Graph–Theoretic Links in Regional Economies

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

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Graph–Theoretic Links in Regional Economies

  1. 1. Graph–Theoretic Linksin Regional Economies Rose Baker David Passmore Institute for Research in Training & Development
  2. 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 & DevelopmentHappy REMI users since the DOS version!
  3. 3. Our aim: Portray regional supply & demand structure graphically
  4. 4. Examined graphicalrepresentation ofbinary input–outputanalysisAt 2011 Lake Tahoe REMI Conference….
  5. 5. Graphical representationof interindustry links& final demand bysourceToday….
  6. 6. First, a reminder about some input–output accounting relationships
  7. 7. A hypothetical input–output table fromMiernyk, Elements of Input–Output Analysis
  8. 8. Focus on a portion of input-output table
  9. 9. X + y = xFocus on a portion of input-output table
  10. 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 yInput–output accounting identities
  11. 11. Input–output accounts as graphs
  12. 12. A collection of nodes or vertices… A graph  with a collection of edges that connect the nodesGraph theory: Study of mathematical structuresof pairwise relations between objects
  13. 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 rowTransactions matrix as an adjacency matrix:Representing transactions as a graph
  14. 14. X y xThe Miernyk table, again
  15. 15. XThe Miernyk table: Interindustry transactions
  16. 16. X AThe Miernyk table: Matrix A
  17. 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 WTransforming real–valued transactionsmatrix to Boolean adjacency matrix
  18. 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 WTransforming real–valued transactionsmatrix to Boolean adjacency matrix
  19. 19. X A W wij for aij > 0.1The Miernyk table: Matrix W for aij > 0.1
  20. 20. D A X E C B F WThe Miernyk table: Directed graph ofmatrix W for aij > 0.1
  21. 21. X y xThe Miernyk table, again
  22. 22. yThe Miernyk table: Final demand
  23. 23. Proportion of total GRP by source of final demandThe Miernyk table: Final demand by source
  24. 24. Capital Inventory Exports Government Households Formation A B C D E FThe Miernyk table: Directed graph of finaldemand by source with wij > 0.20
  25. 25. Capital Inventory Exports Government Households Formation A B C D E FThe Miernyk table: Directed graph of finaldemand by source with wij > 0.20
  26. 26. Capital Inventory Exports Government Households Formation A B C D E FThe Miernyk table: Directed graph of exportswith wij > 0.20
  27. 27. Capital Inventory Exports Government Households Formation A B C D E FThe Miernyk table: Directed graph of capitalformation with wij > 0.20
  28. 28. Capital Inventory Exports Government Households Formation A B C D E FThe Miernyk table: Directed graph of finaldemand by source with wij > 0.20
  29. 29. Graphs & their metrics
  30. 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 edgesSome graph metrics that describe the structureof transactions in a regional economy
  31. 31. D DENSITY = 0.28 A 10 edges among 36 E C possible B F WThe Miernyk table: Directed graph metricsfor matrix W for aij > 0.1
  32. 32. D CENTRALITY A Nodes A & C are central E C Both have 3 edges B F WThe Miernyk table: Directed graph metricsfor matrix W for aij > 0.1
  33. 33. IN– & OUT–DEGREE D A Nodes A & C have high in–degree E C Node B has high out–degree B F WThe Miernyk table: Directed graph metricsfor matrix W for aij > 0.1
  34. 34. D ISOLATION A Node E E C is almost an an isolate B F WThe Miernyk table: Directed graph metricsfor matrix W for aij > 0.1
  35. 35. Construct a graph & associated metrics of interindustry transactions for Centre County, Pennsylvania Explore graphing relationships between industries & categories of final demand within regionWhat we will do within a largersocial networking project
  36. 36. Graph–Theoretic Linksin Regional Economies Rose Baker David Passmore Institute for Research in Training & Development

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