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Graph–Theoretic Links in Regional Economies
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Graph–Theoretic Links in Regional Economies

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Slides displayed during presentation the 27th Annual REMI Users’ Conference

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 Graph–Theoretic Links in Regional Economies Presentation Transcript

  • Graph–Theoretic Linksin Regional Economies Rose Baker David Passmore 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 & Professor of Operations Research / Director of Institute for Research in Training & DevelopmentHappy REMI users since the DOS version!
  • 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 by total row output, x x = Ax + y Add an identity matrix and rearrange terms x = (I – A)-1 yInput–output accounting identities
  • Input–output accounts as graphs
  • 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
  • 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
  • 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 0 1 0 0 “Small” entries in A are filtered as “0” in WTransforming real–valued transactionsmatrix to Boolean adjacency matrix
  • 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
  • X A W wij for aij > 0.1The Miernyk table: Matrix W for aij > 0.1
  • D A X E C B F WThe Miernyk table: Directed graph ofmatrix W for aij > 0.1
  • X y xThe Miernyk table, again
  • yThe Miernyk table: Final demand
  • Proportion of total GRP by source of final demandThe Miernyk table: Final demand by source
  • Capital Inventory Exports Government Households Formation A B C D E FThe Miernyk table: Directed graph of finaldemand by source with wij > 0.20
  • Capital Inventory Exports Government Households Formation A B C D E FThe Miernyk table: Directed graph of finaldemand by source with wij > 0.20
  • Capital Inventory Exports Government Households Formation A B C D E FThe Miernyk table: Directed graph of exportswith wij > 0.20
  • Capital Inventory Exports Government Households Formation A B C D E FThe Miernyk table: Directed graph of capitalformation with wij > 0.20
  • Capital Inventory Exports Government Households Formation A B C D E FThe Miernyk table: Directed graph of finaldemand by source with wij > 0.20
  • Graphs & their metrics
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • Graph–Theoretic Linksin Regional Economies Rose Baker David Passmore Institute for Research in Training & Development