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.

In this paper, based on susceptible–infected–susceptible (SIS) scheme, we introduce a framework that allows us to describe the spreading dynamics of two interacting diseases with active nodes. Different from previous studies, the two different diseases, propagating concurrently on the same population, can interact with each other by modifying their transmission rates. Meanwhile, according to certain probabilities, each node on the complex networks rotates between active state and inactive state. Based on heterogeneous mean-field approach, we analyze the epidemic thresholds of the two diseases and compute the temporal evolution characterizing the spreading dynamics. In addition, we validate these theoretical predictions by numerical simulations with phase diagrams. Results show that the secondary thresholds for the two opposite scenarios (mutual enhancement scenario and mutual impairment scenario) are different. We also find that the value of critical threshold and the final size of spreading dynamics are reduced as the node activity rate decreases.

Studying more realistic propagation mechanisms of rumors is crucial to controlling their spreading. Considering the reality of people’s forgetting and losing interest in the process of rumor spreading, the oblivion-recall mechanism and the loss-interest mechanism are both introduced in this paper to construct a novel susceptible-infected-removed (SIR) model. In our SIR model, the forgetting is regarded as an independent state type of the population, and we use the forgetting factor $\alpha$ and the recall factor $\beta$ to characterize the oblivion-recall mechanism. The mean-field equations are established respectively to describe the transmission dynamics of rumors in homogeneous networks and inhomogeneous networks. By performing stable state analysis, the relationship between these two parameters $\alpha$, $\beta$ and the propagation critical thresholds ${\lambda }_c$ is investigated. It shows that ${\lambda }_c$ is directly proportional to $\alpha$, which indicates that the loss-interest mechanism makes ${\lambda }_c$ exist, and the oblivion-recall mechanism increases the value of ${\lambda }_c$. Thus, the oblivion-recall mechanism reduces the outbreak probability of rumor spreading. It also reveals that when considering the impact of oblivion-recall mechanism, both the final rumor size and the propagation velocity of rumors decrease. Moreover, in the case of considering the existence of oblivion-recall mechanism, it is still found that the network topology is an important factor affecting the spread of rumors. We wish that our study can offer a new angle of view on the issue of the spread of rumors.

We use the Hoshen–Kopelman algorithm with the Nakanashi method of recycling redundant labels to measure the fraction of spanning configurations, R(p_c), at and near p_c, for random site percolation in two and three dimensions with different boundary conditions. For the square and cubic lattices we find that R(p_c) is 0.50 and 0.28 for free edges and 0.64 (2-d) and 0.56 (3-d) for both helical and periodic boundary conditions. The error bars are of the order of ±0.01 for these results.

Controlling the virus spreading in complex networks is very important to the research fields ranging from social science to technology. However, recent researches only consider information transmission and virus spreading as independent and uncorrelated, which neglect their interactive process in real systems. To study the interactive process between information transmission and virus spreading, in this paper, we propose a new virus information interactive spreading model on multiplex network. The multiplex networks can be divided into information transmission layer and virus spreading layer. The interactive spreading model considers the inherent characteristics of information by introducing the information forgetting and information loss mechanisms. Specially, by utilizing the mean field theory, the information transmission and virus process are unified to deduce the critical threshold of virus breakout. It shows that the critical threshold is not only related to the spreading properties of virus, but also influenced by the information density of individuals. We also find that the critical threshold is independent of the information generation probability, and the loss of the information will lead to a decrease. Finally, to verify the theoretical derivations, we apply it on both artificial network and real social network. The experimental results show that the simulations perfectly coincide with theoretical results, which verified the effectiveness of our model.

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.

This paper is concerned with the critical threshold phenomenon for one-dimensional damped, pressureless Euler–Poisson equations with electric force induced by a constant background, originally studied in [S. Engelberg and H. Liu and E. Tadmor, Indiana Univ. Math. J.50 (2001) 109–157]. A simple transformation is used to linearize the characteristic system of equations, which allows us to study the geometrical structure of critical threshold curves for three damping cases: overdamped, underdamped and borderline damped through phase plane analysis. We also derive the explicit form of these critical curves. These sharp results state that if the initial data is within the threshold region, the solution will remain smooth for all time, otherwise it will have a finite time breakdown. Finally, we apply these general results to identify critical thresholds for a non-local system subjected to initial data on the whole line.

We study the stabilizing effect of rotational forcing in the nonlinear setting of two-dimensional shallow-water equations. The pressureless version of these equations admit global smooth solutions for a large set of sub-critical initial configurations. But what happens with more realistic models, in the presence of pressure? It is shown that when rotational forcing dominates the pressure, it prolongs the life-span of such sub-critical solutions, for a time period ln(1/δ) ≫ 1 dictated by the ratio δ =Rossby number/squared Froude number. Our study reveals a "nearby" periodic-in-time approximate solution in the small δ regime, upon which hinges the long time existence of the exact smooth solution. These results are in agreement with the close-to-periodic dynamics observed in the "near inertial oscillation" (NIO) regime which follows oceanic storms.