Degree correlations in scale-free random graph models

2019 ◽  
Vol 56 (3) ◽  
pp. 672-700 ◽  
Author(s):  
Clara Stegehuis
Keyword(s):  
Random Graph ◽  
Power Law ◽  
Configuration Model ◽  
Large Network ◽  
Graph Models ◽  
Scale Free ◽  
Random Graph Models ◽  
Degree Correlations ◽  
Law Degree ◽  
Inhomogeneous Random Graph

AbstractWe study the average nearest-neighbour degree a(k) of vertices with degree k. In many real-world networks with power-law degree distribution, a(k) falls off with k, a property ascribed to the constraint that any two vertices are connected by at most one edge. We show that a(k) indeed decays with k in three simple random graph models with power-law degrees: the erased configuration model, the rank-1 inhomogeneous random graph, and the hyperbolic random graph. We find that in the large-network limit for all three null models, a(k) starts to decay beyond $n^{(\tau-2)/(\tau-1)}$ and then settles on a power law $a(k)\sim k^{\tau-3}$, with $\tau$ the degree exponent.

scholarly journals Interplay between $$k$$-core and community structure in complex networks

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Irene Malvestio ◽  
Alessio Cardillo ◽  
Naoki Masuda
Keyword(s):  
Community Structure ◽  
Complex Networks ◽  
Random Graph ◽  
Shell Structure ◽  
Crucial Role ◽  
Configuration Model ◽  
Graph Models ◽  
Random Graph Models ◽  
Empirical Networks

Abstract The organisation of a network in a maximal set of nodes having at least k neighbours within the set, known as $$k$$ k -core decomposition, has been used for studying various phenomena. It has been shown that nodes in the innermost $$k$$ k -shells play a crucial role in contagion processes, emergence of consensus, and resilience of the system. It is known that the $$k$$ k -core decomposition of many empirical networks cannot be explained by the degree of each node alone, or equivalently, random graph models that preserve the degree of each node (i.e., configuration model). Here we study the $$k$$ k -core decomposition of some empirical networks as well as that of some randomised counterparts, and examine the extent to which the $$k$$ k -shell structure of the networks can be accounted for by the community structure. We find that preserving the community structure in the randomisation process is crucial for generating networks whose $$k$$ k -core decomposition is close to the empirical one. We also highlight the existence, in some networks, of a concentration of the nodes in the innermost $$k$$ k -shells into a small number of communities.

GROWING HIERARCHICAL SCALE-FREE NETWORKS BY MEANS OF NONHIERARCHICAL PROCESSES

2007 ◽  
Vol 17 (07) ◽  
pp. 2447-2452 ◽  
Author(s):  
S. BOCCALETTI ◽  
D.-U. HWANG ◽  
V. LATORA
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Rate Equations ◽  
Scale Free ◽  
Scale Free Networks ◽  
Degree Correlations ◽  
Law Degree ◽  
Hierarchical Scale

We introduce a fully nonhierarchical network growing mechanism, that furthermore does not impose explicit preferential attachment rules. The growing procedure produces a graph featuring power-law degree and clustering distributions, and manifesting slightly disassortative degree-degree correlations. The rigorous rate equations for the evolution of the degree distribution and for the conditional degree-degree probability are derived.

scholarly journals Weighted Exponential Random Graph Models: Scope and Large Network Limits

2018 ◽  
Vol 173 (3-4) ◽  
pp. 704-735 ◽  
Author(s):  
Shankar Bhamidi ◽  
Suman Chakraborty ◽  
Skyler Cranmer ◽  
Bruce Desmarais
Keyword(s):  
Random Graph ◽  
Large Network ◽  
Exponential Random Graph Models ◽  
Graph Models ◽  
Random Graph Models ◽  
Exponential Random Graph

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 Closure coefficients in scale-free complex networks

2020 ◽  
Vol 8 (3) ◽  
Author(s):  
Clara Stegehuis
Keyword(s):  
Complex Networks ◽  
Random Graph ◽  
Clustering Coefficient ◽  
Graph Models ◽  
Variable Model ◽  
Scale Free ◽  
Random Graph Models ◽  
Local Closure ◽  
Head Node ◽  
High Degree

Abstract The formation of triangles in complex networks is an important network property that has received tremendous attention. The formation of triangles is often studied through the clustering coefficient. The closure coefficient or transitivity is another method to measure triadic closure. This statistic measures clustering from the head node of a triangle (instead of from the centre node, as in the often studied clustering coefficient). We perform a first exploratory analysis of the behaviour of the local closure coefficient in two random graph models that create simple networks with power-law degrees: the hidden-variable model and the hyperbolic random graph. We show that the closure coefficient behaves significantly different in these simple random graph models than in the previously studied multigraph models. We also relate the closure coefficient of high-degree vertices to the clustering coefficient and the average nearest neighbour degree.

scholarly journals Greed is Good for Deterministic Scale-Free Networks

Algorithmica ◽  
2020 ◽  
Vol 82 (11) ◽  
pp. 3338-3389
Author(s):  
Ankit Chauhan ◽  
Tobias Friedrich ◽  
Ralf Rothenberger
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
High Probability ◽  
Maximum Independent Set ◽  
Graph Models ◽  
Random Graph Models ◽  
Graph Classes

Abstract Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung–Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical $$\textsf {NP}$$ NP -hard optimization problems on PLB networks. It is known that on general graphs with maximum degree $$\Delta$$ Δ , a greedy algorithm, which chooses nodes in the order of their degree, only achieves a $$\Omega (\ln \Delta )$$ Ω ( ln Δ ) -approximation for Minimum Vertex Cover and Minimum Dominating Set, and a $$\Omega (\Delta )$$ Ω ( Δ ) -approximation for Maximum Independent Set. We prove that the PLB-U property with $$\beta >2$$ β > 2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are -hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless = . Furthermore, we prove that all three problems are in if the PLB-U property holds.

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.

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