3.3 Network-Centric Community Detection

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: • is block indicator matrix with if node belongs to th block. • is matrix indicating the block (group) interaction density • 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: ) s.t.and • : the complement of • : the number of edges between and • The smaller ratio/normalized cut is preferable.

5. 3.3 Network-Centric Community Detection • Spectral Clustering • Ratio cut & Normalized cut - Examples • ) • , • , • ) • , • ,

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 Laplacianis defined as follows: • , …, ) • Then, corresponds to the top eigenvectors of 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 nodes and edges, the expected number of edges between nodes and is • 9 nodes and 14 edges • The expected number of edges between nodes 1 and 2 is 3×2/(2×14) = 3/14. • So, measures how far the true network interaction between nodes and deviates from the expected random connections

8. 3.3 Network-Centric Community Detection • Modularity Maximization • Given a group of nodes , 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 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 • where is a vector of each node’s degree. • Let be a community indicator matrix with if node belongs to community , and the th column of • Modularity can be reformulated as • The optimal can be computed as the top k eigenvectors of the modularity matrix with the maximum eigenvalues.