The asymptotic number of labeled graphs with given degree sequences

We generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs. Among others, this class contains the Poissonian random graph, the expected degree random graph and the generalized random graph (including the classical Erdős-Rényi graph).In the paper we assign to each node a deterministic capacity and the probability that there exists an edge between a pair of nodes is equal to a function of the product of the capacities of the pair divided by the total capacity of all the nodes. We consider capacities which are such that the degrees of a node have uniformly bounded moments of order strictly larger than two, so that, in particular, the degrees have finite variance. We prove that the graph distance grows like log ν N , where the ν depends on the capacities and N denotes the size of the graph. In addition, the random fluctuations around this asymptotic mean log ν N are shown to be tight. We also consider the case where the capacities are independent copies of a positive random with P ( > x) ≤ cx 1−τ , for some constant c and τ > 3, again resulting in graphs where the degrees have finite variance.The method of proof of these results is to couple each member of the class to the Poissonian random graph, for which we then give the complete proof by adapting the arguments of van der Hofstad et al. (Random Struct. Algorithms 27(2):76-123, 2005).

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“…Let W (1) and W (2) be two independent copies of W in (1.15), then we can identify the limit random variables (R a ) a∈ (−1,0] as follows:…”

Section: Resultsmentioning

confidence: 99%

Universality for the Distance in Finite Variance Random Graphs

2008

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“….). We then construct the network describing the global structure of the population accoring to the 'configuration model' (see [12,13]). This model works by assigning each individual in the population a number of 'half-edges' (that individual's degree in the global network) according to independent samples from some distribution D with P(D = k) = p k (k = 0, 1, .…”

Section: Modelmentioning

confidence: 99%

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

Ball

Sirl

Trapman

2010

Mathematical Biosciences

134

148

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This paper is concerned with a stochastic SIR (susceptible → infective → removed) model for the spread of an epidemic amongst a population of individuals, with a random network of social contacts, that is also partitioned into households. The behaviour of the model as the population size tends to infinity in an appropriate fashion is investigated. A threshold parameter which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak is obtained, as are the probability that a major outbreak occurs and the expected proportion of the population that are ultimately infected by such an outbreak, together with methods for calculating these quantities. Monte Carlo simulations demonstrate that these asymptotic quantities accurately reflect the behaviour of finite populations, even for only moderately sized finite populations. The model is compared and contrasted with related models previously studied in the literature. The effects of the amount of clustering present in the overall population structure and the infectious period distribution on the outcomes of the model are also explored.

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“…The top curve in the graph depicts the highest measured diameter for each value of N , where the x axis is given 7 We did not analyze the average distance and connectivity for the experiments with N = 10, 000. in logarithmic scale. Note that this value does not necessarily increase when we increase the group size, and hence there are plateaus in this curve.…”

Section: Overlay Properties and Scalabilitymentioning

confidence: 99%

“…Steger and Wormald [61] propose a faster algorithm based on Bollobas's [11] and Bender and Canfield's [7] constructions. This algorithm creates a perfect matching that does not contain self-loops and parallel edges, and hence the resulting graph is always simple.…”

Section: Centralized Constructions Of K-regular Random Graphsmentioning

confidence: 99%

“…Previous algorithms for generating k-regular random graphs were centralized and static. For example, Bollobas [11] and Bender and Canfield [7] give a centralized construction of a k-regular random graph on N vertices, which works roughly as follows: it duplicates each vertex k times and creates a uniform random perfect matching 3 on these N k copies of vertices. The resulting graph contains an edge between two vertices, i and j, if the matching contains an edge between copies of i and j.…”

Section: Centralized Constructions Of K-regular Random Graphsmentioning

confidence: 99%

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Araneola: A scalable reliable multicast system for dynamic environments

Melamed

Keidar

2008

Journal of Parallel and Distributed Computing

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This paper presents Araneola 1 , a scalable reliable application-level multicast system for highly dynamic wide-area environments. Araneola supports multi-point to multi-point reliable communication in a fully distributed manner while incurring constant load (in terms of message and space complexity) on each node. For a tunable parameter k ≥ 3, Araneola constructs and dynamically maintains a basic overlay structure in which each node's degree is either k or k + 1, and roughly 90% of the nodes have degree k. Empirical evaluation shows that Araneola's basic overlay achieves three important mathematical properties of k-regular random graphs (i.e., random graphs in which each node has exactly k neighbors) with N nodes: (i) its diameter grows logarithmically with N ; (ii) it is generally k-connected; and (iii) it remains highly connected following random removal of linear-size subsets of edges or nodes. The overlay is constructed and maintained at a low cost: each join, leave, or failure is handled locally, and entails the sending of only about 3k messages in total, independent of N . Moreover, this cost decreases as the churn rate increases.The low degree of Araneola's basic overlay structure allows for allocating plenty of additional bandwidth for specific application needs. In this paper, we give an example for such a need -communicating with nearby nodes; we enhance the basic overlay with additional links chosen according to geographic proximity and available bandwidth. We show that this approach, i.e., a combination of random and nearby links, reduces the number of physical hops messages traverse without hurting the overlay's robustness as compared to completely random Araneola overlays (in which all the links are random) with the same average node degree.Given Araneola's overlay, we sketch out several message dissemination techniques that can be implemented on top of this overlay. We present a full implementation and evaluation of a gossip-based multicast scheme with up to 10,000 nodes. We show that compared to a (non-overlay-based) gossipbased multicast protocol, gossiping over Araneola achieves substantial improvements in load, reliability, and latency.

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The asymptotic number of labeled graphs with given degree sequences

Cited by 833 publications

References 2 publications

Universality for the Distance in Finite Variance Random Graphs

Universality for the Distance in Finite Variance Random Graphs

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

Araneola: A scalable reliable multicast system for dynamic environments

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