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European Journal of Operational Research 234 (2014) 583–596

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European Journal of Operational Research

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Invited Review

Evolution of social networks

0377-2217/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ejor.2013.08.022

⇑ Corresponding author. Tel.: +49 521 106 67256.E-mail addresses: [email protected] (T. Hellmann), mathias.stau-

[email protected] (M. Staudigl).

1 ‘‘On-line’’ models are models in which the population is growing over timportant class of models contains the very popular preferential attachmen(Barabási & Albert, 1999), which we are not touching in this survey.(technical) summaries of these models can be found in Newman (2003b) arigorously, in Chung et al. (2006) and Durrett (2007).

Tim Hellmann, Mathias Staudigl ⇑Center for Mathematical Economics (IMW), Bielefeld University, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 October 2012Accepted 18 August 2013Available online 30 August 2013

Keywords:Evolution of networksGame theoryStochastic processesRandom graphs

Modeling the evolution of networks is central to our understanding of large communication systems, andmore general, modern economic and social systems. The research on social and economic networks istruly interdisciplinary and the number of proposed models is huge. In this survey we discuss a smallselection of modeling approaches, covering classical random graph models, and game-theoretic modelsto analyze the evolution of social networks. Based on these two basic modeling paradigms, we introduceco-evolutionary models of networks and play as a potential synthesis.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

The importance of network structure in social and economicsystems is by now very well understood. In sociology and appliedstatistics the study of social ties among actors is a classical field,known as social network analysis (Wasserman & Faust, 1994).More recently a large scientific community, including game theo-rists, economists, as well as computer scientists and physicists, rec-ognized the importance of network structure. In particular, thedynamic evolution of networks became an important question of re-search. Of course all these subjects put different emphasis on whatis considered to be a ‘‘good’’ model of network formation. Tradi-tionally, economists are used to interpreting observed networkstructure as equilibrium outcomes. Naturally, game theory is thepredominant tool used in this literature. Computer scientists, onthe other hand, prefer to think of network formation in terms ofdynamic algorithms. Finally, physicists tend to think of networksas an outgrowth of complex system analysis, where the main inter-est is to understand and characterize the statistical regularities oflarge stochastic networks. Given this interdisciplinary characterof the subject, the number of publications is enormous, and it isimpossible to provide a concise survey covering the plethora ofmodels developed in each of the above mentioned disciplines.For this reason, we have decided to focus in this survey on two,in our opinion, particular promising approaches to model the evo-lution of social and economic networks. We concentrate on dy-namic models of network formation, using elements fromrandom graph theory and game theory. These two approaches ma-

tured over the years, and some recent efforts have been made tocombine them. This article summarizes a small body of thesetwo streams of literature; it is our aim to convince the reader thatrandom graph dynamics and (evolutionary) game theory havemany elements in common, and we hope that this survey providessome ideas for future research on this young and interdisciplinarytopic. However, before jumping into the details, let us give a shortoverview of topics which this survey covers, a pointer to the fur-ther relevant literature, and an acknowledgment of the literaturewhich we shamefully exclude.

Section 2 starts with a short discussion of random graph mod-els. These models are the basis for the statistical analysis of net-works and have had a large impact on theoretical models ofnetwork evolution. Random graph models have a long traditionin social network analysis, and are the foundation of the recent lit-erature on network evolution in computer science, mathematicsand physics. Following the terminology of Chung et al. (2006),we focus on ‘‘off-line’’ models. Hence, we consider network forma-tion models in which the number of nodes (the size of the popula-tion) is a given parameter.1 The most general random graph modelintroduced in this survey belongs to the family of inhomogeneousrandom graph models (Bollobás, Janson, & Riordan, 2007). This classof random graphs is rather rich. It contains well-known stochasticblock-models (Karrer & Newman, 2011), variants of the importantexponential random graph (Snijders, Pattison, Robins, & Handcock,2006), and the classical Bernoulli random graph model as specialcases. Since these models are very well documented in the literature,

ime. Thist modelsExcellentnd, more

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584 T. Hellmann, M. Staudigl / European Journal of Operational Research 234 (2014) 583–596

we are satisfied with a short summary of their main properties. Themain purpose of this section is to introduce concepts, and to famil-iarize the reader with our notation. Nevertheless, we consider ran-dom graph models as the building block to synthesize randomgraph models with dynamic game theoretic ideas, and thereforewe think it is useful to introduce them already at the beginning.

Section 3 presents an alternative approach of modeling net-works, mainly developed by economic theorists. It uses game the-oretical concepts to interpret network structures as equilibriumphenomena of strategically acting players who create and destroylinks according to their own incentives. Our discussion is centeredaround two particularly important concepts: The semi-cooperativesolution concept of pairwise-stability (Jackson & Wolinsky, 1996),and various modifications of Nash equilibrium. These concepts arestatic equilibrium notions, and the natural question which comesto one’s mind is whether there are natural dynamic models sup-porting predictions based on such equilibrium concepts. This ques-tion leads us to consider an evolutionary approach to networkformation. These models are used as an introduction to a particularinteresting class of dynamic network formation models, called co-evolutionary processes of networks and play in Staudigl (2010). Sec-tion 4 is devoted to illustrate these types of models, which is ourmodest attempt to synthesize the strategic approach of networkformation with a random graph approach. Section 5 summarizesthe main points contained in this article and discusses some ideasfor potential future research.

1.1. What this survey does not cover

Given the enormous number of network formation models, itwas necessary to be selective in writing this survey. Hence, thereare many important network models which we were not able tocover. Some notable omissions are the following; we restrict thediscussion to network models where the size of the graph is fixed(‘‘off-line’’ models). Networks with variable number of nodes(‘‘on-line’’ models) are of course important, but they require differ-ent mathematical tools to be analyzed successfully. In particular,we believe that a different game theoretic approach would beneeded to study such models.2 Readers interested in the mechanicsof growing networks should consult the book by Dorogovtsev andMendes (2003), and the survey article by Albert and Barabási(2002). Another class of networks which we will not consider areweighted graphs. These models are very important for applications,and the physics community provides many interesting approachesto model such networks (see e.g. Barthélemy, Barrat, Pastor-Satorras,& Vespignani (2005) and Kumpula, Onnela, Saramäki, Kaski, & Ker-tész (2007), and the references therein). There are also somegame–theoretic models on the formation of weighted networks,see e.g. Bloch and Dutta (2009). These are just preliminary studies,and we have the feeling that much more work on these kinds of net-works will be needed before they should be included in a survey. Fi-nally, we would like to point out that our main focus will be onevolutionary models of undirected networks. This does not mean thatwe think directed networks are less important. In fact, many real-world networks, such as traffic networks or the world-wide-web,are more naturally interpreted as directed graphs. However mostof the game-theoretic concepts emphasize bilateral externalities,and thus admit a cleaner interpretation when links are undirected.

2 For instance, a basic strategic decision in a network model with variable numberof players is when and whether a single player should enter or leave the network. SeeDutta, Ghosal, and Ray (2005) for a model in this direction. In evolutionary gametheory, the standard model assumes a constant population size. However, it seems tobe likely that allowing for population growth introduces for new phenomena absentfrom the stationary world. See Sandholm and Pauzner (1998) for an interesting studyin this direction. A recent model of economic network formation in a non-stationaryenvironment is Jackson and Rogers (2007).

Nevertheless, we provide some discussion of directed networks. Inparticular, the random graph models of Section 2 can be used tomodel the evolution of directed, as well as undirected networks,after straightforward modifications.

1.2. Related literature

Most of the models which found no space in this article havebeen surveyed elsewhere. We just provide some cross referencesto the literature, and urge the reader to consult the referencestherein. Recent textbooks containing in-depth discussions on dy-namic network formation are Chung et al. (2006) and Durrett(2007). These books focus on the mathematical aspects, and onlymention rudimentary applications. With a slight bias towards eco-nomics applications, we recommend the beautiful books by Goyal(2007), Vega-Redondo (2007), Jackson (2008), and Easley andKleinberg (2010). Additionally to textbook treatments, the readermay want to consult one of the numerous survey articles available.We just mention Jackson (2003), Newman (2003b), Jackson (2005),Van den Nouweland (2005), Goyal (2005), Goldenberg, Zheng,Fienberg, and Airoldi (2009). At the heart of our discussion is theevolution of networks, and we use the language of stochastic pro-cesses and/or game theory to formalize our ideas. Additionally wetry to highlight the potential connections between these twoseemingly separate modeling strategies. We hope that this inte-grated perspective on the evolution of networks makes this surveya good contribution to the literature and will inspire some peopleto work on this fascinating topic.

2. Stochastic models of network evolution

A statistical analysis of networks is usually based on elementsof random graph theory. Random graph models are very flexiblemathematical representations of interdependency relations. In-deed, the motivation for such classical models as the Markov graphmodel of Frank and Strauss (1986) was to develop a tractable rep-resentation of complicated interdependencies in empirical data.Random graph models in general provide a flexible framework toconstruct statistical ensembles of networks (i.e. probability spaces),which are parsimonious enough to get pointed predictions, andrich enough to be able to reproduce as many stylized facts the re-searcher is aiming to model. In this section our discussion is cen-tered around a rather general class of a random graph process,which will be used as one pillar in our development of co-evolu-tionary processes of networks and play, starting in Section 4.

2.1. Random graphs

In this survey we use the terms ‘‘networks’’ and ‘‘graphs’’ inter-changeably. A graph is a pair G = ([N],E), where [N] :¼ {1, 2, . . . , N}is the set of vertices (or nodes), and E � E(G) � [N] � [N] is the setof edges (or links). This notation applies to directed as well as undi-rected networks. If G is a directed graph, then (i, j) 2 E means thatthere is a directed edge from i to j. If G is undirected, then(i, j) 2 E(G) if and only if (j, i) 2 E(G). In such a symmetric setting itwill be convenient to use the shortened notation ij for the link con-necting node j and i. We denote the collection of graphs on N ver-tices by G½N�. Elements of this set can either be directed orundirected networks, depending on the context.

A random graph is a probability space ðG½N�;2G½N�;PÞ. The proba-bility measure P : 2G½N� ! ½0;1� assigns to each graph a weight,which should reflect the likelihood that a certain graph G is drawnfrom the set G½N�, when performing a statistical experiment withdistribution P. Thus, the underlying reference measure P it ischosen by the modeler. A historically very important random

T. Hellmann, M. Staudigl / European Journal of Operational Research 234 (2014) 583–596 585

graph model is the Erdös–Rényi graph (or Bernoulli graph).3 TheBernoulli random graph is built on the assumption that edges areformed independently with constant probability p 2 [0,1]. Thus, letG½N� be the set of undirected graphs constructable on N vertices.Then the random graph model ðG½N�;2G½N�;PÞ is determined by theprobability measure

PðfGgÞ :¼YNi¼1

Yj>i

pAijðGÞð1� pÞ1�AijðGÞ; ð1Þ

where Aij(G) = 1 if and only if (i, j) 2 E(G), and 0 otherwise.A(G) = [Aij(G); 1 6 i, j 6 N] is the adjacency matrix of the graph G.4

The advantage of the Bernoulli graph model is its simplicity. Infact, the complete random graph is described by two parameters:the population size (N), and the edge-success probability (p). Thismodel is the benchmark of random graph theory and is by nowvery well understood. A detailed analysis can be found in Bollobás(2008). It comes with little surprise that such a simple model israrely a good description of a real-world network. In particular,the lack of correlation across links is a well-known deficit of the Ber-noulli graph. However, it still constitutes an important benchmark(Newman et al., 2003a). Compared to random graphs, real worldnetworks are often observed to have similarly small average pathlength (an effect coined small worlds phenomenon, or six degreesof separation),5 higher clustering (friends of friends are more likelyto be friends),6 exhibit homophily (connections between nodes ofsimilar kind are more likely),7 and often exhibit a power law degreedistribution (more nodes with very high and very low degree com-pared to Bernoulli graphs, Newman, 2003b). For a more detailed dis-cussion we refer the reader to Vega-Redondo (2007) and Jackson(2008).

A different example for a commonly used random graph mea-sure is the exponential random graph (ERG). This class of randomgraph models builds on the Markov graph models of Frank andStrauss (1986), and is also known as p⁄-models (Robins, Pattison,Kalish, & Lusher, 2007). This class of models is very rich, and itsmain strength are its tractability and ability to reproduce networkswith arbitrary degree distributions, and clustering. Let us give asimple example for an exponential random graph model. For moreinformation we refer to the excellent paper by Snijders et al.(2006), where a state-of-the-art survey can be found.8 A simpleversion of the exponential random graph model is given by the expo-nential family

PbðfGgÞ :¼ expXk

i¼1

biTiðGÞ � wðbÞ !

; ð2Þ

where ~b ¼ ðb1; . . . ;bkÞ is a vector of real parameters, Ti(G) are func-tions on the space of graphs, i = 1, . . . , k for some k 2 N, and w(b) isthe normalizing constant (the ‘‘partition function’’). The functions Ti

are usually taken to be simple statistics of the graph, such as the de-gree sequence of the graph, or the number of edges. Depending onthe sign of the parameter bi, the value of the function Ti(G) deter-mines the probability that the graph G is drawn from the wholespace of graphs when using the measure Pb. Similar exponential

3 Erdös and Rényi (1959, 1960) introduced the slightly different model in which thenumber of vertices and edges are given parameters. The Bernoulli graph model is dueto Gilbert (1959).

4 It is clear that if G is undirected, then A(G) is symmetric. For directed graphs thissymmetry may not hold.

5 A classical reference is the famous letter experiment of Milgram (1967). Otherstudies include Dodds, Muhamad, and Watts (2003) and Goyal, Van Der Leij, andMoraga-González (2006).

6 See e.g. Watts and Strogatz (1998), Watts (1999) and Newman (2003b, 2004).7 See McPherson, Smith-Lovin, and Cook (2001) and Golub (2012), among others.8 Despite its huge importance, the ERG model has technical problems. See

Chatterjee and Diaconis (2011).

families for graph measures will appear in Section 4 of this survey.As a final word on this family of models, let us show that in fact theERG contains the Bernoulli graph. To wit, let G½N� be our space ofundirected networks, and take T1(G) :¼ jE(G)j the number of edges,and b1 = b, bi = 0 for i P 2. Setting p :¼ eb/(1 + eb), one immediatelysees that

PbðfGgÞ ¼1

expðwðbÞÞ exp bXi;j>i

AijðGÞ !

¼YNi¼1

Yj>i

pAijðGÞð1� pÞ1�AijðGÞ:

2.2. Network formation as a stochastic process

A classical approach in social network theory is to view the evo-lution of a network as a stochastic process (see Wasserman (1980)for an early contribution). Snijders (2001) constructs a dynamicERG which can be analytically solved, and numerically simulatedvia MCMC-methods. In the following we will describe a fairly gen-eral dynamic network formation model in terms of continuous-time Markov jump processes.

In a dynamic model of network formation we would like to cap-ture two things: First, the network should be volatile; i.e. links aredeleted and formed over time. Second, the likelihood that a link isformed or destroyed, should depend on the attributes of the verti-ces in the graph. The following network formation algorithm cap-tures both of these requirements. We are given some probabilityspace ðX;F ;PÞ. The sample space X might be larger than the setof graphs.9 We call a Markov jump process fcðtÞgt2Rþ a random graphprocess if each c(t) is a G½N�-valued random variable, measurablewith respect to some given filtration fF tgtP0 containing the filtrationgenerated by the random variables {c(t)}tP0. We model the dynamicevolution of the network via the following steps:

Link creation: With a constant rate k P 0 the network isallowed to expand. Let W : G½N� ! RN�N

þ be a bounded matrix-valued function, whose components wij(G) define the intensitiesof link formation between vertex i and j. The function W will becalled the attachment mechanism of the process.Link destruction: Let n P 0 denote the constant rate of linkdestruction. Let V : G½N� ! RN�N

þ be a bounded matrix-valuedfunction, whose components vij(G) define the intensities of linkdestruction between vertex i and j. The function V is calledthe volatility mechanism of the process.

In some models of network formation the rate of link destruc-tion n has been interpreted as environmental volatility (Ehrhardt,Marsili, & Vega-Redondo, 2006; Marsili, Vega-Redondo, & Slanina,2004). This is also the motivation behind our definition of a volatil-ity mechanism. In principle the intensities of link creation anddestruction can be asymmetric, i.e. we do not require in the con-struction that wij(G) = wji(G), or vij(G) � vji(G), respectively. Hence,in principle the network formation process can be used to modelthe formation of directed as well as undirected networks. This sim-ple model of an evolving network, will be the building block of ourconstruction of a co-evolutionary model of networks and play inSection 4.

2.2.1. Inhomogeneous random graphsLet G½N� be the set of all undirected graphs. Suppose that the

intensities of link creation and link destruction are given by thefunctions

wijðGÞ ¼ ð1� AijðGÞÞjij; and v ijðGÞ ¼ AijðGÞdij: ð3Þ

9 This will be necessary in order to model the co-evolution of networks and play inSection 4.

11 In this figure, the star is presented as an undirected network. When consideringdirected networks, there are several stars: we will call a star network a center-sponsored star if all links are directed from the center to the periphery players.Likewise, in a periphery sponsored star, links are directed from the peripheral playersto the center.

12


The scalars jij, dij are positive and symmetric, meaning that jij � jji

and dij � dji for all i,j 2 [N]. Additionally we assume that k = n = 1.This essentially says that the processes of link creation and linkdestruction run on the same time scale. Then the following generalpicture emerges.

Theorem 2.1 Staudigl (2013). Consider the random graph processfcðtÞgt2Rþ with attachment and volatility mechanism W and V givenby the functions specified in Eq. (3). Then the graph process is ergodicwith unique invariant measure

PðfGgÞ ¼YN

i¼1

Yj>i

ðpijÞAijðGÞð1� pijÞ

1�AijðGÞ; ð4Þ

where pij ¼jij

jijþdijis the edge-success probability of vertex i and j.

The random graph measure (4) describes the probability spaceof an inhomogeneous random graph. The wonderful work of Bol-lobás et al. (2007) studies this model in quite some detail.10 It con-tains the Erdös–Rényi model as a special case by setting jij � j anddij � d. Moreover, it generalizes certain networks based on clusteringnodes according to some notion of ‘‘similarity’’, as explained in thenext subsection.

2.2.2. Multi-type random networksIn general, networks are complex objects and therefore difficult

to analyze. However, it is often the case that vertices in a networkcan be classified to belong to certain groups. Indeed, a prevalentfact in social networks is the phenomenon of homophily, meaningthat vertices of similar characteristics are more likely to be con-nected with each other. Therefore, Fienberg, Meyer, and Wasser-man (1985) introduced blockmodels, where nodes are categorizedto belong to certain subgroups. Each subgroup may have its ownlaw of network formation. Recently, this type of networks has alsobeen used in economic theory (Golub, 2012), where it has beencalled a multi-type random network.

Suppose that the set of vertices can be partitioned into finitelymany types k 2 {1, . . . , m}. The number of nodes of type k is nk,where 0 6 nk 6 N. The vector ~n ¼ ðn1; . . . ;nmÞ defines the partitionof the population of nodes into its types. The number nk -2 {0, 1, . . . , N} can be either deterministically given, or representthe realization of a random variable. Assume that the intensitiesof link formation and link destruction are wij(G) = (1 � Aij(G))jrl,vij(G) = Aij(G)drl, whenever vertex i is a member of group r, and ver-tex j is a member of group l for 1 6 r, l 6m. The model then reducesto a special case of the inhomogeneous random graph. The edge-success probabilities between members of group r and group lare given by prl ¼

jrljrlþdrl

. This grouping, of course, reduces the com-plexity of the random graph model tremendously. Compared to theinhomogeneous random graph, the multi-type random graph iscompletely specified by the partitioning of the vertices, and thegroup-specific edge-success probabilities.

2.3. Random graphs versus strategic network formation

Our proposed strategy to model dynamic network formation israther parsimonious. The stochastic process is entirely specified bythe attachment and the volatility mechanism. Particularly interest-ing examples are the inhomogeneous random graph model, con-taining the multi-type random network. All these models rely ona special choice of the intensities of link creation and link destruc-tion. These rates can in principle be fitted from data. Thus, the sta-tistical models are useful to describe how networks form and

10 See also Söderberg (2002) and Park and Newman (2004) for earlier treatments ofthis model.

evolve from an observer’s point of view. They are not able toexplain why networks form and evolve, without adding a comple-mentary theory of network formation. Thus, to complete the studyof network formation, we have to go to the micro level and under-stand the forces that drive the nodes to connect to each other. Eco-nomic and game theoretic reasoning provides the natural languagefor this task, and Section 3 summarizes this field to some extend.Using these game theoretic concepts, Section 4 presents somemodels which combine elements from random graph theory andgame theory.

3. Game theoretic models of network evolution

In social and economic network a link is often interpreted as aform of revealed preference: Two players are connected with eachother, because it is in their self-interest to have the link. With otherwords, individuals are connected with each other because a payoffor utility is derived from these connections. Frequently the linkingdecisions of a single individual affects the utilities of other mem-bers of the society. To understand networked systems in situationswith interdependent preferences, game theory has been devel-oped. Using game theoretic reasoning, networks are interpretedas equilibrium outcomes implemented by rational players. Thismodeling approach is potentially useful as it allows us to rational-ize predicted or observed network structure in terms of socioeco-nomic behavior. A purely statistical approach to networkformation cannot provide us with such a ‘‘micro-foundation’’ of ob-served networks. Moreover, the game theoretic approach allowsthe researcher to understand the mechanisms underlying the for-mation of the network. As a simple example, consider a situationwhere a statistician observes a networked system with a center-sponsored star architecture, such as the one depicted in Fig. 1(i).11

Having collected these data, a game theoretic model can now(potentially) be used to postulate preferences (i.e. utility func-tions), rationalizing this observed interaction structure.12

3.1. Networks and utilities

A utility-based approach to network formation is ratherstraightforward. For each player, there is a utility functionui : G½N� ! R. An n-tuple of functions u = (u1, . . . , uN) representsthe preferences of the society over network architectures. Let usstart with some concrete examples.

3.1.1. The connections modelResearch in sociology has emphasized the role of centrality in

social networks.13 Centrality measures are based on an assessmentof the graph distance between two nodes. The following examplesuses a version of this distance measure, called decay centrality, toconstruct a model of strategic network formation. Consider a givengraph G = (V,E). Let dij(G) denote the geodesic distance between ver-tex i and j, i.e. the length of the shortest path connecting i and j. Ifnode i cannot be connected to j, then set dij(G) =1. A measure ofcloseness should then be decreasing in distances. Thus, let d 2 (0,1)be a given parameter. The decay centrality between vertex i and jis then defined as CLijðGÞ :¼ ddijðGÞ. It has a natural interpretation interms of a probabilistic communication model. Let d 2 (0,1) be the

See Echenique, Lee, Shum, and Yenmez (2013) for a very recent paper in thisdirection.

13 See, e.g. Freeman (1979) or Bonacich (1987) for an intriguing discussion ofcentrality measures.

Fig. 1. Some important network architectures.


probability that communication between adjacent nodes is success-ful. Then, if the signal transfer is independent, the probability thatthe signal reaches vertex j from vertex i with dij(G) = k is on the orderof dk. Jackson and Wolinsky (1996) introduce a model were a player’sutility is increasing in closeness centrality, but there is a cost to bepaid for each maintained link. Specifically, let wij be the intrinsic va-lue of a communication of player i with player j, and cij > 0 be a costwhich player i has to pay when a direct connection with j is main-tained. The utility function for player i 2 [N] is given by

uCoi ðGÞ ¼

Xj–i

wijddijðGÞ �

Xj2NiðGÞ

cij ð5Þ

where Ni(G) :¼ {j 2 N:ij 2 G} is the set of i0s neighbors. This is a ver-sion of the connections model due to Jackson and Wolinsky (1996).

3.1.2. Local complementarities and Bonacich centralityBonacich (1987) introduced a parametric family of centrality

measures in order to formulate the intuitive idea that the centralityof a single node in a network should depend on the centrality of itsdirect neighbors. This self-referential definition of centrality leadsto an eigenvector-based measure, which can be derived from basicutility-maximization ideas, as shown by Ballester, Calvó-Armengol,and Zenou (2006). Let A be the adjacency matrix of a given net-work. The powers of this matrix give us information of the connec-tivity structure of the network.14 Let b > 0 be a given parameter,chosen in such a way that the following matrix power series exists15:Bðb;AÞ ¼

P1n¼0Anbn ¼ ½I � bA��1. The centrality index proposed by

Bonacich (1987) is defined as

bða; b;AÞ ¼ Bðb;AÞa; ð6Þ

where a> = (a1, . . . , aN) is a given vector of individual characteristicsof the vertices.16 Ballester et al. (2006) show that this centralitymeasure is actually a Nash equilibrium of an interesting class ofnon-cooperative games. There are N agents who are involved in ateam production problem. Each player chooses a non-negative quan-tity xi P 0, interpreted as efforts invested in the team production.High efforts are costly, and the level of effort invested by the other

14 Indeed, A~1 is a N � 1 matrix whose entries are just the degree of the individualnodes. The vector A2~1 counts the number of walks of length 2 starting from theindividual nodes, and so on.

15 The necessary condition for this to be the case is that 0 < b < k1(A)�1, where k1(A)is the eigenvalue of A having largest modulus.

16 The original Bonacich centrality index is defined as c½I � bA��1A~1. Formula (6) is atrue generalization by setting a ¼ cA~1.

players affects the utility of player i. To capture these effects, we as-sume that the payoff of player i from an effort profile x = (x1, . . . , xN)is given by uiðx1; . . . ; xNÞ ¼ aixi � 1

2 x2i þ bxi

PNj¼1Aijxj. The players

choose their efforts independently, and in a utility maximizingway. It can be shown that this game has a unique Nash equilibriumx⁄ given by x⁄ = b(a,b,A). Hence, the equilibrium effort invested byplayer i depends only on his centrality in the network with adja-cency matrix A. Given the network G, and parameter vectorða;bÞ 2 RNþ1, so that (6) is well defined, we can then compute theequilibrium payoff of player i as

uBCi ðGÞ ¼

12

biða; b;AðGÞÞ2 ð7Þ

This utility function expresses now preferences over a set of possi-ble network architectures underlying the team production problem.

3.1.3. R&D collaborations between firmsThe above example made the interesting point that equilibrium

structure of non-cooperative games can be closely linked to theunderlying interaction structure of the agents. Another examplewhere this structure is clearly visible is the model due to Goyaland Joshi (2003). In this model the nodes in the graph are firmswhich produce a homogeneous product and compete in quantitieson a single market. Additionally firms can agree to form bilateralR&D collaborations which lower their marginal cost of producingthe output. Let us assume that marginal costs of production aregiven by wi(G) = c0 � c1gi(G), with parameters c0; c1 2 Rþ such thatc1 <

c0N�1 and gi(G) :¼ jNij denoting the firm i’s degree. The output of

firm i is denoted by xi, and the market is completely described by acommonly known inverse demand function P(X) = max[0,a � X],where X P 0 is the total output of the industry, and a P 1 is thesize of the market. The profit function of firm i is then given by

uiðx1; . . . ; xNÞ ¼ xi Pðxi þP

j–ixjÞ � wiðGÞ� �

. For a given network of

R& D collaborations, it can be shown that the quantity choice gamehas a unique Nash equilibrium in which each firm producesx�i ðGÞ ¼ a� X� � wi, with X� ¼ Na

Nþ1� 1Nþ1

Pjwj. The resulting equilib-

rium profit of firm i is

uR&Di ðGÞ :¼

ða� c0Þ þ Nc1giðGÞ � c1

Xj–i

gjðGÞ

N þ 1

0BB@

1CCA

2

ð8Þ


Again we see that the equilibrium industry performance can be as-sessed by knowing the collaboration structure of the firms.

3.2. Pairwise stability and refinements

One of the most important equilibrium notations in the strategicapproach to network formation is the concept of pairwise stability.This concept considers the differential impact an added link has onthe utility of the involved parties. Thus, let us define player i’s utilitydifferential of a set of links l # E(G) deleted from the network G as

muiðG; lÞ :¼ uiðGÞ � uiðG� lÞ: ð9Þ

where G � l denotes the network obtained by deleting the set oflinks l # E(G) from network G. Similarly, G l denotes the networkobtained by adding the set of links l # E(Gc)nE(G) to G. In the fol-lowing definition Gc denotes the complete undirected network.

Definition 3.1 Jackson and Wolinsky (1996). A network G 2 G½N�is pairwise stable (PS) if

(i) for all ij 2 E(G):mui(G, ij) P 0 and muj(G, ij) P 0, and(ii) for all ij 2 E(Gc)nE(G):mui(G ij,ij) > 0)muj(G ij, ij) < 0.

Thus, a network is pairwise stable, if (i) no player has an incen-tive to delete one of her links and (ii) there does not exists a pair ofplayers who want to form a mutually beneficial link. Pairwise sta-bility can be reformulated in terms of a simple test performed onthe set of all possible links. Define

ADDðGÞ :¼ fij 2 EðGcÞ n EðGÞjmuiðG ij; ijÞ> 0 & mujðG ij; ijÞP 0g; ð10Þ

the set of links that can be added to G, and similarly

DELðGÞ :¼ ij 2 EðGÞj9k 2 fi; jg : mukðG; ijÞ < 0f g; ð11Þ

the set of links that can be deleted from G. Then G is a pairwise sta-ble network if and only if ADD (G) = DEL (G) = ;.

One potential limitation of Pairwise stability is that it only con-siders single link changes in each updating step. A natural refine-ment is thus to consider the formation of mutually beneficiallinks as well as the deletion of more than one link in one period.Such an extended stability concepts has been used in the literatureas well, running under the name of Pairwise Nash Stability.17

Let Ei(G) :¼ {ij 2 E(G)jj 2 [N]} denote the set of player i’s links in G,and E�i(G) :¼ E(G)nEi(G) denote the set of links in G in which playeri is not involved. A network is pairwise Nash stable (PNS) if

(i) ui(G) P ui(G � li) "li � Ei(G), and(ii) mui(G ij, ij) > 0)muj(G ij, ij) < 0 "ij 2 E(Gc)nE(G).

Thus, a network that is pairwise Nash stable is also robustagainst the deletion of a set of links by any player, while pairwisestability only considers one link at a time.

Beyond these commonly used stability notions, several refine-ments have been introduced in the literature. Some concepts,which we do not discuss in this survey, are the following:

strong and weak stability (Dutta & Mutuswami, 1997), bilateral stability (Goyal & Vega-Redondo, 2007), pairwise stability with transfers (Bloch & Jackson, 2007), strict pairwise stability (Chakrabarti & Gilles, 2007),

17 Pairwise Nash stability was first discussed in Jackson and Wolinsky (1996). It hasbeen used (without formal definition) in Goyal and Joshi (2003), Belleflamme andBloch (2004) and has been defined in, e.g. Gilles (2005), Bloch and Jackson (2006), andCalvó-Armengol and Ilkiliç (2009).

unilateral stability (Buskens & van de Rijt, 2008) cognitive stability (Gallo, 2012), and (strict) Nash stability (supported by (strict) Nash equilibrium in

a link formation game due to Myerson (1991)).

3.3. Existence of stable networks

The concept of pairwise stability is one of the most importantequilibrium concepts used in modern network theory. In this sec-tion we address the question of existence of PS-networks usingtwo different techniques. The first technique uses the potentialfunction method (Monderer & Shapley, 1996; Nisan, Roughgarden,Tardos, & Vazirani, 2007). The second technique emphasizes theincentive structure directly. We start with the important conceptof an improving path.

Definition 3.2. An improving path from network G to network G0 isa sequence of networks (G1, . . . , GK) such that G1 = G, GK = G0, andfor all 1 6 k < K, it holds that either

(i) Gk+1 = Gk � ij and ij 2 DEL (Gk), or(ii) Gk+1 = Gk ij and ij 2 ADD (Gk).

Hence, in an improving path each graph Gk+1 is obtained byeither deleting a single link, contained in DEL (Gk), from the prede-cessor Gk, or adding a single link, contained in ADD (Gk), to the pre-decessor Gk. The algorithm is running as long as there exists aplayer who wants to delete a link, or there exists a pair of playerswho want to form a link connecting them. It is clear that if thisalgorithm converges, then the limit network must be pairwise sta-ble. Therefore, the set of improving paths emanating from a PS net-work is empty. Based on this concept, Jackson and Watts (2001)define a cycle C as an improving path (G1, . . . , GK) such thatG1 = GK. They call a cycle C closed, if for all networks G 2 C theredoes not exists an improving path leading out of it. Jackson andWatts (2001) show that for any profile of utility functions u, thereexists at least one pairwise stable network or a closed cycle of net-works. The idea to prove existence of a PS network is now to ex-clude the existence of cycles. A well-known sufficient conditionruling out cyclic behavior in non-cooperative games is the exis-tence of a potential function (Monderer & Shapley, 1996). In a sim-ilar way, Jackson and Watts (2001) rule out the existence ofimproving cycles by imposing the existence of a function w whichcould be called a ‘‘potential function’’ as well.

Proposition 3.3 Jackson and Watts (2001). If there exists a functionw : G½N� ! R such that ij 2 ADD (G) if and only if w(G ij) > w(G) andij 2 DEL (G) if and only if w(G � ij) > w(G) for all G 2 G½N�, then thereexist no cycles. If u exhibits no indifference then there exist no cyclesonly if such a function exists.

The ‘‘no-indifference’’-condition ensures that for two networksG, G0, separated by one link, the two players affected by that linkhave a strict preference ordering between the two networks. Thefunction w acts as an ordinal potential function.18 Ordinal potentialsmust respect the incentive structure of the underlying utility func-tions, and thus not every game will have an ordinal potential. Asan example, the connections model of Section 3.1.1 has an ordinalpotential only for certain parameter values (see Jackson & Watts,2001).

As already indicated, the existence of an ordinal potentialfunction is stronger than necessary to show existence of pairwisestable networks. Moreover, it is not easy to check whether a utility

18 Existence of such a function is implied by the existence of an ordinal networkpotential. See also Chakrabarti and Gilles (2007) and Tardos et al. (2007).


function delivers such a function.19 Externality conditions onmarginal utilities are, however, easy to check. Hellmann (2013) usessuch externality conditions directly to prove an existence result ofPS-networks. Say a profile of utility function exhibits ordinal convex-ity in own links if

muiðG ij; ijÞP 0 )muiðG li ij; ijÞP 0

for all G 2 G½N�; li # EiðGcÞ n EiðGÞ and for all ij 2 Ei(Gc)nEi(G li).20 Inwords, if in G the link ij is profitable for player i, then it remainsprofitable in the graph G0 = G li which is obtained from G by addinga set of links li. Say a utility function exhibits ordinal strategiccomplements (Goyal & Joshi, 2006; Hellmann, 2013) if, for allG 2 G½N�, for all l�i # E�i(Gc)nE�i(G), and for all ij 2 Ei(Gc)nEi(G) one has

muiðG ij; ijÞP 0)muiðG l�i ij; ijÞP 0:

In an environment in which players’ utility functions exhibitordinal convexity in own links and the ordinal strategic comple-ments property, Hellmann (2013) reports the following result.

Proposition 3.4 Hellmann (2013). If a profile of utility functionsu = (u1, . . . , un) satisfies ordinal convexity in own links and the ordinalstrategic complements property, then:

(1) There does not exist a closed improving cycle.(2) There exists a PNS (and hence also a PS) network.

Compared to Proposition 3.3 cyclic behavior is not ruled outcompletely by assuming ordinal convexity and the ordinal strate-gic complements property. Only the existence of closed cycles isexcluded.

Example 3.5. An example, where the conditions of Proposition 3.4are satisfied, is given by uBC defined in Section 3.1.2, which may alsoinclude linking costs. Applying Proposition 3.4, there exists a PNSnetwork. Moreover, because all players have the same utilityfunction, one can easily conclude that if the empty network is notPNS then the complete network is uniquely PNS and if the completenetwork is not PNS then the empty network is uniquely PNS.

Remark 3.6. When imposing negative externalities (i.e. ordinalconcavity and ordinal strategic substitutes), existence of a PS net-work cannot be guaranteed, but several uniqueness conditionsare satisfied. Hellmann (2013) shows that no pairwise stable net-work can contain or can be contained in another pairwise stablenetwork. Furthermore conditions for existence of a globally uniquestable network are given (for details see Hellmann, 2013).

3.4. Stable networks in the connections model

To derive structural properties of stable networks beyond exis-tence and uniqueness, more assumptions have to be imposed onthe players’ utility functions. A particular nice environment isdescribed by the connections model (Jackson & Wolinsky, 1996).Let us focus on the special case, known as the symmetric connectionsmodel

uCoi ðGÞ ¼

Xj–i

ddijðGÞ � cgiðGÞ: ð12Þ

19 For recent efforts to characterize the class of potential games see Sandholm(2010b) and Candogan, Menache, Ozdaglar, and Parrilo (2011).

20 For the various definitions of convexity of a network utility function the reader isreferred to Goyal and Joshi (2006), Bloch and Jackson (2007), Calvó-Armengol andIlkiliç (2009), Hellmann (2013). It is shown in Hellmann (2013) that all thesedefinitions are equivalent. The weaker ordinal version is defined in Hellmann (2013).

For this utility function Jackson and Wolinsky (1996) partially char-acterize the set of pairwise stable networks.21

Proposition 3.7 Jackson and Wolinsky (1996). Suppose the utilityfunction is given by (12). Any pairwise stable network has at most onenon-singleton component. For c < d � d2 the complete network isuniquely pairwise stable, for d � d2 < c < d the star is among the stablenetworks and for c > d the empty network is among the stablenetworks and no stable network can contain a player with only onelink.

The star network, as visualized in Fig. 1(i), is the network whereone player (the center) has links to all other players (the periphery)and those players do not have any other links. Beside the star net-work, many more networks can be pairwise stable.

For the general utility function (5), Jackson and Wolinsky(1996) do not report any stability results. Johnson and Gilles(2000) study a version of the connections models, allowing for het-erogeneity in costs, but homogeneity in benefits. In their settingplayers are ordered by name from 1, . . . , N such that the cost of alink between i to j is linearly increasing in the number of playerslij :¼ j{k 2 [N]:i < k 6 j}j. Thus, in Eq. (5) cij is replaced by lij � c. Forhigh costs c similar features of PS networks as in Proposition 3.7can be observed. If instead costs are low (c < d � d2), then there al-ways exist k � networks where players form links to all playersj 2 [N] such that lij 6 k and no other links are formed. Thus, in thesek � networks links remain local implying that clustering is quitehigh, similar to the Small-World architectures displayed inFig. 1(iii) (Watts, 1999; Watts & Strogatz, 1998).

A different way of introducing cost heterogeneity is the islandmodel by Jackson and Rogers (2005). There it is assumed that theplayer set [N] is partitioned into islands of equal size J such thatcost of linking is small cij = c if i and j are from the same island,but are high c < C = cij if i and j belong to different islands. More-over, benefits are only obtained for small distances, i.e. wij = 1 ifdij(G) 6 D and wij = 0 else. The interesting case occurs whenc < d � d2 and C < d + (J � 1)d2. Then, in any PS network the islandsare (internally) completely connected and there is a path betweenany two islands. Moreover, stable networks have low diameter(bounded above by D + 1) and high clustering coefficients. Thus,this model matches some key statistics empirically observed inreal word networks.

Persitz (2010) allows for heterogeneous benefits wij in the con-nections model. In his model the players are characterized byeither one of two types ti 2 {a,b} such that benefits depend onthe types: wij = w1 if ti = tj = a, wij = w2 if ti = a, tj = b, and wij = w3 ifti = tj = b such that w1 > w2 > w3. Thus all players prefer the connec-tion to a type-a player over the connection to a type-b player.While the trivial networks (complete respectively empty) are sta-ble for very low respectively high costs, stable networks for med-ium cost levels have the core periphery architecture where alla � players (the core) are completely connected (if (d � d2)w1 > c)and the b � players (the periphery) have connections to the core.Whether b-players connect to each other or how many links toa � players they have, depends on the exact cost structure. Hence,stable networks in Persitz (2010) have low diameter and high clus-tering coefficients and therefore share some statistical features ofstable networks in Jackson and Rogers (2005).

In an incomplete information setting similar to McBride (2006)(see Section 3.5), Gallo (2012) also shows that stable networks

21 Although the issue of efficient networks and the tension between stable andefficient networks displays a recurrent theme in the literature, it is beyond the scopeof this survey to review the results on efficiency. The reader may be referred e.g. toJackson (2005); Buechel and Hellmann (2012) and textbooks by Goyal (2007) andJackson (2008).


exhibit low diameter (smaller than 5) and high clustering coeffi-cient.22 In fact, Gallo (2012) studies a slightly more general utilityfunction than eq. (5), called distance-based utility function. Thisclass of utility functions were first introduced in Bloch and Jackson(2007) and also studied in Möhlmeier, Rusinowska, and Tanimura(2013). Thus, by generalizing the utility function from relatively sim-ple settings, like the homogeneous connections model, to moresophisticated models, stability predictions may reflect more realisticcharacteristics like low diameter and high clustering.

24 The notion of a minimal connected network depends on whether the network isdirected or undirected. A center–sponsored star and a wheel network are depicted inFig. 1 and (ii). See Bala and Goyal (2000) for precise definitions.

25 In Galeotti et al. (2006), this holds when costs are player specific. However, whencosts are allowed to be heterogeneous with respect to links (called partner specific

3.5. One-sided network formation

Most of the concepts introduced so far were formulated forundirected networks. There are many applications where directednetworks seem to be more appropriate. Examples of directed net-works include citation networks, hyperlinks connecting web-pages, flow of knowledge, or more general any communicationnetwork. When network formation can be done without mutualconsent, it is natural to speak about one-sided link formation. In avery influential paper, Bala and Goyal (2000) introduce a game the-oretic model of one-sided network formation. They distinguish be-tween two basic cases:

One-Way flow: Costs and benefits are only incurred by the ini-tiator of a link.Two-Way flow: Costs are carried by the initiator of a link, whilebenefits are shared by both parties.

A pure strategy for player i is a list si 2 Si = {0,1}Nn{i}, with theinterpretation that player i wants a link with j if and only ifsij = 1. Since no consent is required to link to a given player, strat-egies directly translate into a directed network G 2 G½N� such that(i, j) 2 E(G) if and only if sij = 1.23

In the one-way flow model, each player i receives benefits fromall players who are accessible to i. Define the set Ci(G) :¼ {j 2 [N]j$j1 . . . jk 2 [N]: j1 = j, jk = i, (jl, jl+1) 2 E(G) "1 6 l 6 k � 1}, and callli(G) :¼ jCi(G)j. In the one-way flow model of Bala and Goyal(2000) the utility of player i 2 [N] is given by

uOWi ðGÞ :¼ /ðliðGÞ;g�i ðGÞÞ; ð13Þ

where g�i ðGÞ :¼ jfj 2 ½N�jði; jÞ 2 EðGÞgj is the outdegree of player i, and/(�, �) is strictly increasing in the first argument and strictly decreas-ing in the second argument.

In the two-way flow model both linked players receive benefits,making it necessary to define the closure of a directed network G.This is the network G, defined by the property that ði; jÞ 2 EðGÞ ifand only if (i, j) 2 E(G) or (j, i) 2 E(G). The set of players who areaccessible to i in the two-way flow model, CiðGÞ, is then simplythe component to which player i belongs. Utility in the two-wayflow model is given by

uTWi ðGÞ :¼ /ðliðGÞ;g�i ðGÞÞ; ð14Þ

such that above assumptions on / holds.Since in one-sided network formation consent to link is not re-

quired, the concept of Nash equilibrium is more appealing than inthe undirected case. Formally, a Nash network is a network G suchthat no single player has an incentive to create, respectively delete,

22 Gallo (2012) refines the concepts of pairwise stability to cognitive stability toaccount for the incomplete information setting.

23 Recall that G½N� is used as general representation of the set of possible networks;The direction of links is then clear from the context.

a subset of his links, given the links announced by all other players.Given this definition, some of the findings of Bala and Goyal (2000)are summarized by the following proposition.

Proposition 3.8 Bala and Goyal (2000). Nash networks are eitherminimally connected or empty in both the one-way flow model (13)and the two-way flow model (14). Among those, only the emptynetwork and the wheel network are strict Nash networks for theone-way flow model, and only the empty network and thecenter sponsored star are strict Nash networks for the two-way flowmodel.24

The Bala and Goyal (2000) model has been extended alongvarious directions. In the framework of the one-way flow model(13), Galeotti (2006) allows for heterogeneous benefits and forheterogeneous cost of connecting. Galeotti, Goyal, and Kam-phorst (2006) allow for heterogeneity in the two-way flow model(14). Both papers show that the basic equilibrium architecturesobtained in Bala and Goyal (2000) are quite robust. Even withheterogeneous benefits and cost, Nash networks are either min-imal or empty, and also strict Nash networks share the same ba-sic architectures as the homogeneous case, but can consist ofmore than one non-singleton component such that the differentcomponents consists of the basic equilibrium structures.25 Intro-ducing incomplete information, so that players can only observethose parts of the network within some distance x, McBride(2006) shows that for low enough values of x cycles may occur(i.e. equilibrium networks are not necessarily minimal anymore)and Nash networks may be disconnected in the two-way flowmodel.

The fact that Nash networks are always minimal in the com-plete information setting is driven by the assumption that playersbenefit from all accessible players equally, no matter how longeach path is. Bala and Goyal (2000) therefore introduce decay forboth the one-way flow and two-way flow model in a similar fash-ion as the connections model (see Eq. (12)). For the latter case theydefine

uTWdi ðGÞ ¼ 1þ

Xj2CiðGÞnfig

ddijðGÞ � g�i c; ð15Þ

with c being some positive constant. As a result, Bala and Goyal(2000) get very similar architectures as Jackson and Wolinsky(1996) for the connections model. Strict Nash networks are forsmall costs (c < d � d2) the complete network, for medium costs(d � d2 < c < d) the star among others, and for high costs c > dthe empty network among others. Hojman and Szeidl (2008)extend this approach and assume payoff structure uHS

i ðGÞ ¼f a1d1 þ . . .þ aDdDð Þ � g�i c where dk:¼j{j 2 [N]:dij = k}j such that f isincreasing and concave. Under some additional assumptions, theyshow that the periphery sponsored star and an extended star areunique Nash networks.26 Another extension is due to Feri andMeléndez-Jiménez (2013) where co-evolution of networks and play(also cf. Section 4.2) in the setup of the two-way flow model withdecay (15) is studied. In this paper, the decay is link specific andendogenous since it depends on the actions taken by the (involved)players in a coordination game.

costs), then different structures may emerge as strict Nash equilibria.26 The periphery sponsored star is such that there exists a player i 2 [N] (the center)

such that E(G) = {(j, i)jj 2 [N], j – i}, i.e. all links are directed from the peripheralplayers to the center. In an extended star there is also a central player who does notsupport any links. All peripheral players maintain a single link and from everyperipheral player there is a directed path to the center.


4. Evolutionary and co-evolutionary processes

4.1. Evolution

In the context of the connections model (cf. Sections 3.1.1 and3.4), a natural adjustment dynamics is described by the improvingpaths (Definition 3.2). Watts (2001) investigates the long-runproperties of such a dynamic explicitly. For intermediate costs suchthat the star network is pairwise stable in the static model (i.e.ensuring that the star is always an absorbing state of the dynamicprocess), the likelihood that the process starting from the emptynetwork converges to the star goes to 0 as the number of playersN grows. The reason is that the myopic best-response dynamicsis path dependent. Once two distinct pairs of players form a link(which is beneficial when starting from the empty network), theimproving path will never lead to the star network.

To avoid path dependency Jackson and Watts (2002a) introduceperturbations in the decisions of players in the spirit of evolution-ary game theory (Kandori, Mailath, & Rob, 1993; Young, 1993).These perturbations are interpreted as mistakes the players maymake when evaluating the profitability of link. Formally the timingof the process is as follows:

1. At each point in time t 2 N a network Gt 2 G½N� is given and onelink ij 2 E(Gc) is selected according to a probability distributionwith full support on E(Gc);

2. The link ij is added if ij 2 ADD (Gt); it is deleted if ij 2 DEL (Gt),and it is not changed otherwise;

3. With probability � > 0 the decision is reversed.

For every 0 < � < 1 this process defines an irreducible Markovchain on the set of networks G½N�. It therefore has a unique invari-ant distribution l�, which describes the probability with which agiven network can be observed. From which network the processstarts does not influence this probability. The networks G 2 G½N�such that lim�?0l�(G) > 0 are called stochastically stable. These arethe networks observed most of the time, when the noise limit goesto 0. Naturally, only networks contained in recurrent classes of theMarkov process with zero noise are candidates for stochastic sta-bility. Hence, the concept of stochastic stability is an evolutionaryselection mechanism among the stable networks (and closedcycles).

Jackson and Watts (2002a) show that stochastically stable net-works can be detected by ‘‘counting mistakes’’ involved in the tran-sition from one recurrent class to another.27 However, theconstruction of these trees for every recurrence class, and the calcu-lation of minimal number of mistakes can be quite complex. There-fore, Tercieux and Vannetelbosch (2006) present conditions forpairwise stable networks (or sets of networks) to be stochasticallystable that are easy to check, called p-pairwise stability. A networkG is said to be p-pairwise stable, if altering a fraction p of theN(N � 1) possible links in network G implies that any improving pathstarting from the altered network leads back to G (in particular pair-wise stable networks are at least 0-pairwise stable). It is shown that1/2-pairwise stable networks (if existing) are uniquely stochasticallystable.

Dawid and Hellmann (2012) study the evolution of R&D collab-orations in the model setting of Goyal and Joshi (2003) (seeSection 3.1.3). It has been shown by Goyal and Joshi (2003) thatany PNS network is of ‘‘dominant group architecture’’; i.e. there

27 The arguments used in Jackson and Watts (2002a) are based on the classicalFreidlin and Wentzell (1998) tree constructions to understand the long-run behaviorof dynamical systems subject to random perturbations. These methods have foundnumerous application in game theory. See Young (1998), Sandholm (2010a), andStaudigl (2012).

is one completely connected group and all other players are iso-lated (the same is true for PS networks, see Dawid & Hellmann,2012). The sizes of the dominant group are sensitive to the costof link formation, but there is no unique prediction with respectto the networks which will be observed. When introducing theevolutionary process above it can be shown that the size of thedominant group in stochastically stable networks is generically un-ique and monotonically decreasing in cost of link formation. In thecontext of the evolution of R&D collaborations, these unique pre-dictions give way to interesting comparative static analysis withrespect to welfare, industry profits, and consumer surplus.

In the context of directed networks Bala and Goyal (2000) pres-ent a myopic best response process in discrete time for the one-way flow model (13) and two-way flow model (14). They assumethat at any time step t 2 N each player i 2 [N] chooses a myopicbest response to last periods’ network with probability 0 < pi < 1while with probability 1 � pi exhibits inertia.28 For both modelsthe myopic best response process always converges to the set ofstrict Nash networks (see Proposition 3.8).

Feri (2007) adds random perturbations to this dynamics and re-stricts himself to the two-way flow model with decay (see (15)).Feri (2007) obtains an almost complete characterization of the sto-chastically stable networks. Trivially, for low respectively largecosts, only the complete respectively empty network are stochasti-cally stable. For intermediate costs, and large networks, all starswhere each link is supported by only one player are among the sto-chastically stable networks while if costs are slightly higher, noother network is stochastically stable. These results stand in con-trast to Watts (2001), who shows that for the connections modelin bilateral network formation the likelihood of emergence of thestar decreases in the number of players N for the pure best re-sponse dynamics.

4.2. Co-evolution of networks and play

Having discussed models of pure network formation, we now ex-tend our model to allow the players to control (at least partially) twodimensions of the model. These two dimensions consist of the links aplayer supports in a graph, and an action she may choose, which gen-erates a certain payoff according to an underlying game in normalform. Already the basic examples of Section 3.1 can be used to moti-vate an analysis of such co-evolutionary processes. For instance, inthe local complementarity game of Section 3.1.2, the effort choiceof a player depends on its position in the networks. As new linksare added or deleted, this will have an effect on the effort investedby a player, and vice versa. We now discuss a mathematical modelin which such joint dynamics can be analyzed.

4.2.1. Games with local interaction structureRecall that a normal form game is a tuple h[N], (ui)i2[N], (Ai)i2[N]i,

where [N] = {1, 2, . . . , N} is the set of players, and ui : �i2½N�Ai ! R isthe utility function of player i, assigning a utility index ui(a1, . . . , aN)to each action profile a :¼ (a1, . . . , aN) 2 � i2[N]Ai. In this generaldefinition the interaction structure of the players is hidden in theutility function. In order to make this dependency structure more ex-plicit, it is useful to separate these effects and define a preferencerelation directly on the product set of action profiles and interactionstructures (i.e. networks). To do so, let us redefine the strategic inter-action as a game with local interaction structure (Morris, 1997), whichis a game in normal form, but the preference relation is directlydefined over a product set Ai ¼ Si � G½N�. Such models have a quitelong tradition (early contributions are Blume, 1993; Ellison, 1993).

28 Note that compared to the best-response dynamics with two-sided link formationin Watts (2001) and Jackson and Watts (2002a), the revision process is nowformulated on the individual player level, rather than on the set of edges.

29 Jackson and Watts (2002b) describe a dynamic evolving in discrete periods. Thecontinuous-time version of their model given here differs in no substantial details,because the stochastic law of motion is the one of a Markov jump process. Thediscrete time models can be seen as embedded jump chains of the continuous-timemodel. See, e.g. Stroock (2005) for a general reference, and Staudigl (2010) for anexplicit construction.


4.2.2. Adding network dynamicsStarting from a given local interaction game, we now construct

an explicit dynamic model, which includes a dynamic model ofnetwork formation as well as a dynamic model of action revision.The models we will discuss are based on the following hypothesis:

(i) Agents are myopic in their linking and action decision (cf.Section 4.1).

(ii) The dynamic process is stationary.

Hypothesis (i) is a huge simplification and essentially boilsdown to the hypothesis that players base their decision’s todayonly on the current information they have. They do not attemptto forecast the impact of their behavior on the future evolutionof the state and its consequence on their own payoff. Such anassumption is standard practice in evolutionary game theory (Wei-bull, 1995). Together with hypothesis (ii), it allows us to model aco-evolutionary process as a time-homogeneous Markov processesliving on the product set X ¼ S� G½N�, where S = � i2[N]Si is theset of action profiles. Hence, a co-evolutionary model with noiseis a family of continuous-time Markov jump processes

fX�ðtÞgt2Rþ

n o�2ð0;��

, defined on the finite state space X . The stochas-

tic process X� is indexed by the parameter � > 0. This parameteroften has the interpretation of a behavioral noise parameter(cf. Section 4.1).

A realization {X�(t) = x} defines an action profile r(x) =(ri(x))i2[N] 2 S, and a network cðxÞ 2 G½N�. In concrete examples itis often more convenient to encode the network via its adjacencymatrix. With a slight abuse of notation we denote the adjacencymatrix of the network c(x) by A(x).

The elementary updating steps of the Markov process are thefollowing:

Action adjustments: With constant rate 1 the action profiler(x) is allowed to change. Conditional on this event, playeri 2 [N] is chosen uniformly at random to update his currentaction. If i gets a revision opportunity, he draws an action froma distribution b�i ð�jxÞ 2 DðSiÞ. The mapping b�i : X ! DðSiÞ iscalled the choice function of player i.Link creation: With constant rate k P 0 the process allows thenetwork to expand. As in Section 2 we model this process by amatrix-valued function W� : X ! RN�N , whose elements arebounded non-negative functions w�

ijðxÞ, interpreted as the inten-sity of link creation given the current state x 2 X . For each x 2 Xand � > 0 the matrix W�(x) is symmetric, and called the attach-ment mechanism of the process.Link destruction: With constant rate n P 0 a link becomesdestroyed. The intensity with which the link between player iand j is destroyed is modeled by matrix-valued functionV� : X ! RN�N , whose elements are bounded non-negativefunctions v�ijðxÞ. For each x 2 X and � > 0 the matrix V�(x) is sym-metric, and called the volatility mechanism of the process.

Co-evolutionary processes are general enough to cover mostevolutionary models which have been studied in the literature. Inparticular, evolutionary learning models on fixed interaction struc-tures are obtained by specializing the setting to k = n = 0. Second, itis instructive to see that X� is indeed an extension of the randomgraph process of Section 2. The added feature is that the attach-ment and the volatility mechanism are now functions defined onthe domain X .

4.2.3. A micro-founded model for inhomogeneous random graphsTo give a characterization of the class of networks generated by

co-evolutionary processes, we would like to rely on our knowledge

about general random graph models. In particular we would like touse the general characterization Theorem 2.1. For a given profiles 2 S, let us define the s-section of the state space X as the setX s :¼ fsg � G½N�. An s-conditional random graph process is a con-tinuous-time Markov process {c�(t)}tP0 taking values in the setX s, with attachment mechanisms W�jX s

and V�jX s. On an s-section

the intensities can only vary with the network. Our goal is to derivea probability law defined over the set of possible networks, condi-tional on the event that the players play according to the actionprofile s, which we denote by P(�—s).

Assumption 4.1. The co-evolutionary process X� satisfies thefollowing assumptions:

(i) k, n > 0;(ii) If Aij(x) = 0 and � > 0, then w�

ijðxÞ > 0. If Aij(x) = 1 or i = j, thenw�

ijðxÞ ¼ 0 for all �;(iii) For all pairs of players i,j 2 [N] and states x 2 X we have

w�ijðxÞ ¼ j�ijðrðxÞÞð1� AijðxÞÞ, and v�ijðxÞ ¼ d�ijðrðxÞÞAijðxÞ,

where j�ij and d�ij are positive functions.

The set of assumptions is not minimal in order to be able to ap-ply Theorem 2.1 to characterize the conditional random graphmeasure. Item (ii) is more restrictive than actually needed. Theonly requirement we need is that the generator of the conditionalrandom graph process satisfies the necessary irreducibilityassumptions in order to guarantee the existence of a unique invari-ant measure. However, item (iii) is necessary.

Item (iii) of Assumption 4.1 is arguably the most restrictive one. Itrequires that the intensities of link creation and destruction are func-tions only of the given action profile. One can imagine exampleswhere this assumption makes sense, but it is clear that many exam-ples will not fit this description. Nevertheless we can give the follow-ing characterization of the conditional random graph measure, whichis obtained from a straightforward application of Theorem 2.1.

Proposition 4.2. Consider a co-evolutionary process X� as definedabove whose attachment and volatility mechanism satisfies Assumption4.1. Conditional on an action profile s 2 S, the conditioned randomgraph process {c�(t)}tP0 has a unique invariant graph measure agreeingwith the probability measure of an inhomogeneous random graph

P�ðfGgjsÞ ¼YN

i¼1

Yj>i

ðp�ijðsÞÞAijðGÞð1� p�ijðsÞÞ

1�AijðGÞ;

where p�ijðsÞ :¼j�

ijðsÞ

j�ijðsÞþd�ijðsÞ

is the edge-success probability of vertex i and j.

4.3. Examples of co-evolutionary processes

Let us now present some simple examples of co-evolutionaryprocesses of networks and play. The first example is a version ofa model due to Jackson and Watts (2002b). The second exampleis a version of Staudigl (2011, 2013).

4.3.1. A co-evolutionary model based on pairwise stabilityJackson and Watts (2002b) combine the network formation

model of Jackson and Wolinsky (1996) (see Section 3.2) with thedynamic network formation model due to Watts (2001), (seeSection 4.1).29 The local interaction game is a symmetric 2 � 2


coordination game with action set S1 = S2 � {1,2}. For each linkformed by a player, a fixed marginal cost / > 0 has to be paid. Theutility function of player i is given by

uiðs;GÞ ¼Xj–i

pðsi; sjÞAijðGÞ � /giðGÞ;

where pð1; sÞ ¼ a ifs ¼ 1;b ifs ¼ 2

�and pð2; sÞ ¼ c ifs ¼ 1;

d ifs ¼ 2:

�The tuple

(a, b, c, d) are the payoff parameters of the underlying game,satisfying the ordering a > c, b > d, and min{c,d} > 0.30 Note that,for a fixed profile of actions, the marginal utility of the link ij forplayer i is given exactly by mui(G(x) ij,ij) = p(ri(x), rj(x)) � /. Theco-evolutionary process is set up as follows:

Action adjustment: With rate 1 each player receives the oppor-tunity to revise his action. Conditional on this event he selectsaction s with probability

30 Thisthe bila

31 Thisfor playaction.

b�i ðsjxÞ ¼

1� � iffsg ¼ arg maxs02f1;2g

uiððs0;r�iðxÞÞ;GðxÞÞ;

1� � if riðxÞ ¼ s and friðxÞg � arg maxs02f1;2g

uiððs0;r�iðxÞÞ;GðxÞÞ;

� otherwise:

8>>><>>>:

This choice function says that a player abandons his currently usedaction with relatively high probability 1 � �, if there exists a strictlybetter action. Otherwise he sticks to his action and switches onlywith the relatively small probability �.31Link creation: With rate

k > 0 a link becomes created. Jackson and Watts (2002b) modelnetwork formation in the flavor of pairwise stability as dis-cussed in Section 3. Let

ADDðxÞ ¼ fij 2 EðGcÞ n EðGðxÞÞjpðriðxÞ;rjðxÞÞ> /; and pðrjðxÞ;riðxÞÞP /g

be the set of links that are mutually profitable. Similarly we define

DELðxÞ ¼ fij 2 EðGðxÞÞjpðriðxÞ;rjðxÞÞ < /; or pðrjðxÞ;riðxÞÞ< /g:

Jackson and Watts (2002b) assume that a previously non-existinglink becomes active with probability 1 � � if both players mutuallyagree. With the small probability �all links have a chance to beformed. The rate that a currently non-existing link ij will beadded is

1�� (
8ði;jÞ R EðGðxÞÞ : w�
ijðxÞ :¼jADDðxÞjþ jEðGc�GðxÞÞj if ij2ADDðxÞ;

�jEðGc�GðxÞÞj otherwise:

Link destruction: With rate n > 0 links become destroyed. Con-ditional on this event, pick one edge ij 2 E(G(x)) uniformly atrandom and allow the incident players to re-evaluate the bene-fits arising from this connection. If ij is a link where at least oneplayer is better off after its deletion it is assumed that with largeprobability 1 � � it will be destroyed. With the small probability� every active link can be destroyed. This leads to the followingversion of the volatility mechanism:

8ði; jÞ 2 EðGðxÞÞ : v�i;jðxÞ ¼1��jDELðxÞj þ �

jEðGðxÞÞj if ij 2 DELðxÞ;�

jEðGðxÞÞj otherwise:

(

means that the action profiles (1,1) and (2,2) are (strict) Nash equilibria ofteral game.

choice function uses a tie-breaking rule. Namely, if both actions are optimaler i at x, then the choice function says that i continuous playing his current

Clearly this model is a version of a co-evolutionary process. Theattachment and volatility mechanism depends however in a non-trivial way on the current network and the action chosen by theplayers, so that we cannot apply Proposition 4.2 to this model.However, this model can still be quite well understood in the ex-treme case where �? 0. Taking this limit, Jackson and Watts(2002b) show that the complete network will be observed mostof the time (i.e. it is a stochastically stable state), and the playerscoordinate on a single strongly symmetric action profile. The inter-esting open question is which of the two symmetric equilibria willbe played. Jackson and Watts (2002b) provide an exhaustive char-acterization of the long-run equilibria in this model.32

Proposition 4.3 (Preposition 1, Jackson and Watts (2002b)). Sup-pose that N is sufficiently large. The unique stochastically stable statesof the co-evolutionary model with constant linking costs / > 0 consistof the complete network and the following pattern of play:

(i) If a � / > 0 and d � / > 0 then either all players play action 1 oraction 2.

(ii) If b � / > 0 and c � / > 0 then all players play action 1.(iii) If b � / < 0 and/or d � / < 0, and a � / > 0 and d � / > 0, then

either all play 1 or all players play 2.

Related to the Jackson and Watts (2002b) model is the coordi-nation game scenario investigated by Goyal and Vega-Redondo(2005). Their paper is based on the two-way flow model of Balaand Goyal (2000), but adds the additional layer of complexity byallowing agents to derive utility from a coordination game as inJackson and Watts (2002b). In the spirit of the two-way flow mod-el, the unit of a one-step transition is not the edge, but the singleindividual. Specifically, Goyal and Vega-Redondo (2005) envisiona dynamic co-evolutionary process in which at each time step asingle individual receives a revision opportunity (independent ofthe other players) in which she can simultaneously change heraction played in the coordination and the set of links she would liketo support in the network realized in the next period. Once anetwork is realized, the agents play the coordination game p withall their neighbors. Thus, if G is the closure of the network G(cf. Section 3.5), player i’s total instantaneous payoff isP

j–ipðai; ajÞAijðGÞ. Each initiated link has some costs / > 0, whichhave to be paid only by the initiator. Hence, the utility functionin the Goyal and Vega-Redondo (2005) model is

uiðs;GÞ ¼Xj–i

pðai; ajÞAijðGÞ � /g�i ðGÞ:

Using this payoff specification, the main result of Goyal andVega-Redondo (2005) is the following.

Proposition 4.4 (Theorem 3.1, Goyal and Vega-Redondo(2005)). Suppose the stage game p is a coordination game with dataa + b < c + d, and a > d. There exists a value �/ 2 ðb; dÞ such that for/ < �/ the unique stochastically stable configuration is the completenetwork in which all agents play action 2. If / 2 ð�/; aÞ all players play1 and the network is complete for N sufficiently large.

Contrasted with the Jackson and Watts (2002b) model we seethat one-sided link formation leads to sharper predictions as thenoise in the players decision function is small and the society issufficiently large.

32 In the proposition we require the number of players to be sufficiently large. Thisensures that certain special cases are ruled out. See Jackson and Watts (2002b) for acomplete statement of the Proposition.


4.3.2. A volatility modelIn this example we consider a family of games in which the util-

ity function ui consists of two components. The first component is acommon payoff term, interpreted as the externality a single players’actions exert on the other players. The second component is an idi-osyncratic payoff. As such it depends only on the player’s ownchoice. We assume that all players have the same action set, de-noted by S. Making a slight change in notation, we write SN forthe set of action profiles. A typical element of this set is an N-tuple~s :¼ ðsiÞi2½N�. The common payoff term from bilateral social interac-tion is modeled by a reward function p : S� S! R. In many appli-cations it is conceivable that the common payoff terms displays astrong symmetry property of the form

pðs; s0Þ ¼ pðs0; sÞ 8s; s0 2 S;

which we assume henceforth to hold. The idiosyncratic payoffcomponent is modeled via a function si : S! R, where si is anelement of the set of functions H ¼ fh1; . . . ; hmjhl : S! Rg. Weinterpret this idiosyncratic component as the type of the player.Thus, a type of a player is a function hr : S! R, assigning to eachaction s 2 S an idiosyncratic utility hr(s). We assume that playerslearn their types before the co-evolutionary process starts. Thisassumption allows us to interpret the types of the players as param-eters of the co-evolutionary process.33

Given an action profile~s 2 SN and a profile of types s 2HN, the(ex-post) payoff of player i is assumed to be

uiðs;G; siÞ ¼Xj–i

AijðGÞpðsi; sjÞ þ siðsiÞ:

The co-evolutionary process is specified by the following data.

Action adjustment: Agents use the logit-choice function tochoose actions. This choice function is defined as

33 Momodelstherein.

ð8s 2 SÞ : b�i ðsjx; siÞ ¼exp uiððs;r�iðxÞÞ;GðxÞ; siÞ=�½ �P

s02S exp uiððs0;r�iðxÞÞ;GðxÞ; siÞ=�½ � :

The rate of the transition x ¼ ð~s;GÞ ! x0 ¼ ððs;r�iðxÞÞ;GðxÞÞ is thusb�i ðsjx; siÞ.

Link creation: The attachment mechanism is given by a collec-tion of functions fj�ðs; s0Þgðs;s0 Þ2S�S, defined as

j�ðs; s0Þ ¼ 2N

expðpðs; s0Þ=�Þ 8s; s0 2 S:

The rate of a transition x ¼ ð~s;GÞ ! x0 ¼ ð~s;G ijÞ is then given byk(1 � Aij(x))j�(ri(x),rj(x)).

Link destruction: The volatility mechanism is

v�ijðxÞ ¼ d�si ;sj;

where d�k;l � d�l;k is the volatility rate of a link between a player oftype k and a player of type l. The rate of the transitionx ¼ ð~s;GÞ ! x0 ¼ ð~s;GðxÞ � ijÞ is then given by nAijðxÞd�si ;sj

.

As shown in Staudigl (2013) this model can be completely ana-lyzed using elementary arguments. The (strong) assumptions mak-ing the model tractable are the strong symmetry of the rewardfunction p and the particular specification of the volatility andattachment mechanism. Working with these assumptions, the

dels with similar payoff structure are frequently encountered in economicwith local interactions. See e.g. Brock and Durlauf (2001), and the references

long-run properties of the stochastic dynamics can be completelyunderstood.

Theorem 4.5 (Staudigl (2011, 2013)). The unique invariant distri-bution of the co-evolutionary process fX�;sN ðtÞgtP0 is the Gibbsmeasure

l�;sN ðxÞ ¼expð��1H�

Nðx; sÞÞPx02X expð��1H�

Nðx0; sÞÞ

¼l�;s0;NðxÞ exp ��1qðx; sÞ

� �P

x02Xl�;s0;Nðx0Þ expð��1qðx0; sÞÞ

; ð16Þ

where, for all x ¼ ðs;GÞ 2 X ,

H�Nðx; sÞ :¼ qðx; sÞ þ � logl�;s0;NðxÞ;

l�;s0;NðxÞ :¼YNi¼1

Yj>i

2Nd�si ;sj

!AijðGÞ

;

qðx; sÞ :¼ 12

Xj–i

AijðGÞpðriðxÞ;rjðxÞÞ þX

i

siðsiÞ:

We can use the measure l�;sN to derive a random graph measureover the~s-section X~s. The random graph process {c�,s(t)}tP0 definedon G½N� for a fixed profile of actions~s 2 SN can be formally identifiedwith a birth–death process with ‘‘birth rates’’ of the link ij given by 2exp (p(si,sj)/�), and ‘‘death-rates’’ d�si ;sj

. Calling the rate ratio

u�k;lðs; s0Þ ¼

2� expðpðs; s0Þ=�Þd�k;l

;

for s, s0 2 S and 1 6 k, l 6m, Staudigl (2013) goes on in proving thefollowing result.

Proposition 4.5. Consider the random graph process describedabove. This process is ergodic with unique invariant graph measure

P�;sN ðGj~sÞ ¼YN

i¼1

Yj>i

p�ijð~s; sÞAijðGÞ 1� p�ijð~s; sÞ

� �ð1�AijðGÞÞ;

where the edge-success probabilities are defined as

p�ijð~s; sÞ ¼u�

k;lðs; s0Þu�

k;lðs; s0Þ þ N�if si ¼ s; sj ¼ s0; si ¼ hk; sj ¼ hl ð17Þ

for all i, j 2 [N].The proof of this Proposition is a simple application of Proposi-

tion 4.2. The interpretation of this result is as follows. For a fixedaction profile~s and known type profile, the stationary probabilitydistribution over graphs will be an inhomogeneous random graph(cf. Section 2.2.1), with edge-success probabilities defined as ineq. (17).34

5. Summary and suggestions for future research

In this survey we have tried to give a short overview on a partic-ular class of models designed for the modeling of the dynamic evolu-tion of networks. Random graph models are combined with gametheory to formalize the co-evolution of networks and play. This mod-eling approach reflects the interdisciplinary character of research ondynamic networks (both theoretical and applied). The modelstouched in this survey can and should be considered as only preli-minary steps towards a more general theory. In the following lineswe give some suggestions for future research and their challenges.

34 Staudigl (2013) goes on in characterizing the supports of the invariant measurel�;sN as e ? 0 (the so-called stochastically stable states mentioned in Section 4.1), andalso derives the large deviation rate function of this measure in the limit of largeplayer sets, i.e. where N ?1.


5.1. Weighted networks

As mentioned in Section 1, we have avoided any discussion ofweighted networks in this survey. The simple reason for this is thatwe feel that there is not a satisfactory theory out there in the mo-ment. Much more efforts should be devoted to this very importantclass of networks, which are so prevalent in applications.

5.2. Mechanism design of networks

In many practical situations a system controller would like toensure that a certain network structure will be achieved via decen-tralized decision making of the players. In such a model a control-ler collects a map of the vertices’ preferences before evolutionstarts, and given these data, a tax/subsidy scheme is designed sothat a certain network architecture will be the outcome (at leastwith high probability).35 It is an interesting question how such amodel extends to a dynamic evolutionary setting. We envision aBayesian interaction model as in Section 4.3.2 as a reasonable mod-eling approach for this endeavor. Related to this question, we thinkthat it is important to invest more effort to understand the connec-tion between structural features of the players’ utility functions andthe structure of stable networks (see Sections 3.3, 3.4 and 3.5), butthere is room for improvement. General characterizations of stablenetworks are so far missing (e.g. necessary conditions for existenceand uniqueness or general conditions for emergence of particularnetworks). An important contribution to foster our understandingof ‘‘incentive-driven’’ network formation models is to start with aclass of network games (potential games would be a reasonablestarting point) and investigate the question how equilibrium net-work topologies change when basic properties of the utility functionchange. This is not only an important theoretical question, but mayalso be important for statistical identification.

Acknowledgments

We would like to thank the editor and four anonymous refereesfor their very constructive comments on earlier versions of thissurvey. Mathias Staudigl acknowledges financial from the ViennaScience and Technology Fund (WWTF) under project fund MA09-017.

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