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.

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 Power-law decay of the degree-sequence probabilities of multiple random graphs with application to graph isomorphism

2017 ◽  
Vol 21 ◽  
pp. 235-250 ◽  
Author(s):  
Jefferson Elbert Simões ◽  
Daniel R. Figueiredo ◽  
Valmir C. Barbosa
Keyword(s):  
Random Graphs ◽  
Power Law ◽  
Degree Sequence ◽  
Graph Isomorphism ◽  
Power Law Decay

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 Overlapping Structures Detection in Protein-Protein Interaction Networks Using Community Detection Algorithm Based on Neighbor Clustering Coefficient

2021 ◽  
Vol 12 ◽  
Author(s):  
Yan Wang ◽  
Chen Qiong ◽  
Lili Yang ◽  
Sen Yang ◽  
Kai He ◽  
...  
Keyword(s):  
Community Detection ◽  
Detection Algorithm ◽  
Clustering Coefficient ◽  
Functional Modules ◽  
Seed Selection ◽  
Protein Protein Interaction ◽  
Ppi Networks ◽  
Local Clustering ◽  
Community Detection Algorithm ◽  
Local Clustering Coefficient

With the rapid development of bioinformatics, researchers have applied community detection algorithms to detect functional modules in protein-protein interaction (PPI) networks that can predict the function of unknown proteins at the molecular level and further reveal the regularity of cell activity. Clusters in a PPI network may overlap where a protein is involved in multiple functional modules. To identify overlapping structures in protein functional modules, this paper proposes a novel overlapping community detection algorithm based on the neighboring local clustering coefficient (NLC). The contributions of the NLC algorithm are threefold: (i) Combine the edge-based community detection method with local expansion in seed selection and the local clustering coefficient of neighboring nodes to improve the accuracy of seed selection; (ii) A method of measuring the distance between edges is improved to make the result of community division more accurate; (iii) A community optimization strategy for the excessive overlapping nodes makes the overlapping structure more reasonable. The experimental results on standard networks, Lancichinetti-Fortunato-Radicchi (LFR) benchmark networks and PPI networks show that the NLC algorithm can improve the Extended modularity (EQ) value and Normalized Mutual Information (NMI) value of the community division, which verifies that the algorithm can not only detect reasonable communities but also identify overlapping structures in networks.

scholarly journals Structure and Evolution of the International Pesticide Trade Networks

2021 ◽  
Vol 9 ◽  
Author(s):  
Jian-An Li ◽  
Wen-Jie Xie ◽  
Wei-Xing Zhou
Keyword(s):  
International Trade ◽  
Dynamic Properties ◽  
Network Size ◽  
Clustering Coefficient ◽  
Trade Networks ◽  
The World ◽  
Local Clustering ◽  
Increasing Demand ◽  
Positive Correlations ◽  
Local Clustering Coefficient

To meet the increasing demand for food around the world, pesticides are widely used and will continue to be widely used in agricultural production to reduce yield losses and maintain product quality. International pesticide trade serves to reallocate the distribution of pesticides around the world. We investigate the statistical properties of the international trade networks of five categories of pesticides from the view angle of temporal directed and weighted networks. We observed an overall increasing trend in network size, network density, average in- and out-degrees, average in- and out-strengths, temporal similarity, and link reciprocity, indicating that the rising globalization of pesticides trade is driving the networks denser. However, the distributions of link weights remain unchanged along time for the five categories of pesticides. In addition, all the networks are disassortatively mixed because large importers or exporters are more likely to trade with small exporters or importers. We also observed positive correlations between in-degree and out-degree, in-strength and out-strength, link reciprocity and in-degree, out-degree, in-strength, and out-strength, while node’s local clustering coefficient is negatively related to in-degree, out-degree, in-strength, and out-strength. We show that some structural and dynamic properties of the international pesticide trade networks are different from those of the international trade networks, highlighting the presence of idiosyncratic features of different goods and products in the international trade.

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 ≥ 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 Disparity of clustering coefficients in the Holme‒Kim network model

2018 ◽  
Vol 50 (3) ◽  
pp. 918-943
Author(s):  
R. I. Oliveira ◽  
R. Ribeiro ◽  
R. Sanchis
Keyword(s):  
Random Graph ◽  
Network Model ◽  
Slow Rate ◽  
Clustering Coefficient ◽  
Concentration Inequality ◽  
Free Graph ◽  
Scale Free ◽  
Local Clustering ◽  
Clustering Coefficients ◽  
Local Clustering Coefficient

Abstract The Holme‒Kim random graph process is a variant of the Barabási‒Álbert scale-free graph that was designed to exhibit clustering. In this paper we show that whether the model does indeed exhibit clustering depends on how we define the clustering coefficient. In fact, we find that the local clustering coefficient typically remains positive whereas global clustering tends to 0 at a slow rate. These and other results are proven via martingale techniques, such as Freedman's concentration inequality combined with a bootstrapping argument.

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