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
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
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
Higraphs : combining graphs and Venn diagrams • sets + cartesian product • hypergraphs Hans Vangheluwe hv@cs.mcgill.ca Higraphs 4/22
Blobs: set inclusion, not membership A D E Hans Vangheluwe hv@cs.mcgill.ca Higraphs 5/22
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
• empty space meaningless, identify intersection ( e.g., N = K ∩ W ) Hans Vangheluwe hv@cs.mcgill.ca Higraphs 7/22
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
Meaningless syntactic constructs C A D B Hans Vangheluwe hv@cs.mcgill.ca Higraphs 9/22
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
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
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
Clique Example B C A E D Hans Vangheluwe hv@cs.mcgill.ca Higraphs 13/22
Clique: fully connected semantics A B C D E Hans Vangheluwe hv@cs.mcgill.ca Higraphs 14/22
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
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
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
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
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
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
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
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