Graph–Theoretic Linksin Regional Economies                     Rose Baker                 David Passmore             Insti...
Rose Baker              Assistant Professor of Workforce              Education & Development / Research              Asso...
Our aim: Portray regional supply &         demand structure graphically
Examined graphicalrepresentation ofbinary input–outputanalysisAt 2011 Lake Tahoe REMI Conference….
Graphical representationof interindustry links& final demand bysourceToday….
First, a reminder about some  input–output accounting          relationships
A hypothetical input–output table fromMiernyk, Elements of Input–Output Analysis
Focus on a portion of input-output table
X    +      y   =   xFocus on a portion of input-output table
Total output = interindustry transactions + final demand                         x=X+y    Divide each element in row of X ...
Input–output accounts      as graphs
A collection of nodes or vertices…      A graph            with a collection of edges            that connect the nodesGr...
x = Ax + y                             Columns are                          A             purchasing                      ...
X                y   xThe Miernyk table, again
XThe Miernyk table: Interindustry transactions
X                          AThe Miernyk table: Matrix A
A                        W      0   .2    0              0   1    0     .2   .3   .1              1   1    1     .3    0  ...
A                        W      0   .2    0              0   1    0     .2   .3   .1              1   1    1     .3    0  ...
X                 A                                          W                      wij for aij > 0.1The Miernyk table: Ma...
D                    A                  X  E         C                    B                F                      WThe Mie...
X                y   xThe Miernyk table, again
yThe Miernyk table: Final demand
Proportion of total                             GRP by source of                             final demandThe Miernyk table...
Capital  Inventory       Exports       Government               Households                                             For...
Capital  Inventory       Exports       Government               Households                                             For...
Capital  Inventory       Exports       Government               Households                                             For...
Capital  Inventory       Exports       Government               Households                                             For...
Capital  Inventory       Exports       Government               Households                                             For...
Graphs & their metrics
1                      In–degree — number of                                  incoming edges     2         3             ...
D                   DENSITY = 0.28                    A                            10 edges                            amo...
D                   CENTRALITY                    A                            Nodes A & C                            are ...
IN– & OUT–DEGREE        D                    A       Nodes A & C                            have high in–degree  E        ...
D                   ISOLATION                    A                            Node E  E         C               is almost ...
Construct a graph & associated metrics of interindustry transactions for Centre County, Pennsylvania Explore graphing rela...
Graph–Theoretic Linksin Regional Economies                     Rose Baker                 David Passmore             Insti...
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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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