Graph Theoretic Approach to Describing Regional Economic Structure

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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.

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Graph Theoretic Approach to Describing Regional Economic Structure

  1. 1. A Graph Theoretic Approach to Describing Regional Economic Structure<br />Rose Baker<br />David Passmore<br />Institute for Research in <br />Training & Development<br />
  2. 2. Institute for Research in Training & Development<br />Rose Baker<br />Assistant Professor of Workforce Education & Development / Research Associate in Office of Associate Dean for Research, Outreach, &Technology<br />David Passmore<br />Professor of Workforce Education & Development / Director of Institute<br />
  3. 3. Our aim: Portray supply structure<br />Nothing new…<br />An approach we want to take…<br />Your advice appreciated…<br />
  4. 4. First, the I/O accounting relationship<br />
  5. 5. An I/O table<br />
  6. 6. Focus on portion of I/O table <br />X<br />y<br />+<br />=<br />x<br />
  7. 7. I/O accounting identities<br />Total output = interindustry transactions + final demand<br />x = X + y<br />Divide each element of X by total output<br />x = Ax + y<br />Add an identity matrix and rearrange terms<br />x = (I – A)-1y<br />
  8. 8. Total requirements matrix: Leontief Inverse<br />(I – A)-1<br />One way to calculate the Leontief Inverse: power series<br />(I – A)-1 = (I + A + A2 + A3 +…An)<br />
  9. 9. Relationship of graph theory with I/O<br />
  10. 10. A collection of nodes or vertices…<br />A graph:<br />with a collection of edges <br />that connect nodes<br />Graph theory: Study of mathematical structures of pairwise relations between objects<br />
  11. 11. A<br />Columns are purchasing industries.<br />0 .2 0<br />.2 .3 .1<br />.3 0 0<br />Transactions matrix as an adjacency matrix: Representing transactions as a graph<br />Rows are <br />producing industries.<br />A number entered in a column indicates the proportion of total outlays of the industry purchased from an industry in a row.<br />
  12. 12. Transforming real–valued transactionsmatrix to Boolean adjacency matrix<br />A<br />W(1)<br /><br />0 .2 0<br />.2 .3 .1<br />.3 0 0<br />0 1 0<br /> 1 1 1<br /> 1 0 0<br />A “small” entries in A<br />can be filtered as zero in W(1).<br />
  13. 13. Round 2 of impact from power series:(I – A)-1 = (I + A + A2 + A3 +…An)<br />A<br />A<br />x<br />0 .2 0<br />.2 .3 .1<br />.3 0 0<br />0 .2 0<br />.2 .3 .1<br />.3 0 0<br />=<br />A2<br />.04 .06 .02<br />.09 .13 .03<br /> 0 .06 0<br />
  14. 14. A2<br />W(2)<br /><br />1 1 1<br /> 1 1 1<br />0 1 0<br />.04 .06 .02<br />.09 .13 .03<br /> 0 .06 0<br />Indirect connections indicated by “1” in W(2).<br />Round 2 impacts in A through A2translated to Boolean matrix W(2)<br />
  15. 15. W(1)<br />W(1)<br />x<br />0 1 0<br /> 1 1 1<br /> 1 0 0<br />0 1 0<br /> 1 1 1<br /> 1 0 0<br />=<br />W2<br />1 1 1<br /> 2 2 1<br />0 1 0<br />Recover number of linkages through Round 2<br />
  16. 16. And, graph and its metrics<br />
  17. 17. W(1)<br />1<br />0 1 0<br /> 1 1 1<br /> 1 0 0<br />2<br />3<br />Three–industry digraph from Boolean matrix W(1)<br />
  18. 18. Some graph metrics that describe the structure of supply chain relationships in an economy<br />Density — proportion of edges (links) among all possible edges.<br />Centrality — nodes with highest number of edges.<br />Isolation — proportion of nodes without edges among all nodes.<br />Cohesion — average number of edges required to reach all possible pairs of nodes. <br />Brokerage — a node that connects otherwise unconnected nodes.<br />1<br />2<br />3<br />
  19. 19. What we will do within a larger social networking project<br />Construct a graph and associated metrics of interindustry transactions for Centre County, Pennsylvania.<br />Explore graphing relationships between industries and categories of final demand.<br />
  20. 20. A Graph Theoretic Approach to Describing Regional Economic Structure<br />Rose Baker<br />David Passmore<br />Institute for Research in <br />Training & Development<br />

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