Robustness of Network of Networks with Interdependent and Interconnected links

Robustness of network of networks (NON) has been studied only for dependency coupling (J.X. Gao et. al., Nature Physics, 2012) and only for connectivity coupling (E.A. Leicht and R.M. D Souza, arxiv:0907.0894). The case of network of n networks with both interdependent and interconnected links is more complicated, and also more closely to real-life coupled network systems. Here we develop a framework to study analytically and numerically the robustness of this system. For the case of starlike network of n ER networks, we find that the system undergoes from second order to first order phase transition as coupling strength q increases. We find that increasing intra-connectivity links or inter-connectivity links can increase the robustness of the system, while the interdependency links decrease its robustness. Especially when q=1, we find exact analytical solutions of the giant component and the first order transition point. Understanding the robustness of network of networks with interdependent and interconnected links is helpful to design resilient infrastructures

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 percolation of interacting networks with feedback-dependency links. We find that when nodes of between networks are lowly connected, the system undergoes from second order transition through hybrid order transition to first order transition as coupling strength increases. And, as average degree of each inter-network increases, first order region becomes smaller and second-order region becomes larger but hybrid order region almost keep constant. Especially, the results implies that average degree \bar{k} between intra-networks has a little influence on robustness of system for weak coupling strength, but for strong coupling strength corresponding to first order transition system become robust as \bar{k} increases. However, when average degree k of inter-network is increased, the system become robust for all coupling strength. Additionally, when nodes of between networks are highly connected, the hybrid order region disappears and the system first order region becomes larger and secondorder region becomes smaller. Moreover, we find that the existence of feedback dependency links between interconnecting networks makes the system extremely vulnerable by comparing non-feedback condition for the same parameters.First author draf

Robustness on distributed coupling networks with multiple dependent links from finite functional components

The rapid advancement of technology underscores the critical importance of robustness in complex network systems. This paper presents a framework for investigating the structural robustness of interconnected network models. This paper presents a framework for investigating the structural robustness of interconnected network models. In this context, we define functional nodes within interconnected networks as those belonging to clusters of size greater than or equal to $s$ in the local network, while maintaining at least $M$ significant dependency links. This model presents precise analytical expressions for the cascading failure process, the proportion of functional nodes in the stable state, and a methodology for calculating the critical threshold. The findings reveal an abrupt phase transition behavior in the system following the initial failure. Additionally, we observe that the system necessitates higher internal connection densities to avert collapse, especially when more effective support links are required. These results are validated through simulations using both Poisson and power-law network models, which align closely with the theoretical outcomes. The method proposed in this study can assist decision-makers in designing more resilient reality-dependent systems and formulating optimal protection strategies

Exact results of the limited penetrable horizontal visibility graph associated to random time series and its application

The limited penetrable horizontal visibility algorithm is a new time analysis tool and is a further development of the horizontal visibility algorithm. We present some exact results on the topological properties of the limited penetrable horizontal visibility graph associated with random series. We show that the random series maps on a limited penetrable horizontal visibility graph with exponential degree distribution $P(k)\sim exp[-\lambda (k-2\rho-2)], \lambda = ln[(2\rho+3)/(2\rho+2)],\rho=0,1,2,...,k=2\rho+2,2\rho+3,...$, independent of the probability distribution from which the series was generated. We deduce the exact expressions of the mean degree and the clustering coefficient and demonstrate the long distance visibility property. Numerical simulations confirm the accuracy of our theoretical results. We then examine several deterministic chaotic series (a logistic map, the H$\acute{e}$non map, the Lorentz system, and an energy price chaotic system) and a real crude oil price series to test our results. The empirical results show that the limited penetrable horizontal visibility algorithm is direct, has a low computational cost when discriminating chaos from uncorrelated randomness, and is able to measure the global evolution characteristics of the real time series.Comment: 23 pages, 12 figure

目次 : 『千葉医学雑誌 オープン・アクセス・ペーパー』 92E巻4号 2016年8月

<p>(a) Clustering evolution of oil importers. (b) Evolution of cluster ratios.</p

Percolation of partially interdependent networks under targeted attack

The study of interdependent networks, and in particular the robustness on networks, has attracted considerable attention. Recent studies mainly assume that the dependence is fully interdependent. However, targeted attack for partially interdependent networks simultaneously has the characteristics of generality in real world. In this letter, the comprehensive percolation of generalized framework of partially interdependent networks under targeted attack is analyzed. As $\alpha=0$ and $\alpha=1$, the percolation law is presented. Especially, when $a=b=k$, $p_{1}=p_{2}=p$, $q_{A}=q_{B}=q$, the first and second lines of phase transition coincide with each other. The corresponding phase transition diagram and the critical line between the first and the second phase transition are found. We find that the tendency of critical line is monotone decreasing with parameter $p_{1}$. However, for different $\alpha$, the tendency of critical line is monotone increasing with $\alpha$. In a larger sense, our findings have potential application for designing networks with strong robustness and can regulate the robustness of some current networks.Comment: 6 pages, 9 figure