Deriving an underlying mechanism for discontinuous percolation

Chen, Wei; Zheng, Zhiming; D’Souza, Raissa M.

doi:10.1209/0295-5075/100/66006

Cited by 40 publications

(28 citation statements)

References 33 publications

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“…There is much recent interest in understanding the formation mechanism of multiple giant components in percolation [38,40,54,55]. In Refs.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…The discontinuous transition of the order parameter C 1 has been substantiated [38] and the underlying mechanism accounting for the discontinuous transition has been identified [40]. However, the behavior of C 1 after the largest jump (i.e., in supercritical regime) has not been studied previously.…”

Section: Growth Cessation Of the Largest Componentmentioning

confidence: 97%

“…In [38] they showed that the BFW model exhibits a discontinuous percolation transition leading to the simultaneous emergence of multiple giant components, and that the number of stable giant components can be tuned via the parameter α. In [40] the underlying mechanism leading to the discontinuous phase transition was derived, namely that growth of the largest component is dominated by overtaking when two smaller components merge together to become the new largest component. Such growth by overtaking is a natural growth mechanism observed in many complex systems from ecologies to the business world [41][42][43].…”

Section: Introductionmentioning

confidence: 99%

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Phase transitions in supercritical explosive percolation

et al. 2013

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Percolation describes the sudden emergence of large-scale connectivity as edges are added to a lattice or random network. In the Bohman-Frieze-Wormald model (BFW) of percolation, edges sampled from a random graph are considered individually and either added to the graph or rejected provided that the fraction of accepted edges is never smaller than a decreasing function with asymptotic value of α, a constant. The BFW process has been studied as a model system for investigating the underlying mechanisms leading to discontinuous phase transitions in percolation. Here we focus on the regime αε[0.6,0.95] where it is known that only one giant component, denoted C(1) , initially appears at the discontinuous phase transition. We show that at some point in the supercritical regime C(1) stops growing and eventually a second giant component, denoted C(2), emerges in a continuous percolation transition. The delay between the emergence of C(1) and C(2) and their asymptotic sizes both depend on the value of α and we establish by several techniques that there exists a bifurcation point α(c)=0.763±0.002. For αε[0.6,α(c)), C(1) stops growing the instant it emerges and the delay between the emergence of C(1) and C(2) decreases with increasing α. For αε(α(c),0.95], in contrast, C(1) continues growing into the supercritical regime and the delay between the emergence of C(1) and C(2) increases with increasing α. As we show, α(c) marks the minimal delay possible between the emergence of C(1) and C(2) (i.e., the smallest edge density for which C(2) can exist). We also establish many features of the continuous percolation of C(2) including scaling exponents and relations.

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“…There is much recent interest in understanding the formation mechanism of multiple giant components in percolation [38,40,54,55]. In Refs.…”

Section: Summary and Discussionmentioning

confidence: 99%

Section: Growth Cessation Of the Largest Componentmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Phase transitions in supercritical explosive percolation

et al. 2013

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“…The BFW process gives rise to the simple underlying mechanism of growth by overtaking 51 . The growth of the largest component is severely limited, as it can merge only with isolated nodes.…”

Section: Underlying Mechanismsmentioning

confidence: 99%

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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“…(See Supplementary Material [26] for more details.) Tuning the control parameter α allows for controlling the type and position of the phase transition, as well as the number of giant components that abruptly emerge [23,32]. Fig.…”

Section: O(λx)mentioning

confidence: 99%

Microtransition Cascades to Percolation

et al. 2014

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We report the discovery of a discrete hierarchy of micro-transitions occurring in models of continuous and discontinuous percolation. The precursory micro-transitions allow us to target almost deterministically the location of the transition point to global connectivity. This extends to the class of intrinsically stochastic processes the possibility to use warning signals anticipating phase transitions in complex systems.

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Deriving an underlying mechanism for discontinuous percolation

Cited by 40 publications

References 33 publications

Phase transitions in supercritical explosive percolation

Phase transitions in supercritical explosive percolation

Anomalous critical and supercritical phenomena in explosive percolation

Microtransition Cascades to Percolation

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