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 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.

TOPOLOGICAL ANALYSIS OF SPECIFIC SPATIAL COMPLEX NETWORKS

2009 ◽  
Vol 12 (01) ◽  
pp. 45-71 ◽  
Author(s):  
JUN WANG ◽  
GREGORY PROVAN
Keyword(s):  
Random Graph ◽  
Optimization Model ◽  
Preferential Attachment ◽  
Topological Analysis ◽  
Brain Networks ◽  
Graph Model ◽  
Spatial Networks ◽  
Electronic Circuits ◽  
Random Graph Model ◽  
Attachment Model

Based on analyses of specific spatial networks, we compare the accuracy of three models in capturing topologies of two types of spatial networks: electronic circuits and brain networks. The models analyzed are an optimization model trading off multiple-objective constraints, an extended preferential attachment model with spatial constraints, and the generalized random graph model. First, we find that the optimization model and the spatial preferential attachment model can generate similar topological structures under appropriate parameters. Second, our experiments surprisingly show that the simple generalized random graph model outperforms the two proposed models. Third, we find that a series of spatial networks under global optimization of wire length, including the electronic circuits, brain networks, neuronal networks and transportation networks, have high s-metric values close to those of the corresponding generalized random graph models. These s-metric observations explain why the generalized random graph model can match the electronic circuits and the brain networks well from a probabilistic viewpoint, and distinguish their structures from self-organized spatial networks, such as the Internet.

scholarly journals Limit theorems for the weights and the degrees in anN-interactions random graph model

2016 ◽  
Vol 14 (1) ◽  
pp. 414-424 ◽  
Author(s):  
István Fazekas ◽  
Bettina Porvázsnyik
Keyword(s):  
Asymptotic Behaviour ◽  
Random Graph ◽  
Limit Theorems ◽  
Preferential Attachment ◽  
Graph Model ◽  
Maximal Degree ◽  
Random Graph Model ◽  
Maximal Weight ◽  
Martingale Methods

Abstract A random graph evolution based on interactions of N vertices is studied. During the evolution both the preferential attachment rule and the uniform choice of vertices are allowed. The weight of an M-clique means the number of its interactions. The asymptotic behaviour of the weight of a fixed M-clique is studied. Asymptotic theorems for the weight and the degree of a fixed vertex are also presented. Moreover, the limits of the maximal weight and the maximal degree are described. The proofs are based on martingale methods.

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 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.

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.

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