1. 3.3 Network-Centric Community Detection
Block Model Approximation
The adjacency matrix can be approximated by a block structure.
Each block represents one community
We approximate a given adjacency matrix A as follows:
𝑆 ∈ {0,1} 𝑛×𝑘 is block indicator matrix with 𝑆 𝑖𝑗 =1 if node 𝑖 belongs to 𝑗th block.
Σ is 𝑘 ×𝑘 matrix indicating the block (group) interaction density
𝑘 is the number of blocks

2. 3.3 Network-Centric Community Detection
Block Model Approximation
Natural objective is to minimize the following:
The optimal S corresponds to the top k eigenvectors of the adjacency matrix A with maximum eigenvalues.
K-means clustering can be applied to S to recover the community partition H .

3. 3.3 Network-Centric Community Detection
Spectral Clustering
It is derived from the problem of graph partition.
Graph partition aims to find out a partition such that the cut is minimized.
Cut - the total number of edges between two disjoint sets of nodes
The green cut between two sets of nodes {1,2,3,4} and {5,6,7,8,9} is 2
Community detection problem can be reduced to finding the minimum cut in a network.
This minimum cut problem can be solved efficiently
It often returns imbalanced communities, with one being trivial or a singleton, i.e., a community consisting of only one node.
The minimum cut is 1, between {9} and {1, 2, 3, 4, 5, 6, 7, 8}
Therefore, the group sizes of communities should be considered.

4. 3.3 Network-Centric Community Detection
Spectral Clustering
Ratio cut & Normalized cut
Graph partition: 𝜋=( 𝐶 1 , 𝐶 2 ,…, 𝐶 𝑘 ) s.t. 𝐶 1 ∩ 𝐶 2 =∅ and 𝑖=1 𝑘 𝐶 𝑖 =𝑉
𝐶 𝑖 : the complement of 𝐶 𝑖
𝑐𝑢𝑡 𝐶 𝑖 , 𝐶 𝑖 : the number of edges between 𝐶 𝑖 and 𝐶 𝑖
𝑣𝑜𝑙 𝐶 𝑖 = 𝑣∈ 𝐶 𝑖 𝑑 𝑣
The smaller ratio/normalized cut is preferable.

5. 3.3 Network-Centric Community Detection
Spectral Clustering
Ratio cut & Normalized cut - Examples
𝜋 1 =( 𝐶 1 ={9}, 𝐶 2 ={1,2,3,4,5,6,7,8})
𝑐𝑢𝑡 𝐶 1 , 𝐶 1 =1, 𝑐𝑢𝑡 𝐶 2 , 𝐶 2 =1
𝐶 1 =1, 𝐶 2 =8
𝑣𝑜𝑙 𝐶 1 =1, 𝑣𝑜𝑙 𝐶 2 =27
𝜋 2 =( 𝐶 1 ={1,2,3,4}, 𝐶 2 ={5,6,7,8,9})
𝑐𝑢𝑡 𝐶 1 , 𝐶 1 =2, 𝑐𝑢𝑡 𝐶 2 , 𝐶 2 =2
𝐶 1 =4, 𝐶 2 =5
𝑣𝑜𝑙 𝐶 1 =12, 𝑣𝑜𝑙 𝐶 2 =16

6. 3.3 Network-Centric Community Detection
Spectral Clustering
Finding the minimum ratio cut or normalized cut is NP-hard
Both ratio cut and normalized cut can be formulated as a min-trace problem like below
Graph Laplacian 𝐿 is defined as follows:
𝐷=𝑑𝑖𝑎𝑔( 𝑑 1 , 𝑑 2 , …, 𝑑 𝑛 )
Then, 𝑆 corresponds to the top eigenvectors of 𝐿 with the smallest eigenvalues.

7. 3.3 Network-Centric Community Detection
Modularity Maximization
Modularity
It measure the strength of a community partition for real-world networks by taking into account the degree distribution of nodes
Given a network of 𝑛 nodes and 𝑚 edges, the expected number of edges between nodes 𝑣 𝑖 and 𝑣 𝑗 is 𝑑 𝑖 𝑑 𝑗 /2𝑚
9 nodes and 14 edges
The expected number of edges between nodes 1 and 2 is 3×2/(2×14) = 3/14.
So, 𝐴 𝑖𝑗 − 𝑑 𝑖 𝑑 𝑗 /2𝑚 measures how far the true network interaction between nodes 𝑖 and 𝑗 deviates from the expected random connections

8. 3.3 Network-Centric Community Detection
Modularity Maximization
Given a group of nodes 𝐶, the strength of community effect is defined as
If a network is partitioned into k groups, the overall community effect can be summed up as follows
Therefore, Modularity is defined as
where 1/2𝑚 is introduced to normalize the value between -1 and 1.
Modularity calibrates the quality of community partitions thus can be used as an objective measure to maximize it

9. 3.3 Network-Centric Community Detection
Modularity Maximization
Define a modularity matrix 𝐵
where 𝑑∈ 𝑅 𝑛×1 is a vector of each node’s degree.
Let 𝑆∈ {0, 1} 𝑛×𝑘 be a community indicator matrix with 𝑆 𝑖𝑙 =1 if node 𝑖 belongs to community 𝐶 𝑙 , and 𝑠 𝑙 the 𝑙th column of 𝑆
Modularity can be reformulated as
The optimal 𝑆 can be computed as the top k eigenvectors of the modularity matrix 𝐵 with the maximum eigenvalues.