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 small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

In a previous paper we showed that a random 4-regular graph asymptotically almost surely (a.a.s.) has chromatic number 3. Here we extend the method to show that a random 6-regular graph asymptotically almost surely (a.a.s.) has chromatic number 4 and that the chromatic number of a random d-regular graph for other d between 5 and 10 inclusive is a.a.s. restricted to a range of two integer values: {3, 4} for d = 5, {4, 5} for d = 7, 8, 9, and {5, 6} for d = 10. The proof uses efficient algorithms which a.a.s. colour these random graphs using the number of colours specified by the upper bound. These algorithms are analysed using the differential equation method, including an analysis of certain systems of differential equations with discontinuous right hand sides. 1

Given a sequence of non-negative 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

The k -core of a graph is the largest subgraph with minimum degree at least k . For the Erdos-R'enyi random graph G(n; m) on n vertices, with m edges, it is known that a giant 2-core grows simultaneously with a giant component, that is when m is close to n=2 . We show that for k 3 , with high probability, a giant k -core appears suddenly when m reaches c k n=2 ; here c k = min ?0 = k () and k () = PfPoisson() k \Gamma 1g . In particular, c 3 3:35 . We also demonstrate that, unlike the 2-core, when a k -core appears for the first time it is very likely to be giant, of size p k ( k )n . Here k is the minimum point of = k () and p k ( k ) = PfPoisson( k ) kg . For k = 3 , for instance, the newborn 3-core contains about 0:27n vertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always finds a k -core if the graph has one. 1991 Mathematics Subject Classification. Primary 05C80, 05C85, 60C05; Secondary 60F10, 60G42, 60J10.

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, power-law 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

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this paper we examine an algorithm which, although it does not generate uniformly at random, is provably close to a uniform generator when the degrees are relatively small. Moreover, it is easy to implement and quite fast in practice. The most interesting case is the regular one, when all degrees are equal to d = d(n) say. Moreover, methods for the regular case of this problem usually extend to arbitrary degree sequences, although the analysis can become more complicated and it may be needed to impose restrictions on the variation in the degrees (such as is analyzed by Jerrum et al. [4]). The rst algorithm for generating d-regular graphs uniformly at random was implicit in the paper of Bollobas [2] and also in the approaches to counting regular graphs by Bender and Caneld [1] and in [13] (see also [14] for explicit algorithms). The

strategies for the distribution of objects among a heterogeneous set of servers. Ideally, such a strategy should allow the computation of the position of an object with a low time and space complexity, and it should be able to adapt with a near-minimum amount of replacements of objects to changes in the capabilities of the servers so that objects are always distributed among the servers according to their capabilities. Previous techniques are able to handle these requirements only in part. For example, standard hashing techniques can be used to achieve a non-uniform distribution of objects among a set of servers and the time and space efficient computation of the position of the objects, but they usually do not adapt well to a change in the capabilities. We present two strategies based on hashing that achieve all of the goals above. Furthermore, we give a list of applications for these strategies demonstrating that they can be used efficiently for distributed data management, web caches, and adaptive random graphs, which may be of interest for peer-to-peer networks.

) Ravi Kannan Prasad Tetali y Santosh Vempala z Abstract We consider two problems: randomly generating labeled bipartite graphs with a given degree sequence and randomly generating labeled tournaments with a given score sequence. We analyze simple Markov chains for both problems. For the first problem, we cannot prove that our chain is rapidly mixing in general, but in the (near-) regular case, i.e. when all the degrees are (almost) equal, we give a proof of rapid mixing. Our methods also apply to the corresponding problem for general (nonbipartite) regular graphs which was studied earlier by several researchers. One significant difference in our approach is that our chain has one state for every graph (or bipartite graph) with the given degree sequence; in particular, there are no auxiliary states as in the chain used by Jerrum and Sinclair. For the problem of generating tournaments, we are able to prove that our Markov chain on tournaments is rapidly mixing, if the score seque...

. The asymptotic distribution of the number of Hamilton cycles in a random regular graph is determined. The limit distribution is of an unusual type; it is the distribution of a variable whose logarithm can be written as an infinite linear combination of independent Poisson variables, and thus the logarithm has an infinitely divisible distribution with a certain discrete L'evy measure. Similar results are found for some related problems. These limit results imply that some different models of random regular graphs are contiguous, which means that they are qualitatively asymptotically equivalent. For example, if r 3, then the usual (uniformly distributed) random r-regular graph is contiguous to the one constructed by taking the union of r perfect matchings on the same vertex set (assumed to be of even cardinality), conditioned on there being no multiple edges. Some consequences of contiguity for asymptotic distributions are discussed. 0. Introduction In two remarkable papers, Robinso...