Narrow Results

In this paper, adopting the initial load of a node j to be , where k_j is the degree of the node j and α is a tunable parameter that controls the strength of the initial load of a node, we propose a cascading model with a breakdown probability and explore cascading failures on a typical network, i.e., the Barabási–Albert (BA) network with scale-free property. Assume that a failed node leads only to a redistribution of the load passing through it to its neighboring nodes. According to the simulation results, we find that BA networks reach the strongest robustness level against cascading failures when α = 1 and the robustness of networks has a positive correlation with the average degree 〈k〉, not relating to the different breakdown probabilities. In addition, it is found that the robustness against cascading failures has an inversely proportional relationship with the breakdown probability of an overload node. Finally, the numerical simulations are verified by the theoretical analysis.

Assume the initial load of an edge ij in a network to be L_ij =[(k_i ∑_{a ∈ Γi} k_a)(k_j ∑_{b ∈ Γj} k_b)]^α with k_i and k_j being the degrees of the nodes connected by the edge, where α is a tunable parameter which controls the strength of the edge initial load, and Γ_i and Γ_j are the sets of neighboring nodes of i and j, respectively. We investigate the cascading phenomenon of uncorrelated scale-free networks subject to two different attacking strategies on edges, i.e. attacking on the edges with the highest loads or the lowest loads (LL). By the critical threshold of edge capacity quantifying the network robustness, we numerically discuss the effects of two attacks for the network vulnerability. Interestingly, it is found that the attack on the edge with the LL is highly effective in disrupting the structure of the attacked network when α < 0.5. In the case of α = 0.5, the effects of two attacks for the network robustness against cascading failures are almost identical. We furthermore provide the theoretical prediction support for the numerical simulations. These results may be very helpful for real-life networks to protect the key edges selected effectively to avoid cascading-failure-induced disasters.

Many real-world networks interact with other networks by only several links, for example, transportation networks and aviation networks among cities, internet router network among the different regions, power supply network among cities and so on. Understanding how to protect these coupled networks and improve their robustness against cascading failures is very important. By protecting the edges between coupled networks, we investigate its efficiency on improving the robustness of coupled networks against cascading failures. Fixing the total protective cost of coupled networks, we find that adjusting the capacities of the edges among coupled networks can better improve the robustness of coupled networks against cascading failures and observe that the more uniform the distribution of the edge load, the more effective the protection strategy. In addition, by immunizing the edges among coupled networks, we compare two protecting methods and find that the immunization strategy can better protect coupled networks. Our results are useful not only for how to protect coupled networks from the local perspective, but also for significantly improving the robustness of a single network by protecting some key edges.

In this paper, based on the local preferential redistribution rule of the load after removing a node, we propose a cascading model and explore cascading failures on four typical networks, i.e. the BA with scale-free property, the WS small-world network, the NW network and the ER random network. Assume that a failed node leads only to a redistribution of the load passing through it to its neighboring nodes. We find that all networks reach the strongest robustness level against cascading failures in the case of α=1, which is a tunable parameter in our model, where the robustness is quantified by the critical threshold T_c, at which a phase transition occurs from a normal state to collapse. To a constant network size, we further discuss the correlations between the average degree 〈k〉 and T_c, and draw the conclusion that T_c has a negative correlative with 〈k〉, i.e. the bigger the value of 〈k〉, the smaller the critical threshold T_c. These results may be very helpful for real-life networks to avoid cascading-failure-induced disasters.

In this paper, based on the local preferential redistribution rule of the load after removing a node, we propose a cascading model and explore cascading failures on scale-free networks. Assuming that a failed node leads only to a redistribution of the load passing through it to its neighboring nodes, we study the response of scale-free networks subject to attacks on nodes. The network robustness against cascading failures is quantitatively measured by the critical threshold T_c, at which a phase transition occurs from normal state to collapse. For each case of attacks on nodes, four different attack strategies are used: removal by the descending order of the degree, attack by the ascending order of the degree, random removal of breakdown, and removal by the ascending order of the average degree of neighboring nodes of a broken node. Compared with the previous result, i.e. the robust-yet-fragile property of scale-free networks on random failures of nodes and intentional attacks, our cascading model has totally different and interesting results. On the one hand, as unexpected, choosing the node with the lowest degree is more efficient than the one with the highest degree when α < 1, which is a tunable parameter in our model. On the other hand, the robustness against cascading failures and the harm order of four attack strategies strongly depends on the parameter α. These results may be very helpful for real-life networks to protect the key nodes and avoid cascading-failure-induced disasters.

In this paper, based on the local information about nodes, we propose a new attack strategy, considering the average degree of the node's neighboring nodes. Adopting the cascading model proposed in Chin. Phys. Lett.25(10) (2008) 3826, we investigate the effect of the new attack strategy for the robustness against cascading failures on a typical network, i.e. BA scale-free networks. Compared with two attacks on the nodes with the highest load or the lowest load, numerically we find that our presented attack is the most efficient way to destruct the BA scale-free networks in the case of α ≤ 0.6, where α is a tunable parameter and determines to the load strength of a node. In addition, we also find that the efficiency of the new attack strategy is more obvious near the parameter α = 0.5. We expect our findings to provide insights into the protection of the key nodes for real-life networks to avoid cascading-failure-induced disasters.

Cascading failures can occur in many infrastructure networks. How to protect these networks and improve their robustness against cascading failures has been of great interest. To this end, considering that there exist some monitoring and protection measures in these networks, we propose a new mitigation strategy and investigate its effectiveness on improving the robustness level against cascading failures in Barabáasi-Albert (BA) networks and the power grid. We numerically observe that only by once adopting this strategy the robustness of BA networks and the power grid can be improved dramatically. We additionally find that BA networks and the power grid can reach the strongest robustness against cascading failures in the case of the specific value of the parameter α, which controls the strength of the initial load on a node. And we obtain the correlation between the load distribution and the effectiveness of the mitigation strategy. Our findings can well explain the origin of the stronger robustness against cascading failures of complex networks and may be very useful for guiding the improvement robustness of infrastructure networks and avoiding various cascading-failure-induced disasters in the real world.

The paper devises a Parrondo's game model of biotic population with the network as its spatial carrier, trying to analyze individual's coopetition behavior and investigate the degree distribution of the heterogeneity on the impact of coopetition. The populational Parrondo's game model consists of a zero-sum game among individuals and a negative sum game between individuals and environment. In terms of relations of zero-sum game, four patterns are defined: cooperation, competition, harmony, and poor-competition-rich-cooperation. The simulation result shows that: (1) Cooperation and competition in any forms are adaptive behaviors. Cooperative and competitive behavior could convert the losing games combined into winning. The positive average fitness of the population represents the paradoxical feature that the Parrondo's game is counterintuitive. (2) BA Network is conducive to cooperation. (3) The relationships of individual fitness with node degree and with clustering coefficient are disclosed. As for cooperation and poor-competition-rich-cooperation pattern, the greater the node degree is, the greater the individual fitness is. (4) The heterogeneity has a positive impact on cooperation. (5) Population average fitness is the largest when the probability of playing zero-sum game is 1/3 in the Parrondo's game model.

The paper studies a multi-agent Parrondo's game with history dependence. With the complex networks as the spatial carrier, the adaptation of cooperation and competition (coopetition for short) behaviors is analyzed and the impact of the degree distribution of the heterogeneity on the behavioral adaptation is investigated. The multi-agent Parrondo's game consists of a zero-sum game between individuals and a negative-sum game between individuals and environment based on the history of the game. In terms of relations of the zero-sum game, two behavioral patterns are determined: cooperation and competition. The simulation results show that: (1) Cooperation and competition in any forms are the adaptive behaviors. The coopetition behavior results in the variety of winning and losing states of the history, which makes the population develop toward the beneficial direction where nature affects. The positive average fitness of the population represents the paradoxical feature that the Parrondo's game is counterintuitive, (2) for the cooperation pattern, the average fitness of the population is the largest under Barabási–Albert (BA) network which is conducive to cooperation, (3) the heterogeneity has a positive impact on cooperation.