ESTIMATION OF POWER-LAW EXPONENT OF DEGREE DISTRIBUTION USING MEAN VERTEX DEGREE

2009 ◽  
Vol 23 (17) ◽  
pp. 2073-2088 ◽  
Author(s):  
NOBUTOSHI IKEDA
Keyword(s):  
Numerical Simulations ◽  
Power Law ◽  
Degree Distribution ◽  
Minimum Degree ◽  
Preferential Attachment ◽  
Random Walkers ◽  
Power Law Exponent ◽  
Growing Networks ◽  
The Mean ◽  
Ordinary Integral

The power-law exponent γ describing the form of the degree distribution of networks is related to the mean degree of the networks. This relation provides a useful method to estimate this exponent because mean degrees can easily be obtained by the number of vertices and edges in the networks. We improved the relation obtained only by ordinary integral approximation, and examined its availability for number of vertices, minimum degree considered, and range of γ, taking also into consideration the number of vertices with minimum degree. As a result, we were able to estimate slight deviations of γ from 3, which are usually observed in numerical simulations of growing networks with linear preferential attachment. Furthermore, using this method, we were also able to predict to what extent γ changed by joining pre-existing vertices in growing networks or by imposing restrictions to prevent the gaining of new edges for certain vertices. For cases where γ < 2, we estimated the power-law exponent of degree distributions of networks formed by traces of random walkers from the increased rate of vertices with created edges.

scholarly journals Preferential Attachment with Location-Based Choice: Degree Distribution in the Noncondensation Phase

2021 ◽  
Vol 184 (1) ◽  
Author(s):  
Arne Grauer ◽  
Lukas Lüchtrath ◽  
Mark Yarrow
Keyword(s):  
Phase Transition ◽  
Power Law ◽  
Degree Distribution ◽  
Maximum Degree ◽  
Preferential Attachment ◽  
Critical Value ◽  
Time Step ◽  
Power Law Exponent ◽  
Attachment Model ◽  
Sampling Probability

AbstractWe consider the preferential attachment model with location-based choice introduced by Haslegrave et al. (Random Struct Algorithms 56(3):775–795, 2020) as a model in which condensation phenomena can occur. In this model, each vertex carries an independent and uniformly distributed location. Starting from an initial tree, the model evolves in discrete time. At every time step, a new vertex is added to the tree by selecting r candidate vertices from the graph with replacement according to a sampling probability proportional to these vertices’ degrees. The new vertex then connects to one of the candidates according to a given probability associated to the ranking of their locations. In this paper, we introduce a function that describes the phase transition when condensation can occur. Considering the noncondensation phase, we use stochastic approximation methods to investigate bounds for the (asymptotic) proportion of vertices inside a given interval of a given maximum degree. We use these bounds to observe a power law for the asymptotic degree distribution described by the aforementioned function. Hence, this function fully characterises the properties we are interested in. The power law exponent takes the critical value one at the phase transition between the condensation–noncondensation phase.

scholarly journals Percolation phase transition in weight-dependent random connection models

2021 ◽  
Vol 53 (4) ◽  
pp. 1090-1114
Author(s):  
Peter Gracar ◽  
Lukas Lüchtrath ◽  
Peter Mörters
Keyword(s):  
Phase Transition ◽  
Poisson Process ◽  
Random Graphs ◽  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Dimensional Space ◽  
Geometric Model ◽  
Scale Free ◽  
Power Law Exponent

AbstractWe investigate spatial random graphs defined on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a non-trivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.

scholarly journals The structure of co-publications multilayer network

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Ghislain Romaric Meleu ◽  
Paulin Yonta Melatagia
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Small World ◽  
Generation Model ◽  
Small World Network ◽  
Power Law Distribution ◽  
Multilayer Networks ◽  
Multilayer Network ◽  
Scientific Papers

AbstractUsing the headers of scientific papers, we have built multilayer networks of entities involved in research namely: authors, laboratories, and institutions. We have analyzed some properties of such networks built from data extracted from the HAL archives and found that the network at each layer is a small-world network with power law distribution. In order to simulate such co-publication network, we propose a multilayer network generation model based on the formation of cliques at each layer and the affiliation of each new node to the higher layers. The clique is built from new and existing nodes selected using preferential attachment. We also show that, the degree distribution of generated layers follows a power law. From the simulations of our model, we show that the generated multilayer networks reproduce the studied properties of co-publication networks.

scholarly journals Typical Distances in Ultrasmall Random Networks

2012 ◽  
Vol 44 (2) ◽  
pp. 583-601 ◽  
Author(s):  
Steffen Dereich ◽  
Christian Mönch ◽  
Peter Mörters
Keyword(s):  
Power Law ◽  
Preferential Attachment ◽  
Shortest Paths ◽  
Giant Component ◽  
Configuration Model ◽  
Random Networks ◽  
Power Law Exponent ◽  
Extra Factor ◽  
Typical Distances

We show that in preferential attachment models with power-law exponent τ ∈ (2, 3) the distance between randomly chosen vertices in the giant component is asymptotically equal to (4 + o(1))log log N / (-log(τ − 2)), where N denotes the number of nodes. This is twice the value obtained for the configuration model with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.

Effect of connection on transport between scale free networks

2017 ◽  
Vol 28 (05) ◽  
pp. 1750064 ◽  
Author(s):  
A. Ould Baba ◽  
O. Bamaarouf ◽  
A. Rachadi ◽  
H. Ez-Zahraouy
Keyword(s):  
Numerical Simulations ◽  
Power Law ◽  
Preferential Attachment ◽  
Electrical Conductance ◽  
Global Network ◽  
Scale Free ◽  
Scale Free Networks ◽  
Conductance Distribution

Using numerical simulations, we investigate the effects of the connectivity and topologies of network on the quality of transport between connected scale free networks. Hence, the flow as the electrical conductance between connected networks is calculated. It is found that the conductance distribution between networks follow a power law [Formula: see text] where [Formula: see text] is the exponent of the global Network of network, we show that the transport in the symmetric growing preferential attachment connection is more efficient than the symmetric static preferential attachment connection. Furthermore, the differences of transport and networks communications properties in the different cases are discussed.

SELF-ORGANIZED CORONA GRAPHS: A DETERMINISTIC COMPLEX NETWORK MODEL WITH HIERARCHICAL STRUCTURE

2019 ◽  
Vol 22 (06) ◽  
pp. 1950019
Author(s):  
ROHAN SHARMA ◽  
BIBHAS ADHIKARI ◽  
TYLL KRUEGER
Keyword(s):  
Power Law ◽  
Small World ◽  
Self Organization ◽  
Scale Free ◽  
Self Organized ◽  
Power Law Exponent ◽  
Growing Networks ◽  
Growing Network ◽  
Hierarchical Pattern ◽  
Corona Graphs

In this paper, we propose a self-organization mechanism for newly appeared nodes during the formation of corona graphs that define a hierarchical pattern in the resulting corona graphs and we call it self-organized corona graphs (SoCG). We show that the degree distribution of SoCG follows power-law in its tail with power-law exponent approximately 2. We also show that the diameter is less equal to 4 for SoCG defined by any seed graph and for certain seed graphs, the diameter remains constant during its formation. We derive lower bounds of clustering coefficients of SoCG defined by certain seed graphs. Thus, the proposed SoCG can be considered as a growing network generative model which is defined by using the corona graphs and a self-organization process such that the resulting graphs are scale-free small-world highly clustered growing networks. The SoCG defined by a seed graph can also be considered as a network with a desired motif which is the seed graph itself.

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 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 Typical Distances in Ultrasmall Random Networks

2012 ◽  
Vol 44 (02) ◽  
pp. 583-601 ◽  
Author(s):  
Steffen Dereich ◽  
Christian Mönch ◽  
Peter Mörters
Keyword(s):  
Power Law ◽  
Preferential Attachment ◽  
Shortest Paths ◽  
Giant Component ◽  
Configuration Model ◽  
Random Networks ◽  
Power Law Exponent ◽  
Extra Factor ◽  
Typical Distances

We show that in preferential attachment models with power-law exponent τ ∈ (2, 3) the distance between randomly chosen vertices in the giant component is asymptotically equal to (4 +o(1))log logN/ (-log(τ − 2)), whereNdenotes the number of nodes. This is twice the value obtained for the configuration model with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.

scholarly journals Asymptotic Degree Distribution of a Kind of Asymmetric Evolving Network

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Zhimin Li ◽  
Zhaolin He ◽  
Chunhua Hu
Keyword(s):  
Exponential Distribution ◽  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Degree Sequence ◽  
Tree Structure ◽  
Uniform Model ◽  
Attachment Model

We propose a kind of evolving network which shows tree structure. The model is a combination of preferential attachment model and uniform model. We show that the proportional degree sequencepkk>1obeys power law, exponential distribution, and other forms according to the relation ofkand parameterm.

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