View source for M. Kim and J. Leskovec. ICML'12

This is a scientific paper authored by [http://www.stanford.edu/~mykim/ M. Kim]  and [http://cs.stanford.edu/people/jure/ J. Leskovec], and appeared in [http://icml.cc/2012/ ICML'12]. Below is the paper summary written by [http://malt.ml.cmu.edu/mw/index.php/User:Tahoang Tuan Anh].
== Citation == 

== Online Version ==
[http://cs.stanford.edu/people/jure/pubs/lmmg-icml12.pdf Latent Multi-group Membership Graph Model].

== Summary ==
<p>
This is a paper on block based network analysis and prediction. This work furthers the work by Airoldi. et. al (see Related papers) to the
extent that each node/ user can actually belong to more than one block and node features are modeled in addition to link existence. The generative process is given as below (see the figure)</p>

[[File:KimLeskovec.png]]


<p>On the figure above, <math>N</math>, <math>L</math>, <math>K</math> is the number of users, number of user feature categories, and number of blocks respectively. The block membership probability of node <math>i</math> at bloc <math>k</math> is <math>\phi_{ik}\in [0, 1]</math>. Note that this model does not require <math>\sum_{k}\phi_{ik} = 1</math>
*For each node <math>i</math>, and each block <math>k</math>, <math>\phi_{ik}</math> is generated from [http://en.wikipedia.org/wiki/Beta_distribution Beta distribution]: <math> \phi_{ik}\sim Beta(\alpha_{k1},\alpha_{k2})</math>
*Then
**The truly block membership of node <math>i</math> at block <math>k</math> is a binary indicator which is generated from [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli distribution] <math>z_{ik}\sim Bernoulli(\phi_{ik})</math>
**The features of node <math>i</math> at feature category <math>l</math> is a is generated from [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli distribution] with the parameter is computed from block membership probabilities using [http://en.wikipedia.org/wiki/Logistic_function Logistic function]
***<math>y_{il} = \frac{1}{1+exp(w_l^T\phi_i)}</math>
***<math>F_{il} \sim Bernoulli(y_{il})</math>
**The link between node <math>i</math> and node <math>j</math> is generated from [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli distribution] with the parameter is the product of block-wise linking probabilities <math>p_{ij} = \prod_k\theta_k(z_{ik}, z_{jk})</math> <math>A_{ij} \sim Bernoulli(p_{il})</math>

== Dicussion ==

== Related papers ==
*Probabilistic graph clustering: Airoldi. et. al. [http://jmlr.csail.mit.edu/papers/volume9/airoldi08a/airoldi08a.pdf Mixed Membership Stochastic Blockmodels]. Journal of Machine Learning Research 9 (2008) 1981-2014
*The paper by Hoff on [http://www.csss.washington.edu/Papers/wp54.pdf Multiplicative latent factor models for description and prediction of social networks]