Network analysis with the W -graph model

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Network analysis with the

W

-graph model

(via the Stochastic Block Model)

S. Robin

Joint work with P. Latouche and S. Ouadah INRA / AgroParisTech

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Outline

1 _{Modeling heterogeneity in interaction networks}

2 Statistical inference of latent space models (focus on SBM)

3 From SBM toW-graph: Averaging models

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Modeling heterogeneity in interaction networks

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Modeling heterogeneity in interaction networks Heterogeneity in biological networks

Heterogeneity in biological networks

Biological networks describe interactions between entities: genes, proteins, individuals, species...

Observed networks display heterogeneous topologies, that one would like to decipher and better understand.

Dolphine social network.

[Newman and Girvan (2004)]

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Heterogeneity in biological networks

Biological networks describe interactions between entities: genes, proteins, individuals, species...

Observed networks display heterogeneous topologies, that one would like to decipher and better understand.

Dolphine social network.

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Heterogeneous means ...

... not homogeneous,that is: different from an Erd¨os-Renyi (ER) graph.

Erd¨os-Renyi random graphG(n,p): Considernnodes, node pairs 1≤i<j≤n

are independently connected with same probabilityp:

(Yij) iid, Yij ∼ B(p).

Very intensively studied. Fits very few real-life networks.

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Modeling heterogeneity in interaction networks Latent space models

Latent space models

Latent variablesallow to capture some underlying structure of a network (see review[Matias and R. (2014)]).

General setting for binary graphs. [Bollob´aset al.(2007)]:

A latent (unobserved) variableZi is associated with each node:

{Zi}iid ∼π

EdgesYij =I{i∼j} are independent conditionally to theZi’s:

{Yij} independent|{Zi}: Pr{Yij= 1}=γ(Zi,Zj)

We focus here on model approaches, in contrast with, e.g.

Graph clustering[Girvan and Newman (2002)],[Newman (2004)];

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Latent space models

{Zi}iid ∼π

Graph clustering[Girvan and Newman (2002)],[Newman (2004)];

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Latent space models

{Zi}iid ∼π

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Latent space models

State-space model: principle.

Considernnodes (i= 1..n);

Zi = unobserved position of nodei,

e.g.

{Zi}iid ∼ N(0,I)

Edge{Yij}independent given{Zi},

e.g.

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Latent space models

e.g.

{Zi}iid ∼ N(0,I)

e.g.

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Latent space models

e.g.

{Zi}iid ∼ N(0,I)

e.g.

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Latent space models

e.g.

{Zi}iid ∼ N(0,I)

e.g.

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Latent space models

e.g.

{Zi}iid ∼ N(0,I)

e.g. Pr{Yij= 1}=γ(Zi,Zj). Y =        0 1 1 0 1 . . . 0 0 1 0 1 . . . 0 0 0 0 0 . . . 0 0 0 0 1 . . . 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . ..       

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A variety of state-space models

Latent position models.

[Hoffet al. (2002)]: Zi∈Rd, logitγ(z,z0) =a− |z−z0| [Handcocket al. (2007)]: Zi ∼ X k pkNd(µk, σk2I) [Daudinet al. (2010)]: Zi∈ SK, γ(z,z0) = X k,` zkz`0γk`

In this talk, focus on

the Stochastic Block Model (SBM) and

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A variety of state-space models

Latent position models.

[Hoffet al. (2002)]: Zi∈Rd, logitγ(z,z0) =a− |z−z0| [Handcocket al. (2007)]: Zi ∼ X k pkNd(µk, σk2I) [Daudinet al. (2010)]: Zi∈ SK, γ(z,z0) = X k,` zkz`0γk`

In this talk, focus on

the Stochastic Block Model (SBM) and

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Stochastic Block Model (SBM)

A mixture model for random graphs.

[Nowicki and Snijders (2001)]

Considernnodes (i = 1..n);

Zi= unobserved label of nodei:

{Zi} iid ∼ M(1;π)

π= (π1, ...πK);

EdgeYij depends on the labels:

{Yij} independent given{Zi},

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Stochastic Block Model (SBM)

π= (π1, ...πK);

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π= (π1, ...πK);

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Stochastic Block Model (SBM)

π= (π1, ...πK);

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π= (π1, ...πK);

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W

-graph model

Latent variables: (Zi) iid ∼ U[0,1], Graphon functionγ: γ(z,z0) : [0,1]2→[0,1] Edges: Pr{Yij = 1}=γ(Zi,Zj) Graphon functionγ(z,z0)

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Interpreting the graphon function

The graphon function provides a global picture of the network’s topology.

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Few words about the

W

-graph

Probabilistic point of view.

W-graph have been mostly studied in the probability literature: [Lov´asz and

Szegedy (2006)],[Diaconis and Janson (2008)]

Motif (sub-graph) frequencies are invariant characteristics of aW-graph.

Intrinsic un-identifiability of the graphon functionγis often overcome by

imposing thatu7→R

γ(u,v) dv is monotonous increasing.

Statistical point of view.

Not much attention has been paid to its inference until recently: [Airoldiet al.

(2013)],[Chatterjee et al. (2014)],[Olhede and Wolfe (2014)], ...

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Few words about the

W

-graph

Probabilistic point of view.

W-graph have been mostly studied in the probability literature: [Lov´asz and

Szegedy (2006)],[Diaconis and Janson (2008)]

Motif (sub-graph) frequencies are invariant characteristics of aW-graph.

Intrinsic un-identifiability of the graphon functionγis often overcome by

imposing thatu7→R

γ(u,v) dv is monotonous increasing.

Statistical point of view.

Not much attention has been paid to its inference until recently: [Airoldiet al.

(2013)],[Chatterjee et al. (2014)],[Olhede and Wolfe (2014)], ...

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Modeling heterogeneity in interaction networks Some generalizations of latent space graph models

Some generalizations of latent space graph models

Latent space models can be extended in various directions.

Weighted or directed networks. Edges may have values: count, real,{0,+,−,±}, ... Latent space model can be adapted as

Yij|Zi,Zj ∼ F(γ(Zi,Zj))

whereF is can be any distribution: Poisson, normal, multinomial, etc.

Accounting for covariates. Latent space model can also accommodate for covariates, via a regression term:

Yij|Zi,Zj ∼ F(γ(Zi,Zj) +xij0β)

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Some generalizations of latent space graph models

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Some generalizations of latent space graph models

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Statistical inference of latent space models

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Statistical inference of latent space models Incomplete data models

Incomplete data models

Aim. Based on the observed networkY = (Yij), one want typically to infer

the parameters

θ= (π, γ)

the hidden states

Z = (Zi)

State space models belong to the class of incomplete data models as

the edges (Yij) are observed,

the latent positions (or status) (Zi) are not,

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Incomplete data models

Aim. Based on the observed networkY = (Yij), one want typically to infer

the parameters

θ= (π, γ)

the hidden states

Z = (Zi)

State space models belong to the class of incomplete data models as

the edges (Yij) are observed,

the latent positions (or status) (Zi) are not,

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Frequentist or Bayesian inference

Frequentist inference. θis fixed andZ is random. The aim is then to

provide an estimateθbofθ,

provide the conditional distributionPθ(Z|Y) (for classification purposes and

as a side product of the inference).

Bayesian inference. BothθandZ are random. The aim is then to

provide the joint conditional distributionP(θ,Z|Y).

Whatever the approach, we have to deal with conditional distributions:

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Frequentist or Bayesian inference

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Frequentist or Bayesian inference

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Conditional distributions (1/2)

Graphical modelsdescribe the conditional independences between the random

vari-ables from a model[Lauritzen (1996)].

Frequentist setting:

iidZi’s,

P(Yij|Zi,Zj),

P(Zi,Zj|Y): graph moralization,

this holds for each pair (i,j),

Conditional distribution. The dependency graph of Z given Y is a clique.

→No factorization can be hoped (unlike for HMM).

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Conditional distributions (1/2)

iidZi’s,

P(Yij|Zi,Zj),

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Conditional distributions (1/2)

iidZi’s,

P(Yij|Zi,Zj),

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Conditional distributions (1/2)

iidZi’s,

P(Yij|Zi,Zj),

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Conditional distributions (1/2)

iidZi’s,

P(Yij|Zi,Zj),

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Conditional distributions (1/2)

iidZi’s,

P(Yij|Zi,Zj),

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Conditional distributions (2/2)

Bayesian perspective. Things get

worst becauseθ = (π, γ) is also

random. Model: P(θ) P(Z|π) P(Y|γ,Z) P(θ,Z|Y) is even more involved .

Both frequentist and Bayesian inference require the calculation of conditional distributions that can not be computed.

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Conditional distributions (2/2)

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Conditional distributions (2/2)

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Conditional distributions (2/2)

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Conditional distributions (2/2)

random. Model: P(θ) P(Z|π) P(Y|γ,Z) P(θ,Z|Y) is even more involved.

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Conditional distributions (2/2)

random. Model: P(θ) P(Z|π) P(Y|γ,Z) P(θ,Z|Y) is even more involved.

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Statistical inference of latent space models Variational (Bayes) inference

Variational (Bayes) inference

Variational approximationsaim at replacing an intractable exact distributionP

with a tractable approximate distributionP. Typically:e

Pθ(Z|Y) ≈ Y i e Pθ,Y(Zi) P(θ,Z|Y) ≈ PeY(θ)×PeY(Z) P(θ,Z|Y) ≈ PeY(θ)× Y i e PY(Zi)

Popular strategy: minimize the K¨ullback-Leibler divergence betweenPe andP:

minKL[P(Ze )||Pθ(Z|Y)] or minKL[P(θ,e Z)||P(θ,Z|Y)]

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VBEM inference for SBM:

E. coli

’s operon network

[Picardet al. (2009)]

Meta-graph representation.

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E. coli

’s operon network

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E. coli

’s operon network

[Picardet al. (2009)]

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Accuracy of VBEM estimates for SBM: Simulation study

Credibility intervals: π1: +,γ11: 4, γ12: ◦, γ22: •

Width of the posterior credibility intervals. π1,γ11,γ12,γ22

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Accuracy of VBEM estimates for SBM: Simulation study

Credibility intervals: π1: +,γ11: 4, γ12: ◦, γ22: •

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First half summary

Latent space graph models are useful to describe network heterogeneity. Their statistical inference raises some specific issues.

Variational approximations help to circumvent these issues.

And also

Theoretical justifications of these approximations exist for SBM:[Celisseet al.

(2012)],[Mariadassou and Matias (2014)]

VEM and VBEM algorithms have been specifically developed for SBM:

[Daudinet al. (2008)],[Latoucheet al. (2012)]

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First half summary

Latent space graph models are useful to describe network heterogeneity. Their statistical inference raises some specific issues.

Variational approximations help to circumvent these issues.

And also

Theoretical justifications of these approximations exist for SBM:[Celisseet al.

(2012)],[Mariadassou and Matias (2014)]

VEM and VBEM algorithms have been specifically developed for SBM:

[Daudinet al. (2008)], [Latoucheet al. (2012)]

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From SBM toW-graph: Averaging models

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From SBM toW-graph: Averaging models SBM as aW-graph model

SBM as a

W

-graph model

Latent variables:

(Zi) iid ∼ M(1, π)

Blockwise constant graphon: γ(z,z0) =γk`

Edges:

Pr{Yij = 1}=γ(Zi,Zj)

Graphon functionγSBM

K (z,z0)

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From SBM toW-graph: Averaging models SBM as aW-graph model

Variational Bayes estimation of

γ

(

z

,

z

0

)

VBEM inferenceprovides the approximate posteriors:

(π|Y) ≈ Dir(π∗)

(γk`|Y) ≈ Beta(γk0`∗, γk1∗`)

Estimate ofγ(u,v). Due

to the uncertainty of theπk,

the posterior mean ofγSBM

K

is smooth

(Explicit integration using [Gouda and

Sz´antai (2010)])

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From SBM toW-graph: Averaging models Bayesian model averaging

Bayesian model averaging

Bayesian model averaging (BMA).Consider a series of models 1, . . . ,K, . . . in

which a certain function of the parameterf(θ) can always be defined.

Bayesian inference within each modelK provides the posterior

P(θ|K,Y) → P(f(θ)|K,Y).

BMA[Hoetinget al.(1999)]relies on the marginal posterior off(θ):

P(f(θ)|Y) =X

K

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Bayesian model averaging

P(θ|K,Y) → P(f(θ)|K,Y).

P(f(θ)|Y) =X

K

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Bayesian model averaging

P(θ|K,Y) → P(f(θ)|K,Y).

P(f(θ)|Y) =X

K

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Variational Bayes model averaging

Pushing it further: Consider the modelK as an additional hidden variable:

P(Z, θ,K|Y) ≈ P(Z, θ,e K)

:= P(Ze |K)×P(θe |K)×P(Ke )

Note that no additional independence assumption is needed.

Variational Bayes model averaging (VBMA).The optimal1approximation of P(K|Y) satisfies[Volantet al. (2012)]:

e

P(K)∝P(K)elogP(Y|K)−KL(K)=P(K|Y)e−KL(K)

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Variational Bayes model averaging

Pushing it further: Consider the modelK as an additional hidden variable:

P(Z, θ,K|Y) ≈ P(Z, θ,e K)

:= P(Ze |K)×P(θe |K)×P(Ke )

Note that no additional independence assumption is needed.

Variational Bayes model averaging (VBMA).The optimal1 approximation of P(K|Y) satisfies[Volantet al. (2012)]:

e

P(K)∝P(K)elogP(Y|K)−KL(K)=P(K|Y)e−KL(K)

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From SBM toW-graph: Averaging models Inferring the graphon function

Inferring the graphon function

Model averaging: There is no ’trueK’ in theW-graph model.

Apply VBMA recipe toγ(z,z0). ForK = 1..Kmax, fit an SBM model via VBEM and compute

b

γ_KSBM(z,z0) =Ee[γC(z),C(z0₎|Y,K].

Then perform model averaging as

b γ(z,z0) =_Ee[γC(z),C(z0₎|Y] = X K e P(K)_bγKSBM(z,z0), [Latouche and R. (2013)].

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Inferring the graphon function

Model averaging: There is no ’trueK’ in theW-graph model.

Apply VBMA recipe toγ(z,z0). ForK = 1..Kmax, fit an SBM model via VBEM and compute

b

γ_KSBM(z,z0) =Ee[γC(z),C(z0₎|Y,K].

Then perform model averaging as

b γ(z,z0) =Ee[γC(z),C(z0₎|Y] = X K e P(K)_bγKSBM(z,z0), [Latouche and R. (2013)].

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PPI network

Like many PPI networks,E. coli’s network is highly concentrated around few

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PPI network

Like many PPI networks,E. coli’s network is highly concentrated around few

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Ecological network between fungal species

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Ecological network between fungal species

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Brain network

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Brain network

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Blog network (non-biological)

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Blog network (non-biological)

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Goodness-of-fit

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Goodness-of-fit Motifs frequency

Motifs frequency

Network motifs have a biological (or sociological) interpretation in terms of building blocks of the global network

→Triangles = ’friends of my friends are my friends’.

Latent space graph models only describe binary interactions, conditional on the latent positions

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Moments of motif counts

Moments under SBM:The first moments_EN(m),_VN(m) of the count are known

for exchangeable graph models (incl. SBM)[Picardet al. (2008)]:

ESBMN(m)∝µSBM(m) =:f(θSBM)

whereµSBM(m) is the motif occurrence probability under SBM.

Moments underW-graph: Motif probability under theW -graph can be estimated as b µ(m) =X k e P(K)_Ee(µSBM(m)|X,K)

Estimates ofEWN(m) andVWN(m) can be derived accordingly[Latouche and R.

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Network frequencies in the blog network

Motif Count Mean Std. dev.

(×103) (×103) (×103) 29.7 39.7 8.3 3.8 4.6 1.3 608.7 968.3 336.8 279.8 428.9 154.0 47.4 74.5 35.1 270.5 397.0 177.0 62.1 87.8 47.4 6.5 8.8 5.4

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Goodness-of-fit ’Residual graphon’

Covariates: Tree interaction (valued) network

Data: n= 51 tree species,

Yij= number of shared parasites

[Vacheret al. (2008)].

SBM:GivenZi =k,Zj=`,

Yij∼ P(eγk`),

γk`= log-mean number of shared

parasites.

Results: ICL selects K = 7 groups that are partly related with phylums. eγbk` T1 T2 T3 T4 T5 T6 T7 T1 14.46 4.19 5.99 7.67 2.44 0.13 1.43 T2 14.13 0.68 2.79 4.84 0.53 1.54 T3 3.19 4.10 0.66 0.02 0.69 T4 7.42 2.57 0.04 1.05 T5 3.64 0.23 0.83 T6 0.04 0.06 T7 0.27 b πk 7.8 7.8 13.7 13.7 15.7 19.6 21.6

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Covariates: Tree interaction (valued) network

Yij∼ P(eγk`),

parasites.

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Yij∼ P(eγk`),

parasites.

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Covariates: Tree interaction (valued) network

Yij∼ P(eγk`),

parasites.

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Yij∼ P(eγk`),

parasites.

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Accounting for the taxonomic distance

Model: xij = distance(i,j) Yij∼ P(eγk`+βxij), [Mariadassouet al. (2010)]. Results: βb=−0.317. →forx = 3.82, eβbx ₌_.298

→ The mean number of shared

parasites decreases with taxo-nomic distance. eλb_k_` _T’1 _T’2 _T’3 _T’4 T’1 0.75 2.46 0.40 3.77 T’2 4.30 0.52 8.77 T’3 0.080 1.05 T’4 14.22 b π_k 17.7 21.5 23.5 37.3 b β -0.317

→Groups are no longer associated with the phylogenetic structure.

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Accounting for the taxonomic distance

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Accounting for the taxonomic distance

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Accounting for the taxonomic distance

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Accounting for the taxonomic distance

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’Residual’ graphon

A simple graph model with covariates. When edge covariatesxij are available,

simply fit a logistic regression[Pattison and Robins (2007)]:

(Yij) independent Yij ∼ B(pij)    logitpij =xij0β.

Introducing a residual term. To assess the fit of the model, simply add a residual graphon-like term: (Zi) iidU[0,1] Yij|Zi,Zj∼ B(pij)    logitpij=xij0β+γ(Zi,Zj).

→A VBEM algorithm can be designed to getP(β, θ,e Z)≈P(β, θ,Z|Y):

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’Residual’ graphon

A simple graph model with covariates. When edge covariatesxij are available,

simply fit a logistic regression[Pattison and Robins (2007)]:

(Yij) independent Yij ∼ B(pij)    logitpij =xij0β.

Introducing a residual term. To assess the fit of the model, simply add a residual graphon-like term: (Zi) iidU[0,1] Yij|Zi,Zj ∼ B(pij)    logitpij =xij0β+γ(Zi,Zj).

→A VBEM algorithm can be designed to getP(β, θ,e Z)≈P(β, θ,Z|Y):

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Tree network

Binary version: Links between tree species if they host at least one common fungal parasite.

Regression: covariates = genetic dis-tance, taxonomic disdis-tance, geographic distance

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Tree network

Binary version: Links between tree species if they host at least one common fungal parasite.

Regression: covariates = genetic dis-tance, taxonomic disdis-tance, geographic distance

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Blog network

Blog network: Already shown.

Regression: covariates = same political party, pair includes a journalist

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Blog network

Blog network: Already shown.

Regression: covariates = same political party, pair includes a journalist

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Conclusion & future work

Some conclusions.

The graphon provides a representation of the network topology It can be estimated using variational Bayes inference

→R packages ’mixer’ and ’blockmodels’

It can be combined with covariates as a residual term

Future work.

Formal goodness-of-fit test

Quality of variational Bayes estimates in SBM with covariates

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Conclusion & future work

Some conclusions.

The graphon provides a representation of the network topology It can be estimated using variational Bayes inference

→R packages ’mixer’ and ’blockmodels’

It can be combined with covariates as a residual term

Future work.

Formal goodness-of-fit test

Quality of variational Bayes estimates in SBM with covariates

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