Assortativity decreases the robustness of interdependent networks

It was recently recognized that interdependencies among different networks can play a crucial role in triggering cascading failures and, hence, systemwide disasters. A recent model shows how pairs of interdependent networks can exhibit an abrupt percolation transition as failures accumulate. We report on the effects of topology on failure propagation for a model system consisting of two interdependent networks. We find that the internal node correlations in each of the two interdependent networks significantly changes the critical density of failures that triggers the total disruption of the two-network system. Specifically, we find that the assortativity (i.e., the likelihood of nodes with similar degree to be connected) within a single network decreases the robustness of the entire system. The results of this study on the influence of assortativity may provide insights into ways of improving the robustness of network architecture and, thus, enhance the level of protection of critical infrastructures.

Figure 1

The link-swap procedure consists in deleting two edges $AB$ and $CD$ and adding either the edges $AC$,$BD$ or the edges $AD$,$BD$ respecting constraints like the absence of multiple links. Such a procedure always leaves the number of links attached to each node unchanged and, therefore, it leaves the degrees of the nodes unchanged.

Figure 2

Fraction $\langle s\rangle$ of sites in the giant component as a function of the fraction $x$ of initially undamaged nodes for $N=10000$ scale-free networks. Curves are obtained by averaging 1000 simulations over 100 independent networks for each value of $x$.

Figure 3

Fluctuations $\langle {\left(\delta s\right)}^2\rangle$ of the order parameter $s$ as a function of the fraction $x$ of the initially undamaged nodes for $N=10000$ scale-free networks. The position of the peak for $\langle {\left(\delta s\right)}^2\rangle$ can be used to estimate the critical fraction of nondamaged sites $x_c$. Curves are obtained by averaging 1000 simulations over 100 independent networks for each value of $x$.

Figure 4

Normalized numerical increment $\Delta \langle s\rangle$ of the order parameter $s$ as a function of the fraction $x$ of initially undamaged nodes for $N=10000$ scale-free networks. In conventional percolation $\Delta \langle s\rangle \sim \langle \delta s\rangle$ can be used to estimate the the critical fraction of damaged sites $x_c$. Curves are obtained by averaging 1000 simulations over 100 independent networks for each value of $x$.

Figure 5

Estimated values of the percolation threshold $x_c$ as a function of the assortativity coefficient $r$ for $N=10000$ scale-free networks. The values of $x_c$ are estimated as the maxima of $\langle {\left(\delta s\right)}^2\rangle$, as well as the peaks of the $\Delta \langle s\rangle$ profiles. Note that the two estimates are very close for disassortative nets but differ a bit more for assortative nets.

Figure 6

Number of iterations (NOI) for the IFM algorithm to converge as a function of the initially undamaged node fraction. Peak positions for different assortative coefficients are close to $x_c$ as estimated from $\langle {\left(\delta s\right)}^2\rangle$.

Figure 7

Fraction $\langle s\rangle$ of sites in the largest component as a function of the fraction $x$ of initially undamaged nodes for $N=10000$ Erdős-Rényi networks. Curves are obtained by averaging for each $x$ value ${10}^4$ simulations over 100 independent networks.

Figure 8

Comparison of the percolation thresholds $x_c$ for both the Erdős-Rényi and scale-free interacting networks. Scale-free networks exhibit a significant variation upon a small increase of the assortativity $r$, but Erdős-Rényi networks exhibit only a small variation over the whole possible range of assortativities. (Inset) The estimates of $x_c$ via the normalized increment $\Delta \langle s\rangle$ are very close to the estimates of $x_c$ via the peaks of $\langle {\left(\delta s\right)}^2\rangle$ also for Erdős-Rényi networks.

Figure 9

Fluctuations $\langle {\left(\delta s\right)}^2\rangle$ of the order parameter $s$ as a function of the fraction $x$ of initially undamaged nodes for scale-free networks of different sizes. The peaks on the left correspond to disassortative networks ($J=-10$, i.e., $r$ from $-0.086$ to 0.051) while the peaks on the right correspond to assortative networks ($J=10$, i.e., $r=$ from 0.181 to 0.163). The different curves correspond to sizes $N=5000$ (circles), $N=10000$ (squares), and $N=14000$. For the sizes analyzed, there is no significant evidence for the growth and narrowing of the peaks that would be expected in a second-order transition.