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Generalized relationships between characteristic path length, efficiency, clustering coefficients, and density

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Abstract

Graph theoretic properties such as the clustering coefficient, characteristic (or average) path length, global and local efficiency provide valuable information regarding the structure of a graph. These four properties have applications to biological and social networks and have dominated much of the literature in these fields. While much work has done in applied settings, there has yet to be a mathematical comparison of these metrics from a theoretical standpoint. Motivated by both real-world data and computer simulations, we present asymptotic linear relationships between the characteristic path length, global efficiency, and graph density, and also between the clustering coefficient and local efficiency. In the current literature, these properties are often presented as independent metrics; however, we show in this paper that they are inextricably linked.

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Acknowledgements

The authors are grateful to two anonymous referees for their comments and suggestions that greatly improved this paper. Research was supported by a National Science Foundation Research Experiences for Undergraduates Grant #1358583. Darren Narayan was also supported by a National Science Foundation CCLI Grant #1019532. The authors would like to thank Bradford Z. Mahon and Frank Garcea of the Rochester Center for Brain Imaging at the University of Rochester and Dr. Jeffery Bazarian MD and Kian Merchant-Borna of the Department of Emergency Medicine for providing functional MRI data.

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Authors and Affiliations

  1. Case Western Reserve University, Cleveland, OH, 44106, USA

    Alexander Strang

  2. Rochester Institute of Technology, Rochester, NY, 14623, USA

    Oliver Haynes & Nathan D. Cahill

  3. Rochester Institute of Technology, Rochester, NY, 14623-5604, USA

    Darren A. Narayan

Authors
  1. Alexander Strang
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  2. Oliver Haynes
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  3. Nathan D. Cahill
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  4. Darren A. Narayan
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Correspondence to Darren A. Narayan.

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Strang, A., Haynes, O., Cahill, N.D. et al. Generalized relationships between characteristic path length, efficiency, clustering coefficients, and density. Soc. Netw. Anal. Min. 8, 14 (2018). https://doi.org/10.1007/s13278-018-0492-3

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  • DOI: https://doi.org/10.1007/s13278-018-0492-3

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