Graph Partitioning and Clustering

E={w ij } Set of weighted edges indicating pair-wise similarity between points Similarity Graph Represent dataset as a weighted graph G(V,E) Represent dataset as a weighted graph G(V,E) Example dataset Example dataset V={x i } Set of n vertices representing data points

Graph Partitioning Clustering can be viewed as partitioning a similarity graph Clustering can be viewed as partitioning a similarity graph Bi-partitioning task: Bi-partitioning task: Divide vertices into two disjoint groups (A,B) Divide vertices into two disjoint groups (A,B) A B Relevant Issues: Relevant Issues: How can we define a “good” partition of the graph? How can we define a “good” partition of the graph? How can we efficiently identify such a partition? How can we efficiently identify such a partition?

Clustering Objectives Traditional definition of a “good” clustering: Traditional definition of a “good” clustering: 1. Points assigned to same cluster should be highly similar. 2. Points assigned to different clusters should be highly dissimilar. 2. Minimise weight of between-group connections Maximise weight of within-group connections Apply these objectives to our graph representation Apply these objectives to our graph representation

Graph Cuts Express partitioning objectives as a function of the “edge cut” of the partition. Express partitioning objectives as a function of the “edge cut” of the partition. Cut: Set of edges with only one vertex in a group. Cut: Set of edges with only one vertex in a group A B cut(A,B) = 0.3

Graph Cut Criteria Criterion: Minimum-cut Criterion: Minimum-cut Minimise weight of connections between groups Minimise weight of connections between groups min cut(A,B) Optimal cut Minimum cut Problem: Problem: Only considers external cluster connections Only considers external cluster connections Does not consider internal cluster density Does not consider internal cluster density Degenerate case: Degenerate case:

Graph Cut Criteria (continued) Criterion: Normalised-cut (Shi & Malik,’97) Criterion: Normalised-cut (Shi & Malik,’97) Consider the connectivity between groups relative to the density of each group. Consider the connectivity between groups relative to the density of each group. Normalise the association between groups by volume. Normalise the association between groups by volume. Vol(A) : The total weight of the edges originating from group A. Vol(A) : The total weight of the edges originating from group A. Why use this criterion? Why use this criterion? Minimising the normalised cut is equivalent to maximising normalised association. Minimising the normalised cut is equivalent to maximising normalised association. Produces more balanced partitions. Produces more balanced partitions.

How do we efficiently identify a “good” partition? Problem: Computing an optimal cut is NP-hard