1. by Hyunwoo Park and Kichun Lee Knowledge-Based Systems 60 (2014) 58–72
Dependence clustering, a method revealing community structure with group dependence
by Hyunwoo Park and Kichun Lee
Knowledge-Based Systems 60 (2014) 58–72
Kiburm Song, Ph.D. student
Intelligent Data Systems Lab.
Department of Industrial Engineering
Hanyang University

2. Introduction
Community structure detection in networks
Most similarity based clustering algorithms require to set predefined number of clusters

3. Modularity-based clustering by Newman (2006)1
Introduction
Modularity-based clustering by Newman (2006)1
Modularity measures how precise a division of the network is against a graph with edges placed at random
Drawback
assuming that each node can be linked to any other nodes of the network whether they are large or small regardless of the geometric structure of the network
it cannot adjust the level of resolution or the scale on which the modularity measure relies
1Newman, M. E. (2006). Modularity and community structure in networks. Proceedings of the national academy of sciences, 103(23),

4. Dependence clustering - dependence
: an undirected graph
Assuming
the graph as a Markov chain
the whole chain is ergodic and all transitions follow the Markovian property

5. Dependence clustering - dependence
Figure 1 from Park & Lee (2014)
Park, H. & Lee, K. (2014). Dependence clustering, a method revealing community structure with group dependence. Knowledge-Based Systems 60, 58-72

6. Dependence clustering - dependence
Dependence captures how 𝑥 𝑚 in the initial state is inter-dependent with 𝑥 𝑖 at the 𝑡th step:
Dep < 1: negative dependence
Dep = 1: independence
Dep > 1: positive dependence

7. Dependence clustering – group dependence
captures how much data points in each group are dependent on each other
measures overall coherence in terms of dependence from group assignments for the whole data set at the t-step transition
𝑠 𝑖 =1 if data point 𝑖 belongs to group1
𝑠 𝑖 =−1 if data point 𝑖 belongs to group2
1 2 ( 𝑠 𝑖 𝑠 𝑗 +1) is 1 if 𝑖 and 𝑗 are in the same group
𝒔= 𝑠 1 ,…, 𝑠 𝑛 : group assignment vector

8. Dependence clustering – group dependence
To explain the simplified way to compute 𝐷 𝑡
need to describe transition matrix
𝑊 𝑖,𝑗 : adjacency matrix
consists of nonnegative elements
represent the closeness between point 𝑖 and point 𝑗
𝑃 𝑖,𝑗 : transition matrix ( 𝑃 𝑖,𝑗 =Pr 𝑋 1 =𝑗 𝑋 0 =𝑖), 𝑗 𝑃 𝑖,𝑗 =1 )

9. Dependence clustering – group dependence
To overcome computational hurdle in computing 𝑃 𝑡 , we used one-time spectral decomposition
eigenvector
eigenvalues

10. Dependence clustering – group dependence

11. Dependence clustering – group dependence
Division of data points is made based on the signs of the eigenvector corresponding to the largest positive eigenvalue of 𝐺

12. Dependence clustering – Illustrating example

13. Dependence clustering – Illustrating example

14. Dependence clustering – Multiple groups
a dependence gain parameter 𝛿 𝑑 ∈ 0, 1
minimal dependence gain ∆ 𝑑 = 𝛿 𝑑 𝑔 𝑖𝑗 >0 𝐺
a set of points belonging to a candidate cluster Ω 𝐶 ⊆Ω
The division into the sub-clusters proceeds if 𝐷 𝑡 𝑠𝑖gn( 𝐬 C ′ ) − 𝐷 𝑡 𝒔 𝐶 >𝑁 ∆ 𝑑
𝐬 C ′ : within cluster group after division of Ω 𝐶
𝒔 𝐶 : within cluster group before division of Ω 𝐶
𝑁: size of Ω 𝐶

15. Soft dependence clustering
Dependence clustering provides only a hard clustering and is not designed for handling overlaps
To extend DC scheme, SDC introduces a soft probability interval
Interpreting eigenvector values as generated from two probability distributions
one with a negative mean and the other with a positive mean
𝜹 𝑷 = 𝒑 𝒍 , 𝒑 𝒉 , 𝑝 𝑙 + 𝑝 ℎ =1
lower and upper bounds of the posterior probability that a data point should have in order to be assigned to both clusters.