Overcoming Resolution Limits in MDL Community Detection

L. Karl Branting The MITRE Corporation

2nd SNA-KDD Workshop 24 Aug 2008

Outline



Utility functions in community detection



Resolution limits



MDL-based community detection

– –

Previous: RB and AP New: SGE



Experimental Evaluation



Lessons 2

2nd SNA-KDD Workshop 24 Aug 2008

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)



Utility function: modularity



Search strategy: greedy agglomerative hierarchical clustering (iteratively choose highest modularity merge)

–

Tasgin & Bingol (2006)



Utility function: modularity



Search strategy: genetic algorithm

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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)



Utility function: modularity



Search strategy: greedy agglomerative hierarchical clustering with spectral division 4

2nd SNA-KDD Workshop 24 Aug 2008

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)



Utility function: Minimum Description Length



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.

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Modularity

1 

i

 

m w

(

D ii

) 

l l i l

2  – –

W(D ii ) = number of edges internal to group i l i = 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

2

l

–

Rosvall & Bergstrom (2007) Biased towards same-sized groups

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Resolution Limit



Ring graph R 15,4

–

15 communities

–

4 nodes per community



Community structure that maximizes modularity conflates groups 7

2nd SNA-KDD Workshop 24 Aug 2008

Approaches to modularity’s resolution limit



Apply recursively to large communities (Ruan & Zhang 2007)



Apply locally (Clauset 2005)



Choose a different utility function 8

2nd SNA-KDD Workshop 24 Aug 2008

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

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Components of Description

 

2.

3.

4.

5.

Components (details in paper) 1.

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

2nd SNA-KDD Workshop 24 Aug 2008

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Surprising Experimental Result



RB, AP, and modularity compared as utility functions

–

Applied to ring graphs R m,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 R m,c !

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2nd SNA-KDD Workshop 24 Aug 2008

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 1.

2.

3.

4.

5.

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

2nd SNA-KDD Workshop 24 Aug 2008

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SGE (Sparse Graph Encoding)



Components 1.

2.

3.

4.

5.

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



Follows RB

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Performance of SGE on Ring Graphs

   – –

Correct community structure found for every R m,c for 4 ≤ m ≤ 16 and 3 ≤ c ≤ 9 except R 4,3 R 13,3 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? 14

2nd SNA-KDD Workshop 24 Aug 2008

Uniform random graph

15



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:

– – – – –

32 vertices 4 groups average degree 6 size ratio 1.25

Proportion internal edges 0.67

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 4 initial vertices 3 edges added per time step

–

20 time steps

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Evaluation Criteria



Rand index (Rand 1971)



Adjusted Rand index (Hubert & Arabie 1985)



F-measure – based on same-cluster pairs

–

Recall =

|

proposedPa irs



actualPair s

| |

actualPair s

| –

Precision =

|

proposedPa irs



actualPair s

| |

proposedPa irs

| –

F-measure =

2 *

recall recall

 *

precision precision

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2nd SNA-KDD Workshop 24 Aug 2008

Results: Uniform random graph

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2nd SNA-KDD Workshop 24 Aug 2008

Results: Uniform random graph

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2nd SNA-KDD Workshop 24 Aug 2008

Results: Uniform random graph

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2nd SNA-KDD Workshop 24 Aug 2008

Results: Embedded Barabasi-Albert

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2nd SNA-KDD Workshop 24 Aug 2008

Summary of Evaluation



Random 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

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Conclusion



Narrow

– –

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

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