Presentation by Rose M. Baker and David L. Passmore (Penn State) at the 26th Annual Regional Economic Models, Inc., Users' Conference, Lake Tahoe, Incline Village, Nevada, on October 13, 2011.

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

2. 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

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

4. First, the I/O accounting relationship

5. An I/O table

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

7. 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

8. 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)

9. Relationship of graph theory with I/O

10. 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

11. 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.

12. 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).

13. 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

14. 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)

15. 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

16. And, graph and its metrics

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

18. 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

19. 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.

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