1. Overcoming Resolution Limits in MDL Community Detection
L. Karl Branting
The MITRE Corporation

2. Outline Utility functions in community detection Resolution limits
MDL-based community detection
Previous: RB and AP
New: SGE
Experimental Evaluation
Lessons

3. Utility functions in community detection
Two components of community detection algorithms
Utility function – quality criterion to be optimized
Search strategy – procedure for finding optimal partition
Examples
Garvin & Newman (2003)
Utility function: modularity
Search strategy: greedy divisive hierarchical clustering (iteratively remove highest betweenness edge)
Newman (2003)
Search strategy: greedy agglomerative hierarchical clustering (iteratively choose highest modularity merge)
Tasgin & Bingol (2006)
Search strategy: genetic algorithm

4. Utility functions in community detection
Other search strategies used with modularity
Rattigan, Maier, Jensen (2007)
Utility function: modularity
Search strategy: Greedy divisive hierarchical clustering using a Network Structured Index to approximation edge betweenness
Donetti & Munoz (2004)
Search strategy: greedy agglomerative hierarchical clustering with spectral division

5. Utility functions in community detection
Statistical Approaches
Zhang, Qiu, Giles, Foley, & Yen (2007)
Utility function: log-likelihood (LDA parameters)
Search strategy: fixed-point iteration
Compression-Based Approaches
Rosvall & Bergstrom (2007)
Utility function: Minimum Description Length
Search strategy: simulated annealing
Chakrabarti (2004)
Search strategy: exhaustive search for k, hill-climbing given k
Utility function implicit in search strategy
Raghavan, Albert, & Kumara (2007) – marker passing
Cliques, cores, etc.

6. Modularity
W(Dii) = number of edges internal to group i
li = number of edges incident to vertices in group I
l = total number of edges
Intuitive – expresses intuition that ratio of internal to external edges is greater for groups than for non-groups
Popular
Imperfect
Fortunato & Barthelemy (2007) Resolution limit: groups conflated if number of vertices less than
Rosvall & Bergstrom (2007) Biased towards same-sized groups

7. Resolution Limit Ring graph R15,4
15 communities
4 nodes per community
Community structure that maximizes modularity conflates groups

8. Approaches to modularity’s resolution limit
Apply recursively to large communities (Ruan & Zhang 2007)
Apply locally (Clauset 2005)
Choose a different utility function

9. Description Length
Utility of community structure is sum of bits needed to represent
Community structure +
Graph given community structure
Search strategy attempts to minimize description length
There is no unique bit count
Undecidability of Kolmogorov complexity
Previous approaches
Rosvall & Bergstrom (2007): RB
Handles group size skew better than modularity
Chakrabarti (2004): AP
Comparison
Similar breakdown of bits
Different calculation

10. Components of Description
Components (details in paper)
Bits to represent number of nodes in graph
ignored because not specific to community structure
Bits to represent number of groups
Bits to represent mapping between nodes and groups
Bits needed for number of group-to-group edges
Bits needed for adjacencies between nodes
Purpose
2, 3, 4: represent group structure
1, 5: represent graph as a whole

11. Surprising Experimental Result
RB, AP, and modularity compared as utility functions
Applied to ring graphs Rm,c for 4 ≤ m ≤ 16 and 3 ≤ c ≤ 9
Search strategy: greedy divisive hierarchical clustering (iteratively remove highest betweenness edge)
Unsurprising result. Modularity led to conflated groups for:
m > 8 and c = 3
m > 10 and c = 4
m > 11 and c = 5
m > 13 and c = 6,7
Surprising result.
Both RB and AP conflated at least one pair of groups in every Rm,c!

12. Hypothesis
Both RB and AP require at least one bit per pair of groups in term 4
Perhaps this estimation causes group conflation
Term 4 grows as the square of the number of groups
If graph is sparse, conflating groups may save more in term 4 reduction than it costs in term 5 increase
Components
Bits to represent number of nodes in graph
ignored because not specific to community structure
Bits to represent number of groups
Bits to represent mapping between nodes and groups
Bits needed for number of group-to-group edges
Bits needed for adjacencies between nodes

13. SGE (Sparse Graph Encoding)
Components
Bits to represent number of nodes in graph
Ignored, as in RB and AP
Bits to represent number of groups
Follows RB
Bits to represent mapping between nodes and groups
Similar to AP
Bits needed for number of group to group edges
Split into 2 terms
Which pairs of groups are connected (much less than one bit per pair if pairs sparsely or densely connected)
Number of edges between connected groups
Grows as number of connected pairs, not total number of pairs
Bits needed for adjacencies between nodes

14. Performance of SGE on Ring Graphs
Correct community structure found for every Rm,c for 4 ≤ m ≤ 16 and 3 ≤ c ≤ 9 except
R4,3
R13,3
Results confirm hypothesis that resolution limit in RB and AP is result of over-counting term 4: the bits needed for group-to-group edges
Significance
Ring graphs rare in real world
How does SGE compare on more realistic graphs?

15. Uniform random graph Similar to graphs in Rosvall & Bergstrom (2007)
Test set
32 vertices
4 groups
average degree 6
size ratio {1.0,1.25,1.5,1.75,2.0}
Proportion internal edges {0.6,0.75,0.9}
Example:
size ratio 1.25
Proportion internal edges 0.67

16. Embedded Barabasi-Albert Graphs
Test set
4 communities separately generated by preferential attachment
In each community
4 initial vertices
2-4 edges added per time step
20 time steps
Example
4 communities
3 edges added per time step

17. Evaluation Criteria Rand index (Rand 1971)
Adjusted Rand index (Hubert & Arabie 1985)
F-measure – based on same-cluster pairs
Recall =
Precision =
F-measure =

18. Results: Uniform random graph

19. Results: Uniform random graph

20. Results: Uniform random graph

21. Results: Embedded Barabasi-Albert

22. Summary of Evaluation Random graphs EBA graphs
Community structure is weak
Group sizes are balanced – modularity is best
Group sizes are imbalanced – RS is best (as per Rosvall & Bergstrom 2007)
Community structure is strong
Group sizes are balanced – not much difference
Group sizes are imbalanced – modularity is particularly bad (as per Rosvall & Bergstrom 2007), SGE slightly better than RS and AP
EBA graphs
Sparse – AP and SGE weaker than modularity and RS
Dense – essentially identical accuracy

23. Conclusion Narrow Broad
Conflation of groups by MDL in sparse graphs (e.g., ring graphs) can be avoided by adjusting group-to-group edge counts.
This change doesn’t hurt performance in more common types of graphs.
Compression-based clustering works well, but requires tinkering
Modularity detects weak structure well when graph not too big and groups not too imbalanced
Broad
Still unclear what utility function is best overall
Needed: theory relating graph typology to utility functions