SIAM Review
On Certain Connectivity Properties of the Internet Topology
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
The cover time of sparse random graphs
Random Structures & Algorithms - Proceedings from the 12th International Conference “Random Structures and Algorithms”, August1-5, 2005, Poznan, Poland
Probability Theory with Applications (Mathematics and Its Applications)
Probability Theory with Applications (Mathematics and Its Applications)
Random Graph Dynamics (Cambridge Series in Statistical and Probabilistic Mathematics)
Random Graph Dynamics (Cambridge Series in Statistical and Probabilistic Mathematics)
Convergence Speed in Distributed Consensus and Averaging
SIAM Journal on Control and Optimization
Probability Theory: An Analytic View
Probability Theory: An Analytic View
Randomized consensus algorithms over large scale networks
IEEE Journal on Selected Areas in Communications
Controlling opinion propagation in online networks
Computer Networks: The International Journal of Computer and Telecommunications Networking
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We study a tractable opinion dynamics model that generates long-run disagreements and persistent opinion fluctuations. Our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents: 1 regular agents who update their beliefs according to information that they receive from their social neighbors and 2 stubborn agents who never update their opinions and might represent leaders, political parties, or media sources attempting to influence the beliefs in the rest of the society. When the society contains stubborn agents with different opinions, the belief dynamics never lead to a consensus among the regular agents. Instead, beliefs in the society fail to converge almost surely, the belief profile keeps on fluctuating in an ergodic fashion, and it converges in law to a nondegenerate random vector. The structure of the graph describing the social network and the location of the stubborn agents within it shape the opinion dynamics. The expected belief vector is proved to evolve according to an ordinary differential equation coinciding with the Kolmogorov backward equation of a continuous-time Markov chain on the graph with absorbing states corresponding to the stubborn agents, and hence to converge to a harmonic vector, with every regular agent's value being the weighted average of its neighbors' values, and boundary conditions corresponding to the stubborn agents' beliefs. Expected cross products of the agents' beliefs allow for a similar characterization in terms of coupled Markov chains on the graph describing the social network. We prove that, in large-scale societies, which are highly fluid, meaning that the product of the mixing time of the Markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large, a condition of homogeneous influence emerges, whereby the stationary beliefs' marginal distributions of most of the regular agents have approximately equal first and second moments.