Epidemic centrality — is there an underestimated epidemic impact of network peripheral nodes?

Abstract

In the study of disease spreading on empirical complex networks in SIR model, initially infected nodes can be ranked according to some measure of their epidemic impact. The highest ranked nodes, also referred to as "superspreaders", are associated to dominant epidemic risks and therefore deserve special attention. In simulations on studied empirical complex networks, it is shown that the ranking depends on the dynamical regime of the disease spreading. A possible mechanism leading to this dependence is illustrated in an analytically tractable example. In systems where the allocation of resources to counter disease spreading to individual nodes is based on their ranking, the dynamical regime of disease spreading is frequently not known before the outbreak of the disease. Therefore, we introduce a quantity called epidemic centrality as an average over all relevant regimes of disease spreading as a basis of the ranking. A recently introduced concept of phase diagram of epidemic spreading is used as a framework in which several types of averaging are studied. The epidemic centrality is compared to structural properties of nodes such as node degree, k-cores and betweenness. There is a growing trend of epidemic centrality with degree and k-cores values, but the variation of epidemic centrality is much smaller than the variation of degree or k-cores value. It is found that the epidemic centrality of the structurally peripheral nodes is of the same order of magnitude as the epidemic centrality of the structurally central nodes. The implications of these findings for the distributions of resources to counter disease spreading are discussed. Author SummaryStudies of disease spreading on complex networks have provided a deep insight into the conditions of onset, dynamics and prevention of epidemics in human populations and malicious software propagation in computer networks. Identifying nodes which, when initially infected, on average infect the largest part of the network and ranking them according to their epidemic impact (the portion of the network eventually infected) is a priority for public health policies. In the study of epidemic spreading on empirical complex networks in the Susceptible-Infected-Recovered model, we find that the required ranking depends on the disease spreading regime, i.e. on how fast the disease is transmitted between nodes and how fast the infected node recovers. A measure called epidemic centrality, averaging the epidemic impact over all possible disease spreading regimes, is introduced as a basis of epidemic ranking. We find the epidemic centrality of nodes which are structurally central, to be of the same order of magnitude as the epidemic centrality of structurally peripheral nodes. These findings point to the need to study if the impact of an epidemic starting at structurally peripheral nodes might be considerably underestimated. Network periphery should gain a more prominent role in the study of the allocation of resources in future epidemic preparedness plans.

Vital nodes identification in complex networks

“…In addition, they showed that for small t, degree centrality is better than eigenvector centrality while for very large t, eigenvector centrality is much better. Note that, the Ghanbarnejad-Klemm method [174] is not a really time-aware method since the eigenvector of M contains no temporal information, however, their simulation results have clearly demonstrated the significance of temporal factor, similar to the contribution byŠikić et al [157], who have showed the relevance of dynamical parameters. Liu et al [51] proposed a so-called dynamics-sensitive (DS) centrality to predict the outbreak size at a given time step t, which can be directly applied in quantifying the spreading influences of nodes.…”

“…Sikić et al [157] raised a very important issue, but their solution is not a real solution since if we can accurately calculate the epidemic centrality, we should know every details of this dynamics and we must have done all possible simulations, so that we do not need to know epidemic centrality again because given any parameter pair (β, δ), we already have the corresponding ranking of nodes' influences. Evolutionary games have long been exemplary models for the emergence of cooperation in socioeconomic and biological systems [175,176].…”

Numerical investigation of metrics for epidemic processes on graphs

“…However, the tables give little insight as to why effective resistance outperforms the other quantities. To gain intuition, we look at the heatmaps of the Lesmis network [23] in Figure 4 representing the importance of each node according to the respective graph quantity. We see that effective resistance and the epidemic hitting time are the only two graph quantities that assign relative importance to peripheral nodes of the network.…”

The fastest spreader in SIS epidemics on networks

“…Van Mieghem et al [12] propose that the best conduction node in a resistor network is the minimizer of the diagonal elements of the pseudoinverse matrix Q † of the weighted Laplacian matrix of the graph. In the Susceptible-Infected-Removed (SIR) model [13], Sikić et al [14] show that the ranking of nodal influences is sensitive to the spreading dynamics, which depends on the a e-mail: Z.He@tudelft.nl infection rate and the curing rate. Measured by the cumulative infection probabilities of nodes, the degree centrality can better identify influential spreaders when the spreading rate is very small.…”