Weituo Zhang

Social consensus through the influence of committed minorities.

We show how the prevailing majority opinion in a population can be rapidly reversed by a small fraction p of randomly distributed committed agents who consistently proselytize the opposing opinion and are immune to influence. Specifically, we show that when the committed fraction grows beyond a critical value p(c) ≈ 10%, there is a dramatic decrease in the time T(c) taken for the entire population to adopt the committed opinion. In particular, for complete graphs we show that when p < pc, T(c) ~ exp [α(p)N], whereas for p>p(c), T(c) ~ ln N. We conclude with simulation results for Erdős-Rényi random graphs and scale-free networks which show qualitatively similar behavior.

Analytic treatment of tipping points for social consensus in large random networks.

We introduce a homogeneous pair approximation to the naming game (NG) model by deriving a six-dimensional Open Dynamics Engine (ODE) for the two-word naming game. Our ODE reveals the change in dynamical behavior of the naming game as a function of the average degree {k} of an uncorrelated network. This result is in good agreement with the numerical results. We also analyze the extended NG model that allows for presence of committed nodes and show that there is a shift of the tipping point for social consensus in sparse networks.

Opinion Dynamics and Influencing on Random Geometric Graphs

We investigate the two-word Naming Game on two-dimensional random geometric graphs. Studying this model advances our understanding of the spatial distribution and propagation of opinions in social dynamics. A main feature of this model is the spontaneous emergence of spatial structures called opinion domains which are geographic regions with clear boundaries within which all individuals share the same opinion. We provide the mean-field equation for the underlying dynamics and discuss several properties of the equation such as the stationary solutions and two-time-scale separation. For the evolution of the opinion domains we find that the opinion domain boundary propagates at a speed proportional to its curvature. Finally we investigate the impact of committed agents on opinion domains and find the scaling of consensus time.

Tipping Points of Diehards in Social Consensus on Large Random Networks

We introduce the homogeneous pair approximation to the Naming Game (NG) model, establish a six dimensional ODE for the two-word NG. Our ODE reveals how the dynamical behavior of the NG changes with respect to the average degree of an uncorrelated network and shows a good agreement with the numerical results.We also extend the model to the committed agent case and show the shift of the tipping point on sparse networks.

Naming game with greater stubbornness and unilateral zealots

We examine a modified naming game on a complete network. In this modification, there are multiple intermediate states rather than one, with the intermediate states evenly spaced between commitment to opinion A and commitment to opinion B. The presence of zealots with unshakable support for opinion A was added to the model. The system tends to a 1-dimensional center manifold, either reaching consensus at A or near-consensus near B. This behavior is similar to the standard naming game on the complete graph. Special attention was paid to the case of two intermediate states. It was found that near-consensus at B could only be reached when the number of zealots was below a certain value. In the case of a single intermediate state, this value is 7.2%, of the total population. This is consistent with the work of Zhang and Lim, who found that the tipping point was 8% ± 1%[1] In the case of two intermediate states, this value was 12.1%.

Noisy Naming Games, partial synchronization and coarse-graining in social networks

The Naming Games (NG) are typical agent-based models for agreement dynamics, peer pressure and herding in social networks, and protocol selection in autonomous ad-hoc sensor networks. They form a large class that includes the Voter models and many others. By introducing a rare Poisson noise term to the signaling protocol of the NG, the resulting Markov Chain model called Noisy Naming Game (NNG) is ergodic, in which all partial consensus states are recurrent. By this generic method, any member of a large class of agent-based network models, including the Voter models and Bass models, is easily changed from a totally synchronous system to a partially synchronous one where even after reaching total consensus, the network revisits multi-namestates infinitely often. The method introduced here works on any underlying network topology including small world and scale-free ones. Furthermore, by organizing the partially-synchronized namestates / microstates according to their coarse-grained total probability, which counts the number of namestates associated with a macrostate / community structure (CS) weighted by their microstate probability, the NNG offers a new method for ranking competing CS in social interactions, that is not based entirely on hierarchical modularity or other ways for counting the number of intra vs inter group links in the social network. In fact, the coarse-grained CS incorporates entropic effects since its entropy is enumerated as the logarithm of the relative number of namestates it contains. As such, the Gibbs free energy is taken here to represent a good measure of overall social tension, arising from the ways in which different possibly overlapping subgroups choose and maintain differing opinions. Through simulations the NNG is shown to also successfully resolve the smallest groups.

Social opinion dynamics is not chaotic

Motivated by the research on social opinion dynamics over large and dense networks, a general framework for verifying the monotonicity property of multi-agent dynamics is introduced. This allows a derivation of sociologically meaningful sufficient conditions for monotonicity that are tailor-made for social opinion dynamics, which typically have high nonlinearity. A direct consequence of monotonicity is that social opinion dynamics is nonchaotic. A key part of this framework is the definition of a partial order relation that is suitable for a large class of social opinion dynamics such as the generalized naming games. Comparisons are made to previous techniques to verify monotonicity. Using the results obtained, we extend many of the consequences of monotonicity to this class of social dynamics, including several corollaries on their asymptotic behavior, such as global convergence to consensus and tipping points of a minority fraction of zealots or leaders.

Concentration and Stability of Community-Detecting Functions on Random Networks

We propose a general form of community-detecting functions for finding communities—an optimal partition of a random network—and examine the concentration and stability of the function values using the bounded difference martingale method. We derive LDP inequalities for both the general case and several specific community-detecting functions: modularity, graph bipartitioning, and q-Potts community structure. We also discuss the concentration and stability of community-detecting functions on different types of random networks: sparse and nonsparse networks and some examples such as ER and CL networks.