This thesis investigates opinion dynamics on random graphs through the lens of the voter model, a classical interacting particle system used to describe how consensus emerges in networks. Individuals are represented as nodes in a graph, with directed edges encoding acquaintance relationships. Each node holds one of two opinions and updates randomly over time under the influence of its neighbors.
We analyze the asymptotic behavior of this process as the underlying network size grows. In Chapter 2 we study consensus times in sparse directed configuration models, showing that they scale linearly with network size and depend explicitly on degree distributions via random walk meeting times. Chapter 3 extends this analysis to discordant edges, links between nodes with opposite opinions, deriving precise asymptotics across multiple time scales from initial opinion fragmentation to full consensus. Chapter 4 introduces a nonlinear variant with competing external bias and...
Show moreThis thesis investigates opinion dynamics on random graphs through the lens of the voter model, a classical interacting particle system used to describe how consensus emerges in networks. Individuals are represented as nodes in a graph, with directed edges encoding acquaintance relationships. Each node holds one of two opinions and updates randomly over time under the influence of its neighbors.
We analyze the asymptotic behavior of this process as the underlying network size grows. In Chapter 2 we study consensus times in sparse directed configuration models, showing that they scale linearly with network size and depend explicitly on degree distributions via random walk meeting times. Chapter 3 extends this analysis to discordant edges, links between nodes with opposite opinions, deriving precise asymptotics across multiple time scales from initial opinion fragmentation to full consensus. Chapter 4 introduces a nonlinear variant with competing external bias and stubbornness, proving the existence of a sharp phase transition in consensus behavior. Finally, Chapter 5 explores heterogeneous directed networks, particularly power law degree distributions, combining rigorous results with simulations to assess the validity of mean field approximations.
Overall, the thesis provides new mathematical insights into the mechanisms driving consensus and opinion diversity in complex networks, highlighting the role of network topology and interaction biases in shaping collective behavior.
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