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We explore the geometry of complex networks in terms of an n-dimensionalEuclidean embedding represented by the Moore-Penrose pseudo-inverse of thegraph Laplacian $(\bb L^+)$. The squared distance of a node $i$ to the originin this n-dimensional space $(l^+_{ii})$, yields a topological centrality index$(\mathcal{C}^{*}(i) = 1/l^+_{ii})$ for node $i$. In turn, the sum ofreciprocals 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 Kirchhoff index $(\mathcal{K})$, an overall structuraldescriptor for the network. In addition to this geometric interpretation, weprovide alternative interpretations of the proposed indices to reveal theirtrue topological characteristics: first, in terms of forced detour overheadsand frequency of recurrences in random walks that has an interesting analogy tovoltage distributions in the equivalent electrical network; and then as theaverage connectedness of $i$ in all the bi-partitions of the graph. Theseinterpretations respectively help establish the topological centrality$(\mathcal{C}^{*}(i))$ of node $i$ as a measure of its overall position as wellas its overall connectedness in the network; thus reflecting the robustness ofnode $i$ to random multiple edge failures. Through empirical evaluations usingsynthetic and real world networks, we demonstrate how the topologicalcentrality is better able to distinguish nodes in terms of their structuralroles in the network and, along with Kirchhoff index, is appropriatelysensitive to perturbations/rewirings in the network.

Geometry of Complex Networks and Topological Centrality