higraphs visualising information
play

Higraphs: Visualising Information complex non-quantitative, - PowerPoint PPT Presentation

Higraphs: Visualising Information complex non-quantitative, structural topological, not geometrical Euler Venn diagrams (Jordan curve: inside/outside): enclosure, intersection graphs (nodes, edges: binary relation );


  1. Higraphs: Visualising Information • complex • non-quantitative, structural • topological, not geometrical • Euler – Venn diagrams (Jordan curve: inside/outside): enclosure, intersection – graphs (nodes, edges: binary relation ); hypergraphs David Harel. On Visual Formalisms. Communications of the ACM. Volume 31, No. 5. 1988. pp. 514 - 530. Hans Vangheluwe hv@cs.mcgill.ca Higraphs 1/22

  2. Venn diagrams, Euler circles Q P B A R • topological notions (syntax): enclosure, exclusion, intersection • Used to represent (denote) mathematical set operations: union, difference, intersection Hans Vangheluwe hv@cs.mcgill.ca Higraphs 2/22

  3. Hypergraphs a c b d e f g h i a hypergraph a graph • topological notion (syntax): connectedness • Used to represent (denote) relations between sets. • Hyperedges: non longer binary relation ( ⊆ X × X ): ⊆ 2 X (undirected), ⊆ 2 X × 2 X (directed). Hans Vangheluwe hv@cs.mcgill.ca Higraphs 3/22

  4. Higraphs : combining graphs and Venn diagrams • sets + cartesian product • hypergraphs Hans Vangheluwe hv@cs.mcgill.ca Higraphs 4/22

  5. Blobs: set inclusion, not membership A D E Hans Vangheluwe hv@cs.mcgill.ca Higraphs 5/22

  6. Unique Blobs (atomic sets, no intersection) A C D W B X P K Q R N U L O V F M S P T E • atomic blobs are identifiable sets • other blobs are union of enclosed sets ( e.g., K = L ∪ M ∪ N ∪ O ∪ P ) Hans Vangheluwe hv@cs.mcgill.ca Higraphs 6/22

  7. • empty space meaningless, identify intersection ( e.g., N = K ∩ W ) Hans Vangheluwe hv@cs.mcgill.ca Higraphs 7/22

  8. Unordered Cartesian Product: Orthogonal Components A C D W B X Y P K Q R N U L O V F M S P T G H E K = G × H = H × G = ( L ∪ M ) × ( N ∪ O ∪ P ) Hans Vangheluwe hv@cs.mcgill.ca Higraphs 8/22

  9. Meaningless syntactic constructs C A D B Hans Vangheluwe hv@cs.mcgill.ca Higraphs 9/22

  10. Simple Higraph blobs orthogonal components A B J D C E K L M F G H I Hans Vangheluwe hv@cs.mcgill.ca Higraphs 10/22

  11. Induced Acyclic Graph (blob/orth comp alternation) a g OR level (blob level) � � � � � � � � � � � � � � ���� ���� ������ ������ � � � � � � � � ���� ���� ������ ������ � � ���� ���� ������ ������ � � ���� ���� ������ ������ � � ���� ���� ������ ������ � � ���� ���� ������ ������ � � ���� ���� ������ ������ � � ���� ���� ������ ������ AND level (orthogonal component level) � � ���� ���� ������ ������ � � ���� ���� ������ ������ ����������� ����������� � � �� �� � � � � ������ ������ �� �� ���� ���� � � ����������� ����������� � � �� �� � � � � ������ ������ �� �� ���� ���� � � ����������� ����������� � � ������ ������ �� �� ���� ���� � � ����������� ����������� � � ������ ������ �� �� ���� ���� � � ����������� ����������� � � ������ ������ �� �� ���� ���� � � ����������� ����������� � � ������ ������ �� �� ���� ���� � � ����������� ����������� � � ������ ������ �� �� ���� ���� � � ����������� ����������� � � ������ ������ �� �� ���� ���� � � ����������� ����������� c OR level � � ������ ������ �� �� ���� ���� � � ����������� ����������� b j � � ������ ������ �� �� ���� ���� � � ����������� ����������� � � �� �� �� �� � � � � � i � �� �� h �� �� � � � � � � � � �� �� ��� ��� � � � � �� �� ��� ��� � � � � �� �� ��� ��� � � � � �� �� ��� ��� � � � � �� �� ��� ��� � � � � �� �� ��� ��� � � � � �� �� ��� ��� AND level � � � � �� �� ��� ��� �� �� � � � � �� �� � � � � �� �� ��� ��� �� �� � � � � �� �� �� �� � � � � �� �� ���� ���� � � ��� ��� � � �� �� ��� ��� ���� ���� � � ��� ��� � � �� �� ��� ��� ���� ���� � � ��� ��� � � �� �� ��� ��� ���� ���� � � ��� ��� � � �� �� ��� ��� ���� ���� � � ��� ��� � � �� �� ��� ��� ���� ���� � � ��� ��� � � �� �� ��� ��� ���� ���� � � ��� ��� � � �� �� ��� ��� OR level ���� ���� � � ��� ��� � � �� �� ��� ��� �� �� �� �� �� �� � � � � �� �� ���� ���� � � ��� ��� � � �� �� ��� ��� �� �� �� �� �� �� � � � � �� �� �� �� �� �� �� �� � � � � �� �� m d e k l f Hans Vangheluwe hv@cs.mcgill.ca Higraphs 11/22

  12. Adding (hyper) edges A C D W B X Y P K Q R N U L O V F M S P T E • hyper edges • attach to contour of any blob • inter-level possible ( e.g., denote global variables binding) Hans Vangheluwe hv@cs.mcgill.ca Higraphs 12/22

  13. Clique Example B C A E D Hans Vangheluwe hv@cs.mcgill.ca Higraphs 13/22

  14. Clique: fully connected semantics A B C D E Hans Vangheluwe hv@cs.mcgill.ca Higraphs 14/22

  15. Entity Relationship Diagram ( is-a ) DATES EMPLOYEES PAID ON WORKS FOR IS IS A A SALARIES SECRETARIES CAN FLY AIRCRAFT PILOTS Hans Vangheluwe hv@cs.mcgill.ca Higraphs 15/22

  16. Higraph version of E-R diagram dates works for months ... paid employees on years secretaries ... salaries others arrived in pilots equipment nuts aircraft can fly bolts Hans Vangheluwe hv@cs.mcgill.ca Higraphs 16/22

  17. Extending the E-R diagram works for paid employees on men women secretaries others pilots can fly married Hans Vangheluwe hv@cs.mcgill.ca Higraphs 17/22

  18. Formally (syntax) A higraph H is a quadruple H = ( B,E,σ,π ) B : finite set of all unique blobs E : set of hyperedges ⊆ 2 X × 2 X ⊆ 2 X , ⊆ X × X, The subblob (direct descendants) function σ σ : B → 2 B + ∞ σ 0 ( x ) = { x } , σ i +1 = � σ ( y ) , σ + ( x ) = � σ i ( x ) i =1 y ∈ σ i ( x ) Hans Vangheluwe hv@cs.mcgill.ca Higraphs 18/22

  19. Subblobs + cycle free x �∈ σ + ( x ) The partitioning function π associates equivalence relationship with x π : B → 2 B × B Equivalence classes π i are orthogonal components of x π 1 ( x ) ,π 2 ( x ) ,...,π k x ( x ) k x = 1 means a single orthogonal component (no partitioning) Blobs in different orthogonal components of x are disjoint ∀ y,z ∈ σ ( x ) : σ + ( y ) ∩ σ + ( z ) = ∅ unless in the same equivalence class Hans Vangheluwe hv@cs.mcgill.ca Higraphs 19/22

  20. Simple Higraph blobs orthogonal components A B J D C E K L M F G H I Hans Vangheluwe hv@cs.mcgill.ca Higraphs 20/22

  21. Induced Orthogonal Components B = { A,B,C,D,E,F,C,G,H,I,J,K,L,M } E = { ( I,H ) , ( B,J ) , ( L,C ) } ρ ( A ) = { B,C,H,J } ,ρ ( G ) = { H,I } ,ρ ( B ) = { D,E } ,ρ ( C ) = { E,F } , ρ ( J ) = { K,L,M } ρ ( D ) = ρ ( E ) = ρ ( F ) = ρ ( H ) = ρ ( I ) = ρ ( K ) = ρ ( L ) = ρ ( M ) = ∅ π ( J ) = { ( K,K ) , ( K,L ) , ( L,L ) , ( L,K ) , ( M,M ) } Induces equivalence classes π 1 ( J ) = { K,L } and π 2 ( J ) = { M } , . . . These are the orthogonal components Hans Vangheluwe hv@cs.mcgill.ca Higraphs 21/22

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend


More recommend