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Project Presentation- Simplicial Complex conversion to graph

By: Priya Gangaraju Saloni Tamotia. Project Presentation- Simplicial Complex conversion to graph. Simplex. Convex hull of a set of (n+1) independent points in some Euclidean Space of dimension n or higher. Examples of Simplices : 0-Simplex (1 vertex) 1-Simplex (2 vertices)

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Project Presentation- Simplicial Complex conversion to graph

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  1. By: PriyaGangaraju SaloniTamotia Project Presentation- SimplicialComplex conversion to graph

  2. Simplex • Convex hull of a set of (n+1) independent points in some Euclidean Space of dimension n or higher. • Examples of Simplices: • 0-Simplex (1 vertex) • 1-Simplex (2 vertices) • 2-Simplex(3 vertices)

  3. 0-Simplex 0 1-Simplex 1 2-Simplex 2 Simplices 0-Simplex which is just a node or 1 vertex 1- simplex with 2 vertices; represented as line segment 2-simplex with 3 vertices; represented as triangle

  4. Simplicial Complex • A simplicial complex is constructed by “gluing together” points, line segments, triangles, and simplices.

  5. Keywords assumed as vertices Formation of simplex by connection of keywords provides special meaning. Example of WallSreet Journal Used in search engines Use of Simplicial Complex in Project

  6. Begin with 0-Simplex We have vertices {0,1,2,3,4,5,6,7,8} • Formation of 1-simplex • Pair two vertices of the 0-simplexes • Formation of 2-Simplex • Pair three vertices of the 0-simplexes Formation of Simplexes

  7. A B C AB BC CA ABC This is the formation of 2- simplex Theory: Graphical Representation

  8. Input looks like: 25 6 3 4 0 1 7 8 c0={0,1,2,3,4,5,6,7,8} c1={(0 1),(0 2),(1 2),(3 4),(5 6),(5 7),(6 7),(6 8), (7 8)} c2={(0 1 2),(5 6 7),(6 7 8)} Input Form

  9. Vertices – 0,1,2,3,4,5,6,7,8 (0 1) – vertex 9 (0 2) – vertex 10 (1 2) – vertex 11 …. (0 1 2) – vertex 18 (5 6 7) – vertex 19 (6 7 8) – vertex 20 Representation

  10. Two output forms: • Vertices: which are single nodes {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20} • Edges: connection between nodes { (20,6),(20,7),(20,8),(20,15),(20,16),(20,17), (19,5),(19,6),(19,7),(19,13),(19,14),(19,15), (18,0),(18,1),(18,2),(18,9),(18,10),(18,11)} Expected Output

  11. A counter “Vertices” keeps the count of vertices. An array is used to represent the adjacency matrix. The matrix keeps track of the vertex given to the edge. For example, if the edge between 0 and 1 is represented as vertex 9, then the entry in the matrix will be e[0][1]=9. Implementation in C

  12. Edges are listed as • One edge for each of the vertices. • One edge for each of the combination of the vertices if such a combination exists. • For example, (0 1 2) will be vertex 18. So one set of edges will be, (18,0),(18,1),(18,2). • second set will be (18,(0 1)) which will be (18,9) as (0 1) represents vertex 9. Implementation in c (contd.)

  13. Each of the combinations are verified against the adjacency matrix and an edge is added to the edges list if there is an edge which is represented as vertex. Each of the array elements are initialized to -1. Implementation in c (contd.)

  14. Vertices={0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, 16,17,18,19,20} Edges={(20,6),(20,7),(20,8),(20,15),(20,16),(20,17), (19,5),(19,6),(19,7),(19,13),(19,14),(19,15), (18,0),(18,1),(18,2),(18,9),(18,10),(18,11)} Press any key to continue . . . Sample output produced

  15. Thank You

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