Topologically biased random walks with application for community finding

Vinko ZlatićDep. Of Physics, “Sapienza”, Roma, Italia

Theoretical Physics Division, Institute “Ruđer Bošković”, Zagreb, Croatia

CNR-INFM Centro SMC Dipartimento di Fisica, Universita di Roma “Sapienza”, Italy

Zlatić, Gabrielli, Caldarelli, arXiv:1003.1883

Andrea Gabrielli

Guido Caldarelli

Community Finding

Community: One of many (possibly overlapping) subgraphs1.Has strong internal node-node connections2.Weaker external connections

Q:How to find communities in large networks?

Santo Fortunato “Community detection in graphs”, Eprint arXiv: 0906.0612, Physics Reports 486, 75-174 (2010)

MotivationDistribution of links in the network is locally heterogeneous

1. Finding sets of nodes with similar function

2. Visualization

3. Classification

4. Hierarchical organization

Useful in: Systems Biology, Sociology, Computer sciences, Physics ...

Community Finding(2)

Algorithms:S. Fortunato, arXiv:0906.0612, Physics Reports 486, (2010) pp.75

•Newman-Girvan, removal of links with high beetweeness•Different algorithms to maximise Newman modularity•Radicchi et al. ,removal of links based on local properties•Cfinder, Markov cluster algorithm,Potts model ...

Spectral Methods●Idea: distance in the M dimensional space spanned by eigenvectors associated with random walks●M corresponds to number of eigenvectors used●Then we can apply standard clustering techniques Manhattan distance, angle distance, etc●Graph Laplacian

L. Donetti and M. A. Munoz, J. Stat. Mech. P10012 (2004).

ProblemCommunity structure can be very hard to detectMixing ParameterDifferent performances of algorithms (detectability, speed, size of networks)

A. Lancichinetti, S. Fortunato, Phys. Rev. E,80, 056117 (2009).

Donetti and Munoz is one of better algorithms

Biased random walks IDEA●New idea: Different topological quantities have different frequencies in between different communities

Why not use this additional information to improve spectral methods???

Example: Edge multiplicity

Other possibilities:Shortest path betweeness, subgraph frequencies, degree, clustering, even eigenvectors of nonbiased random walk.

Biased Random walksUnbiased transition operator (Frobenius-Perron operator)

Biased transition operator (Frobenius-Perron operator)

Exponential familly of transition probabilities

Asymptotic probabilities Detailed balance condition

Symmetrization(1)

Symmetrization leeds to hermitian operator.

Orthogonal eigenvectors

Symmetrization(2)

Parametric equations of motion

D. A. Mazziotti, et al, Journal of Physical Chemistry 99, 112-117 (1995).

SpectraSpectra contains important information on community structure

N separate graphs considered as one have N-fold degeneracy of the first eigenvector

Characteristic time to approach stationary distribution is related to spectral gap

N communities should produce N-1 close eigenvalues!

Application to community finding(1)

Tuning of biases in such a way to maximize the community gap

Indeed!!!Close large eigenvalues form clear “community band” for modest values of mixing parameterThere is a clear separation between N-1 eigenvalues associated with community structure and rest of the spectraSeparation beetween Nth eigenvalue and N+1th eigenvalue we name “community gap”

Application to community finding(2)

Every network has its own optimal parameters

Tetrahedric structure – description with angles

Conclusion1.This topic is still “hot”

2. Possibility to include variables related to dynamics on networks

3. Promissing preliminary results (outperform DM 0.60 vs. 0.45 at mixing parameter =0.5)

To do: Test different topological variables as basis for biasesDevelop better clustering algorithm based on distances between nodes.

Thanks for your attention