Introduction Axioms Modularity Adaptive Modularity Conclusion Axioms for graph clustering objective functions Twan van Laarhoven Institute for Computing and Information Sciences Radboud University Nijmegen, The Netherlands 28th June 2013 1 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Outline Introduction Axioms Modularity Adaptive Modularity Conclusion 2 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion The motivation • There is no strict definition of clustering. • Can we formalize our intuition? • Previous work is about distance based clustering (hierarchical clustering, K-means, etc.) • What about graphs? 3 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion The setting Definition (Graph) A symmetric weighted graph is a pair ( V , E ) of • a finite set V of nodes , and • a function E : V × V → R ≥ 0 of edge weights , such that E ( i , j ) = E ( j , i ) for all i , j ∈ V . • Larger weight = stronger connection. • We allow self loops. 4 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion The setting (cont.) Definition (Clustering) A clustering C of a graph G = ( V , E ) is a partition of its nodes. Definition (Clustering function) A graph clustering function f is a function from graphs G to clusterings of G . Definition (Objective function) A graph clustering objective function Q is a function from graphs G and clusterings of G to R . • Larger objective value = better. 5 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Outline Introduction Axioms Modularity Adaptive Modularity Conclusion 6 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion The form of axioms Things that define clusterings Form Notation 1 Clustering function f ( G ) = argmax C Q ( G , C ) 2 Objective function Q ( G , C ) Q ( G , C ) ≥ Q ( G , D ) or C ≥ G D 3 Objective relation 7 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Basic axioms Axiom 1: Scale invariance (first form) A graph clustering objective function Q is scale invariant if • for all graphs G = ( V , E ), • all constants α > 0, f ( G ) = f ( α G ). (where α G = ( V , ( i , j ) �→ α E ( i , j )).) Example b d b d b d a a = a = f f e e e c c c 8 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Basic axioms Axiom 1: Scale invariance (second form) A graph clustering objective function Q is scale invariant if • for all graphs G = ( V , E ), • all constants α > 0, • all clusterings C of G , Q ( G , C ) = Q ( α G , C ). (where α G = ( V , ( i , j ) �→ α E ( i , j )).) Example b d b d a a = Q Q e e c c 8 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Basic axioms Axiom 1: Scale invariance (second form) A graph clustering objective function Q is scale invariant if • for all graphs G = ( V , E ), • all constants α > 0, • all clusterings C of G , Q ( G , C ) = α Q ( α G , C ) ??? (where α G = ( V , ( i , j ) �→ α E ( i , j )).) Example b d b d a a = α Q Q e e c c 8 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Basic axioms Axiom 1: Scale invariance (third form) A graph clustering objective function Q is scale invariant if • for all graphs G = ( V , E ), • all constants α > 0, • all clusterings C 1 , C 2 of G , Q ( G , C 1 ) ≥ Q ( G , C 2 ) if and only if Q ( α G , C 1 ) ≥ Q ( α G , C 2 ). (where α G = ( V , ( i , j ) �→ α E ( i , j )).) Example � � � � � � � � Q ≥ Q ⇐ ⇒ Q ≥ Q 8 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Basic axioms Axiom 2: permutation invariance A graph clustering objective function Q is permutation invariant if • for all graphs G = ( V , E ) and • all isomorphisms f : V → V ′ , it is the case that Q ( G , C ) = Q ( f ( G ) , f ( C )). (where f is extended to graphs and clusterings in the obvious way.) Example x y b d a z = Q Q e c u v 9 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Basic axioms Axiom 3: Richness A graph clustering objective function Q is rich if • for all sets V and • all partitions C ∗ of V , there is • a graph G = ( V , E ) • such that C ∗ is the optimal clustering of G . Intuition: • No trivial objective functions. • No fixed number of clusters. 10 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Basic axioms Definition (Consistent improvement) Let • G = ( V , E ) and G ′ = ( V , E ′ ) be graphs, and • C be a clustering of G and G ′ . Then G ′ is a C-consistent improvement of G if • E ′ ( i , j ) ≥ E ( i , j ) for all i ∼ C j and • E ′ ( i , j ) ≤ E ( i , j ) for all i �∼ C j . Intuition: • Consistent improvements make a clustering fit better. 11 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Basic axioms Axiom 4: Monotonicity A graph clustering objective function Q is monotonic if • for all graphs G , • all clusterings C of G and • all C -consistent improvements G ′ of G it is the case that Q ( G ′ , C ) ≥ Q ( G , C ). Example b d b d a a ≥ Q Q e e c c 12 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Local changes Definition (agreement) Let • G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) be two graphs and • V a ⊆ V 1 ∩ V 2 . The graphs agree on V a if E 1 ( i , j ) = E 2 ( i , j ) for all i , j ∈ V a . Definition (agreement on neighborhood) The graphs also agree on the neighborhood of V a if E 1 ( i , j ) = E 2 ( i , j ) for all i ∈ V a , j ∈ V 1 ∩ V 2 , and E 1 ( i , j ) = 0 for all i ∈ V a , j ∈ V 1 \ V 2 , and E 2 ( i , j ) = 0 for all i ∈ V a , j ∈ V 2 \ V 1 . What this means: • For nodes/clusters in V a , all incident edges are the same. 13 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Local changes Definition (agreement) Let • G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) be two graphs and • V a ⊆ V 1 ∩ V 2 . The graphs agree on V a if E 1 ( i , j ) = E 2 ( i , j ) for all i , j ∈ V a . Definition (agreement on neighborhood) The graphs also agree on the neighborhood of V a if E 1 ( i , j ) = E 2 ( i , j ) for all i ∈ V a , j ∈ V 1 ∩ V 2 , and E 1 ( i , j ) = 0 for all i ∈ V a , j ∈ V 1 \ V 2 , and E 2 ( i , j ) = 0 for all i ∈ V a , j ∈ V 2 \ V 1 . What this means: • For nodes/clusters in V a , all incident edges are the same. 13 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Local changes Axiom 5: Locality A graph clustering objective function Q is local if • for all graphs G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) that agree on a set V a and its neighborhood, • for all clusterings C 1 of V 1 \ V a , C 2 of V 2 \ V a and C a , D a of V a . if Q ( G 1 , C a ∪ C 1 ) ≥ Q ( G 1 , D a ∪ C 1 ) then Q ( G 2 , C a ∪ C 2 ) ≥ Q ( G 2 , D a ∪ C 2 ). 14 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Local changes Example · · b b a a Q ≥ Q · · · · c c � · · b b a a Q ≥ Q · · · · c c 15 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Local changes Special cases • G 1 = G 2 : change part of a clustering. In practice: optimize parts separately (divide and conquer). • V a = ∅ : union of two disjoint graphs. 16 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Interlude: Related work Theorem (Kleinberg 2002) There is no clustering function that is permutation invariant, scale invariant, monotonic and rich. Theorem (Ackerman, Ben-David 2008) There is a clustering quality function that is permutation invariant, scale invariant, monotonic and rich. 17 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Discontinuity is magic Theorem There is a graph clustering function that is scale invariant, permutation invariant, monotonic, rich and local. Connected components f coco ( G ) = the connected components of G Q coco ( G , C ) = 1 [ C are the connected components of G ] Huh!?!? • Doesn’t this contradict Kleinberg’s theorem? • No: edge weight 0 = distance ∞ . 18 / 32
Introduction Axioms Modularity Adaptive Modularity Conclusion Discontinuity is magic Theorem There is a graph clustering function that is scale invariant, permutation invariant, monotonic, rich and local. Connected components f coco ( G ) = the connected components of G Q coco ( G , C ) = 1 [ C are the connected components of G ] Huh!?!? • Doesn’t this contradict Kleinberg’s theorem? • No: edge weight 0 = distance ∞ . 18 / 32
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