The Largest Component in Subcritical Inhomogeneous Random Graphs

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

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The natural connectivity of colored random graphs

Creative Mathematics and Informatics ◽

10.37193/cmi.2011.02.11 ◽

2011 ◽

Vol 20 (2) ◽

pp. 197-202

Author(s):

YILUN SHANG ◽

Keyword(s):

Complex Network ◽

Random Graph ◽

Random Graphs ◽

Graph Model ◽

Graph Spectrum ◽

Random Graph Model ◽

Robustness Measure ◽

Theoretical Results ◽

Inhomogeneous Random Graph

The natural connectivity as a robustness measure of complex network has been proposed recently. It can be regarded as the average eigenvalue obtained from the graph spectrum. In this paper, we introduce an inhomogeneous random graph model, G(n, {ci}, {pi}), and investigate its natural connectivity. Binomial random graph ... . Simulations are performed to validate our theoretical results.

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Joint Vertex Degrees in the Inhomogeneous Random Graph Model ℊ(n,{pij})

Advances in Applied Probability ◽

10.1239/aap/1331216648 ◽

2012 ◽

Vol 44 (1) ◽

pp. 139-165

Author(s):

Kaisheng Lin ◽

Gesine Reinert

Keyword(s):

Random Graph ◽

Power Law ◽

Graph Model ◽

Multivariate Normal ◽

Random Graph Model ◽

Inhomogeneous Random Graphs ◽

Vertex Degrees ◽

Law Degree ◽

Process Approximation ◽

Inhomogeneous Random Graph

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

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Random graphs

10.1093/oso/9780198805090.003.0011 ◽

2018 ◽

Author(s):

Mark Newman

Keyword(s):

Random Graph ◽

Random Graphs ◽

Large Scale ◽

Random Network ◽

Graph Model ◽

Clustering Coefficient ◽

Giant Component ◽

Random Graph Model ◽

Definition Of ◽

Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

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The Impact of Edge Correlations in Random Networks

Complex Systems ◽

10.25088/complexsystems.30.4.525 ◽

2021 ◽

Vol 30 (4) ◽

pp. 525-537

Author(s):

András Faragó ◽

Keyword(s):

Random Graph ◽

Random Graphs ◽

Real Life ◽

Graph Model ◽

Random Networks ◽

Random Graph Model ◽

Special Cases ◽

Graph Parameters ◽

The Impact ◽

Good Number

Random graphs are frequently used models of real-life random networks. The classical Erdös–Rényi random graph model is very well explored and has numerous nontrivial properties. In particular, a good number of important graph parameters that are hard to compute in the deterministic case often become much easier in random graphs. However, a fundamental restriction in the Erdös–Rényi random graph is that the edges are required to be probabilistically independent. This is a severe restriction, which does not hold in most real-life networks. We consider more general random graphs in which the edges may be dependent. Specifically, two models are analyzed. The first one is called a p-robust random graph. It is defined by the requirement that each edge exist with probability at least p, no matter how we condition on the presence/absence of other edges. It is significantly more general than assuming independent edges existing with probability p, as exemplified via several special cases. The second model considers the case when the edges are positively correlated, which means that the edge probability is at least p for each edge, no matter how we condition on the presence of other edges (but absence is not considered). We prove some interesting, nontrivial properties about both models.

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A Note on the Warmth of Random Graphs with Given Expected Degrees

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2014/749856 ◽

2014 ◽

Vol 2014 ◽

pp. 1-4 ◽

Cited By ~ 2

Author(s):

Yilun Shang

Keyword(s):

Statistical Mechanics ◽

Chromatic Number ◽

Random Graph ◽

Lower Bounds ◽

Random Graphs ◽

Upper Bound ◽

Degree Sequence ◽

Graph Model ◽

Upper And Lower Bounds ◽

Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

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First Passage Percolation on Inhomogeneous Random Graphs

Advances in Applied Probability ◽

10.1239/aap/1435236989 ◽

2015 ◽

Vol 47 (2) ◽

pp. 589-610 ◽

Cited By ~ 4

Author(s):

István Kolossváry ◽

Júlia Komjáthy

Keyword(s):

Limit Theorem ◽

Graph Model ◽

Exponential Weight ◽

First Passage Percolation ◽

First Passage ◽

Full Generality ◽

Random Graph Model ◽

Minimal Weight ◽

Inhomogeneous Random Graphs ◽

Inhomogeneous Random Graph

In this paper we investigate first passage percolation on an inhomogeneous random graph model introduced by Bollobás et al. (2007). Each vertex in the graph has a type from a type space, and edge probabilities are independent, but depend on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distribution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal-weight path, properly normalized, follows a central limit theorem. We handle the cases where the average number of neighbors λ̃n of a vertex tends to a finite λ̃ in full generality and consider λ̃ = ∞ under mild assumptions. This paper is a generalization of the paper of Bhamidi et al. (2011), where first passage percolation is explored on the Erdős-Rényi graphs.

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Analysis of E-Commerce Product Graphs

10.36227/techrxiv.12814244.v1 ◽

2020 ◽

Author(s):

Shalin Shah

Keyword(s):

Random Graph ◽

Random Graphs ◽

Power Law ◽

Real World ◽

Degree Distribution ◽

Graph Model ◽

Graph Analysis ◽

Random Graph Model ◽

Product Graphs ◽

Clustering Coefficients

Consumer behavior in retail stores gives rise to product graphs based on copurchasingor co-viewing behavior. These product graphs can be analyzed usingthe known methods of graph analysis. In this paper, we analyze the product graphat Target Corporation based on the Erd˝os-Renyi random graph model. In particular,we compute clustering coefficients of actual and random graphs, and we find thatthe clustering coefficients of actual graphs are much higher than random graphs.We conduct the analysis on the entire set of products and also on a per categorybasis and find interesting results. We also compute the degree distribution andwe find that the degree distribution is a power law as expected from real worldnetworks, contrasting with the ER random graph.

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Phase transition in random distance graphs on the torus

Journal of Applied Probability ◽

10.1017/jpr.2017.63 ◽

2017 ◽

Vol 54 (4) ◽

pp. 1278-1294 ◽

Cited By ~ 1

Author(s):

Fioralba Ajazi ◽

George M. Napolitano ◽

Tatyana Turova

Keyword(s):

Phase Transition ◽

General Theory ◽

Random Graphs ◽

Connected Component ◽

Subcritical Case ◽

Distance Graphs ◽

Inhomogeneous Random Graphs ◽

Supercritical Phase

Abstract In this paper we consider random distance graphs motivated by applications in neurobiology. These models can be viewed as examples of inhomogeneous random graphs, notably outside of the so-called rank-1 case. Treating these models in the context of the general theory of inhomogeneous graphs helps us to derive the asymptotics for the size of the largest connected component. In particular, we show that certain random distance graphs behave exactly as the classical Erdős–Rényi model, not only in the supercritical phase (as already known) but in the subcritical case as well.

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Analysis of E-Commerce Product Graphs

10.36227/techrxiv.12814244 ◽

2020 ◽

Author(s):

Shalin Shah

Keyword(s):

Random Graph ◽

Random Graphs ◽

Power Law ◽

Real World ◽

Degree Distribution ◽

Graph Model ◽

Graph Analysis ◽

Random Graph Model ◽

Product Graphs ◽

Clustering Coefficients

Consumer behavior in retail stores gives rise to product graphs based on copurchasingor co-viewing behavior. These product graphs can be analyzed usingthe known methods of graph analysis. In this paper, we analyze the product graphat Target Corporation based on the Erd˝os-Renyi random graph model. In particular,we compute clustering coefficients of actual and random graphs, and we find thatthe clustering coefficients of actual graphs are much higher than random graphs.We conduct the analysis on the entire set of products and also on a per categorybasis and find interesting results. We also compute the degree distribution andwe find that the degree distribution is a power law as expected from real worldnetworks, contrasting with the ER random graph.

Download Full-text