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Santo Fortunato SNAP Stanford Network Analysis Project
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I INTRODUCTION, The origin of graph theory dates back to Euler s solu. tion of the puzzle of Ko nigsberg s bridges in 1736 Euler. 1736 Since then a lot has been learned about graphs. and their mathematical properties Bollobas 1998 In. the 20th century they have also become extremely useful 00. 11111 0000, as representation of a wide variety of systems in different. areas Biological social technological and information. networks can be studied as graphs and graph analysis. has become crucial to understand the features of these. systems For instance social network analysis started in. the 1930 s and has become one of the most important. topics in sociology Scott 2000 Wasserman and Faust. 1994 In recent times the computer revolution has pro. vided scholars with a huge amount of data and computa. tional resources to process and analyze these data The. size of real networks one can potentially handle has also. grown considerably reaching millions or even billions of 0. vertices The need to deal with such a large number of. units has produced a deep change in the way graphs are. approached Albert and Baraba si 2002 Barrat et al FIG 1 A simple graph with three communities enclosed. 2008 Boccaletti et al 2006 Mendes and Dorogovtsev by the dashed circles Reprinted figure with permission from. 2003 Newman 2003 Pastor Satorras and Vespignani Ref Fortunato and Castellano 2009 2009. c by Springer, Graphs representing real systems are not regular like. e g lattices They are objects where order coexists with. disorder The paradigm of disordered graph is the ran like online communities Indeed social communities have. dom graph introduced by P Erdo s and A Re nyi Erdo s been studied for a long time Coleman 1964 Freeman. and Re nyi 1959 In it the probability of having an 2004 Kottak 2004 Moody and White 2003 Communi. edge between a pair of vertices is equal for all possible ties also occur in many networked systems from biology. pairs see Appendix In a random graph the distribu computer science engineering economics politics etc. tion of edges among the vertices is highly homogeneous In protein protein interaction networks communities are. For instance the distribution of the number of neigh likely to group proteins having the same specific function. bours of a vertex or degree is binomial so most ver within the cell Chen and Yuan 2006 Rives and Galitski. tices have equal or similar degree Real networks are 2003 Spirin and Mirny 2003 in the graph of the World. not random graphs as they display big inhomogeneities Wide Web they may correspond to groups of pages deal. revealing a high level of order and organization The de ing with the same or related topics Dourisboure et al. gree distribution is broad with a tail that often follows 2007 Flake et al 2002 in metabolic networks they may. a power law therefore many vertices with low degree be related to functional modules such as cycles and path. coexist with some vertices with large degree Further ways Guimera and Amaral 2005 Palla et al 2005. more the distribution of edges is not only globally but in food webs they may identify compartments Krause. also locally inhomogeneous with high concentrations of et al 2003 Pimm 1979 and so on. edges within special groups of vertices and low concen Communities can have concrete applications Cluster. trations between these groups This feature of real net ing Web clients who have similar interests and are ge. works is called community structure Girvan and New ografically near to each other may improve the perfor. man 2002 or clustering and is the topic of this review mance of services provided on the World Wide Web in. for earlier reviews see Refs Danon et al 2007 Fortu that each cluster of clients could be served by a dedi. nato and Castellano 2009 Newman 2004a Porter et al cated mirror server Krishnamurthy and Wang 2000. 2009 Schaeffer 2007 Communities also called clusters Identifying clusters of customers with similar interests. or modules are groups of vertices which probably share in the network of purchase relationships between cus. common properties and or play similar roles within the tomers and products of online retailers like e g. graph In Fig 1 a schematic example of a graph with www amazon com enables to set up efficient recommen. communities is shown dation systems Reddy et al 2002 that better guide. Society offers a wide variety of possible group organi customers through the list of items of the retailer and. zations families working and friendship circles villages enhance the business opportunities Clusters of large. towns nations The diffusion of Internet has also led graphs can be used to create data structures in order. to the creation of virtual groups that live on the Web to efficiently store the graph data and to handle naviga. tional queries like path searches Agrawal and Jagadish systems Simon 1962 The generation and evolution of. 1994 Wu et al 2004 Ad hoc networks Perkins 2001 a system organized in interrelated stable subsystems are. i e self configuring networks formed by communication much quicker than if the system were unstructured be. nodes acting in the same region and rapidly changing cause it is much easier to assemble the smallest subparts. because the devices move for instance usually have first and use them as building blocks to get larger struc. no centrally maintained routing tables that specify how tures until the whole system is assembled In this way. nodes have to communicate to other nodes Grouping the it is also far more difficult that errors mutations occur. nodes into clusters enables one to generate compact rout along the process. ing tables while the choice of the communication paths The aim of community detection in graphs is to iden. is still efficient Steenstrup 2001 tify the modules and possibly their hierarchical orga. Community detection is important for other reasons nization by only using the information encoded in the. too Identifying modules and their boundaries allows for graph topology The problem has a long tradition and it. a classification of vertices according to their structural has appeared in various forms in several disciplines The. position in the modules So vertices with a central posi first analysis of community structure was carried out by. tion in their clusters i e sharing a large number of edges Weiss and Jacobson Weiss and Jacobson 1955 who. with the other group partners may have an important searched for work groups within a government agency. function of control and stability within the group ver The authors studied the matrix of working relationships. tices lying at the boundaries between modules play an im between members of the agency which were identified by. portant role of mediation and lead the relationships and means of private interviews Work groups were separated. exchanges between different communities alike to Cser by removing the members working with people of differ. mely s creative elements Csermely 2008 Such clas ent groups which act as connectors between them This. sification seems to be meaningful in social Burt 1976 idea of cutting the bridges between groups is at the ba. Freeman 1977 Granovetter 1973 and metabolic net sis of several modern algorithms of community detection. works Guimera and Amaral 2005 Finally one can Section V Research on communities actually started. study the graph where vertices are the communities and even earlier than the paper by Weiss and Jacobson Al. edges are set between clusters if there are connections be ready in 1927 Stuart Rice looked for clusters of people. tween some of their vertices in the original graph and or in small political bodies based on the similarity of their. if the modules overlap In this way one attains a coarse voting patterns Rice 1927 Two decades later George. grained description of the original graph which unveils Homans showed that social groups could be revealed by. the relationships between modules 1 Recent studies indi suitably rearranging the rows and the columns of matri. cate that networks of communities have a different degree ces describing social ties until they take an approximate. distribution with respect to the full graphs Palla et al block diagonal form Homans 1950 This procedure is. 2005 however the origin of their structures can be ex now standard Meanwhile traditional techniques to find. plained by the same mechanism Pollner et al 2006 communities in social networks are hierarchical cluster. ing and partitional clustering Sections IV B and IV C. Another important aspect related to community struc where vertices are joined into groups according to their. ture is the hierarchical organization displayed by most mutual similarity. networked systems in the real world Real networks are. usually composed by communities including smaller com Identifying graph communities is a popular topic in. munities which in turn include smaller communities etc computer science too In parallel computing for in. The human body offers a paradigmatic example of hier stance it is crucial to know what is the best way to. archical organization it is composed by organs organs allocate tasks to processors so as to minimize the commu. are composed by tissues tissues by cells etc Another nications between them and enable a rapid performance. example is represented by business firms who are char of the calculation This can be accomplished by splitting. acterized by a pyramidal organization going from the the computer cluster into groups with roughly the same. workers to the president with intermediate levels corre number of processors such that the number of physi. sponding to work groups departments and management cal connections between processors of different groups is. Herbert A Simon has emphasized the crucial role played minimal The mathematical formalization of this prob. by hierarchy in the structure and evolution of complex lem is called graph partitioning Section IV A The first. algorithms for graph partitioning were proposed in the. early 1970 s, In a seminal paper appeared in 2002 Girvan and New. 1 Coarse graining a graph generally means mapping it onto a man proposed a new algorithm aiming at the identifica. smaller graph having similar properties which is easier to handle tion of edges lying between communities and their suc. For this purpose the vertices of the original graph are not nec cessive removal a procedure that after some iterations. essarily grouped in communities Gfeller and De Los Rios have leads to the isolation of the communities Girvan and. proposed coarse graining schemes that keep the properties of dy. namic processes acting on the graph like random walks Gfeller Newman 2002 The intercommunity edges are detected. and De Los Rios 2007 and synchronization Gfeller and De Los according to the values of a centrality measure the edge. Rios 2008 betweenness that expresses the importance of the role. of the edges in processes where signals are transmitted club members Zachary 1977 a well known graph reg. across the graph following paths of minimal length The ularly used as a benchmark to test community detection. paper triggered a big activity in the field and many new algorithms Section XV A It consists of 34 vertices the. methods have been proposed in the last years In partic members of a karate club in the United States who were. ular physicists entered the game bringing in their tools observed during a period of three years Edges connect. and techniques spin models optimization percolation individuals who were observed to interact outside the ac. random walks synchronization etc became ingredients tivities of the club At some point a conflict between. of new original algorithms The field has also taken ad the club president and the instructor led to the fission of. vantage of concepts and methods from computer science the club in two separate groups supporting the instruc. nonlinear dynamics sociology discrete mathematics tor and the president respectively indicated by squares. In this manuscript we try to cover in some detail the and circles The question is whether from the original. work done in this area We shall pay a special atten network structure it is possible to infer the composition. tion to the contributions made by physicists but we shall of the two groups Indeed by looking at Fig 2a one. also give proper credit to important results obtained by can distinguish two aggregations one around vertices 33. scholars of other disciplines Section II introduces com and 34 34 is the president the other around vertex 1. munities in real networks and is supposed to make the the instructor One can also identify several vertices. reader acquainted with the problem and its relevance In lying between the two main structures like 3 9 10 such. Section III we define the basic elements of community vertices are often misclassified by community detection. detection i e the concepts of community and parti methods. tion Traditional clustering methods in computer and Fig 2b displays the largest connected component of. social sciences i e graph partitioning hierarchical a network of collaborations of scientists working at the. partitional and spectral clustering are reviewed in Sec Santa Fe Institute SFI There are 118 vertices repre. tion IV Modern methods divided into categories based senting resident scientists at SFI and their collaborators. on the type of approach are presented in Sections V Edges are placed between scientists that have published. Santo Fortunato Complex Networks and Systems Lagrange Laboratory ISI Foundation Viale S Severo 65 10133 Torino I ITALY The modern science of networks has brought signi cant advances to our understanding of complex systems One of the most relevant features of graphs representing real systems is community

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