Listing all triangles is a fundamental graph operation. Triangles can have important interpretations in real-world graphs, especially social and other interaction networks. Despite the lack of provably efficient (linear, or slightly super-linear) worst-case algorithms for this problem, practitioners run simple, efficient heuristics to find all triangles in graphs with millions of vertices. How are these heuristics exploiting the structure of these special graphs to provide major speedups in running time? We study one of the most prevalent algorithms used by practitioners. A trivial algorithm enumerates all paths of length @math , and checks if each such path is incident to a triangle. A good heuristic is to enumerate only those paths of length @math where the middle vertex has the lowest degree. It is easily implemented and is empirically known to give remarkable speedups over the trivial algorithm. We study the behavior of this algorithm over graphs with heavy-tailed degree distributions, a defining feature of real-world graphs. The erased configuration model (ECM) efficiently generates a graph with asymptotically (almost) any desired degree sequence. We show that the expected running time of this algorithm over the distribution of graphs created by the ECM is controlled by the @math -norm of the degree sequence. Norms of the degree sequence are a measure of the heaviness of the tail, and it is precisely this feature that allows non-trivial speedups of simple triangle enumeration algorithms. As a corollary of our main theorem, we prove expected linear-time performance for degree sequences following a power law with exponent @math , and non-trivial speedup whenever @math .
Configuration models for generating random graphs with given degree sequences have a long history. Bender and Canfield @cite_21 study this model for counting graphs with a given degree sequence. Wormald @cite_5 looks at the connectivity of these graphs. Molloy and Reed @cite_9 @cite_20 study various properties like the largest connected component of this graph distribution. Physicists studying complex networks have also paid attention to this model @cite_40 . Britton, Deijfen, and Martin-L " o f @cite_3 show that the simple graph generated by the ECM asymptotically matches the desired degree sequence. Aiello, Chung, and Lu @cite_10 give a model for power-law graphs, where edge @math are independent inserted with probability @math . This was studied for more general degree sequences in subsequent work by Chung, Lu, and Vu @cite_27 @cite_28 . Mihail and Papadimitriou @cite_15 independently discuss this model. Most of this work focused on eigenvalues and average distances in these graphs. Newman @cite_26 gives an excellent survey of these models, their similarities, and applications.
@cite_26 @cite_28 @cite_9 @cite_21 @cite_3 @cite_40 @cite_27 @cite_5 @cite_15 @cite_10 @cite_20
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Random graphs with a given degree sequence are a useful model capturing several features absent in the classical Erd˝os-Renyi model, such as dependent edges and non-binomial degrees. In this paper, we use a characterization due to Erd˝os and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows surprisingly ecient sequential importance sampling. Applications are given, in- cluding simulating a biological network and estimating the number of graphs with a given degree sequence. 1. Introduction. Random graphs with given vertex degrees have recently attracted great interest as a model for many real-world complex networks, including the World Wide Web, peer-to-peer networks, social networks, and biological networks. Newman (58) contains an excellent survey of these networks, with extensive references. A common approach to simulating these systems is to study (empirically or theoretically) the degrees of the vertices in instances of the network, and then to generate a random graph with the appropriate degrees. Graphs with prescribed degrees also appear in random matrix theory and string theory, which can call for large simulations based on random k-regular graphs. Throughout, we are concerned with generating simple graphs, i.e., no loops or multiple edges are allowed (the problem becomes considerably easier if loops and multiple edges are allowed). The main result of this paper is a new sequential importance sampling algorithm for generating random graphs with a given degree sequence. The idea is to build up the graph sequentially, at each stage choosing an edge from a list of candidates with probability proportional to the degrees. Most previously studied algorithms for this problem sometimes either get stuck or produce loops or multiple edges in the output, which is handled by starting over and trying again. Often for such algorithms, the probability of a restart being needed on a trial rapidly approaches 1 as the degree parameters grow, resulting in an enormous number of trials being needed on average to obtain a simple graph. A major advantage of our algorithm is that it never gets stuck. This is achieved using the Erd˝os- Gallai characterization, which is explained in Section 2, and a carefully chosen order of edge selection. 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 inhomogeneous 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 property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real-world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007 Consider the random graph models G(n, #edges = M) and G(n, Prob(edge) = p) with M = M(n) = (1 + )n-1 3)n 2 and p = p(n) = (1 +An-1 3) n . For 1 > -1 define an i-component of a random graph as a component which has exactly I more edges than vertices. Call an i-component with I > 1 a complex component. For both models, we show that when A is constant, the expected number of complex components is bounded, almost surely (a.s.) each of these components (if any exist) has size of order n2 3, and the maximum value of I is bounded in probability. We prove that a.s. the largest suspended tree in each complex component has size of order n2 3, and deletion of all suspended trees results in a "smoothed" graph of size of order n 3, with the maximum vertex degree 3. The total number of branching vertices, i.e., of degree 3, is bounded in probability. Thus, each complex component is almost surely topologically equivalent to a 3-regular multigraph of a uniformly bounded size. Lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n 3. We find a relatively simple integral formula for the limit distribution of the numbers of complex components, which implies, in particular, that all values of the "complexity spectrum" have positive limiting probabilities. We also answer questions raised by Erdos and Renyi back in 1960. It is proven that there exists p(A), the limiting planarity probability, with 0 < p(A) < 1, p(-oo) = 1, p(oo) = 0. In particular, G(n, M) (G(n, p), resp.) is almost surely nonplanar iff (M n 2)n-213 -* 00 ((np I)n-1 3) -+ oo, resp.). Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the worldwide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph. We investigate social networks of characters found in cultural works such as novels and films. These character networks exhibit many of the properties of complex networks such as skewed degree distribution and community structure, but may be of relatively small order with a high multiplicity of edges. Building on recent work of Beveridge and Shan [4], we consider graph extraction, visualization, and network statistics for three novels: Twilight by Stephanie Meyer, Steven King’s The Stand, and J.K. Rowling’s Harry Potter and the Goblet of Fire. Coupling with 800 character networks from films found in the http: moviegalaxies.com database, we compare the data sets to simulations from various stochastic complex networks models including random graphs with given expected degrees (also known as the Chung-Lu model), the configuration model, and the preferential attachment model. Using machine learning techniques based on motif (or small subgraph) counts, we determine that the Chung-Lu model best fits character networks and we conjecture why this may be the case. One of the most influential recent results in network analysis is that many natural networks exhibit a power-law or log-normal degree distribution. This has inspired numerous generative models that match this property. However, more recent work has shown that while these generative models do have the right degree distribution, they are not good models for real-life networks due to their differences on other important metrics like conductance. We believe this is, in part, because many of these real-world networks have very different joint degree distributions, that is, the probability that a randomly selected edge will be between nodes of degree k and l. Assortativity is a sufficient statistic of the joint degree distribution, and it has been previously noted that social networks tend to be assortative, while biological and technological networks tend to be disassortative. We suggest understanding the relationship between network structure and the joint degree distribution of graphs is an interesting avenue of further research. An important tool for such studies are algorithms that can generate random instances of graphs with the same joint degree distribution. This is the main topic of this article, and we study the problem from both a theoretical and practical perspective. We provide an algorithm for constructing simple graphs from a given joint degree distribution, and a Monte Carlo Markov chain method for sampling them. We also show that the state space of simple graphs with a fixed degree distribution is connected via endpoint switches. We empirically evaluate the mixing time of this Markov chain by using experiments based on the autocorrelation of each edge. These experiments show that our Markov chain mixes quickly on these real graphs, allowing for utilization of our techniques in practice. Random key graphs form a class of random intersection graphs that are naturally induced by the random key predistribution scheme of Eschenauer and Gligor for securing wireless sensor network (WSN) communications. Random key graphs have received much attention recently, owing in part to their wide applicability in various domains, including recommender systems, social networks, secure sensor networks, clustering and classification analysis, and cryptanalysis to name a few. In this paper, we study connectivity properties of random key graphs in the presence of unreliable links. Unreliability of graph links is captured by independent Bernoulli random variables, rendering them to be on or off independently from each other. The resulting model is an intersection of a random key graph and an Erdős–Renyi graph, and is expected to be useful in capturing various real-world networks; e.g., with secure WSN applications in mind, link unreliability can be attributed to harsh environmental conditions severely impairing transmissions. We present conditions on how to scale this model’s parameters so that: 1) the minimum node degree in the graph is at least @math and 2) the graph is @math -connected, both with high probability as the number of nodes becomes large. The results are given in the form of zero-one laws with critical thresholds identified and shown to coincide for both graph properties. These findings improve the previous results by Rybarczyk on @math -connectivity of random key graphs (with reliable links), as well as the zero-one laws by Yagan on one-connectivity of random key graphs with unreliable links. We consider the issue of protection in very large networks displaying randomness in topology. We employ random graph models to describe such networks, and obtain probabilistic bounds on several parameters related to reliability. In particular, we take the case of random regular networks for simplicity and consider the length of primary and backup paths in terms of the number of hops. First, for a randomly picked pair of nodes, we derive a lower bound on the average distance between the pair and discuss the tightness of the bound. In addition, noting that primary and protection paths form cycles, we obtain a lower bound on the average length of the shortest cycle around the pair. Finally, we show that the protected connections of a given maximum finite length are rare. We then generalize our network model so that different degrees are allowed according to some arbitrary distribution, and show that the second moment of degree over the first moment is an important shorthand for behavior of a network. Notably, we show that most of the results in regular networks carry over with minor modifications, which significantly broadens the scope of networks to which our approach applies. We present as an example the case of networks with a power-law degree distribution. We show that large deviation properties of Erdos-Renyi random graphs can be derived from the free energy of the q-state Potts model of statistical mechanics. More precisely the Legendre transform of the Potts free energy with respect to ln q is related to the component generating function of the graph ensemble. This generalizes the well-known mapping between typical properties of random graphs and the q→ 1 limit of the Potts free energy. For exponentially rare graphs we explicitly calculate the number of components, the size of the giant component, the degree distributions inside and outside the giant component, and the distribution of small component sizes. We also perform numerical simulations which are in very good agreement with our analytical work. Finally we demonstrate how the same results can be derived by studying the evolution of random graphs under the insertion of new vertices and edges, without recourse to the thermodynamics of the Potts model. Consider the Erdős–Renyi random graph on (n ) vertices where each edge is present independently with probability ( n ), with ( >0 ) fixed. For large (n ), a typical random graph locally behaves like a Galton–Watson tree with Poisson offspring distribution with mean ( ). Here, we study large deviations from this typical behavior within the framework of the local weak convergence of finite graph sequences. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported on trees. We also establish large deviations for other commonly studied random graph ensembles such as the uniform random graph with given number of edges growing linearly with the number of vertices, or the uniform random graph with given degree sequence. To prove our results, we introduce a new configuration model which allows one to sample uniform random graphs with a given neighborhood distribution, provided the latter is supported on trees. We also introduce a new class of unimodular random trees, which generalizes the usual Galton Watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth. These generalized Galton Watson trees turn out to be useful in the analysis of unimodular random trees and may be considered to be of interest in their own right. Many graphs arising in various information networks exhibit the "power law" behavior — the number of vertices of degree k is proportional to k −β for some positive β. We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1+ o(1)) √ m where m is the maximum degree. Moreover, the k largest eigenvalues of a random power law graph with exponent β have power law distribution with exponent 2β − 1 if the maximum degree is sufficiently large, where k is a function depending on β, m and d, the average degree. When 2< β < 2.5, the largest eigenvalue is heavily concentrated at cm 3−β for some constant c depending on β and the average degree. This result follows from a more general theorem which shows that the largest eigenvalue of a random graph with a given expected degree sequence is determined by m, the maximum degree, and ˜ d, the weighted average of the squares of the expected degrees. We show that the k-th largest eigenvalue is almost surely (1+ o(1)) √ m k where mk is the k-th largest expected degree provided mk is large enough. These results have implications on the usage of spectral techniques in many areas related to pattern detection and information retrieval.