Crackling noise in fractional percolation

“…The devil's staircase rule is a three-vertex rule that preferentially merges components of equal (or similar) size or adds an intra-cluster edge 41,42 . Hence, regardless of how many of the chosen vertices reside in C, it is impossible that C merges with a smaller component, meaning P gr = 0.…”

Section: Stochastic Staircases In the Supercritical Regimementioning

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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“…[28]) while in other cases it is for p c < 1 (like in the NEP model and, e.g., Ref. [29]). Also, bootstrap percolation models have been found to exhibit discontinuous phase transitions, although it remains unclear whether these are critical in 2D [30][31][32].…”

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confidence: 90%

Anomalous Discontinuity at the Percolation Critical Point of Active Gels

et al. 2015

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We develop a percolation model motivated by recent experimental studies of gels with active network remodeling by molecular motors. This remodeling was found to lead to a critical state reminiscent of random percolation (RP), but with a cluster distribution inconsistent with RP. Our model not only can account for these experiments, but also exhibits an unusual type of mixed phase transition: We find that the transition is characterized by signatures of criticality, but with a discontinuity in the order parameter. DOI: 10.1103/PhysRevLett.114.098104 PACS numbers: 87.16.Ka, 64.60.ah, 64.60.Bd, 64.60.De Percolation theory has become pervasive in a number of fields ranging from physics to mathematics and even computer science [1]. In particular, it successfully describes connectivity and elastic properties of polymer networks [2,3]. The simplest percolation model is the random percolation (RP) model, consisting of a collection of nodes with controlled connectivity, p, representing the fraction of occupied bonds between the nodes. As a function of p, the order parameter-the mass fraction of the largest clusterbecomes finite above the percolation threshold p c . The nature of the transition is of special interest because the system properties are highly tunable at this point, especially if the transition is discontinuous; in that case, just a few bonds can have a significant impact, even for very large systems [4]. Usually, however, percolation transitions are second order, with a continuous variation of the order parameter and various critical signatures. More specialized percolation models can exhibit different phase behavior, including discontinuous transitions between the two phases (see discussion below).Here, we present a simple model based on random percolation that develops a discontinuous jump in the order parameter in the thermodynamic limit, while exhibiting other features of criticality in such quantities as the correlation length and susceptibility. Interestingly, the transition we observe occurs for the same p c < 1 as for random percolation. Moreover, our model can account for recent experimental results on active biopolymer gels that have been shown to self-organize towards a critical connectivity point [5]. The experimentally observed cluster properties at this point were found to be inconsistent with the ordinary random percolation model.In these experiments, we studied a model cytoskeletal system, composed of actin filaments, fascin cross-links and myosin motors in a quasi-2D chamber of dimensions 3 mm × 2 mm × 80 μm [5] (see Supplemental Material [6]). We observed a motor-driven collapse of the network into disjointed clusters (see Fig. 1(a) and movie of the collapse in the Supplemental Material [6]). The configuration of the clusters prior to the collapse is obtained by analyzing the timereversed movie [see Fig. 1(b)] and their masses were estimated from their initial areas. We found that, over a wide range of the experimental parameters, the number n s of clusters of mass s exhibits a power-l...

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Crackling noise in fractional percolation

Cited by 43 publications

References 52 publications

Fragmentation of Fractal Random Structures

Fragmentation of Fractal Random Structures

Anomalous critical and supercritical phenomena in explosive percolation

Anomalous Discontinuity at the Percolation Critical Point of Active Gels

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