Statistical mechanics of complex networks

  title={Statistical mechanics of complex networks},
  author={R{\'e}ka Albert and A L Barabasi},
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… 

Evolution of networks

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.

Structure and dynamics of evolving complex networks

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.

Hierarchy in directed random networks: analytical and numerical results

  • Enys Mones
  • Computer Science
    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.

Spin models on random graphs

This thesis studies two models: the ferromagnetic Ising model on power-law random graphs and the antiferromagnetic Potts model on the Erd?os-Renyi random graph, and identifies the thermodynamic limit of the magnetization, internal energy and susceptibility.

Random Graphs and Complex Networks

  • R. Hofstad
  • Computer Science
    Cambridge Series in Statistical and Probabilistic Mathematics
  • 2016
This chapter explains why many real-world networks are small worlds and have large fluctuations in their degrees, and why Probability theory offers a highly effective way to deal with the complexity of networks, and leads us to consider random graphs.

Exploring biological network structure with clustered random networks

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.

Self-similarity of complex networks

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.

Evolving Properties of Growing Networks

  • J. Wu
  • Computer Science
  • 2009
The objective of the thesis is to understand the evolving properties of growing networks by comparison of topological metrics with different number of nodes and links and concludes evolving properties based on both empirical and analytical results.

Hierarchy Measure for Complex Networks

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.

Theory of Random Networks and Their Role in Communications Networks

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.



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

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.

Random networks created by biological evolution.

  • F. SlaninaM. Kotrla
  • Computer Science
    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.

Dynamics of directed graphs: the world-wide Web

Collective dynamics of ‘small-world’ networks

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.

Emergence of scaling in random networks

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.

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

  • S. JainS. Krishna
  • Biology
    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.

Random graphs with arbitrary degree distributions and their applications.

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.

Scaling properties of scale-free evolving networks: continuous approach.

It is shown that permanent random damage to a growing scale-free network-a permanent deletion of some links-radically changes the values of the scaling exponents, and the limits of their validity are indicated.