World Library  
Flag as Inappropriate
Email this Article

Reciprocity (network science)

Article Id: WHEBN0008447449
Reproduction Date:

Title: Reciprocity (network science)  
Author: World Heritage Encyclopedia
Language: English
Subject: Network science, Erdős–Rényi model, Efficiency (Network Science), Computer network, Boolean network
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Reciprocity (network science)

Theoretical efforts have been made to study the nontrivial properties of complex networks, such as clustering, scale-free degree distribution, community structures, etc. Here Reciprocity is another quantity to specifically characterize directed networks. Link reciprocity measures the tendency of vertex pairs to form mutual connections between each other.[1]

Motivation

In real network problems, people are interested in determining the [1]

How is it defined?

Traditional definition

A traditional way to define the reciprocity r is using the ratio of the number of links pointing in both directions L^{<->} to the total number of links L [6] r = \frac {L^{<->}}{L}

With this definition, r = 1 is for a purely bidirectional network while r = 0 for a purely unidirectional one. Real networks have an intermediate value between 0 and 1.

However, this definition of reciprocity has some defects. It cannot tell the relative difference of reciprocity compared with purely random network with the same number of vertices and edges. The useful information from reciprocity is not the value itself, but whether mutual links occur more or less often than expected by chance. Besides, in those networks containing self-linking loops (links starting and ending at the same vertex), the self-linking loops should be excluded when calculating L.

Garlaschelli and Loffredo's definition

In order to overcome the defects of the above definition, Garlaschelli and Loffredo defined reciprocity as the correlation coefficient between the entries of the adjacency matrix of a directed graph (a_{ij} = 1 if a link from i to j is there, and a_{ij} = 0 if not):

\rho \equiv \frac {\sum_{i \neq j} (a_{ij} - \bar{a}) (a_{ji} - \bar{a})}{\sum_{i \neq j} (a_{ij} - \bar{a})^2},

where the average value \bar{a} \equiv \frac {\sum_{i \neq j} a_{ij}} {N(N-1)} = \frac {L} {N(N-1)}.

\bar{a} measures the ratio of observed to possible directed links (link density), and self-linking loops are now excluded from L because of i not equal to j.

The definition can be written in the following simple form:

\rho = \frac {r - \bar{a}} {1- \bar{a}}

The new definition of reciprocity gives an absolute quantity which directly allows one to distinguish between reciprocal (\rho > 0) and antireciprocal (\rho < 0) networks, with mutual links occurring more and less often than random respectively.

If all the links occur in reciprocal pairs, \rho = 1; if r=0, \rho = \rho_{min}. \rho_{min} \equiv \frac {- \bar{a}} {1- \bar{a}}

This is another advantage of using \rho, because it incorporates the idea that complete antireciprocal is more statistical significant in the networks with larger density, while it has to be regarded as a less pronounced effect in sparser networks.

References

  1. ^ a b Diego Garlaschelli, and Maria I. Loffredo. Patterns of Link Reciprocity in Directed Networks. Phys. Rev. Lett. 93, 268701 (2004).
  2. ^ M. E. J. Newman, S. Forrest, and J. Balthrop, Phys. Rev. E 66, 035101(R) (2002).
  3. ^ R. Albert, H. Jeong, and A.-L. Baraba´si, Nature (London) 401, 130 (1999).
  4. ^ D. Garlaschelli and M. I. Loffredo, Phys. Rev. Lett. 93, 188701 (2004).
  5. ^ V. Zlatic, M. Bozicevic, H. Stefancic, and M. Domazet, Phys. Rev. E 74, 016115 (2006)
  6. ^ M. E. J. Newman, S. Forrest, and J. Balthrop, Phys. Rev. E 66, 035101(R) (2002).
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 



Copyright © World Library Foundation. All rights reserved. eBooks from World Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.