scholarly journals The Degree Distribution of Thickened Trees

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Michael Drmota ◽  
Bernhard Gittenberger ◽  
Alois Panholzer
Keyword(s):  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Combinatorial Structure ◽  
Random Real ◽  
Asymptotic Power ◽  
Scale Free ◽  
International Audience ◽  
Recursive Trees

International audience We develop a combinatorial structure to serve as model of random real world networks. Starting with plane oriented recursive trees we substitute the nodes by more complex graphs. In such a way we obtain graphs having a global tree-like structure while locally looking clustered. This fits with observations obtained from real-world networks. In particular we show that the resulting graphs are scale-free, that is, the degree distribution has an asymptotic power law.

scholarly journals A Spatial Biosurveillance Synthetic Data Generator in R

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
Drew Levin ◽  
Patrick Finley
Keyword(s):  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Transportation Network ◽  
Synthetic Data ◽  
Data Sets ◽  
Real World Data ◽  
Data Set ◽  
Scale Free ◽  
Data Generator

ObjectiveTo develop a spatially accurate biosurveillance synthetic datagenerator for the testing, evaluation, and comparison of new outbreakdetection techniques.IntroductionDevelopment of new methods for the rapid detection of emergingdisease outbreaks is a research priority in the field of biosurveillance.Because real-world data are often proprietary in nature, scientists mustutilize synthetic data generation methods to evaluate new detectionmethodologies. Colizza et. al. have shown that epidemic spread isdependent on the airline transportation network [1], yet current datagenerators do not operate over network structures.Here we present a new spatial data generator that models thespread of contagion across a network of cities connected by airlineroutes. The generator is developed in the R programming languageand produces data compatible with the popular `surveillance’ softwarepackage.MethodsColizza et. al. demonstrate the power-law relationships betweencity population, air traffic, and degree distribution [1]. We generate atransportation network as a Chung-Lu random graph [2] that preservesthese scale-free relationships (Figure 1).First, given a power-law exponent and a desired number of cities,a probability mass function (PMF) is generated that mirrors theexpected degree distribution for the given power-law relationship.Values are then sampled from this PMF to generate an expecteddegree (number of connected cities) for each city in the network.Edges (airline connections) are added to the network probabilisticallyas described in [2]. Unconnected graph components are each joinedto the largest component using linear preferential attachment. Finally,city sizes are calculated based on an observed three-quarter power-law scaling relationship with the sampled degree distribution.Each city is represented as a customizable stochastic compartmentalSIR model. Transportation between cities is modeled similar to [2].An infection is initialized in a single random city and infection countsare recorded in each city for a fixed period of time. A consistentfraction of the modeled infection cases are recorded as daily clinicvisits. These counts are then added onto statically generated baselinedata for each city to produce a full synthetic data set. Alternatively,data sets can be generated using real-world networks, such as the onemaintained by the International Air Transport Association.ResultsDynamics such as the number of cities, degree distribution power-law exponent, traffic flow, and disease kinetics can be customized.In the presented example (Figure 2) the outbreak spreads over a 20city transportation network. Infection spreads rapidly once the morepopulated hub cities are infected. Cities that are multiple flights awayfrom the initially infected city are infected late in the process. Thegenerator is capable of creating data sets of arbitrary size, length, andconnectivity to better mirror a diverse set of observed network types.ConclusionsNew computational methods for outbreak detection andsurveillance must be compared to established approaches. Outbreakmitigation strategies require a realistic model of human transportationbehavior to best evaluate impact. These actions require test data thataccurately reflect the complexity of the real-world data they wouldbe applied to. The outbreak data generated here represents thecomplexity of modern transportation networks and are made to beeasily integrated with established software packages to allow for rapidtesting and deployment.Randomly generated scale-free transportation network with a power-lawdegree exponent ofλ=1.8. City and link sizes are scaled to reflect their weight.An example of observed daily outbreak-related clinic visits across a randomlygenerated network of 20 cities. Each city is colored by the number of flightsrequired to reach the city from the initial infection location. These generatedcounts are then added onto baseline data to create a synthetic data set forexperimentation.KeywordsSimulation; Network; Spatial; Synthetic; Data

scholarly journals Width of a scale-free tree

2005 ◽  
Vol 42 (03) ◽  
pp. 839-850 ◽  
Author(s):  
Zsolt Katona
Keyword(s):  
Theoretical Result ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Random Graph Model ◽  
Scale Free ◽  
Slight Generalization ◽  
Free Tree ◽  
Law Degree

Consider the random graph model of Barabási and Albert, where we add a new vertex in every step and connect it to some old vertices with probabilities proportional to their degrees. If we connect it to only one of the old vertices then this will be a tree. These graphs have been shown to have a power-law degree distribution, the same as that observed in some large real-world networks. We are interested in the width of the tree and we show that it is at the nth step; this also holds for a slight generalization of the model with another constant. We then see how this theoretical result can be applied to directory trees.

scholarly journals Width of a scale-free tree

2005 ◽  
Vol 42 (3) ◽  
pp. 839-850 ◽  
Author(s):  
Zsolt Katona
Keyword(s):  
Theoretical Result ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Random Graph Model ◽  
Scale Free ◽  
Slight Generalization ◽  
Free Tree ◽  
Law Degree

Consider the random graph model of Barabási and Albert, where we add a new vertex in every step and connect it to some old vertices with probabilities proportional to their degrees. If we connect it to only one of the old vertices then this will be a tree. These graphs have been shown to have a power-law degree distribution, the same as that observed in some large real-world networks. We are interested in the width of the tree and we show that it is at the nth step; this also holds for a slight generalization of the model with another constant. We then see how this theoretical result can be applied to directory trees.

EVOLVING NETWORKS DRIVEN BY NODE DYNAMICS

2004 ◽  
Vol 18 (17n19) ◽  
pp. 2540-2546 ◽  
Author(s):  
ZHENGPING FAN ◽  
GUANRONG CHEN
Keyword(s):  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Point Of View ◽  
Scale Free ◽  
Evolving Networks ◽  
Dynamical Behaviors ◽  
Scale Free Networks ◽  
Theory Point ◽  
Energy Signal

In this paper, a new model of evolving networks is proposed based on the dynamical behaviors of nodes in the network. The probability that an existing node in the current network receives a new link from the newly added node is proportional to the defined "activity" of that particular node, which is equivalent to the energy signal of the system from a control theory point of view. This network is found to have the scale-free feature with the degree distribution in a power-law form. This model provides another explanation for the emergence of scale-free networks in many real-world examples.

scholarly journals Power-law distribution of degree–degree distance: A better representation of the scale-free property of complex networks

2020 ◽  
Vol 117 (26) ◽  
pp. 14812-14818 ◽  
Author(s):  
Bin Zhou ◽  
Xiangyi Meng ◽  
H. Eugene Stanley
Keyword(s):  
Complex Networks ◽  
Complex Network ◽  
Power Law ◽  
Real World ◽  
Preferential Attachment ◽  
Statistical Tests ◽  
Finite Size ◽  
Distance Distribution ◽  
Scale Free ◽  
Degree Distance

Whether real-world complex networks are scale free or not has long been controversial. Recently, in Broido and Clauset [A. D. Broido, A. Clauset,Nat. Commun.10, 1017 (2019)], it was claimed that the degree distributions of real-world networks are rarely power law under statistical tests. Here, we attempt to address this issue by defining a fundamental property possessed by each link, the degree–degree distance, the distribution of which also shows signs of being power law by our empirical study. Surprisingly, although full-range statistical tests show that degree distributions are not often power law in real-world networks, we find that in more than half of the cases the degree–degree distance distributions can still be described by power laws. To explain these findings, we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted, two-way selection process. The model does not always produce solid power-law distributions but predicts that the degree–degree distance distribution exhibits stronger power-law behavior than the degree distribution of a finite-size network, especially when the network is dense. We test the strength of our model and its predictive power by examining how real-world networks evolve into an overly dense stage and how the corresponding distributions change. We propose that being scale free is a property of a complex network that should be determined by its underlying mechanism (e.g., preferential attachment) rather than by apparent distribution statistics of finite size. We thus conclude that the degree–degree distance distribution better represents the scale-free property of a complex network.

scholarly journals The use of the power law for designing wireless networks

2018 ◽  
Vol 21 ◽  
pp. 00012
Author(s):  
Andrzej Paszkiewicz
Keyword(s):  
Wireless Networks ◽  
Power Law ◽  
Degree Distribution ◽  
Limited Resources ◽  
Random Networks ◽  
New Approach ◽  
Available Bandwidth ◽  
Scale Free ◽  
Scale Free Networks ◽  
Node Removal

The paper concerns the use of the scale-free networks theory and the power law in designing wireless networks. An approach based on generating random networks as well as on the classic Barabási-Albert algorithm were presented. The paper presents a new approach taking the limited resources for wireless networks into account, such as available bandwidth. In addition, thanks to the introduction of opportunities for dynamic node removal it was possible to realign processes occurring in wireless networks. After introduction of these modifications, the obtained results were analyzed in terms of a power law and the degree distribution of each node.

scholarly journals Ultra-small scale-free geometric networks

2006 ◽  
Vol 43 (3) ◽  
pp. 665-677 ◽  
Author(s):  
J. E. Yukich
Keyword(s):  
Long Range ◽  
Power Law ◽  
Degree Distribution ◽  
Small Scale ◽  
Graph Distance ◽  
Power Law Distribution ◽  
Scale Free

We consider a family of long-range percolation models (Gp)p>0on ℤdthat allow dependence between edges and have the following connectivity properties forp∈ (1/d, ∞): (i) the degree distribution of vertices inGphas a power-law distribution; (ii) the graph distance between pointsxandyis bounded by a multiple of logpdlogpd|x-y| with probability 1 -o(1); and (iii) an adversary can delete a relatively small number of nodes fromGp(ℤd∩ [0,n]d), resulting in two large, disconnected subgraphs.

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 Generation of 2-mode scale-free graphs for link-level internet topology modeling

PLoS ONE ◽  
2020 ◽  
Vol 15 (11) ◽  
pp. e0240100
Author(s):  
Khalid Bakhshaliyev ◽  
Mehmet Hadi Gunes
Keyword(s):  
Power Law ◽  
Degree Distribution ◽  
Autonomous Systems ◽  
Node Degree ◽  
Bipartite Network ◽  
Research Issues ◽  
Scale Free ◽  
Internet Topology ◽  
Scale Free Graphs ◽  
Multi Access

Comprehensive analysis that aims to understand the topology of real-world networks and the development of algorithms that replicate their characteristics has been significant research issues. Although the accuracy of newly developed network protocols or algorithms does not depend on the underlying topology, the performance generally depends on the topology. As a result, network practitioners have concentrated on generating representative synthetic topologies and utilize them to investigate the performance of their design in simulation or emulation environments. Network generators typically represent the Internet topology as a graph composed of point-to-point links. In this study, we discuss the implications of multi-access links on the synthetic network generation and modeling of the networks as bi-partite graphs to represent both subnetworks and routers. We then analyze the characteristics of sampled Internet topology data sets from backbone Autonomous Systems (AS) and observe that in addition to the commonly recognized power-law node degree distribution, the subnetwork size and the router interface distributions often exhibit power-law characteristics. We introduce a SubNetwork Generator (SubNetG) topology generation approach that incorporates the observed measurements to produce bipartite network topologies. In particular, generated topologies capture the 2-mode relation between the layer-2 (i.e., the subnetwork and interface distributions) and the layer-3 (i.e., the degree distribution) that is missing from the current network generators that produce 1-mode graphs. The SubNetG source code and experimental data is available at https://github.com/netml/sonet.

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.

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