Geometry of complex networks and topological centrality

• A geometric approach to centrality using the Moore–Penrose pseudo-inverse of the Laplacian.
• Topological centrality of a node is determined in terms of the position vector and the robustness of the overall network in terms of the Kirchhoff index.
• Interpretations provided in terms of detour overheads, recurrence probabilities and connectedness in bi-partitions.
• Empirical analysis shows how these indices reflect structural roles of nodes in the network and their sensitivity to perturbations.

We explore the geometry of complex networks in terms of an nn-dimensional Euclidean embedding represented by the Moore–Penrose pseudo-inverse of the graph Laplacian (L+). The squared distance of a node ii to the origin in this nn-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)(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 ii in all the bi-partitions of the graph. These interpretations respectively help establish the topological centrality (C∗(i))(C∗(i)) of node ii as a measure of its overall position as well as its overall connectedness   in the network; thus reflecting the robustness of ii 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.