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

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.

scholarly journals A theoretical and empirical comparison of the temporal exponential random graph model and the stochastic actor-oriented model

2019 ◽  
Vol 7 (1) ◽  
pp. 20-51 ◽  
Author(s):  
Philip Leifeld ◽  
Skyler J. Cranmer
Keyword(s):  
Network Analysis ◽  
Random Graph ◽  
Graph Model ◽  
Real Data ◽  
The Other ◽  
Exponential Random Graph Model ◽  
Empirical Comparison ◽  
Random Graph Model ◽  
Exponential Random Graph ◽  
Longitudinal Network Analysis

AbstractThe temporal exponential random graph model (TERGM) and the stochastic actor-oriented model (SAOM, e.g., SIENA) are popular models for longitudinal network analysis. We compare these models theoretically, via simulation, and through a real-data example in order to assess their relative strengths and weaknesses. Though we do not aim to make a general claim about either being superior to the other across all specifications, we highlight several theoretical differences the analyst might consider and find that with some specifications, the two models behave very similarly, while each model out-predicts the other one the more the specific assumptions of the respective model are met.

scholarly journals SYNCHRONIZABILITY AND SYNCHRONIZATION DYNAMICS OF WEIGHED AND UNWEIGHED SCALE FREE NETWORKS WITH DEGREE MIXING

2007 ◽  
Vol 17 (07) ◽  
pp. 2419-2434 ◽  
Author(s):  
FRANCESCO SORRENTINO ◽  
MARIO DI BERNARDO ◽  
FRANCO GAROFALO
Keyword(s):  
Power Law ◽  
Biological Networks ◽  
Optimization Strategy ◽  
Power Law Distribution ◽  
Scale Free ◽  
Degree Correlation ◽  
Topological Features ◽  
Low Degree ◽  
Correlation Properties ◽  
Law Degree

We study the synchronizability and the synchronization dynamics of networks of nonlinear oscillators. We investigate how the synchronization of the network is influenced by some of its topological features such as variations of the power law degree distribution exponent γ and the degree correlation coefficient r. Using an appropriate construction algorithm based on clustering the network vertices in p classes according to their degrees, we construct networks with an assigned power law distribution but changing degree correlation properties. We find that the network synchronizability improves when the network becomes disassortative, i.e. when nodes with low degree are more likely to be connected to nodes with higher degree. We consider the case of both weighed and unweighed networks. The analytical results reported in the paper are then confirmed by a set of numerical observations obtained on weighed and unweighed networks of nonlinear Rössler oscillators. Using a nonlinear optimization strategy we also show that negative degree correlation is an emerging property of networks when synchronizability is to be optimized. This suggests that negative degree correlation observed experimentally in a number of physical and biological networks might be motivated by their need to synchronize better.

scholarly journals A Random Graph Model for Power Law Graphs

2001 ◽  
Vol 10 (1) ◽  
pp. 53-66 ◽  
Author(s):  
William Aiello ◽  
Fan Chung ◽  
Linyuan Lu
Keyword(s):  
Random Graph ◽  
Power Law ◽  
Graph Model ◽  
Random Graph Model

Geometric-Edge Random Graph Model for Image Representation

2012 ◽  
pp. 42-57
Author(s):  
Jiang Bo ◽  
Tang Jing ◽  
Luo Bin
Keyword(s):  
Random Graph ◽  
Image Matching ◽  
Graph Matching ◽  
Image Representation ◽  
Graph Model ◽  
Experimental Results ◽  
Structural Variations ◽  
Matching Algorithm ◽  
Random Graph Model ◽  
Dot Product

This chapter presents a random graph model for image representation. The first contribution the authors propose is a Geometric-Edge (G-E) Random Graph Model for image representation. The second contribution is that of casting image matching into G-E Random Graph matching by using the random dot product graph based matching algorithm. Experimental results show that the proposed G-E Random Graph model and matching algorithm are effective and robust to structural variations.

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