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.

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 Scale-Free Property for Degrees and Weights in a Preferential Attachment Random Graph Model

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
István Fazekas ◽  
Bettina Porvázsnyik
Keyword(s):  
Asymptotic Behaviour ◽  
Random Graph ◽  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Graph Model ◽  
Random Graph Model ◽  
Scale Free ◽  
Attachment Model ◽  
Law Degree

A random graph evolution mechanism is defined. The evolution studied is a combination of the preferential attachment model and the interaction of four vertices. The asymptotic behaviour of the graph is described. It is proved that the graph exhibits a power law degree distribution; in other words, it is scale-free. It turns out that any exponent in(2,∞)can be achieved. The proofs are based on martingale methods.

scholarly journals Width of a scale-free tree

2005 ◽  
Vol 42 (03) ◽  
pp. 839-850 ◽  
Author(s):  
Zsolt Katona
Keyword(s):  
Theoretical Result ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Random Graph Model ◽  
Scale Free ◽  
Slight Generalization ◽  
Free Tree ◽  
Law Degree

Consider the random graph model of Barabási and Albert, where we add a new vertex in every step and connect it to some old vertices with probabilities proportional to their degrees. If we connect it to only one of the old vertices then this will be a tree. These graphs have been shown to have a power-law degree distribution, the same as that observed in some large real-world networks. We are interested in the width of the tree and we show that it is at the nth step; this also holds for a slight generalization of the model with another constant. We then see how this theoretical result can be applied to directory trees.

scholarly journals Snowdrift Game on Topologically Alterable Complex Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Zhe Wang ◽  
Hong Yao ◽  
Jun Du ◽  
Xingzhao Peng ◽  
Chao Ding
Keyword(s):  
Complex Networks ◽  
Power Law ◽  
Network Models ◽  
Average Degree ◽  
Clustering Coefficient ◽  
Topological Parameters ◽  
Snowdrift Game ◽  
Cooperation Level ◽  
Optimal Value ◽  
Power Law Exponent

In order to study the influence of network’s structure on cooperation level of repeated snowdrift game, in the frame of two kinds of topologically alterable network models, the relation between the cooperation density and the topological parameters was researched. The results show that the network’s cooperation density is correlated reciprocally with power-law exponent and positively with average clustering coefficient; in other words, the more homogenous and less clustered a network, the lower the network’s cooperation level; and the relation between average degree and cooperation density is nonmonotonic; when the average degree deviates from the optimal value, the cooperation density drops.

scholarly journals The Power of D-hops in Matching Power-Law Graphs

2021 ◽  
Vol 5 (2) ◽  
pp. 1-43
Author(s):  
Liren Yu ◽  
Jiaming Xu ◽  
Xiaojun Lin
Keyword(s):  
Power Law ◽  
Graph Matching ◽  
Graph Model ◽  
Real Data ◽  
Random Graph Model ◽  
Small Constant ◽  
Low Degree ◽  
Improved Performance ◽  
Common Parent ◽  
Law Degree

This paper studies seeded graph matching for power-law graphs. Assume that two edge-correlated graphs are independently edge-sampled from a common parent graph with a power-law degree distribution. A set of correctly matched vertex-pairs is chosen at random and revealed as initial seeds. Our goal is to use the seeds to recover the remaining latent vertex correspondence between the two graphs. Departing from the existing approaches that focus on the use of high-degree seeds in $1$-hop neighborhoods, we develop an efficient algorithm that exploits the low-degree seeds in suitably-defined D-hop neighborhoods. Specifically, we first match a set of vertex-pairs with appropriate degrees (which we refer to as the first slice) based on the number of low-degree seeds in their D-hop neighborhoods. This approach significantly reduces the number of initial seeds needed to trigger a cascading process to match the rest of graphs. Under the Chung-Lu random graph model with n vertices, max degree Θ(√n), and the power-law exponent 2<β<3, we show that as soon as D> 4-β/3-β, by optimally choosing the first slice, with high probability our algorithm can correctly match a constant fraction of the true pairs without any error, provided with only Ω((log n)4-β) initial seeds. Our result achieves an exponential reduction in the seed size requirement, as the best previously known result requires n1/2+ε seeds (for any small constant ε>0). Performance evaluation with synthetic and real data further corroborates the improved performance of our algorithm.

scholarly journals Width of a scale-free tree

2005 ◽  
Vol 42 (3) ◽  
pp. 839-850 ◽  
Author(s):  
Zsolt Katona
Keyword(s):  
Theoretical Result ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Random Graph Model ◽  
Scale Free ◽  
Slight Generalization ◽  
Free Tree ◽  
Law Degree

Consider the random graph model of Barabási and Albert, where we add a new vertex in every step and connect it to some old vertices with probabilities proportional to their degrees. If we connect it to only one of the old vertices then this will be a tree. These graphs have been shown to have a power-law degree distribution, the same as that observed in some large real-world networks. We are interested in the width of the tree and we show that it is at the nth step; this also holds for a slight generalization of the model with another constant. We then see how this theoretical result can be applied to directory trees.

scholarly journals Artificial Benchmark for Community Detection (ABCD)—Fast random graph model with community structure

2021 ◽  
pp. 1-26
Author(s):  
Bogumił Kamiński ◽  
Paweł Prałat ◽  
François Théberge
Keyword(s):  
Community Structure ◽  
Complex Networks ◽  
Community Detection ◽  
Random Graph ◽  
Power Law ◽  
Main Parameter ◽  
Graph Model ◽  
Machine Learning Algorithms ◽  
Random Graph Model ◽  
Mixing Parameter

Abstract Most of the current complex networks that are of interest to practitioners possess a certain community structure that plays an important role in understanding the properties of these networks. For instance, a closely connected social communities exhibit faster rate of transmission of information in comparison to loosely connected communities. Moreover, many machine learning algorithms and tools that are developed for complex networks try to take advantage of the existence of communities to improve their performance or speed. As a result, there are many competing algorithms for detecting communities in large networks. Unfortunately, these algorithms are often quite sensitive and so they cannot be fine-tuned for a given, but a constantly changing, real-world network at hand. It is therefore important to test these algorithms for various scenarios that can only be done using synthetic graphs that have built-in community structure, power law degree distribution, and other typical properties observed in complex networks. The standard and extensively used method for generating artificial networks is the LFR graph generator. Unfortunately, this model has some scalability limitations and it is challenging to analyze it theoretically. Finally, the mixing parameter μ, the main parameter of the model guiding the strength of the communities, has a non-obvious interpretation and so can lead to unnaturally defined networks. In this paper, we provide an alternative random graph model with community structure and power law distribution for both degrees and community sizes, the Artificial Benchmark for Community Detection (ABCD graph). The model generates graphs with similar properties as the LFR one, and its main parameter ξ can be tuned to mimic its counterpart in the LFR model, the mixing parameter μ. We show that the new model solves the three issues identified above and more. In particular, we test the speed of our algorithm and do a number of experiments comparing basic properties of both ABCD and LFR. The conclusion is that these models produce graphs with comparable properties but ABCD is fast, simple, and can be easily tuned to allow the user to make a smooth transition between the two extremes: pure (independent) communities and random graph with no community structure.

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

2012 ◽  
Vol 44 (01) ◽  
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.

scholarly journals On the Largest Component of a Hyperbolic Model of Complex Networks

10.37236/4958 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Michel Bode ◽  
Nikolaos Fountoulakis ◽  
Tobias Müller
Keyword(s):  
Complex Networks ◽  
Uniform Distribution ◽  
Random Graph ◽  
Power Law ◽  
High Probability ◽  
Degree Sequence ◽  
Hyperbolic Model ◽  
Hyperbolic Distance ◽  
Component Structure ◽  
Two Parameters

We consider a model for complex networks that was introduced by Krioukov et al.  In this model, $N$ points are chosen randomly inside a disk on the hyperbolic plane and any two of them are joined by an  edge if they are within a certain hyperbolic distance.  The $N$ points are distributed according to a quasi-uniform distribution, which is a distorted version of  the uniform distribution. The model turns out to behave similarly to the well-known Chung-Lu model, but without the independence between the edges. Namely, it exhibits a power-law degree sequence and small distances but, unlike the Chung-Lu model and many other well-known models for complex networks, it also exhibits clustering. The model is controlled by two parameters $\alpha$ and $\nu$ where, roughly speaking, $\alpha$ controls the exponent of the power-law and $\nu$ controls the average degree. The present paper focuses on the evolution of the component structure of the random graph.  We show that (a) for $\alpha > 1$ and $\nu$ arbitrary, with high probability, as the number of vertices grows, the largest component of the random graph has sublinear order; (b) for $\alpha < 1$ and $\nu$ arbitrary with high probability there is a "giant" component  of linear order,  and (c) when $\alpha=1$ then there is a non-trivial phase transition for the existence of a linear-sized component in terms of $\nu$. A corrigendum was added to this paper 29 Dec 2018.

scholarly journals Complex Network Theory and Its Application Research on P2P Networks

2016 ◽  
Vol 1 (1) ◽  
pp. 45-52 ◽  
Author(s):  
Feng Jian ◽  
Shi Dandan
Keyword(s):  
Complex Networks ◽  
Complex Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Research Network ◽  
P2p Networks ◽  
Simple Description ◽  
Random Graph Model ◽  
Complex Network Theory ◽  
Network Mechanism

AbstractAdvances in complex networks of Peer-to-Peer (P2P) networks were reviewed and summarized. The paper outlines some important topological properties such as degree, average path length and clustering coefficient at first, and then three kinds of most important network mechanism models are introduced, including random graph model, small world model and scale-free model. A simple description about research status for P2P networks based on complex networks is made from three aspects: positive research, network mechanism model, network broadcast and control. Some developing prospects of complex networks of P2P are pointed out finally. Complex network provides new ideas and methods to deal with many complex problems including P2P networks.

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