Autocorrelation of the susceptible-infected-susceptible process on networks

In this paper, we focus on the autocorrelation of the susceptible-infected-susceptible (SIS) process on networks. The $N$-intertwined mean-field approximation (NIMFA) is applied to calculate the autocorrelation properties of the exact SIS process. We derive the autocorrelation of the infection state of each node and the fraction of infected nodes both in the steady and transient states as functions of the infection probabilities of nodes. Moreover, we show that the autocorrelation can be used to estimate the infection and curing rates of the SIS process. The theoretical results are compared with the simulation of the exact SIS process. Our work fully utilizes the potential of the mean-field method and shows that NIMFA can indeed capture the autocorrelation properties of the exact SIS process.

Figure 1

A randomly selected node is evaluated in an Erdős-Rényi (ER) network with the link connecting probability 0.4 and $N=50$. The autocorrelation is approximately constant for different value of $\delta$. The cross correlation between the node with one of its neighbors is also plotted, which is almost zero.

Figure 2

The autocorrelation of the infection state of a 26-regular graph with $N=50$. The results are similar to those of the ER graph. The autocorrelation is invariant to $\delta$.

Figure 3

The autocorrelation of the infection state of the hub and a leaf node in a star graph with $N=50$. The NIMFA autocorrelation shows a very good approximation and the cross correlation between hub and leaf nodes is approximately 0.

Figure 4

The correlation of the infection state of a node and the prevalence in a cycle graph with $N=50$. Initially all nodes are infected to prevent the inaccuracy caused by early die-out [24]. The NIMFA autocorrelation is much smaller than the exact one, and the cross correlations between neighbors and second-hop neighbors are very large.

Figure 5

The autocorrelation of the fraction of infected nodes $I\left(t\right)$ in the metastable state.

Figure 6

The joint expectation of the infection state $E\left[X_j\left(0\right)X_j\left(h\right)\right]$ and the corresponding NIMFA approximation $E\left[V_j\left(0\right)V_j\left(h\right)\right]$ of the SIS process on the star graph.

Figure 7

The autocorrelation of the fraction of infected nodes $R_I(0.5,0.5+h)$ and the corresponding NIMFA approximation $R_{I^{\left(1\right)}}(0.5,0.5+h)$ of the SIS process on the star graph.

Figure 8

The estimation of the infection rate $\beta$ and the curing rate $\delta$ using (23) and (22) for the star graph corresponding to Fig. 3. The curves are $-(1-v_{j\infty })ln\left[R_{j\infty }\left(h\right)\right]$ and $-\frac{v_{j\infty }ln\left[R_{j\infty }\left(h\right)\right]}{{\sum }_{i=1}^N{a}_{ij}{v}_{i\infty }}$ of a leaf node versus $h$. Both the estimated $\beta$ and $\delta$ are 1.00, while the real values of rates are both 1.