We focus on the majority model in a topology consisting of two coupled fully connected networks, thereby mimicking the existence of communities in social networks. We show that a transition takes place at a value of the interconnectivity parameter. Above this value, only symmetric solutions prevail, where both communities agree with each other and reach consensus. Below this value, in contrast, the communities can reach opposite opinions and an asymmetric state is attained. The importance of the interface between the subnetworks is shown. DOI: 10.1103/PhysRevE.75.030101 PACS number͑s͒: 02.50.Ϫr, 87.23.Ge, 05.40.Ϫa It is well known that social networks exhibit modular structure of weakly coupled clusters ͓1͔. Namely, these networks are composed of many communities of nodes, where the nodes of the same community are highly connected, while there are few links between the nodes of different communities. It is therefore of particular interest to understand how well-known models of opinion formation ͓2-4͔ behave on these realistic geometries. It is an obvious fact ͓5͔ that the opinion or taste of users may strongly differ from one cluster to another due to the lack of interactions between the communities. In an economic context, for instance, it is well known that small cliques of core users can form a niche ͓6,7͔ and have a different behavior than the majority. As an example, one may think of Mac users who concentrate in niche markets like education, graphic arts, and multimedia creation ͓8͔. From a marketing point of view, the propagation of opinion between different communities is also an important problem, due to the growing popularity of viral marketing techniques in online social networks ͓9͔. In order to address this problem, we will focus in this Rapid Communication on the majority model ͓4͔ ͑MR, for majority rule͒ applied on a simplified geometry that mimicks community structure. MR is defined as follows. The network is composed of N T agents. Each of them has an opinion ␣ or ␤ ͑equivalently spin͒ about some question. E.g., will you vote for the Republicans or the Democrats ͓10͔? Do you believe in Darwin's theory? Which of these two products do you want to buy? … At each time step, G = 3 contiguous nodes are selected ͑the main requirement is that G be an odd number͒ and the agents in this selection all adopt the state of the local majority. This model rests on the tendency of social agents to copy the opinion of their neighbors and friends ͓11͔ and has been shown to lead to rich collective behaviors ͓4͔. For the sake of clarity, let us first focus on MR on a fully connected network-i.e., any pair of nodes is connected. When N T → ϱ, it is straightforward to show that the average total number of nodes having opinion ␣, denoted A t , evolves as where a t = A t / N T is the average proportion of nodes having opinion ␣. Equation ͑1͒ comes from the fact that the probability that two nodes ␣ ͑␤͒ and one node ␤ ͑␣͒ are selected is a 2 ͑1−a͒ ͑a͑1−a͒ 2 ͒, so that the total contribution to the evolution of A t is It is easy to show that Eq. ͑1͒ possesses two stable stationary solutions a =0 or a = 1, which correspond to consensus states; i.e., all agents in the system have the same opinion. The mixed solution a =1/2, where nodes with different opinions coexist, is also stationary but is unstable. Let us also insist on the fact that MR does not involve temperaturelike parameters that randomize the opinion dynamics. In order to generalize the fully connected network and to account for communitylike structures, we introduce coupled fully connected networks ͑CFCN's͒ defined as follows. The system is composed of two fully connected clusters 1 and 2. These two fully connected clusters are composed of N 1 T and N 2 T nodes, respectively. The connection between the two structures is ensured by interface nodes, denoted 0, that belong to each of the subnetworks. By construction, the number of interface nodes verifies N 0 ഛ min͑N 1 T , N 2 T ͒ and the case N 0 Ӷ min͑N 1 T , N 2 T ͒ corresponds to a network composed of two sparsely connected communities. In the following, we denote by N 1 and N 2 the number of core nodes in the clusters 1 and 2, respectively, where core nodes are those that belong to only one cluster. By construction, the above quantities satisfy For the sake of clarity, we focus on equipopulated clusters Let us also note that the total number of nodes is N is a measure of the interconnectivity between the communities. By construction, the following relations hold: N 0 = N and N 1 = N 2 = ͑1−͒N. Some typical realizations of CFCN's can be viewed in Fig. 1 . In this Rapid Communication, we will answer the following question: are there values of the interconnectivity such that the coexistence of two disagreeing populations is possible? In the limiting case → 1, each agent in 1 also belongs to 2 and inversely; i.e., all the agents are interface agents N 0 = N T = N, N 1 = N 2 = 0 and the network reduces to one fully PHYSICAL REVIEW E 75, 030101͑R͒ ͑2007͒
Finally, a similar perspective to ours has been adopted in REF , where the authors show a phase-transition in a mean-field analysis of the behavior of deterministic majority in an asynchronous-update model, in which only few nodes update their opinion at each round, on "coupled fully connected networks" composed of overlapping complete graphs.
Majority model on a network with communities
Physical review. E, Statistical, nonlinear, and soft matter physics
Physical review. E, Statistical, nonlinear, and soft matter physics
Medicine Mathematics Physics