Emergence of coexisting percolating clusters in networks

Faqeeh, Ali; Melnik, Sergey; Colomer-de-Simón, Pol; Gleeson, James P.

doi:10.1103/physreve.93.062308

Cited by 9 publications

(6 citation statements)

References 39 publications

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“…Furthermore, the emergence of such noncritical power laws introduces new challenges for accurate detection of criticality in experimental setups due to the finite size of the commonly used samples. In addition to the importance of these findings in the context of neuronal systems, the insights on percolation properties of networks that this Rapid Communication provides may find applications in topics such as network robustness [56,66], epidemic spreading [50,67,68], and stability of biological systems [69].…”

Section: Discussionmentioning

confidence: 98%

Emergence of power laws in noncritical neuronal systems

et al. 2019

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Experimental and computational studies provide compelling evidence that neuronal systems are characterized by power-law distributions of neuronal avalanche sizes. This fact is interpreted as an indication that these systems are operating near criticality, and, in turn, typical properties of critical dynamical processes, such as optimal information transmission and stability, are attributed to neuronal systems. The purpose of this Rapid Communication is to show that the presence of power-law distributions for the size of neuronal avalanches is not a sufficient condition for the system to operate near criticality. Specifically, we consider a simplistic model of neuronal dynamics on networks and show that the degree distribution of the underlying neuronal network may trigger power-law distributions for neuronal avalanches even when the system is not in its critical regime. To certify and explain our findings we develop an analytical approach based on percolation theory and branching processes techniques.

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Section: Discussionmentioning

confidence: 98%

Emergence of power laws in noncritical neuronal systems

et al. 2019

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“…5 (a)]. A message passing method for modular networks is suggested in [29], however, to use it, [25] and an Amazon copurchase network [26]. In all cases the simulation results differ significantly from the message passing solutions.…”

Section: Accuracy and Limitations Of The Message Passing Approachmentioning

confidence: 99%

Nonbacktracking expansion of finite graphs

Timár¹,

Costa²,

Dorogovtsev³

et al. 2017

Phys. Rev. E

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Message passing equations yield a sharp percolation transition in finite graphs, as an artifact of the locally treelike approximation. For an arbitrary finite, connected, undirected graph we construct an infinite tree having the same local structural properties as this finite graph, when observed by a nonbacktracking walker. Formally excluding the boundary, this infinite tree is a generalization of the Bethe lattice. We indicate an infinite, locally treelike, random network whose local structure is exactly given by this infinite tree. Message passing equations for various cooperative models on this construction are the same as for the original finite graph, but here they provide the exact solutions of the corresponding cooperative problems. These solutions are good approximations to observables for the models on the original graph when it is sufficiently large and not strongly correlated. We show how to express these solutions in the critical region in terms of the principal eigenvector components of the nonbacktracking matrix. As representative examples we formulate the problems of the random and optimal destruction of a connected graph in terms of our construction, the nonbacktracking expansion. We analyze the limitations and the accuracy of the message passing algorithms for different classes of networks and compare the complexity of the message passing calculations to that of direct numerical simulations. Notably, in a range of important cases, simulations turn out to be more efficient computationally than the message passing.

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“…In addition, the network of networks is also studied as interacting networks [319,451,452], in which the connections between the network layers are ordinary links. A system of this kind is therefore equivalent to a single modular network, for which the system can be easily separated into isolated modules, and different modules might percolate at different thresholds [389,[453][454][455]. It should mention that there are some special reviews on the resilience of network of networks [456,457].…”

Section: Network Of Networkmentioning

confidence: 99%

Percolation on complex networks: Theory and application

Li,

Liu,

Lü

et al. 2021

Preprint

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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. As a paradigm for random and semi-random connectivity, percolation model plays a key role in the development of network science and its applications. 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. 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. 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. So far, the percolation theory has already percolated into the researches of structure analysis and dynamic modeling in network science. 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. 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.

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Emergence of coexisting percolating clusters in networks

Cited by 9 publications

References 39 publications

Emergence of power laws in noncritical neuronal systems

Emergence of power laws in noncritical neuronal systems

Nonbacktracking expansion of finite graphs

Percolation on complex networks: Theory and application

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