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43
The phase transition in inhomogeneous random graphs, preprint available from http://www.arxiv.org/abs/math.PR/0504589
"... Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, powerlaw degree distributions. ..."
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Cited by 98 (30 self)
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Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, powerlaw degree distributions. Thus there has been a lot of recent interest in defining and studying ‘inhomogeneous ’ random graph models. One of the most studied properties of these new models is their ‘robustness’, or, equivalently, the ‘phase transition ’ as an edge density parameter is varied. For G(n, p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogenous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this already, and others can be approximated by models with
Coalescent Random Forests
 J. COMBINATORIAL THEORY A
, 1998
"... Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ..."
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Cited by 38 (18 self)
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Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...
The Birth of the Giant Component
, 1993
"... Limiting distributions are derived for the sparse connected components that are present when a random graph on n vertices has approximately 1 n edges. In particular, we show that such a graph consists entirely of trees, 2 unicyclic components, and bicyclic components with probability approaching ..."
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Cited by 31 (5 self)
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Limiting distributions are derived for the sparse connected components that are present when a random graph on n vertices has approximately 1 n edges. In particular, we show that such a graph consists entirely of trees, 2 unicyclic components, and bicyclic components with probability approaching
Random subgraphs of finite graphs: I. The scaling window under the triangle condition
, 2003
"... We study random subgraphs of an arbitrary finite connected transitive graph G obtained by independently deleting edges with probability 1−p. Let V be the number of vertices in G, and let Ω be their degree. We define the critical threshold pc = pc(G,λ) to be the value of p for which the expected clus ..."
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Cited by 28 (10 self)
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We study random subgraphs of an arbitrary finite connected transitive graph G obtained by independently deleting edges with probability 1−p. Let V be the number of vertices in G, and let Ω be their degree. We define the critical threshold pc = pc(G,λ) to be the value of p for which the expected cluster size of a fixed vertex attains the value λV 1/3, where λ is fixed and positive. We show that for any such model, there is a phase transition at pc analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold pc. In particular, we show that the largest cluster inside a scaling window of size p −pc  = Θ(Ω −1 V −1/3) is of size Θ(V 2/3), while below this scaling window, it is much smaller, of order O(ǫ −2 log(V ǫ 3)), with ǫ = Ω(pc −p). We also obtain an upper bound O(Ω(p − pc)V) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order Θ(Ω(p − pc)). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the ncube and certain Hamming cubes, as well as the spreadout ndimensional torus for n> 6.
The Birth Of The Infinite Cluster: FiniteSize Scaling In Percolation
 Commun. Math. Phys
"... We address the question of finitesize scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d = 2, we obtain a complete characterization of finitesize scaling. In dimensions d > 2, we establish the same results ..."
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Cited by 26 (6 self)
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We address the question of finitesize scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d = 2, we obtain a complete characterization of finitesize scaling. In dimensions d > 2, we establish the same results under a set of hypotheses related to socalled scaling and hyperscaling postulates which are widely believed to hold up to d = 6.
Random subgraphs of finite graphs: III. The phase transition for the ncube
, 2003
"... We study random subgraphs of the ncube , where nearestneighbor edges are occupied with probability p. Let p c (n) be the value of p for which the expected cluster size of a fixed vertex attains the value #2 , where # is a small positive constant. Let # = n(p p c (n)). In two previous ..."
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Cited by 23 (11 self)
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We study random subgraphs of the ncube , where nearestneighbor edges are occupied with probability p. Let p c (n) be the value of p for which the expected cluster size of a fixed vertex attains the value #2 , where # is a small positive constant. Let # = n(p p c (n)). In two previous papers, we showed that the largest cluster inside a scaling window given by # = #(2 n/3 ) is of size #(2 2n/3 ), below this scaling window it is at most 2(log 2)n# 2 , and above this scaling window it is at most O(#2 ). In this paper, we prove that for p the size of the largest cluster is at least #(#2 ), which is of the same order as the upper bound. This provides an understanding of the phase transition that goes far beyond that obtained by previous authors. The proof is based on a method that has come to be known as "sprinkling," and relies heavily on the specific geometry of the ncube.
Counting connected graphs insideout
 J. Comb. Th. B
, 2005
"... The theme of this work is an “insideout ” approach to the enumeration of graphs. It is based on a wellknown decomposition of a graph into its 2core, i.e. the largest subgraph of minimum degree 2 or more, and a forest of trees attached. Using our earlier (asymptotic) formulae for the total number ..."
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Cited by 18 (6 self)
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The theme of this work is an “insideout ” approach to the enumeration of graphs. It is based on a wellknown decomposition of a graph into its 2core, i.e. the largest subgraph of minimum degree 2 or more, and a forest of trees attached. Using our earlier (asymptotic) formulae for the total number of 2cores with a given number of vertices and edges, we solve the corresponding enumeration problem for the connected 2cores. For a subrange of the parameters, we also enumerate those 2cores by using a deeper insideout notion of a kernel of a connected 2core. Using this enumeration result in combination with Caley’s formula for forests, we obtain an alternative and simpler proof of the asymptotic formula of Bender, Canfield and McKay for the number of connected graphs with n vertices and m edges, with improved error estimate for a range of m values. As another application, we study the limit joint distribution of three parameters of the giant component of a random graph with n vertices in the supercritical phase, when the difference between average vertex degree and 1 far exceeds n −1/3. The three parameters are defined in terms of the 2core of the giant component, i.e. its largest subgraph of minimum degree 2 or more. They are the number of vertices in the 2core, the excess (#edges − #vertices) of the 2core, and the number of vertices not in the 2core. We show that the limit distribution is jointly Gaussian throughout the whole supercritical phase. In particular, for the first time, the 2core size is shown to be asymptotically normal, in the widest possible range of the average vertex degree. 1
The Size of the Largest Strongly Connected Component of a Random Digraph With a Given Degree Sequence
, 2002
"... We give results on the strong connectivity for spaces of sparse random digraphs specified by degree sequence. A full characterization is provided, in probability, of the fanin and fanout of all vertices including the number of vertices with small (o(n)) and large (cn) fanin or fanout. We also gi ..."
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Cited by 17 (7 self)
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We give results on the strong connectivity for spaces of sparse random digraphs specified by degree sequence. A full characterization is provided, in probability, of the fanin and fanout of all vertices including the number of vertices with small (o(n)) and large (cn) fanin or fanout. We also give the size of the giant strongly connected component, if any, and the structure of the bowtie digraph induced by the vertices with large fanin or fanout. Our results follow a direct analogy of the extinction probabilities of classical branching processes.
Phase transition and finitesize scaling for the integer partitioning problem
, 2001
"... Dedicated to D. E. Knuth on the occasion of his 64th birthday. Abstract. We consider the problem of partitioning n randomly chosen integers between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if ..."
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Cited by 17 (2 self)
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Dedicated to D. E. Knuth on the occasion of his 64th birthday. Abstract. We consider the problem of partitioning n randomly chosen integers between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if the optimum discrepancy is 0 when the sum of all n integers in the original set is even, or 1 when the sum is odd. Parameterizing the random problem in terms of κ = m/n, we prove that the problem has a phase transition at κ = 1, in the sense that for κ < 1, there are many perfect partitions with probability tending to 1 as n → ∞, while for κ> 1, there are no perfect partitions with probability tending to 1. Moreover, we show that this transition is firstorder in the sense the derivative of the socalled entropy is discontinuous at κ = 1. We also determine the finitesize scaling window about the transition point: κn = 1 − (2n) −1 log 2 n + λn/n, by showing that the probability of a perfect partition tends to 1, 0, or some explicitly computable p(λ) ∈ (0, 1), depending on whether λn tends to −∞, ∞, or λ ∈ (−∞, ∞), respectively. For λn → − ∞ fast enough, we show that the number of perfect partitions is Gaussian in the limit. For λn → ∞, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is Θ(2 λn). Within the window, i.e., if λn  is bounded, we prove that the optimum discrepancy is bounded. Both for λn → ∞ and within the window, we find the limiting distribution of the (scaled) discrepancy. Finally, both for the integer partitioning problem and for the continuous partitioning problem, we find the joint distribution of the k smallest discrepancies above the scaling window.