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Community structure in
complex networks at
different resolution levels
Sergio Gómez Universitat Rovira i Virgili, Tarragona (Spain)
Graph-TA 2014
2
Our research team: Alephsys
Details
Dept. Enginyeria Informàtica i Matemàtiques,
Universitat Rovira i Virgili, Tarragona (Spain)
http://deim.urv.cat/~alephsys/
People
Alex Arenas, Sergio Gómez
Manlio De Domenico, Albert Solé, Per Sebastian Skardal
Pau Erola, Clara Granell
Joan Matamalas, José Magaña, Antonio González
Roger Guimerà, Jordi Duch, Sergio Lozano, Albert
Fernández, Javier Borge, etc.
Community structure at different resolution levels
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Interests: Complex Networks
Structure of complex networks
Community structure
Multiplex networks
Mathematical formalism
Descriptors: centrality, assortativity, etc.
Dynamics on complex networks
Synchronization
Epidemic spreading
Diffusion
Evolutionary games
Interplay between structure and dynamics
Community structure at different resolution levels
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Complex networks at different scales
Microscale Nodes and links
URV (Tarragona, Spain) email network
Community structure at different resolution levels
Schools
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Complex networks at different scales
Macroscale Statistical properties
URV (Tarragona, Spain) email network
Community structure at different resolution levels
Schools
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Complex networks at different scales
Mesoscale Sub-structure
URV (Tarragona, Spain) email network
Community structure at different resolution levels
Schools
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What are mesoscales?
Sub-structures involving sets of nodes
Communities
Finding communities
Optimization of global quality function, e.g. modularity
Local optimization of communities by growth
Probabilistic and Information-based approaches
Many different clustering algorithms
Set of communities
Partitions
Overlapping communities
Hierarchical structure
Community structure at different resolution levels
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Finding communities
Optimization of Modularity (Newman & Girvan (2004) Phys Rev E 69, 026113)
More links inside the communities than in a random (null case) network preserving nodes’ strengths
Community structure at different resolution levels
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Problems with modularity
Unexpected partitions
Zachary karate club:
Q(best) = 0.419790
Four communities found
Two communities expected
Community structure at different resolution levels
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Problems with modularity
Community structure at different resolution levels
Resolution limit (Fortunato & Barthelemy (2007) PNAS 104, 36)
Modules not separable if internal strength
Km
Km
Km
Km
Km
Km
Km
Km
Km
Km
Kp
Km
Kp
Km
Modularity only gives one mesoscale
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Why these problems?
Community structure at different resolution levels
Macroscale Microscale Mesoscales
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Scanning the whole mesoscale
Criteria for the scanning method:
Recovery of macroscale One community formed by all nodes
Recovery of microscale Each node in its own community
Preserve the semantics of Modularity In particular, recovery of modularity mesoscale
Comment:
We do not impose hierarchical mesoscales
Community structure at different resolution levels
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Scanning the whole mesoscale
How?
For instance, how can we isolate a node?
Community structure at different resolution levels
How?
Tune the resistance of nodes to join communities, adding self-loops
The self-loop increases the internal strength
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Scanning the whole mesoscale
Community structure at different resolution levels
Multiple resolution method (Arenas, Fernandez & Gómez (2008) New J Phys 10, 053039)
Add a common resistance (self-loop) to all nodes
Optimize modularity
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Scanning the whole mesoscale
Community structure at different resolution levels
Multiple resolution method (Arenas, Fernandez & Gómez (2008) New J Phys 10, 053039)
Run the resistance in the interval
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Scanning the whole mesoscale
Community structure at different resolution levels
Macroscale
Smallest positive value which satisfies
Microscale
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Scanning the whole mesoscale
Multiple resolution method (Arenas, Fernandez & Gómez (2008) New J Phys 10, 053039)
Comments: Topological properties not affected:
Strength distribution
Vertex to vertex correlations
Spectra
Laplacian
Resolution limit skipped
Macroscale and Microscale recovered
Semantics of Modularity preserved
Community structure at different resolution levels
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Validation
Circle
Community structure at different resolution levels
-1.0 -0.4 0.67 6.0
0.0
Ma
cro
sca
le
Mic
rosca
le
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Validation
Homogeneous with two hierarchical levels (Arenas, Diaz-Guilera & Perez-Vicente (2006) Phys Rev Lett 96, 114102)
Community structure at different resolution levels
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Validation
Hierarchical scale-free (Ravasz & Barabasi (2003) Phys Rev E 67, 026112)
Community structure at different resolution levels
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Validation
Resolution limit (Fortunato & Barthelemy (2007) PNAS 104, 36)
Community structure at different resolution levels
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Social networks
Zachary karate club (Zachary (1977) J Anthropol Res 33, 452)
Community structure at different resolution levels
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Social networks
Dolphins (Lusseau et al (2003) Behav Ecol Sociobiol 54, 396)
Community structure at different resolution levels
Merge and split problem (Lancichinetti & Fortunato (2011) Phys Rev E 84, 066122)
Cannot be solved by changing the resolution Modularity is not appropriate if communities of very
different sizes coexist
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Problems with modularity
Community structure at different resolution levels
Kp
Km
Kp
Kp
Km
Kp
Small cliques merge or large clique splits
Hierarchical multiresolution (Granell, Gómez & Arenas (2012) Int J Bifurcat Chaos 22, 1250171)
Divisive method, generates hierarchy of communities
Based in the optimization of (signed) modularity
No resolution limit problem
No merge and split problem
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Solution
Community structure at different resolution levels
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Hierarchical multiresolution
Community structure at different resolution levels
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Discussion
All community detection (clustering) algorithms
Have an (implicit or explicit) definition of clusters
The “shape” of the clusters and the distribution of sizes
depend on this definition
Have an (implicit or explicit) scale of resolution
A parameter needed to scan all the mesoscale
A rule needed to find the relevant scales of resolution
Have an equivalent to the resolution limit problem
It is possible to build counter-examples of community
structures which cannot be solved by the algorithm
Depending on the network they can be harmless
Community structure at different resolution levels
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Conclusions
Do not trust algorithms which just give a partition
You need to know how to “tune” the resolution
Analyze the whole mesoscale
Large computing times
Look for the right scales of resolution
Stability, sizes, number of communities, etc.
Resolution limit-like problems may appear
Not always a problem
Try with different algorithms
Consensus communities
Community structure at different resolution levels
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Community structure at different resolution levels
http://mediterraneanschoolcomplex.net/
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Thank you for your attention!
Info
Contact [email protected]
http://deim.urv.cat/~sgomez/
Software: Radatools
http://deim.urv.cat/~sgomez/radatools.php
References
A. Arenas, A. Fernández, S. Gómez,
Analysis of the structure of complex networks at different resolution levels,
New Journal of Physics 10 (2008) 053039
C. Granell, S. Gómez, A. Arenas, Hierarchical multiresolution method to overcome the resolution limit in complex
networks,
Int. Journal of Bifurcation and Chaos 22 (2012) 1250171
Community structure at different resolution levels
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