Construction of and efficient sampling from the simplicial configuration model

Presented at HONS 2017 (http://complexdata.businesscatalyst.com/2017/)

Paper: https://doi.org/10.1103/PhysRevE.96.032312
arXiv: https://arxiv.org/abs/1705.10298
Code: https://github.com/jg-you/scm/

Abstract
=======
Simplicial complexes are now a popular alternative to networks when it comes to describing the structure of complex systems, primarily because they encode multi-node interactions explicitly. With this new description comes the need for principled null models that allow for easy comparison with empirical data. We propose a natural candidate, the simplicial configuration model. The core of our contribution is an efficient and uniform Markov chain Monte Carlo sampler for this model. We demonstrate its usefulness in a short case study by investigating the topology of three real systems and their randomized counterparts (using their Betti numbers). For two out of three systems, the model allows us to reject the hypothesis that there is no organization beyond the local scale.

Transcript

1. 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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2. /
   There are multi-agent interactions
   in complex systems
   Simplicial complexes track this explicitly
   

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3. /
   Why simplicial complexes?
   No loss of information upon projection
   

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4. /
   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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9. /
   Outline
   Simplicial complexes and null models
   Simplicial configuration model
   The sampling problem and its solution
   

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10. /
    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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11. T
    /
    

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    The simplicial configuration model : basic definitions
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    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
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    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
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    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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17. S SCM
    /
    

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    Sampling : Change of representation
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    S : B
    Factor graph ensemble with degree sequences (d, s) ...
    

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    Sampling : Change of representation
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    S : B
    Factor graph ensemble with degree sequences (d, s) ...
    

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

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21. /
    Sampling : Constraints
    Input sequences : (d, s) ([2, 2, 1, 2, 1], [3, 3, 2])
    Correct mapping
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    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
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    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
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    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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29. /
    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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30. 2
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    MCMC
    

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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., , ( )]
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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., , ( )]
    MCMC is uniform over Ω(d, s)
    Move set yields aperiodic chain
    ∗
    Move set connects the space
    100 101 102 103 104
    Lmax
    0
    2500
    5000
    7500
    Edit distance
    = 0
    = 1
    = +1
    (a)
    

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36. 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
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    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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    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?
    Reference : arxiv.org/1705.10298
    Software : github.com/jg-you/scm
    

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59. /
    Reference : arxiv.org/1705.10298
    Software : github.com/jg-you/scm
    info@jgyoung.ca jgyoung.ca @_jgyou
    

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