O U S B SYNS : I’ ... Jean-Gabriel Young Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI, USA jgyoung.ca @_jgyou May th, Based on joint work with Guillaume St-Onge, Patrick Desrosiers, Louis J. Dubé

“I made this...” projects

“I made this...” projects Marie Kondo projects

Political Books co-purchase network V. Krebs, unpublished, http://www.orgnet.com/

Political Books co-purchase network V. Krebs, unpublished, http://www.orgnet.com/

Political Books co-purchase network V. Krebs, unpublished, http://www.orgnet.com/

Political Books co-purchase network V. Krebs, unpublished, http://www.orgnet.com/

Political Books co-purchase network V. Krebs, unpublished, http://www.orgnet.com/

Political Books co-purchase network V. Krebs, unpublished, http://www.orgnet.com/

Political Books co-purchase network V. Krebs, unpublished, http://www.orgnet.com/

Political Books co-purchase network V. Krebs, unpublished, http://www.orgnet.com/

Political Books co-purchase network V. Krebs, unpublished, http://www.orgnet.com/

Our goal : Organize MPE algorithms.

Mesoscopic pattern extraction (MPE) with objective functions MPE Algorithm

Mesoscopic pattern extraction (MPE) with objective functions What truly matters : the objective function. Explore Partition Space Objective Function E : Propose partitions σ1 , σ2 , ... O : Order partitions H(σo ) ≥ H(σi ) ∀i

A large class of MPE objectives G The class of objective functions that can be decomposed as H(σ, A; f ) (ij) f (aij , , σi , σj ), with f (aij , σi , σj ) a score function for node pair (i, j). f (1 , , ) f (0, , ) 2 1 3 W.W. Zachary, J. Anthropol. Res. ( ), http://www-persona .umich.edu/∼mejn/netdata/

Two-part algorithms in practice Core-periphery Greedy (Kernighan-Lin type)

Two-part algorithms in practice Core-periphery Greedy (Kernighan-Lin type) Modularity Greedy (Kernighan-Lin type)

Two-part algorithms in practice Core-periphery Greedy (Kernighan-Lin type) Modularity Greedy (Kernighan-Lin type) Spectral + Gaussian Mixture

Our goal : Organize MPE algorithms

Our goal : Organize MPE algorithms using their objective functions

Organization tools relationships to organize objective functions

Organization tools relationships to organize objective functions E H1 ∼ H2 Order partitions σA , σB the same way

Organization tools relationships to organize objective functions E H1 ∼ H2 Order partitions σA , σB the same way S H1 ⇓ H2 Restrict the “expressiveness” of H1 to get H2.

We’ll use equivalence and specialization to construct a hierarchy.

Stochastic Block Model Near the top of the hierarchy sits...

Stochastic Block Model Near the top of the hierarchy sits... T A Generative model that creates network based on a hidden partition σ, a density matrix ω P(A|σ, ω) ij P(Aij |σi , σj , ω) ij (ωσi σj )Aij Aij e−ωσi σj 2 1 3 ω

Stochastic Block Model To find communities we : maximize the likelihood w.r.t. to σ. L SBM P(A|σ, ω) ij (ωσi σj )Aij Aij e−ωσi σj

Stochastic Block Model To find communities we : maximize the likelihood w.r.t. to σ. L SBM P(A|σ, ω) ij (ωσi σj )Aij Aij e−ωσi σj Since argmax f (x) argmax log f (x), we can also use log P(σ|A, ω) ∼ ij [Aij log ωσi σj − ωσi σj ]

Stochastic Block Model To find communities we : maximize the likelihood w.r.t. to σ. L SBM P(A|σ, ω) ij (ωσi σj )Aij Aij e−ωσi σj Since argmax f (x) argmax log f (x), we can also use log P(σ|A, ω) ∼ ij [Aij log ωσi σj − ωσi σj ] But! That’s suspiciously close to : H(σ, A; f ) (ij) f (aij , σi , σj )

By restricting ω : we get a whole hierarchy!

By restricting ω : we get a whole hierarchy! Max. likelihood of the SISBM aij log ωσiσj − ωσiσj λij ωσiσj ≥ 0, λij ≥ 0 Max. likelihood of the SBM aij log ωσiσj − ωσiσj ωσiσj ≥ 0 Max. likelihood of the SIGMGM xσiσj aij + γλij λij ≥ 0, xσiσj ∈ {0, 1}, γ < 0 General modularities xσiσj aij + γλij λij ≥ 0, γ < 0 Modularities δσiσj aij + γλij λij ≥ 0, γ < 0 Max. likelihood of the GMGM xσiσj aij + γ xσiσj ∈ {0, 1}, γ < 0 Edge counts with quadratic size constraints xσiσj aij + γ xσiσj ∈ {0, 1}, γ < 0 Balanced cut δσiσj aij + γ γ < 0 Balanced coloring (1 − δσiσj ) aij + γ γ < 0 Core-periphery e.g.: δσi1 δσj 1 aij + γ γ < 0 *we actually need to use a more general SBM

The Stochastic Block Model is in that it is at the top of the hierarchy

Why does this matter? . S ! There is good code that will fit the SBM.

Why does this matter? . S ! There is good code that will fit the SBM. . S MPE ! You can see them as fitting a model -> statistically principled.

Why does this matter? . S ! There is good code that will fit the SBM. . S MPE ! You can see them as fitting a model -> statistically principled. . D MPE ? They are fitting ill-defined models.

Why does this matter? . S ! There is good code that will fit the SBM. . S MPE ! You can see them as fitting a model -> statistically principled. . D MPE ? They are fitting ill-defined models. . D ’ ! If algorithm A and B find the same partition, not necessarily a sign of consensus.

Why does this matter? . S ! There is good code that will fit the SBM. . S MPE ! You can see them as fitting a model -> statistically principled. . D MPE ? They are fitting ill-defined models. . D ’ ! If algorithm A and B find the same partition, not necessarily a sign of consensus. . T ! Trivial reduction for NP-hardness. Detectability is a general phenomenon. And more?

Further reading O ( ) J.-G. Young, . St-Onge and P. Desrosiers, Louis J. Dubé Phys. Rev. E, ( ) Inspiration / related work ( ) J. Reichardt and S. Bornholdt, Phys. Rev. E ( ) ( ) M.E.J Newman, Phys. Rev. E ( ) ( ) S. C. Olhede and P. J. Wolfe, PNAS , ( )

Take-home message Objective function = important part of MPE algorithms Organization : via equivalence and specialization Max. likelihood SBM is almost universal as MPE goes Reference : Phys. Rev. E, ( ) Software : github.com/jg-you/sbm_canonica _mcmc

Reference : Phys. Rev. E, ( ) arXiv : 1806.04214 jgyou@umich.edu jgyoung.ca @_jgyou