A Novel Self-Adaptive SIS Model Based on the Mutual Interaction between a Graph and its Line Graph (Short Version)

In the standard Susceptible-Infected-Susceptible (SIS) epidemic model on networks the initial probabilities for nodes to be infected evolve over time according to a dynamic model that depends on the infection rate β, the recovery rate γ and, assuming a weighted network, on a static assignment of weights to the edges. The weight of each edge conveys how likely that edge is to be a channel for the spread of the infection. Hence, the potential of an edge to transmit the infection may vary from edge to edge due to the intrinsic and topological features of the network. However, this capability may also vary over time as a result of the spread of the infection itself. For instance, in a transportation network in which nodes are airports and edges are routes, the weight assigned to edges can be the number of flights per day along each route. The volume of flights and passengers is the most natural proxy of how likely the contagion will move along that route and propagate from one airport to another. Suppose, however, that, at one of (or both) the two airports which are the ends of a given route, there is a high probability to detect infected individuals and node isolation policy is in place. In the absence of external interventions, that route increases the probability to be a channel of transmission, independently of the number of actual flights. In other words, the probability that an edge is a channel for the transmission of an infection is not independent of the probability that its ends are infected. This is the perfect counterpart to the fact that the probability attribute that a node is infected at time t is not independent of the probability that the link would transmit the epidemic, which is usually condensed in the weight edge. The most natural way to take into account this idea is to consider an auxiliary, or dual, process in which the shock propagates among edges through the nodes; that is, a process occurring in a new network in which edges become the new nodes and nodes become the new edges. This network is usually defined in the literature as line graph. The key idea is then to design a model in which the weights assigned to the edges can adapt according to the changing ability of the edges themselves to transmit the disease. The updated values of the edge weights are computed as outcomes of a dual process on the line graph.