  Time-varying community structures widely exist in various real-world
networks. However, the spreading dynamics on this kind of network has not been
fully studied. To this end, we systematically study the effects of time-varying
community structures on social contagions. We first propose a non-Markovian
social contagion model on time-varying community networks based on the activity
driven network model, in which an individual adopts a behavior if and only if
the accumulated behavioral information it has ever received reaches a
threshold. Then, we develop a mean-field theory to describe the proposed model.
From theoretical analyses and numerical simulations, we find that behavior
adoption in the social contagions exhibits a hierarchical feature, i.e., the
behavior first quickly spreads in one of the communities, and then outbreaks in
the other. Moreover, under different behavioral information transmission rates,
the final behavior adoption proportion in the whole network versus the
community strength shows one of the patterns, which are a monotone increasing
pattern, a non-monotonic changing pattern, and a monotone decreasing pattern.
An optimal community strength maximizing the final behavior adoption can be
found in a suitable range of behavioral information transmission rate. Finally,
for a given average degree, increasing the number of edges generated by active
nodes is more beneficial to the social contagions than increasing the average
activity potential.
