Structure Formaton of Socal Network DU Nan 1, FENG Hu 2, HUANG Zgang 3, Sally MAKI 4, WANG Ru(Ruby) 5, and ZHAO Hongxa (Melssa) 6 1 Bejng Unversty of Posts and Telecommuncatons, Chna 2 Fudan Unversty, Chna 3 Lanzhou Unversty, Chna 4 Unversty of Calforna, Berkeley, USA 5 Central Chna Normal Unversty, Chna 6 Insttute of Automaton, Chnese Academy of Scences, Chna Abstract: We consder how ndvdual s behavors affect structure formaton of socal network. A dynamc model s desgned, n whch nodes move and buld nterpersonal relatonshp wth other nodes accordng to ts personalty factor. Agent-Based modelng shows evolutonal structure wth small-world and power-law propertes. Keywords: Socal network; complex networks; Agent-Based modelng 1 Introducton Socal network s the backbone of socal relatonshp, whch s an attractng nterdscplnary research feld wthn socal scence, statstc and mathematcs [1]. The structure of socal networks has dstnct propertes from random networks and regular networks. The well-known sx-degree experment done by Mlgram n 1969 [2] revealed the astonshng short average dstance compare wth the large scale of socal network, whch depcts the small-world network [3], as well as large clusterng coeffcent 1. Besdes small-world property, many natural complex networks present scale-free dstrbuton [4], e.g. socal network, bologcal network, WWW, n whch the degree dstrbuton of network follows power-law. It s also dfferent from the sngle-scale dstrbuton n random networks and regular networks. Although several computatonal models could generate small-world or scale-free networks [3]-[5], these models do not consder detaled backgrounds of dfferent networks ownng smlar propertes. For nstance, statstcal methods have gven some common characterstc of socal network, whle what are the reasons behnd the behnd those characterstc. [6]-[8] use game theory to evaluate the nfluence on network structure by node par s game. In [9], lnks are bult mtatng socal communcaton rules wthn node par selected randomly. However, random parng methods do not consder the contnuous space-tme evoluton procedure n real socal scene. The dfferences of nodes are not consdered, ether. We desgned an evolutonary model to evaluate the nfluence of ndvdual s behavors on formaton of socal network structure, n whch personalty factor s key pont on agents movng and ntercommuncaton. 1 Large clusterng coeffcent means one nodes neghbors are more lke to connect wth each other.
2 Evoluton model of socal network 2.1 The weghted graph model of socal network Weghted graph G = (N, L, W) s proper to descrbe nterpersonal relatonshp wth a vew of dfferent socal ntmacy among peoples, n whch N s the set of nodes, L s the set of lnks and W s the set of weghts on lnks. We use N N adjacent matrx to represent the lnk relatonshp between nodes, n whch a j =1 f there s a lnk present between node and j, a j =0 f node and j are not adjacent. There s no self-crcle n our model, n other words, all dagonal elements of A are zero. Smlarly, we use N N weghts matrx W for weghted graph, n whch w j s the weght of the lnk connectng from node to node j. w j = 0 f node and j are not connected, and a postve nteger s assgned to w j f node and j are adjacent. Each node has three propertes: personalty, step-sze and energy. The personalty s a real number n [0, 1], whch represents how extroverted (small personalty factor) or ntroverted (large personalty factor) one acts n socal communcaton. Extroverted nodes prefer to makng new frends, whle ntroverted nodes would lke to stay wth old frends. Nodes move nsde an area wth perodc boundary and would meet other nodes 2. New lnk a j wll be bult f node and j meet each other and no lnk before them. The ntal weght s set to 1. w j wll be added by 1 3 at each tme when node and j meet each other. We gve energy factor E to node, whch wll be consumed by cost j = cost = p p when new lnk between node and j s bult. Each j j node s neghbors wll not be added f resdual energy cannot afford new lnks, then only weghts update all the tme. The cost of new lnk depends on mutual personalty factor. Extroverted people have more frends and less cost of buldng new lnk, whch follows socal common sense. 2.2 Dynamc evoluton rules Dscrete model s sutable for computer smulaton. Nodes are dstrbuted randomly n some area and move around accordng to the state of last epoch. The movng method depends on angle and step sze. new new The movng angle of node ponts to targeted poston ( x, y ) consstng of determnstc part and random part 4 as Equ.(1). The determnstc part s the weghted average poston of nearest neghbors, whch stands for the nfluence from frends. The random part, x random and y random, are the random poston nsde the area, whch expresses the desre for explorng new socal relatonshp. x y new new where = = [ p x + ( 1 p ) xrandom ] [ p y + ( 1 p ) y ] random x = wj x j wj, y = wj y j j N j N, (1) j N j N w j, and N s the set of node s nearest neghbors. The more extroverted a node s, the more randomly t moves and t would lke to make new frends. The step-sze s of node also depends on ts personalty factor p, extroverted nodes behave 2 Two or more nodes meet each other f they locate the same grd n smulaton zone. 3 Smlar count method as n [10] 4 Smlar desgn as lnear Vcsek model [11].
actvely n socal communcaton and take more aggressve move, as step ( 1 p ) stepmax =, (2) n whch step max s the preset maxmum step sze. 3 Smulaton of socal network formaton We ddn t gve rgorous dynamcs of the teratve model n Secton 2.2, n that t s too dffcult to deduce n mathematcal way. Instead, we use Agent-Based smulaton method nvented n 1940s, whch dd not become a conventonal method n research on socal network, dstrbuted processng, macro-economcs etc. untl 1990s subject to computaton capablty. Each agent n Agent-Based smulaton follows certan rules and move accordng to local nformaton. By lots of teraton, some statstcal result may emerge. In ths work, we use NetLogo 5, a wdely used Agent-Based tool, to smulate our model, whch can provde frendly vsualzaton. Fgure 1 shows the ntal state of once smulaton. Fgure 1 Intal state of once smulaton of 100 nodes for E =1 and Step max =5 n 30*30 perodc boundary 6 rectangle. The personalty factor p has great effects on network evoluton n our model. To llustrate an ntutve way, we present two smulaton snapshots as Fgure 2, where all p s fxed to ones and zeros. In all ones case, all the nodes prefer to stay wth ther nearest neghbors and don t lke to make new frends, so we can observe many small groups n the snapshot. The all zeros case gves the opposte phenomenon, where each node has random lnks wth others. Snce extreme cases never happen n real world, we use lmted Gaussan dstrbuton 7 nstead n later smulaton. Fgure 2 gves once smulaton snapshot wth Gaussan personalty factor, whch presents topology status between two extreme cases. 5 http://ccl.northwestern.edu/netlogo/ 6 When an object passes through one boundary of the area, t reappears on the opposte boundary wth the same velocty. Perodc boundary condtons are partcularly useful for smulatng a part of a bulk system wth no surfaces present. 7 Ths dstrbuton s derved from a small survey over about 40 students n CSSS2008. Not strctly, of course.
(a) (b) Fgure 2 Evoluton snapshots wth fxed personalty factor, the other parameters are same as n Fgure 1. (a) s the snapshot n T=100 wth all p=1; (b) s the snapshot n T=100 wth all p=0. Fgure 3 Smulaton snapshot wth lmted Gaussan personalty between 0.4 and 1, the other parameters are same as n Fgure 1. 4 Results and analyss 4.1 Evaluaton metrcs for weghted graph We use specal metrcs [13][14] for weghted graph to evaluate the network structure, as well as some metrcs [15] for non-weghted graph f necessary. 1) weghted clusterng coeffcent The weghted clusterng coeffcent of node s defned as c = ( wj + wm ) 1 aj a jm am. s (k 1) j,m 2 The weghted clusterng coeffcent of whole graph s the average of all the nodes. (3)
C = 1 c (4) N 2) weghted dstance Frst, we defne the length l j between node and j as the recprocal of the weght, l j =1/w j. It s then possble to defne the weghted dstance d j as the smallest sum of the edge lengths throughout all the possble paths n the graph from node to j. The dstance n weghted graph can be calculated usng well-known Djkstra algorthm. 3) Degree dstrbuton The degree k of a node s the number of edges ncdent wth the node, and s defned n terms of the adjacency matrx A as k =, (5) a j j Ν where N s the set of the nearest neghbors of node. The degree dstrbuton P(k) s defned as the fracton of nodes n the graph havng degree k. 4) Strength Dstrbuton In weghted graph, the strength of node s the sum of weghts between node and ts nearest neghbors, defned as s =. (6) w j j Ν The degree dstrbuton P(s) s defned as the fracton of nodes n the graph havng strength s. 5) Weght Dstrbuton The degree dstrbuton P(w) s defned as the fracton of nodes n the graph havng strength w. 4.2 Analyss Fgure 4 shows the average dstance and clusterng coeffcent as functons of smulaton epochs. When topology become steady, we observe typcal small-world propertes wth short dstance and large coeffcent. (a) (b) Fgure 4 Small-world propertes of 500 nodes smulaton results for E =0.5, the other parameters are same as n Fgure 1. (a) Average dstance decreases wth smulaton epochs. The steady shortest path length s rather small. (b) The clusterng coeffcent does not change a lot wth smulaton epochs and remans a rather large number.
Fgure 5 shows the power-law dstrbuton of weghts and strength, whch becomes more clearly as ncreasng number of nodes. (a) (b) Fgure 5 Scale-free dstrbuton of (a) weghts and (b) strength when topology becomes steady, where the blue lne s the fttng power law. All the parameters are same as n Fgure 4. 5 Concluson Our work ntends to smulate the nfluence of nterpersonal relatonshp on socal network structure. We observe large clusterng coeffcent, short average dstance, and power-law dstrbuted weghts and strength, whch presents certan regular herarchy and communty structure concdng wth the real world. However, ths model s a toy rather than real research. Some assumpton s too arbtrary to convncng, such as the energy lmtaton, maxmum step sze and so on. And we dd not mplement large-scale smulaton due to the capablty of smulaton tool, so the result s not suffcent n statstcal way. Also we dd not have enough tme to valdate the communty structure and modularty [16] observed on smulaton, nor the jon-and-qut mechansm. 6 Acknowledgements Ths work was done n the Complex System Summer School (CSSS) held by Santa Fe Insttute and Insttute of Theoretcal Physcs, Chnese Academy of Scence n 2008. We thank Aaron Clauset for hs project advce. We also thank the organzers and sponsors of ths summer school. References: [1] J. Scott, Socal network analyss: A handbook, Sage, 2000. [2] Travers, Jeffrey and Stanley Mlgram. "An Expermental Study of the Small World Problem." Socometry, Vol. 32, No. 4, 1969, pp. 425-443. [3] D.J. Watts and S.H. Strogatz, Collectve dynamcs of `small-world' networks, Nature, vol. 393, Jun. 1998, p. 440. [4] A. Barabás and R. Albert, Emergence of Scalng n Random Networks, Scence, vol. 286, 1999, pp. 509-512. [5] M.E.J. Newman and D.J. Watts, Renormalzaton group analyss of the small-world network
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