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### Convergence law for random graphs with specified degree sequence

James F. Lynch
2005 ACM Transactions on Computational Logic
A random graph with specified asymptotic degree sequence D is a random graph on n with degree sequence d 0 (n), . . . , d n−1 (n) for some n ∈ ω. If • James F.  ...  Our convergence law also extends the convergence law for classical random graphs with edge probability c/n, c constant [Lynch 1992] or with specified number of edges cn/2.  ...

### Convergence law for random graphs with specified degree sequence

J.F. Lynch
18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings.
Our main result is a convergence law for random graphs with degree sequences approximated by some sequence λ 0 , λ 1 , . . . .  ...  A random graph with degree sequence d 0 , . . . , d n−1 is a randomly selected member of the set of graphs on {1, . . . , n} with that degree sequence, all choices being equally likely.  ...  A random graph with specified asymptotic degree sequence D is a random graph on n with degree sequence d 0 (n), . . . , d n−1 (n) for some n ∈ ω. If • James F.  ...

### Convergence law for hyper-graphs with prescribed degree sequences [article]

Nans Lefebvre
2015 arXiv   pre-print
It defines a random hyper-multigraph specified by two distributions, one for the degrees of the vertices, and one for the sizes of the hyper-edges.  ...  Convergence laws of other models follow, and in particular for the classical Erdős-Rényi graphs and k-uniform hyper-graphs.  ...  graphs (including the graphs with specified degree sequences), the¨-graph can be a sparse graph with nontrivial clustering coefficient.  ...

### Local weak convergence for PageRank [article]

Alessandro Garavaglia and Remco van der Hofstad and Nelly Litvak
2018 arXiv   pre-print
We start from the definition of local weak convergence for sequences of (random) undirected graphs, and extend this notion to directed graphs.  ...  One of the intriguing empirical properties of PageRank is the so-called 'power-law hypothesis': in a scale-free network the PageRank scores follow a power law with the same exponent as the (in-)degrees  ...  The work of RvdH is further supported by the Netherlands Organisation for Scientific Research (NWO) through VICI grant 639.033.806.  ...

### Fast Generation of Sparse Random Kernel Graphs

Aric Hagberg, Nathan Lemons, Wen-Bo Du
2015 PLoS ONE
As a practical example we show how to generate samples of power-law degree distribution graphs with tunable assortativity.  ...  We specify a class of inhomogeneous random graph models, called random kernel graphs, that produces sparse graphs with tunable graph properties, and we develop an efficient generation algorithm to sample  ...  Acknowledgments We would like to thank Terry Haut, Joel Miller, and Pieter Swart for helpful comments and suggestions. Author Contributions Conceived and designed the experiments: AH NL.  ...

### Counting triangles in power-law uniform random graphs [article]

Pu Gao, Remco van der Hofstad, Angus Southwell, Clara Stegehuis
2018 arXiv   pre-print
We count the asymptotic number of triangles in uniform random graphs where the degree distribution follows a power law with degree exponent τ∈(2,3).  ...  Interestingly, uniform random graphs contain more triangles than erased configuration models with the same degree sequence.  ...  Other methods for analyzing uniform random graphs rely on asymptotic enumeration of graphs with specified degree sequences.  ...

### Counting Triangles in Power-Law Uniform Random Graphs

Pu Gao, Remco Van der Hofstad, Angus Southwell, Clara Stegehuis
2020 Electronic Journal of Combinatorics
We count the asymptotic number of triangles in uniform random graphs where the degree distribution follows a power law with degree exponent $\tau\in(2,3)$.  ...  Interestingly, uniform random graphs contain more triangles than erased configuration models with the same degree sequence.  ...  We thank an anonymous referee for pointing out how the integrals in (4.32) and (5.32) could be computed.  ...

### Tracking a Markov-Modulated Stationary Degree Distribution of a Dynamic Random Graph

Maziyar Hamdi, Vikram Krishnamurthy, George Yin
2014 IEEE Transactions on Information Theory
This paper considers a Markov-modulated duplication-deletion random graph where at each time instant, one node can either join or leave the network; the probabilities of joining or leaving evolve according  ...  First, motivated by social network applications, the asymptotic behavior of the degree distribution is analyzed.  ...  Fig. 1 . 1 The degree distribution of the duplication-deletion random graph satisfies a power law. The parameters are specified in Example 1 of Sec.VI.  ...

### Universality for distances in power-law random graphs

Remco van der Hofstad, Gerard Hooghiemstra
2008 Journal of Mathematical Physics
Since many real network have been empirically shown to have power-law degree sequences, these random graphs can be seen as more realistic models for real complex networks.  ...  We focus on inhomogeneous random graphs, the configuration model and affine preferential attachment models, and pay special attention to setting where these random graphs have a power-law degree sequence  ...  Indeed, as proved in [17] , the generalized random graph, conditioned on its degree sequence, is a uniform simple random graph with that degree sequence.  ...

### Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees [article]

Victor M. Preciado, M. Amin Rahimian
2017 arXiv   pre-print
Under some technical conditions on the expected degree sequence, we show that with probability one, F_n(·) converges weakly to a deterministic distribution F(·).  ...  In this paper, we analyze the limiting spectral distribution of the adjacency matrix of a random graph ensemble, proposed by Chung and Lu, in which a given expected degree sequence w_n^^T = (w^(n)_1,..  ...  Then, consider a Chung-Lu random graph with the expected degree sequence specified as w (n) i = ∆ n e −αxi , i ∈ [n] .  ...

### Sampling random graphs with specified degree sequences [article]

Upasana Dutta, Bailey K. Fosdick, Aaron Clauset
2022 arXiv   pre-print
The configuration model is a standard tool for uniformly generating random graphs with a specified degree sequence, and is often used as a null model to evaluate how much of an observed network's structure  ...  We develop an algorithm, based on the assortativity of the sampled graphs, for estimating the gap between effectively independent MCMC states, and a computationally efficient gap-estimation heuristic derived  ...  The authors thank Joel Nishimura for providing the Python code for checking if the loopy graph space is connected or not.  ...

### Random-walk models of network formation and sequential Monte Carlo methods for graphs

Benjamin Bloem-Reddy, Peter Orbanz
2018 Journal of The Royal Statistical Society Series B-statistical Methodology
Theoretical properties, including the limiting degree sequence, are studied analytically. If the entire history of the graph is observed, parameters can be estimated by maximum likelihood.  ...  We introduce a class of generative network models that insert edges by connecting the starting and terminal vertices of a random walk on the network graph.  ...  A random sequence G 1:T can then be specified as a pair (Π, G T ), for a random permutation Π of the edges in G T , with G t = Π t (G T ).  ...

### Resolvent of large random graphs

Charles Bordenave, Marc Lelarge
2010 Random structures & algorithms (Print)
We illustrate our results on the uniform regular graphs, Erdös-Rényi graphs and graphs with a given degree sequence.  ...  We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph.  ...  Acknowledgment The authors thank Noureddine El Karoui for fruitful discussions and his interest in this work.  ...

### Resolvent of Large Random Graphs [article]

Charles Bordenave, Marc Lelarge
2009 arXiv   pre-print
We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph.  ...  We apply these results to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs.  ...  Acknowledgment The authors thank Noureddine El Karoui for fruitful discussions and his interest in this work.  ...

### Graph Signal Processing: Filter Design and Spectral Statistics [article]

Stephen Kruzick, José M. F. Moura
2018 arXiv   pre-print
An application considered in this paper, convergence acceleration filters for distributed average consensus may be viewed as lowpass graph filters periodically applied to the states.  ...  With respect to the graph shift operator, polynomial functions of the shift matrix perform filtering.  ...  the convergence rate for large scale constant (with respect to time iterations) random networks.  ...
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