1 / 25

Dense Subgraph Extraction with Application to Community Detection

Dense Subgraph Extraction with Application to Community Detection. Author: Jie chen and Yousef Saad IEEE transactions of knowledge and data engineering. Outline. Introduction Assumption of proposed method The types of graph The method Undirected graph Directed graph Bipartite graph

elu
Download Presentation

Dense Subgraph Extraction with Application to Community Detection

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Dense Subgraph Extraction with Application to Community Detection Author: Jiechen and YousefSaad IEEE transactions of knowledge and data engineering

  2. Outline • Introduction • Assumption of proposed method • The types of graph • The method • Undirected graph • Directed graph • Bipartite graph • Experiment

  3. Introduction • A challenging problem in the analysis of graph structures is the dense subgraph problem, where given a sparse graph, the objective is to identify a set of meaningful dense subgraphs. • The dense subgraphs are often interpreted as “communities”, based on the basic assumption that a network system consists of a number of communities, among with the connections are much fewer than those inside the same community.

  4. The drawbacks of general partition methods • The number k of partitions is mandatory input parameter, and the partitioning result is sensitive to the change of k. • Most of partition methods yield a complete clustering of the data. • Many graph partitioning techniques favor balancing, i.e., sizes of different partitions should not vary too much.

  5. The assumption of the proposed method by author • The adjacency matrix A is a sparse matrix. • The entries of A are either 0 or 1, since the weights of the edges are not taken into account for the density of a graph. • The diagonal of A is empty, since it does not allow self-loops.

  6. The types of graphs • Undirected graph-G(V,E) • V is the vertex set and E is the edge set. Adjacency matrix-symmetric 1 4 2 5 3 • The definition of the undirected graph density is

  7. The types of graphs • Bipartite graph-G(V,E) • Undirected graph • V is the vertex set and E is the edge set. Adjacency matrix 1 4 B 2 5 BT 3 • The definition of the bipartite graph density is

  8. The types of graphs • Directed graph-G(V,E) • V is the vertex set and E is the edge set. Adjacency matrix-nonsymmetric 1 4 2 5 3 • The definition of the directed graph density is

  9. Undirected graph • We construct a adjacency matrix A with G(V,E). • We use to build matrix M that stores the cosines between any two columns of the adjacency matrix A. • Then we construct a weight graph G’(V,E’) whose weighted adjacency matrix M is defined as M(i,j).

  10. Undirected graph • A top-down hierarchical clustering of the vertex set V is performed by successively deleting the edges e’ ∈ E’, in ascending order of the edge weights. When G’ first becomes disconnected, V is partitioned in two subsets, each of which corresponds to a connected component of G’. • The termination will take place when the density of the partition passes a certain density threshold dmin.

  11. 1 2 3 5 4 8 6 7 9

  12. 1 6 9 7 2 4 8 d(Gt)=0.6 d(Gs)=1 d(Gt)=0.8 dmin=0.75 3 5

  13. 7 8 9 1 2 6 3 5 4

  14. Directed graph • The adjacency matrix A of a directed graph is square but not symmetric. When Algorithm is applied to a nonsymmetric adjacency matrix, it will result in two different dendrograms, depending on whether M is computed as the cosines of the columns of A, or the rows of A. • We symmetrize the matrix A (i.e., replacing A by the pattern matrix of A+AT ) and use the resulting symmetric adjacency matrix to compute the similarity matrix M.

  15. Directed graph transform into undirected graph • Remove the direction of the edges and combine the duplicated resulting edges, then it yields an undirected graph • Or use the adjacency matrix AA+AT 1 1 4 4 2 2 5 5 3 3

  16. Bipartite graphs • Without any edge removal of the graph G’ (using M as the weighted adjacency matrix), the vertex set is already partitioned into two subsets: V1 and V2. • Any subsequent hierarchical partitioning will only further subdivide these two subsets separately.

  17. Augment the bipartite graph • A reasonable strategy for this purpose is to augment the original bipartite graph by adding edges between some of the vertices that are connected by a path of length 2. is obtained by erasing the diagonal of M1.

  18. dmin=0.5

  19. Experiments-accuracy

  20. Experiments-Polblogs • The graph contain 1490 vertices, among which the first 758 are liberal blogs, and remaining 732 are conservative. • The edge in the graph indicates the existence of citation between the two blogs.

  21. Experiments • Comparisons with the Clauset, Newman, and Moore(CNM) approach. • CNM approach: bottom-up hierarchical clustering. • Dataset: foldoc-G(13356, 120238) • It extracted from the online dictionary of computing

More Related