  The susceptible--infected--susceptible (SIS) epidemic process on complex
networks can show metastability, resembling an endemic equilibrium. In a
general setting, the metastable state may involve a large portion of the
network, or it can be localized on small subgraphs of the contact network.
Localized infections are not interesting because a true outbreak concerns
network--wide invasion of the contact graph rather than localized infection of
certain sites within the contact network. Existing approaches to localization
phenomenon suffer from a major drawback: they fully rely on the steady--state
solution of mean--field approximate models in the neighborhood of their phase
transition point, where their approximation accuracy is worst; as statistical
physics tells us. We propose a dispersion entropy measure that quantifies the
localization of infections in a generic contact graph. Formulating a maximum
entropy problem, we find an upper bound for the dispersion entropy of the
possible metastable state in the exact SIS process. As a result, we find
sufficient conditions such that any initial infection over the network either
dies out or reaches a localized metastable state. Unlike existing studies
relying on the solution of mean--field approximate models, our investigation of
epidemic localization is based on characteristics of exact SIS equations. Our
proposed method offers a new paradigm in studying spreading processes over
complex networks.
