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Presentation at NetSci 2011, Budapest, April 8, 2011.

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- 1. Community Detection Resolution Limit Deﬁnition of resolution-free Results Resolution-free community detection V.A. Traag1, P. Van Dooren1, Y.E. Nesterov2 1ICTEAM Universit´e Catholique de Louvain 2CORE Universit´e Catholique de Louvain 8 April 2011
- 2. Community Detection Resolution Limit Deﬁnition of resolution-free Results Outline 1 Community Detection 2 Resolution Limit 3 Deﬁnition of resolution-free 4 Results
- 3. Community Detection Resolution Limit Deﬁnition of resolution-free Results Community Detection • Detect ‘natural’ communities in network. • Modularity approach: ‘relatively’ many links inside communities
- 4. Community Detection Resolution Limit Deﬁnition of resolution-free Results Community Detection (formal) • In general, commmunities should have relatively many present links (beneﬁt), few missing links (cost) Minimize H = − ij (aij Aij − bij (1 − Aij ))δ(σi , σj ), • Compare to random null-model pij (RB) aij = wij − bij and bij = γRBpij HRB = − ij (Aij wij − γRBpij )δ(σi , σj ). • Modularity (NG): use conﬁguration null model pij = ki kj 2m . Reichardt and Bornholdt. Phys Rev E (2006) 74:1,016110 Newman and Girvan. Phys Rev E (2004) 69:2,026113
- 5. Community Detection Resolution Limit Deﬁnition of resolution-free Results Resolution limit • Modularity might miss ‘small’ communities. • Merge two cliques in ring of cliques when γRB < q nc(nc − 1) + 2 . • Depends on the total size of the graph. • Number of communities scales as √ γRBm. • For general null model, problem remains since ij pij = 2m. Fortunato and Barthlemy PNAS (2007) 104:1, pp. 36 Kumpala et al. Eur Phys J B (2007) 56, pp. 41-45
- 6. Community Detection Resolution Limit Deﬁnition of resolution-free Results Evading the resolution limit • New model (RN) suggested aij = wij bij = γRN HRN = − ij (Aij (wij + γRN) − γRN)δ(σi , σj ). • Claim: no resolution limit, as merge depends only on ‘local’ variables γRN < 1 n2 c − 1 . • But, take pij = ki kj (rescale γRB by 2m), we obtain γRB < 1 2(nc(nc − 1) + 2)2 , also only ‘local’ variables. Hence, also no resolution limit? Ronhovde and Nussinov. Phys Rev E (2010) 81:4,046114.
- 7. Community Detection Resolution Limit Deﬁnition of resolution-free Results Problems remain Subgraph • Assume pij = ki kj (rescale γRB by 2m) • Then separate in large graph when γRB > 1 2(nc(nc − 1) + 2)2 • But merged in subgraph when γRB < 1 2(nc(nc − 1) + 1)2
- 8. Community Detection Resolution Limit Deﬁnition of resolution-free Results Resolution limit revisited Resolution-limit Resolution-free • Problem is not merging per s´e. • Rather, cliques separate in subgraph, but merge in large graph (or vice versa). • Suggests following deﬁnition.
- 9. Community Detection Resolution Limit Deﬁnition of resolution-free Results Resolution limit revisited Resolution-limit Resolution-free Deﬁnition (Resolution-free) Objective function H is called resolution-free if, whenever partition C optimal for G, then subpartition D ⊂ C also optimal for subgraph H(D) ⊂ G induced by D.
- 10. Community Detection Resolution Limit Deﬁnition of resolution-free Results Deﬁning resolution-free Deﬁnition (Resolution-free) Objective function H is called resolution-free if, whenever partition C optimal for G, then subpartition D ⊂ C also optimal for subgraph H(D) ⊂ G induced by D. • Implicitly deﬁnes resolution limit: method is not resolution-free. • Some nice properties of resolution-free methods: Replace optimal subpartitions Never split cliques (unless in single nodes) Main questions • Do such methods exist? • What conditions to impose?
- 11. Community Detection Resolution Limit Deﬁnition of resolution-free Results General framework General community detection H = − ij (aij Aij − bij (1 − Aij ))δ(σi , σj ), RB model Set aij = wij − bij , bij = γRBpij . RN model Set aij = wij , bij = γRN. Simpler alternative CPM Set aij = wij − bij and bij = γ. Leads to H = − ij (Aij wij − γ)δ(σi , σj ). Clear interpretation: γ is minimum density of a community H = − c ec − γn2 c.
- 12. Community Detection Resolution Limit Deﬁnition of resolution-free Results Main result Do resolution-free methods exists? Yes: Both RN and CPM are resolution-free, results from general theorem. What conditions to impose? Suﬃcient condition: aij and bij should be ‘local’. Deﬁnition (Local weights) Weights aij , bij called local whenever for every subgraph H ⊂ G, weights remain similar, i.e. aij (G) ∼ aij (H) and bij (G) ∼ bij (H). • Implies local weigths aij and bij can only depend on node i and node j, nothing further. • RN and CPM use local weights, hence resolution-free. • Not necessary condition, but seem to be few exceptions. • So, RN and CPM (almost) only sensible deﬁnitions.
- 13. Community Detection Resolution Limit Deﬁnition of resolution-free Results Performance (directed networks) µ0 0.2 0.4 0.6 0.8 1.0 NMI 0.25 0.5 0.75 1 CPM Infomap Modularity ER n = 103 n = 104
- 14. Community Detection Resolution Limit Deﬁnition of resolution-free Results Conclusions • Provided deﬁnition of resolution-free. • Methods using local weights are resolution-free. • Clariﬁes link between ‘local’ methods and resolution limit. • Only few resolution-free methods. • Tested CPM, performs superbly. Thank you for your attention. Questions? Traag, Van Dooren and Nesterov arXiv:1104.3083v1

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