Topology of wireless networks L. Decreusefond Institut Also starring (by chronological order of Mines-Telecom appearance) P. Martins, E. Ferraz, F. Yan, A. Vergne, I. Flint, N.K. Le, A. Vasseur, (T. Bonis, B. Robert) GANDI: Graphs ANalysis for Data and Information
Algebraic topology Poisson homologies Persistence Applications : intelligent vehicle, agriculture, house, ... 2/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Outline Algebraic topology Poisson homologies Persistence 3/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Coverage 4 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Mathematical framework Geometry leads to a combinatorial object Combinatorial object is equipped with a Linear algebra structure Coverage and connectivity reduce to compute the rank of a matrix Localisation of hole: reduces to the computation of a basis of a vector matrix, obtained by matrix reduction (as in Gauss algorithm). 5/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex 6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex 6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex 6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex 6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex 6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex 6/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex d e b c a Vertices : a, b, c, d, e Edges : ab, bc, ca, be, ec, ed Triangles : bec Tetrahedron : ; 7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex d e b c a Vertices : a, b, c, d, e Edges : ab, bc, ca, be, ec, ed Triangles : bec Tetrahedron : ; 7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex d e b c a Vertices : a, b, c, d, e Edges : ab, bc, ca, be, ec, ed Triangles : bec Tetrahedron : ; 7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex d e b c a Vertices : a, b, c, d, e Edges : ab, bc, ca, be, ec, ed Triangles : bec Tetrahedron : ; 7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex d e b c a Vertices : a, b, c, d, e Edges : ab, bc, ca, be, ec, ed Triangles : bec Tetrahedron : ; 7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex d e b c a Vertices : { a, b, c, d, e } = C 0 Edges : { ab, bc, ca, be, ec, ed } = C 1 Triangles : { bec } = C 2 Tetrahedron : ; = C 3 Comput. Rips 7/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Hypergraphs A simplicial complex = hypergraph = boolean monotone function 8/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Hypergraphs A simplicial complex = hypergraph = boolean monotone function The Embedded Homology of Hypergraphs and Applications Stephane Bressan, Shiquan Ren, Jie Wu arXiv:1610.00890 8/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Cech complex k -simplices [ { [ x 0 , · · · , x k � 1 ] , x i 2 ! , \ k C k = i = 0 B ( x i , ✏ ) 6 = ; } Nerve theorem We can read some topological properties of S x 2 ! B ( x , ✏ ) on ( C k , k � 0 ) I Same nb of connected components I Same nb of holes I Same Euler characteristic 9/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Boundary operator Definition @ k : C k � ! C k � 1 k X ( � 1 ) j [ v 0 , · · · , ˆ [ v 0 , · · · , v k � 1 ] 7� ! v j , · · · ] j = 0 10/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Boundary operator Definition @ k : C k � ! C k � 1 k X ( � 1 ) j [ v 0 , · · · , ˆ [ v 0 , · · · , v k � 1 ] 7� ! v j , · · · ] j = 0 Example @ 2 ( bec ) = ec � bc + be 10/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Boundary operator Definition @ k : C k � ! C k � 1 k X ( � 1 ) j [ v 0 , · · · , ˆ [ v 0 , · · · , v k � 1 ] 7� ! v j , · · · ] j = 0 Example @ 2 ( bec ) = ec � bc + be @ 1 @ 2 ( bec ) = c � e � ( c � b ) + e � b = 0 10/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Theorem @ k � @ k + 1 = 0 Consequence Im @ k + 1 ⇢ ker @ k Definition H k = ker @ k / Im @ k + 1 and � k = dim ker @ k � range @ k + 1 11/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Interpretation : The magic I � 0 : number of connected components I � 1 : number of holes I � 2 : number of voids I to be continued 12/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Example 0 � 1 0 1 � 1 0 0 1 � 1 � 1 1 0 0 0 B C B C @ 0 ⌘ 0 , @ 1 = 0 1 � 1 0 1 0 B C B C 0 0 0 0 0 1 @ A 0 0 0 1 � 1 0 Nb of connected components dim ker @ 0 = 5 , range @ 1 = 4 hence � 0 = 1 13/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Number of holes 0 0 1 � 1 B C B C 0 B C @ 2 = B C 1 B C B C 1 @ A 0 Nb of holes dim ker @ 1 = 2 , range @ 2 = 1 hence � 1 = 1 14/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Polygons=cycles � 1 = Nb of independent polygons � Nb of independent triangles . 15/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Polygons=cycles � 1 = Nb of independent polygons � Nb of independent triangles . 15/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Polygons=cycles � 1 = Nb of independent polygons � Nb of independent triangles . � 1 = 2 � 1 = 1 . 15/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Polygons=cycles � 1 = Nb of independent polygons � Nb of independent triangles . � 1 = 2 � 2 = 0 . 15/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Open question What is the interpretation of the Betti numbers for hypergraphs or boolean monotone functions ? 16/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Open question What is the interpretation of the Betti numbers for hypergraphs or boolean monotone functions ? Find the single minimal triangulation = construct the minimum weight basis of H 2 16/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Euler characteristic ( S � A + F ) Definition d X ( � 1 ) j � j � = j = 0 Discrete Morse inequality � |C k � 1 | + |C k | � |C k + 1 | � k |C k | 17/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Euler characteristic ( S � A + F ) Definition d 1 ( � 1 ) j |C j | X ( � 1 ) j � j = X � = j = 0 j = 0 Discrete Morse inequality � |C k � 1 | + |C k | � |C k + 1 | � k |C k | 17/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
Algebraic topology Poisson homologies Persistence Alternative complex Cech complex ) \ k [ v 0 , · · · , v k � 1 ] 2 C k ( j = 0 B ( x j , ✏ ) 6 = ; Rips-Vietoris complex [ v 0 , · · · , v k � 1 ] 2 R k ( ) B ( x j , ✏ ) \ B ( x l , ✏ ) 6 = ; k simplex = clique of k + 1 points 18/47 February, 2017 Institut Mines-Telecom Topology of wireless networks
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