A Research Of Non-Markovian Spreading Dynamics Analysis On Complex Networks

The spreading processes of epidemic in the real world depends on human behaviors,and the event occurrence processes of its spreading dynamics cannot be simply described as Poisson random processes.The corresponding event time is not exponentially distributed,so it has a typical non-Markovian property.So far,we still lack a suitable theoretical framework to analyze and understand the non-Markovian spreading processes.To solve this problem,we propose a first-order mean field theory,which can analyze and deal with the spreading dynamics of non-Markovian SIS model with non-exponential distributions of infection and recovery times on complex networks.Our theory firstly analyses the probability density distribution about single node's state age on SIS spreading dynamics.Meanwhile,we distinguish the two different edge activation mechanisms in non-Markovian processes,get a set of partial differential equations which could predict the corresponding distributions at any time,and depict precisely the two mechanisms in the equations.It is worthy noting that,the mechanism without temporal correlations on active edges needs higher dimensional description,and it is dealt with by dimension reduction in mathematics which will bring about less computational complexity without lower prediction accuracy.Simulations show that the mean field theory can accurately predict the transient processes and steady state of non-Markovian dynamics on artificial networks and real networks.It is worth noting that this theory is of great significance for solving practical problems,that is,under what conditions non-Markovian spreading dynamics and Markovian spreading dynamics are equivalent.In non-Markovian spreading dynamics,equivalence depends on the activation mechanism of active edge.In particular,when there exists no temporal correlation on active edges,equivalence can be established,which greatly promotes the analysis and understanding of non-Markovian dynamics.When correlation cannot be ignored,there is no exact equivalence relation.However,if the infection density is large,the corresponding non-Markovian processes can be approximated as Markovian processes.Moreover,we summarize the synchronization and asynchronization updating algorithms for simulating the non-Markovian dynamic processes,and give the theoretical basis of the two algorithms.According to different algorithms,asynchronization update methods can be divided into two categories:One is to assign random event time to random events,and the other is to assign random events to each potential event.In addition,we determined the main source of the dynamics correlation of SIS models under different edge activation mechanisms,and find that there exists no dynamics correlation between infected nodes.The dynamics correlation between susceptible nodes is mainly due to the susceptible nodes' disability to spread epidemic.The dynamics correlation between infected nodes and susceptible nodes are different under different edge activation mechanisms.For the edge activation mechanism with temporal correlation,the dynamics correlation comes from the temporal correlation on the activated edge.For the edge activation mechanism without temporal correlation,the dynamics correlation derives from the causal correlation between the disease transmission processes on the active edge and the infection processes of the susceptible node.Our theory provides a new direction for developing a more comprehensive theoretical framework,and our results provide a general analytical method for the spread of epidemic in the real world.