scholarly journals Random complex networks

2014 ◽  
Vol 1 (3) ◽  
pp. 357-367 ◽  
Author(s):  
Michael Small ◽  
Lvlin Hou ◽  
Linjun Zhang
Keyword(s):  
Complex Networks ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Small World ◽  
Node Degree ◽  
Scale Free ◽  
Wide Range ◽  
Random Complex ◽  
Law Degree ◽  
Attachment Process

Abstract Exactly what is meant by a ‘complex’ network is not clear; however, what is clear is that it is something other than a random graph. Complex networks arise in a wide range of real social, technological and physical systems. In all cases, the most basic categorization of these graphs is their node degree distribution. Particular groups of complex networks may exhibit additional interesting features, including the so-called small-world effect or being scale-free. There are many algorithms with which one may generate networks with particular degree distributions (perhaps the most famous of which is preferential attachment). In this paper, we address what it means to randomly choose a network from the class of networks with a particular degree distribution, and in doing so we show that the networks one gets from the preferential attachment process are actually highly pathological. Certain properties (including robustness and fragility) which have been attributed to the (scale-free) degree distribution are actually more intimately related to the preferential attachment growth mechanism. We focus here on scale-free networks with power-law degree sequences—but our methods and results are perfectly generic.

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 SZNAJD MODEL WITH SYNCHRONOUS UPDATING ON COMPLEX NETWORKS

2005 ◽  
Vol 16 (07) ◽  
pp. 1149-1161 ◽  
Author(s):  
YU-SONG TU ◽  
A. O. SOUSA ◽  
LING-JIANG KONG ◽  
MU-REN LIU
Keyword(s):  
Complex Networks ◽  
Small World ◽  
System Size ◽  
The State ◽  
Clustering Coefficient ◽  
Square Lattice ◽  
Node Degree ◽  
Scale Free ◽  
Free Network ◽  
Formation Step

We analyze the evolution of Sznajd Model with synchronous updating in several complex networks. Similar to the model on square lattice, we have found a transition between the state with nonconsensus and the state with complete consensus in several complex networks. Furthermore, by adjusting the network parameters, we find that a large clustering coefficient does not favor development of a consensus. In particular, in the limit of large system size with the initial concentration p =0.5 of opinion +1, a consensus seems to be never reached for the Watts–Strogatz small-world network, when we fix the connectivity k and the rewiring probability ps; nor for the scale-free network, when we fix the minimum node degree m and the triad formation step probability pt.

scholarly journals Polygon-Based Hierarchical Planar Networks Based on Generalized Apollonian Construction

Physics ◽  
2021 ◽  
Vol 3 (4) ◽  
pp. 998-1014
Author(s):  
Mikhail Tamm ◽  
Dmitry Koval ◽  
Vladimir Stadnichuk
Keyword(s):  
Complex Networks ◽  
Power Law ◽  
Degree Distribution ◽  
Small World ◽  
Scale Free ◽  
Model Network ◽  
Suggested Procedure ◽  
Similar Construction ◽  
Scale Free Graphs ◽  
Planar Networks

Experimentally observed complex networks are often scale-free, small-world and have an unexpectedly large number of small cycles. An Apollonian network is one notable example of a model network simultaneously having all three of these properties. This network is constructed by a deterministic procedure of consequentially splitting a triangle into smaller and smaller triangles. In this paper, a similar construction based on the consequential splitting of tetragons and other polygons with an even number of edges is presented. The suggested procedure is stochastic and results in the ensemble of planar scale-free graphs. In the limit of a large number of splittings, the degree distribution of the graph converges to a true power law with an exponent, which is smaller than three in the case of tetragons and larger than three for polygons with a larger number of edges. It is shown that it is possible to stochastically mix tetragon-based and hexagon-based constructions to obtain an ensemble of graphs with a tunable exponent of degree distribution. Other possible planar generalizations of the Apollonian procedure are also briefly discussed.

scholarly journals Detecting different topologies immanent in scale-free networks with the same degree distribution

2019 ◽  
Vol 116 (14) ◽  
pp. 6701-6706 ◽  
Author(s):  
Dimitrios Tsiotas
Keyword(s):  
Complex Networks ◽  
Power Law ◽  
Network Topology ◽  
Degree Distribution ◽  
Growth Process ◽  
Preferential Attachment ◽  
Self Similarity ◽  
Scale Free ◽  
Scale Free Networks ◽  
Definition Of

The scale-free (SF) property is a major concept in complex networks, and it is based on the definition that an SF network has a degree distribution that follows a power-law (PL) pattern. This paper highlights that not all networks with a PL degree distribution arise through a Barabási−Albert (BA) preferential attachment growth process, a fact that, although evident from the literature, is often overlooked by many researchers. For this purpose, it is demonstrated, with simulations, that established measures of network topology do not suffice to distinguish between BA networks and other (random-like and lattice-like) SF networks with the same degree distribution. Additionally, it is examined whether an existing self-similarity metric proposed for the definition of the SF property is also capable of distinguishing different SF topologies with the same degree distribution. To contribute to this discrimination, this paper introduces a spectral metric, which is shown to be more capable of distinguishing between different SF topologies with the same degree distribution, in comparison with the existing metrics.

scholarly journals A detailed characterization of complex networks using Information Theory

2021 ◽  
Author(s):  
CGS Freitas ◽  
ALL Aquino ◽  
HS Ramos ◽  
Alejandro Frery ◽  
OA Rosso
Keyword(s):  
Information Theory ◽  
Complex Networks ◽  
Complex Network ◽  
Preferential Attachment ◽  
Configuration Model ◽  
Complex Network Theory ◽  
Scale Free ◽  
Network Entropy ◽  
Law Degree

Understanding the structure and the dynamics of networks is of paramount importance for many scientific fields that rely on network science. Complex network theory provides a variety of features that help in the evaluation of network behavior. However, such analysis can be confusing and misleading as there are many intrinsic properties for each network metric. Alternatively, Information Theory methods have gained the spotlight because of their ability to create a quantitative and robust characterization of such networks. In this work, we use two Information Theory quantifiers, namely Network Entropy and Network Fisher Information Measure, to analyzing those networks. Our approach detects non-trivial characteristics of complex networks such as the transition present in the Watts-Strogatz model from k-ring to random graphs; the phase transition from a disconnected to an almost surely connected network when we increase the linking probability of Erdős-Rényi model; distinct phases of scale-free networks when considering a non-linear preferential attachment, fitness, and aging features alongside the configuration model with a pure power-law degree distribution. Finally, we analyze the numerical results for real networks, contrasting our findings with traditional complex network methods. In conclusion, we present an efficient method that ignites the debate on network characterization.

scholarly journals A detailed characterization of complex networks using Information Theory

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Cristopher G. S. Freitas ◽  
Andre L. L. Aquino ◽  
Heitor S. Ramos ◽  
Alejandro C. Frery ◽  
Osvaldo A. Rosso
Keyword(s):  
Information Theory ◽  
Complex Networks ◽  
Complex Network ◽  
Preferential Attachment ◽  
Configuration Model ◽  
Complex Network Theory ◽  
Scale Free ◽  
Network Entropy ◽  
Law Degree

Abstract Understanding the structure and the dynamics of networks is of paramount importance for many scientific fields that rely on network science. Complex network theory provides a variety of features that help in the evaluation of network behavior. However, such analysis can be confusing and misleading as there are many intrinsic properties for each network metric. Alternatively, Information Theory methods have gained the spotlight because of their ability to create a quantitative and robust characterization of such networks. In this work, we use two Information Theory quantifiers, namely Network Entropy and Network Fisher Information Measure, to analyzing those networks. Our approach detects non-trivial characteristics of complex networks such as the transition present in the Watts-Strogatz model from k-ring to random graphs; the phase transition from a disconnected to an almost surely connected network when we increase the linking probability of Erdős-Rényi model; distinct phases of scale-free networks when considering a non-linear preferential attachment, fitness, and aging features alongside the configuration model with a pure power-law degree distribution. Finally, we analyze the numerical results for real networks, contrasting our findings with traditional complex network methods. In conclusion, we present an efficient method that ignites the debate on network characterization.

scholarly journals Dense Networks With Mixture Degree Distribution

2021 ◽  
Vol 9 ◽  
Author(s):  
Xiaomin Wang ◽  
Fei Ma ◽  
Bing Yao
Keyword(s):  
Complex Networks ◽  
Degree Distribution ◽  
Metabolic Networks ◽  
First Passage Time ◽  
Small World ◽  
Passage Time ◽  
Friendship Networks ◽  
Scale Free ◽  
Dense Networks ◽  
Protein Protein Interaction

Complex networks have become a powerful tool to describe the structure and evolution in a large quantity of real networks in the past few years, such as friendship networks, metabolic networks, protein–protein interaction networks, and software networks. While a variety of complex networks have been published, dense networks sharing remarkable structural properties, such as the scale-free feature, are seldom reported. Here, our goal is to construct a class of dense networks. Then, we discover that our networks follow the mixture degree distribution; that is, there is a critical point above which the cumulative degree distribution has a power-law form and below which the exponential distribution is observed. Next, we also prove the networks proposed to show the small-world property. Finally, we study random walks on our networks with a trap fixed at a vertex with the highest degree and find that the closed form for the mean first-passage time increases logarithmically with the number of vertices of our networks.

scholarly journals A detailed characterization of complex networks using Information Theory

2021 ◽  
Author(s):  
CGS Freitas ◽  
ALL Aquino ◽  
HS Ramos ◽  
Alejandro Frery ◽  
OA Rosso
Keyword(s):  
Information Theory ◽  
Complex Networks ◽  
Complex Network ◽  
Preferential Attachment ◽  
Configuration Model ◽  
Complex Network Theory ◽  
Scale Free ◽  
Network Entropy ◽  
Law Degree

Understanding the structure and the dynamics of networks is of paramount importance for many scientific fields that rely on network science. Complex network theory provides a variety of features that help in the evaluation of network behavior. However, such analysis can be confusing and misleading as there are many intrinsic properties for each network metric. Alternatively, Information Theory methods have gained the spotlight because of their ability to create a quantitative and robust characterization of such networks. In this work, we use two Information Theory quantifiers, namely Network Entropy and Network Fisher Information Measure, to analyzing those networks. Our approach detects non-trivial characteristics of complex networks such as the transition present in the Watts-Strogatz model from k-ring to random graphs; the phase transition from a disconnected to an almost surely connected network when we increase the linking probability of Erdős-Rényi model; distinct phases of scale-free networks when considering a non-linear preferential attachment, fitness, and aging features alongside the configuration model with a pure power-law degree distribution. Finally, we analyze the numerical results for real networks, contrasting our findings with traditional complex network methods. In conclusion, we present an efficient method that ignites the debate on network characterization.

scholarly journals MODELLING COLLABORATION NETWORKS BASED ON NONLINEAR PREFERENTIAL ATTACHMENT

2007 ◽  
Vol 18 (02) ◽  
pp. 297-314 ◽  
Author(s):  
TAO ZHOU ◽  
BING-HONG WANG ◽  
YING-DI JIN ◽  
DA-REN HE ◽  
PEI-PEI ZHANG ◽  
...  
Keyword(s):  
Empirical Data ◽  
Degree Distribution ◽  
Alternative Model ◽  
Free Parameter ◽  
Preferential Attachment ◽  
Average Distance ◽  
Small World ◽  
Clustering Coefficient ◽  
Collaboration Networks ◽  
Scale Free

In this paper, we propose an alternative model for collaboration networks based on nonlinear preferential attachment. Depending on a single free parameter "preferential exponent", this model interpolates between networks with a scale-free and an exponential degree distribution. The degree distribution in the present networks can be roughly classified into four patterns, all of which are observed in empirical data. And this model exhibits small-world effect, which means the corresponding networks are of very short average distance and highly large clustering coefficient. More interesting, we find a peak distribution of act-size from empirical data which has not been emphasized before. Our model can produce the peak act-size distribution naturally that agrees with the empirical data well.

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