High-Accuracy Approximation of Binary-State Dynamics on Networks

Binary-state dynamics (such as the susceptible-infected-susceptible (SIS) model of disease spread, or Glauber spin dynamics) on random networks are accurately approximated using master equations. Standard mean-field and pairwise theories are shown to result from seeking approximate solutions of the master equations. Applications to the calculation of SIS epidemic thresholds and critical points of nonequilibrium spin models are also demonstrated.

Figure 1

(a) Infected fraction $\rho (t)$ in the SIS disease-spread model on 3-regular random graphs, with transmission rate $\lambda =1$ and recovery rate $\mu =1.4$. (b) Steady-state fraction of infected nodes as a function of the nondimensional recovery rate $\mu /\lambda$. The arrow marks the epidemic threshold predicted from the linearized master equations [top row of Table (a)].

Figure 2

Infected fraction $\rho (t)$ (i.e., fraction of $+1$ spins) for zero-temperature Glauber dynamics on (a) networks with truncated power-law degree distribution: $P_k\propto k^{-2.5}$ for $3\le k\le 20$, and (b) 3-regular random graphs. In each case the initial condition is $\rho (0)=0.4$.