Spectral influence in networks: An application to Input-Output analysis

TL;DR

A new clustering algorithm is proposed that identifies communities with high cyclicality and interdependence, allowing for overlaps, and is applied to input-output analysis within the context of the Moroccan economy.

Abstract

This paper introduces the concepts of spectral influence and spectral cyclicality, both derived from the largest eigenvalue of a graph's adjacency matrix. These two novel centrality measures capture both diffusion and interdependence from a local and global perspective respectively. We propose a new clustering algorithm that identifies communities with high cyclicality and interdependence, allowing for overlaps. To illustrate our method, we apply it to input-output analysis within the context of the Moroccan economy.

Figures (11)

• Figure 1: The whole is not the sum of the parts: for $S_1=\{1,2\}$ and $S_2=\{4,5\}$, $\sum_{s\in S_1}\left(\rho(W)- \rho(W(s))\right)=0.42 < \rho(W)-\rho(W(S_1))=1.74$ while $\sum_{s\in S_2}\left(\rho(W)- \rho(W(s))\right)=0.08 > \rho(W)-\rho(W(S_2))=0.06$.
• Figure 2: The spectral cyclicality for different dispositions of a three vertices unweighted and connected digraph: a) perfect diffusion on a cycle, b) diffusion slowed down by a circuit, c) diffusion slowed down by a loop, d) slowest diffusion in a complete graph, e) partial diffusion, f) one way diffusion.
• Figure 3: Evolution of Moroccan spectral cyclicality from $1995$ to $2018$.
• Figure 4: The five divisive clusters of $2018$ Moroccan input-output table.
• Figure 5: The six agglomerative clusters of $2018$ Moroccan input-output table.

Theorems & Definitions (19)

• Proposition 2.1
• proof
• Definition 2.1: Dominant cycle
• Theorem 2.2: Perron-Frobenius Horn2013 ch.8
• Corollary 2.3: Perron-Frobenius Horn2013 ch.8
• Theorem 2.4: Horn2013 p.539
• Definition 3.1: Spectral influence
• Theorem 3.1: Spectral radius, closed walks and dominant cycle
• proof
• Remark 3.1