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.

Random graphs

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.

scholarly journals The diameter of KPKVB random graphs

2019 ◽  
Vol 51 (2) ◽  
pp. 358-377 ◽  
Author(s):  
Tobias Müller ◽  
Merlijn Staps
Keyword(s):  
Complex Networks ◽  
Power Law ◽  
Degree Sequence ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Maximum Diameter ◽  
Hyperbolic Distance ◽  
Random Graph Model ◽  
Power Law Exponent ◽  
Law Degree

AbstractWe consider a random graph model that was recently proposed as a model for complex networks by Krioukov et al. (2010). In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has previously been shown that this model has various properties associated with complex networks, including a power-law degree distribution and a strictly positive clustering coefficient. The model is specified using three parameters: the number of nodes N, which we think of as going to infinity, and $\alpha, \nu > 0$, which we think of as constant. Roughly speaking, $\alpha$ controls the power-law exponent of the degree sequence and $\nu$ the average degree. Earlier work of Kiwi and Mitsche (2015) has shown that, when $\alpha \lt 1$ (which corresponds to the exponent of the power law degree sequence being $\lt 3$), the diameter of the largest component is asymptotically almost surely (a.a.s.) at most polylogarithmic in N. Friedrich and Krohmer (2015) showed it was a.a.s. $\Omega(\log N)$ and improved the exponent of the polynomial in $\log N$ in the upper bound. Here we show the maximum diameter over all components is a.a.s. $O(\log N),$ thus giving a bound that is tight up to a multiplicative constant.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
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.

scholarly journals Counting Triangles in Power-Law Uniform Random Graphs

10.37236/9239 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Pu Gao ◽  
Remco Van der Hofstad ◽  
Angus Southwell ◽  
Clara Stegehuis
Keyword(s):  
Random Graphs ◽  
Power Law ◽  
Degree Sequence ◽  
Clustering Coefficient ◽  
Linear Functions ◽  
Configuration Model ◽  
Local Clustering ◽  
Asymptotic Number ◽  
Multiple Edges ◽  
Local Clustering Coefficient

We count the asymptotic number of triangles in uniform random graphs where the degree distribution follows a power law with degree exponent $\tau\in(2,3)$. We also analyze the local clustering coefficient $c(k)$, the probability that two random neighbors of a vertex of degree $k$ are connected. We find that the number of triangles, as well as the local clustering coefficient, scale similarly as in the erased configuration model, where all self-loops and multiple edges of the configuration model are removed. Interestingly, uniform random graphs contain more triangles than erased configuration models with the same degree sequence. The number of triangles in uniform random graphs is closely related to that in a version of the rank-1 inhomogeneous random graph, where all vertices are equipped with weights, and the probabilities that edges are present are moderated by asymptotically linear functions of the products of these vertex weights.

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.

scholarly journals A Complex Networks Approach to Analysing the Erdős-Straus Conjecture and Related Problems

2019 ◽  
Author(s):  
Vikhyath Mondreti
Keyword(s):  
Complex Networks ◽  
Degree Sequence ◽  
Clustering Coefficient ◽  
Configuration Model ◽  
Directed Network ◽  
Length Unit ◽  
Strongly Connected ◽  
Network Metrics ◽  
Positive Integers ◽  
Made In

For any positive integer n &ge; 2, the Erdős-Straus Conjecture claims that the Diophantine equation 4/n = 1/x + 1/y + 1/z has a solution where x, y, z are also positive integers. In this paper, a directed network based on this equation is generated, with properties such as its average clustering coefficient, average path length, degree distributions, and largest strongly connected component analysed to reveal some underlying trends about the nature of the conjecture. Potential connections between different numbers, that result from satisfying a source-solution relationship for this equation, are described using the appropriate number-theoretic interpretations wherever possible, while conjectures backed by these trends are made in other instances. Additionally, a directed configuration model is used to show that the origin of several results is the degree sequence of the network. Metrics relating to the prime number nodes, specifically their in and out degrees, are also explored to yield some intriguing observations. On the whole, the aim is to highlight the viability of complex networks as a computational tool to study this general class of problems pertaining to fixed-length unit fraction splits.

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 Tweeting #RamNavami: A Comparison of Approaches to Analyzing Bipartite Networks

2021 ◽  
pp. 227797522110180
Author(s):  
Michael T. Heaney
Keyword(s):  
Graph Model ◽  
Data Availability ◽  
Bipartite Networks ◽  
International Treaty ◽  
Random Graph Model ◽  
Affiliation Networks ◽  
Coalition Structures ◽  
Estimated Parameters ◽  
Exponential Random Graph ◽  
Event Model

Bipartite networks, also known as two-mode networks or affiliation networks, are a class of networks in which actors or objects are partitioned into two sets, with interactions taking place across but not within sets. These networks are omnipresent in society, encompassing phenomena such as student-teacher interactions, coalition structures and international treaty participation. With growing data availability and proliferation in statistical estimators and software, scholars have increasingly sought to understand the methods available to model the data-generating processes in these networks. This article compares three methods for doing so: (a) Logit (b) the bipartite Exponential Random Graph Model (ERGM) and (c) the Relational Event Model (REM). This comparison demonstrates the relevance of choices with respect to dependence structures, temporality, parameter specification and data structure. Considering the example of Ram Navami, a Hindu festival celebrating the birth of Lord Ram, the ego network of tweets using #RamNavami on 21April 2021 is examined. The results of the analysis illustrate that critical modelling choices make a difference in the estimated parameters and the conclusions to be drawn from them.

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.

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