Presented by Yuhua Jiao
2012-12-4

• Limitation of some network clustering methods
• Hierarchical Agglomerative Clustering
– Method
– Performance evaluation
• Results and Discussion
– Data preparation
– Empirical evaluation
– Multi-resolution view of a physical interaction network

• Challenges in biological network analysis
– Inference of structure of subgroups of related vertices
– Prediction of possible links not represented in data
• Network clustering is a valuable approach for
– summarizing the structure in large networks,
– predicting unobserved interactions
– predicting functional annotations

• Poor resolution of top-level clusters
– Stochastic block models
• Over-splitting of bottom-level clusters
– Hierarchical network model
• Requirements to pre-define the number of clusters prior to analysis
– Stochastic block models
• An inability to jointly cluster over multiple interaction types

• Hierarchical Agglomerative Clustering
– An approximation for optimizing a network probability motivated by CMN.
– Interactions with vertices outside a group often provide more information than within-group interactions.
– Power Graph Analysis is a lossless transformation of biological networks into a compact, less redundant representation, exploiting the abundance of cliques and bicliques as elementary topological motifs.

• Notation
– Graph
– Groups
{
C k
: k
Î
1, … , K
– Edges between groups
} e ij e ii
=
=
u
Î i ,v
Î j u
< v
Î i e uv e uv for i
¹ j e uv
– Total possible connections
=
ìì
1
0 t ij
= n i n j for i
¹ j
– Number of holes t ii
= n i
A n edge between u and v
No edge between u and v
( n i
-
1
)
2 h ij
= t ij
e ij

• For a given pair of group i and j , edges between groups are result of t ij independent Bernoulli trials.
• The probability of observed edges, conditioned on parameter θ ij
P ij
( )
= q ij e ij q ij h ij
• The maximum likelihood estimate of θ ij is q ˆ ij
= e ij t ij
• The maximum likelihood value of P ij
( θ ij
) is
P ij
ML = e ij e ij h ij h ij t ij t ij

• Given two groups: n i
= 5
• Probability density is n j
= 4
P ij
ML = e ij e ij h ij h ij t t ij ij
• The likelihood of the flat model
L
( )
= Õ i
£ j
P ij
ML

• Generalization to hierarchical model
– Binary dendrogram T
– Each node r in the dendrogram represents the joining of vertices in left sub-tree L(r) and vertices in right sub-tree R(r).
– Er and hr are numbers of edges and holes crossing between the left and right sub-trees.
L
( )
= Õ r
£ ¢ Î top
P
ML r r
¢
Õ r
P r
ML

• Maximum likelihood guide tree
– K top-level clusters
– R total tree nodes
– Merge clusters 1 and 2 into cluster 1’, defining a new model
M’ l
ML
12
=
L
( )
( )
¢
=
K Õ k
=
3
P
1
¢
ML k
P
1 k
ML
P
2
ML k
1
1’
2
Current top level

• During the merging process, if clusters 1 and 2 are selected for merging and are both collapsed, the probability ratio is calculated, where the subscripts indicate edges and holes within and between groups.
l
C
12
=
Beta
(
Õ
å
2
2 i
£ j
=
1 e ij i
£ j
=
1
Beta
(
+
1,
å e ij
2 i
£ j
=
1
+
1, h ij h ij
+
1
)
+
1,
)
• The merged cluster is collapsed if λ c ≥1 .
• Clusters of two vertices are always merged because λ c = 1.

• Data preparation
– BioGRID database (http://thebiogrid.org)
– The graph is undirected and unweighted with no self edges.
• Other methods
– Fast Modularity (CNM)
– Variational Bayes modularity (VBM)
– Graph Diffusion Kernel (GDK)
– Heuristic merging scores
• Edge density (HAC-E)
• Combined edge density and shared neighbor density (HAC-ES)
• Decomposed Newman modularity Q from CNM (HAC-Q)

• Starting with a real-world network, training networks are generated by deleting a specified fraction of edges.
• A test set is defined by the held-out edges and a random choice of an equal number of holes.
• The trained group structure provides maximum likelihood estimates for edges within and between clusters (Eq. 9). For VBM and CNM, we estimated edge densities between all pairs of clusters and within all clusters. For hierarchical models, we estimated densities between all left and right clusters at all tree levels. For GDK, each pair’s diffusion was directly used to rank pairs.
• Finally we assessed precision and recall of pairs in the test set ranked by link probability or GDK score.

• Data Preparation

• Extending HAC to dynamic networks is limited:
– A solution is required to the identifiability problem: how complexes inferred at one time point correspond to complexes inferred at other time points.
– Transitions of a protein from one complex to another must be permitted by the model, requiring dynamical coupling between network snapshots.
• Dynamical Hierarchical Agglomerative Clustering (DHAC)
– Maximum likelihood is converted to fully Bayesian statistics
– The likelihood modularity is ‘kernelized’ with an adaptive bandwidth to couple network clusters at nearby time points.
– Matching clusters across time points is solved with a new belief propagation method that extends Expectation-Maximization and belief propagation for bipartite matching to consistently match multiple time-evolving clusters.