Eur. Phys. J. B (2014) 87: 62DOI: 10.1140/epjb/e2014-41008-7

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Coevolutionary dynamics of opinion propagationand social balance: The key role of small-worldness

Yan Chen1, Lixue Chen2, Xian Sun1, Kai Zhang3, Jie Zhang4, and Ping Li1,a

1 Center for Networked System, School of Computer Science, Southwest Petroleum University, Chengdu 610500, P.R. China2 School of Computer Science, Southwest Petroleum University, Chengdu 610500, P.R. China3 NEC Laboratories America, Inc. 4 Independence Way, Suite 200, Princeton, NJ 08540, USA4 Center for Computational Systems Biology, Fudan University, Shanghai 200433, P.R. China

Received 3 July 2013 / Received in final form 14 November 2013Published online 12 March 2014 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2014

Abstract. The propagation of various opinions in social networks, which influences human inter-relationships and even social structure, and hence is a most important part of social life. We have in-corporated social balance into opinion propagation in social networks are influenced by social balance. Theedges in networks can represent both friendly or hostile relations, and change with the opinions of indi-vidual nodes. We introduce a model to characterize the coevolutionary dynamics of these two dynamicalprocesses on Watts-Strogatz (WS) small-world network. We employ two distinct evolution rules (i) opinionrenewal; and (ii) relation adjustment. By changing the rewiring probability, and thus the small-worldnessof the WS network, we found that the time for the system to reach balanced states depends criticallyon both the average path length and clustering coefficient of the network, which is different than othernetworked process like epidemic spreading. In particular, the system equilibrates most quickly when theunderlying network demonstrates strong small-worldness, i.e., small average path lengths and large clus-tering coefficient. We also find that opinion clusters emerge in the process of the network approaching theglobal equilibrium, and a measure of global contrariety is proposed to quantify the balanced state of asocial network.

1 Introduction

As far back as 1946, Heider put forward a conception ofbalance to deal with the interactions between two personsand an entity [1,2]. His analysis focused on the P-O-Xunit of cognitive field consisting of P (one person), O (an-other person), and X (an impersonal entity). The relationamong parts of this unit is described with rules of cogni-tive balance. For example, if P has a relation of affectionwith O and if O is seen as responsible for X, then therewill be a tendency for P to like or approve X and the wholeunit is balanced. Otherwise, the state is imbalanced if Pdislikes X.

Previous work [2,3] shows that the P-O-X unit has thetendency to get balanced in the real world, which resultsin the balance of macroscopic social structure [4]. That is,the whole social network achieves the global balance whenall the P-O-X units get balanced (see Fig. 1). This trend ofbalance has been posited as a fundamental social processand used to account for the structure of affective relationsof social actors towards one another [5,6]. It should benoted, however, that the driving force for the evolution ofsocial balance is human interaction, through which ideas

a e-mail: dping.li@gmail.com

and opinions spread over the whole social networks. Thepropagation of opinions can influence the affective rela-tionship between two persons and hence have an effect onthe balanced state of social networks. Similarly, the pair-wise relationships as well as cognitive balance can also in-fluence the spreading of opinion [7]. Opinion dynamics hasreceived a great deal of attention in recent years [8–10].In the literature, much effort has been devoted to model-ing opinion diffusion in static networks [11–15]. However,it still remains largely unknown how opinion spreading isaffected by the state of the connection (i.e., the affectionof the relation between two persons, which is subject tosocial balance) in the network. In this paper, we systemat-ically study the interaction between opinion propagationand changes of human inter-relationship [16,17] using theWS small-world network. To understand how the under-lying structure of the network, in particular the small-worldness, can have an impact on the opinion dynamicsunder the rules of social balance, we study the time neededfor the system to reach balance on small-world networkswith variable rewiring probability (p) and evolution rules.We find that the fastest convergence under such coevo-lutionary dynamics is achieved on the critical point p∗(as p increases), at which the small-world property ofthe network becomes most prominent. Furthermore, we

Page 2 of 5 Eur. Phys. J. B (2014) 87: 62

friendadversaryagreement to Xopposition to X

+-

+

-

-

+

+

+

-

-

-

+

a

b

+ + -+ --

+ + -+ --

Balanced P-O-X units

Unbalanced P-O-X units

Fig. 1. (a) Balanced and unbalanced P-O-X units where P andO correspond to the two neighboring agents, and their opinionsof X are marked with “+” and “–”. (b) Global balance for asigned social network. Once all the P-O-X units get balanced,the whole network will approach to the global balance state.

investigate the opinion distribution in the balanced state.We find that multiple opinion groups will be observed,and the contrariety among them can be defined by theconnections within and outside the groups. According tothe cognitive balance theory, people usually tend to ac-cept their friends’ opinion and object to their enemies’.However, it is also possible for some maverick persons toadopt their enemy’s opinion. As a result, the local P-O-Xrelation will get unbalanced and therefore the relationshipbetween the maverick person and his (her) enemy will pos-sibly change. For example, hostile relation might changeto be friendly if two enemies have the same view to an en-tity and vice visa. Based on this consideration, we built amodel to capture the interplay between opinion propaga-tion and interrelationship evolution [18–21]. In this model,social networks are described by signed graphs, in whichthe nodes denote the individuals and the links representthe acquaintances between them. The sign of a link (+1or −1) depends on if two individuals are friends or en-emies [22]. The opinions and interrelationships coevolveaccording to the rules as follows.

(1) Opinion renewal (OR). At each time step, every nodeupdates its opinion according to the opinion pressuresfrom its neighbors. More precisely, for a given nodei, every of its neighbor j imposes an opinion pressurethat will affect the opinion of the node i. Let sj be theopinion of node j and γij the relationship between iand j, then sj · γij is the opinion pressure from nodej to node i.Let Ni denote the neighborhood of node i, and N+

i bethe neighbors exerting positive opinion pressure to i(j ∈ N+

i if sj · γij > 0) and N−i be those who exert

negative opinion pressure. Consider the positive opin-ion pressure θ+

i and the negative opinion pressure θ−i

from the neighbors to node i in equation (1), then theprobability that node i adopts the positive or negativeopinion can be computed by equation (2).

⎧⎪⎨

⎪⎩

θ+i =

∑

j∈N+i

sj · γij

θ−i =∑

j∈N−i

sj · γij(1)

P+i =

θ+i

θ+i + θ−i

, P−i = 1 − P+

i . (2)

(2) Relation adjustment (RA). At each time step after thesystem deploying OR rule, the sign of edges will bemodified with probability pq if the edge is involved inan unbalanced P-O-X relation (as shown in Fig. 1a).This way, the balance of the whole network can beimproved.

Initially, all the nodes in the system are in the ignorant sta-tus (si = 0 for every node i). To simulate the co-evolutionof people’s opinions and their interaction relations in thenetwork, we randomly pick one node o as the origin ofopinion spreading (so = 1 or so = −1), and update thesystem using the OR and RA rules. By doing this, thesystem gradually reaches a steady state, i.e., the signs ofall nodes and edges no longer change.

During the evolution of the system, the OR rule im-plies both the influence of cognitive balance and the localfield opinions. That is, people tend to accept their friends’opinion and oppose their enemies’, which tends to makethe networks converge to a configuration determined bythe probabilities P+

i ’s and P−i ’s. Compared with the rule

that forces node i to have the majority opinion of its neigh-bors, the OR rule allows nodes to accept minority opinionwith some chance, which leads to more unbalanced rela-tions as observed in the real world. On the other hand,the RA rule characterizes the evolution of the interrela-tionships influenced by opinion propagation. The possi-bility of correcting the unbalanced relations is given bypq which reflects the time scale of the changes in unbal-anced relationship as well. A small pq indicates a stablerelationship with high tolerance for inconsistent opinions,while a large pq indicates a strong tendency to adjust theunbalanced relation.

2 Simulation results and discussions

Under the OR and the RA rules, the dynamical systemshows a tendency to get balanced during its evolution.The energy of the system defined by equation (3) [23,24]will decrease gradually during this process. At last, thesystem will ultimately reach a “harmonious state” undera nonzero pq, in which all the P-O-X relations are bal-anced in the social network. When the system reaches thebalanced state, it has the lowest energy which equals thenegative value of the number of edges. Therefore, equa-tion (3) can be used to verify whether the system has

Eur. Phys. J. B (2014) 87: 62 Page 3 of 5

200 400 600 8000

20

40

60

80

100

N

T

(a)

pq = 0.3pq = 0.4pq = 0.8

200 400 600 800

40

60

80

100

120

140

N

T

(b)

k = 10k = 14

Fig. 2. The dependence of convergence time T on networksize N ; (a) for different pq with average degree k being 10 and(b) for different average degree k while pq = 0.3.

reached the balanced state.

h(s) = −∑

i,j

γijsisj . (3)

In order to investigate the dynamical process towardsthe balanced state, we study the number of time stepsneeded for the system to reach balance, denoted by T .We first study the dependence of T on the network size(N) under different values of pq’s (note that for pq = 0convergence cannot be achieved). We have employed theWatts-Strogatz (WS) small-world network model [25] inour work, as real social networks have been found to ex-hibit “six degree of separation” feature [26] and “small-world” property. In the simulations, we use the rewiringprobability p = 0.02 and the average degree k = 10 as themodel parameters to represent a social network [27].

As can be seen from Figure 2, T increases with the net-work size N in most cases. Figure 2a shows the effect ofbalancing parameter pq. Larger pq leads to a smaller T , asthe P-O-X unit gets balanced faster. For very small pq, ittakes much more time for the system to get balanced. Fur-ther more, it is shown in Figure 2b that loosely connectednetworks corresponding to small k are more faster to ap-proach equilibrium than densely ones, which just coincideswith our intuition.

We are also interested in the relation between the con-vergence time T and the network’s topological structures,such as the small-world properties. Here we use the WSsmall-world model again to perform simulations. In Fig-ure 3, we produced a number of networks with the averagedegree k = 6, 8 and 10, respectively. For each of the net-works, we adjust the small worldness of the network bytuning the rewiring probability p used in the WS model.At last, we use pq = 0.3 so that all the individuals in thesystem have relatively high tolerance to unbalanced rela-tion, namely, it is a slow process for people in the networkto change their inter-relations.

We first examine the network property such as theclustering coefficient and the average path length in Fig-ure 4a. We can see that the clustering coefficient remainsalmost constant as p increases, and falls off at relativelylarger p (p > 0.01), while the average path length falls

0

0.5

1C(p)/C(0)

L(p)/L(0)

10−4

10−3

10−2

10−1

1000

50

100

p

T

0.02 0.4

k = 6k = 8k = 10

Fig. 3. The correlation between small-worldness and conver-gence time, the upper subplot shows the average clusteringcoefficient (C(p)) and the shortest path (L(p)) varying in p insmall-world networks with N = 500 and k = 10. The data havebeen normalized by the values C(0) and L(0) for the regularlattice. The lower subplot is the relationship between T and p.

1. Dynamical process in a network with high clustering coefficient

a. two unbalanced rela�ons exist between one nodes and its two short-range neighbors

b. unbalanced state transferred to local

neighbors

c. unbalanced mo�f

2. Dynamical process in a network with low clustering coefficient

a�er adjustment

a. two unbalanced rela�ons exist between one nodes and its two long-range neighbors

b. unbalanced state transferred to remote

neighbors

c. unbalanced mo�f

node with unbalanced rela�ons unbalanced rela�on

+-

+- ++

++

- --

++ + -

+-

+- -+

- - -+

+ +-+

- -

+

+ ++

--

--

+-

+

+ +

+-- -

-

++ --+

+- +

-

+-

+

+ +

+

+

+

-

+--

a�er adjustment

+

- +

Fig. 4. Comparison of the networks with high clustering coef-ficient and low clustering coefficient on their evolution process.

rapidly, reaching a constant at about p = 0.02. In par-ticular, when p = 0.02, the small-world characteristic ismost evident, i.e., the average clustering coefficient andthe shortest path have the largest disparity.

We then plot T as a function of the rewiring prob-ability p in the WS-model in Figure 3. It is interestingto note that T reaches its minimum at p = 0.02 regard-less of the average degree k of the WS-model. Since theaverage path length of the WS-model drops to almost aconstant at p = 0.02, and a shorter average path can fa-cilitate propagation of opinions, thus there is a drop of Tbefore p = 0.02. As p further increases and the averageshortest path becomes almost constant, the key factor thatdetermines T is the average clustering coefficient.

Another observation is that for 0.4 > p > 0.02 in whichthe clustering coefficient C(p)/C(0) decreases monotoni-cally, the time T need for the system to get balanced alsoincreases in a monotonic manner. For p > 0.4, both theclustering coefficient and average path length are closeto constant, therefore T remains unchanged (the lower

Page 4 of 5 Eur. Phys. J. B (2014) 87: 62

subplot in Fig. 3). Moreover, we can find that the averagedegree k which reflects network density can affect the con-vergence time T . This is in agreement with the observationin Figure 2b.

In summary, from Figure 3, we can see that the under-lying network structure plays a vital role in affecting theopinion dynamics taking place on the network. In particu-lar, the clustering coefficient and the average path lengthare two key factors that influence the opinion propagationwith social balance on networks.

The opinion propagation studied here is distinct fromother dynamical processes taking place on networks, suchas epidemic spreading, in which a small average pathlength is important for disease propagation (i.e., for WSnetworks, the propagation time decreases monotonouslywith p). Here, the opinion dynamics that is coupled withsocial balance depends on both the average path lengthand the clustering coefficient, especially at the range0.02 < p < 0.4 in which the small-worldness is mostprominent.

Next we elaborate in more detail on how the cluster-ing coefficients affect T . Consider the process of a sys-tem getting balanced under the rule of OR. Remind thatnodes in the network typically have multiple neighbors,which can lead to different opinion pressures. As the nodechooses its opinion under a probabilistic manner as de-fined by equation (2), there probably are some unbalancedP-O-X relations. Changing these unbalanced P-O-X willdrive nearby nodes to adjust themselves to get balancedagain (e.g. change the sign of the node or the edge). FromFigure 4 we can see that the difference between a networkwith high clustering coefficient and one with low cluster-ing coefficient (other conditions being the same) is thatthere are more triangle structures in the former (Fig. 41-c). This triangular structure is more likely to get bal-anced than the tree-like structures shown in Figure 4 2-cbecause there is a possibility that both of the two unbal-anced nodes in the triangle change their opinion and thetriangle gets balanced. Therefore, when the average clus-tering coefficient C(p) is large, the unbalanced relationscan always get flipped to balance within a short time. IfC(p) is small, vibrations in the network will always sweepthrough long-range edges in a large domain and can pre-vent the system from getting balanced effectively. Conse-quently, a network with better small-world features canget balanced quicker.

Besides the convergence time T , it is also noteworthyto inspect the distribution of different opinions. In struc-turally balanced networks, it is common that many opin-ion groups will take place and the network can thereforebe split into communities: opinions within a communityare monolithic and antipodal to those of other groups con-necting to it [28]. The edges within a community all havethe sign of +1 and the edges among the communities allhave the sign of −1, as shown in Figure 5.

To further understand the emergence of opiniongroups, we generate a series of networks with N = 500and pq = 0.3 for different k, and set the proportion ofnegative edges in the network as 20%, approximating real

friendadversary

agreementopposition

Fig. 5. A schematic diagram of opinion groups in a socialnetwork.

5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

k

ρc

Fig. 6. The relationship between the number of opinion groupsρc and average degree k in balanced small-world networkswith their rewiring probability p = 0.02 for the networks andpq = 0.3 for the coevolutionary model.

social networks [23]. By extensive numerical simulations, itis found that the number of opinion groups in the networkconverges to 2 when k is larger than 20, while the densityof the network approaches to 0.2. In Figure 6, we show therelation between the number of opinion groups with regardto the connection density of the network. It can be notedthat if the density is large enough, the social network willalways form two opposite opinion groups. Considering theopposition among the adjacent opinion groups, we definea so-called “Social Contrariety” (SC) to reflect the globalcontrariety of the social network, as:

SC =

∑ρc

i=1μi

2νi+μi

ρc. (4)

Here νi and μi are the number of internal and externaledges of opinion group i, respectively; the number of in-ternal edge stubs (i.e., one of the endpoints of an edge)is then 2νi and the number of external edge stubs is μi.The social contrariety is defined as the average ratio ofexternal/internal edge stubs for the communities, namelythe average ratio of adversaries outside to friends insidefor the communities. Larger value of SC indicates sharperglobal contradiction among groups in the social network.

Eur. Phys. J. B (2014) 87: 62 Page 5 of 5

5 10 15 20 25 30 35 40 45 500.30

0.35

0.40

0.45

0.50

0.55

k

SC

Fig. 7. The relationship between social contrariety SC and av-erage degree k with rewiring probability p = 0.02 for the small-world networks and pq = 0.3 for the coevolutionary model.

Figure 7 shows the dependence of SC on k, the av-eraged degree in balanced networks. We can see that fora small k, SC is prominent while the contrariety is alle-viated as networks become more and more dense. It hasbeen observed from Figure 6 that there are more opiniongroups for k < 20, which makes the denominator of equa-tion (4) larger. However, the sparsity of the network canbring on some very small groups which have few edges in-side and makes SC even larger (e.g., for the group witha single node, there are no internal but 1 external edge,which will make the item of the accumulation in the nu-merator equal to 1). Note that for k > 20, SC are almostinvariant. The reason is that the community number doesnot increase with the average degree anymore, and the twogroups have almost the same ui and νi when the networksapproach to the balanced state.

3 Conclusion

In summary, we have investigated the coevolutionary dy-namics of signed social networks with different opinionspropagating in the network. We proposed a model to de-pict the interaction between opinion propagation and so-cial structural balance by the OR and RA rules. By study-ing the coevolutionary dynamics in the WS small-worldnetwork with varying rewiring probabilities, we found thatthe time needed for a social network to get balanced isstrongly affected by the topological characteristics of thenetwork, namely, the small-world properties reflected interms of the average path length and the clustering coef-ficient. We find that when the small-world networks havea relatively small average path length and larger clus-tering coefficient, the time needed to achieve the globalbalance is the shortest. Furthermore, we investigated thedistribution of opinion groups in the balanced networksand found that apparent correlation exits between thenumber of opinion groups and the tightness of the opin-ion groups. A compactly connected network with highconnection density has minimum social contrariety.

Yan Chen acknowledges the support from MOE (Ministry ofEducation in China) Project of Humanities and Social Sci-ences, 12YJA790167. Ping Li is supported by National NaturalScience Foundation of China 61104224 and Foundations ofSiChuan Educational Committee (13ZB0198). Xian Sun ac-knowledges the support from Graduate Innovation Foundationof SouthWest Petroleum University, SGIFSWPU GIFSS0727.Jie Zhang acknowledges the financial support from the NNSFC(Nos. 61004104 and 61104143).

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