We introduce the concept of natural connectivity as a measure of structural robustness in complex networks. The natural connectivity characterizes the redundancy of alternative routes in a network by quantifying the weighted number of closed walks of all lengths. This definition leads to a simple mathematical formulation that links the natural connectivity to the spectrum of a network. The natural connectivity can be regarded as an average eigenvalue that changes strictly monotonically with the addition or deletion of edges. We calculate both analytically and numerically the natural connectivity of three typical networks: regular ring lattices, random graphs, and random scale-free networks. We also compare the proposed natural connectivity to other structural robustness measures within a scenario of edge elimination and demonstrate that the natural connectivity provides sensitive discrimination of structural robustness that agrees with our intuition.

Dealing with the protection of critical infrastructures, many game-theoretic methods have been developed to study the strategic interactions between defenders and attackers. However, most game models ignore the interrelationship between different components within a certain system. In this paper, we propose a simultaneous-move attacker-defender game model, which is a two-player zero-sum static game with complete information. The strategies and payoffs of this game are defined on the basis of the topology structure of the infrastructure system, which is represented by a complex network. Due to the complexity of strategies, the attack and defense strategies are confined by two typical strategies, namely, targeted strategy and random strategy. The simulation results indicate that in a scale-free network, the attacker virtually always attacks randomly in the Nash equilibrium. With a small cost-sensitive parameter, representing the degree to which costs increase with the importance of a target, the defender protects the hub targets with large degrees preferentially. When the cost-sensitive parameter exceeds a threshold, the defender switches to protecting nodes randomly. Our work provides a new theoretical framework to analyze the confrontations between the attacker and the defender on critical infrastructures and deserves further study.

Robustness and small-world effect are two crucial structural features of complex networks and have attracted increasing attention. However, little is known about the relation between them. Here we demonstrate that, there is a conflicting relation between robustness and small-world effect for a given degree sequence. We suggest that the robustness-oriented optimization will weaken the small-world effect and vice versa. Then, we propose a multi-objective trade-off optimization model and develop a heuristic algorithm to obtain the optimal trade-off topology for robustness and small-world effect. We show that the optimal network topology exhibits a pronounced core-periphery structure and investigate the structural properties of the optimized networks in detail.