# Adjacency Matrix

## Contents

## Definition

An **Adjacency Matrix** in the context of Input-Output Analysis is a boolean matrix (composed of zeros and ones) representing the Supply Chain network connecting economic activities without specific regard to the actual magnitudes (e.g., volumes or monetary values) being exchanged.

*Adjacency* is the graph theoretic representation of the fact that two entities, represented by graph nodes, are directly related, tied, or connected with one another. The sectoral dependencies represented in an input-output table (IOT) can be converted into an adjacency matrix representing the supply chain network that connects economic activities.

## Formula

The general approach^{[1]} for the construction of an adjacency matrix is to perform is a binary (Boolean) transformation on either the transactions matrix Z or the coefficients matrix (A) to generate matrices with “1” in cells for which (or ) and “0” elsewhere (where H is a suitable threshold).

## Usage

Given entities i and j in a set of agents (nodes, entities, sectors) N, and arcs (edges, links) denoting the existence of relations from i to j, agents i and j are adjacent if there exist either of the two arcs from i to j or from j to i. The adjacency matrix G can then be analysed using techniques from graph and network theory to reveal economic structure and other features of the system being analysed.

## See Also

## References

- ↑ R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009