scholarly journals Ordered increasing $k$-trees: Introduction and analysis of a preferential attachment network model

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Alois Panholzer ◽  
Georg Seitz
Keyword(s):  
Network Model ◽  
Preferential Attachment ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Tree Model ◽  
Random Graph Model ◽  
Local Clustering ◽  
Tree Models ◽  
International Audience ◽  
Local Clustering Coefficient

International audience We introduce a random graph model based on $k$-trees, which can be generated by applying a probabilistic preferential attachment rule, but which also has a simple combinatorial description. We carry out a precise distributional analysis of important parameters for the network model such as the degree, the local clustering coefficient and the number of descendants of the nodes and root-to-node distances. We do not only obtain results for random nodes, but in particular we also get a precise description of the behaviour of parameters for the $j$-th inserted node in a random $k$-tree of size $n$, where $j=j(n)$ might grow with $n$. The approach presented is not restricted to this specific $k$-tree model, but can also be applied to other evolving $k$-tree models.

scholarly journals Sublinear Random Access Generators for Preferential Attachment Graphs

2021 ◽  
Vol 17 (4) ◽  
pp. 1-26
Author(s):  
Guy Even ◽  
Reut Levi ◽  
Moti Medina ◽  
Adi Rosén
Keyword(s):  
Random Graph ◽  
Preferential Attachment ◽  
Random Access ◽  
Graph Model ◽  
Efficient Estimation ◽  
Sublinear Algorithms ◽  
Tree Model ◽  
Random Graph Model ◽  
Time Space ◽  
Attachment Model

We consider the problem of sampling from a distribution on graphs, specifically when the distribution is defined by an evolving graph model, and consider the time, space, and randomness complexities of such samplers. In the standard approach, the whole graph is chosen randomly according to the randomized evolving process, stored in full, and then queries on the sampled graph are answered by simply accessing the stored graph. This may require prohibitive amounts of time, space, and random bits, especially when only a small number of queries are actually issued. Instead, we propose a setting where one generates parts of the sampled graph on-the-fly, in response to queries, and therefore requires amounts of time, space, and random bits that are a function of the actual number of queries. Yet, the responses to the queries correspond to a graph sampled from the distribution in question. Within this framework, we focus on two random graph models: the Barabási-Albert Preferential Attachment model (BA-graphs) ( Science , 286 (5439):509–512) (for the special case of out-degree 1) and the random recursive tree model ( Theory of Probability and Mathematical Statistics , (51):1–28). We give on-the-fly generation algorithms for both models. With probability 1-1/poly( n ), each and every query is answered in polylog( n ) time, and the increase in space and the number of random bits consumed by any single query are both polylog( n ), where n denotes the number of vertices in the graph. Our work thus proposes a new approach for the access to huge graphs sampled from a given distribution, and our results show that, although the BA random graph model is defined by a sequential process, efficient random access to the graph’s nodes is possible. In addition to the conceptual contribution, efficient on-the-fly generation of random graphs can serve as a tool for the efficient simulation of sublinear algorithms over large BA-graphs, and the efficient estimation of their on such graphs.

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.

Clustering in preferential attachment random graphs with edge-step

2021 ◽  
Vol 58 (4) ◽  
pp. 890-908
Author(s):  
Caio Alves ◽  
Rodrigo Ribeiro ◽  
Rémy Sanchis
Keyword(s):  
Preferential Attachment ◽  
Graph Model ◽  
Positive Function ◽  
Clique Number ◽  
Geometric Graph ◽  
Clustering Coefficient ◽  
Concentration Inequality ◽  
Random Graph Model ◽  
Graph Properties ◽  
Attachment Model

AbstractWe prove concentration inequality results for geometric graph properties of an instance of the Cooper–Frieze [5] preferential attachment model with edge-steps. More precisely, we investigate a random graph model that at each time $t\in \mathbb{N}$ , with probability p adds a new vertex to the graph (a vertex-step occurs) or with probability $1-p$ an edge connecting two existent vertices is added (an edge-step occurs). We prove concentration results for the global clustering coefficient as well as the clique number. More formally, we prove that the global clustering, with high probability, decays as $t^{-\gamma(p)}$ for a positive function $\gamma$ of p, whereas the clique number of these graphs is, up to subpolynomially small factors, of order $t^{(1-p)/(2-p)}$ .

scholarly journals Local clustering coefficient of spatial preferential attachment model

2019 ◽  
Author(s):  
Lenar Iskhakov ◽  
Bogumił Kamiński ◽  
Maksim Mironov ◽  
Paweł Prałat ◽  
Liudmila Prokhorenkova
Keyword(s):  
Preferential Attachment ◽  
Clustering Coefficient ◽  
Local Clustering ◽  
Attachment Model ◽  
Local Clustering Coefficient

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.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

A WEIGHTED LOCAL-WORLD EVOLVING NETWORK MODEL BASED ON THE EDGE WEIGHTS PREFERENTIAL SELECTION

2013 ◽  
Vol 27 (12) ◽  
pp. 1350039 ◽  
Author(s):  
PING LI ◽  
QINGZHEN ZHAO ◽  
HAITANG WANG
Keyword(s):  
Network Model ◽  
Preferential Attachment ◽  
Online Social Network ◽  
Evolutionary Model ◽  
Strength Distribution ◽  
Clustering Coefficient ◽  
Time Step ◽  
Preferential Selection ◽  
C Elegans ◽  
Attachment Mechanism

In this paper, we use the edge weights preferential attachment mechanism to build a new local-world evolutionary model for weighted networks. It is different from previous papers that the local-world of our model consists of edges instead of nodes. Each time step, we connect a new node to two existing nodes in the local-world through the edge weights preferential selection. Theoretical analysis and numerical simulations show that the scale of the local-world affect on the weight distribution, the strength distribution and the degree distribution. We give the simulations about the clustering coefficient and the dynamics of infectious diseases spreading. The weight dynamics of our network model can portray the structure of realistic networks such as neural network of the nematode C. elegans and Online Social Network.

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.

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