AVALANCHES IN A SHORT-MEMORY EXCITABLE NETWORK

Abstract

We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced in [24], and is mathematically equivalent to an endemic variation of the Reed-Frost epidemic model introduced in [28]. Two types of heuristic approximation are frequently used for models of this type in applications, a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network. In this paper we prove several results concerning the exact relation between the avalanche model and these limits, including rates of convergence and rigorous bounds for common characteristics of the model.

Keywords: 60J85; 60K40; 90B15; Primary 60J10; Secondary 92D25; branching processes; cascading failures; complex networks; criticality; dynamic graphs.

Figure 1:

Plot of the function $f\left(i_0\right)=\mathbb{E}\left(T|X_0=i_0\right)$ for n = 100 and several values of the parameter c = np ranging from c = 0.9 to c = 1.3.

Figure 2:

Plot of the function $f\left(i_0\right)=\mathbb{E}\left(T|X_0=i_0\right)$ for n = 1000 and several values of the parameter c = np ranging from c = 0.9 to c = ‘1.3.

Figure 3:

Graph of the function g_α(x) = (1 − x)(1 − e^−αx) for several values of the parameter α, including the critical branching value α = 1, for which $g_1^{\prime }(0)=1$, and α = 2.46742 which is a close approximation to the transitional value α_tr.