scholarly journals Flooding and diameter in general weighted random graphs

2020 ◽  
Vol 57 (3) ◽  
pp. 956-980
Author(s):  
Thomas Mountford ◽  
Jacques Saliba
Keyword(s):  
Asymptotic Behavior ◽  
Graph Model ◽  
Configuration Model ◽  
Tail Behavior ◽  
First Passage Percolation ◽  
Exponential Tail ◽  
First Passage ◽  
Random Graph Model ◽  
Optimal Paths ◽  
Flooding Time

AbstractIn this paper we study first passage percolation on a random graph model, the configuration model. We first introduce the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists in describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distribution G has an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.

scholarly journals First Passage Percolation on Inhomogeneous Random Graphs

2015 ◽  
Vol 47 (2) ◽  
pp. 589-610 ◽  
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.

scholarly journals First Passage Percolation on Inhomogeneous Random Graphs

2015 ◽  
Vol 47 (02) ◽  
pp. 589-610 ◽  
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 Bollobáset 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 λ̃nof 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 Bhamidiet al.(2011), where first passage percolation is explored on the Erdős-Rényi graphs.

The configuration model

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Degree Sequence ◽  
Graph Model ◽  
Network Models ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Configuration Model ◽  
Bipartite Networks ◽  
Small Component ◽  
Random Graph Model ◽  
Average Size

A discussion of the most fundamental of network models, the configuration model, which is a random graph model of a network with a specified degree sequence. Following a definition of the model a number of basic properties are derived, including the probability of an edge, the expected number of multiedges, the excess degree distribution, the friendship paradox, and the clustering coefficient. This is followed by derivations of some more advanced properties including the condition for the existence of a giant component, the size of the giant component, the average size of a small component, and the expected diameter. Generating function methods for network models are also introduced and used to perform some more advanced calculations, such as the calculation of the distribution of the number of second neighbors of a node and the complete distribution of sizes of small components. The chapter ends with a brief discussion of extensions of the configuration model to directed networks, bipartite networks, networks with degree correlations, networks with high clustering, and networks with community structure, among other possibilities.

scholarly journals Configuration Models as an Urn Problem: The Generalized Hypergeometric Ensemble of Random Graphs

2021 ◽  
Author(s):  
Giona Casiraghi ◽  
Vahan Nanumyan
Keyword(s):  
Large Scale ◽  
Data Science ◽  
Graph Model ◽  
Configuration Model ◽  
World Systems ◽  
Fundamental Issue ◽  
Random Graph Model ◽  
Large Scale Data ◽  
Standard Configuration ◽  
Scale Data

Abstract A fundamental issue of network data science is the ability to discern observed features that can be expected at random from those beyond such expectations. Configuration models play a crucial role there, allowing us to compare observations against degree-corrected null-models. Nonetheless, existing formulations have limited large-scale data analysis applications either because they require expensive Monte-Carlo simulations or lack the required flexibility to model real-world systems. With the generalized hypergeometric ensemble, we address both problems. To achieve this, we map the configuration model to an urn problem, where edges are represented as balls in an appropriately constructed urn. Doing so, we obtain a random graph model reproducing and extending the properties of standard configuration models, with the critical advantage of a closed-form probability distribution.

The random triangle model

1999 ◽  
Vol 36 (3) ◽  
pp. 852-867 ◽  
Author(s):  
Johan Jonasson
Keyword(s):  
Asymptotic Behavior ◽  
Complete Graph ◽  
Cluster Model ◽  
Graph Model ◽  
Random Cluster Model ◽  
Random Graph Model ◽  
Random Cluster ◽  
Markov Random ◽  
Probability Mass ◽  
Edge Probability

The random triangle model is a Markov random graph model which, for parameters p ∊ (0,1) and q ≥ 1 and a graph G = (V,E), assigns to a subset, η, of E, a probability which is proportional to p|η|(1-p)|E|-|η|qt(η), where t(η) is the number of triangles in η. It is shown that this model has maximum entropy in the class of distributions with given edge and triangle probabilities.Using an analogue of the correspondence between the Fortuin-Kesteleyn random cluster model and the Potts model, the asymptotic behavior of the random triangle model on the complete graph is examined for p of order n−α, α > 0, and different values of q, where q is written in the form q = 1 + h(n) / n. It is shown that the model exhibits an explosive behavior in the sense that if h(n) ≤ c log n for c < 3α, then the edge probability and the triangle probability are asymptotically the same as for the ordinary G(n,p) model, whereas if h(n) ≥ c' log n for c' > 3α, then these quantities both tend to 1. For critical values, h(n) = 3α log n + o(log n), the probability mass divides between these two extremes.Moreover, if h(n) is of higher order than log n, then the probability that η = E tends to 1, whereas if h(n) = o(log n) and α > 2/3, then, with a probability tending to 1, the resulting graph can be coupled with a graph resulting from the G(n,p) model. In particular these facts mean that for values of p in the range critical for the appearance of the giant component and the connectivity of the graph, the way in which triangles are rewarded can only have a degenerate influence.

scholarly journals Extreme value theory, Poisson-Dirichlet distributions, and first passage percolation on random networks

2010 ◽  
Vol 42 (03) ◽  
pp. 706-738 ◽  
Author(s):  
Shankar Bhamidi ◽  
Remco van der Hofstad ◽  
Gerard Hooghiemstra
Keyword(s):  
Dirichlet Distribution ◽  
Random Network ◽  
Mean Field ◽  
Network Models ◽  
Value Theory ◽  
Configuration Model ◽  
Exponential Weight ◽  
First Passage Percolation ◽  
First Passage ◽  
Minimal Weight

We study first passage percolation (FPP) on the configuration model (CM) having power-law degrees with exponent τ ∈ [1, 2) and exponential edge weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of edges on the minimal-weight path, both of which can be computed in terms of the Poisson-Dirichlet distribution. We explicitly describe these limits via construction of infinite limiting objects describing the FPP problem in the densely connected core of the network. We consider two separate cases, the original CM, in which each edge, regardless of its multiplicity, receives an independent exponential weight, and the erased CM, for which there is an independent exponential weight between any pair of direct neighbors. While the results are qualitatively similar, surprisingly, the limiting random variables are quite different. Our results imply that the flow carrying properties of the network are markedly different from either the mean-field setting or the locally tree-like setting, which occurs as τ &gt; 2, and for which the hopcount between typical vertices scales as log n. In our setting the hopcount is tight and has an explicit limiting distribution, showing that information can be transferred remarkably quickly between different vertices in the network. This efficiency has a down side in that such networks are remarkably fragile to directed attacks. These results continue a general program by the authors to obtain a complete picture of how random disorder changes the inherent geometry of various random network models; see Aldous and Bhamidi (2010), Bhamidi (2008), and Bhamidi, van der Hofstad and Hooghiemstra (2009).

scholarly journals Configuration models as an urn problem

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Giona Casiraghi ◽  
Vahan Nanumyan
Keyword(s):  
Large Scale ◽  
Data Science ◽  
Graph Model ◽  
Configuration Model ◽  
World Systems ◽  
Fundamental Issue ◽  
Random Graph Model ◽  
Large Scale Data ◽  
Standard Configuration ◽  
Scale Data

AbstractA fundamental issue of network data science is the ability to discern observed features that can be expected at random from those beyond such expectations. Configuration models play a crucial role there, allowing us to compare observations against degree-corrected null-models. Nonetheless, existing formulations have limited large-scale data analysis applications either because they require expensive Monte-Carlo simulations or lack the required flexibility to model real-world systems. With the generalized hypergeometric ensemble, we address both problems. To achieve this, we map the configuration model to an urn problem, where edges are represented as balls in an appropriately constructed urn. Doing so, we obtain the generalized hypergeometric ensemble of random graphs: a random graph model reproducing and extending the properties of standard configuration models, with the critical advantage of a closed-form probability distribution.

Nonuniversality of weighted random graphs with infinite variance degree

2017 ◽  
Vol 54 (1) ◽  
pp. 146-164 ◽  
Author(s):  
Enrico Baroni ◽  
Remco van der Hofstad ◽  
Júlia Komjáthy
Keyword(s):  
Continuous Time ◽  
Degree Sequence ◽  
Optimal Path ◽  
Configuration Model ◽  
Infinite Variance ◽  
Finite Variance ◽  
First Passage Percolation ◽  
First Passage ◽  
Continuous Time Processes ◽  
Graph Distances

AbstractWe prove nonuniversality results for first-passage percolation on the configuration model with independent and identically distributed (i.i.d.) degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of the weight distribution, we use an example-based approach and show that rather different behaviours are possible. When the weights are almost surely larger than a constant, the weight and number of edges in the graph grow proportionally to log log n, as for the graph distances. On the other hand, when the continuous-time branching process describing the first-passage percolation exploration through the graph reaches infinitely many vertices in finite time, the weight converges to the sum of two i.i.d. random variables representing the explosion times of the continuous-time processes started from the two sources. This nonuniversality is in sharp contrast to the setting where the degree sequence has a finite variance, Bhamidi et al. (2012).

The random triangle model

1999 ◽  
Vol 36 (03) ◽  
pp. 852-867 ◽  
Author(s):  
Johan Jonasson
Keyword(s):  
Asymptotic Behavior ◽  
Complete Graph ◽  
Cluster Model ◽  
Graph Model ◽  
Random Cluster Model ◽  
Random Graph Model ◽  
Random Cluster ◽  
Markov Random ◽  
Probability Mass ◽  
Edge Probability

The random triangle model is a Markov random graph model which, for parameters p ∊ (0,1) and q ≥ 1 and a graph G = (V,E), assigns to a subset, η, of E, a probability which is proportional to p |η|(1-p)|E|-|η| q t(η), where t(η) is the number of triangles in η. It is shown that this model has maximum entropy in the class of distributions with given edge and triangle probabilities. Using an analogue of the correspondence between the Fortuin-Kesteleyn random cluster model and the Potts model, the asymptotic behavior of the random triangle model on the complete graph is examined for p of order n −α, α &gt; 0, and different values of q, where q is written in the form q = 1 + h(n) / n. It is shown that the model exhibits an explosive behavior in the sense that if h(n) ≤ c log n for c &lt; 3α, then the edge probability and the triangle probability are asymptotically the same as for the ordinary G(n,p) model, whereas if h(n) ≥ c' log n for c' &gt; 3α, then these quantities both tend to 1. For critical values, h(n) = 3α log n + o(log n), the probability mass divides between these two extremes. Moreover, if h(n) is of higher order than log n, then the probability that η = E tends to 1, whereas if h(n) = o(log n) and α &gt; 2/3, then, with a probability tending to 1, the resulting graph can be coupled with a graph resulting from the G(n,p) model. In particular these facts mean that for values of p in the range critical for the appearance of the giant component and the connectivity of the graph, the way in which triangles are rewarded can only have a degenerate influence.

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