Graph Theoretic Approach to Describing Regional Economic Structure

A Graph Theoretic Approach to Describing Regional Economic Structure
Rose Baker
David Passmore
Institute for Research in
Training & Development

Institute for Research in Training & Development
Rose Baker
Assistant Professor of Workforce Education & Development / Research Associate in Office of Associate Dean for Research, Outreach, &Technology
David Passmore
Professor of Workforce Education & Development / Director of Institute

Our aim: Portray supply structure
Nothing new…
An approach we want to take…
Your advice appreciated…

First, the I/O accounting relationship

An I/O table

Focus on portion of I/O table
X
y
+
=
x

I/O accounting identities
Total output = interindustry transactions + final demand
x = X + y
Divide each element of X by total output
x = Ax + y
Add an identity matrix and rearrange terms
x = (I – A)-1y

Total requirements matrix: Leontief Inverse
(I – A)-1
One way to calculate the Leontief Inverse: power series
(I – A)-1 = (I + A + A2 + A3 +…An)

Relationship of graph theory with I/O

A collection of nodes or vertices…
A graph:
with a collection of edges
that connect nodes
Graph theory: Study of mathematical structures of pairwise relations between objects

A
Columns are purchasing industries.
0 .2 0
.2 .3 .1
.3 0 0
Transactions matrix as an adjacency matrix: Representing transactions as a graph
Rows are
producing industries.
A number entered in a column indicates the proportion of total outlays of the industry purchased from an industry in a row.

Transforming real–valued transactionsmatrix to Boolean adjacency matrix
A
W(1)

0 .2 0
.2 .3 .1
.3 0 0
0 1 0
1 1 1
1 0 0
A “small” entries in A
can be filtered as zero in W(1).

Round 2 of impact from power series:(I – A)-1 = (I + A + A2 + A3 +…An)
A
A
x
0 .2 0
.2 .3 .1
.3 0 0
0 .2 0
.2 .3 .1
.3 0 0
=
A2
.04 .06 .02
.09 .13 .03
0 .06 0

A2
W(2)

1 1 1
1 1 1
0 1 0
.04 .06 .02
.09 .13 .03
0 .06 0
Indirect connections indicated by “1” in W(2).
Round 2 impacts in A through A2translated to Boolean matrix W(2)

W(1)
W(1)
x
0 1 0
1 1 1
1 0 0
0 1 0
1 1 1
1 0 0
=
W2
1 1 1
2 2 1
0 1 0
Recover number of linkages through Round 2

And, graph and its metrics

W(1)
1
0 1 0
1 1 1
1 0 0
2
3
Three–industry digraph from Boolean matrix W(1)

Some graph metrics that describe the structure of supply chain relationships in an economy
Density — proportion of edges (links) among all possible edges.
Centrality — nodes with highest number of edges.
Isolation — proportion of nodes without edges among all nodes.
Cohesion — average number of edges required to reach all possible pairs of nodes.
Brokerage — a node that connects otherwise unconnected nodes.
1
2
3

What we will do within a larger social networking project
Construct a graph and associated metrics of interindustry transactions for Centre County, Pennsylvania.
Explore graphing relationships between industries and categories of final demand.

A Graph Theoretic Approach to Describing Regional Economic Structure
Rose Baker
David Passmore
Institute for Research in
Training & Development

Graph Theoretic Approach to Describing Regional Economic Structure

by Penn State

Accessibility

Categories

Upload Details

Usage Rights

Statistics

Graph Theoretic Approach to Describing Regional Economic Structure Presentation Transcript