Slides displayed during presentation the 27th Annual REMI Users’ Conference Washington, DC, October 18, 2012.

1. Graph–Theoretic Links
in Regional Economies
Rose Baker
David Passmore
Institute for Research in
Training & Development

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!

3. Our aim: Portray regional supply &
demand structure graphically

4. Examined graphical
representation of
binary input–output
analysis
At 2011 Lake Tahoe REMI Conference….

5. Graphical representation
of interindustry links
& final demand by
source
Today….

6. First, a reminder about some
input–output accounting
relationships

7. A hypothetical input–output table from
Miernyk, Elements of Input–Output Analysis

8. Focus on a portion of input-output table

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

11. Input–output accounts
as graphs

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

14. X y x
The Miernyk table, again

15. X
The Miernyk table: Interindustry transactions

16. X
A
The Miernyk table: Matrix A

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

21. X y x
The Miernyk table, again

22. y
The Miernyk table: Final demand

23. Proportion of total
GRP by source of
final demand
The Miernyk table: Final demand by source

24. Capital
Inventory Exports Government Households
Formation
A B C D E F
The Miernyk table: Directed graph of final
demand by source with wij > 0.20

25. Capital
Inventory Exports Government Households
Formation
A B C D E F
The Miernyk table: Directed graph of final
demand by source with wij > 0.20

26. Capital
Inventory Exports Government Households
Formation
A B C D E F
The Miernyk table: Directed graph of exports
with wij > 0.20

27. Capital
Inventory Exports Government Households
Formation
A B C D E F
The Miernyk table: Directed graph of capital
formation with wij > 0.20

28. Capital
Inventory Exports Government Households
Formation
A B C D E F
The Miernyk table: Directed graph of final
demand by source with wij > 0.20

29. Graphs & their metrics

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

36. Graph–Theoretic Links
in Regional Economies
Rose Baker
David Passmore
Institute for Research in
Training & Development