scholarly journals On Symmetry of Uniform and Preferential Attachment Graphs

10.37236/4415 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Abram Magner ◽  
Svante Janson ◽  
Giorgos Kollias ◽  
Wojciech Szpankowski
Keyword(s):  
High Probability ◽  
Natural Variation ◽  
Dynamic Models ◽  
Preferential Attachment ◽  
Graph Structure ◽  
Constant Number ◽  
Network Growth ◽  
Graph Size ◽  
Source Models ◽  
Attachment Model

Motivated by the problem of graph structure compression under realistic source models, we study the symmetry behavior of preferential and uniform attachment graphs. These are two dynamic models of network growth in which new nodes attach to a constant number $m$ of existing ones according to some attachment scheme. We prove symmetry results for $m=1$ and $2$, and we conjecture that for $m\geq 3$, both models yield asymmetry with high probability. We provide new empirical evidence in terms of graph defect. We also prove that vertex defects in the uniform attachment model grow at most logarithmically with graph size, then use this to prove a weak asymmetry result for all values of $m$ in the uniform attachment model. Finally, we introduce a natural variation of the two models that incorporates preference of new nodes for nodes of a similar age, and we show that the change introduces symmetry for all values of $m$.

WEIGHTED EVOLVING NETWORKS WITH INTRINSIC STRENGTH

2007 ◽  
Vol 18 (09) ◽  
pp. 1435-1442 ◽  
Author(s):  
XIANMIN GENG ◽  
GUANGHUI WEN
Keyword(s):  
Preferential Attachment ◽  
Dynamical Evolution ◽  
Intrinsic Property ◽  
Network Growth ◽  
Evolving Networks ◽  
Attachment Model ◽  
Analytical Expressions ◽  
Power Law Distributions ◽  
Intrinsic Strength ◽  
Good Agreement

In this paper, we introduce the concept of intrinsic strength which is used to describe the node's intrinsic property. Furthermore, we present a single preferential attachment model for the evolution of weighted networks in which the network growth is coupled with dynamical evolution of weights and intrinsic strength. The model yields a nontrivial time evolution of nodes' properties and generalized power law distributions for the weight, strength and degree, as confirmed in many real networks. The numerical simulation results are in good agreement with the analytical expressions.

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.

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.

scholarly journals Likelihood-based approach to discriminate mixtures of network models that vary in time

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Naomi A. Arnold ◽  
Raul J. Mondragón ◽  
Richard G. Clegg
Keyword(s):  
Network Model ◽  
Change Point ◽  
Network Science ◽  
Preferential Attachment ◽  
Network Models ◽  
Explanatory Models ◽  
Growth Mechanisms ◽  
Artificial Data ◽  
Network Growth ◽  
Over Time

AbstractDiscriminating between competing explanatory models as to which is more likely responsible for the growth of a network is a problem of fundamental importance for network science. The rules governing this growth are attributed to mechanisms such as preferential attachment and triangle closure, with a wealth of explanatory models based on these. These models are deliberately simple, commonly with the network growing according to a constant mechanism for its lifetime, to allow for analytical results. We use a likelihood-based framework on artificial data where the network model changes at a known point in time and demonstrate that we can recover the change point from analysis of the network. We then use real datasets and demonstrate how our framework can show the changing importance of network growth mechanisms over time.

Clustering Coefficient of a Spatial Preferential Attachment Model

2018 ◽  
Vol 98 (1) ◽  
pp. 304-307 ◽  
Author(s):  
L. N. Iskhakov ◽  
M. S. Mironov ◽  
L. A. Prokhorenkova ◽  
B. Kamiński ◽  
P. Prałat
Keyword(s):  
Preferential Attachment ◽  
Clustering Coefficient ◽  
Attachment Model

scholarly journals Nonuniform Distribution of Nodes in the Spatial Preferential Attachment Model

2015 ◽  
Vol 12 (1-2) ◽  
pp. 121-144 ◽  
Author(s):  
Jeannette Janssen ◽  
Paweł Prałat ◽  
Rory Wilson
Keyword(s):  
Preferential Attachment ◽  
Nonuniform Distribution ◽  
Attachment Model
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