In contrast to models without recovery, incorporating spontaneous recovery in failure propagation better describes the dynamics in real-world networks. There are two types of spontaneous recovery processes: Markovian with instantaneous recovery (i.e., without memory) and the non-Markovian type with time delays in the recovery process (i.e., with memory), where the former is idealized and the latter is ubiquitous in the real world. How does the non-Markovian characteristic in nodal recovery affect failure propagation in complex networks? 

We study both types of models incorporating two failure mechanisms: failures due to internal causes at the nodal level and external failures due to an adverse environment. In the idealized Markovian model, recovery of a failed node occurs instantaneously with a probability, while in the more realistic non-Markovian model, recovery occurs after a time delay. For each model, we develop a mean-field theory and a pair approximation (PA) analysis, where we find that the PA analysis gives better predictions. In both models, there are low and high failure stationary states, with the latter corresponding to large scale failures that can significantly compromise the functioning of the network. How the network evolves towards a high-failure state with the MR or NMR mechanism is thus highly pertinent to the resilience of the system against large scale failures. To reveal the effect of MR versus that of NMR on the transition in a concrete and clear manner, we set the rate of internal failures as a control or bifurcation parameter. For any given fraction of the initially failed nodes, denoted as  (), as the internal failing rate is increased through a critical value, say , the network will approach a high-failure state. A larger value of is thus indication that the network is more resilient to large scale failures and is desired. A meaningful, effective and accurate way to assess and distinguish the impacts of Markovian and non-Markovian processes is thus to examine the dependence of on , denoted as . We find drastically and characteristically different behaviors of for MR and NMR processes. 

Our theoretical analyses and numerical simulations on many aspects of the failure propagation dynamics lead to a striking phenomenon: memory in the nodal recovery can counter-intuitively make the network more resilient against large scale failures. Besides, we have carried out a systematic study of the effects of Markovian versus non-Markovian recovery on network synchronization using the paradigmatic Kuramoto network model, with the main finding that nonMarkovian recovery makes the network more resilient against large-scale breakdown of synchronization. The implications of the study are two. In natural systems, the intrinsic non-Markovian characteristic of nodal recovery may be one reason for the resilience of these networks. In engineering design, incorporating certain non-Markovian features into the network may be beneficial to equipping it with a strong resilient capability to resist large scale failures.

Figure 1. Comparison of simulation results with predictions from PA analysis and mean-field theory for the MR model. a Time evolution of the fraction of inactive nodes. b Phase diagram on the () parameter plane. c Dependence of for reaching a high failure state on the initial value of , with .

Figure 2. Benefit of non-Markovian recovery to making the network more resilient against large scale failures for the NMR model.

For more details, please see our recent work in Nature Communications: Non-Markovian recovery makes complex networks more resilient against large scale failures.