A new metric for robustness with respect to virus spread

The robustness of a network is depending on the type of attack we are considering. In this paper we focus on the spread of viruses on networks. It is common practice to use the epidemic threshold as a measure for robustness. Because the epidemic threshold is inversely proportional to the largest eigenvalue of the adjacency matrix, it seems easy to compare the robustness of two networks. We will show in this paper that the comparison of the robustness with respect to virus spread for two networks actually depends on the value of the effective spreading rate tau. For this reason we propose a new metric, the viral conductance, which takes into account the complete range of values tau can obtain. In this paper we determine the viral conductance of regular graphs, complete bi-partite graphs and a number of realistic networks.

💡 Research Summary

The paper addresses a fundamental shortcoming in the way network robustness against virus-like spreading processes is traditionally measured. In most studies, the epidemic threshold τ_c of the susceptible‑infected‑susceptible (SIS) model is used as a proxy for robustness. Because τ_c = 1/λ₁, where λ₁ is the largest eigenvalue of the adjacency matrix, researchers often compare two networks simply by looking at λ₁: a smaller λ₁ (hence a larger τ_c) is interpreted as a more robust structure. The authors argue that this approach is intrinsically limited because it only reflects the behavior of the system at a single point – the critical spreading rate – while real epidemics can operate over a wide range of effective spreading rates τ = β/δ (infection probability over recovery probability).

To overcome this limitation, the authors introduce a new metric called viral conductance (VC). VC is defined as the integral of the steady‑state infection prevalence f_G(τ) over the entire admissible range of τ:

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