Abstract
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 DeGroot 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.
Highlights
One can represent the structure of many natural, societal, and engineered systems as networks [1]
We examine the convergence speed of opinion dynamics that arise from a time-varying combinatorial Laplacian matrix for tie-decay networks that we construct from empirical data
We examined the convergence speeds of continuous-time Laplacian dynamics on tie-decay networks that we constructed from empirical social-contact data by calculating the spectral gap of the matrix M(tn), which maps the initial opinions of the nodes of a network to the opinions of the nodes at time tn
Summary
One can represent the structure of many natural, societal, and engineered systems as networks [1]. It is important to examine dynamical processes, such as opinion dynamics [16,17,18,19], on temporal networks In such scenarios, both the structure of a network and the states of its nodes and/or edges change in time. If one is not careful about examining the relative time scales of dynamical processes on a network and temporal changes in network structure (or if multiple time scales or burstiness are relevant to one or more of these processes), aggregating network dynamics into discrete time windows may lead to qualitatively incorrect conclusions. We examine the convergence speed of opinion dynamics that arise from a time-varying combinatorial Laplacian matrix for tie-decay networks that we construct from empirical data.
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