Interdependent networks with correlated degrees of mutually dependent nodes

We study a problem of failure of two interdependent networks in the case of correlated degrees of mutually dependent nodes. We assume that both networks (A and B) have the same number of nodes $N$ connected by the bidirectional dependency links establishing a one-to-one correspondence between the nodes of the two networks in a such a way that the mutually dependent nodes have the same number of connectivity links, i.e. their degrees coincide. This implies that both networks have the same degree distribution $P(k)$. We call such networks correspondently coupled networks (CCN). We assume that the nodes in each network are randomly connected. We define the mutually connected clusters and the mutual giant component as in earlier works on randomly coupled interdependent networks and assume that only the nodes which belong to the mutual giant component remain functional. We assume that initially a $1-p$ fraction of nodes are randomly removed due to an attack or failure and find analytically, for an arbitrary $P(k)$, the fraction of nodes $\mu(p)$ which belong to the mutual giant component. We find that the system undergoes a percolation transition at certain fraction $p=p_c$ which is always smaller than the $p_c$ for randomly coupled networks with the same $P(k)$. We also find that the system undergoes a first order transition at $p_c>0$ if $P(k)$ has a finite second moment. For the case of scale free networks with $2<\lambda \leq 3$, the transition becomes a second order transition. Moreover, if $\lambda<3$ we find $p_c=0$ as in percolation of a single network. For $\lambda=3$ we find an exact analytical expression for $p_c>0$. Finally, we find that the robustness of CCN increases with the broadness of their degree distribution.Comment: 18 pages, 3 figure

Strong Connectivity in Real Directed Networks

In many real, directed networks, the strongly connected component of nodes which are mutually reachable is very small. This does not fit with current theory, based on random graphs, according to which strong connectivity depends on mean degree and degree-degree correlations. And it has important implications for other properties of real networks and the dynamical behaviour of many complex systems. We find that strong connectivity depends crucially on the extent to which the network has an overall direction or hierarchical ordering -- a property measured by trophic coherence. Using percolation theory, we find the critical point separating weakly and strongly connected regimes, and confirm our results on many real-world networks, including ecological, neural, trade and social networks. We show that the connectivity structure can be disrupted with minimal effort by a targeted attack on edges which run counter to the overall direction. And we illustrate with example dynamics -- the SIS model, majority vote, Kuramoto oscillators and the voter model -- how a small number of edge deletions can utterly change dynamical processes in a wide range of systems.Comment: 16 pages, 6 figure

Avoiding catastrophic failure in correlated networks of networks

Networks in nature do not act in isolation but instead exchange information, and depend on each other to function properly. An incipient theory of Networks of Networks have shown that connected random networks may very easily result in abrupt failures. This theoretical finding bares an intrinsic paradox: If natural systems organize in interconnected networks, how can they be so stable? Here we provide a solution to this conundrum, showing that the stability of a system of networks relies on the relation between the internal structure of a network and its pattern of connections to other networks. Specifically, we demonstrate that if network inter-connections are provided by hubs of the network and if there is a moderate degree of convergence of inter-network connection the systems of network are stable and robust to failure. We test this theoretical prediction in two independent experiments of functional brain networks (in task- and resting states) which show that brain networks are connected with a topology that maximizes stability according to the theory.Comment: 40 pages, 7 figure