# Complex network

In the context of network theory, a

**complex network**is a graph network with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in graphs modelling of real systems The study of complex networks is a young and active area of scientific research since 2000 inspired largely by the empirical study of real-world networks such as computer networks, technological networks, brain networks and social networks

## Contents

- 1 Definition
- 2 Scale-free networks
- 3 Small-world networks
- 4 See also
- 5 Books
- 6 References

## Definition

Most social, biological, and technological networks display substantial non-trivial topological features, with patterns of connection between their elements that are neither purely regular nor purely random Such features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure, and hierarchical structure In the case of directed networks these features also include reciprocity, triad significance profile and other features In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features The most complex structures can be realized by networks with a medium number of interactions[1] This corresponds to the fact that the maximum information content entropy is obtained for medium probabilities

Two well-known and much studied classes of complex networks are scale-free networks[2] and small-world networks,[3][4] whose discovery and definition are canonical case-studies in the field Both are characterized by specific structural features—power-law degree distributions for the former and short path lengths and high clustering for the latter However, as the study of complex networks has continued to grow in importance and popularity, many other aspects of network structure have attracted attention as well

Recently, the study of complex networks has been expanded to networks of networks[5] If those networks are interdependent, they become significantly more vulnerable to random failures and targeted attacks and exhibit cascading failures and first-order percolation transitions[6]

Furthermore, the collective behavior of a network in the presence of nodes failure and recovery has been studied[7] it was found that such a network can have spontaneous failures and spontaneous recoveries

The field continues to develop at a brisk pace, and has brought together researchers from many areas including mathematics, physics, biology, climate, computer science, sociology, epidemiology, and others[8] Ideas from network science and engineering have been applied to the analysis of metabolic and genetic regulatory networks; the modeling and design of scalable communication networks such as the generation and visualization of complex wireless networks;[9] the development of vaccination strategies for the control of disease; and a broad range of other practical issues Research on networks are regularly published in the most visible scientific journals and obtain vigorous funding in many countries Network theory was found recently useful to identify bottlenecks in city traffic[10] Network science is the topic of many conferences in a variety of different fields, and has been the subject of numerous books both for the lay person and for the expert

## Scale-free networks

Main article: Scale-free networksA network is named scale-free[2] if its degree distribution, ie, the probability that a node selected uniformly at random has a certain number of links degree, follows a particular mathematical function called a power law The power law implies that the degree distribution of these networks has no characteristic scale In contrast, networks with a single well-defined scale are somewhat similar to a lattice in that every node has roughly the same degree Examples of networks with a single scale include the Erdős–Rényi ER random graph and hypercubes In a network with a scale-free degree distribution, some vertices have a degree that is orders of magnitude larger than the average - these vertices are often called "hubs", although this is a bit misleading as there is no inherent threshold above which a node can be viewed as a hub If there were such a threshold, the network would not be scale-free

Interest in scale-free networks began in the late 1990s with the reporting of discoveries of power-law degree distributions in real world networks such as the World Wide Web, the network of Autonomous systems ASs, some networks of Internet routers, protein interaction networks, email networks, etc Most of these reported "power laws" fail when challenged with rigorous statistical testing, but the more general idea of heavy-tailed degree distributions—which many of these networks do genuinely exhibit before finite-size effects occur -- are very different from what one would expect if edges existed independently and at random ie, if they followed a Poisson distribution There are many different ways to build a network with a power-law degree distribution The Yule process is a canonical generative process for power laws, and has been known since 1925 However, it is known by many other names due to its frequent reinvention, eg, The Gibrat principle by Herbert A Simon, the Matthew effect, cumulative advantage and, preferential attachment by Barabási and Albert for power-law degree distributions Recently, Hyperbolic Geometric Graphs have been suggested as yet another way of constructing scale-free networks

Some networks with a power-law degree distribution and specific other types of structure can be highly resistant to the random deletion of vertices—ie, the vast majority of vertices remain connected together in a giant exponent[11] Such networks can also be quite sensitive to targeted attacks aimed at fracturing the network quickly When the graph is uniformly random except for the degree distribution, these critical vertices are the ones with the highest degree, and have thus been implicated in the spread of disease natural and artificial in social and communication networks, and in the spread of fads both of which are modeled by a percolation or branching process While random graphs ER have an average distance of order log N[3] between nodes, where N is the number of nodes, scale free graph can have a distance of log log N Such graphs are called ultra small world networks[12]

## Small-world networks

Main article: Small-world networkA network is called a small-world network[3] by analogy with the small-world phenomenon popularly known as six degrees of separation The small world hypothesis, which was first described by the Hungarian writer Frigyes Karinthy in 1929, and tested experimentally by Stanley Milgram 1967, is the idea that two arbitrary people are connected by only six degrees of separation, ie the diameter of the corresponding graph of social connections is not much larger than six In 1998, Duncan J Watts and Steven Strogatz published the first small-world network model, which through a single parameter smoothly interpolates between a random graph and a lattice Their model demonstrated that with the addition of only a small number of long-range links, a regular graph, in which the diameter is proportional to the size of the network, can be transformed into a "small world" in which the average number of edges between any two vertices is very small mathematically, it should grow as the logarithm of the size of the network, while the clustering coefficient stays large It is known that a wide variety of abstract graphs exhibit the small-world property, eg, random graphs and scale-free networks Further, real world networks such as the World Wide Web and the metabolic network also exhibit this property

In the scientific literature on networks, there is some ambiguity associated with the term "small world" In addition to referring to the size of the diameter of the network, it can also refer to the co-occurrence of a small diameter and a high clustering coefficient The clustering coefficient is a metric that represents the density of triangles in the network For instance, sparse random graphs have a vanishingly small clustering coefficient while real world networks often have a coefficient significantly larger Scientists point to this difference as suggesting that edges are correlated in real world networks

## See also

- Community structure
- Complex adaptive system
- Complex systems
- Dual-phase evolution
- Dynamic network analysis
- Interdependent networks
- Network theory
- Network science
- Percolation theory
- Random graph
- Scale-free networks
- Small world networks
- Spatial network

## Books

- Niloy Ganguly Editor, Andreas Deutsch Editor and Animesh Mukherjee Editor, Dynamics On and Of Complex Networks Applications to Biology, Computer Science, and the Social Sciences, 2009, ISBN 978-0-8176-4750-6
- Albert-László Barabási, Linked: How Everything is Connected to Everything Else, 2004, ISBN 0-452-28439-2
- Alain Barrat, Marc Barthelemy, Alessandro Vespignani, Dynamical processes on complex networks, Cambridge University Press, 2008, ISBN 978-0-521-87950-7
- Stefan Bornholdt Editor and Heinz Georg Schuster Editor, Handbook of Graphs and Networks: From the Genome to the Internet, 2003, ISBN 3-527-40336-1
- Guido Caldarelli, Scale-Free Networks Oxford University Press, 2007, ISBN 978-0-19-921151-7
- Guido Caldarelli, Michele Catanzaro, Networks: A Very Short Introduction Oxford University Press, 2012, ISBN 978-0-19-958807-7
- E Estrada, "The Structure of Complex Networks: Theory and Applications", Oxford University Press, 2011, ISBN 978-0-199-59175-6
- Reuven Cohen and Shlomo Havlin, Complex Networks: Structure, Robustness and Function, Cambridge University Press, 2010, ISBN 978-0-521-84156-6
- SN Dorogovtsev and JFF Mendes, Evolution of Networks: From biological networks to the Internet and WWW, Oxford University Press, 2003, ISBN 0-19-851590-1
- Mark Newman, Networks: An Introduction, Oxford University Press, 2010, ISBN 978-0-19-920665-0
- Mark Newman, Albert-László Barabási, and Duncan J Watts, The Structure and Dynamics of Networks, Princeton University Press, Princeton, 2006, ISBN 978-0-691-11357-9
- R Pastor-Satorras and A Vespignani, Evolution and Structure of the Internet: A statistical physics approach, Cambridge University Press, 2004, ISBN 0-521-82698-5
- Duncan J Watts, Six Degrees: The Science of a Connected Age, W W Norton & Company, 2003, ISBN 0-393-04142-5
- Duncan J Watts, Small Worlds: The Dynamics of Networks between Order and Randomness, Princeton University Press, 2003, ISBN 0-691-11704-7
- T Lewis, Network Science, Wiley 2009,

## References

This article includes a list of references, but its sources remain unclear because it has insufficient inline citations Please help to improve this article by introducing more precise citations August 2008 Learn how and when to remove this template message |

**^**T Wilhelm, J Kim 2008 "What is a complex graph" Physica A**387**: 2637–2652 Bibcode:2008PhyA3872637K doi:101016/jphysa200801015- ^
**a****b**A Barabasi, E Bonabeau May 2003 "Scale-Free Networks" Scientific American: 50–59 - ^
**a****b****c**S H Strogatz, D J Watts 1998 "Collective dynamics of 'small-world' networks" Nature**393**6684: 440–442 Bibcode:1998Natur393440W doi:101038/30918 PMID 9623998 **^**HE Stanley, LAN Amaral, A Scala, M Barthelemy 2000 "Classes of small-world networks" PNAS**97**21: 11149–52 arXiv:cond-mat/0001458 Bibcode:2000PNAS9711149A doi:101073/pnas200327197 PMC 17168 PMID 11005838**^**Buldyrev, Sergey V; Parshani, Roni; Paul, Gerald; Stanley, H Eugene; Havlin, Shlomo 2010 "Catastrophic cascade of failures in interdependent networks" Nature**464**7291: 1025–1028 arXiv:09071182 Bibcode:2010Natur4641025B doi:101038/nature08932 ISSN 0028-0836 PMID 20393559**^**Parshani, Roni; Buldyrev, Sergey V; Havlin, Shlomo 2010 "Interdependent Networks: Reducing the Coupling Strength Leads to a Change from a First to Second Order Percolation Transition" Physical Review Letters**105**4: 048701 arXiv:10043989 Bibcode:2010PhRvL105d8701P doi:101103/PhysRevLett105048701 ISSN 0031-9007 PMID 20867893**^**Majdandzic, Antonio; Podobnik, Boris; Buldyrev, Sergey V; Kenett, Dror Y; Havlin, Shlomo; Eugene Stanley, H 2013 "Spontaneous recovery in dynamical networks" Nature Physics**10**1: 34–38 doi:101038/nphys2819 ISSN 1745-2473**^**AE Motter, R Albert 2012 "Networks in Motion" Physics Today**65**4: 43–48 arXiv:12062369 Bibcode:2012PhT65d43M doi:101063/pt31518**^**Mouhamed Abdulla "On the Fundamentals of Stochastic Spatial Modeling and Analysis of Wireless Networks and its Impact to Channel Losses" PhD Dissertation, Dept of Electrical and Computer Engineering, Concordia Univ, Montréal, Québec, Canada, Sep 2012: Ch4 develops algorithms for complex network generation and visualization**^**Li, Daqing; Fu, Bowen; Wang, Yunpeng; Lu, Guangquan; Berezin, Yehiel; Stanley, H Eugene; Havlin, Shlomo 2015 "Percolation transition in dynamical traffic network with evolving critical bottlenecks" Proceedings of the National Academy of Sciences**112**3: 669–672 doi:101073/pnas1419185112 ISSN 0027-8424**^**Cohen, Reuven; Erez, Keren; ben-Avraham, Daniel; Havlin, Shlomo 2000 "Resilience of the Internet to Random Breakdowns" Physical Review Letters**85**21: 4626–4628 doi:101103/PhysRevLett854626 ISSN 0031-9007**^**R Cohen, S Havlin 2003 "Scale-free networks are ultrasmall" Phys Rev Lett**90**: 058701 arXiv:cond-mat/0205476 Bibcode:2003PhRvL90e8701C doi:101103/physrevlett90058701

- D J Watts and S H Strogatz 1998 "Collective dynamics of 'small-world' networks" Nature
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