We study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in article amsmath,amsfonts empty . We focus on the cases when an edge between x and y is added with probability decaying with the Euclidean distance as |x − y|−s+o(1) when |x − y| → ∞. For s ∈ (d, 2d) we show that the graph diameter for the graph reduced to a box of side L scales like (log L)Δ+o(1) where Δ−1 := log2(2d s). In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance L. We also show that a ball of radius r in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius exp r1 Δ+o(1) in the Euclidean metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 210-227, 2011 (Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.)
Recently, long-range percolation has been invoked as a fruitful source of graphs with non-trivial growth properties. Our interest was stirred by the work of Benjamini and Berger @cite_13 who proposed (and studied) long-range percolation as a model of social networks. It is this context where the graph distance scaling, and volume growth, are particularly of much interest. Thanks to numerous contributions that followed @cite_13 , this scaling is now known for most values of @math and @math . Explicitly, for @math , a corollary to the main result of Benjamini, Kesten, Peres and Schramm @cite_0 asserts that almost surely. As @math , the right-hand side tends to infinity and so, at @math , we expect @math . And, indeed, the precise growth rate in this case has been established by Coppersmith, Gamarnik and Sviridenko @cite_12 , where @math '' means that the ratio of left and right-hand side is a random variable that is bounded away from zero and infinity with probability tending to one.
@cite_0 @cite_13 @cite_12
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We consider long-range percolation in dimension @math , where distinct sites @math and @math are connected with probability @math . Assuming that @math is translation invariant and that @math with @math , we show that the graph distance is at least linear with the Euclidean distance. In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one non-trivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale governs the cross-over from large- to small-world behavior in the model, as well as the number of vertices in a neighborhood of given radius on the network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the network, and, using series expansion and Pade approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of small-world graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on small-world networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model. Appears in Phys. Rev. E 60, 7332-7342 (1999). Bounds for the diameter and expansion of the graphs created by long-range percolation on the cycle ℤ Nℤ are given. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 102–111, 2001 We consider the following long-range percolation model: an undirected graph with the node set 0, 1, ..., N d, has edges (x, y) selected with probability ≈ β ||x -y||s if ||x - y|| ' η1 > η2 > 1, it is at most Nη2 when s = 2d, and is at least Nη1 when d = 1, s = 2, β > 1 or when s > 2d. We also provide a simple proof that the diameter is at most log O(1) N with high probability, when d > s > 2d, established previously in [2]. In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs (G n ). Let λn be the largest eigenvalue of the adjacency matrix of G n , and let G n (p n ) be the random subgraph of G n obtained by keeping each edge independently with probability p n . We show that the appearance of a giant component in G n (p n ) has a sharp threshold at p n = 1 λ n . In fact, we prove much more: if (G n ) converges to an irreducible limit, then the density of the largest component of G n (c n) tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lovasz, S6s and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollobas, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs. The percolation phase transition and the mechanism of the emergence of the giant component through the critical scaling window for random graph models, has been a topic of great interest in many dierent communities ranging from statistical physics, combinatorics, computer science, social networks and probability theory. The last few years have witnessed an explosion of models which couple random aggregation rules, that specify how one adds edges to existing congurations, and choice, wherein one selects from a ‘limited’ set of edges at random to use in the conguration. These models exhibit fascinating new phenomenon, ranging from delaying or speeding up the emergence of the giant component, to percolation", where the diameter of the scaling window is several orders of magnitude smaller than that for standard random graph models. While an intense study of such models has ensued, understanding the actual emergence of the giant component and merging dynamics in the critical scaling window has remained impenetrable to a rigorous analysis. In this work we take an important step in the analysis of such models by studying one of the standard examples of such processes, namely the Bohman-Frieze model, and provide rst results on the asymptotic dynamics, through the critical scaling window, that lead to the emergence of the giant component for such models. We identify the scaling window and show that through this window, the component sizes properly rescaled converge to the standard multiplicative coalescent. Proofs hinge on a careful analysis of certain innite-type branching processes with types taking values in the space of RCLL paths, and stochastic analytic techniques to estimate susceptibility functions of the components all the way through the scaling window where these functions explode. Previous approaches for analyzing random graphs at criticality have relied largely on classical breadth-rst search techniques that exploit asymptotic connections with Brownian excursions. For dynamic random graph models evolving via general Markovian rules, such approaches fail and we develop a quite dierent set of tools that can potentially be used for the study of critical dynamics for all bounded size rules. We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied in Le Jan (C R Math Acad Sci Paris 350(13–14):643–646, 2012, Ill J Math 57(2):525–558, 2013). It is a model with long range correlations with two parameters ( ) and ( ). The non-negative parameter ( ) measures the amount of loops, and ( ) plays the role of killing on vertices penalizing ( ( >0 )) or favoring ( ( <0 )) appearance of large loops. It was shown in Le Jan (Ill J Math 57(2):525–558, 2013) that for any fixed ( ) and large enough ( ), there exists an infinite cluster in the loop percolation on ( Z ^d ). In the present article, we show a non-trivial phase transition on the integer lattice ( Z ^d ) ( (d 3 )) for ( =0 ). More precisely, we show that there is no loop percolation for ( =0 ) and ( ) small enough. Interestingly, we observe a critical like behavior on the whole sub-critical domain of ( ), namely, for ( =0 ) and any sub-critical value of ( ), the probability of one-arm event decays at most polynomially. For (d 5 ), we prove that there exists a non-trivial threshold for the finiteness of the expected cluster size. For ( ) below this threshold, we calculate, up to a constant factor, the decay of the probability of one-arm event, two point function, and the tail distribution of the cluster size. These rates are comparable with the ones obtained from a single large loop and only depend on the dimension. For (d=3 ) or 4, we give better lower bounds on the decay of the probability of one-arm event, which show importance of small loops for long connections. In addition, we show that the one-arm exponent in dimension 3 depends on the intensity ( ). Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability @math , independently of each other, and a deterministic spreading rule with a fixed parameter @math : if a vacant site has at least @math occupied neighbours at a certain time step, then it becomes occupied in the next step. This process is well studied on @math ; here we investigate it on regular and general infinite trees and on non-amenable Cayley graphs. The critical probability is the infimum of those values of @math for which the process achieves complete occupation with positive probability. On trees we find the following discontinuity: if the branching number of a tree is strictly smaller than @math , then the critical probability is 1, while it is @math on the @math -ary tree. A related result is that in any rooted tree @math there is a way of erasing @math children of the root, together with all their descendants, and repeating this for all remaining children, and so on, such that the remaining tree @math has branching number @math . We also prove that on any @math -regular non-amenable graph, the critical probability for the @math -rule is strictly positive. Theoretical attempts proposed so far to describe ordinary percolation processes on real-world networks rely on the locally treelike ansatz. Such an approximation, however, holds only to a limited extent, because real graphs are often characterized by high frequencies of short loops. We present here a theoretical framework able to overcome such a limitation for the case of site percolation. Our method is based on a message passing algorithm that discounts redundant paths along triangles in the graph. We systematically test the approach on 98 real-world graphs and on synthetic networks. We find excellent accuracy in the prediction of the whole percolation diagram, with significant improvement with respect to the prediction obtained under the locally treelike approximation. Residual discrepancies between theory and simulations do not depend on clustering and can be attributed to the presence of loops longer than three edges. We present also a method to account for clustering in bond percolation, but the improvement with respect to the method based on the treelike approximation is much less apparent. We study the number @math of open paths of length @math in supercritical oriented percolation on @math , with @math . We prove that on the percolation event @math , @math almost surely converges to a positive deterministic constant. We also study the existence of directional limits. The proof relies on the introduction of adapted sequences of regenerating times, on subadditive arguments and on the properties of the coupled zone in supercritical oriented percolation. In this paper, we provide general conditions on a one parameter family of random infinite subsets of Zd to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distance. In addition, we show that these conditions also imply a shape theorem for the corresponding infinite connected component. By verifying these conditions for specific models, we obtain novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. As a byproduct, we obtain alternative proofs to the corresponding results for random interlacements in the work of Cerný and Popov [“On the internal distance in the interlacement set,” Electron. J. Probab. 17(29), 1–25 (2012)], and while our main interest is in percolation models with long-range correlations, we also recover results in the spirit of the work of Antal and Pisztora [“On the chemical distance for supercritical Bernoulli percolation... The problem of long-range percolation in one dimension is proposed. The authors consider a one-dimensional bond percolation system with bonds connecting an infinite number of neighbours where the occupation probability for the nth nearest-neighbour bond pn varies as p1 ns. Using the transfer-matrix method, they find that when s>2 only the short-range percolation exists; namely the system percolates only when p1=1. A transition to long-range percolation is found at s=2 where the percolation threshold drops suddenly from the short-range value p1c=1 to the long-range value p1c=0. Many complex networks in nature have directed links, a property that affects the network’s navigability and large-scale topology. Here we study the percolation properties of such directed scale-free networks with correlated in and out degree distributions. We derive a phase diagram that indicates the existence of three regimes, determined by the values of the degree exponents. In the first regime we regain the known directed percolation mean field exponents. In contrast, the second and third regimes are characterized by anomalous exponents, which we calculate analytically. In the third regime the network is resilient to random dilution, i.e., the percolation threshold is pc!1. Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike. For a natural number k, define an oriented site percolation on ℤ2 as follows. Let x i , y j be independent random variables with values uniformly distributed in 1, …, k . Declare a site (i, j) ∈ℤ2closed if x i = y j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i0, j0)(i1, j1) · · · such that 0 = i0≤i1≤· · ·, 0 = j0≤j1≤· · ·, and each site (i n , j n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x i ) and (y j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods. We consider a growing network, whose growth algorithm is based on the preferential attachment typical for scale-free constructions, but where the long-range bonds are disadvantaged. Thus, the probability of getting connected to a site at distance d is proportional to @math where is a tunable parameter of the model. We show that the properties of the networks grown with @math are close to those of the genuine scale-free construction, while for @math the structure of the network is quite different. Thus, in this regime, the node degree distribution is no longer a power law, and it is well represented by a stretched exponential. On the other hand, the small-world property of the growing networks is preserved at all values of . We consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on @math vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy @math vertices, for some @math . The value of @math is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. The motivation for this work comes from the study of viral marketing on social networks. The described processes can be viewed as two competing products spreading through a social network (random regular graph). Considering the processes which grow at different rates (corresponding to different attraction levels of the two products) or starting at different times (the first to market advantage) allows to model aspects of real competition. The results obtained can be interpreted as one of the two products taking the lion share of the market. We compare these results to the same process run on @math dimensional grids where we show that in the generic situation the two products will have a linear fraction of the market each. We prove large deviation estimates at the correct order for the graph distance of two sites lying in the same cluster of an independent percolation process. We improve earlier results of Gartner and Molchanov and Grimmett and Marstrand and answer affirmatively a conjecture of Kozlov. We study signal-to-interference plus noise ratio (SINR) percolation for Cox point processes, i.e., Poisson point processes with a random intensity measure. SINR percolation was first studied by in the case of a two-dimensional Poisson point process. It is a version of continuum percolation where the connection between two points depends on the locations of all points of the point process. Continuum percolation for Cox point processes was recently studied by Hirsch, Jahnel and Cali. We study the SINR graph model for a stationary Cox point process in two or higher dimensions. We show that under suitable moment or boundedness conditions on the path-loss function and the intensity measure, this graph has an infinite connected component if the spatial density of points is large enough and the interferences are sufficiently reduced (without vanishing). This holds in all dimensions larger than 1 if the intensity measure is asymptotically essentially connected, and in two dimensions also if the intensity measure is only stabilizing but the connection radius is large. A prominent example of the intensity measure is the two-dimensional Poisson-Voronoi tessellation. We show that its total edge length in a given square has some exponential moments. We conclude that its SINR graph has an infinite cluster if the path-loss function is bounded and has a power-law decay of exponent at least 3. We study percolation on networks, which is used as a model of the resilience of networked systems such as the Internet to attack or failure and as a simple model of the spread of disease over human contact networks. We reformulate percolation as a message passing process and demonstrate how the resulting equations can be used to calculate, among other things, the size of the percolating cluster and the average cluster size. The calculations are exact for sparse networks when the number of short loops in the network is small, but even on networks with many short loops we find them to be highly accurate when compared with direct numerical simulations. By considering the fixed points of the message passing process, we also show that the percolation threshold on a network with few loops is given by the inverse of the leading eigenvalue of the so-called non-backtracking matrix.