W. Wang, H. E. Stanley, L. A. Braunstein, New J. Phys. 20, 013034 (2018)
Time-delays are pervasive in such real-world complex networks as social contagions and biological systems, and they radically alter the evolution of the dynamic processes in networks.We use a non-Markovian spreading threshold model to study the effects of time-delays on social contagions. Using extensive numerical simulations and theoretical analyses we find that relatively long time-delays induce a microtransition in the evolution of a fraction of recovered individuals, i.e., the fraction of recovered individuals versus time exhibits multiple phase transitions. The microtransition is sharper and more obvious when high-degree individuals have a higher probability of experiencing time delays, and the microtransition is obscure when the time-delay distribution reaches heterogeneity.We use an edge-based compartmental theory to analyze our research and find that the theoretical results agree well with our numerical simulation results.