have shown that authoridentity is an important signal people use when judg-ing the authoritativeness and credibility of messages onsocial networks [25].

In the user studies described in the next section, wewill focus on measuring the efficacy of the Pathos andLogos modes of persuasion in various contexts. We fo-cus on these two modes because of their ability to beexpressed solely through the text of a message (whereasEthos depends on contextual signals and prior knowl-edge about the author of a message). Note that wetreat these two persuasion strategies as being orthog-onal — for example, some messages make both logicaland emotional appeals, while other appeals make useneither strategy. Furthermore, there are many persua-sion techniques that fall outside of the scope of thistaxonomy, such as the use of deception or force, andmore complex multi-step strategies of persuasion, suchas inoculation. We leave a more complete study of per-suasion techniques for future work.

4. ONLINE USER STUDIESWe will now present experiments to establish the im-

portance of the role of message properties on its even-tual adoption by network users. Specifically, we confirmthat there is a complex relationship between the mea-surable properties of a message, the domain of discus-sion, and the message’s persuasive impact on recipients.Toward this goal, we ran two sets of experiments thatcenter around the sender and recipient respectively. Wechoose the dimensions of emotional (pathos) and logical(logos) in a message to represent two of the three impor-tant persuasion techniques. We then measure the effectof these dimensions on the adoption of the message un-der different polarities (i.e., positive and negative sen-timents). Further, we also vary the topics or entitiesabout which the messages are drawn from to tease outthe effect of the topic on the choice of dimensions of amessage. The results are reported for three topics, viz.,Hyundai, Coca Cola, and organic food.

4.1 Effect of message type on the recipientWe begin with the creation of the data set used in

this experiment. We manually selected a set of 200messages, O, for each of the three different topics.

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Message Sentiment Label

The hyundai has more crashes per vehicle than average. Its not worth the risk Negative Logical & EmotionalI had to replace the rear brakes on my hyundai after just 21k miles. Ridiculous! Negative Logical & Not Emotionalugh. My hyundai has cost me so much in repairs. Stay away! Negative Not Logical & EmotionalI dont know why exactly, but I prefer honda and ford cars over hyundai Negative Not Logical & Not Emotionalif consumer reports says are safe and a good value, thats all I need to know Positive Logical & Emotionalthe hyundai fits my 6’1” frame. Roomy and drives well Positive Logical & Not EmotionalMy hyundai is like a member of the family Positive Not Logical & EmotionalI like how the hyundai looks Positive Not Logical & Not Emotional

Table 1: A sample of messages related to Hyundai Cars

In the first step, we gave the messages in O to judgeson Amazon Mechanical Turk and asked them to labelthem along two dimensions: logical or not logical andemotional or not emotional. Each message was pre-sented to 15 judges. We then slotted a message in one ofthe four quadrants resulting from the {logical, not logical}and {emotional, not emotional} labels based on whata majority of judges chose. The inter-annotator agree-ment for this step was 0.75 showing a large agreementbetween the judges. In addition, we also asked 15 judgesto label each message as containing either a positive ornegative sentiment. Table 1 shows a sample of the mes-sages we presented to the user and their categorizationinto the label space.

In the second step, we aim to quantify the effect ofa message’s quadrant on the likelihood of its adoptionby a user with a certain predisposition (positive or neg-ative) to the topic. For this purpose, we carried outanother study in which the user was initially asked tostate her message on a given topic. Based on her re-sponse on a five-point scale (very negative=0 to verypositive=4), she was presented 10 messages of the op-posite sentiment, all chosen from a quadrant which waschosen uniformly at random. After being shown the10 messages, she was asked for her message again. Thisexperiment was repeated for all the aforementioned top-ics. Figure 1 illustrates the relative change in a user’smessage before and after she is exposed to the messagesof the opposite sentiment is topic dependent. In fact,in the case of both Hyundai cars as well as Coke, thereis a increase in the negative sentiment after the treat-ment and corresponding (although by not as much) de-crease in positive sentiment. This suggests that userspredisposed with a positive sentiment are more likelyto change to a negative sentiment after seeing messageswith negative sentiment than the other way around. Inorder to understand the same effect at a more granularlevel, we computed the average change from a particu-lar sentiment (e.g., very negative) to another sentimentand measured the change as the difference between thetwo sentiment levels. For example, we compute the dis-tance between very negative and neutral to be 2. Inother words, we compute the average shift from eachsentiment level for each topic and present the results ofthis experiment in Figure 2. In this figure, for Organic

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OrganicFood (A)

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Very Negative NegativeNeutral PositiveVery Positive

Figure 1: Distribution of the sentiment valuesbefore(B) and after(A)

Food, we observe that users with a negative sentimentare more likely to change to a positive sentiment (high-lighted by the large average shift out of the very negativesentiment. On the other hand, users with an initial pos-itive sentiment do not shift their sentiment by as muchsuggesting that they are more sticky or stubborn withtheir original message. In the next experiment, we willdelve into more detail and try to tease out the actualmessage type that can likely cause the shifts we are ob-serving in Figure 2.

Toward this end, we analyzed the effect of differentdimensions on the change in sentiment. Thus, in eachtreatment, we presented messages belonging to only onequadrant to the user. We remind the reader that themessages were of the opposite sentiment as the user’s.Table 2 summarizes the change in the distribution ofsentiments before and after a particular treatment forthe Hyundai cars topic. Clearly, there is a change in thesentiment distribution after the user sees the treatment.In fact, both logical and emotional messages induce big-ger changes in the sentiments values corresponding tothe treatments without such messages.

Finally, we computed the average shift from a priorsentiment which measures the overall drift away fromthat sentiment as a result of the treatment. For exam-ple, a change of 2 for a very negative sentiment impliesthat on the average, a user moved up two levels in her

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Initial FinalTreatment (−−)% (−)% Neutral% (+)% (++)% (−−)% (−)% Neutral% (+)% (++)%

Logical 1.0% 3.96% 24.75% 62.38% 7.92% 2.97% 17.8% 26.73% 45.54% 6.93%Not Logical 1.18% 2.35% 23.53% 49.41% 23.53% 4.71% 5.88% 27.06% 49.41% 12.94%Emotional 2.08% 2.08% 22.92% 60.42% 12.50% 5.21% 13.54% 30.21% 42.71% 8.33%

Not Emotional 0.00% 4.44% 25.56% 52.22% 17.78% 2.22% 11.11% 23.33% 52.22% 11.11%Logical & Emotional 2.04% 2.04% 28.57% 65.31% 2.04% 6.12% 20.41% 26.53% 42.86% 4.08%

Neither 0% 2.63% 31.58% 42.11% 23.68% 5.26% 5.26% 18.42% 57.89% 13.16%Logical Only 0% 5.77% 21.15% 59.62% 13.46% 0% 15.38% 26.92% 48.08% 9.62%

Emotional Only 2.13% 2.13% 17.02% 55.32% 23.4% 4.26% 6.38% 34.04% 42.55% 12.77%

Table 2: The effect of properties of the message on a user for the topic Hyundai cars

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Coke Hyundai Organic Food

Very Negative Negative

Neutral Positive

Very Positive

Figure 2: Distribution of the change in senti-ment values

sentiment to around positive. We note that we also con-sider combinations of the quadrants in the set of treat-ments. For example, a treatment of logical implies alltreatments that include logical, i.e., logical and logicaland emotional. Figure 3 shows the average change frominitial sentiment for all three topics.

Let us go through the results for each topic. In thecase of Hyundai cars, we observe that there is movementin both the directions, i.e., users with a positive senti-ment change their value downwards while users withnegative sentiment move upwards. However, we qual-ify the moves as follows. While the positive sentimentsmove for all treatments, albeit in different measures,the presence of logical message in the treatment inducedsignificant shifts in the negative sentiments (toward thepositive). This becomes clear when one observes theshifts in the treatments Not Logical, Not Emotional,and Neither Logical nor Emotional. In the first andthe third treatments where there are no logical mes-sages in the treatment, there is no shift in the negativesentiments. Finally, we observe that the biggest shift(in both directions) comes when the user is presentedwith both logical and emotional messages and the small-est shift is associated when the user is presented withmessages that are neither logical nor emotional.

In terms of the effect of logical messages on the shift

in negative sentiments, we observe the same behavior inthe case of Organic Food. Indeed, all three cases, viz.,Logical, Not Emotional, and Only Logical produce thelargest shift in the negative sentiment with Only Logi-cal producing a shift of 2 levels - well into the positivelevels! This suggests, that users with negative predis-positions to organic food generally are more open tochanging their sentiment when presented with logicalexplanations or messages while the change in the otherdirection is less pronounced. This could be attributedto the fact that people with positive predisposition toorganic food are those that have already done their “re-search” and are therefore less likely to get swayed byany kind of message from others.

We observe a slightly different shifting behavior inthe case of Coke. While the logical messages still havemore effect on the shift than emotional messages, in thiscase the shift is more pronounced for positive sentimentscompared to the topics.

Finally, we measure the relative standard deviationin the average shift in the receiver’s sentiments. Thismeasure gives us an insight into the variability in theunderlying characteristics associated with a receiver. Inparticular, a high value of this measure corresponds toa large variability in the population. Figure 4 preciselyillustrates such a phenomenon. A large variability al-lows us to infer different users are persuaded to differentextent for a given type of message. Indeed, we will usethis observation to motivate our model in Section 5.

4.2 Message selection by sendersIn our second set of experiments, we ask people to

write a message about a domain-specific item with thespecific purpose of either persuading or charming therecipient. We measure the degree of emotional and log-ical content chosen by the sender and find that it does,indeed, vary by domain.

Then, we analyze and compare the distribution ofmessage attributes chosen by our senders and find that,on average, senders do select the set of attributes thatare best aligned for a specific domain and desired effect.

Similar to the recipient experiments, we ran user stud-ies by asking a set of 200 users to create messages ona particular topic (same as the three used in the previ-

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Figure 3: Average change in shift from a sentiment after a particular treatment

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Figure 4: Relative standard deviation in the av-erage shift in receiver sentiments

ous set of experiments) with the goal of maximizing itsreach. We created a HIT1 on Amazon Mechanical Turkasking each judge to do the following -

“Your task is to write a short message (limited toone or two sentences) about why you like or dislike acar brand, such that you can convince as many of yourfriends on Facebook/Twitter who read your message toagree with your opinion. For example, if you have areason to like Hyundai cars, your message should bewritten such that it also convinces your friends to likeHyundai cars. On the other hand, if you do not likeHyundai cars, your message should be written such thatit persuades your friends to not prefer Hyundai cars.”

We then went ahead and got each of the message la-beled by 15 users in another user study, again using theAmazon Mechanical Turk platform. Table 3 shows asample of the messages crafted by the sender and thesubsequent categorization of the message. Finally, Fig-ure 5 illustrates the distribution of message types thatthe senders choose to send in response to our HIT. Weobserved that the senders showed a clear preference fora specific message type based on the topic. Hence, sug-

1Human Intelligence Task

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Hyundai Coke Organic Food

Only Logical

Only Emotional

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Neither

Figure 5: Distribution of the message types cho-sen by the senders

gesting that the senders are indeed strategic in theirchoice of the message. Further, they exhibited a generalpreference to logical messages over emotional messagesacross the three topics.

5. MODEL OF SOCIAL EFFECTS ON MES-SAGES

In this section, we describe a simple model of theindividual-level behaviors of senders and receivers thatcapture the two core observations from our crowdsourc-ing experiments. This model focuses on the interactionbetween a sender and a receiver: namely, how does thesender of a message reason about the persuasive effectthat a message will have on recipients, and how doesthis influence what messages are actually sent and con-sequently adopted by users in a social network?

5.1 Motivation for a new modelWe make the following observations from our user

studies that existing models of information diffusionsuch as the Linear Threshold and Independent Cascademodels [22] do not incorporate.

• Message selection by sender: Our first observa-tion is that senders exhibit relatively consistent

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Opinion Sentiment Label

Hyundai cars just suck. Mine broke down right after their guarantee period ended. Negative Logical & EmotionalHyundai cars are not giving good milage compared to other car brands Negative Logical & Not EmotionalHyundai should come up with some new designs for their cars Negative Not Logical & EmotionalHyundai cars are unreliable and of cheap quality Negative Not Logical & Not EmotionalHyundai offers a sweet, stylish ride for less money Positive Logical & EmotionalHyundai cars are best comes with good mileage and pickup. Great value for our money. Positive Logical & Not Emotionalhyundai cars are very good to look and drive Positive Not Logical & EmotionalIt looks good in design Positive Not Logical & Not Emotional

Table 3: A sample of opinions expressed by the senders related to Hyundai Cars

preferences for certain types of messages over oth-ers when trying to persuade the recipients of themessage. In particular, we observed that acrossall the three domains in our user studies, mostsenders preferred composing messages with high-logical content and low-emotional content whenasked to persuade their friends on Twitter or Fac-book to “adopt” the message. This suggests thatsenders in a social networks are strategic in se-lecting their messages to maximize the persuasiveeffect of the messages, and consequently the adop-tion of the message in the social network.

• Effect of message type on persuation of recipient:Our second observation is that the ability of a mes-sage containing a positive (c.f. negative) opiniontowards an entity to persuade a recipient holding anegative (c.f. positive) opinion to change her mindvaries greatly depending on the type of messsage.In particular, messages about an entity with highlogical and emotional content were more likely topersuade people to change their opinion than othertypes of messages. This suggests that the extentto which a recepient of a message is persuaded toadopt the message’s opinion depends to a largeextent on the type of the message.

• Heterogeneous user interests resulting in differentpreferences and thresholds at which they are will-ing to be persuaded (i.e., adopt the sender’s mes-sage and propagate it to their neighbors). Thismakes the propagation dynamics depend both thepreferences of the recipient as well as the sender’smessage.

Existing diffusion models do not take into accountthe role played by the type of a message or opinionwhen estimating its spread in the social network. Thismotivates the following model.

5.2 Basic ModelWe represent a message as a vector in d dimensional

space, where every dimension corresponds to an at-tribute of the message such as logic, emotion etc. Forsimplicity, we assume that the range for each dimensionlies in [0, 1]. Hence every message m is a vector in [0, 1]d

We consider an online social network graph G =(V,E) with nodes {v1, v2, . . . , vn} ∈ V . For ease ofnotation, we will frequently interchange node vi andindex i. The vertices vi correspond to individuals, andedges E = [eij ], denote social ties between the individ-uals. Every vertex vi has a d-dimensional unit-lengthpreference vector mi ∈ Rd and a persuasion thresholdti ∈ R. Vertices in the graph can propagate messagesalong their edges to their neighbors. For a node vi, itsset of neighbors is denoted by N(i) = {j : eij = 1}, andits degree is denoted by di = |Ni|. We denote the totalnumber of edges in the graph as |E|.

The dynamics of message propagation in the graphworks as follows: Every vertex in the graph can actas a sender or a receiver of messages. A sender in thegraph can compose a d-dimensional message vector mand broadcasts it to its neighboring vertices. A receiv-ing vertex is “persuaded” to adopt an incoming mes-sage only if the incoming message is ”close” (in someLp norm) to its preference vector. In particular, when-ever a (receiver) vertex vr receives a message m that ithas not previously adopted, it adopts the message andpropagates the same message to all other neighbors inN(r) iff ‖m−mr‖p < tr. Otherwise it drops the mes-sage, and does not adopt it. We typically assume p tobe 1 (Manhattan distance) or 2 (Euclidean distance).Figure 6 illustrates an example of message propagationin a graph based on this model.

Note that unlike in the case of traditional informationdiffusion models such as Linear Threshold and Indepen-dent Cascade models, the adoption of a message com-posed by a sender node depends not only on the graphstructure but also on the message vector itself. Indeed,as we saw in our user studies, most senders strategi-cally compose their messages to have high logical andemotional content when given a task of persuading theirfriends to adopt the message. Based on this observation,and using the above model, a natural question is thento understand what message a given sender should com-pose in order to persuade as many nodes in the graphto adopt the message.

In the next section, we propose an optimization prob-lem that arise from the above question.

5.3 Optimization Problem

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Figure 6: Message Propagation Model: Ex-ample of message propagation in a graph with1-dimensional message vectors, where v1 is asender sending a message m = 3. Every ver-tex vi has a preference vector mi and persua-sion threshold ti. The green vertices are nodesthat have been persuaded to adopt the mes-sage, while grey vertices are nodes that have notadopted the message (either they did not receivethe message from any of their neighbors, or theyreceived the message but dropped it because themessage did not fall within a distance ti of theirpreference vector mi.

More formally, we define the following optimizationproblem: given a Graph G = V,E, preference vectorsmi and thresholds ti of each node vi ∈ V , and a sendernode vs ∈ V , choose a message vector m that vs shouldcompose that maximizes the number of vertices in Gthat adopt the message.

5.3.1 Complete GraphWe first consider the above problem for the special

case of the complete graph. Since all nodes are directlyreachable from a sender node, the problem can be re-duced to the following version:

argmaxm∈[0,1]d

|v ∈ V : ‖m−mv‖p ≤ tv| (1)

This is equivalent to the following depth problem incomputational geometry []: given n hyperspheres (orhypercubes, depending on the choice of p for the Lpnorm) in d dimensions, find the point that has the max-imum depth, i.e. lies inside the maximum number ofhyperspheres (hypercubes). Figure 7 illustrates an ex-ample of the depth problem corresponding to the op-timization problem for message selection in a completegraph with 8 vertices.

We have the following results for two special cases ofthis problem:

The first lemma corresponds to the case when p = 1.

Lemma 1 For p = 1 and constant d, given a complete

m8

m2

m3

m4

m5

m6

m7

t2

t7

t8

t3

t5

t6t4

Depth=4

Figure 7: Optimization problem for messageselection in a complete graph with vertices{v1, v2, . . . , v8}, where v1 is the sender node seek-ing to choose a message that maximizes adop-tion. Every vertex vi has a preference vectormi ∈ R2 and persuation threshold ti, which corre-sponds to a hypersphere centered at mi with ra-dius ti. The problem of selecting a message thatis adopted by as many nodes as possible is equiv-alent to selecting a point in R2 that lies in theintersection of as many of the hyperspheres (cor-responding to {v2, v3, . . . , v8}) as possible, or al-ternatively, a point with maximum depth. Thiscorresponds to any point in the green region,which has a depth of 4.

graph G = (V,E), there is a polynomial time exact algo-rithm for finding a message m ∈ [0, 1]d that maximizesthe number of vertices that adopt the message. The al-gorithm runs in O(2nd) time.

Proof. We note that each of the hypercubes centeredaround the preference vectors of the vertices are axis-aligned with each other. Furthermore, it is known [5]that the intersection graph of axis-aligned hypercubessatisfies the Helly property. A family of sets F satisfythe Helly property if whenever a subfamily S ⊆ F hasnon-empty intersection for every pair of sets in S everypair of sets in S, then the whole subfamily has non-empty total intersection. Hence, finding the largest sub-set of hypercubes with a non-empty total intersectionis equivalent to the problem of finding the maximum-clique in the intersection graph of the hypercubes. Thisis solvable exactly in time O((2n)d) [9, 8]

An immediate corollary of this result gives us an ap-proximation for the case when p = 2.

Corollary 2 For p = 2 and constant d, given a com-plete graph G = (V,E) and a threshold vector t ={t1, t2, . . . , tn}, let m∗(G, t) be the optimal message in

8

[0, 1]d that maximizes the number of vertices adoptingthe message. Then, there is a polynomial time approxi-mation algorithm that finds a message m′ ∈ [0, 1]d suchthat N(m′) ≥ N(m∗(G, t ·

√d)) vertices, where N(m)

denote the number of vertices adopting a message m.The algorithm runs in O((2n)d) time.

The next lemma corresponds to the special case whenp = 2 and d = 2.

Lemma 3 For p = 2, given a complete graph G =(V,E), there is a 1+ε approximation algorithm for find-ing a message m ∈ [0, 1]2 that maximizes the number ofvertices that adopt the message. The algorithm runs inO(nε−2 log n) expected time.

Proof. The proof follows directly from Theorem 4.6of [2].

5.3.2 General GraphIn the case of a general graph, dependency constraints

arise for each node which are of the form - “a node seesa message only if there exists at least one path from thesender to the node along which all the nodes adopt andpropagate the message”. Thus, for the general graphcase with sender vertex vs, the optimization problemcan be formulated as follows:

argmaxm∈[0,1]d

|vt ∈ V : ‖m−mvt‖p ≤ tv and AdoptionPath(s, t) = 1|

(2)where AdoptionPath(s, t) = 1 if there exists a pathP = {vs, vp1 , . . . , vpk , vt} in G such that for all vpi ∈ P ,‖m−mvpi

‖p ≤ tvpiFor this problem, we provide an approximation algo-

rithm using epsilon-nets.

Lemma 4 Given a general graph G = (V,E), a sendervs ∈ V , and a threshold vector t = {t1, t2, . . . , tn}, letm∗(G, t) be the optimal message in [0, 1]d that maxi-mizes the number of vertices adopting the message. Forconstant d, there is a polynomial time approximationalgorithm that finds a message m′ ∈ [0, 1]d such thatN(m′) ≥ N(m∗(G, t + ε

√d · 1)) vertices, where N(m)

denote the number of vertices adopting a message m,and 1 denotes the all ones vector. The algorithm runs

in time O(|E| 1εd).

Proof. We first uniformly discretize the hypercube[0, 1]d. It is easy to see that the space [0, 1]d can be

covered with 1ε

dgrid hypercubes of size ε. We consider

message vectors corresponding to the centers of eachof the grid hypercubes. For every such message vec-tor, we can compute the number of vertices in G thatadopt the message by simulating the diffusion processover G in O(|E|) steps. Thus, by exhaustively enumer-ating over the centers of all the grid hypercubes, we

can find the optimal message vector m′ among them in

time O(|E| 1εd). Since each grid hypercube is of size ε,

the true optimal message m∗(G, t) has distance at mostε√d from m′. The result follows.

Next, we study through simulations the emergent be-havior implied by our core model and extensions.

6. SIMULATION EXPERIMENTSIn this section, we explore the large-scale cascades

that emerge from the micro-behavior model of inter-play between message senders and receivers. Using themodel presented in Section 5.2, our simulation tracksthe propagation of a message represented as a d-dimensionalvector through a social network. Given the network ofindividuals, each associated with its own d-dimensionalpreference vector and threshold t, and an initial startingnode, a message’s propagation is deterministic.

In the rest of this section, we first present details ofour simulation setup. Then, we characterize the rela-tionships between key parameters of the model and thevirality rate of information cascades. Finally, we studythe structural virality of information cascades generatedwithin our model, and find that our model capturesproperties of structural virality that have been empiri-cally observed but not captured by previous models [15].

6.1 Simulation DetailsIn each of our simulation experiments, we build a

1 million node network and execute 100k informationcascades. Our simulation model is governed by 2 pa-rameters that control the creation of the network struc-ture and the propagation of messages: d specifies thedimensionality of the latent message space and prefer-ence vectors; and t the maximum threshold acceptancedistance for re-broadcasting messages.

There are two key stages to building and runningan information dissemination model. The first stage isthe initialization of the network of connections amongnodes. The nodes of the network are each randomly as-signed a desired message vector and a random threshold≤ t. The network is initialized as an asymmetric net-work. Every node’s in-degree is assigned according to apower law distribution parameterized by α = 2.3.

The association of a preference vector to nodes allowsus to experiment with peer influence and homophilywithin our model. To do so, we add two additional pa-rameters to control the network initialization. To repre-sent the effect of homophily—that an individual is morelikely to create a social tie to a user similar to herself,we replace the random selection of a neighbor with apreferential selection, where the closest neighbor in thed-space is selected from a random candidate pool of size

9

0

0.5

1

1 10 100 1000

n=1

n=2

n=10

n=20

n=100

n=200

Figure 8: The out-bound degree distribution ofnodes.

n, and the candidate pool is fully replaced for each se-lection. While the distribution of out-bound degreesis not directly controlled, this process does generate aheavy-tailed distribution, as shown in Figure 8.

To represent peer influence, s controls the smoothingof preference vectors between a vertex in the graph andits neighbors. More formally, each node’s preferencevector mi ← (1− s) ∗mi + s ∗ avg(Ni) where Ni is theset of vertices neighboring i.

The second stage of the simulation—propagating amessage through the nodes of this network—maps di-rectly to the microbehavior of a sender and a recipientdescribed in Section 5.2. We represent each messageas a vector in the same latent message space associ-ated with the nodes in our network. Following the ba-sic model, whether or not a node will re-broadcast amessage it has received is determined by whether themessage lies within the threshold distance of the node’sown desired message vector.

6.2 Virality RateIn running our simulations, we validate that our model

generates information cascades that are broadly con-sistent with past models and observations of cascadeevents. We find that for certain parameterizations, namely,alpha ≈ 2.3, t ≈ 0.45, s ≈ 0.45, and 8 ≤ k ≤ 10, ourcascades fit previously empirically observed data. Forexample, we find that, consistent with past empiricalobservations, most cascades remain very small, and therate of large-scale diffusion events (i.e., events reachinggreater than 100 nodes) is roughly 1 in 1000 [15, 16].

Exploring the relationship between the four key pa-rameters of our model, we confirm that as the dimen-sionality d factor for the latent space increases, the vi-rality rate goes down (Figure 9(a)). The intuitive expla-nation is that as d increases, the latent space becomesmore sparse, and the likelihood of a message satisfyingthe preference vector and threshold of a sufficient num-ber of connected vertices decreases. Furthermore, as

the acceptance threshold t increases, the virality rateincreases as well (Figure 9(b)). Increasing either thesmoothing factor s or neighbor candidate pool size n,representing peer influence and homophily, we find thatthe virality rate increases (Figure 9(c) and 9(d)).

It is interesting to observe that in our simulationssuch relatively low values of d produce realistic cas-cades. An implication for the semantic interpretationof our model is that the variety of properties of mes-sages that people care about and examine when de-ciding whether to post or re-post a message may becorrespondingly small. While this may be intuitive, itis also possible that suitable parameterizations of ourmodel (higher values of t for example, or more struc-tured networks) might act as a counter-weight to enablehigher values of d to produce realistic cascades as well.The manifold of semantically meaningful combinationsof message properties within this space, and the resul-tant sparsity may also play a role. We leave furtherexperimentation in this area for future work.

6.3 Structural viralityStructural virality is a measure proposed by Goel et al. [15]

to measure and distinguish between information dissem-inations that occur primarily through broadcast mech-anisms (one sender broadcasting a message to a largenumber of people, with relatively few or no indepen-dent decisions to rebroadcast or spread the message);and propagations where no one sender is responsible formost of the dissemination (i.e., multigenerational “vi-ral” propagation). Specifically, structural virality v(T )over a diffusion tree T is defined to be the average dis-tance between all pairs of nodes:

v(T ) = 1/n(n− 1)

n∑i=1

n∑j=1

dij (3)

where distance dij is the shortest path length throughthe tree T between nodes i and j.

One of the elements of empirically observed informa-tion cascades that previous models have not capturedis the variance in structural virality observed in infor-mation cascades. In their large-scale study of informa-tion diffusion events in Twitter, Goel et al. find thatstructural virality is weakly correlated (0.36) with thesize of a cascade— smaller cascades are more likely tohave a broadcast-like spread, and larger cascades aremore likely to have a viral-like spread. However, as thecorrelation is weak, there are many large cascades thathave a broadcast-like spread, and many small cascadesthat have a viral-like spread.

Figure 10(a) and 10(b) plots on a log-scale the struc-tural virality of information cascades, binned by the sizeof the cascade under two information diffusion modes.Figure 10(a) shows structural virality under our model

10

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

7 8 9 10 11

vira

lity

rate

k, latent message space dimensionality

(a) Relationship between dimensionality d and viralityrate. n = 0.5,t = 0.5

00.020.040.060.08

0.10.120.140.160.18

0.35 0.45 0.55

rate

of

vira

lity

Acceptance threshold, t

(b) Relationship between acceptance threshold and vi-rality rate. d = 8,t = 0.5

0

0.02

0.04

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0.08

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0.16

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

rate

of

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lity

Smoothing factor

k=8

k=9

(c) Relationship between smoothing factor and viralityrate. d = 8,t = 0.5

0

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150

rate

of

vira

lity

neighbor candidate pool size

(d) Relationship between candidate neighbor pool sizevirality rate. d = 7,t = 0.5

Figure 9: Relationship between virality rate andmodel parameters

10

30

100

100 300 1000 3000 10000 30000

(a) Structural virality under a persuasion model. d =7,s = 0.45,t = 0.45

10

30

100

100 300 1000 3000

(b) Structural virality under a simple contagion modelwith uniform infection probability. r = 0.7

Figure 10: A boxplot of the structural viralityof cascade size in simulations. The x-axis plotscascade size (number of message recipients) andthe y-axis shows the structural virality. The boxplot demarcates the quartiles of the structuralvirality.

of persuasion effects. Overall, we see that structuralvirality ranges from broadcast-like (low values) to veryviral (high values). Within each bin of cascades, wealso see a high variance in structural virality. Fig-ure 10(b) shows structural virality under a simple con-tagion model, where the likelihood that any given nodewill propagate a message is β. We select β such thatr = βd; where r = 0.7, and d is the average degree ofthe network. Under this contagion model, we find thatcascades are less likely to have a highly viral structureand, as cascade size increases, the variance of structuralvirality decreases significantly.

7. CONCLUSIONS AND FUTURE WORK

11

We present a first study into analyzing the effectsof persuasion on well-studied social network phenom-ena such as information diffusion and propagation. Weperformed large-scale online user studies to motivatethe need for a model of persuasion that we then use toexplain structural properties of information diffusion.We leave the analysis of the model both in terms ofthe characterization of the resulting dynamics as wellas the performance of algorithms for message selection(e.g., see Section 5.3) to future work.

In addition to the analysis of the current model, thereare rich set of open questions we leave to future work.One deals with making the model richer to distinguishamong individual nodes or messages based on their vec-tor representation in the latent space allows us to asknew kinds of questions. These need to be asked and val-idated against empirical observations we make in Sec-tion 4. One direction would be to better explore impli-cations of homophily and peer influence; for example,model whether the same message in a different networklocation would likely generate the same cascade. (i.e.,to what degree are cascades caused by the message, theuser or the network structure? or all of the above! An-other example would be to broadly validate the modelthrough more and deeper user studies (e.g., user studiesacross more domains); explore model susceptibility tospecific persuasion techniques such as the authority ofthe sender, message repetition etc.

Finally, understanding the synergies between the senderand the recipient in terms of message content has widerimplications for for social systems engineering. For ex-ample, how do individuals learn what messages (i.e.,persuasion strategies) are effective? how does the feed-back they receive through social systems features (news-feeds, retweets, likes, comments, etc.) influence theirlearning?

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