Geometry of Complex Networks and Structural Centrality

Details Geometry of Complex Networks and Structural Centrality Geometry of Complex Networks and Structural Centrality

2011-07-05

arxiv.org/abs/1107.0989v2

Computer Science

Discrete Mathematics

14 pages, 7 figures, 2 tables

Scientific paper

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 $(\bb L^+)$. The squared distance of a node $i$ to the origin in this n-dimensional space $(l^+_{ii})$, yields a structural centrality index $(\mathcal{C}^{*}(i) = 1/l^+_{ii})$ for node $i$. In turn, the sum of reciprocals of individual node structural centralities, $\sum_{i}1/\mathcal{C}^*(i) = \sum_{i} l^+_{ii}$, i.e. the trace of $\bb L^+$, yields the well-known Kirchoff index $(\mathcal{K})$, an overall structural descriptor for the network. In addition to this geometric interpretation, we provide alternative interpretations of the proposed indices to reveal their true topological characteristics: first, in terms of 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 structural centrality $(\mathcal{C}^{*}(i))$ as a measure of both the overall position as well as the overall connectedness of node $i$ in the network which in turn reflects the robustness of node $i$ to random multiple edge failures. Through empirical evaluations using synthetic and real world networks, we demonstrate how the structural centrality is better able to distinguish nodes in terms of their structural roles in the network and, along with Kirchoff index, is appropriately sensitive to perturbations/rewirings in the network.

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