An Efficient Method of Generating Deterministic Small-World and Scale-Free Graphs for Simulating Real-World Networks

Abstract Many real-world networks are intrinsically directed. Such networks include activation of genes, hyperlinks on the internet and the network of followers on Twitter among many others. The challenge, however, is to create a network model that has many of the properties of real-world networks such as power-law degree distributions and the small-world property. To meet these challenges, we introduce the Directed Random Geometric Graph (DRGG) model, which is an extension of the random geometric graph model. We prove that it is scale-free with respect to the indegree distribution, has binomial outdegree distribution, has a high clustering coefficient, has few edges and is likely small-world. These are some of the main features of aforementioned real-world networks. We also empirically observed that word association networks have many of the theoretical properties of the DRGG model.

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Complex networks

10.1093/oxfordhb/9780198744191.013.43 ◽

2018 ◽

Author(s):

Graziano Vernizzi ◽

Henri Orland

Keyword(s):

Complex Networks ◽

Small World ◽

Laplacian Matrix ◽

Square Lattice ◽

Small World Networks ◽

Scale Free ◽

Scale Free Networks ◽

Free Network ◽

Scale Free Graphs ◽

Regular Networks

This article deals with complex networks, and in particular small world and scale free networks. Various networks exhibit the small world phenomenon, including social networks and gene expression networks. The local ordering property of small world networks is typically associated with regular networks such as a 2D square lattice. The small world phenomenon can be observed in most scale free networks, but few small world networks are scale free. The article first provides a brief background on small world networks and two models of scale free graphs before describing the replica method and how it can be applied to calculate the spectral densities of the adjacency matrix and Laplacian matrix of a scale free network. It then shows how the effective medium approximation can be used to treat networks with finite mean degree and concludes with a discussion of the local properties of random matrices associated with complex networks.

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Real-World Networks Are Not Always Fast Mixing

The Computer Journal ◽

10.1093/comjnl/bxaa150 ◽

2020 ◽

Author(s):

Yi Qi ◽

Wanyue Xu ◽

Liwang Zhu ◽

Zhongzhi Zhang

Keyword(s):

Real World ◽

Transition Matrix ◽

Mixing Time ◽

Real Life ◽

Small World ◽

Design Of Algorithms ◽

Scale Free ◽

Analytical Research ◽

Second Largest Eigenvalue ◽

Recursive Relations

Abstract The mixing time of random walks on a graph has found broad applications across both theoretical and practical aspects of computer science, with the application effects depending on the behavior of mixing time. It is extensively believed that real-world networks, especially social networks, are fast mixing with their mixing time at most $O(\log N)$ where $N$ is the number of vertices. However, the behavior of mixing time in the real-life networks has not been examined carefully, and exactly analytical research for mixing time in models mimicking real networks is still lacking. In this paper, we first experimentally evaluate the mixing time of various real-world networks with scale-free small-world properties and show that their mixing time is much higher than anticipated. To better understand the behavior of the mixing time for real-world networks, we then analytically study the mixing time of the Apollonian network, which is simultaneously scale-free and small-world. To this end, we derive the recursive relations for all eigenvalues, especially the second largest eigenvalue modulus of the transition matrix, based on which we deduce a lower bound for the mixing time of the Apollonian network, which approximately scales sublinearly with $N$. Our results indicate that real-world networks are not always fast mixing, which has potential implications in the design of algorithms related to mixing time.

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The Evolution of Price Competition Game on Complex Networks

Complexity ◽

10.1155/2018/9649863 ◽

2018 ◽

Vol 2018 ◽

pp. 1-13

Author(s):

Feng Jie Xie ◽

Jing Shi

Keyword(s):

Complex Networks ◽

Real World ◽

Price Competition ◽

Small World ◽

Square Lattice ◽

Small World Network ◽

Scale Free Network ◽

The Real ◽

Scale Free ◽

Free Network

The well-known “Bertrand paradox” describes a price competition game in which two competing firms reach an outcome where both charge a price equal to the marginal cost. The fact that the Bertrand paradox often goes against empirical evidences has intrigued many researchers. In this work, we study the game from a new theoretical perspective—an evolutionary game on complex networks. Three classic network models, square lattice, WS small-world network, and BA scale-free network, are used to describe the competitive relations among the firms which are bounded rational. The analysis result shows that full price keeping is one of the evolutionary equilibriums in a well-mixed interaction situation. Detailed experiment results indicate that the price-keeping phenomenon emerges in a square lattice, small-world network and scale-free network much more frequently than in a complete network which represents the well-mixed interaction situation. While the square lattice has little advantage in achieving full price keeping, the small-world network and the scale-free network exhibit a stronger capability in full price keeping than the complete network. This means that a complex competitive relation is a crucial factor for maintaining the price in the real world. Moreover, competition scale, original price, degree of cutting price, and demand sensitivity to price show a significant influence on price evolution on a complex network. The payoff scheme, which describes how each firm’s payoff is calculated in each round game, only influences the price evolution on the scale-free network. These results provide new and important insights for understanding price competition in the real world.

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PERFORMANCE AND ROBUSTNESS OF CELLULAR AUTOMATA COMPUTATION ON IRREGULAR NETWORKS

Advances in Complex Systems ◽

10.1142/s0219525907001124 ◽

2007 ◽

Vol 10 (supp01) ◽

pp. 85-110 ◽

Cited By ~ 16

Author(s):

CHRISTIAN DARABOS ◽

MARIO GIACOBINI ◽

MARCO TOMASSINI

Keyword(s):

Information Diffusion ◽

Small World ◽

Superior Performance ◽

Small World Networks ◽

Scale Free ◽

Irregular Networks ◽

Scale Free Networks ◽

The Face ◽

Scale Free Graphs ◽

Synchronization Problems

We investigate the performances of collective task-solving capabilities and the robustness of complex networks of automata using the density and synchronization problems as typical cases. We show by computer simulations that evolved Watts–Strogatz small-world networks have superior performance with respect to several kinds of scale-free graphs. In addition, we show that Watts–Strogatz networks are as robust in the face of random perturbations, both transient and permanent, as configuration scale-free networks, while being widely superior to Barabási–Albert networks. This result differs from information diffusion on scale-free networks, where random faults are highly tolerated by similar topologies.

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Searching for small-world and scale-free behaviour in long-term historical data of a real-world power grid

Scientific Reports ◽

10.1038/s41598-021-86103-7 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Bálint Hartmann ◽

Viktória Sugár

Keyword(s):

Real World ◽

Power Grid ◽

Electrical Power ◽

Small World ◽

Network Evolution ◽

Power Grids ◽

Infrastructure Development ◽

Real World Data ◽

Scale Free

AbstractSince the introduction of small-world and scale-free properties, there is an ongoing discussion on how certain real-world networks fit into these network science categories. While the electrical power grid was among the most discussed examples of these real-word networks, published results are controversial, and studies usually fail to take the aspects of network evolution into consideration. Consequently, while there is a broad agreement that power grids are small-world networks and might show scale-free behaviour; although very few attempts have been made to find how these characteristics of the network are related to grid infrastructure development or other underlying phenomena. In this paper the authors use the 70-year-long historical dataset (1949–2019) of the Hungarian power grid to perform complex network analysis, which is the first attempt to evaluate small-world and scale-free properties on long-term real-world data. The results of the analysis suggest that power grids show small-world behaviour only after the introduction of multiple voltage levels. It is also demonstrated that the node distribution of the examined power grid does not show scale-free behaviour and that the scaling is stabilised around certain values after the initial phase of grid evolution.

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Polygon-Based Hierarchical Planar Networks Based on Generalized Apollonian Construction

Physics ◽

10.3390/physics3040063 ◽

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.

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The structure of real-world networks

10.1093/oso/9780198805090.003.0010 ◽

2018 ◽

Author(s):

Mark Newman

Keyword(s):

Power Law ◽

Real World ◽

Small World ◽

Component Structure ◽

Scale Free ◽

Scale Free Networks ◽

Path Lengths ◽

Power Law Distributions

This chapter brings together the ideas and techniques developed in previous chapters, applying them to a range of real-world networks to describe and understand the structure of those networks. Topics discussed include the observed component structure of networks, average path lengths between nodes and the small-world effect, degree distributions including power-law distributions and scale-free networks, clustering and transitivity, and assortative mixing.

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Three-layered supernetwork evolution model and the application for China-world's top 500 enterprises supernetwork

International Journal of Modern Physics C ◽

10.1142/s0129183114400038 ◽

2014 ◽

Vol 25 (05) ◽

pp. 1440003 ◽

Cited By ~ 2

Author(s):

Qiang Liu ◽

Jin-Qing Fang ◽

Yong Li

Keyword(s):

Real World ◽

Small World ◽

Evolution Model ◽

The Real ◽

Network Characteristics ◽

Scale Free ◽

Relevant Theory ◽

Free Model ◽

Cooperative Competition ◽

Definition Of

Network of network (NON) or so-called supernetwork extensively exists in the real world. However, so far the definition of NON is not mutually recognized, relevant theory is rather lacking. In order to reveal certain characteristics of NON, we proposed four kinds of three-layered supernetwork evolution models (TLSEM) based on WS small-world and BA scale-free model, and defined two kinds of layer cross-degrees as new measures of cooperative-competition relationship for different layer nodes. The idea and methods of TLSEM are applied to the construction and analysis of China-world's top 500 enterprises supernetworks as a typical empirical example. The analytical results show that the layer cross-degree is better description than other network characteristics, and TLSEM may lay a certain foundation and extend to study more multilevel supernetworks.

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SYNCHRONIZABILITY OF COMPLEX NETWORKS WITH COMMUNITY STRUCTURE

International Journal of Modern Physics C ◽

10.1142/s0129183112500295 ◽

2012 ◽

Vol 23 (04) ◽

pp. 1250029 ◽

Cited By ~ 5

Author(s):

MAHDI JALILI

Keyword(s):

Community Structure ◽

Real World ◽

Order Parameter ◽

Small World ◽

Laplacian Matrix ◽

Small World Networks ◽

Scale Free ◽

Community Connections ◽

Scale Free Networks ◽

Kuramoto Oscillators

Many real-world networks show community structure characterized by dense intra-community connections and sparse inter-community links. In this paper we investigated the synchronization properties of such networks. In this work we constructed such networks in a way that they consist of a number of communities with scale-free or small-world structure. Furthermore, with a probability, the intra-community connections are rewired to inter-community links. Two synchronizability measures were considered as the eigenratio of the Laplacian matrix and the phase order parameter obtained for coupled nonidentical Kuramoto oscillators. We found a power-law relation between the eigenratio and the inter-community rewiring probability in which as the rewiring probability increased, the eigenratio decreased, and hence, the synchronizability enhanced. The phase order parameter also increased by increasing the rewiring probability. Also, small-world networks with community structure showed better synchronization properties as compared to scale-free networks with community structure.

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