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Edge Importance in Complex Networks

Silvia Noschese, Lothar Reichel

TL;DR

This paper compares two approaches to estimate the importance of edges of medium-sized to large networks by computes partial derivatives of the total communicability of the weights of the edges.

Abstract

Complex networks are made up of vertices and edges. The latter connect the vertices. There are several ways to measure the importance of the vertices, e.g., by counting the number of edges that start or end at each vertex, or by using the subgraph centrality of the vertices. It is more difficult to assess the importance of the edges. One approach is to consider the line graph associated with the given network and determine the importance of the vertices of the line graph, but this is fairly complicated except for small networks. This paper compares two approaches to estimate the importance of edges of medium-sized to large networks. One approach computes partial derivatives of the total communicability of the weights of the edges, where a partial derivative of large magnitude indicates that the corresponding edge may be important. Our second approach computes the Perron sensitivity of the edges. A high sensitivity signals that the edge may be important. The performance of these methods and some computational aspects are discussed. Applications of interest include to determine whether a network can be replaced by a network with fewer edges with about the same communicability.

Edge Importance in Complex Networks

TL;DR

This paper compares two approaches to estimate the importance of edges of medium-sized to large networks by computes partial derivatives of the total communicability of the weights of the edges.

Abstract

Complex networks are made up of vertices and edges. The latter connect the vertices. There are several ways to measure the importance of the vertices, e.g., by counting the number of edges that start or end at each vertex, or by using the subgraph centrality of the vertices. It is more difficult to assess the importance of the edges. One approach is to consider the line graph associated with the given network and determine the importance of the vertices of the line graph, but this is fairly complicated except for small networks. This paper compares two approaches to estimate the importance of edges of medium-sized to large networks. One approach computes partial derivatives of the total communicability of the weights of the edges, where a partial derivative of large magnitude indicates that the corresponding edge may be important. Our second approach computes the Perron sensitivity of the edges. A high sensitivity signals that the edge may be important. The performance of these methods and some computational aspects are discussed. Applications of interest include to determine whether a network can be replaced by a network with fewer edges with about the same communicability.
Paper Structure (18 sections, 3 theorems, 89 equations, 2 figures, 8 tables)

This paper contains 18 sections, 3 theorems, 89 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Network modifications based on the gradient
  3. Methods for small to medium-sized networks
  4. Network simplification by edge removal
  5. Network modification to increase or decrease total communicability
  6. Network modification by inclusion of new edges
  7. Methods for large networks
  8. Network modifications based on Perron root sensitivity
  9. Perron communicability for small to medium-sized networks
  10. Network simplification by edge removal
  11. Network modification by edge-weight tuning
  12. Network modification by inclusion of new edges
  13. Network modification criteria for large-scale networks
  14. Computed examples
  15. A synthetic example
  16. ...and 3 more sections

Key Result

theorem 1

(Schweitzer Sc) Let $\mathbf{A}\in\mathbb{R}^{n\times n}$ and $u,v\in\mathbb{R}^n\backslash\{0\}$, and assume that $f$ is Fréchet differentiable at $\mathbf{A}$. Define $\mathbf{E}_{ij}=e_ie_j^T$, where $e_k=[0,\ldots,0,1,0,\ldots,0]^T\in\mathbb{R}^n$ denotes the $k$th column of the identity matrix.

Figures (2)

  • Figure 1: Example \ref{['ex0']}. Structure of $L_{\exp_0}(\mathbf{A},ee^T)$ (left picture (a)) and of $\mathbf{W}$ (right picture (b)) for the matrix $\mathbf{A}\in \mathbb{R}^{50\times 50}$ in \ref{['Atoep']} with $\sigma=1$.
  • Figure 2: Example \ref{['ex2']}. The vertices (marked in red) that are connected by the edges to be removed in order to simplify the network according to ${\mathcal{E}}_{L_{f}}$ for (a), according to ${\widetilde{\mathcal{E}}}_{L_{f}}$ for (b), and according to ${\mathcal{E}}_{\rho}$ for (c).

Theorems & Definitions (5)

  • theorem 1
  • theorem 2
  • proof
  • proposition 1
  • proof