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26
The structure and function of complex networks
 SIAM REVIEW
, 2003
"... Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, ..."
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Cited by 1407 (9 self)
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Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the smallworld effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
A Random Graph Model for Massive Graphs
, 2000
"... We propose a random graph model which is a special case of sparse random graphs with given degree sequences. This model involves only a small number of parameters, called logsize and loglog growth rate. These parameters capture some universal characteristics of massive graphs. Furthermore, from the ..."
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Cited by 335 (26 self)
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We propose a random graph model which is a special case of sparse random graphs with given degree sequences. This model involves only a small number of parameters, called logsize and loglog growth rate. These parameters capture some universal characteristics of massive graphs. Furthermore, from these parameters, various properties of the graph can be derived. For example, for certain ranges of the parameters, we will compute the expected distribution of the sizes of the connected components which almost surely occur with high probability. We will illustrate the consistency of our model with the behavior of some massive graphs derived from data in telecommunications. We will also discuss the threshold function, the giant component, and the evolution of random graphs in this model.
The Size Of The Giant Component Of A Random Graph With A Given Degree Sequence
 COMBIN. PROBAB. COMPUT
, 2000
"... Given a sequence of nonnegative real numbers 0 ; 1 ; : : : which sum to 1, we consider a random graph having approximately i n vertices of degree i. In [12] the authors essentially show that if P i(i \Gamma 2) i ? 0 then the graph a.s. has a giant component, while if P i(i \Gamma 2) i ! 0 ..."
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Cited by 127 (0 self)
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Given a sequence of nonnegative real numbers 0 ; 1 ; : : : which sum to 1, we consider a random graph having approximately i n vertices of degree i. In [12] the authors essentially show that if P i(i \Gamma 2) i ? 0 then the graph a.s. has a giant component, while if P i(i \Gamma 2) i ! 0 then a.s. all components in the graph are small. In this paper we analyze the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine
Distributed Construction of Random Expander Networks
 In IEEE Infocom
, 2003
"... We present a novel distributed algorithm for constructing random overlay networks that are composed of d Hamilton cycles. The protocol is completely decentralized as no globallyknown server is required. The constructed topologies are expanders with O(log d n) diameter with high probability. ..."
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Cited by 77 (0 self)
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We present a novel distributed algorithm for constructing random overlay networks that are composed of d Hamilton cycles. The protocol is completely decentralized as no globallyknown server is required. The constructed topologies are expanders with O(log d n) diameter with high probability.
A Random Graph Model for Power Law Graphs
 Experimental Math
, 2000
"... We propose a random graph m del which is a special case of sparse random graphs with given degree sequences which satisfy a power law. Thism odel involves only asm all num ber of param eters, called logsize and loglog growth rate. These param eters capturesom e universal characteristics ofm assive ..."
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Cited by 73 (4 self)
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We propose a random graph m del which is a special case of sparse random graphs with given degree sequences which satisfy a power law. Thism odel involves only asm all num ber of param eters, called logsize and loglog growth rate. These param eters capturesom e universal characteristics ofm assive graphs. Furtherm re, from these paramfi ters, various properties of the graph can be derived. Forexam)(( for certain ranges of the paramJ?0CM we willcom?C7 the expected distribution of the sizes of the connectedcom onents which almJC surely occur with high probability. We will illustrate the consistency of our m del with the behavior of so m m ssive graphs derived from data in telecom unications. We will also discuss the threshold function, the giant com ponent, and the evolution of random graphs in thism del. 1
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
Randomly Sampling Molecules
 SIAM Journal on Computing
, 1996
"... We give the first polynomialtime algorithm for the following problem: Given a degree sequence in which each degree is bounded from above by a constant, select, uniformly at random, an unlabelled connected multigraph with the given degree sequence. We also give the first polynomialtime algorithm ..."
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Cited by 10 (2 self)
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We give the first polynomialtime algorithm for the following problem: Given a degree sequence in which each degree is bounded from above by a constant, select, uniformly at random, an unlabelled connected multigraph with the given degree sequence. We also give the first polynomialtime algorithm for the following related problem: Given a molecular formula, select, uniformly at random, a structural isomer having the given formula.
Random Regular Graphs of High Degree
"... Random dregular graphs have been well studied when d is fixed and the number of vertices goes to infinity. We obtain results on many of the properties of a random dregular graph when d = d(n) grows more quickly than p n. These properties include connectivity, hamiltonicity, independent set size, c ..."
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Cited by 9 (5 self)
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Random dregular graphs have been well studied when d is fixed and the number of vertices goes to infinity. We obtain results on many of the properties of a random dregular graph when d = d(n) grows more quickly than p n. These properties include connectivity, hamiltonicity, independent set size, chromatic number, choice number, and the size of the second eigenvalue, amongst others.
Weakly Pancyclic Graphs
, 1996
"... In generalizing the concept of a pancyclic graph, we say that a graph is `weakly pancyclic' if it contains cycles of every length between the length of a shortest and a longest cycle. In this paper it is shown that in many cases the requirements on a graph which ensure that it is weakly pancyclic ar ..."
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Cited by 6 (0 self)
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In generalizing the concept of a pancyclic graph, we say that a graph is `weakly pancyclic' if it contains cycles of every length between the length of a shortest and a longest cycle. In this paper it is shown that in many cases the requirements on a graph which ensure that it is weakly pancyclic are considerably weaker than those required to ensure that it is pancyclic. This sheds some light on the content of a famous metaconjecture of Bondy. From the main result of this paper it follows that 2connected nonbipartite graphs of sufficiently large order n with minimum degree exceeding 2n/7 are weakly pancyclic; and that graphs with minimum degree at least n/4 + 250 are pancyclic, if they contain both a triangle and a hamiltonian cycle.
Computation in Permutation Groups: Counting and Randomly Sampling Orbits
"... Let be a finite set and let G be a permutation group acting on The permutation group G partitions into orbits. This survey focuses on three related computational problems, each of which is defined with respect to a particular input set I. The problems, given an input ( ; G) 2 I, are (1) count the or ..."
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Cited by 6 (0 self)
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Let be a finite set and let G be a permutation group acting on The permutation group G partitions into orbits. This survey focuses on three related computational problems, each of which is defined with respect to a particular input set I. The problems, given an input ( ; G) 2 I, are (1) count the orbits (exactly), (2) approximately count the orbits, and (3) choose an orbit uniformly at random. The goal is to quantify the computational diculty of the problems. In particular, we would like to know for which input sets I the problems are tractable.