Social networks are not static but rather constantly evolve in time. One of the elements thought to drive the evolution of social network structure is homophily - the need for individuals to connect with others who are similar to them. In this paper, we study how the spread of a new opinion, idea, or behavior on such a homophily-driven social network is affected by the changing network structure. In particular, using simulations, we study a variant of the Axelrod model on a network with a homophilic rewiring rule imposed. First, we find that the presence of homophilic rewiring within the network, in general, impedes the reaching of consensus in opinion, as the time to reach consensus diverges exponentially with network size $N$. We then investigate whether the introduction of committed individuals who are rigid in their opinion on a particular issue, can speed up the convergence to consensus on that issue. We demonstrate that as committed agents are added, beyond a critical value of the committed fraction, the consensus time growth becomes logarithmic in network size $N$. Furthermore, we show that slight changes in the interaction rule can produce strikingly different results in the scaling behavior of $T_c$. However, the benefit gained by introducing committed agents is qualitatively preserved across all the interaction rules we consider.
A dynamical process occurring on a network can be strongly influenced by the evolution of the network's structure itself. Furthermore, if the dynamical process on the network directly affects the network's structural evolution, a complex feedback process arises. In the context of the spread of opinions, behaviors, or ideas on a social network, such an interplay between individual states and the network's structural evolution is expected on the basis of the theory of homophily. Homophily, introduced by Lazarsfeld and Merton [1,2], describes the tendency of individuals to form social connections with those who are similar to them. Complementarily, the persistence of ties is also thought to depend strongly on the similarity of the individuals they connect [3,4]. If the traits of individuals are unchanging, then we expect that the structure of the network will stabilize when each link connects a pair of individuals who are sufficiently similar. However, if individuals influence one another to adopt new behaviors, opinions, or ideas, and thereby affect each other's attributes, then the mutual similarities between pairs of individuals can be thought of as continuously evolving entities. Thus, one can envision the structure of a social network as being in a constant state of flux: links between dissimilar individuals decay with time while new ties between similar individuals form at some rate. This continuous death and birth of links is presumably balanced in such a way that on average, at any given time, the mean number of connections ascribed to any individ- * Corresponding author: sreens@rpi.edu ual is roughly constant, or at least, bounded from above [5].
There have been few empirical studies which track the simultaneous co-evolution of network structure and individual behaviors. Notably, Lazer et al. have recently studied [6] how political views of students in a public policy program, and the network structure of their interactions evolved over a two-semester observation period. The main finding of this study was that in the process of making connections to other individuals, homophilic selection on the basis of political views was weak, while race and religion-based selection was comparatively strong. Furthermore, the study found that an individual’s political views at the end of the observation period were significantly correlated with the mean affiliation of his/her neighborhood at the beginning of the observation period (controlling for the individual’s own initial views), an indication, possibly, of social influence.
In contrast, models of networks where social opinions and network structure co-evolve have been studied extensively in previous literature [7]. Benczik et al. [8,9] studied a two-parameter voter model that could be tuned to study all cases between two extremes where nodes preferentially interacted with other nodes holding the same opinion, or those holding the opposite opinion. They demonstrated that three outcomes were possible depending on the parameter values -a consensus state, a disordered state, or a frozen, polarized state. Holme et al. [10] studied a single-parameter model of a co-evolving network and demonstrated the existence of a non-equilibrium phase transition between a steady state with diverse co-existing opinions and a consensus state. Nardini et al. [11] studied a variant of the voter model where an individual, with a certain probability, either severs a tie with a neighbor whose state differs from its own and forms a new tie with another node, or otherwise adopts the neighbor’s state. They showed how small changes to the interaction rules, such as the order of choosing the interacting individuals, as well as the introduction of an intermediate state in the voter model can dramatically affect the probability of consensus as well as consensus times. Finally, Vazquez et al. [12] studied a model where nodes are assigned attributes and undergo influence as per the Axelrod model [13], while existing links are rewired with a probability proportional to the dissimilarity between the nodes they connect. They demonstrated that there arise three phases characterized by differences in the steady state network structure, as the number of possible traits per attribute is varied. A model similar to the one presented in [12] was studied in [14], where the authors showed how cultural diversity can be stably maintained despite the presence of cultural drift.
The network model we consider in this paper is similar to the latter two studies [12,14] in its use of an Axelrodtype measure of the similarity between individuals which in turn dictates how social influence and link rewiring occur. However, the central motivation of our work is to understand how a fast consensus to a particular attribute can be ensured on such evolving networks. In particular, we investigate how the introduction of committed agents -individuals who are selectively immune to influence on a given issue, and who hold th
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