Deterministic scale-free networks

Barabási, Albert‐László; Ravasz, Erzsébet; Vicsek, Tamás

doi:10.1016/s0378-4371(01)00369-7

Cited by 423 publications

(344 citation statements)

References 28 publications

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“…In particular, within Escherichia coli, the observed hierarchy coincides with known metabolic functions. Incorporating hierarchy into graph theoretical models allows the simultaneous description of two distinct systems-level features of real world networks, power-law degree distribution and modular topology 34 .…”

Section: Hierarchymentioning

confidence: 99%

The art of community detection

Gulbahce

Lehmann

2008

BioEssays

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SUMMARYNetworks in nature possess a remarkable amount of structure. Via a series of datadriven discoveries, the cutting edge of network science has recently progressed from positing that the random graphs of mathematical graph theory might accurately describe real networks to the current viewpoint that networks in nature are highly complex and structured entities. The identification of high order structures in networks unveils insights into their functional organization. Recently, Clauset, Moore, and Newman 1 , introduced a new algorithm that identifies such heterogeneities in complex networks by utilizing the hierarchy that necessarily organizes the many levels of structure. Here, we anchor their algorithm in a general community detection framework and discuss the future of community detection. STRUCTURE EVERYWHEREThe view that networks are essentially random was challenged in 1999 when it was discovered that the distribution of number of links per node (degree) of many real networks (internet, metabolic network, sexual contacts, airports, etc) is different from what is expected in random networks 2 . In a large random network node degrees are distributed according to the normal distribution, but in many manmade and biological networks the degree distribution follows a power-law. In the human protein-protein interaction networks 3,4 , for instance, some proteins act as hubs, they are highly connected, and interact with more than 200 other proteins contrary to most proteins that interact with only a few other proteins.Various local to global measures have been introduced to unveil the organizational principles of complex networks 5,6,7,8,9 . Maslov and Sneppen 10 discovered that who links to whom can depend on node degree; in many biological networks, high degree nodes systematically link to nodes of low degree. This disassortativity decreases the likelihood of cross talk between functional modules inside the cell and increases overall robustness. Other networks, for example social networks 11 , are highly assortative -in these networks nodes with similar degree tend to link to each other. The scales of organization of complex networks. The illustrations on the left show how to break down the "hairball" that arises when we plot the entire network. On the smallest scale, the degree provides information about single nodes. The notion of assortativity enters when we discuss pairs of nodes. With three or more nodes, we are in the realm of motifs. Larger groups of nodes are called modules or communities. Hierarchy describes how the various structural elements are combined; how nodes a linked to form motifs, motifs are combined to form communities, and communities are joined into the entire network.Going beyond the properties of single nodes and pairs of nodes, the natural next step is to consider structures that include several nodes. Interestingly, a few select motifs of three to four nodes are ubiquitous in real networks 12 while most others occur only as often as they would at random, or, are actively suppressed....

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Section: Hierarchymentioning

confidence: 99%

The art of community detection

Gulbahce

Lehmann

2008

BioEssays

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“…That is, new nodes connect using a probabilistic rule to the nodes already present in the system. But as mentioned by Barabási et al, the randomness, while in line with the major features of real-life networks, makes it harder to gain a visual understanding of how networks are shaped, and how do different nodes relate to each other [15]. In addition, the probabilistic analysis techniques and random placement or addition of edges used in most previous studies are not appropriate for communication networks that have fixed interconnections, such as neural networks, computer networks, electronic circuits, and so on.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the probabilistic analysis techniques and random placement or addition of edges used in most previous studies are not appropriate for communication networks that have fixed interconnections, such as neural networks, computer networks, electronic circuits, and so on. Therefore, it would be not only of major theoretical interest but also of great practical significance to construct models that lead to scale-free networks [15,16,17,18,19,20,21,22,23,24,25] and small-world net-works [26,27] in deterministic fashions. A strong advantage of deterministic networks is that it is often possible to compute analytically their properties, for example, degree distribution, clustering coefficient, average path length, diameter and adjacency matrix whose eigenvalue spectrum characterizes the topology.…”

Section: Introductionmentioning

confidence: 99%

A deterministic small-world network created by edge iterations

Zhang

Rong

Guo

2006

Physica A: Statistical Mechanics and its Applications

113

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Small-world networks are ubiquitous in real-life systems. Most previous models of small-world networks are stochastic. The randomness makes it more difficult to gain a visual understanding on how do different nodes of networks interact with each other and is not appropriate for communication networks that have fixed interconnections. Here we present a model that generates a small-world network in a simple deterministic way. Our model has a discrete exponential degree distribution. We solve the main characteristics of the model.

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“…In particular, it is possible to construct small-world deterministic graphs with different degree distributions matching the distribution of real networks, see [5,2,7,4]. We introduce in this paper a simple deterministic model, a toy model, to show analytically, that the observed eigenvalue power law of the Internet may be a direct consequence of the degree distribution of a star based structure (whose power law can be justified considering, for example, preferential attachment [1] or duplication [6] models).…”

Section: Introductionmentioning

confidence: 99%