Xufei Wang, ICDM, 2010

Citation

Xufei Wang. 2010. Discovering Overlapping Groups in Social Media, the 10th IEEE International Conference on Data Mining (ICDM 2010).

Online Version

http://dmml.asu.edu/users/xufei/Papers/ICDM2010.pdf

Databases

BlogCatalog [1]

Delicious [2]

Summary

In this paper, the authors propose a novel co-clustering framework, which takes advantage of networking information between users and tags in social media, to discover these overlapping communities. The basic ideas are:

• To discover overlapping communities in social media. Diverse interests and interactions that human beings can have in online social life suggest that one person often belongs more than one community.

• To use user-tag subscription information instead of user-user links. Metadata such as tags become an important source in measuring the user-user similarity. The paper shows that more accurate community structures can be obtained by scrutinizing tag information.

• To obtain clusters containing users and tags simultaneously. Existing co-clustering methods cluster users/tags separately. Thus, it is not clear which user cluster corresponds to which tag cluster. But the proposed method is able to find out user/tag group structure and their correspondence

Problem Statement

In this paper, the concept of community is generalized to include both users and tags. Tags of a community imply the major concern of people within it.

Let $\mu =\left(\mu _{1},\mu _{2},...,\mu _{m}\right)$ denote the user set, $\tau =\left(\tau _{1},\tau _{2},...,\tau _{n}\right)$ the tay set. A community $C_{i}\left(1\leq i\leq k\right)$ is a subset of user and tags, where k is the number of communities. As mentioned above, communities usually overlap, i.e., $C_{i}\bigcap C_{j}\neq \emptyset \left(1\leq i,j\leq k\right)$.On the other hand, users and their subscribed tags form a user-tag matrix M, in which each entry $M_{ij}\in \left\{0,1\right\}$ indicates whether user $u_{i}$ subscribes to tag $t_{j}$. So it is reasonable to view a user as a sparse vector of tags, and each tag as a sparse vector of users.

Given notations above, the overlapping co-clustering problem can be stated formally as follows:

Input:

• A user-tag subscription matrix $M_{N_{\mu }\times N_{t}}$, where $N_{\mu }$ and $N_{t}$ are the numbers of users and tags.

• The number of communities k.

Output:

• k overlapping communities which consist of both users and tags.

Brief description of the method

Communities that aggregate similar users and tags together can be detected by maximizing intra-cluster similarity, which is shown below: $argmax{\frac {1}{k}}\sum _{i=1}^{k}\sum _{x_{j}\in C_{i}}^{}S_{C}\left(x_{j},c_{i}\right)$ where k is the number of communities, x is the edges and c is the centroid of community. This formulation can be solved by a k-means variant.

This paper uses different methods to solve the problem of overlapping communities:

A. Independent Learning

If two tags are different, their similarity can be defined as 0, and 1 if they are the same. their cosine similarity can be rewritten as: $S_{e}\left(e,{e}'\right)={\frac {1}{2}}\left(\delta \left(u_{i},u_{j}\right)+\delta \left(t_{p},t_{q}\right)\right)$

B. Normalized Learning

Let $d_{u_{i}}$ denote the degree of the user $u_{i}$,and $d_{t_{p}}$ represent the degree of tag $t_{p}$ in a user-tag network. their cosine similarity can be rewritten as: $S_{e}\left(e,{e}'\right)={\frac {d_{t_{p}}d_{t_{q}}\delta \left(u_{i},u_{j}\right)+d_{u_{i}}d_{u_{j}}\delta \left(t_{p},t_{q}\right)}{{\sqrt {d_{u_{i}}^{2}+d_{t_{p}}^{2}}}{\sqrt {d_{u_{j}}^{2}+d_{t_{q}}^{2}}}}}$

C. Correlational Learning

The singular value decomposition of user-tag network M is given by $M=U\Sigma V^{T}$, where columns of U and V are the left and right singular vectors and $\Sigma$ is the diagonal matrix whose elements are singular values.

${\vec {u}}_{i}({\vec {t}}_{1},{\vec {t}}_{2},...,{\vec {t}}_{m})=u_{i}(t_{1},t_{2},...t_{n})V$

So we can get $S_{e}\left(e,{e}'\right)=\alpha S_{u}\left(u_{i},u_{j}\right)+\left(1-\alpha \right)S_{t}\left(t_{p},t_{q}\right)$

where $S_{u}(u_{i},u_{j})={\frac {{\vec {u}}_{i}{\vec {u}}_{j}}{\left\|{\vec {u}}_{i}\right\|\left\|{\vec {u}}_{j}\right\|}}$ and $S_{t}(t_{i},t_{j})={\frac {{\vec {t}}_{i}{\vec {t}}_{j}}{\left\|{\vec {t}}_{i}\right\|\left\|{\vec {t}}_{j}\right\|}}$