Weakly explosive percolation in directed networks

Squires, Shane; Sytwu, Katherine; Alcala, Diego A.; Antonsen, Thomas M.; Ott, Edward; Girvan, Michelle

doi:10.1103/physreve.87.052127

Cited by 19 publications

(35 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Ordinary percolation on hierarchical lattices leads to an explosive percolation transition 53 and may also show interesting connections to community structures and clustering phenomena. There is also very limited work concerning explosive percolation on directed networks, with work thus far focused on m-edge Achlioptas processes 79 .…”

Section: Future Directionsmentioning

confidence: 99%

Anomalous critical and supercritical phenomena in explosive percolation

2015

View full text Add to dashboard Cite

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...

show abstract

Section: Future Directionsmentioning

confidence: 99%

Anomalous critical and supercritical phenomena in explosive percolation

2015

View full text Add to dashboard Cite

show abstract

“…( for each equation of (14), the first term is the probability that the root site of a sub-branch is not occupied and the second term is the probability that the root site of the sub-branch is occupied but no child sub-branch connects to the percolating cluster. If ≤ p p c , equation (14) has only the trivial solution = = Q i j 1, , 1, 2 ij , then from (13) Fig.…”

Section: Figure 3 Average Cluster Size χ(P) Ofmentioning

confidence: 99%

“…Site percolation is more general than bond percolation because every bond model may be reformulated as a site model (on a different graph) and the converse is in general not true 3 . The appeal of percolation is the occurrence of a critical phenomenon, which has attracted attention for a wide range of applications: liquid flows in porous media 4,5 , epidemic spread [6][7][8] , granular and composite materials [9][10][11][12] , forest fires [13][14][15] and fracture patterns and earthquakes in rocks 16 .…”

mentioning

confidence: 99%

How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application

Ren

Zhang

Siegmund³

2016

Sci Rep

View full text Add to dashboard Cite

Inhomogeneous percolation, for its closer relationship with real-life, can be more useful and reasonable than homogeneous percolation to illustrate the critical phenomena and dynamical behaviour of complex networks. However, due to its intricacy, the theoretical framework of inhomogeneous percolation is far from being complete and many challenging problems are still open. In this paper, we first investigate inhomogeneous site percolation on Bethe Lattices with two occupation probabilities, and then extend the result to percolation with m occupation probabilities. The critical behaviour of this inhomogeneous percolation is shown clearly by formulating the percolation probability P p ( ) ∞ with given occupation probability p, the critical occupation probability, and the average cluster size χ p ( ) where p is subject to P p ( ) = 0 ∞ . Moreover, using the above theory, we discuss in detail the diffusion behaviour of an infectious disease (SARS) and present specific disease-control strategies in consideration of groups with different infection probabilities.Percolation (for reviews see Stauffer 1979 1 , Essam 1980 2 ) is the random occupation of sites or bonds on lattices or networks, named as site percolation or bond percolation, respectively. Site percolation is more general than bond percolation because every bond model may be reformulated as a site model (on a different graph) and the converse is in general not true 3 . The appeal of percolation is the occurrence of a critical phenomenon, which has attracted attention for a wide range of applications: liquid flows in porous media 4,5 , epidemic spread 6-8 , granular and composite materials 9-12 , forest fires 13-15 and fracture patterns and earthquakes in rocks 16 .The research originated from homogeneous percolation, i.e., percolation with a single occupation probability. For example, Fisher and Essam (1961) solved homogeneous percolation problems on Bethe lattices 17 ; Sykes and Essam (1964) studied the exact critical occupation probabilities in two dimensions 18 ;Gerald (et al., 2001) derived the value of critical occupation probabilities for a four-dimensional percolation problem on hyper-cubic lattices 19 .Building on those results, researchers began to study inhomogeneous percolation [20][21][22][23][24][25][26] , in which sites (or bonds) may have different occupation probabilities. In 1982, Kesten 20 obtained a critical surface of occupation probabilities for inhomogeneous percolation on square lattices. In 2013, Grimmett 21 extended the result of inhomogeneous bond percolation to triangular and hexagonal lattices utilizing Russo-Seymour-Welsh (RSW) theory of box-crossings. Recently, Radicchi 27 studied percolation on not necessarily infinite graphs and used graph decomposition methods to identify abrupt and continuous changes in percolation.In the current paper, we focus on inhomogeneous percolation on Bethe lattices. The interest in a good understanding of this inhomogeneous percolation process is twofold. Firstly, Bethe lattices might hold fractal st...

show abstract

“…Our method is based on the fact that, according to Eqs. (17), (22) and (23), the susceptibilities are determined explicitly by the mean size of the finite individual in-and out-components of vertices in the corresponding network parts, i.e., IN , OU T , and F . We found analytically the susceptibilities in directed uncorrelated random networks by use of the generating function method.…”

Section: Discussionmentioning

confidence: 99%

“…According to Eqs. (22) and (23), these susceptibilities are determined by the statistics of the individual finite in-and out-components of vertices in IN and OU T , respectively. The susceptibilities are related with the probability distribution functions Π …”

Section: (S)mentioning

confidence: 99%

Sensitivity of directed networks to the addition and pruning of edges and vertices

2017

View full text Add to dashboard Cite

We study the sensitivity of directed complex networks to the addition and pruning of edges and vertices and introduce the susceptibility, which quantifies this sensitivity. We show that topologically different parts of a directed network have different sensitivity to the addition and pruning of edges and vertices and, therefore, they are characterized by different susceptibilities. These susceptibilities diverge at the critical point of the directed percolation transition, signaling the appearance (or disappearance) of the giant strongly connected component in the infinite size limit. We demonstrate this behavior in randomly damaged real and synthetic directed complex networks, such as the World Wide Web, Twitter, the Caenorhabditis elegans neural network, directed Erdős-Rényi graphs, and others. We reveal a non-monotonous dependence of the sensitivity to random pruning of edges or vertices in the case of Caenorhabditis elegans and Twitter that manifests specific structural peculiarities of these networks. We propose the measurements of the susceptibilities during the addition or pruning of edges and vertices as a new method for studying structural peculiarities of directed networks.

show abstract

Weakly explosive percolation in directed networks

Cited by 19 publications

References 31 publications

Anomalous critical and supercritical phenomena in explosive percolation

Anomalous critical and supercritical phenomena in explosive percolation

How Inhomogeneous Site Percolation Works on Bethe Lattices: Theory and Application

Sensitivity of directed networks to the addition and pruning of edges and vertices

Contact Info

Product

Resources

About