The SIS process on Erdös-Rényi graphs: determining the infected fraction

TL;DR

A new method to determine the infected fraction in sparse graphs, based on degree-pairs, does take into account correlations and gives accurate estimates, which is very feasible and can easily be done even for large networks.

Abstract

There are many methods to estimate the quasi-stationary infected fraction of the SIS process on (random) graphs. A challenge is to adequately incorporate correlations, which is especially important in sparse graphs. Methods typically are either significantly biased in sparse graphs, or computationally very demanding already for small network sizes. The former applies to Heterogeneous Mean Field and to the N-intertwined Mean Field Approximation, the latter to most higher order approximations. In this paper we present a new method to determine the infected fraction in sparse graphs, which we test on Erdős-Rényi graphs. Our method is based on degree-pairs, does take into account correlations and gives accurate estimates. At the same time, computations are very feasible and can easily be done even for large networks.

Figures (23)

• Figure 1: Number of infected nodes over time for the complete graph and an Erdős-Rényi graph. In both cases $n=1000$ and the global infection rate $p\tau$ is equal to $2/n$, but the ER graph has a lower equilibrium.
• Figure 2: Probability distribution of the number of infected nodes in the complete graph (red) and in 10 realizations of the Erdős-Rényi graph (blue), all with $n=1000$ and global infection rate $2/n$. Dashed lines indicate means.
• Figure 3: Left: Fixing an Erdős-Rényi graph ($p=\log(n)/n$, $p\tau = 2/n$), the metastable number of infected has mean (blue) and variance (red) both close to $n/2$. Varying the graph, the means themselves also fluctuate with a variance of order $n$ (yellow). Right: Approximate minimal standard error for any annealed method to estimate a quenched infected fraction.
• Figure 4: Simulated infected fraction compared to different estimates. In all cases, $n=1000$. Annealed methods are smooth, quenched methods follow the fluctuations of the simulation.
• Figure 5: Comparison of computation times for Erdős-Rényi graphs with $p=\log(n)/n$ and $\lambda = 2$. NIMFA needs a lot of time. Quenched methods and simulation need a lot of memory.