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Considering the real-world scenarios that there are interactions between edges in different networks and each network has different topological structure and size, we introduce a model of interdependent networks with arbitrary edge-coupling strength, in which q A and q B are used to represent the edge-coupling strength of network A and network B respectively. A mathematical framework using generating functions is developed based on self-consistent probabilities approach, which is verified by computer simulations. In particular, we carry out this mathematical framework on the Erdös–Rényi edge-coupled interdependent networks to calculate the values of phase transition thresholds and the critical coupling strengths which distinguish different types of transitions. Moreover, as contrast to the corresponding node-coupled interdependent networks, we find that for edge-coupled interdependent networks the critical coupling strengths are smaller, and the critical thresholds as well, which means the robustness of partially edge-coupled interdependent networks is better than that of partially node-coupled interdependent networks. Furthermore, we find that network A will have hybrid percolation behaviors as long as the coupling strength q A belongs to a certain range, and the range does not affected by average degree of network A . Our findings may fill the gap of understanding the robustness of edge-coupled interdependent networks with arbitrary coupling strength, and have significant meaning for network security design and optimization.

In interdependent networks, nodes are connected to each other with respect to their failure dependency relations. As a result of this dependency, a failure in one of the nodes of one of the networks within a system of several interdependent networks can cause the failure of the entire system. Diagnosing the initial source of the failure in a collapsed system of interdependent networks is an important problem to be addressed. We study an online failure diagnosis problem defined on a collapsed system of interdependent networks where the source of the failure is at an unknown node ( v ). In this problem, each node of the system has a positive inspection cost and the source of the failure is diagnosed when v is inspected. The objective is to provide an online algorithm which considers dependency relations between nodes and diagnoses v with minimum total inspection cost. We address this problem from worst-case competitive analysis perspective for the first time. In this approach, solutions which are provided under incomplete information are compared with the best solution that is provided in presence of complete information using the competitive ratio (CR) notion. We give a lower bound of the CR for deterministic online algorithms and prove its tightness by providing an optimal deterministic online algorithm. Furthermore, we provide a lower bound on the expected CR of randomized online algorithms and prove its tightness by presenting an optimal randomized online algorithm. We prove that randomized algorithms are able to obtain better CR compared to deterministic algorithms in the expected sense for this online problem.

In interdependent networks, it is usually assumed, based on percolation theory, that nodes become nonfunctional if they lose connection to the network giant component. However, in reality, some nodes, equipped with alternative resources, together with their connected neighbors can still be functioning after disconnected from the giant component. Here, we propose and study a generalized percolation model that introduces a fraction of reinforced nodes in the interdependent networks that can function and support their neighborhood. We analyze, both analytically and via simulations, the order parameter—the functioning component—comprising both the giant component and smaller components that include at least one reinforced node. Remarkably, it is found that, for interdependent networks, we need to reinforce only a small fraction of nodes to prevent abrupt catastrophic collapses. Moreover, we find that the universal upper bound of this fraction is 0.1756 for two interdependent Erdős–Rényi (ER) networks: regular random (RR) networks and scale-free (SF) networks with large average degrees. We also generalize our theory to interdependent networks of networks (NONs). These findings might yield insight for designing resilient interdependent infrastructure networks.

Increasing evidence shows that real-world systems interact with one another via dependency connectivities. Failing connectivities are the mechanism behind the breakdown of interacting complex systems, e.g., blackouts caused by the interdependence of power grids and communication networks. Previous research analyzing the robustness of interdependent networks has been limited to undirected networks. However, most real-world networks are directed, their in-degrees and out-degrees may be correlated, and they are often coupled to one another as interdependent directed networks. To understand the breakdown and robustness of interdependent directed networks, we develop a theoretical framework based on generating functions and percolation theory. We find that for interdependent Erdős–Rényi networks the directionality within each network increases their vulnerability and exhibits hybrid phase transitions. We also find that the percolation behavior of interdependent directed scale-free networks with and without degree correlations is so complex that two criteria are needed to quantify and compare their robustness: the percolation threshold and the integrated size of the giant component during an entire attack process. Interestingly, we find that the in-degree and out-degree correlations in each network layer increase the robustness of interdependent degree heterogeneous networks that most real networks are, but decrease the robustness of interdependent networks with homogeneous degree distribution and with strong coupling strengths. Moreover, by applying our theoretical analysis to real interdependent international trade networks, we find that the robustness of these real-world systems increases with the in-degree and out-degree correlations, confirming our theoretical analysis.

Catastrophic and major disasters in real-world systems, such as blackouts in power grids or global failures in critical infrastructures, are often triggered by minor events which originate a cascading failure in interdependent graphs. We present here a self-consistent theory enabling the systematic analysis of cascading failures in such networks and encompassing a broad range of dynamical systems, from epidemic spreading, to birth–death processes, to biochemical and regulatory dynamics. We offer testable predictions on breakdown scenarios, and, in particular, we unveil the conditions under which the percolation transition is of the first-order or the second-order type, as well as prove that accounting for dynamics in the nodes always accelerates the cascading process. Besides applying directly to relevant real-world situations, our results give practical hints on how to engineer more robust networked systems.

Coevolution between strategy and network structure is established as a means to arrive at the optimal conditions needed to resolve social dilemmas. Yet recent research has highlighted that the interdependence between networks may be just as important as the structure of an individual network. We therefore introduce the coevolution of strategy and network interdependence to see whether this can give rise to elevated levels of cooperation in the prisoner's dilemma game. We show that the interdependence between networks self-organizes so as to yield optimal conditions for the evolution of cooperation. Even under extremely adverse conditions, cooperators can prevail where on isolated networks they would perish. This is due to the spontaneous emergence of a two-class society, with only the upper class being allowed to control and take advantage of the interdependence. Spatial patterns reveal that cooperators, once arriving at the upper class, are much more competent than defectors in sustaining compact clusters of followers. Indeed, the asymmetric exploitation of interdependence confers to them a strong evolutionary advantage that may resolve even the toughest of social dilemmas.

Interdependent networks (IN) are collections of non-trivially interrelated graphs that are not physically connected, and provide a more realistic representation of real-world networked systems as compared to traditional isolated networks. In particular, they are an efficient tool to study the evolution of cooperative behavior from the viewpoint of statistical physics. Here, we consider a prisoner dilemma game taking place in IN, and introduce a simple rule for the calculation of fitness that incorporates individual popularity, which in its turn is represented by one parameter . We show that interdependence between agents in different networks influences the cooperative behavior trait. Namely, intermediate values guarantee an optimal environment for the evolution of cooperation, while too high or excessively low values impede cooperation. These results originate from an enhanced synchronization of strategies in different networks, which is beneficial for the formation of giant cooperative clusters wherein cooperators are protected from exploitation by defectors.

Cascading failures caused by interdependencies make modern coupled systems extremely fragile to failures. In existing network robustness enhancing methods, maximizing interlayer degree-degree correlations has been proven to be an effective way to improve the robustness of interdependent networks under random failures. Here, we propose a portion of nodes positively correlated strategy (PNC) to improve network robustness by positively correlating a portion of nodes, in which the nodes that are positively correlated are selected in descending order of degree, starting at a cutoff value. Based on percolation theory, we verify the effectiveness of PNC on different networks. And find that, when the nodes with the highest degree are preferentially correlated, i.e. the cutoff value takes the maximum degree, this strategy achieves a state-of-the-art optimization effect. In particular, for interdependent scale-free networks with power-law exponent γ that satisfies 2 < γ < 3 , we theoretically demonstrate that the highest degree preferentially correlated mode can maximize network robustness by changing the coupling state of the near 0 proportion of nodes. For γ  = 3, such a mode can make the network turn into a second-order phase transition at the collapse point. Finally, we discuss the relationship between the robustness of optimized networks and common links in real-world networks.

Evolutionary game theory in the realm of network science appeals to a lot of research communities, as it constitutes a popular theoretical framework for studying the evolution of cooperation in social dilemmas. Recent research has shown that cooperation is markedly more resistant in interdependent networks, where traditional network reciprocity can be further enhanced due to various forms of interdependence between different network layers. However, the role of mobility in interdependent networks is yet to gain its well-deserved attention. Here we consider an interdependent network model, where individuals in each layer follow different evolutionary games, and where each player is considered as a mobile agent that can move locally inside its own layer to improve its fitness. Probabilistically, we also consider an imitation possibility from a neighbor on the other layer. We show that, by considering migration and stochastic imitation, further fascinating gateways to cooperation on interdependent networks can be observed. Notably, cooperation can be promoted on both layers, even if cooperation without interdependence would be improbable on one of the layers due to adverse conditions. Our results provide a rationale for engineering better social systems at the interface of networks and human decision making under testing dilemmas.

The models of k -core percolation and interdependent networks (IN) have been extensively studied in their respective fields. A recent study has revealed that they share several common critical exponents. However, several newly discovered exponents in IN have not been explored in k -core percolation, and the origin of the similarity still remains unclear. Thus, in this paper, by considering k -core percolation on random networks, we first verify that the two newly discovered exponents (fractal fluctuation dimension, d f ′ , and correlation length exponent, ν ′ ) observed in d -dimensional IN spatial networks also exist with the same values in k -core percolation. That is, the fractality of the k -core giant component fluctuations is manifested by a fractal fluctuation dimension, d ˜ f = 3 / 4 , within a correlation size N ʹ that scales as N ′ ∝ ( p − p c ) − ν ˜ , with ν ˜ = 2 . Here we define, ν ˜ ≡ d ⋅ ν ′ and d ˜ f ≡ d f ′ / d . This implies that both models, IN and k -core, feature the same scaling behaviors with the same critical exponents, further reinforcing the similarity between the two models. Furthermore, we suggest that these two models are similar since both have two types of interactions: short-range (SR) connectivity links and long-range (LR) influences. In IN the LR are the influences of dependency links while in k -core we find here that for k  = 1 and k  = 2 the influences are SR and in contrast for k ⩾ 3 the influence is LR. In addition, analytical arguments for a universal hyper-scaling relation for the fractal fluctuation dimension of the k -core giant component and for IN as well as for any mixed-order transition are established. Our analysis enhances the comprehension of k -core percolation and supports the generalization of the concept of fractal fluctuations in mixed-order phase transitions.