Avalanche Collapse of Interdependent Networks

We reveal the nature of the avalanche collapse of the giant viable component in multiplex networks under perturbations such as random damage. Specifically, we identify latent critical clusters associated with the avalanches of random damage. Divergence of their mean size signals the approach to the hybrid phase transition from one side, while there are no critical precursors on the other side. We find that this discontinuous transition occurs in scale-free multiplex networks whenever the mean degree of at least one of the interdependent networks does not diverge.

Figure 1

A small network with two kinds of edges (left). Applying the algorithm described in the text, nonviable vertices are removed leaving two viable clusters (right).

Figure 2

Size of the giant viable cluster $\mathcal{S}$ as a function of the fraction $p$ of vertices remaining undamaged for two symmetric power-law distributed networks with, from right to left, $\gamma =2.8$, 2.5, and 2.1. The height of the jump becomes very small as $\gamma$ approaches 2, but is not zero, as seen in the inset, which is $\mathcal{S}$ vs $p$ on a logarithmic vertical scale for $\gamma =2.1$.

Figure 3

Viable and critical viable vertices for two interdependent networks. (a) A vertex is in the giant viable cluster if it has connections to giant viable subtrees (represented by infinity symbols) of both kinds. (b) A critical viable vertex of type $a$ has exactly 1 connection to a giant subtree of type $a$.

Figure 4

Diagrammatic representation of Eq. (1) in a system of two interdependent networks $a$ and $b$. The probability $X_a$, represented by a shaded infinity symbol, can be written recursively as a sum of second-neighbor probabilities. Open infinity symbols represent the equivalent probability $X_b$ for network $b$, which obeys a similar recursive equation.

Figure 5

A critical cluster. Removal of any of the shown viable vertices will result in the removal of all downstream critical viable vertices. Removal of the vertex labeled 1 will result in all of the shown vertices being removed (becoming nonviable), while removal of vertex 4 results only in vertex 5 also being removed.