the simplicial configuration model - alice patania
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The simplicial configuration model
Alice Patania, Network Science Institute, Indiana University
A Sampling Method
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A simplicial complex is a collection of simplices (non-empty finite sets ordered under inclusion).It can be uniquely identified by its maximal simplices under inclusion (facets)
A simplicial complex
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A simplicial complex is a collection of simplices (non-empty finite sets ordered under inclusion).It can be uniquely identified by its maximal simplices under inclusion (facets)
A simplicial complex
To every simplicial complex we can assign two sequences:
Degree sequenceNumber of maximal simplices under inclusion incident on a node
Size sequenceNumber of nodes contained in a maximal simplex
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Simplicial degree and Facet size
To every simplicial complex we can assign two sequences:
Degree sequenceNumber of maximal simplices under inclusion incident on a node
Size sequenceNumber of nodes contained in a maximal simplex
d = (2,2,1,2,1)
A simplicial complex is a collection of simplices (non-empty finite sets ordered under inclusion).It can be uniquely identified by its maximal simplices under inclusion (facets)
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Simplicial degree and Facet size
To every simplicial complex we can assign two sequences:
Degree sequenceNumber of maximal simplices under inclusion incident on a node
Size sequenceNumber of nodes contained in a maximal simplex
d = (2,2,1,2,1)s = (3,3,2)
A simplicial complex is a collection of simplices (non-empty finite sets ordered under inclusion).It can be uniquely identified by its maximal simplices under inclusion (facets)
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Bipartite graphs and Simplicial complexes
Factor graph
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Bipartite graphs and Simplicial complexes
One-mode projection
Factor graph
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Bipartite graphs and Simplicial complexes
One-mode projection
Factor graph
IdeaUse existing random models for bipartite graphs to construct a sampling method for SCM.
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*this list is not exhaustive
Erdos-Renyi inspired:Random pure simplicial complexes [Linial-Meshulam (2006)]Random simplicial complexes [Kahle (2009)]Multi-parameter random simplicial complexes [Costa-Farber (2015)]
Exponential random graphs inspired:Exponential random simplicial complex [Zuev et al. (2015)]
Preferential attachment for simplicial complexes:Network geometry with flavor [Bianconi-Rahmede (2016)]
Configuration model:Configuration model for pure simplicial complexes [Courtney-Bianconi (2016)]
Existing approaches
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Configuration model
The configuration model is a generative model for random graphs with fixed number of nodes and degree sequence.
IdeaUse existing random models for bipartite graphs to construct a sampling method for SCM.
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Configuration model
The configuration model is a generative model for random graphs with fixed number of nodes and degree sequence.
This implies the following are fixed:number of vertices (0-simplices) and number of maximal simplicesvertices’ degree sequence and maximal simplices’ size sequence
IdeaUse existing random models for bipartite graphs to construct a sampling method for SCM.
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Simplicial degree and Facet size
This approach generalizes a method by[Courtney and Bianconi, Phys. Rev. E 93, (2016)]
Pr(K; d, s) = 1/|Ω(d, s)|The simplicial configuration model (SCM) is the distribution
Where Ω(d, s): number of simplicial complexes with sequences (d, s)
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Pr(K; d, s) = 1/|Ω(d, s)|
Bipartite graphs and Simplicial complexes
NOT VA
LID
The simplicial configuration model (SCM) is the distribution
Where Ω(d, s): number of simplicial complexes with sequences (d, s)
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Adding constraints
NOT VALID
First constraint: No multi-edges Multi-edges decrease the size of the maximal simplices.
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Adding constraints
Second constraint: No included neighborhoodsIncluded neighborhoods violate the maximality assumption of the facets.
NOT VALID
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Adding constraints
Constraints: No multi-edges No included neighborhoods
Then the acceptable configurations for the toy example are the following:
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Pr(K; d, s) = 1/|Ω(d, s)|
Bipartite graphs and Simplicial complexes
REJ
ECTED
The simplicial configuration model (SCM) is the distribution
Where Ω(d, s): number of simplicial complexes with sequences (d, s)
ACCEPTED
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Adding constraints
! Problem with rejection sampling: Far too many rejections!
Loose upper bound :
Pr[reject] > exp[-0.5(〈d2〉/〈d〉− 1) (〈s2〉/〈s〉− 1)]
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Markov Chain Monte Carlo sampling
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Markov Chain Monte Carlo sampling
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Markov Chain Monte Carlo sampling
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MCMC sampling: The details
Move set
1. Pick L∼P random edges in bipartite graphP can be arbitrary, we use Pr[L = l] = exp[λl]/Z
2. Rewire edges. If multi-edge or included neighbors, reject.
Similar to [Miklós–Erdős–Soukup, Electron. J. Combin., 20, (2013)]
● MCMC is uniform over Ω(d, s)● Move set yields aperiodic chain● Move set connects the space
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Results - True systems
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Results - Random instances
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Building a null model
How to assess the significanceof the properties of a simplicial complex?
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Concept for a null model
Null modelIs the quantity f(X) close to f(K) for random simplicial complexes X∼SCM[d(K), s(K)] ?
Pr[| f(K) − f(X)| < ] ≈ 1
K is typical, the local quantities (d,s) explain f.
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Concept for a null model
Pr[| f(K) − f(X)| < ] ≪ 1
Null modelIs the quantity f(X) close to f(K) for random simplicial complexes X∼SCM[d(K), s(K)] ?
K is atypical, K is organized beyond the local scale.
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Results on Betti numbers of real data sets
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Results on Betti numbers of real data sets
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Our contribution(1) J.-G. Young, G. Petri, F. Vaccarino and A. Patania, Phys. Rev. E 96 (3), 032312 (2017)
Equilibrium random ensembles(2) O. Courtney and G. Bianconi, Phys. Rev. E 93, (2016)(3) K. Zuev, O. Eisenberg and K. Krioukov, J. Phys. A 48, (2015)
Sampling(4) B. K. Fosdick, et al., arXiv :1608.00607 (2016)
Code : github.com/jg-you/scm
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Thank You for the Attention
Our contribution(1) J.-G. Young, G. Petri, F. Vaccarino and A. Patania, Phys. Rev. E 96 (3), 032312 (2017)
Code : github.com/jg-you/scm