scholarly journals An $L^p$ theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions

2019 ◽  
Vol 372 (5) ◽  
pp. 3019-3062 ◽  
Author(s):  
Christian Borgs ◽  
Jennifer T. Chayes ◽  
Henry Cohn ◽  
Yufei Zhao
Keyword(s):  
Random Graph ◽  
Power Law ◽  
Sparse Graph ◽  
Graph Models ◽  
Random Graph Models ◽  
Graph Convergence ◽  
Power Law Distributions

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.

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 Exponential random graph models for little networks

2021 ◽  
Vol 64 ◽  
pp. 225-238
Author(s):  
George G. Vega Yon ◽  
Andrew Slaughter ◽  
Kayla de la Haye
Keyword(s):  
Random Graph ◽  
Exponential Random Graph Models ◽  
Graph Models ◽  
Random Graph Models ◽  
Exponential Random Graph

Modularity in several random graph models

2017 ◽  
Vol 61 ◽  
pp. 947-953 ◽  
Author(s):  
Liudmila Ostroumova Prokhorenkova ◽  
Paweł Prałat ◽  
Andrei Raigorodskii
Keyword(s):  
Random Graph ◽  
Graph Models ◽  
Random Graph Models

scholarly journals Random graph models for directed acyclic networks

2009 ◽  
Vol 80 (4) ◽  
Author(s):  
Brian Karrer ◽  
M. E. J. Newman
Keyword(s):  
Random Graph ◽  
Graph Models ◽  
Random Graph Models
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