1. Discovering Functional Communities in Social Media
Brian Thompson, Linda Ness, David Shallcross, Devasis Bassu
Workshop on Data Mining in Social Networks ICDM 2015

2. Discovering Functional Communities in Social Media
Problem Description
Setup:
Social media users share or endorse content such as products, artists, news, and music
Goal:
Identify functional communities of users who share or endorse common content
Challenges:
Scalability – there may be many users and lots of content
Mixed membership – each user may belong to more than one community
Discovering Functional Communities in Social Media

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Examples
memes shared by users of online social media
products endorsed by users of a recommender system
[credit: Yannick Feder]
[credit:
Discovering Functional Communities in Social Media

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Data Representation A B C D
Muthu
a bicluster 𝐵 is formed by the intersection of a set of rows and a set of columns
Danfeng
Paul
Rebecca
A bicluster is formed by the intersection of a set of rows with a set of columns.
Hanghang
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Data Representation A B C D
Muthu
permuting rows and columns can reveal latent structure
Danfeng
Paul
Rebecca
A bicluster is formed by the intersection of a set of rows with a set of columns.
Hanghang
Discovering Functional Communities in Social Media

6. Discovering Functional Communities in Social Media
Data Representation A C B D
Muthu
permuting rows and columns can reveal latent structure
Rebecca
Paul
Danfeng
Hanghang
Discovering Functional Communities in Social Media

7. Discovering Functional Communities in Social Media
Co-Clustering
Goal: Given a matrix, cluster the rows and columns simultaneously to reveal hidden structure
Challenges:
Don’t know the number or sizes of clusters a priori
Number of possible co-clusterings is exponential in the size of the matrix R1 R2 C1 C2
Discovering Functional Communities in Social Media

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Related Work
Spectral methods use linear algebraic techniques such as SVD to fit a block diagonal structure
Usually require number of clusters to be pre-specified
Likely to perform well on the matrix on the left, but not the one on the right:
Discovering Functional Communities in Social Media

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Our Approach
Define a quality metric for co-clusterings that rewards large, dense biclusters
Find a co-clustering that maximizes the metric value
NP-hard in general, so need efficient heuristics
Others have taken a similar approach. The novelty of our work is (1) the metrics we consider reward large, dense clusters, as opposed to other metrics which optimize encoding cost or other objectives; (2) our approach allows any number of clusters and run-time is not dependent on output; (3) we do not require a block diagonal structure; (4) our algorithm efficiently traverses a wide range of possible co-clusterings, without getting stuck at local optima.
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Choosing a Metric
Motivated by two desired properties:
We propose the following class of metrics:
Proposition: 𝜇 𝛼 satisfies P1 and P2 for all 𝛼≥0.
Property P1
Property P2
𝜇 𝛼 = 𝐵∈𝛱 𝑎 𝐵 2 𝑠(𝐵) ⋅ 𝑤 𝐵 𝑎 𝐵 2+𝛼
large
dense
𝑎 𝐵 = area
𝑠 𝐵 = semiperimeter
𝑤 𝐵 = weight
where
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The CC-MACS Algorithm
Co-Clustering via Maximal Anti-Chain Search
Build randomized k-d trees on the rows and columns
Initialize maximal anti-chains as the leaves of each tree
Traverse the trees simultaneously from the bottom up, greedily merging the rows or columns that result in the greatest increase in the metric value
Output the co-clustering with the best metric value
Complexity: 𝑂 𝑁⋅ log 2 𝑚𝑛 time for an 𝑚×𝑛 matrix, where 𝑁≪𝑚𝑛 is the number of non-zero values
Discovering Functional Communities in Social Media

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Traverse all the way to the top, instead of stopping at a local optimum.
Discovering Functional Communities in Social Media

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Output the best co-clustering found.
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Experiments: Synthetic Data
1024×1024 matrix with dense biclusters of size 4×4
Compare to Cross-Association algorithm (Chakrabarti et al., KDD ’04) via 𝐹1-score: 𝐹 1 =2⋅ 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 × 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦 + 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛
Averaged over 10 trials. Cross-Association algorithm of Chakrabarti et al. [KDD ’04]
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Experiments: Visual Comparison
Matrices with known structure (NIST Matrix Market repository)
Original Matrix
Randomly Permuted
Cross-Association
CC-MACS ( 𝝁 𝟐 )
Constructed from the domains of finite element modeling (FIDAP005, top, and FIDAPM05, middle) and quantum chemistry (QC324, bottom)
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Experiments: Web Memes
Meme-Tracker dataset (Leskovec et al., KDD ’09)
~70k domains, ~50k memes, ~4m non-zeros (~0.1%)
Top biclusters returned by the CC-MACS algorithm:
# of Domains
# of Memes
Density
Topic 21 26
98.2%
St. Jude Children’s Hospital 5
178
96.1%
Brazilian news 6 39
98.7%
Spanish news 20
99.2%
Tech news 17
100.0%
Politics
Meme-Tracker dataset: ~70k rows/domains, ~50k columns/memes/”phrase clusters”, ~4m non-zeros (~0.1%)
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23. Discovering Functional Communities in Social Media
Summary of Contributions
A new class of co-clustering metrics that reward large, dense biclusters
The CC-MACS algorithm, which efficiently searches the space of possible co-clusterings for one which maximizes the value of a given metric
Advantages over existing methods:
Do not need to specify number of clusters in advance
Not limited to matrices with a block diagonal structure
Discovering Functional Communities in Social Media

24. Discovering Functional Communities in Social Media
Future Work
Consider a more general class of bicluster partitions (not just those formed by co-clustering rows and columns)
Bound the approximation factor of our heuristic algorithm
Adapt our approach for distributed computation
Related Projects
Infer pairwise influence between nodes based on the times of their respective activity (MILCOM)
Detect correlated activity in comm networks (DaMNet)
Discover cascades when textual content is not available
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Contact Info:
Brian Thompson
Discovering Functional Communities in Social Media