Growth dominates choice in network percolation

Vijayaraghavan, Vikram S.; Noël, Pierre-André; Waagen, Alex; D’Souza, Raissa M.

doi:10.1103/physreve.88.032141

Cited by 10 publications

(6 citation statements)

References 44 publications

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“…This leads to an infinite-order percolation transition 70 . Following the same procedure, but using the 'adjacent edge' Achlioptas process 24 for edge addition, considerably delays the onset of the percolation transition, but retains the smooth, infinite-order transition 71 . Thus, network growth via node arrival allows a significantly delayed percolation transition yet can mitigate the abrupt, explosive nature that typically results from delay interventions 71,72 .…”

Section: Disordered Mediamentioning

confidence: 98%

Anomalous critical and supercritical phenomena in explosive percolation

2015

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The emergence of large-scale connectivity on an underlying network or lattice, the so-called percolation transition, has a profound impact on the system's macroscopic behaviours. There is thus great interest in controlling the location of the percolation transition to either enhance or delay its onset and, more generally, in understanding the consequences of such control interventions. Here we review explosive percolation, the sudden emergence of large-scale connectivity that results from repeated, small interventions designed to delay the percolation transition. These transitions exhibit drastic, unanticipated and exciting consequences that make explosive percolation an emerging paradigm for modelling real-world systems ranging from social networks to nanotubes.T he percolation transition, named for the prototypical mathematical problem of pouring liquid through a porous material, describes the onset of large-scale connectivity on an underlying network or lattice. At times, ensuring large-scale connectivity is essential: a transportation network (such as the world-wide airline network) or a communication system (such the Internet) is useful only if a large fraction of the nodes are connected. Yet, in other contexts, large-scale connectivity is a liability: under certain conditions, a virus spreading on a well-connected social or computer network can reach enough nodes to cause an epidemic. Thus, percolation theory is a theoretical underpinning across a range of fields 1,2 and the desire to enhance or delay the onset of percolation has been of interest for many years. The consequences of delaying the transition have only recently been established and here we review explosive percolation (EP), the phenomenon that usually results from repeated, small interventions designed to delay the percolation transition. The onset can indeed be significantly delayed, but once the percolation transition is inevitably reached, large-scale connectivity emerges suddenly.The traditional approach for constructing a random graph, the Erdős-Rényi model, considers a collection of N isolated nodes, with each possible edge between two distinct nodes added to the graph with probability p (refs 3-5). This is a static formulation with no dependence on the history of how edges have been added to the graph. A mathematically equivalent kinetic formulation is initialized with N originally isolated nodes with a randomly sampled edge added at each discrete time step 6 . Letting T denote the number of steps, the process is parameterized by the relative number of introduced edges t = T /N , and typically analysed in the thermodynamic limit of infinite size N . Below some critical t = t c the resulting graph is disjoint, consisting of small isolated clusters (or components) of connected nodes. (See Fig. 1c for an illustration of distinct components.) Let C denote the largest component and |C| its size. For the Erdős-Rényi model, the order parameter |C| undergoes a second-order transition at t c = 1/2 where, below t c , |C| is logarithmic in N and, a...

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Section: Disordered Mediamentioning

confidence: 98%

Anomalous critical and supercritical phenomena in explosive percolation

2015

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“…Another approach for a model of this class was used in Ref. [25] to estimate the percolation threshold position imposing the strong assumption that the cluster size distribution P (s, t) ∝ s 1−τ e −cs , where c is time-dependent, and c(t c ) = 0. In this way, after solving 10 5 evolution equations, they achieved the same precision of t c (or even worse) which our method provides with only 10 equations.…”

Section: The Approachmentioning

confidence: 99%

Critical exponents of the explosive percolation transition

et al. 2014

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In a new type of percolation phase transition, which was observed in a set of non-equilibrium models, each new connection between vertices is chosen from a number of possibilities by an Achlioptaslike algorithm. This causes preferential merging of small components and delays the emergence of the percolation cluster. First simulations led to a conclusion that a percolation cluster in this irreversible process is born discontinuously, by a discontinuous phase transition, which results in the term "explosive percolation transition". We have shown that this transition is actually continuous (second-order) though with an anomalously small critical exponent of the percolation cluster. Here we propose an efficient numerical method enabling us to find the critical exponents and other characteristics of this second order transition for a representative set of explosive percolation models with different number of choices. The method is based on gluing together the numerical solutions of evolution equations for the cluster size distribution and power-law asymptotics. For each of the models, with high precision, we obtain critical exponents and the critical point.

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“…Instead of connecting nodes randomly, other typical mechanisms for generating networks can also be incorporated into the growing network model, such as BA network model [367] and the explosive percolation model [308,333,368,369]. In these models, when a new node enters the system, with probability p links are inserted following the given rule, otherwise, do nothing.…”

Section: Variants and Related Modelsmentioning

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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Growth dominates choice in network percolation

Cited by 10 publications

References 44 publications

Anomalous critical and supercritical phenomena in explosive percolation

Anomalous critical and supercritical phenomena in explosive percolation

Critical exponents of the explosive percolation transition

Percolation on complex networks: Theory and application

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