Evolution of networks

doi:10.1080/00018730110112519

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Evolution of networks

Journal Article•DOI•

Evolution of networks

Sergey N. Dorogovtsev, José F. F. Mendes

01 Jun 2002-Advances in Physics (Taylor & Francis Group)-Vol. 51, Iss: 4, pp 1079-1187

TL;DR: The recent rapid progress in the statistical physics of evolving networks is reviewed, and how growing networks self-organize into scale-free structures is discussed, and the role of the mechanism of preferential linking is investigated.

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Abstract: We review the recent rapid progress in the statistical physics of evolving networks. Interest has focused mainly on the structural properties of complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of this kind have recently been created, which opens a wide field for the study of their topology, evolution, and the complex processes which occur in them. Such networks possess a rich set of scaling properties. A number of them are scale-free and show striking resilience against random breakdowns. In spite of the large sizes of these networks, the distances between most of their vertices are short - a feature known as the 'small-world' effect. We discuss how growing networks self-organize into scale-free structures, and investigate the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation and disease spread on these networks. We present a number of models demonstrat...

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Journal Article•DOI•

The Structure and Function of Complex Networks

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Mark Newman

01 Jan 2003-Siam Review

TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

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Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

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17,647 citations

Cites background or methods from "Evolution of networks"

...Albert and Barabási [13] and Dorogovtsev and Mendes [120] have given extensive pedagogical reviews focusing on the physics literature....
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...Albert and Barabási [13] and Dorogovtsev and Mendes [120] have given extensive pedagogical reviews focusing on the physics literature....
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...Thorough reviews of master-equation methods for grown graph models have been given by Dorogovtsev and Mendes [120] and Krapivsky and Redner [248]....
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...Dorogovtsev and Mendes [122] have expanded their above-mentioned review into a book, which again focuses on models of growing graphs....
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...Networks with power-law degree distributions have been the focus of a great deal of attention in the literature [13, 120, 387]....
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Journal Article•DOI•

Modularity and community structure in networks

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Mark Newman¹•Institutions (1)

University of Michigan¹

06 Jun 2006-Proceedings of the National Academy of Sciences of the United States of America

TL;DR: In this article, the modularity of a network is expressed in terms of the eigenvectors of a characteristic matrix for the network, which is then used for community detection.

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Abstract: Many networks of interest in the sciences, including social networks, computer networks, and metabolic and regulatory networks, are found to divide naturally into communities or modules. The problem of detecting and characterizing this community structure is one of the outstanding issues in the study of networked systems. One highly effective approach is the optimization of the quality function known as “modularity” over the possible divisions of a network. Here I show that the modularity can be expressed in terms of the eigenvectors of a characteristic matrix for the network, which I call the modularity matrix, and that this expression leads to a spectral algorithm for community detection that returns results of demonstrably higher quality than competing methods in shorter running times. I illustrate the method with applications to several published network data sets.

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10,137 citations

Journal Article•DOI•

Complex networks: Structure and dynamics

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Stefano Boccaletti, Vito Latora¹, Vito Latora², Yamir Moreno³, Mario Chavez⁴, Dong-Uk Hwang - Show less +2 more•Institutions (4)

University of Catania¹, Istituto Nazionale di Fisica Nucleare², University of Zaragoza³, Centre national de la recherche scientifique⁴

01 Feb 2006-Physics Reports

TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.

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9,441 citations

Cites background or methods from "Evolution of networks"

...In fact, as pointed out by Dorogovtsev and Mendes [7], it is a particular case of the general n-state Potts model....
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...Albert and Barabási [2], and Dorogovtsev and Mendes [3,7] have mainly focused their reviews on models of growing graphs, from the point of view of statistical mechanics....
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Journal Article•DOI•

Community detection in graphs

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Santo Fortunato¹•Institutions (1)

Institute for Scientific Interchange¹

03 Jun 2009-arXiv: Physics and Society

TL;DR: A thorough exposition of community structure, or clustering, is attempted, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists.

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Abstract: The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.

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9,057 citations

Cites background from "Evolution of networks"

...…degree distributions have for the structure and function of real networks (Albert and Barabási, 2002; Barrat et al., 2008; Boccaletti et al., 2006; Dorogovtsev and Mendes, 2002; Newman, 2003; Pastor-Satorras and Vespignani, 2004), it is preferable to go for a null model with the same degree…...
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...On graphs with strong community structure, that quickly break into communities, the recalculation step needs to be performed only within the connected component including the last removed edge (or the two components bridged by it if the removal of the edge splits a subgraph), as the edge…...
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Journal Article•DOI•

Community detection in graphs

[...]

Santo Fortunato¹•Institutions (1)

Institute for Scientific Interchange¹

01 Feb 2010-Physics Reports

TL;DR: A thorough exposition of the main elements of the clustering problem can be found in this paper, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.

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8,432 citations

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References

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Journal Article•DOI•

Collective dynamics of small-world networks

[...]

Duncan J. Watts¹, Steven H. Strogatz¹•Institutions (1)

Cornell University¹

04 Jun 1998-Nature

TL;DR: Simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder are explored, finding that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs.

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Abstract: Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.

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39,297 citations

"Evolution of networks" refers background in this paper

...In spite of large sizes of these networks, the distances between most their vertices are short — a feature known as the “smallworld” effect....
[...]
...In July of 2000, there were about 150 000 routers in the Internet [104]....
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...The degree distribution of this network was reported to be of a power-law form, P (k) ∝ k−γ where γ ≈ 2.2 (November of 1997 – 2.15, April of 1998 – 2.16, and December of 1998 – 2.20) [5]....
[...]

Journal Article•DOI•

Emergence of Scaling in Random Networks

[...]

Albert-László Barabási¹, Réka Albert¹•Institutions (1)

University of Notre Dame¹

15 Oct 1999-Science

TL;DR: A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.

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Abstract: Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.

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33,771 citations

"Evolution of networks" refers background in this paper

...Thus, the WWW growth is much more complex process than the growth of citation networks....
[...]
...Web documents are accessible through the Internet (wires and hardware), and this determines the relation between the Internet and the WWW....
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...The large size of the Internet and WWW and their extensive and easily accessible documentation allow reliable and informative experimental investigation of their structure and properties....
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...More complex models and estimates for the WWW 30 IX G. Types of preference providing scale-free networks 33 IXH....
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...Structure of the WWW 10 V D. Biological networks 12 V D1....
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Book•

The Fractal Geometry of Nature

[...]

Benoit B. Mandelbrot

01 Jan 1982

TL;DR: This book is a blend of erudition, popularization, and exposition, and the illustrations include many superb examples of computer graphics that are works of art in their own right.

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Abstract: "...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

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24,199 citations

"Evolution of networks" refers background in this paper

...nnects nother points uphill, can be introduced. In fact, nis the size of the basin connected to some point, and P(n) is the distribution of basin sizes. For river networks forming a fractal structure [110], this distribution is of a power-law form, P(n) ∝ n−τ, where values of the τexponent are slightly lower 3/2 [111]. For the Internet, it was found that τ= 1.9±0.1 [109]. 2. Structure of the WWW Let us...
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Journal Article•DOI•

Statistical mechanics of complex networks

[...]

Réka Albert¹, Albert-László Barabási¹•Institutions (1)

University of Notre Dame¹

01 Jan 2001-Reviews of Modern Physics

TL;DR: In this paper, a simple model based on the power-law degree distribution of real networks was proposed, which was able to reproduce the power law degree distribution in real networks and to capture the evolution of networks, not just their static topology.

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Abstract: The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.

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18,415 citations

Book•

Social Network Analysis: Methods and Applications

[...]

Stanley Wasserman, Katherine Faust¹•Institutions (1)

University of South Carolina¹

25 Nov 1994

TL;DR: This paper presents mathematical representation of social networks in the social and behavioral sciences through the lens of Dyadic and Triadic Interaction Models, which describes the relationships between actor and group measures and the structure of networks.

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Abstract: Part I. Introduction: Networks, Relations, and Structure: 1. Relations and networks in the social and behavioral sciences 2. Social network data: collection and application Part II. Mathematical Representations of Social Networks: 3. Notation 4. Graphs and matrixes Part III. Structural and Locational Properties: 5. Centrality, prestige, and related actor and group measures 6. Structural balance, clusterability, and transitivity 7. Cohesive subgroups 8. Affiliations, co-memberships, and overlapping subgroups Part IV. Roles and Positions: 9. Structural equivalence 10. Blockmodels 11. Relational algebras 12. Network positions and roles Part V. Dyadic and Triadic Methods: 13. Dyads 14. Triads Part VI. Statistical Dyadic Interaction Models: 15. Statistical analysis of single relational networks 16. Stochastic blockmodels and goodness-of-fit indices Part VII. Epilogue: 17. Future directions.

...read moreread less

17,104 citations

"Evolution of networks" refers background in this paper

...In spite of large sizes of these networks, the distances between most their vertices are short — a feature known as the “smallworld” effect....
[...]