SIS epidemics on Triadic Random Graphs

It has been shown in the past that many real-world networks exhibit community structures and non-trivial clustering which comes with the occurrence of a notable number of triangular connections. Yet the influence of such connection patterns on the dynamics of disease transmission is not fully understood. In order to study their role in the context of Susceptible-Infected-Susceptible (SIS) epidemics we use the Triadic Random Graph (TRG) model to construct small networks (N=49) from distinct, closed, directed triadic subpatterns. We compare various global properties of TRGs and use the N-intertwined mean-field approximation (NIMFA) model to perform numerical simulations of SIS-dynamics on TRGs. The results show that the infection spread on undirected TRGs displays very similar behavior to TRGs with an abundance of (directed) feed-back-loops, while using (directed) feed-forward-loops as network-entities significantly slows down the epidemic and lowers the number of infected individuals in the endemic state. Moreover, we introduce a novel stochastic approach for modelling the SIS-epidemics on TRGs based on characterizing nodes according to their set of $\left(k_{in},k_{out}\right)$ within triads. Within this model, the topology of the network is given by the number and the local structure of directed triadic motifs and not by the adjacency matrix. Nevertheless, the outcome of simulations is qualitatively similar to the results of the NIMFA model.