Rejection-Based Simulation of Non-Markovian Agents on Complex Networks

Abstract

Stochastic models in which agents interact with their neighborhood according to a network topology are a powerful modeling framework to study the emergence of complex dynamic patterns in real-world systems. Stochastic simulations are often the preferred-sometimes the only feasible-way to investigate such systems. Previous research focused primarily on Markovian models where the random time until an interaction happens follows an exponential distribution. In this work, we study a general framework to model systems where each agent is in one of several states. Agents can change their state at random, influenced by their complete neighborhood, while the time to the next event can follow an arbitrary probability distribution. Classically, these simulations are hindered by high computational costs of updating the rates of interconnected agents and sampling the random residence times from arbitrary distributions. We propose a rejection-based, event-driven simulation algorithm to overcome these limitations. Our method over-approximates the instantaneous rates corresponding to inter-event times while rejection events counterbalance these over-approximations. We demonstrate the effectiveness of our approach on models of epidemic and information spreading.Here, c i , c r ∈ R ≥0 denote the infection and recovery rate constants, respectively. Note that the infection rate is proportional to the number of infected neighbors whereas the rate of recovery is independent from neighboring agents. Moreover,

Importance of Interaction Structure and Stochasticity for Epidemic Spreading: A COVID-19 Case Study

“…We used contact networks with n = 1000 nodes (except for the complete graph where we used 100 nodes). To generate samples of the stochastic spreading process, we utilized event-driven simulation (similar to the rejection-free version in [16]). The simulation started with three random seeds nodes in compartment C (and with an initial fraction of 3/1000 for the ODE model).…”

“…We used contact networks with n = 1000 nodes (except for the complete graph where we used 100 nodes). To generate samples of the stochastic spreading process, we used event-driven simulation (similar to the rejection-free version in [17]). Specifically, we utilized a simulation scheme, where all future events (i.e., transitions of nodes) were sorted in a priority queue (according to their application time).…”

“…For future work, it would be interesting to investigate the influence of non-Markovian dynamics. ODE-models naturally correspond to an exponentially distributed residence times in each compartment [49,17]. Moreover, it would be interesting to reconstruct more realistic contact networks.…”

Efficient simulation of non-Markovian dynamics on complex networks

“…Here, we shortly summarize the relevant algorithms in order to lay the grounds for our RED algorithm which was first introduced in [26]. We present an adaptation of the classical Gillespie method for networked processes as well as the non-Markovian Gillespie algorithm (nMGA) and its adaptation, the Laplace-Gillespie algorithm (LGA).…”