Opinion dynamics on tie-decay networks

In social networks, interaction patterns typically change over time. We study opinion dynamics on tie-decay networks in which tie strength increases instantaneously when there is an interaction and decays exponentially between interactions. Specifically, we formulate continuous-time Laplacian dynamics and a discrete-time DeG-root model of opinion dynamics on these tie-decay networks, and we carry out numerical computations for the continuous-time Laplacian dynamics. We examine the speed of convergence by studying the spectral gaps of combinatorial Laplacian matrices of tie-decay networks. First, we compare the spectral gaps of the Laplacian matrices of tie-decay networks that we construct from empirical data with the spectral gaps for corresponding randomized and aggregate networks. We find that the spectral gaps for the empirical networks tend to be smaller than those for the randomized and aggregate networks. Second, we study the spectral gap as a function of the tie-decay rate and time. Intuitively, we expect small tie-decay rates to lead to fast convergence because the influence of each interaction between two nodes lasts longer for smaller decay rates. Moreover, as time progresses and more interactions occur, we expect eventual convergence. However, we demonstrate that the spectral gap need not decrease monotonically with respect to the decay rate or increase monotonically with respect to time. Our results highlight the importance of the interplay between the times that edges strengthen and decay in temporal networks.

Automated summary

Opinion dynamics on tie-decay networks involve interaction patterns that change over time, with tie strength increasing instantaneously and decaying exponentially between interactions. Numerical computations for continuous-time Laplacian dynamics reveal the spectral gaps of combinatorial Laplacian matrices of tie-decay networks. The study compares spectral gaps of empirical tie-decay networks with randomized and aggregate networks, showing smaller gaps in empirical networks, and explores the relationship between spectral gap, tie-decay rate, and time. The results emphasize the importance of the interplay between edge strengthening and decaying in temporal networks.