2. 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 & Professor of Operations
Research / Director of Institute for
Research in Training & Development
Happy REMI users since the DOS version!
9. X + y = x
Focus on a portion of input-output table
10. Total output = interindustry transactions + final demand
x=X+y
Divide each element in row of X by total row output, x
x = Ax + y
Add an identity matrix and rearrange terms
x = (I – A)-1 y
Input–output accounting identities
12. A collection of nodes or vertices…
A graph
with a collection of edges
that connect the nodes
Graph theory: Study of mathematical structures
of pairwise relations between objects
13. x = Ax + y Columns are
A purchasing
industries
0 .2 0
.2 .3 .1 Rows are
.3 0 0 producing
industries
A number entered in a column is the
proportion of total outlays of the industry
purchased from an industry in a row
Transactions matrix as an adjacency matrix:
Representing transactions as a graph
17. A W
0 .2 0 0 1 0
.2 .3 .1 1 1 1
.3 0 0 1 0 0
“Small” entries in A
are filtered as “0” in W
Transforming real–valued transactions
matrix to Boolean adjacency matrix
18. A W
0 .2 0 0 1 0
.2 .3 .1 1 1 1
.3 0 0 1 0 0
Other entries in A
are filtered as “1” in W
Transforming real–valued transactions
matrix to Boolean adjacency matrix
19. X A
W
wij for aij > 0.1
The Miernyk table: Matrix W for aij > 0.1
20. D
A X
E C
B
F W
The Miernyk table: Directed graph of
matrix W for aij > 0.1
30. 1 In–degree — number of
incoming edges
2 3 Out–degree — number
of outgoing edges
Density — proportion of Isolation — proportion
edges (links) among all of nodes without edges
possible edges among all nodes
Centrality — nodes with
highest number of edges
Some graph metrics that describe the structure
of transactions in a regional economy
31. D DENSITY = 0.28
A
10 edges
among 36
E C possible
B
F W
The Miernyk table: Directed graph metrics
for matrix W for aij > 0.1
32. D CENTRALITY
A
Nodes A & C
are central
E C
Both have 3 edges
B
F W
The Miernyk table: Directed graph metrics
for matrix W for aij > 0.1
33. IN– & OUT–DEGREE
D
A Nodes A & C
have high in–degree
E C Node B has high
out–degree
B
F W
The Miernyk table: Directed graph metrics
for matrix W for aij > 0.1
34. D ISOLATION
A
Node E
E C is almost an
an isolate
B
F W
The Miernyk table: Directed graph metrics
for matrix W for aij > 0.1
35. Construct a graph & associated
metrics of interindustry
transactions for Centre County,
Pennsylvania
Explore graphing relationships
between industries & categories of
final demand within region
What we will do within a larger
social networking project