# Self-similarity of complex networks

@article{Song2005SelfsimilarityOC, title={Self-similarity of complex networks}, author={Chaoming Song and Shlomo Havlin and Hern{\'a}n A. Makse}, journal={Nature}, year={2005}, volume={433}, pages={392-395} }

Complex networks have been studied extensively owing to their relevance to many real systems such as the world-wide web, the Internet, energy landscapes and biological and social networks. A large number of real networks are referred to as ‘scale-free’ because they show a power-law distribution of the number of links per node. However, it is widely believed that complex networks are not invariant or self-similar under a length-scale transformation. This conclusion originates from the ‘small…

## 1,159 Citations

Self-similar scaling of density in complex real-world networks

- Computer Science, PhysicsArXiv
- 2011

Multifractal characterisation and analysis of complex networks

- Computer Science
- 2011

This thesis proposes a new box covering algorithm for multifractal analysis of complex networks and confirms the existence of multifractality in scale-free networks and PPI networks, while the multifractals behaviour is not confirmed for small-world networks and random networks.

Complex systems: Romanesque networks

- Computer ScienceNature
- 2005

A new analysis of ‘Scale-free’ networks, in which nodes are partitioned into boxes of different sizes, reveals that they share the surprising feature of self-similarity, which may help explain how the scale-free property of such networks arises.

Connectivity in biological networks: Are they really scale-free?

- Computer Science
- 2020

This research systematically analyzes 35 biological networks containing gene and protein interaction data to determine whether these networks exhibit power law, and thereby scale-free, and found that although the power law model had a robust fit, the models frequently yielded values out of the desired range.

Power-Hop: A Pervasive Observation for Real Complex Networks

- Computer SciencePloS one
- 2016

This work identifies another power-law pattern that describes the relationship between the fractions of node pairs C(r) within r hops and the hop count r, and considers the hop-count r to be the underlying distance metric between two vertices of the network, and finds that this relationship follows aPower-law in real networks within the range 2 ≤ r ≤ d.

Scale-free networks are rare

- Computer ScienceNature Communications
- 2019

A severe test of their empirical prevalence using state-of-the-art statistical tools applied to nearly 1000 social, biological, technological, transportation, and information networks finds robust evidence that strongly scale-free structure is empirically rare, while for most networks, log-normal distributions fit the data as well or better than power laws.

The Fractal Dimensions of Complex Networks

- Computer Science
- 2009

This work finds that the average 'density' (ρ(r)) of complex networks follows a better power-law function as a function of distance r with the exponent df, which is defined as the fractal dimension, in some real complex networks.

The Polynomial Volume Law of Complex Networks in the Context of Local and Global Optimization

- Computer ScienceScientific Reports
- 2018

In this article neighbourhoods are demonstrated to share a common local structure in many real complex networks, manifested by a polynomial volume law, which turns out that the law is robust against the coexistence of such global structures.

Scale-free networks revealed from finite-size scaling

- Computer ScienceArXiv
- 2019

Finite size scaling provides a powerful framework for analyzing self-similarity and the scale free nature of empirical networks and is a useful tool for analyzing deviations from power law behavior in the vicinity of a critical point in a physical system arising due to a finite correlation length.

On General Laws of Complex Networks

- Computer ScienceComplex
- 2009

It is shown that when vertex degrees of large networks follow a scale-free power-law distribution with the exponent γ ≥ 2, the number of degree-1 vertices, when nonzero, is of the same order as the network size N and the average degree is of order less than log N.

## References

SHOWING 1-10 OF 66 REFERENCES

Complex systems: Romanesque networks

- Computer ScienceNature
- 2005

A new analysis of ‘Scale-free’ networks, in which nodes are partitioned into boxes of different sizes, reveals that they share the surprising feature of self-similarity, which may help explain how the scale-free property of such networks arises.

Statistical mechanics of complex networks

- Computer ScienceArXiv
- 2001

A simple model based on these two principles was able to reproduce the power-law degree distribution of real networks, 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.

Origins of fractality in the growth of complex networks

- Computer Science
- 2006

It is shown that the key principle that gives rise to the fractal architecture of networks is a strong effective ‘repulsion’ (or, disassortativity) between the most connected nodes (that is, the hubs) on all length scales, rendering them very dispersed.

Emergence of scaling in random networks

- Computer ScienceScience
- 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.

Collective dynamics of ‘small-world’ networks

- Computer ScienceNature
- 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.

Evolution of Networks: From Biological Nets to the Internet and WWW (Physics)

- Physics
- 2003

The aim of the text is to understand networks and the basic principles of their structural organization and evolution, so even students without a deep knowledge of mathematics and statistical physics will be able to rely on this as a reference.

The Structure and Function of Complex Networks

- Computer ScienceSIAM Rev.
- 2003

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.

Complex networks

- Computer Science
- 2004

How complexity requires both new tools and an augmentation of the conceptual framework--including an expanded definition of what is meant by a “quantitative prediction” is discussed.

Statistical ensemble of scale-free random graphs.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2001

A thorough discussion of the statistical ensemble of scale-free connected random tree graphs is presented: methods borrowed from field theory are used to define the ensemble and its properties are studied analytically and possible generalizations are discussed.

Optimal paths in disordered complex networks.

- Computer SciencePhysical review letters
- 2003

For strong disorder, where the maximal weight along the path dominates the sum, l(opt) approximately N(1/3) in both Erdos-Rényi (ER) and Watts-Strogatz (WS) networks.