Epidemic Spreading in Networks - Variance of the Number of Infected Nodes

Abstract. The Susceptible Infected Susceptible (SIS) model is one of the basic models and it is applied for different networks and services in telecommunications. For more detailed prediction of the epidemic, it is necessary to examine the higher order moments, namely the variance of the number of infected nodes. Also, the predictability of mean-field models depends on the variations around the mean. However, the variance of epidemic spread on networks has so far received insufficient attention. Epidemics spread in significantly different topologies from power law to complete graphs, thus the model should be independent of the underlying topology. We use the N-intertwined model which captures the topology influences of the finite graph defined by adjacency matrix A to determine the variance. We also determine upper and lower bounds on the variance as a function of effective spreading rate τ and show that for some spreading conditions, the variance is highly dependent on the degree distribution of the underlying network and not on other topological properties. Further, we apply our findings to two types of graphs: complete and complete bipartite graphs. For the complete bipartite graph, we derive the probability distribution function of the number of infected nodes. Finally, we provide deeper understanding of the structural properties expressed via the second smallest eigenvalue of the Laplacian matrix and link this parameter to our N-intertwined model.