Construction of and efficient sampling from the simplicial configuration model

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HONS C J.-G. Young , G. Petri , F. Vaccarino , , A. Patania , June rd, Département de physique, de génie physique, et d’optique, Université Laval, Québec, Canada ISI Foundation, Turin, Italy Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino, Italy

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/ There are multi-agent interactions in complex systems Simplicial complexes track this explicitly

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/ Why simplicial complexes? No loss of information upon projection

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/ Why simplicial complexes? No loss of information upon projection

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/ Why simplicial complexes? Eﬃcient compression of the structure

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/ Why simplicial complexes? Eﬃcient compression of the structure

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/ Disease regulation dataset (facets : genes, nodes : human diseases) [Goh et al., PNAS, , ( )] Diseasome - facets column 1

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/ Problem we address : How to assess the signiﬁcance of the properties of simplicial complexes?

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/ Outline Simplicial complexes and null models Simplicial conﬁguration model The sampling problem and its solution

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/ Outline Simplicial complexes and null models Simplicial conﬁguration model The sampling problem and its solution The shape of real complex systems : random or organized? Homology and Betti numbers

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T /

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/ The simplicial conﬁguration model : basic deﬁnitions 2 3 4 5 1 F Degree sequence : d (2, 2, 1, 2, 1) Size sequence : s (3, 3, 2)

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/ The simplicial conﬁguration model : basic deﬁnitions 2 3 4 5 1 F Degree sequence : d (2, 2, 1, 2, 1) Size sequence : s (3, 3, 2)

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/ The simplicial conﬁguration model : basic deﬁnitions 2 3 4 5 1 F Degree sequence : d (2, 2, 1, 2, 1) Size sequence : s (3, 3, 2)

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/ The simplicial conﬁguration model : the ensemble The Simplicial Conﬁguration Model (SCM) is the distribution : Pr(K; d, s) 1/|Ω(d, s)| Ω(d, s) : number of simplicial complexes with sequences (d, s) 2 3 4 5 1 Generalizes [Courtney and Bianconi, Phys. Rev. E , ( )]

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/ The simplicial conﬁguration model : the ensemble The Simplicial Conﬁguration Model (SCM) is the distribution : Pr(K; d, s) 1/|Ω(d, s)| Ω(d, s) : number of simplicial complexes with sequences (d, s) Generalizes [Courtney and Bianconi, Phys. Rev. E , ( )]

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S SCM /

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/ Sampling : Change of representation 2 3 4 5 1 S : B Factor graph ensemble with degree sequences (d, s) ...

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/ Sampling : Change of representation 2 3 4 5 1 S : B Factor graph ensemble with degree sequences (d, s) ...

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/ Sampling : Change of representation 2 3 4 5 1 S : B Factor graph ensemble with degree sequences (d, s) ... ... and two additional constraints (mapping bĳective) ◦ No ◦ No

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/ Sampling : Constraints Input sequences : (d, s) ([2, 2, 1, 2, 1], [3, 3, 2]) Correct mapping 2 3 4 5 1 2 3 4 5 1 Bipartite graph Simplicial complex Output sequences : (d, s) ([2, 2, 1, 2, 1], [3, 3, 2])

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/ Sampling : Constraints Input sequences : (d, s) ([2, 2, 1, 2, 1], [3, 3, 2]) Constraint not respected : No multiedges 2 3 4 5 1 2 3 4 5 1 Bipartite graph Simplicial complex Output sequences : (d, s) ([1, 2, 1, 1, 1], [2, 2, 2])

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/ Sampling : Constraints Input sequences : (d, s) ([2, 2, 1, 2, 1], [3, 3, 2]) Constraint not respected : No included neighborhoods 2 3 4 5 1 2 3 4 5 1 Bipartite graph Simplicial complex Output sequences : (d, s) ([1, 1, 1, 1, 1], [3, 0, 2])

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/ Consequence : Uniform distribution over bipartite graphs Uniform distribution over simplicial complexes

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/ Consequence : Uniform distribution over bipartite graphs Uniform distribution over simplicial complexes

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/ Sampling : Possible sampling strategies

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/ Sampling : Possible sampling strategies Rejection sampling (stub matching + rejection) All bipartite graphs with sequences (d,s) No constraints violated Reject

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/ Sampling : Possible sampling strategies Rejection sampling (stub matching + rejection) P : Far too many rejections! Loose upper bound : Pr[reject] > exp −1 2 d2 / d − 1 s2 / s − 1 All bipartite graphs with sequences (d,s) Reject

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/ Sampling : Possible sampling strategies Rejection sampling (stub matching + rejection) P : Far too many rejections! Loose upper bound : Pr[reject] > exp −1 2 d2 / d − 1 s2 / s − 1 Markov Chain Monte Carlo The natural choice!

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2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 4 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 MCMC

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2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 4 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1

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2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 4 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1

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2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 4 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1

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/ Sampling : MCMC Details M Pick L ∼ P random edges in bipartite graph P can be , we use Pr[L ] exp[λ ]/Z Rewire edges. If multiedge or included neighbors, reject. Similar to [Miklós–Erdős–Soukup, Electron. J. Combin., , ( )] 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1

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T : ? /

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/ SCM has a null model : concept N : Is the quantity f (X) close to f (K) for random simplicial complexes X ∼ SCM[d(K), s(K)]?

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/ SCM has a null model : concept N : Is the quantity f (X) close to f (K) for random simplicial complexes X ∼ SCM[d(K), s(K)]? 50 100 150 200 10 3 10 2 10 1 100 Distribution Property I Pr[| f (K) − f (X)| < ] ≈ 1 Yes : K is typical, the local quantities d, s explain f.

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/ SCM has a null model : concept N : Is the quantity f (X) close to f (K) for random simplicial complexes X ∼ SCM[d(K), s(K)]? 50 100 150 200 10 3 10 2 10 1 100 Distribution Property I Pr[| f (K) − f (X)| < ] 1 No : K is atypical, K is organized beyond the local scale.

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/ So ... are real systems organized?

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/ Disease regulation dataset (true system) (facets : genes, nodes : human diseases) [Goh et al., PNAS, , ( )] Diseasome - facets column 1

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/ Disease regulation dataset (random instance) (facets : genes, nodes : human diseases) [Goh et al., PNAS, , ( )] Diseasome - SCM (facets column 1)

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/ Crimes in St-Louis (true system) (facets : people, nodes : crimes) [Rosenfeld et al., ( )] Moreno crime - facets column 1

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/ Crimes in St-Louis (random instance) (facets : people, nodes : crimes) [Rosenfeld et al., ( )] Moreno crime - SCM (facets column 1)

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/ So ... are real systems random?

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/ So ... are real systems random? Visibly not.

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/ How to quantify this : Homology in seconds 2 3 4 5 1 2 3 4 5 1 Q Done with homology. Results summarized with B β 0,β 1,... βk : counts the number of dimension k holes

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/ Real systems : organized or random? Diseases Crime Diseasome - facets column 1 Moreno crime - facets column 1 0 50 10 3 10 2 10 1 100 Distribution 350 400 450 500 k 1 0 100 101 10 2 100 Degree Size (b) 50 100 150 200 k 10 3 10 2 10 1 100 Distribution 0 1 100 101 10 2 100 Degree Size (c) F – β0 , β1 in the SCM (symbol) vs real systems (horizontal lines)

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/ Real systems : organized or random? Pollinators (real) Pollinators (random) Pollonators - facets column 0 Pollinators - SCM (facets column 0) 10 20 30 40 50 k 10 3 10 2 10 1 100 Distribution 0 1 101 10 2 10 1 100 Degree Size (a) F – β0 , β1 in the SCM (symbol) vs real systems (horizontal lines)

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/ Software and tutorials : github.com/jg-you/scm

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/ Software and tutorials : github.com/jg-you/scm

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/ Software and tutorials : github.com/jg-you/scm

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/ Selected references O ( ) J.-G. Young, G. Petri, F. Vaccarino and A. Patania, arxiv : . ( ) Equilibrium random ensembles ( ) O. Courtney and G. Bianconi, Phys. Rev. E , ( ) ( ) K. Zuev, O. Eisenberg and K. Krioukov, J. Phys. A , ( ) Sampling ( ) B. K. Fosdick, et al., arXiv : . ( )

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/ Take-home message SCM : random simplicial complexes with ﬁxed (d, s).

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/ Take-home message SCM : random simplicial complexes with ﬁxed (d, s). Eﬃcient sampling with MCMC.

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/ Take-home message SCM : random simplicial complexes with ﬁxed (d, s). Eﬃcient sampling with MCMC. Real system are not not always organized.

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/ Take-home message SCM : random simplicial complexes with ﬁxed (d, s). Eﬃcient sampling with MCMC. Real system are not not always organized. Many open questions! Simpliciality, best distribution P, connectivity?

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/ Reference : arxiv.org/1705.10298 Software : github.com/jg-you/scm info@jgyoung.ca jgyoung.ca @_jgyou