Competitive Dynamics on Complex Networks
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COMPLEX NETWORKSAPPLIED MATHEMATICSAPPLIED PHYSICSCorrespondence and requests for materials should be addressed toX.F.W ([email protected])
Competitive Dynamics on Complex Networks
Jiuhua Zhao, Qipeng Liu & Xiaofan Wang * Department of Automation, Shanghai Jiao Tong University, and Key Laboratory ofSystem Control and Information Processing, Ministry of Education of China, Shanghai200240, China.
We consider a dynamical network model in which two competitors have fixedand different states, and each normal agent adjusts its state according to adistributed consensus protocol. The state of each normal agent converges toa steady value which is a convex combination of the competitors' states, andis independent of the initial states of agents. This implies that thecompetition result is fully determined by the network structure and positionsof competitors in the network. We compute an Influence Matrix (IM) inwhich each element characterizing the influence of an agent on anotheragent in the network. We use the IM to predict the bias of each normal agentand thus predict which competitor will win. Furthermore, we compare the IMcriterion with seven node centrality measures to predict the winner. We findthat the competitor with higher Katz Centrality in an undirected network orhigher PageRank in a directed network is much more likely to be the winner.These findings may shed new light on the role of network structure incompetition and to what extent could competitors adjust network structureso as to win the competition. ompetition among a set of competitors for obtaining a maximumnumber of votes from other agents in a social network is a bothimportant and common phenomena in real world. The competitorscould be candidates in numerous leader-selection cases, ranging from head–election in a small group to president-election in a whole country . They couldalso be those who have different proposals or promote different brands of aproduct such as mobile phone and car . There have also been some researcheson, for example, how the fractions of speakers of several competing languages C evolve in time and even how the emerging Bitcoins appear to be a possiblecompetitor to usual currencies .The most well-known model in social dynamics for the competition of speciesis the voter model , which has also later on been used for the analysis ofdiffusion of innovations and consumption decisions. In its simplest form, eachagent in the voter model holds one of the two states. At each time step, arandomly selected agent takes the state of one of its neighbors. Over the years,many modifications and extensions of the original voter model have beenproposed . Voter-like dynamics on networks with different topologies and theinterplay between topology and dynamics have also been investigated .However, many of such models, including the voter model, Sznajd model ,Deffuant model , Hegselmann-Krause model and so on have been focused onwhether full consensus can be reached.A nature way to consider the existence of competitors in a network is to viewthem as zealots or stubborn agents with fixed and different states. Forexample, it is shown that the existence of competing zealots in the voter modelprevents convergence and results in fluctuations in regular lattices andcomplete graphs . Competitive dynamics with continuous states in the stochasticgossip model is investigated in Ref.16, in which long-run disagreements andpersistent fluctuations appear. Influence of network structure and locations ofstubborn agents on the fluctuation of final states in a binary opinion formationmodel is studied in Ref. 17. In Ref. 18 , given one set of stubborn agents as mis-informers (agents who spread misinformation), the placement of the other set ofstubborn agents (named information disseminators) is formulated as anoptimization problem.The question we address in this work is: How do positions of competitors in anetwork affect voting outcome? That is, can we predict which competitor will win in the sense that majority of agents in the network will eventually support thecompetitor? Can we predict which competitor a normal agent will support basedon the network structure?
Intuitively, the problem of which competitor will winshould be related to the relative impact of the competitors in a network. How tocharacterize the impact or importance of an individual (or even a community) in anetwork is a question of great importance and applications in network analysis.Traditionally, identifying such influential nodes usually relies on concepts ofcentralities, including degree (DC), betweenness (BC) , closeness (CC) ,eigenvector centrality (EC) , Katz centrality (KC) , PageRank (PR) , and so on.Recently, a lot of researches have also been focused on identifying influentialnodes in dynamical processes on networks. For example, Kitsak et al. haveargued that there are circumstances in which a node with the highest DC or thehighest BC has little effect, and the most efficient spreaders are those locatedwithin the core of the network as identified by the k-shell decomposition .However, till now, we still lack an understanding on which of these measurescould best predict the winner among competitors in a network. Results
A dynamic model for competition.
We consider a directed and weighted networkwith N agents and M links. The agent set is denoted as {1, 2, , } V N and thetopology of the network is described by a coupling matrix ( ) kl N N A a : if agent k isdirectly influenced by agent l , then there is a link from agent k to agent l and kl a ; otherwise, kl a . For simplicity, we assume that there are just twocompetitors in the network, denoted as agents i and j , which have fixed anddifferent states as follows: ( ) 1, ( ) 1, 0 i j x t x t t . (1) Every other agent (called normal agent) / { , } k V i j has an initial state randomlychosen from [+1,-1] and updates its state as follows: ( 1) ( ) ( ) ( ) k k k kl l kl N x t x t a x t x t , (2)where ( ) k x t is the state of agent k at time t ; the parameter captures the level ofneighbors’ influence; { | 0} k kl N l V a is the set of neighboring agents of agent k that can directly influence agent k . Note that Eq. (2) belongs to a set of distributedconsensus protocols, which can be traced back to the classical model of DeGroot .However, the existence of competitors in the network prohibits global consensus.Instead, we have the following convergence result:Suppose that1) Each normal agent has a path connecting to at least one competitor;2) D , where max D is the largest out-degree of agents in the network.Then the state of each normal agent will eventually reach a steady value, i.e., as t ,
1( ) ( ) 1 norm i j
X t X D A c c , (3)where Nnorm
X R represents the state vector of all normal agents, and D , A and i j c c can all be derived from the network coupling matrix A . Furthermore, if (0) [-1, +1] k x , / { , } k V i j , then ( ) [-1, +1] k x t , t . The detailed analysis can befound in Methods. k x ( k x ) implies that agent k will finally support competitor i ( j ), and | | k x corresponds to the degree of supporting. k x implies that agent k will be a neutralagent which does not support any competitor. Denote /{ , } sgn( ) ij kk V i j x , (4)where sgn() is the sign function. If ij , then competitor i will win in the sensethat more normal agents will support him; if ij , competitor j will win; if ij ,the competition ends up with a draw. An illustration example.
Fig. 1 shows the competitive dynamics on three simpleundirected networks which have the same number of agents but different couplingstructures. We take agent 1 and agent 10 as two competitors in each network withfixed states x and x . Steady states of normal agents are computedaccording to Eq. (3). An red (blue) node represents an agent with positive (negative)state. The darker the color the larger the absolute value of the state. Nodes withwhite color represent neutral agents. (a)(b) (c) Figure 1 | An example of how network structure influences the competitionresult. (a) A simple undirected network of 10 agents with each edge of unit weight.The competition between agent 1 and agent 10 ends up as draw. (b) The network isderived from (a) by adding one edge between agent 2 and 6, which results in agent10 being the winner. (c) The network has the same structure as network (a) but withdifferent edge weights, which leads to agent 1 being the winner.
For network (a), , hence the competition ends up as draw. Network (b) isderived from network (a) by just adding one edge between agents 2 and 6, whichresults in and agent 10 being the winner. By changing weights of edges innetwork (a), we get network (c), which leads to and agent 1 winning thecompetition. We can see that both network structure and coupling weights influencethe competition results. In the following, we will focus on unweighted networks in thesense that the weight of every link in a network is one. Verification on a real network.
To see whether Eqs. (1)-(2) could properly modelcompetition in real social networks, we test it on a commonly used benchmark modelin social network analysis---the Zachary’s karate club network as shown in Fig. 2(a),which is a network of friendships between 34 members of a karate club at a USuniversity in the 1970s. Due to the confliction between the manager (agent 34) andthe coach (agent 1), the club finally splits into two communities, centered at themanager and the coach, respectively, as depicted by the vertical dashed line in Fig.2(a).In simulation, we fix the states of agents 1 and agent 34 at +1 and -1,respectively. The state of every other agent evolves according to Eq. (2). Fig 2(b)shows the steady states of all agents in the network, in which red agents aresupporters of agent 1 and blue agents are supporters of agent 34. It is surprising tonote that this splitting result completely matches the real situation as shown in Fig.2(a). Furthermore, Fig. 2(b) also reveals the degree of supporting of each normalagent, represented by the darkness of the color. For example, agent 9 has thesmallest absolute value of steady state among those supporters of agent 34, whichimplies that agent 9 is the weakest supporter of agent 34. This is also consistent withthe reality that individual 9 is indeed the weakest political supporter of the manager .Therefore, although our model is a very simplified version of the very complex real-world competition, it might be a reasonable mechanism for the competitive dynamicsin some real social networks. Note that many network community detection methods can correctly reveal the two communities in the karate network , however, they donot explicitly use the information of the two competitors in the network and cannotreveal the degree of supporting of each agent towards the corresponding competitor.(a)(b) Figure 2 | Verification of the model on Zachary’s karate club network. (a) Tworeal communities in the network led by agent 1 and agent 34, respectively, as dividedby the dashed line in the figure. (b) Two communities derived from our model. Redcommunity consists of supporters of agent 1 and blue community consists ofsupporters of agent 34. Darkness of the color represents the degree of supporting.
Influence Matrix Criterion.
From the steady states expression in Eq. (3),competition results are fully determined by network structure and positions of the competitors in the network. However, directly computing the steady states accordingto Eq. (3) is computational inefficient for large-scale networks, since for everydifferent pair of competitors, we have to re-compute the steady states. In thefollowing, we compute the Influence Matrix (IM), in which each element characterizesthe impact of one agent on another. Note that if there is a link from agent k to l , i.e., kl a , then agent l has a direct impact on agent k . If there is a link from agents k to m , and a link from agent m to l , then agent l has an indirect impact on agent k via agent m . Intuitively, such an indirect impact should be weaker than the directimpact. Taking into account the fact that the number of paths of length r from agent k to l is ( ) r kl A in the unweighted network case, we define IM as a sum of theexponentially decreasing impact of increasingly paths: F I A A (5)where (0, 1) is an attenuation factor. If (0, ) , where is the largesteigenvalue of matrix A , then the above series converges and we have: ( ) F I A . (6)Let ki f be the entry of F on k th row and i th column. Denote /{ , } ( ) ij ki kjk V i j sgn f f , (7)We have the following IM criterion: Which competitor will a normal agent support: If ki kj f f ( ki kj f f ), thenagent k will support competitor i ( j ); If ki kj f f , then agent k is a neutralagent; Which competitor will win: If ij ( ij ), then competitor i ( j ) willwin; If ij , the competition ends up with a draw. Although different choice of in Eq. (7) may generally result in different IM, wefind that the IM criterion is robust with respect to , in the sense that the criteriongives similar qualitative prediction for different choice of [0.5 , 0.9 ) (seeSupplementary Figure S1). In the following simulations, we set . Who will you support, and who will win from IM criterion.
For the Zachary’skarate club network, Fig. 3 shows the difference ,1 ,34 k k f f between the influences oftwo competitors (agent 1 and agent 34) on a normal agent k . Comparing Fig. 3 withFig. 2(b), we can see that ,1 ,34 k k f f ( ,1 ,34 k k f f ) if and only if k x ( k x ),which implies that the competition result can be fully predicted by the IM criterion inthis case. Figure 3 | Application of the IM criterion to Zachary’s karate club network.
Agent 1 and agent 34 are two competitors. A normal agent is colored red (blue) if theinfluence difference k k f f ( k k f f ). We dye all the nodes according to theirnormalized difference. The darker the color the larger the absolute difference is.In general, for a given pair of competitors i and j in a network, we use the IMcriterion to predict the bias of each normal agent and calculate the success rate ofprediction as follows: /( , ) ( ) ( ) ( ) ( ) +g ( ) ( )1( 2) 2 ik jk k ik jk k ik jk kij k V i j sgn f f sgn x sgn f f sgn x sgn f f sgn xN , (8) where ( ) 2 g x , if x ; otherwise ( ) 0 g x . The average success rate of predictionon the bias of normal agents over all the ( 1) / 2 N N possible pairs of competitors in anetwork is denoted as . Similarly, the success rate of prediction on who will winas the fraction of correct prediction over all the possible pairs of competitors can beformulated as follows: , ij ij ij ij ij iji j sgn sgn sgn sgn sgn sgnN N . (9)Table I shows the value of and for 15 real social networks. The maximumvalue of is 91.6%, the minimum is 74.0% and the average is 83.6%. isalmost always larger than 80%: the maximum is 96.9%, the minimum is 79.9% andthe average is 86.0%. These results verify the validity of the IM criterion. Weconjecture through simulation that for most pairs of competitors the prediction of anormal agent's bias being incorrect is because two competitors have very similarinfluence on the normal agent (see Supplementary Figure S2). Comparison with centrality-based criteria.
Given a pair of competitors, we canpredict which competitor will win by the IM criterion. Intuitively, the winner should bemore important or have higher impact on the network than the loser. Over the years,a number of centrality measures have been proposed to characterize the “importance”or “impact” of a node in a network. However, one difficulty in applying thesecentrality measures is that it is often unclear which of the many measures should beused in a particular circumstance. Here, we compare the IM criterion with criteriabased on several common-used node centrality measures, including betweeness (BC),closeness (CC), degree (DC), eigenvector (EC), Katz (KC), K-Shell (KS) and PageRank(PR) (see Methods for the computation of these measures).
Centrality-based criterion:
The competitor with higher centrality value will win.Competitors with the same centrality value will end up with a draw. For each criterion, we calculate the success rate of prediction as the fraction ofcorrect prediction of who will win over all ( 1) / 2
N N possible pairs of competitors. Fig.4 and Fig. 5 show the success rate of prediction for 8 real undirected networks and 7real directed networks, respectively. According to the average success rate overundirected and directed networks, we have the following order: For undirected networks: KC (84.8%), IM (84.4%), EC(79.7%), PR (78.4%), DC(77.8%), BC (69.4%), KS (61.4%), CC (39.6%). For directed networks: PR (93.0%), KC (88.3%), IM (88.0%), EC(87.0%), DC(79.8%), BC (77.3%), KS (61.0%), CC (37.5%).We can see that criteria based on KC, PR, IM and EC are always better than thecriteria based on the other four centralities. For undirected networks, KC criterion hasthe best performance: It provides highest success rate of prediction in 5 of 8networks. On the other hand, PR criterion is always the best for each of the 7 directednetworks. From the definition of KC, PR and EC, these results imply that whether acompetitor could win depends to a large extent on both the number and importanceof those agents that the competitor could directly influence.In fact, the KC of node i can be directly defined from IM as the influence of node i on the whole network: i kik V KC f . (10)The KC-based prediction criterion can be derived from the IM criterion by justchanging the order of summation and sign function in Eq. (7): /{ , } ( ) ( ) ( ) i j ij ki kjk V i j sgn KC KC sgn KC sgn f f , (11)where i KC is the KC value of node i (For a directed network, we just need to add onemore term ( ) ji ij f f in the sum). Directly summing up the influence errors in Eq. (11) may help reduce perturbation, and thus result in more robust criterion. This mightexplanation why KC criterion is better to predict the winner than the IM criterion.PageRank is basically a variant of Katz centrality which is widely used for rankingnodes in directed networks such as WWW . Although IM criterion is not the best, anadvantage of IM criterion over node-centrality based criteria is that it could alsopredict the bias of each normal agent, in addition to predict the winner.Degree (DC) is certainly the simplest criterion to predict the winner. However, it isa bit surprising to see that DC criterion provides as high as 80% success rate ofprediction and performs even better than criteria based on BC, KS and CC. Thisimplies that the number of agents that competitors could directly influence is still arelatively important factor. On the other hand, CC turns out to be the poorestcriterion to predict the winner: the corresponding average success rate is just a littlebit better than that of the completely random guessing (33.3%). Note that CC of anode captures how long it will take to spread information from the node to all othernodes sequentially. Our results show that this score has little effect on thecompetition. (a) (b) Figure 4 | The success rate of prediction of competition result on 8 realundirected networks.
Here we compare the IM criterion with 7 centrality-basedcriteria. (a) the success rate of prediction for each network. (b) the average successrate of prediction of each criterion over 8 networks.(a) (b) Figure 5 | The success rate of prediction of competition result on 7 realdirected networks.
Here we compare the IM criterion with 7 centrality-based criteria.(a) the success rate of prediction for each network. (b) the average success rate ofprediction of each criterion over 7 networks.
Discussion
In summary, we study a model of competitive dynamics in which two competitorshave fixed and different states, and each normal agent adjusts its state according to adistributed consensus protocol. The steady states of normal agents are fullydetermined by the network structure and positions of competitors in the network.Although real world competition involves a number of complex factors, we find thatthis very simple model can completely reveals the competition result in the well-known Zachary’s karate club network. We investigate the Influence Matrix (IM)criterion to predict which competitor a normal agent will support and whichcompetitor will win. We further compare the IM criterion with seven well-known nodecentrality measures. We find that Katz centrality (KC) and PageRank (PR) providebest prediction for undirected and directed networks, respectively.These findings suggest that competitors in a network might use techniques suchas PageRank optimization to adjust network structure in order to win the competition. Although we assume that there are only two competitors in the model,the above analysis can also be generalized to the case with two sets of competitors ina network, and a nature way to deal with this case is to view all agents in a set as asuper-agent. However, a key challenge here is that there does not existing a simplerelationship between the sum of the centrality values of all agents in a set in theoriginal network and the centrality score of the super-agent in the new network. Allthese issues will be considered in future work. Methods
Theoretical analysis of the model.
Eqs. (1)-(2) can be can be rewritten in thefollowing matrix form: ( 1) ( ) ( )= ( ) N X t I H L X t TX t , (12)where N I is an identity matrix; L D A is the Laplacian matrix, D is the diagonalmatrix of agents’ out-degrees; H is an indicative diagonal matrix with ( , ) 0 H s s ifagent s is a competitor, and ( , ) 1 H s s otherwise. Obviously, the sum of each row ofmatrix T equals to 1.For convenience, we reorder the agents so that the two competitors come last. Thus,we have i j DD d d and i jij AA c crr , (13)where i d and j d denote the out-degrees of competitor i and j , respectively; vectors i c , j c , i r , and j r contain the corresponding elements in the reordered couplingmatrix.Hence, Eq.(12) can be rewritten as ( 1) ( )( 1) ( )1 0( 1) ( )0 1 norm normi ij j X t X tQ Bx t x tx t x t , (14)where Nnorm
X R represents the state vector of all normal agents; ( ) N Q I D A and i j B c c . Thus, ( 1)( ) ( 1) ( 1)(0) (0) (0) inorm norm jt it knorm k j x tX t QX t B x t xQ X Q B x (15)If each normal agent has a path connecting to at least one competitor, then ( ) N D A R is invertible . Since D , we can show from Geršgorin disktheorem that the spectral radius of Q is less than 1. Thus, as t , we have
12 12 21 (0)( ) ( - ) (0) (0) ( - ) (0)1 ( ) .1 inorm N j iN N i j ji j xX t I Q B x xI I D A xD A c cc c (16)According to Lemma 4 in Ref. 32, each entry of ( ) i j D A c c is nonnegativeand each row sum of ( ) i j D A c c is equal to one. Thus, the steady state of eachnormal agent is a convex combination of +1 and -1. Computation of centrality measures.
As in our definition of the network structure,an link from agent k to agent l means agent k could be directly influenced by agent l . Hence, for directed networks, we compute the centrality measures as follows(which can be also applied to undirected networks):In-BC : the in-betweenness centrality of node l is computed by ( )( ) kmin k m km g lCB l g ,where km g is the number of geodesics from node k to node m and ( ) km g l is thenumber of geodesics that node l is on;In-CC : the in-closeness centrality of node l is computed by
1( ) ( , ) in k l
CC l d k l ,where ( , ) d k l is the shortest distance from node k to node l ;In-DC: the in-degree of a node is the number of agents that an agent could directlyinfluence;In-EC : the eigenvector centrality is a natural extension of degree by consideringboth the number and the importance of those agents that an agent could directlyinfluence. The EC of a network is equal to the eigenvector corresponding to thelargest eigenvalue of the coupling matrix. According to the definition of the networkstructure, we use T A to compute the In-EC;In-KC: the Katz Centrality is a variation of EC, by adding an initial importance to eachagent. The In-Katz-Centrality of a network is computed by ( ) T I A , where is a vector with all ones of an appropriate size, and the attenuation factor. Relatedstudie shows that there is no significant change in ranking of nodes based on KatzCentrality with [0.5 , 0.9 ) . In simulations, we set ;In-KS : nodes are assigned to different in-shells according to their remaining in-degrees, which is obtained by successive pruning of nodes with in-degree smallerthan the current in-k-shell value. We start by removing all nodes with in-degree in k , until that all nodes left are with in-degree larger than 1. The removed nodes,along with the corresponding links, form an in-k-shell with index ins k . In a similarfashion, we iteratively remove the next in-k-shell. As a result, each node is associatedwith one ins k index;PageRank: the algebraic expression of the page rank can be formulated as
1( ) T PR I A D N , where is the dampening factor. We use the powermethod to compute the page rank value, and set . The Page Rank is avariation on the Katz Centrality by dividing the importance of those agents whichcould directly influenced by an agent, by their out-degrees.1. Chatterjee, A., Mitrovic, M., Fortunato, S. Universality in voting behavior: anempirical analysis.
Sci. Rep. J. Management
Adv. Complex Syst.
On thedynamics of competing crypto-currencies. arXiv:
Biometrika
Ann. Probab. Rev. Mod. Phys.
591 (2009).8. Suchecki, K., Eguíluz, V. M. & San Miguel, M. Voter model dynamics in complexnetworks: Role of dimensionality, disorder, and degree distribution.
Phys. Rev. E
J. Stat. Phys.
Int. J. Mod.Phys. C
Physica A
Adv. Complex Syst. J. Artif. Soc. Soc. Simul. Phys. Rev. E
J.Stat. Mech.-Theory Exp.
P08029 (2007).16. Acemoglu, D., Como, G., Fagnani, F. & Ozdaglar, A. Opinion fluctuations anddisagreement in social networks.
Math. Oper. Res.
17. Yildiz, E., Ozdaglar, A., Acemoglu, D., Saberi, A. & Scaglione, A. Binary opiniondynamics with stubborn agents.
ACM Tran. Econ. and Comput.
19 (2013).18. Fardad, M., Zhang, X., Lin, F. & Jovanovic, M.R. On the optimal dissemination ofinformation in social networks.
Proceedings of the 51st IEEE Conference ofDecision and Control (CDC’12), pp. 2539-2544 (2012).19. Freeman, L. C. A set of measures of centrality based on betweenness.
Sociometry,
J. Acoust. Soc. Am.
Tran. Inst. Br.Geogr.
Psychometrika
Technical Report, Computer Science Department,Stanford University , (1998).24. Kitsak, M., Gallos, L.K., Havlin, S., Liljeros, F., Muchnik, L., Stanley, H.E. & Makse,H.A. Identification of influential spreaders in complex networks.
Nat. Phys. J. Am. Stat. Assoc.
J.Anthropol. Res.
Phys. Rep.
The algebraic eigenvalue problem.
Oxford Univ. Press, New York,(1965).29. Franceschet, M. PageRank: Standing on the shoulders of giants.
Commun. ACM
Discrete Appl. Math.
Automatica
Automatica
Soc. Networks
Psychometrika arXiv:
Adv. Data Anal. Classif. TechnicalReport, Uppsala University, (2004).38. Leskovec, J., Kleinberg, J. & Faloutsos, C. Graph evolution: Densification andshrinking diameters.
ACM Trans. Knowl. Discov. Data Behav. Ecol. Sociobiol.
Phys. Rev. E Proceedings of the 25th Conference on Advances in Neural Information ProcessingSystems (NIPS’12), pp. 548-556 (2012).42. Girvan, M. & Newman, M.E. Community structure in social and biological networks.
Proc. Natl. Acad. Sci. USA
Phys. Rev. E (2013).
45. Massa, P., Salvetti, M. & Tomasoni, D. Bowling alone and trust decline in socialnetwork sites.
Proceedings of the 8th IEEE International Conference onDependable, Autonomic and Secure Computing (DASC’09), pp. 658-663 (2009).46. Opsahl, T. & Panzarasa, P. Clustering in weighted networks.
Soc. Networks
ACM Trans. Knowl. Discov. Data Proceedings of the 3rd International Workshop on LinkDiscovery, pp. 36-43 (2005).49. Michalski, R., Palus, S. & Kazienko, P. Matching organizational structure and socialnetwork extracted from email communication.
Lecture Notes in BusinessInformation Processing
Proceedings of the 4th International AAAI Conference on Weblogs andSocial Media (ICWSM’10), pp. 34-41 (2010).51. Leskovec, J., Huttenloche, D. & Kleinberg, J. Signed networks in social media.
Proceedings of the 28th ACM Conference on Human Factors in Computing Systems(CHI’10), pp. 1361-1370 (2010).
Acknowledgments
This work was supported by the National Natural Science Foundation of China underGrant Nos. 61374176 and 61104137, the Science Fund for Creative Research Groupsof the National Natural Science Foundation of China (No. 61221003), and the NationalKey Basic Research Program (973 Program) of China (No. 2010CB731403).
Author contribution
X.W. envisioned the study. J.Z., Q.L. and X. W. conceived the theoretical analysis. J.Z.designed the experiments and performed the computational analysis. J.Z., Q.L. andX.W. wrote the manuscript.
Additional information.
Competing financial interests:
The authors declare no competing financial interests.
License:
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareALike 3.0 Unported License. To view a copy of this license, visithttp://creativecommons.org/licenses/by-nc-sa/3.0/
Figure Legends
Figure 1 | An example of how network structure influences the competitionresult. (a) A simple undirected network of 10 agents with each edge of unit weight.The competition between agent 1 and agent 10 ends up as draw. (b) The network isderived from (a) by adding one edge between agent 2 and 6, which results in agent10 being the winner. (c) The network has the same structure as network (a) but withdifferent edge weights, which leads to agent 1 being the winner.
Figure 2 | Verification of the model on Zachary’s karate club network. (a) Tworeal communities in the network led by agent 1 and agent 34, respectively, as dividedby the dashed line in the figure. (b) Two communities derived from our model. Redcommunity consists of supporters of agent 1 and blue community consists ofsupporters of agent 34. Darkness of the color represents the degree of supporting. Figure 3 | Application of the IM criterion to Zachary’s karate club network.
Agent 1 and agent 34 are two competitors. A normal agent is colored red (blue) if theinfluence difference k k f f ( k k f f ). We dye all the nodes according to theirnormalized difference. The darker the color the larger the absolute difference is. Figure 4 | The success rate of prediction of competition result on 8 realundirected networks.
Here we compare the IM criterion with 7 centrality-basedcriteria. (a) the success rate of prediction for each network. (b) the average successrate of prediction of each criterion over 8 networks.
Figure 5 | The success rate of prediction of competition result on 7 realdirected networks.
Here we compare the IM criterion with 7 centrality-based criteria.(a) the success rate of prediction for each network. (b) the average success rate ofprediction of each criterion over 7 networks.
Table I | The average success rate of prediction of the IM criterion on 15 realnetworks.
For each network, we show its type and name; number of nodes (N) andlinks (M) of the largest strongly connected component; the average success rate ofprediction on the bias of normal agents ( ) and the success rate of prediction onwho will win ( ).Type Name N M Undirected ca-GrQc
62 159 0.878 0.878Undirected email
115 4120 0.834 0.821Undirected karate
34 78 0.902 0.872Undirected netsci
379 1828 0.842 0.829Undirected polbook
105 441 0.821 0.853Directed advogato
793 15781 0.853 0.929Directed rado-email
126 5639 0.916 0.969Directed twitter51