Table Of Content

New approaches to model and study social networks 7 0 0 2 P.G.Lind n ICP,Universita¨tStuttgart,Pfaffenwaldring27,D-70569Stuttgart,Germany a CFTC,UniversidadedeLisboa,Av.Prof.GamaPinto2,1649-003Lisbon,Portugal J 9 H.J.Herrmann ] h ICP,Universita¨tStuttgart,Pfaffenwaldring27,D-70569Stuttgart,Germany p - IfB,HIFE12,ETHHo¨nggerberg,CH-8093Zu¨rich,Switzerland c DepartamentodeF´ısica,UniversidadeFederaldoCeara´,60451-970Fortaleza,Brazil o s . s Abstract. We describe and develop three recent novelties in network research which are c i particularlyusefulforstudyingsocialsystems. Thefirstoneconcernsthediscoveryofsome s y basic dynamical laws that enable the emergence of the fundamental features observed in h socialnetworks, namely the nontrivialclustering properties,the existence of positivedegree p correlationsandthesubdivisionintocommunities.Toreproduceallthesefeatureswedescribe [ a simple modelof mobilecollidingagents, whosecollisionsdefine theconnectionsbetween 1 the agents which are the nodes in the underlying network, and develop some analytical v considerations. The second point addresses the particular feature of clustering and its 7 0 relationshipwithglobalnetworkmeasures,namelywiththedistributionofthesizeofcycles 1 inthenetwork. Sinceinsocialbipartitenetworksitisnotpossibletomeasuretheclustering 1 from standard procedures, we propose an alternative clustering coefficient that can be used 0 7 toextractan improvednormalizedcycledistributioninanynetwork. Finally, thethirdpoint 0 addressesdynamicalprocessesoccurringonnetworks,namelywhenstudyingthepropagation / s ofinformationinthem.Inparticular,wefocusontheparticularfeaturesofgossippropagation c whichimposesomerestrictionsinthepropagationrules. Tothisendweintroduceaquantity, i s the spread factor, which measures the average maximalfraction of nearest neighborswhich y getin contactwith the gossip, andfind the strikingresultthatthereis an optimalnon-trivial h p numberof friends for which the spread factor is minimized, decreasing the danger of being : gossiped. v i X r a PACSnumbers: 89.65.-s,89.75.Fb,89.75.Hc,89.75.Da Socialnetworks: modelsandmeasures 2 1. Introduction Contrary to what maybeperceived at afirst glance, social and physicalmodelswere brought togetherseveraltimes,duringthelastfourcenturies. Infact,notonlyMaxwellandBoltzmann were inspired by the statistical approaches in social sciences to develop the kinetic theory of gases, but one can even cite the English philosopher Thomas Hobbes, who already in the seventeenthcentury,usingamechanical approach,tried toexplainhowpeopleacquaintances andbehaviorsmaycontributetotheevolutiontowardsastableabsolutemonarchy[1,2]. More thanmakingahistoricalperspectiveiftheseapproaches weresuccessfulandcorrect ornot,it isalmostunquestionablethat,atacertainlevel,therearesocialphenomenathatcouldbemore deeply understood by using approaches of statistical and physical models. Recently[3, 4, 5], this perspective gained considerable strength from the increased interest on - and in several senses well-succeed - network approach, where one describes complex systems by mapping themonagraph(network)ofnodesandlinksandstudiestheirstructureanddynamicswiththe helpofsomestatisticaland topologicaltoolsfromstatisticalphysicsandgraph theory[6, 7]. When addressing the specific case of a social system, nodes represent individuals and the connections between them represent social relations and acquaintances of a certain kind. Social networks were studied in different contexts[8, 9, 10, 11, 12, 13, 14], ranging from epidemics spreading and sexual contacts to language evolution and vote elections. However, althoughtheyareubiquitous,socialnetworksdifferfrommostothernetworks,yieldingastill broad spectrum of unanswered questions and improvements to be done when studying their statistical and topological properties. In this paper, we will address three fundamental open questionsrelated to thetypicalstructureand dynamicsassociated tosocial networks. The first open question has to do with the modeling of social networks. The recent broad study of empirical social networks has shown that they have three fundamental features common to all of them[15]. First, they present the small-world effect[16] with small average path lengths between nodes and high clustering coefficients meaning that neighbors tend to be connected with each other. Second, they have positive correlations: the highly (poorly) connected nodes tend to connected to other highly (poorly) connected nodes. Third and last, invariably one observes an organization of the network into some subsets of nodes (communities)more densely connected between each other. Althoughthere areargumentspointingoutthatallthesefeaturescouldbeconsequencefromoneanother[15], the modeling of specific social networks reproducing quantitativelyall these features has not been successful. Usingarecent approach to constructnetworks,basedon asystemofmobile agents,itispossibletoreproduceallthesefeatures. InSection2wewillfurthershowthatthe degree distributions characterizing social networks typically follow a specific one-parameter distribution,so-called Brody distribution. The second question is related to the intrinsic nature of the nodes. For certain social networks there are intrinsic features of the individuals which must be considered in the analysis. For instance, the gender in networks of sexual contacts[14] or the hierarchical position in a network of social contacts inside some enterprise. From the network point of viewthisdistinctionmeanstointroducemultipartivityinthenetwork,biasingthepreferential Socialnetworks: modelsandmeasures 3 attachment between nodes that tend to connect with nodes of a certain type. When there are two types of nodes, e.g. men and women, and the connections between them is strongly relatedtothistype,e.g.mencan onlymatchwomenandvice-versa,thestandardmeasures to analyzenetworkstructurefails. Inparticular,thestandardclusteringcoefficient[16],isunable to quantify the connectedness of broader neighborhoods that typically appear in multipartite networks. InSec. 3wewillrevisitsomeoftheclusteringcoefficientsusedtostudyclustering in bipartite networks, and show how the combination of both clustering coefficients can yield good estimates of normalized cycle distributions. Moreover, we will discuss a general theoreticalpictureofaglobalmeasureofincreasingorderofclusteringcoefficientsaccording tosomesuitableexpansion. Thethirdopenquestionhastodowiththeheterogeneityofnodesinwhatconcernstheir influence in theconnections and therefore in the propagationphenomenaon social networks. Inrumorpropagation[17],forinstance,oneusuallytreatsallconnectionsequallyinthespread of some signal (opinion, rumor, etc). This is a suitable assumption for situations like the spread of an opinion which is equally interesting to all nodes in the network, for example political opinions to some vote election. However, there are also several social situations where the signal is not equally interesting to all nodes, as the case of spreading of some gossip about some common friend. In these cases there are connections which will be more probablyused to spread thesignalthan others,sincenot allourfriends are alsofriends ofthe particular person which is being gossiped about and therefore, either we tend to not tell the gossip to them or they tend to not spread it even if they hear it. In Sec. 4 we will present a simple model for gossip propagation and describe some striking features. Namely, that there isanoptimalnumberoffriends,dependingonthedegreedistributionanddegreecorrelations oftheentirenetwork, forwhichthedangertobegossipedisminimized. Finally, in Sec. 5 we make final conclusions, giving an overview of future questions which could be studied in social networks arising from the topics studied throughout the paper. 2. Modeling socialnetworks: an approach basedonmobileagents Sincethestudyofsocialnetworksismainlyconcernedwithtopologicalandstatisticalfeatures of people’s acquaintances[8, 9], the modeling of such networks has been done within the frameworkofgraphtheoryusingsuitableprobabilisticlawsforthedistributionofconnections betweenindividuals[3,4,5,7]. Thisapproachprovedtobesuccessfulinseveralcontexts,for instanceto describecommunityformation[18, 19]andtheirgrowth[20]. However,they present two majordrawbacks. First, thegraph approach may be suited to describe the structure of social contacts and acquaintances, but lacks to give insight into the socialdynamicallawsunderlyingit. Second,thesemodelsseemtobeunabletoreproduceall the main features characteristic of social networks, at least at the fundamental level. In this context, it was pointed out that[21, 22, 23] dynamical processes based on local information shouldbealsoconsideredwhenmodelingthenetwork. Ourrecentproposaltoovercomethese shortcomings was to construct networks, from a system of mobile agents following a simple Socialnetworks: modelsandmeasures 4 (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) t=P(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)02(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)P1(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)P4(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)P3 (cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) (cid:0)(cid:1) P (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)P(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)2(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)P1(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)P3 P2(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)1P(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)3(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)P t=1 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)P(cid:0)(cid:0)(cid:1)(cid:1)4(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) t=2 (cid:0)(cid:1)(cid:0)(cid:1)4 (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1) Figure 1. Illustration of the two-dimensional mobile agents system. Initially there are no connectionsbetweennodesandnodesmovewithsomeinitialvelocityv inarandomlychosen 0 direction(arrows). Att = 1twonodes,P andP collideandaconnectionbetweenthemis 1 2 introduced(solidline),velocitiesareupdatedincreasingtheirmagnitudeandchoosinganew randomdirection.Att=2twoothercollisionsoccur,betweennodesP andP andbetween 2 4 nodesP andP . Inthiswayanetworkofnodesandconnectionsbetweenthememergesasa 1 3 straightforwardconsequenceoftheirmotion(seetext). motion law[13, 14]. Here, we briefly review this model and further present the analytical expression that fits the obtained degree distributions. In particular we show that the degree distributiontypicallyfollowsaBrody distribution[24]. 2.1. The model The model is given by a system of particles (agents) that move and collide with each other, forming through those collisions the acquaintances between individuals. Consequently, the network results directly from the time evolution of the system and is parameterized by two single parameters, the density ρ of agents characterizing the system composition and the maximal residence time T controlling its evolution. Each agent i is characterized by its ℓ number k of links and by its age A . When initialized each agent has a randomly chosen i i age, position and moving direction with velocity v and one sets k = 0. While moving, 0 i the individuals follow ballistic trajectories till they collide. As a first approximation we assumethatsocialcontactsdo notdeterminewhichsocialcontactwilloccurnext. Therefore, after collisions, the total momentum should not be conserved, with the two agents choosing completely random new moving directions. Figure 1 sketches consecutive stages of the evolutionofsuchasystemofmobileagents. Assuming that large number of acquaintances tend to favor the occurrence of new contacts,thevelocityshouldincreasewithdegreek, namely ~v(k ) = (v¯kα +v )~ω, (1) i i 0 Socialnetworks: modelsandmeasures 5 9 200 8 (b) 7 150 6 T /t5 λ l 100 4 3 2 (a) 50 1 0 0 10 20 30 40 50 60 700 10 20 30 40 50 60 70 <k> <k> Figure2. Bridgingbetweenrealsocialnetworkswith averagedegreehki andthesystem of mobileagentsthat reproducetheir topologicaland statistical features. In (a)the normalized maximalresidencetimeofagentsisplottedasafunctionoftheaveragedegree,whilein(b) oneplotsthecollisionrateλwhichisauniquefunctionoftheresidencetime,andscaleswith hki. where v¯ = 1 m/s is a constant to assure dimensions of velocity, ~ω = (~e cosθ + ~e sinθ) x y with θ a random angle and e~ and e~ are unit vectors. The exponent α in Eq. (1) controls x y the velocity update after each collision. Here, we consider α = 1. Further, the removal of agentsconsideredhereissimplyimposedbysomethresholdT intheageoftheagents: when ℓ A = T agent i leaves the system and a new agent j replaces it with k = 0, v = v and i ℓ j j 0 randomlychosenmovingdirection. TheselectedvaluesforT mustbeoftheorderofseveral ℓ times the characteristic time τ between collisions, in order to avoid either premature death of nodes. Too large values of T are also inappropriate since in that case each node may on ℓ averagecollidewithall othernodes yieldingafullyconnected network. Similarly to other systems[25, 26], this finite T enables the entire system to reach a ℓ non-trivial quasi-stationary state[13]. In fact, only by tuning T within an acceptable range ℓ of small density values, one reproduces networks of social contacts. In Fig. 2a one sees the normalized residence time T /τ as a strictly monotonic function of the average degree ℓ hki. From the residence time it is also possible to define a collision rate, as the fraction betweentheaverageresidencetimeT −hA(0)i = T /2andthecharacteristictimeτ,namely ℓ ℓ λ = T /(2τ) = hviT /(2v τ ), where τ is the characteristic time of the system at the ℓ ℓ 0 0 0 beginningwhen allagents havevelocityv . Figure2bshowsclearly thatλ = 2hki. 0 By looking at Fig. 2 one now understands the main strength of the mobile agent model here described: when taking a real network of social contacts and measuring the average degree hki the correspondence sketched in Fig. 2 straightforwardly returns the suitable value ofT thatreproduces thetopologicaland statisticalfeatures. ℓ It was already reported[13, 27] that empirical networks extracted from a survey among 84Americanschoolsareeasilyreproducedwiththismobileagentmodel,inwhatconcernsthe degree distribution, second-order correlations, community structure, average path length and clustering coefficient. As an illustration,Fig. 3 showsthedegree distributionP(k) of nineof such schools (symbols). Such distributionsare well fitted by Brody distributions(solid lines) Socialnetworks: modelsandmeasures 6 0.1 P(k) 0.01 N=1405 N=2152 N=1974 0.001 <k>=7.02 <k>=7.42 <k>=4.9 0.1 P(k) 0.01 N=1638 N=1877 N=2539 0.001 <k>=6.40 <k>=7.49 <k>=8.24 0.1 P(k) 0.01 N=1545 N=1710 N=1630 0.001 <k>=5.18 <k>=5.15 <k>=8.36 0.1 1 0.1 1 0.1 1 k/<k> k/<k> k/<k> Figure 3. Degree distributions of nine different schools (symbols) from an in-school questionnaireinvolvinga totalof 90118studentswhich respondedto it in a surveybetween 1994 and 1995. Each school comprehends a number N of interviewed students and from their questionnaires an average number hki of acquaintances is extracted. With solid lines we represent the fit obtained with a Brody distribution, Eq. (2), whose parameter value is computedinFig.4. defined as[24]: 1 P (k¯) = (β +1)ηk¯βexp(−ηk¯β+1), (2) B B withk¯ = k/hki and β+1 β +2 η = Γ . (3) (cid:18)β +1(cid:19) and B a normalization constant. Roughly, the Brody distribution in Eq. (2) is, apart some special constants, the product of a power of k with an exponential with a negative exponent proportional to a higher power of k. For the particular case β = 0, the Brody distribution reduces to theexponentialdistributionhavingalways anon-positivederivative. The distributions in Fig. 3 were obtained with values of β slightly above zero, namely between zero and one as shown in Fig. 4. In this case one is able to obtain the non trivial positive slope which is typically observed for small k values in the degree distribution of such social networks. Interestingly, Fig. 4 also shows a linear trend between the average degreehkiinthenetworkandthecorrespondingvalueofβ whichfitsthedegreedistribution. This guarantees that distribution in Eq. (2) has indeed one single parameter. How such a distributioncanbeobtainedfromananalyticalapproachtothemodelofmobileagentsisstill an open questionand willbediscussedin detailelsewhere. Socialnetworks: modelsandmeasures 7 1 0.9 0.8 β 0.7 0.6 0.5 0.4 4 5 6 7 8 9 <k> Figure4.Thelineardependencebetweentheparameterβ oftheBrodydistributioninEq.(2) with the average numberhki of connections. Each bullet correspondsto one of the schools whosedegreedistributionisplottedinFig.3.Thesolidlineyieldsthefitβ =0.094hki+0.078. 3. Particularmeasures forsocialnetworks Tomeasure“thecliquishnessofatypicalneighborhood”inanetwork,WattsandStrogatz[16] introduced a simple coefficient, called the clustering coefficient, which counts the number of pairs of neighbors of a certain node which are connected with each other, forming a cycle of size s = 3. While such tool enables one to access the structure of complex networks arising in many systems [4, 7], helping to characterize small-world networks [16], to understand synchronization in scale-free networks of oscillators [28] and to characterize chemicalreactions[29]andnetworksofsocialrelationships[30,31],thereareothersituations wherethismeasuredoesnotsuit. Namely,whenthenetworkpresentsamultipartitestructure. For instance, when there are two different kinds of nodes and connections link only nodes of different type, the network is bipartite[30, 31, 32] and the bipartite structure does not allow theoccurrence ofcycles withoddsize, inparticularwiths = 3. Bipartite networks are quite common for social systems[32, 33] where the two different kindsofnodesrepresente.g.thetwogenders. Whilethestandardclusteringcoefficientinsuch networks is always zero, they have in general non vanishing clustering properties[31] and therefore more appropriate quantities to access such networks have been proposed, namely coefficients counting larger cycles. In this Section, we will discuss how these different clustering coefficients are related to each other and how one can use them to improve the knowledgeofthenetwork structure. The standard clustering coefficient C is usually defined[16] as the fraction between the 3 numberof cycles of sizes = 3 (triangles)observed in the network out of the total numberof possibletriangleswhich mayappear, namely 2t C (i) = i . (4) 3 k (k −1) i i wheret isthenumberofexistingtrianglescontainingnodeiandk isthenumberofneighbors i i ofnodei, yieldingamaximalnumberk (k −1)/2oftriangles. i i To access the cliquishness in bipartite networks one has proposed[21, 31, 32, 34] a clusteringcoefficientC (i),sometimescalled thegrid coefficient[34], defined as thequotient 4 Socialnetworks: modelsandmeasures 8 betweenthenumberofcyclesofsizes = 4(squares)andthetotalnumberofpossiblesquares. Explicitly,fora givennodei withtwoneighbors,saym andn, thiscoefficient yields[21] q C (i) = imn , (5) 4,mn (k −η )(k −η )+q m imn n imn imn where q is the number of common neighbors between m and n (not counting i) and imn η = 1+q +θ withθ = 1ifneighborsmandnareconnectedwitheachotherand imn imn mn mn 0 otherwise. After averaging over the nodes, the coefficients C and C characterize the contribution 3 4 of the first and second neighbors, respectively, for the network cliquishness. In order to be a suitable quantity to measure the cliquishness of bipartite networks compared to their monopartitecounterparts, C must behave the same way as C when the network parameters 4 3 arechanged,asitisindeedthecaseforhC icomputedfromEq.(5). SeeRef.[21]fordetails. 4 One should notice that in most m-partite networks, it is always possible to have cycles of size s = 4, indicatingthat C is in some sensea more general clustering measure than C . 4 3 However, it could be the case that for a larger number of partitions forming the network, the contribution of larger cycles increases. This is the case, for instance, of trophic relations in an ecological network ofdifferent individualsfrom different species, where large cycles tend to be abundant, namely the ones ranging from the higherpredators to the plants at thelowest trophic level. In such cases, a general clustering coefficient counting the fraction of possible cycles of arbitrary size n may be needed. The generalization is straightforward yielding a clustering coefficient C = E /L , where E is the number of existing cycles with size n, n n n n L the maximal numberof such cycles that is possibleto be attained and n = 3,...,N for a n networkofN nodes. Having C for the required values of n, one is able to introduce a general clustering n measureofthenetwork, givenby thesumofall thesecontributions,namely N N E C = α C = α n, (6) n n n L Xn=3 Xn=3 n where α is a coefficient that weights the contribution of each different clustering order n n and obeys the normalization condition N α = 1. In general one can write E and L in n=3 n n n Eq.(6) as P En = NP(k1)q(k1,k2)NP(k2)q(k2,k3)...NP(kn−1)q(kn−1,kn) (7) X k1,...,kn N! L = BNn! = (8) n n (N −n)! whereBN arethetotalcombinationsofnelementsoutofN,P(k)isthefractionofnodeswith n k neighborsandq(k ,k )isthecorrelationdegreedistribution,i.e.thefractionofconnections 1 2 linkinganodewithk neighborstoanodewithk neighbors. 1 2 FromEq.(7)onecanassumeapproximatelythatE ∼ (hPihqiN)nwithhPiandhqithe n averagefractionsofP(k)andq(k ,k )respectively. SinceL increasesalsoasNn,apossible 1 2 n suitablechoiceforα wouldbeaconstant,namelyα = 1/(N −2)obeyingthenormalization Socialnetworks: modelsandmeasures 9 (a) (b) (c) Figure5. Illustrativeexamplesofcycles(size s = 6)wherethemostconnectednode(◦)is connectedto(a)alltheothernodescomposingthecycle,formingfouradjacenttriangles. In (b)themostconnectednodeisconnectedtoallothernodesexceptone,formingtwotriangles andonesub-cycleofsizes=4,whilein(c)thesamecycles=6enclosestwosub-cyclesof sizes=4andnotriangles(seetext). conditionabove. Havingpresented this general scenario, wenowconcentrate on thetwo first clusteringcoefficients,C andC ,to addressthecyclesizedistribution. 3 4 We first show an estimate introduced in Ref. [35], which considers only the degree distributionP(k) and the distributionof the standard clustering coefficient C (k). One starts 3 by considering the set of cycles with a central node, i.e. cycles with one node connected to all other nodes composing the cycle, as illustrated in Fig. 5a. The central node composes one triangle with each pair of connected neighbors. Due to this fact, the number of cycles with size s can be easily estimated, since the number of different possible cycles to occur is n (s,k) = Bk (s−1)!, for a central node with k neighbors and the corresponding fraction of 0 s−1 2 these cycles which is expected to occur is p (s,k) = C (k)s−2, yielding a total number of 0 3 s-cycles givenby kmax N = Ng P(k)n (s,k)p (s,k), (9) s s 0 0 kX=s−1 whereg isafactorwhich takesintoaccount thenumberofcyclescounted morethan once. s TheestimateinEq. (9)is alowerboundforthetotalnumberofcycles sinceit considers only cycles with a central node. Further, this estimate only accounts for cycles up to size s ≤ k + 1, with k the maximal degree and is not suited for bipartite networks where max max C (k) = 0 for all k. Bipartite networks are typically composed of a set of nodes as those 3 illustratedin Fig. 5c, where nocentral nodeexists. By usingadditionallythecoefficient C (k) in a similarestimate,one isnow ableto take 4 intoaccount several cycleswithoutcentral nodes. One first considerstheset ofcycles ofsize s with one node connected to all the others except one, as illustrated in Fig. 5b. Assuming that this node has k neighbors, s−2 of them belonging to the cycle one is counting for, one hasn (s,k) = Bk (s−2)!/2differentpossiblecyclesofsizes. Thecorrespondingfraction 1 s−2 of such cycles which is expected to occur is givenby p (s,k) = C (k)s−4C (k)(1−C (k)). 1 3 4 3 WritinganequationsimilartoEq.(9),whereinsteadofn (s,k)andp (s,k)onehasn (s,k) 0 0 1 and p (s,k) respectively and the sum starts at s − 2 instead of s − 1, one has an additional 1 numberN′ ofestimatedcycleswhichis notconsidered inestimate(9). s Socialnetworks: modelsandmeasures 10 To improve the estimate further one repeats the same approach, taking out each time oneconnectionto theinitialcentral node, increasing by onethenumberofelementary cycles of size s = 4. Figure 5c illustrates a cycle of size s = 6 composed by two elementary cycles of size 4. In general, for cycles composed by q sub-cycles of size 4 one finds n (s,k) = (s−q−1)!Bk possible cycles of size s looking from a node with k neighbors q 2 s−q−1 and a fraction p (s,k) = C (k)s−2q−2C (k)q(1 −C (k))q of them which are expected to be q 3 4 3 observed. Summingupoverk and q yieldsourfinal expression [s/2]−1 kmax N = Ng P(k)n (s,k)p (s,k). (10) s s q q Xq=0 k=Xs−q−1 where[x]denotestheintegerpartofx. Inparticular,thefirstterm(q = 0)isthesuminEq.(9) and the upper limit [s/2]−1 of the first sum is obtained by imposing the exponent of C (k) 3 inp (s,k) tobenon-negative. q TheestimateinEq.(10)notonlyimprovestheestimatednumbercomputedfromEq.(9), but also enables the estimate of cycles up to a larger maximal size[21], namely up to s = 2k where k isthemaximalnumberofneighborsinthenetwork. max max The estimate in Eq. (10) has also the advantage of being able to estimate cycles in bipartite networks. Since for bipartite networks C (k) = 0, all terms in Eq. (10) vanish 3 except those for which the exponent of C (k) is zero, i.e. for s = 2(q +1) with q an integer, 3 whichnaturallyshowstheabsenceofcyclesofoddsizein suchnetworks. For highly connected networks, both estimates should neverthelessyield similarresults, since in that case there is a very large number of both triangles and squares. For instance, the so-called pseudo-fractal network[36] is a deterministic scale-free network, constructed 106 0.8 (a) (b) 1020 (c) 105 1016 1012 108 0.6 104 104 g)s 0 20 40 60 80 100100 g)s N/(N 103 0.4N/(N 102 0.2 101 1000 20 40 60 80 20 40 60 80 0 s s Figure6. (a)ThefractionN /Ng ofthenumberofcyclesestimatedfromEqs.(9),dashed s s lines, and (10), solid lines, compared with (b) the exact number of cycles as a function of the size for the pseudo-fractal network [36]. From small to large curves one has pseudo- fractal networks with m = 2,3,4,5 generations (see text). In (c) one sees the comparison betweenbothestimatesinascale-freenetworkwithdegreedistributionP(k) = P k−γ with 0 P = 0.737and γ = 2.5, and coefficientdistributionsC (k) = C(0)k−α with C(0) = 2, 0 3,4 3,4 3 C(0) =0.33andα=0.9. 4