  We propose a new paradigm to design a network-based self-adaptive epidemic
model that relies on the interplay between the network and its line graph. We
implement this proposal on a Susceptible-Infected-Susceptible model in which
both nodes and edges are considered susceptible and their respective
probabilities of being infected result in a real-time re-modulation of the
weights of both the graph and its line graph. The new model can be considered
as an appropriate perturbation of the standard Susceptible-Infected-Susceptible
model, and the coupling between the graph and its line graph is interpreted as
a reinforcement factor that fosters diffusion through a continuous adjustment
of the parameters involved. We study the existence and stability conditions of
the endemic and disease-free states for general network topologies. Moreover,
we introduce, through the asymptotic values in the endemic steady states, a new
type of eigenvector centrality where the score of a node depends on both the
neighboring nodes and the edges connected to it. We also investigate the
properties of this new model on some specific synthetic graphs, such as cycle,
regular, and star graphs. Finally, we perform a series of numerical simulations
and prove their effectiveness in capturing some empirical evidence on
behavioral adoption mechanisms.
