Solution of the voter model by spectral analysis

Inferring models of opinion dynamics from aggregated jury data

Jury deliberations provide a quintessential example of collective decision-making, but few studies have probed the available data to explore how juries reach verdicts. We examine how features of jury dynamics can be better understood from the joint distribution of final votes and deliberation time. To do this, we fit several different decision-making models to jury datasets from different places and times. In our best-fit model, jurors influence each other and have an increasing tendency to stick to their opinion of the defendant’s guilt or innocence. We also show that this model can explain spikes in mean deliberation times when juries are hung, sub-linear scaling between mean deliberation times and trial duration, and unexpected final vote and deliberation time distributions. Our findings suggest that both stubbornness and herding play an important role in collective decision-making, providing a nuanced insight into how juries reach verdicts, and more generally, how group decisions emerge.

Solution to urn models of pairwise interaction with application to social, physical, and biological sciences

We investigate a family of urn models that correspond to one-dimensional random walks with quadratic transition probabilities that have highly diverse applications. Well-known instances of these two-urn models are the Ehrenfest model of molecular diffusion, the voter model of social influence, and the Moran model of population genetics. We also provide a generating function method for diagonalizing the corresponding transition matrix that is valid if and only if the underlying mean density satisfies a linear differential equation and express the eigenvector components as terms of ordinary hypergeometric functions. The nature of the models lead to a natural extension to interaction between agents in a general network topology. We analyze the dynamics on uncorrelated heterogeneous degree sequence networks and relate the convergence times to the moments of the degree sequences for various pairwise interaction mechanisms.

Analysis of the high-dimensional naming game with committed minorities

The naming game has become an archetype for linguistic evolution and mathematical social behavioral analysis. In the model presented here, there are N individuals and K words. Our contribution is developing a robust method that handles the case when K=O(N). The initial condition plays a crucial role in the ordering of the system. We find that the system with high Shannon entropy has a higher consensus time and a lower critical fraction of zealots compared to low-entropy states. We also show that the critical number of committed agents decreases with the number of opinions and grows with the community size for each word. These results complement earlier conclusions that diversity of opinion is essential for evolution; without it, the system stagnates in the status quo [S. A. Marvel et al., Phys. Rev. Lett. 109, 118702 (2012)PRLTAO0031-900710.1103/PhysRevLett.109.118702]. In contrast, our results suggest that committed minorities can more easily conquer highly diverse systems, showing them to be inherently unstable.

Solution of the multistate voter model and application to strong neutrals in the naming game

We consider the voter model with M states initially in the system. Using generating functions, we pose the spectral problem for the Markov transition matrix and solve for all eigenvalues and eigenvectors exactly. With this solution, we can find all future probability probability distributions, the expected time for the system to condense from M states to M-1 states, the moments of consensus time, the expected local times, and the expected number of states over time. Furthermore, when the initial distribution is uniform, such as when M=N, we can find simplified expressions for these quantities. In particular, we show that the mean and variance of consensus time for M=N are 1/N(N-1)(2) and 1/3(π(2)-9)(N-1)(2), respectively. We verify these claims by simulation of the model on complete and Erdős-Rényi graphs and show that the results also hold on these sparse networks.