1. Properties of Minimum Rigidity Graphs Associated with a Clustering System ( How to account for internal structure of clusters? ) Gentian GUSHO LUSSI department – ENST Bretagne, FRANCE

2. CLASSIFICATION : binarity – rigidity – stability INTRODUCTION Minimum Spanning Tree (MST) ( Prim algorithm ) Single Linkage Hierarchy (SLH) ( algorithm of AHC ) Each class of SLH is connected in every MST MST is not unique and all the MSTs have the same length MST has a minimum number of edges as a connected graph

3. CLASSIFICATION : binarity – rigidity – stability CLUSTERING Clustering and rigidity Let X be a set of objects and S a set of subsets of X called clusters. Clustering system (CS) ( Folklore ) Minimum Rigidity Graph (MRG) associated with a CS ( Flament et al.(1976) ) Let G = ( X, E ) be a connected graph on X and S a CS. Definition : S is called a clustering system if : -   S -  x  X, { x }  S and X  S ( trivial clusters ) Definition : G is called a minimum rigidity graph of S if : - each cluster of S is a connected class of G - the cardinal of E is minimum for this property

4. CLASSIFICATION : binarity – rigidity – stability DEFINITION OF A CLASS Classes of a dissimilarity ( Jardine & Sibson (1971) the index of a class is the smallest threshold of the graph in which it appears as a maximal clique. { 1 }, { 2, 3 } and { 4, 5, 6 } are classes of d Every maximal clique of a threshold graph is called a class of d. The set of all the classes of d denoted by C d, is a CS. the index is exactly the diameter of the class

5. CLASSIFICATION : binarity – rigidity – stability DEFINITION OF A CLASS Balls and 2 balls of a dissimilarity B( 3, 5 ) = { 3, 5, 6 } B( 3, d( 3, 5 ) ) = { 2, 3, 5, 6 } and B( 5, d( 3, 5 ) ) = { 3, 4, 5, 6 } B(x, r) = { y: d(x, y)  r } Ball of d with centre x and radius r 2-ball of d generated by x and y B(x, y) = { z: max {d(x,z), d(y,z)  d(x,y) } We denote B d the set of all the balls of d and B 2d the set of its 2-balls. They both form a CS. B(x, y) = B(x, d(x, y)  B(y, d(x, y))

6. CLASSIFICATION : binarity – rigidity – stability DEFINITION OF A CLASS Realizations of a dissimilarity (Brucker, 2003) R( x, y)  B( x, y) : R( 3, 5 ) = { 3, 5 }, B( 3, 5 ) = { 3, 5, 6} R(x, y) =  { C  C d : x, y  C } Realization of a pair x, y R(x, y) =  { B  B d : x, y  B and diam d ( B )  d( x, y) } Properties - closest elements to x and y Another definition Property - diam d ( R( x, y) ) = d( x, y) The set of all the realizations denoted by R d, is a CS. - same behaviours relative to the other objects

7. CLASSIFICATION : binarity – rigidity – stability COMPLEXITY Computational complexity Proposition: ( Capobianco and Molluzo (1978) ) : The maximum number of maximal cliques of a graph with n nodes is limited exponentially by n ( O( 3 n/3 ) ). Enumeration issues Complexity issues Note: The number of the balls of d is limited by O ( n) and the number of the 2-balls and the realizations is limited by O ( n 2 ) Maximal cliques: exponential time Balls: polynomial time O ( n 2 ) 2-balls: polynomial time O ( n 3 ) Realizations: polynomial time O ( n 5 )

8. Some approaches in classification TRENDS to approach data by a classification model to extract classes from the data as they are partition complete graph supportminimum rigidity graph articulation point clusterconnected cluster clique hierarchyquasi-hierarchy CLASSIFICATION : binarity – rigidity – stability

9. PROPERTIES Minimum Rigidity Graph of R d The realizations of d ( R d ) The MRG of the realizations is not unique and all are of same length Properties of the MRG of the realizations R d 45, 46, 56, 23, 35, 15, 356, 345, 145, 1456, 123456 Every MRG of the realizations contains at least one MST of d Every MST of an MRG of the realizations is an MST of d The MRG of the realizations is computed in a polynomial time O ( n 5 )

10. EXAMPLE CLASSIFICATION : binarity – rigidity – stability

11. EXAMPLE SLH 45 456 23 17 23456 X R 45 46 56 23 17 35 356 47 15 27 345 14 467 14567 1457 123457 13457 X CLASSIFICATION : binarity – rigidity – stability SLH 45 456 23 23456 X R 45 46 56 23 35 356 15 345 145 1456 12345 1345 X Adding an object

12. EXAMPLE Realizations and quasi-hierarchy - An example ( but an open problem ) R 45 46 56 23 17 35 356 47 15 27 345 14 467 14567 1457 123457 13457 X CLASSIFICATION : binarity – rigidity – stability QH 45 456 23 3456 15 145 1456 X R 45 46 56 23 35 356 15 345 145 1456 12345 1345 X QH 45 46 456 23 17 3456 47 15 27 14567 X

13. Conclusions CONCLUSIONS R d represents the data as they are MRG provides information about the internal structures of the classes R d is computed in a polynomial time CLASSIFICATION : binarity – rigidity – stability The MRG of R d contains at least one MST of d The MRG of R d is not unique, but … ( Osswald 2003 )

14. Perspectives PERSPECTIVES realizations and other classificatory models stability of the method ( noising of the data ) other relationships between MSTs and MRGs study of “the” minimum hypergraph of rigidity CLASSIFICATION : binarity – rigidity – stability

15. CLASSIFICATION : binarité – rigidité – stabilité THANK YOU ! MERCI