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Geometry of complex networks and topological centrality

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  • Ranjan, Gyan
  • Zhang, Zhi-Li

Abstract

We explore the geometry of complex networks in terms of an n-dimensional Euclidean embedding represented by the Moore–Penrose pseudo-inverse of the graph Laplacian (L+). The squared distance of a node i to the origin in this n-dimensional space (lii+), yields a topological centrality index, defined as C∗(i)=1/lii+. In turn, the sum of reciprocals of individual node centralities, ∑i1/C∗(i)=∑ilii+, or the trace of L+, yields the well-known Kirchhoff index (K), an overall structural descriptor for the network. To put into context this geometric definition of centrality, we provide alternative interpretations of the proposed indices that connect them to meaningful topological characteristics — first, as forced detour overheads and frequency of recurrences in random walks that has an interesting analogy to voltage distributions in the equivalent electrical network; and then as the average connectedness of i in all the bi-partitions of the graph. These interpretations respectively help establish the topological centrality (C∗(i)) of node i as a measure of its overall position as well as its overall connectedness in the network; thus reflecting the robustness of i to random multiple edge failures. Through empirical evaluations using synthetic and real world networks, we demonstrate how the topological centrality is better able to distinguish nodes in terms of their structural roles in the network and, along with Kirchhoff index, is appropriately sensitive to perturbations/re-wirings in the network.

Suggested Citation

  • Ranjan, Gyan & Zhang, Zhi-Li, 2013. "Geometry of complex networks and topological centrality," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(17), pages 3833-3845.
  • Handle: RePEc:eee:phsmap:v:392:y:2013:i:17:p:3833-3845
    DOI: 10.1016/j.physa.2013.04.013
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    References listed on IDEAS

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    1. Réka Albert & Hawoong Jeong & Albert-László Barabási, 2000. "Error and attack tolerance of complex networks," Nature, Nature, vol. 406(6794), pages 378-382, July.
    2. Göbel, F. & Jagers, A. A., 1974. "Random walks on graphs," Stochastic Processes and their Applications, Elsevier, vol. 2(4), pages 311-336, October.
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    Cited by:

    1. László Csató, 2017. "Measuring centrality by a generalization of degree," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 25(4), pages 771-790, December.
    2. Ding, Xiao & Wang, Huan & Zhang, Xi & Ma, Chuang & Zhang, Hai-Feng, 2024. "Dual nature of cyber–physical power systems and the mitigation strategies," Reliability Engineering and System Safety, Elsevier, vol. 244(C).
    3. Ariel L. Wirkierman & Monica Bianchi & Anna Torriero, 2022. "Leontief Meets Markov: Sectoral Vulnerabilities Through Circular Connectivity," Networks and Spatial Economics, Springer, vol. 22(3), pages 659-690, September.

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