Analytical approach for opinion dynamics on social networks

Abstract

Opinion dynamics is a class of agent based models arising from the study of the origins and evolutions of languages and now attracts a lot of interest. These models consist of a large number of individuals updating their states through pairwise interactions. These models provide powerful tools to simulate and investigate the collective behavior in complex systems in sociology, physics and computer science.

The thesis is organized as follows. Chapter 2 introduces two basic models based on coarse-graining approach: the ODE model and the random walk model, and analyzes a prototypical opinion dynamics, the voter model, as a random walk. Chapter 3 further applies the random walk model to the 2-word Naming Game on complete graphs, calculates the expectation of the total consensus time T_c and the expected time spent on each macrostate before consensus T(n_A,n_B) in three cases: (i)purely symmetric case; (ii)with biased central influence; (iii)with committed agents. The tipping point of committed fraction and different dynamical behavior below and above the tipping point are also discussed. Chapter 4 analyzes the distribution of consensus time T_c of the 2-word Naming Game on complete graphs in several ways, calculates the variance of T_c through a martingale approach, and provides a path integral approximation for the variance of T_c. Chapter 5 develops an improved ODE model using the homogeneous pairwise assumption, a variation of mean field assumption, which works for sparse networks with narrow degree distribution and no community structures. This model shows the dependence of the dynamic behavior on the average degree < k >.

The main object of this thesis is to provide a framework for analyzing the opinion dynamics, in particular the 2-word Naming Game(NG). We develop one random walk model which works for complete graph and highly connected networks without community structures. We also develop an improved ODE model which works for sparse networks without community structures. Both models are based on a so-called coarse-graining approach which requires an underlining mean field assumption. We derive analytical results according to these models and confirm our results by numerical simulations of NG dynamics.