The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform

Kevin E Bassler - One of the best experts on this subject based on the ideXlab platform.

  • efficient and exact sampling of simple graphs with given arbitrary degree sequence
    PLOS ONE, 2010
    Co-Authors: Charo I Del Genio, Hyunju Kim, Zoltan Toroczkai, Kevin E Bassler
    Abstract:

    Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet Modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large , and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.

Charo I Del Genio - One of the best experts on this subject based on the ideXlab platform.

  • efficient and exact sampling of simple graphs with given arbitrary degree sequence
    PLOS ONE, 2010
    Co-Authors: Charo I Del Genio, Hyunju Kim, Zoltan Toroczkai, Kevin E Bassler
    Abstract:

    Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet Modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large , and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.

Svante Janson - One of the best experts on this subject based on the ideXlab platform.

  • Component structure of the Configuration Model: barely supercritical case
    Random Structures & Algorithms, 2019
    Co-Authors: Remco Van Der Hofstad, Svante Janson, Malwina J. Luczak
    Abstract:

    We study near-critical behavior in the Configuration Model. Let D-n be the degree of a random vertex and nu n=E[Dn(Dn-1)]/E[Dn]; we consider the barely supercritical regime, where nu(n)-> 1 as n ...

  • component structure of the Configuration Model barely supercritical case
    arXiv: Probability, 2016
    Co-Authors: Remco Van Der Hofstad, Svante Janson, Malwina J. Luczak
    Abstract:

    We study near-critical behavior in the Configuration Model. Let $D_n$ be the degree of a random vertex. We let $\nu_n={\mathbb E} [D_n(D_n-1)]/{\mathbb E}[D_n]$ and, assuming that $\nu_n \to 1$ as $n \to \infty$, we write $\varepsilon_n=\nu_n-1$. We call the setting where $\varepsilon_n n^{1/3}/({\mathbb E}[D_n^3])^{2/3} \to \infty$ the {\it barely supercritical} regime. We further assume that the variance of $D_n$ is uniformly bounded as $n \to \infty$. Let $D_n^*$ denote the size-biased version of $D_n$. We prove that there is a unique giant component of size $n \rho_n {\mathbb E} D_n (1+o(1))$, where $\rho_n$ denotes the survival probability of a branching process with offspring distribution $D_n^*-1$. This extends earlier results of Janson and Luczak~\cite{JanLuc07}, as well as those of Janson, Luczak, Windridge and House~\cite{SJ300} to the case where the third moment of $D_n$ is unbounded, filling the gap in the literature. We further study the size of the largest component in the \emph{critical} regime, where $\varepsilon_n = O(n^{-1/3} ({\mathbb E} D_n^3)^{2/3})$, extending and complementing results of Hatami and Molloy~\cite{HatamiMolloy}.

  • the greedy independent set in a random graph with given degrees
    arXiv: Probability, 2015
    Co-Authors: Graham Brightwell, Svante Janson, Malwina J. Luczak
    Abstract:

    We analyse the size of an independent set in a random graph on $n$ vertices with specified vertex degrees, constructed via a simple greedy algorithm: order the vertices arbitrarily, and, for each vertex in turn, place it in the independent set unless it is adjacent to some vertex already chosen. We find the limit of the expected proportion of vertices in the greedy independent set as $n \to \infty$, expressed as an integral whose upper limit is defined implicitly, valid whenever the second moment of a random vertex degree is uniformly bounded. We further show that the random proportion of vertices in the independent set converges to the jamming constant as $n \to \infty$. The results hold under weaker assumptions in a random multigraph with given degrees constructed via the Configuration Model.

  • the probability that a random multigraph is simple ii
    Journal of Applied Probability, 2014
    Co-Authors: Svante Janson
    Abstract:

    Consider a random multigraph G * with given vertex degrees d 1 ,…, d n , constructed by the Configuration Model. We give a new proof of the fact that, asymptotically for a sequence of such multigraphs with the number of edges the probability that the multigraph is simple stays away from 0 if and only if The new proof uses the method of moments, which makes it possible to use it in some applications concerning convergence in distribution. Corresponding results for bipartite graphs are included.

  • the probability that a random multigraph is simple ii
    arXiv: Combinatorics, 2013
    Co-Authors: Svante Janson
    Abstract:

    Consider a random multigraph with given vertex degrees constructed by the Configuration Model. We give a new proof of the fact that, asymptotically for a sequence of such multigraphs with the number of edges tending to infinity, the probability that the multigraph is simple stays away from 0 if and only if $\sum d_i^2 = O(\sum d_i)$, where $d_i$ are the vertex degrees. The new proof uses the method of moments, which makes it possible to use it in some applications concerning convergence in distribution. Corresponding results for bipartite graphs are included.

Julia Komjathy - One of the best experts on this subject based on the ideXlab platform.

  • fixed speed competition on the Configuration Model with infinite variance degrees unequal speeds
    Electronic Journal of Probability, 2015
    Co-Authors: Enrico Baroni, Remco Van Der Hofstad, Julia Komjathy
    Abstract:

    We study competition of two spreading colors starting from single sources on the Configuration Model with i.i.d. degrees following a power-law distribution with exponent $\tau\in (2,3)$. In this Model two colors spread with a fixed but not necessarily equal speedon the unweighted random graph.  We show that if the speeds are not equal, then the faster color paints almost all vertices, while the slower color can paint only a random subpolynomial fraction of the vertices.We investigate the case when the speeds are equal and typical distances in a follow-up paper.

Remco Van Der Hofstad - One of the best experts on this subject based on the ideXlab platform.

  • critical percolation on scale free random graphs new universality class for the Configuration Model
    Communications in Mathematical Physics, 2021
    Co-Authors: Souvik Dhara, Remco Van Der Hofstad, Johan S H Van Leeuwaarden
    Abstract:

    In this paper, we study the critical behavior of percolation on a Configuration Model with degree distribution satisfying an infinite second-moment condition, which includes power-law degrees with exponent $$\tau \in (2,3)$$ . It is well known that, in this regime, many canonical random graph Models, such as the Configuration Model, are robust in the sense that the giant component is not destroyed when the percolation probability stays bounded away from zero. Thus, the critical behavior is observed when the percolation probability tends to zero with the network size, despite of the fact that the average degree remains bounded. In this paper, we initiate the study of critical random graphs in the infinite second-moment regime by identifying the critical window for the Configuration Model. We prove scaling limits for component sizes and surplus edges, and show that the maximum diameter the critical components is of order $$\log n$$ , which contrasts with the previous universality classes arising in the literature. This introduces a third and novel universality class for the critical behavior of percolation on random networks, that is not covered by the multiplicative coalescent framework due to Aldous and Limic (Electron J Probab 3(3):1–59, 1998). We also prove concentration of the component sizes outside the critical window, and that a unique, complex giant component emerges after the critical window. This completes the picture for the percolation phase transition on the Configuration Model.

  • critical percolation on scale free random graphs new universality class for the Configuration Model
    arXiv: Probability, 2019
    Co-Authors: Souvik Dhara, Remco Van Der Hofstad, Johan S H Van Leeuwaarden
    Abstract:

    In this paper, we study the critical behavior of percolation on a Configuration Model with degree distribution satisfying an infinite second-moment condition, which includes power-law degrees with exponent $\tau \in (2,3)$. It is well known that, in this regime, many canonical random graph Models, such as the Configuration Model, are robust in the sense that the giant component is not destroyed when the percolation probability stays bounded away from zero. Thus, the critical behavior is observed when the percolation probability tends to zero with the network size, despite of the fact that the average degree remains bounded. In this paper, we initiate the study of critical random graphs in the infinite second-moment regime by identifying the critical window for the Configuration Model. We prove scaling limits for component sizes and surplus edges, and show that the maximum diameter the critical components is of order $\log n$, which contrasts with the previous universality classes arising in the literature. This introduces a third and novel universality class for the critical behavior of percolation on random networks, that is not covered by the multiplicative coalescent framework due to Aldous and Limic (1998). We also prove concentration of the component sizes outside the critical window, and that a unique, complex giant component emerges after the critical window. This completes the picture for the percolation phase transition on the Configuration Model.

  • Limit laws for self-loops and multiple edges in the Configuration Model
    Annales de l'Institut Henri Poincaré Probabilités et Statistiques, 2019
    Co-Authors: Omer Angel, Remco Van Der Hofstad, Cecilia Holmgren
    Abstract:

    We consider self-loops and multiple edges in the Configuration Model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the Configuration Model, conditioned on being simple, is a uniform random graph with prescribed degrees. Simplicity corresponds to the absence of self-loops and multiple edges. We show that the number of self-loops and multiple edges converges in distribution to two independent Poisson random variables when the second moment of the empirical degree distribution converges. We also provide estimations on the total variation distance between the numbers of self-loops and multiple edges and their limits, as well as between the sum of these values and the Poisson random variable to which this sum converges to. This revisits previous works of Bollobas, of Janson, of Wormald and others. The error estimates also imply sharp asymptotics for the number of simple graphs with prescribed degrees. The error estimates follow from an application of the Stein-Chen method for Poisson convergence, which is a novel method for this problem. The asymptotic independence of self-loops and multiple edges follows from a Poisson version of the Cramer-Wold device using thinning, which is of independent interest. When the degree distribution has infinite second moment, our general results break down. We can, however, prove a central limit theorem for the number of self-loops, and for the multiple edges between vertices of degrees much smaller than the square root of the size of the graph. Our results and proofs easily extend to directed and bipartite Configuration Models.

  • Component structure of the Configuration Model: barely supercritical case
    Random Structures & Algorithms, 2019
    Co-Authors: Remco Van Der Hofstad, Svante Janson, Malwina J. Luczak
    Abstract:

    We study near-critical behavior in the Configuration Model. Let D-n be the degree of a random vertex and nu n=E[Dn(Dn-1)]/E[Dn]; we consider the barely supercritical regime, where nu(n)-> 1 as n ...

  • component structure of the Configuration Model barely supercritical case
    arXiv: Probability, 2016
    Co-Authors: Remco Van Der Hofstad, Svante Janson, Malwina J. Luczak
    Abstract:

    We study near-critical behavior in the Configuration Model. Let $D_n$ be the degree of a random vertex. We let $\nu_n={\mathbb E} [D_n(D_n-1)]/{\mathbb E}[D_n]$ and, assuming that $\nu_n \to 1$ as $n \to \infty$, we write $\varepsilon_n=\nu_n-1$. We call the setting where $\varepsilon_n n^{1/3}/({\mathbb E}[D_n^3])^{2/3} \to \infty$ the {\it barely supercritical} regime. We further assume that the variance of $D_n$ is uniformly bounded as $n \to \infty$. Let $D_n^*$ denote the size-biased version of $D_n$. We prove that there is a unique giant component of size $n \rho_n {\mathbb E} D_n (1+o(1))$, where $\rho_n$ denotes the survival probability of a branching process with offspring distribution $D_n^*-1$. This extends earlier results of Janson and Luczak~\cite{JanLuc07}, as well as those of Janson, Luczak, Windridge and House~\cite{SJ300} to the case where the third moment of $D_n$ is unbounded, filling the gap in the literature. We further study the size of the largest component in the \emph{critical} regime, where $\varepsilon_n = O(n^{-1/3} ({\mathbb E} D_n^3)^{2/3})$, extending and complementing results of Hatami and Molloy~\cite{HatamiMolloy}.