We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erd\Hnyi (ER) edge-coupled interdependent networks to calculate the values of phase transition thresholds and the critical coupling strengths which distinguish different types of transitions. This finding also provides a potential approach for determining the critical threshold and fill the gap between finite components and GC on the percolation process. In particular, we find a new general scaling relationship between 1s and 1pmax, p max represents the value of p (non-failure fraction of initial nodes) corresponding to the peak point of π s in the network. We find theoretically and via simulation that finite components all show the peak shape which is different from GC for random networks including random regular network, Erdős–R e´ nyi networks and scale-free networks. Here we focus on the percolation behaviors of small component π s with the size s = 1, 2, 3, … under different failure scenarios such as random attack, localized attack, target attack and intentional attack with limited knowledge. Different from the percolation behaviors of giant component (GC), the finite components make one more clearly explore network percolation behaviors and critical phenomena from a microscopic perspective, especially for large-scale network systems. Percolation behavior is of wide applicability and provides insight into functional structure of complex networks. The intention of this paper is to offer an overview of these applications, as well as the basic theory of percolation transition on network systems. Understanding the percolation theory should help the study of many fields in network science, including the still opening questions in the frontiers of networks, such as networks beyond pairwise interactions, temporal networks, and network of networks. So far, the percolation theory has already percolated into the researches of structure analysis and dynamic modeling in network science. Meanwhile, network science also brings some new issues to the percolation theory itself, such as percolation of strong heterogeneous systems, topological transition of networks beyond pairwise interactions, and emergence of a giant cluster with mutual connections. On the other hand, the insights into the percolation theory also facilitate the understanding of networked systems, such as robustness, epidemic spreading, vital node identification, and community detection. On the one hand, the concepts and analytical methods, such as the emergence of the giant cluster, the finite-size scaling, and the mean-field method, which are intimately related to the percolation theory, are employed to quantify and solve some core problems of networks. As a paradigm for random and semi-random connectivity, percolation model plays a key role in the development of network science and its applications. In the last two decades, network science has blossomed and influenced various fields, such as statistical physics, computer science, biology and sociology, from the perspective of the heterogeneous interaction patterns of components composing the complex systems. This can be achieved by using Lawler’s method of loop erasure. Hence, it remains to transform γ into a γ ′ path with that property. Next, we identify subsets of with y j = y j ′, then the claim follows from the observation that k ≤ 2 # S ≤ 2 c ′ 1 # U ( i ) m. In other words, our results do not depend on the question whether the base stations are scattered at random in the Euclidean plane or are aligned according to a grid that is viewed from a random reference point. ![]() For instance, they can be applied to homogeneous Poisson point processes as well as randomly shifted lattices. Since we only assume stationarity and ergodicity, our results are valid under quite weak conditions on the spatial distribution of base stations. Here, r ≥ 0 is some scaling parameter controlling the intensity of base stations. We assume that they are of the form Y = r Y ( 1 ), where Y ( 1 ) is assumed to be a stationary and ergodic point process that is independent of X and has a finite and positive intensity λ ′. The base stations constitute the second component. They are modeled by a homogeneous Poisson point process X in R d, d ≥ 2 with some intensity λ ∈ ( 0, ∞ ). The first component is formed by network users. It consists of two types of network components. Next, we provide a precise definition of the wireless spatial telecommunication network under consideration.
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