Run-Ran Liu

Critical effects of overlapping of connectivity and dependence links on percolation of networks

In a recent work Parshani et al (2011 Proc. Natl Acad. Sci. USA 108 1007), dependence links have been introduced to the percolation model and used to study the robustness of the networks with such links, which shows that the networks are more vulnerable than the classical networks with only connectivity links. This model usually demonstrates a first order transition, rather than the second order transition found in classical network percolation. In this paper, considering the real situation that the interdependent nodes are usually connected, we study the cascading dynamics of networks when dependence links partially overlap with connectivity links. We find that the percolation transitions are not always sharpened by making nodes interdependent. For a high fraction of overlapping, the network is robust for random failures, and the percolation transition is second order, while for a low fraction of overlapping, the percolation process shows a first order transition. This work demonstrates that the crossover between two types of transitions does not only depend on the density of dependence links but also on the overlapping fraction of connectivity and dependence links. Using generating function techniques, we present exact solutions for the size of the giant component and the critical point, which are in good agreement with the simulations.

Cascading failures on networks with asymmetric dependence

Networks with mutual dependence have been shown to be much more vulnerable to random failures and targeted attacks than those without. However, in real networks, the dependence between two nodes is not always mutual. Periphery nodes may depend on hub nodes, yet the converse is not necessarily true. Considering this asymmetric dependence, we propose a model of cascading dynamics of networks, where the dependence between nodes is determined by their degrees. We find that the asymmetric dependence makes networks more robust than the symmetric one, and the percolation transition point is not sensitive to the number of the asymmetric dependence nodes. Furthermore, scale-free networks with asymmetric dependence can still be very robust to random failures, rather than extremely fragile as the one with mutual dependence. We also develop an approach to analyse this model and obtain the exact solutions for the size of the giant component and the critical point. Both simulation and analytical results reveal the existence of the crossover between the first- and the second-order percolation transitions in our model.

Cascading failures in coupled networks with both inner-dependency and inter-dependency links.

We study the percolation in coupled networks with both inner-dependency and inter-dependency links, where the inner- and inter-dependency links represent the dependencies between nodes in the same or different networks, respectively. We find that when most of dependency links are inner- or inter-ones, the coupled networks system is fragile and makes a discontinuous percolation transition. However, when the numbers of two types of dependency links are close to each other, the system is robust and makes a continuous percolation transition. This indicates that the high density of dependency links could not always lead to a discontinuous percolation transition as the previous studies. More interestingly, although the robustness of the system can be optimized by adjusting the ratio of the two types of dependency links, there exists a critical average degree of the networks for coupled random networks, below which the crossover of the two types of percolation transitions disappears, and the system will always demonstrate a discontinuous percolation transition. We also develop an approach to analyze this model, which is agreement with the simulation results well.

Cascading failures in coupled networks: The critical role of node-coupling strength across networks

The robustness of coupled networks against node failure has been of interest in the past several years, while most of the researches have considered a very strong node-coupling method, i.e., once a node fails, its dependency partner in the other network will fail immediately. However, this scenario cannot cover all the dependency situations in real world, and in most cases, some nodes cannot go so far as to fail due to theirs self-sustaining ability in case of the failures of their dependency partners. In this paper, we use the percolation framework to study the robustness of interdependent networks with weak node-coupling strength across networks analytically and numerically, where the node-coupling strength is controlled by an introduced parameter α. If a node fails, each link of its dependency partner will be removed with a probability 1−α. By tuning the fraction of initial preserved nodes p, we find a rich phase diagram in the plane p−α, with a crossover point at which a first-order percolation transition changes to a second-order percolation transition.

Percolation on interacting networks with feedback-dependency links

When real networks are considered, coupled networks with connectivity and feedback-dependency links are not rare but more general. Here, we develop a mathematical framework and study numerically and analytically the percolation of interacting networks with feedback-dependency links. For the case that all degree distributions of intra- and inter- connectivity links are Poissonian, we find that for a low density of inter-connectivity links, the system undergoes from second order to first order through hybrid phase transition as coupling strength increases. It implies that the average degree k of inter-connectivity links has a little influence on robustness of the system with a weak coupling strength, which corresponds to the second order transition, but for a strong coupling strength corresponds to the first order transition. That is to say, the system becomes robust as k increases. However, as the average degree k of each network increases, the system becomes robust for any coupling strength. In addition, we find that one can take less cost to design robust system as coupling strength decreases by analyzing minimum average degree kmin of maintaining system stability. Moreover, for high density of inter-connectivity links, we find that the hybrid phase transition region disappears, the first order region becomes larger and second order region becomes smaller. For the case of two coupled scale-free networks, the system also undergoes from second order to first order through hybrid transition as the coupling strength increases. We find that for a weak coupling strength, which corresponds to the second order transitions, feedback dependency links have no effect on robustness of system relative to no-feedback condition, but for strong coupling strength which corresponds to first order or hybrid phase transition, the system is more vulnerable under feedback condition comparing with no-feedback condition. Thus, for designing resilient system, designers should try to avoid the feedback dependency links, because the existence of feedback-dependency links makes the system extremely vulnerable and difficult to defend.

Percolation on networks with weak and heterogeneous dependency

In real networks, the dependency between nodes is ubiquitous; however, the dependency is not always complete and homogeneous. In this paper, we propose a percolation model with weak and heterogeneous dependency; i.e., dependency strengths could be different between different nodes. We find that the heterogeneous dependency strength will make the system more robust, and for various distributions of dependency strengths both continuous and discontinuous percolation transitions can be found. For Erdős-Rényi networks, we prove that the crossing point of the continuous and discontinuous percolation transitions is dependent on the first five moments of the dependency strength distribution. This indicates that the discontinuous percolation transition on networks with dependency is determined not only by the dependency strength but also by its distribution. Furthermore, in the area of the continuous percolation transition, we also find that the critical point depends on the first and second moments of the dependency strength distribution. To validate the theoretical analysis, cases with two different dependency strengths and Gaussian distribution of dependency strengths are presented as examples.

Dependence of evolutionary cooperation on the additive noise to the enhancement level in the spatial public goods game

Either in societies or economic cycles, the benefits of a group can be affected by various unpredictable factors. We study effects of additive spatiotemporal random variations on the evolution of cooperation by introducing them to the enhancement level of the spatial public goods game. Players are located on the sites of a two-dimensional lattice and gain their payoffs from games with their neighbors by choosing cooperation or defection. We observe that a moderate intensity of variations can best favor cooperation at low enhancement levels, which resembles classical coherence resonance. Whereas for high enhancement levels, we find that the random variations cannot increase the cooperation level, but hamper cooperation instead. This discrepancy is attributed to the different roles the additive variations played in the early and late stages of evolution. In the early stage of evolution, the additive variations increase the survival probability of the players with lower average payoffs. However, in the late stage of evolution, the additive variations can promote defectors to destroy the cooperative clusters that have been formed. Our results indicate that additive spatiotemporal noise may not be as universally beneficial for cooperation as the spatial prisoners dilemma game.

Corrigendum: Cascading failures in coupled networks with both inner-dependency and inter-dependency links

Scientific Reports 6: Article number: 25294; published online: 04 May 2016; updated: 10 October 2016. This Article contains typographical errors. In the Results section under subheading ‘Scale-free networks’, “We keep R B constant in eq. (4), and check the behaviours of the order parameter R A ”

Diverse strategy-learning styles promote cooperation in evolutionary spatial prisoner's dilemma game

Observational learning and practice learning are two important learning styles and play important roles in our information acquisition. In this paper, we study a spacial evolutionary prisoners dilemma game, where players can choose the observational learning rule or the practice learning rule when updating their strategies. In the proposed model, we use a parameter p controlling the preference of players choosing the observational learning rule, and found that there exists an optimal value of p leading to the highest cooperation level, which indicates that the cooperation can be promoted by these two learning rules collaboratively and one single learning rule is not favor the promotion of cooperation. By analysing the dynamical behavior of the system, we find that the observational learning rule can make the players residing on cooperative clusters more easily realize the bad sequence of mutual defection. However, a too high observational learning probability suppresses the players to form compact cooperative clusters. Our results highlight the importance of a strategy-updating rule, more importantly, the observational learning rule in the evolutionary cooperation.