  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.
