COMPLEX NETWORKS: TOPOLOGY, DYNAMICS AND SYNCHRONIZATION

2002 ◽  
Vol 12 (05) ◽  
pp. 885-916 ◽  
Author(s):  
XIAO FAN WANG
Keyword(s):  
Complex Networks ◽  
Network Models ◽  
Small World ◽  
Dynamical Networks ◽  
Scale Free ◽  
Epidemic Dynamics ◽  
The Past ◽  
The Relationship

Dramatic advances in the field of complex networks have been witnessed in the past few years. This paper reviews some important results in this direction of rapidly evolving research, with emphasis on the relationship between the dynamics and the topology of complex networks. Basic quantities and typical examples of various complex networks are described; and main network models are introduced, including regular, random, small-world and scale-free models. The robustness of connectivity and the epidemic dynamics in complex networks are also evaluated. To that end, synchronization in various dynamical networks are discussed according to their regular, small-world and scale-free connections.

Research on Edge-Growing Models Related with Scale-Free Small-World Networks

2014 ◽  
Vol 513-517 ◽  
pp. 2444-2448 ◽  
Author(s):  
Bing Yao ◽  
Ming Yao ◽  
Xiang En Chen ◽  
Xia Liu ◽  
Wan Jia Zhang
Keyword(s):  
Complex Networks ◽  
Topological Structure ◽  
Spanning Trees ◽  
Network Models ◽  
Small World ◽  
Small World Networks ◽  
Scale Free ◽  
Scale Free Networks ◽  
Growing Network

Understanding the topological structure of scale-free networks or small world networks is required and useful for investigation of complex networks. We will build up a class of edge-growing network models and provide an algorithm for finding spanning trees of edge-growing network models in this article.

scholarly journals Quantitative Controllability Index of Complex Networks

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Lifu Wang ◽  
Yali Zhang ◽  
Jingxiao Han ◽  
Zhi Kong
Keyword(s):  
Complex Networks ◽  
Condition Number ◽  
Random Network ◽  
Small World ◽  
Scale Free Network ◽  
Scale Free ◽  
Network Topologies ◽  
Free Network ◽  
The Relationship ◽  
Controllability Index

In this paper, the controllability issue of complex network is discussed. A new quantitative index using knowledge of control centrality and condition number is constructed to measure the controllability of given networks. For complex networks with different controllable subspace dimensions, their controllability is mainly determined by the control centrality factor. For the complex networks that have the equal controllable subspace dimension, their different controllability is mostly determined by the condition number of subnetworks’ controllability matrix. Then the effect of this index is analyzed based on simulations on various types of network topologies, such as ER random network, WS small-world network, and BA scale-free network. The results show that the presented index could reflect the holistic controllability of complex networks. Such an endeavour could help us better understand the relationship between controllability and network topology.

scholarly journals Eigenvalue-based entropy in directed complex networks

PLoS ONE ◽  
2021 ◽  
Vol 16 (6) ◽  
pp. e0251993
Author(s):  
Yan Sun ◽  
Haixing Zhao ◽  
Jing Liang ◽  
Xiujuan Ma
Keyword(s):  
Complex Networks ◽  
Complex Network ◽  
Network Models ◽  
Small World ◽  
Laplacian Matrix ◽  
Directed Network ◽  
Small World Network ◽  
Scale Free Network ◽  
Scale Free ◽  
Free Network

Entropy is an important index for describing the structure, function, and evolution of network. The existing research on entropy is primarily applied to undirected networks. Compared with an undirected network, a directed network involves a special asymmetric transfer. The research on the entropy of directed networks is very significant to effectively quantify the structural information of the whole network. Typical complex network models include nearest-neighbour coupling network, small-world network, scale-free network, and random network. These network models are abstracted as undirected graphs without considering the direction of node connection. For complex networks, modeling through the direction of network nodes is extremely challenging. In this paper, based on these typical models of complex network, a directed network model considering node connection in-direction is proposed, and the eigenvalue entropies of three matrices in the directed network is defined and studied, where the three matrices are adjacency matrix, in-degree Laplacian matrix and in-degree signless Laplacian matrix. The eigenvalue-based entropies of three matrices are calculated in directed nearest-neighbor coupling, directed small world, directed scale-free and directed random networks. Through the simulation experiment on the real directed network, the result shows that the eigenvalue entropy of the real directed network is between the eigenvalue entropy of directed scale-free network and directed small-world network.

scholarly journals On the Distributions of Subgraph Centralities in Complex Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Faxu Li ◽  
Liang Wei ◽  
Haixing Zhao ◽  
Feng Hu
Keyword(s):  
Complex Networks ◽  
Probability Distributions ◽  
Network Connectivity ◽  
Network Models ◽  
Small World ◽  
Small World Networks ◽  
Power Law Distribution ◽  
Scale Free ◽  
Scale Free Networks ◽  
Complex Network Models

Subgraph centrality measure characterizes the participation of each node in all subgraphs in a network. Smaller subgraphs are given more weight than large ones, which makes this measure appropriate for characterizing network motifs. This measure is better in being able to discriminate the nodes of a network than alternate measures. In this paper, the important issue of subgraph centrality distributions is investigated through theory-guided extensive numerical simulations, for three typical complex network models, namely, the ER random-graph networks, WS small-world networks, and BA scale-free networks. It is found that these three very different types of complex networks share some common features, particularly that the subgraph centrality distributions in increasing order are all insensitive to the network connectivity characteristics, and also found that the probability distributions of subgraph centrality of the ER and of the WS models both follow the gamma distribution, and the BA scale-free networks exhibit a power-law distribution with an exponential cutoff.

CONGESTION AND EFFICIENCY IN COMPLEX TRAFFIC NETWORKS

2013 ◽  
Vol 24 (10) ◽  
pp. 1350072 ◽  
Author(s):  
JIAN-FENG ZHENG ◽  
ZHI-HONG ZHU ◽  
HAO-MING DU ◽  
ZI-YOU GAO
Keyword(s):  
Complex Networks ◽  
Small World ◽  
Traffic Networks ◽  
Flow State ◽  
Traffic Demand ◽  
Scale Free ◽  
Link Cost ◽  
Network Topologies ◽  
The Impact ◽  
The Relationship

This paper investigates the degree of congestion and efficiency in complex traffic networks, by introducing congestion effects, which can be described by flow-based link cost functions. Different network topologies including random networks, small-world networks and scale-free networks are explored. The impact of different distributions of capacity and origin-destination traffic demand on the degree of congestion and efficiency in complex networks is mainly studied. A phase transition from free flow state to traffic jams can be uncovered. The relationship between congestion and efficiency in complex networks is also discussed.

SNAM

2016 ◽  
pp. 215-236 ◽  
Author(s):  
Bassant Youssef ◽  
Scott F. Midkiff ◽  
Mohamed R. M. Rizk
Keyword(s):  
Community Structure ◽  
Complex Networks ◽  
Degree Distribution ◽  
Network Models ◽  
Small World ◽  
Clustering Coefficient ◽  
Network Nodes ◽  
Scale Free ◽  
Complex Network Models ◽  
Novel Model

Complex networks are characterized by having a scale-free power-law (PL) degree distribution, a small world phenomenon, a high average clustering coefficient, and the emergence of community structure. Most proposed models did not incorporate all of these statistical properties and neglected incorporating the heterogeneous nature of network nodes. Even proposed heterogeneous complex network models were not generalized for different complex networks. We define a novel aspect of node-heterogeneity which is the node connection standard heterogeneity. We introduce our novel model “settling node adaptive model” SNAM which reflects this new nodes' heterogeneous aspect. SNAM was successful in preserving PL degree distribution, small world phenomenon and high clustering coefficient of complex networks. A modified version of SNAM shows the emergence of community structure. We prove using mathematical analysis that networks generated using SNAM have a PL degree distribution.

scholarly journals VORONOI AND FRACTAL COMPLEX NETWORKS AND THEIR CHARACTERIZATION

2004 ◽  
Vol 15 (01) ◽  
pp. 175-183 ◽  
Author(s):  
LUCIANO DA FONTOURA COSTA
Keyword(s):  
Complex Networks ◽  
Metric Spaces ◽  
Spatial Scales ◽  
Shortest Paths ◽  
Network Models ◽  
Small World ◽  
Effective Length ◽  
Scale Free ◽  
Fractal Network ◽  
Natural Means

Real world networks are often characterized by spatial constraints such as the relative position and adjacency of nodes. The present work describes how Voronoi tessellations of the space where the network is embedded provide not only a natural means for relating such networks with metric spaces, but also a natural means for obtaining fractal complex networks. A series of comprehensive measurements closely related to spatial aspects of these networks is proposed, which includes the effective length, adjacency, as well as the fractal dimension of the network in terms of the spatial scales defined by successive shortest paths starting from a specific node. The potential of such features is illustrated with respect to the random, small-world, scale-free and fractal network models.

PINNING IMPULSIVE SYNCHRONIZATION OF COMPLEX DYNAMICAL NETWORKS

2012 ◽  
Vol 22 (10) ◽  
pp. 1250239 ◽  
Author(s):  
NARIMAN MAHDAVI ◽  
MOHAMMAD B. MENHAJ ◽  
JÜRGEN KURTHS ◽  
JIANQUAN LU ◽  
AHMAD AFSHAR
Keyword(s):  
Efficient Algorithm ◽  
Small World ◽  
Complex Dynamical Networks ◽  
Dynamical Networks ◽  
Scale Free ◽  
Dynamical Network ◽  
Impulsive Synchronization ◽  
Strongly Connected ◽  
Successful Control ◽  
The Relationship

In this paper, the problem of pinning impulsive synchronization for complex dynamical networks with directed or undirected but a strongly connected topology is investigated. To remedy this problem, we propose an efficient algorithm to find certain suitable nodes to be controlled via pinning and these selected nodes, in general, could be different at distinct impulsive time instants. The proposed algorithm guarantees the efficiency of the designed pinning impulsive strategy for the global exponential synchronization of state-coupled dynamical networks under an easily-verified condition. In other words, impulsive controllers and an efficient algorithm are designed to control a small fraction of the nodes, which successfully controls the whole dynamical network. Furthermore, we also estimate the upper bound of the number of pinning nodes, which is shown to be closely related to the impulsive intervals. The relationship implies that the required number of pinning nodes, which should be controlled for the successful control of the whole dynamical network, can be greatly reduced by reducing the impulses interval. Finally, simulations of scale-free and small-world networks are given to illustrate the effectiveness of the theoretical results.

STABILITY OF TWO TYPICAL COMPLEX DYNAMICAL NETWORKS

2008 ◽  
Vol 22 (05) ◽  
pp. 553-560 ◽  
Author(s):  
WU-JIE YUAN ◽  
XIAO-SHU LUO ◽  
PIN-QUN JIANG ◽  
BING-HONG WANG ◽  
JIN-QING FANG
Keyword(s):  
Numerical Simulation ◽  
Dissipative System ◽  
Small World ◽  
Complex Dynamical Networks ◽  
Dynamical Networks ◽  
Scale Free ◽  
Scale Free Networks ◽  
General Complex ◽  
The Stability ◽  
Theoretical Results

When being constructed, complex dynamical networks can lose stability in the sense of Lyapunov (i. s. L.) due to positive feedback. Thus, there is much important worthiness in the theory and applications of complex dynamical networks to study the stability. In this paper, according to dissipative system criteria, we give the stability condition in general complex dynamical networks, especially, in NW small-world and BA scale-free networks. The results of theoretical analysis and numerical simulation show that the stability i. s. L. depends on the maximal connectivity of the network. Finally, we show a numerical example to verify our theoretical results.

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