# Statistical mechanics of complex networks

@article{Albert2001StatisticalMO, title={Statistical mechanics of complex networks}, author={R{\'e}ka Albert and A. L. Barabasi}, journal={ArXiv}, year={2001}, volume={cond-mat/0106096} }

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… Expand

#### Figures, Tables, and Topics from this paper

figure 1 figure 10 figure 11 figure 12 figure 13 figure 14 figure 15 figure 16 figure 17 figure 18 figure 19 figure 2 figure 20 figure 21 figure 22 figure 23 figure 24 figure 25 figure 26 figure 27 figure 28 figure 29 figure 3 figure 30 figure 31 figure 32 figure 33 figure 34 figure 35 figure 4 figure 5 figure 6 figure 7 figure 8 figure 9 table I table II table III

#### 17,288 Citations

Scale-free networks as entropy competition.

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2008

By identifying the connectivities of a node with the number of its nearest neighbors, it is shown that the power law is the most probable degree distribution that both nodes and neighbors, in a reciprocal competition, assume when the respective entropy functions reach their maxima, under mutual constraint. Expand

Evolution of networks

- Physics, Biology
- 2002

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. Expand

Structure and dynamics of evolving complex networks

- Computer Science
- 2014

This thesis introduces various novel processes which dictate the development of a network on a small scale, and uses techniques learned from statistical physics to derive the dynamical and structural properties of the network on the macroscopic scale. Expand

Hierarchy in directed random networks: analytical and numerical results

- Computer Science, Physics
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2013

It is shown that the hierarchical structure can be drastically different if there are one-point correlations in the network and there is an optimal level of nonzero correlations maximizing the level of hierarchy. Expand

Spin models on random graphs

- Mathematics
- 2008

In the past decades complex networks and their behavior have attracted much attention. In the real world many of such networks can be found, for instance as social, information, technological and… Expand

Exploring biological network structure with clustered random networks

- Computer Science, Medicine
- BMC Bioinformatics
- 2009

A new Markov chain simulation algorithm is developed and implemented to generate simple, connected random graphs that have a specified degree sequence and level of clustering, but are random in all other respects, allowing for systematic study of the impacts of connectivity and redundancies on network function and dynamics. Expand

Self-similarity of complex networks

- Medicine, Physics
- Nature
- 2005

A power-law relation is identified between the number of boxes needed to cover the network and the size of the box, defining a finite self-similar exponent to explain the scale-free nature of complex networks and suggest a common self-organization dynamics. Expand

Evolving Properties of Growing Networks

- Mathematics
- 2009

Complex networks describe a wide range of systems and structures in the world. Any real network can be modeled as graph, expressed by an adjacency matrix or list. In many complex networks, when a… Expand

Hierarchy Measure for Complex Networks

- Physics, Medicine
- PloS one
- 2012

This work develops an approach and proposes a quantity (measure) which is simple enough to be widely applicable, reveals a number of universal features of the organization of real-world networks and is capable of capturing the essential Features of the structure and the degree of hierarchy in a complex network. Expand

Theory of Random Networks and Their Role in Communications Networks

- Computer Science
- Web Dynamics
- 2004

The goal of this review is to show from a physicist’s point of view the many problems addressed by such networks and the present understanding within the field. Expand

#### References

SHOWING 1-10 OF 372 REFERENCES

Spectra of "real-world" graphs: beyond the semicircle law.

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2001

Methods to determine the eigenvalues of networks comparable in size to real systems are developed, obtaining several surprising results on the spectra of adjacency matrices corresponding to models of real-world graphs. Expand

Random networks created by biological evolution.

- Mathematics, Physics
- Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 2000

The model is a generalization of the Bak-Sneppen model of biological evolution, with the modification that the underlying network can evolve by adding and removing sites, and various geometrical properties of the network are studied. Expand

Dynamics of directed graphs: the world-wide Web

- Mathematics, Physics
- 2001

We introduce and simulate a growth model of the world-wide Web based on the dynamics of outgoing links that is motivated by the conduct of the agents in the real Web to update outgoing links… Expand

Evolution of the social network of scientific collaborations

- Sociology, Computer Science
- 2002

The results indicate that the co-authorship network of scientists is scale-free, and that the network evolution is governed by preferential attachment, affecting both internal and external links, and a simple model is proposed that captures the network's time evolution. Expand

A model for the emergence of cooperation, interdependence, and structure in evolving networks.

- Physics, Medicine
- Proceedings of the National Academy of Sciences of the United States of America
- 2001

A simple mathematical model is described for the evolution of an idealized chemical system to study how a network of cooperative molecular species arises and evolves to become more complex and structured. Expand

Collective dynamics of ‘small-world’ networks

- Computer Science, Medicine
- Nature
- 1998

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. Expand

Emergence of scaling in random networks

- Computer Science, Physics
- Science
- 1999

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. Expand

Random graphs with arbitrary degree distributions and their applications.

- Mathematics, Physics
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2001

It is demonstrated that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph. Expand

Mean-field theory for scale-free random networks

- Computer Science, Mathematics
- 1999

A mean-field method is developed to predict the growth dynamics of the individual vertices of the scale-free model, and this is used to calculate analytically the connectivity distribution and the scaling exponents. Expand

Organization of growing random networks.

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2001

The organizational development of growing random networks is investigated, and the combined age and degree distribution of nodes shows that old nodes typically have a large degree. Expand