Models of network formation

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Degree Distribution ◽  
Network Formation ◽  
Preferential Attachment ◽  
Network Size ◽  
Large Network ◽  
Citation Networks ◽  
Airline Networks ◽  
Attachment Model ◽  
Delivery Networks

This chapter describes models of the growth or formation of networks, with a particular focus on preferential attachment models. It starts with a discussion of the classic preferential attachment model for citation networks introduced by Price, including a complete derivation of the degree distribution in the limit of large network size. Subsequent sections introduce the Barabasi-Albert model and various generalized preferential attachment models, including models with addition or removal of extra nodes or edges and models with nonlinear preferential attachment. Also discussed are node copying models and models in which networks are formed by optimization processes, such as delivery networks or airline networks.

Directed preferential attachment models: Limiting degree distributions and their tails

2020 ◽  
Vol 57 (1) ◽  
pp. 122-136
Author(s):  
Tom Britton
Keyword(s):  
Explicit Expression ◽  
Empirical Evidence ◽  
Explicit Form ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Outgoing Edge ◽  
Tail Probabilities ◽  
Attachment Model ◽  
Birth Processes

AbstractThe directed preferential attachment model is revisited. A new exact characterization of the limiting in- and out-degree distribution is given by two independent pure birth processes that are observed at a common exponentially distributed time T (thus creating dependence between in- and out-degree). The characterization gives an explicit form for the joint degree distribution, and this confirms previously derived tail probabilities for the two marginal degree distributions. The new characterization is also used to obtain an explicit expression for tail probabilities in which both degrees are large. A new generalized directed preferential attachment model is then defined and analyzed using similar methods. The two extensions, motivated by empirical evidence, are to allow double-directed (i.e. undirected) edges in the network, and to allow the probability of connecting an ingoing (outgoing) edge to a specified node to also depend on the out-degree (in-degree) of that node.

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.

The Influence of the Randomness on Average Path Length

2011 ◽  
Vol 219-220 ◽  
pp. 791-794
Author(s):  
Xin Chun Wang ◽  
Yu Bo Jiang
Keyword(s):  
Degree Distribution ◽  
Path Length ◽  
Network Formation ◽  
Preferential Attachment ◽  
Mean Field ◽  
Average Path Length ◽  
Current Structure ◽  
Field Approach ◽  
Evolving Model

The evolving rule includes random attachment and preferential attachment and so on. It forms different network formation by different attachment. In order to analyze this character, this paper put forward an evolving model. It contains two kinds of attachment: some are connected at random, and others are connected based on current structure of the network, and in the model the deleted links are considered. The degree distribution and degree exponent are obtained by mean-field approach. Then it discusses the influence of randomness on the average path length.

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.

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 EMERGENCE OF MULTISCALING IN HETEROGENEOUS COMPLEX NETWORKS

2007 ◽  
Vol 18 (10) ◽  
pp. 1591-1607 ◽  
Author(s):  
A. SANTIAGO ◽  
R. M. BENITO
Keyword(s):  
Complex Networks ◽  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Threshold Model ◽  
Specific Form ◽  
Numerical Evidence ◽  
Attachment Model ◽  
Attachment Mechanism

In this paper we provide numerical evidence of the richer behavior of the connectivity degrees in heterogeneous preferential attachment networks in comparison to their homogeneous counterparts. We analyze the degree distribution in the threshold model, a preferential attachment model where the affinity between node states biases the attachment probabilities of links. We show that the degree densities exhibit a power-law multiscaling which points to a signature of heterogeneity in preferential attachment networks. This translates into a power-law scaling in the degree distribution, whose exponent depends on the specific form of heterogeneity in the attachment mechanism.

scholarly journals Network growth with preferential attachment and without “rich get richer” mechanism

2015 ◽  
Vol 27 (02) ◽  
pp. 1650020
Author(s):  
A. Lachgar ◽  
A. Achahbar
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Target Node ◽  
Network Growth ◽  
Perfect Agreement ◽  
Equation Solution ◽  
Attachment Model ◽  
Law Degree ◽  
Growing Network

We propose a simple preferential attachment model of growing network using the complementary probability of Barabási–Albert (BA) model, i.e. [Formula: see text]. In this network, new nodes are preferentially attached to not well connected nodes. Numerical simulations, in perfect agreement with the master equation solution, give an exponential degree distribution. This suggests that the power law degree distribution is a consequence of preferential attachment probability together with “rich get richer” phenomena. We also calculate the average degree of a target node at time t[Formula: see text] and its fluctuations, to have a better view of the microscopic evolution of the network, and we also compare the results with BA model.

scholarly journals Nonstandard regular variation of in-degree and out-degree in the preferential attachment model

2016 ◽  
Vol 53 (1) ◽  
pp. 146-161 ◽  
Author(s):  
Gennady Samorodnitsky ◽  
Sidney Resnick ◽  
Don Towsley ◽  
Richard Davis ◽  
Amy Willis ◽  
...  
Keyword(s):  
Growth Model ◽  
Degree Distribution ◽  
Regular Variation ◽  
Joint Distribution ◽  
Preferential Attachment ◽  
Directed Edge ◽  
Network Growth ◽  
Regularly Varying ◽  
Attachment Model

Abstract For the directed edge preferential attachment network growth model studied by Bollobás et al. (2003) and Krapivsky and Redner (2001), we prove that the joint distribution of in-degree and out-degree has jointly regularly varying tails. Typically, the marginal tails of the in-degree distribution and the out-degree distribution have different regular variation indices and so the joint regular variation is nonstandard. Only marginal regular variation has been previously established for this distribution in the cases where the marginal tail indices are different.

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.

What Influences Relationship Formation in a Global Civil Society Network? An Examination of Valued Multiplex Relations

2021 ◽  
pp. 009365022110161
Author(s):  
Adam J. Saffer ◽  
Andrew Pilny ◽  
Erich J. Sommerfeldt
Keyword(s):  
Communication Network ◽  
Civil Society ◽  
Real World ◽  
Network Formation ◽  
Preferential Attachment ◽  
Interorganizational Networks ◽  
Communication Research ◽  
Relationship Formation ◽  
Interorganizational Communication ◽  
Strength Of Ties

Recent interorganizational communication research has taken up the question: why are networks structured the way they are? This line of inquiry has advanced communication network research by helping explain how and why networks take on certain structures or why certain organizations become positioned advantageously (or not). Yet, those studies assume relationships among organizations are either present or absent. This study considers how the strength of ties and multiplex relationships among organizations may reveal a more complex explanation for why networks take on certain structures. Our results challenge some long held assumptions about the mechanisms that influence network formation. For instance, our results offer important insights into the consequences of closure mechanisms, the applicability of preferential attachment to real-world networks, and the nuances of homophily in network formation on multidimensional relationships in a communication network. Implications for interorganizational networks are discussed.

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