In social networks, interaction patterns typically change over time. We study opinion dynamics in 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 in  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 gap of combinatorial Laplacian matrices of tie-decay networks. First, we examine the spectral gaps of tie-decay networks from empirical data . Intuitively, we expect small tie-decay rates to lead to fast convergence, because the influence of each interaction event between a pair of nodes lasts longer for smaller decay rates. Moreover, as time progresses and more interaction events occur, we expect to observe convergence to proceed. However, we demonstrate that the spectral gaps need not decrease monotonically with respect to the decay rate or increase monotonically with respect to time. Second, we compare the spectral gaps of these empirical networks with corresponding randomized and aggregate networks. We find that the spectral gaps of the empirical networks tend to be smaller than those of the randomized and aggregate networks. This highlights the importance of the interplay between the times that edges arise and decay in temporal networks.
In social networks, interaction patterns typically change over time. We study opinion dynamics in 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 in 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 gap of combinatorial Laplacian matrices of tie-decay networks. First, we compare the spectral gaps of tie-decay networks from empirical data with those for corresponding randomized and aggregate networks. We find that the spectral gaps of the empirical networks tend to be smaller than those of 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 event between two nodes lasts longer for smaller decay rates. Moreover, as time progresses and more interaction events occur, we expect to observe convergence to proceed. However, we demonstrate that the spectral gaps 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 arise and decay in temporal networks.
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