Structural transition in interdependent networks with regular interconnections

Networks are often made up of several layers that exhibit diverse degrees of interdependencies. An interdependent network consists of a set of graphs $G$ that are interconnected through a weighted interconnection matrix $B$, where the weight of each intergraph link is a non-negative real number $p$. Various dynamical processes, such as synchronization, cascading failures in power grids, and diffusion processes, are described by the Laplacian matrix $Q$ characterizing the whole system. For the case in which the multilayer graph is a multiplex, where the number of nodes in each layer is the same and the interconnection matrix $B=pI$, $I$ being the identity matrix, it has been shown that there exists a structural transition at some critical coupling $p^{*}$. This transition is such that dynamical processes are separated into two regimes: if $p>p^{*}$, the network acts as a whole; whereas when $p<p^{*}$, the network operates as if the graphs encoding the layers were isolated. In this paper, we extend and generalize the structural transition threshold $p^{*}$ to a regular interconnection matrix $B$ (constant row and column sum). Specifically, we provide upper and lower bounds for the transition threshold $p^{*}$ in interdependent networks with a regular interconnection matrix $B$ and derive the exact transition threshold for special scenarios using the formalism of quotient graphs. Additionally, we discuss the physical meaning of the transition threshold $p^{*}$ in terms of the minimum cut and show, through a counterexample, that the structural transition does not always exist. Our results are one step forward on the characterization of more realistic multilayer networks and might be relevant for systems that deviate from the topological constraints imposed by multiplex networks.

Figure 1

The figure depicts two interdependent networks that only differ in their interlayer couplings: the left panel represents the case studied in [10], which is however not commonly found in real systems as it constrains all interlayer couplings to follow a one-to-one interconnection pattern. On the contrary, the right panel represents a scenario in which the interlayer connectivity follows a $k$-to-$k$ coupling scheme, and although still not fully realistic given the regularity—i.e., homogeneity—of the interconnection pattern, it is more complex and representative of real interdependent networks. (a) one-to-one interconnection and (b) $k$-to-$k$ interconnection (here $k=2$).

Figure 2

Accuracy of the upper and lower bounds for the transition threshold $p^{*}$ in interdependent networks consisting of (a) two Erdős-Rényi graphs and (b) two Barabási-Albert graphs with $n=500$ and average degree $d_{av}=6$. The interconnection pattern is a 2-to-2 scheme, i.e., $k=2$.

Figure 3

Example topology with $k$-to-$k$ ($k=3$) interconnections whose transition threshold is calculated exactly by (13).

Figure 4

The figure illustrates how the example topology in Fig. 3 can be obtained from the Cartesian product of subgraphs.

Figure 5

When $p\le p^{*}$, the minimum cut is achieved by cutting all the interdependent links and the minimized cut weight follows $R_{min}=\frac{N}{4}{\mu }_{N-1}$.

Figure 6

Exact transition threshold for special structures including (a) coupled circulant graphs, (b) fully coupled Erdős-Rényi graphs, and (c),(d) a star graph fully coupled with its complementary graph.