Markov Random Geometric Graph (MRGG): A Growth Model for Temporal Dynamic Networks

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Markov Random Geometric Graph (MRGG): A Growth Model for Temporal Dynamic Networks

Yohann de Castro, Quentin Duchemin

To cite this version:

Yohann de Castro, Quentin Duchemin. Markov Random Geometric Graph (MRGG): A Growth Model for Temporal Dynamic Networks. 2021. �hal-02865542v2�

A Growth Model for Temporal Dynamic Networks

Yohann De Castro¹ Quentin Duchemin²

Abstract

We introduce Markov Random Geometric Graphs (MRGGs), a growth model for temporal dynamic networks. It is based on a Markovian latent space dynamic: consecutive latent points are sampled on the Euclidean Sphere using an unknown Markov kernel; and two nodes are connected with a prob- ability depending on a unknown function of their latent geodesic distance.

More precisely, at each stamp-timekwe add a latent pointX_ksampled by jumping from the pre- vious oneX_k−1in a direction chosen uniformly Ykand with a lengthrkdrawn from an unknown distribution called thelatitude function. The con- nection probabilities between each pair of nodes are equal to theenvelope functionof the distance between these two latent points. We provide theo- retical guarantees for the non-parametric estima- tion of the latitude and the envelope functions.

We propose an efficient algorithm that achieves those non-parametric estimation tasks based on an ad-hoc Hierarchical Agglomerative Cluster- ing approach. As a by product, we show how MRGGs can be used to detect dependence struc- ture in growing graphs and to solve link prediction problems.

1. Introduction

In Random Geometric Graphs (RGG), nodes are sampled in- dependently in latent spaceR^d. Two nodes are connected if their distance is smaller than a threshold. A thorough prob- abilistic study of RGGs can be found in (Penrose,2003).

RGGs have been widely studied recently due to their abil- ity to provide a powerful modeling tool for networks with

1Institut Camille Jordan, ´Ecole Centrale de Lyon, Lyon, France

2LAMA, Univ Gustave Eiffel, Univ Paris Est Cr´eteil, CNRS F-77447 Marne-la-Vall´ee, France.. Correspondence to: ydc

<yohann.de-castro@ec-lyon.fr>, qd<quentin.duchemin@univ- eiffel.fr>.

spatial structure. We can mention applications in bioinfor- matics (Higham et al.,2008) or analysis of social media (Hoff et al.,2002). One main feature is to uncover hidden representation of nodes using latent space and to model interactions by relative positions between latent points.

Furthermore, nodes interactions may evolve with time. In some applications, this evolution is given by the arrival of new nodes as in online collection growth (Lo et al.,2017), online social network growth (Backstrom et al.,2006;Jin et al.,2001), or outbreak modeling (Ugander et al.,2012) for instance. The network is growing as more nodes are entering. Other time evolution modelings have been studied, we refer to (Rossetti and Cazabet,2018) for a review.

A natural extension of RGG consists in accounting this time evolution. In (D´ıaz et al.,2008), the expected length of connectivity and dis-connectivity periods of the Dynamic Random Geometric Graph is studied: each node choose at random an angle in [0,2π) and make a constant step size move in that direction. In (Staples,2009), a random walk model for RGG on the hypercube is studied where at each time step a vertex is either appended or deleted from the graph. Their model falls into the class of Geometric Markovian Random Graphs that are generally defined in (Clementi et al.,2009).

As far as we know, there is no extension of RGG to growth model for temporal dynamic networks. For the first time, in this paper, we consider a Markovian dynamic on the latent space where the new latent point is drawn with respect to the latest latent point and some Markov kernel to be estimated.

Estimation of graphon in RGGs: the Euclidean sphere case Random graphs with latent space can be defined us- ing agraphon, see (Lov´asz,2012). A graphon is a kernel function that defines edge distribution. In (Tang et al.,2013), Tang and al. prove that spectral method can recover the ma- trix formed by graphon evaluated at latent points up to an orthogonal transformation, assuming that graphon is a posi- tive definite kernel (PSD). Going further, algorithms have been designed to estimate graphons, as in (Klopp et al., 2017) which provide sharp rates for the Stochastic Block Model (SBM). Recently, the paper (Castro et al., 2020) provides a non-parametric algorithm to estimate RGGs on

Figure 1.Graphical model of the MRGG model: Markovian dy- namics on Euclidean sphere where we jump fromXkontoXk+1. TheYkencodes direction of jump whilerkencodes its distance, see (2).

Euclidean spheres, without PSD assumption.

We present here RGG on Euclidean sphere. Givennpoints X1, X2, . . . , Xnon the Euclidean sphereS^d−1, we set an edge between nodes i and j (where i, j ∈ [n], i 6= j) with independent probabilityp(hXi, Xji). The unknown functionp: [−1,1]→[0,1]is called theenvelope function.

This RGG is a graphon model with a symmetric kernelW given byW(x, y) =p(hx, yi). Once the latent points are given, independently draw the random undirected adjacency matrixAby

Ai,j∼ B(p(hXi, Xji)), i < j

with Bernoulli r.v. drawn independently (set zero on the diagonal and complete by symmetry), and set

T_n:= 1

n(p(hX_i, X_ji))_i,j∈[n] and Tˆ_n:= 1 nA, (1) We do not observe the latent point and we have to estimate the envelopepfromAonly. A standard strategy is to remark thatTˆnis a random perturbation ofTnand to dig intoTnto uncoverp.

One important feature of this model is that the interactions between nodes is depicted by a simple object: the enve- lope functionp. The envelope summarises how individuals connect each others given their latent positions. Standard examples (Bubeck et al.,2016) are given byp_τ(t) =1{t≥τ}

where one connects two points as soon as their geodesic distance is below some threshold. The non-parametric es- timation ofpis given by (Castro et al.,2020) where the authors assume that latent pointsXiare independently and uniformly distributed on the sphere, which will not be the case in the present paper.

A new growth model: the latent Markovian dynamic Consider RGGs where latent points are sampled with Markovian jumps, the Graphical Model under consider- ation can be found in Figure 1. Namely, we samplen pointsX₁, X₂, . . . , X_non the Euclidean sphereS^d−1using

(a) Envelope function (b) Latitude function

Figure 2.Non-parametric estimation of envelope and latitude func- tions using algorithms of Sections2and3. We built a graph of 1500nodes sampled on the sphereS² and using envelope and latitude (dot orange curves) defined in Section5by (10). The estimated envelope is thresholded to get a function in[0,1]and the estimated latitude function is normalized with integral1(plain blue lines).

a Markovian dynamic. We start by sampling randomlyX₁ onS^d−1. Then, for anyi∈ {2, . . . , n}, we sample

• a unit vectorY_i∈S^d−1uniformly, orthogonal toX_i−1.

• a realri∈[−1,1]encoding the distance betweenX_i−1 and Xi, see (3). ri is sampled from a distribution f_L : [−1,1]→[0,1], called thelatitude function.

thenXiis defined by

Xi =ri×X_i−1+ q

1−r²_i ×Yi. (2) This dynamic can be pictured as follows. Consider that X_i−1 is the north pole, then chose uniformly a direction (i.e., a longitude) and, in a independent manner, randomly move along the latitudes (the longitude being fixed by the previous step). The geodesic distanceγ_idrawn on the lati- tudes satisfies

γi= arccos(ri), (3) where random variabler_i=hXi, X_i−1ihas densityf_L(r_i).

The resulting model will be referred to as the Markov Ran- dom Geometric Graph (MRGG) and is described with Fig- ure1.

Temporal Dynamic Networks: MRGG estimation strat- egy Seldom growth models exist for temporal dynamic network modeling, see (Rossetti and Cazabet,2018) for a review. In our model, we add one node at a time making a Markovian jump from the previous latent position. It results in

the observation of(Ai,j)1≤j≤i−1at timeT =i , as pictured in Figure1. Namely, we observe how a new node connects to the previous ones. For such dynamic, we aim at estimating the model, namely envelopepand

respectively latitudef_L. These functions capture in a simple function onΩ = [−1,1]the range of interaction of nodes (represented byp) and respectively the dynamic of the jumps in latent space (represented byf_L), where, in abscissaΩ, valuesr = hXi, Xji near 1 corresponds to close point Xi'Xjwhile values close to−1corresponds to antipodal pointsXi' −Xj. These functions may be non-parametric.

From snapshots of the graph at different time steps, can we recover envelope and latitude functions?This paper proves that it is possible under mild conditions on the Markovian dynamic of the latent points and our approach is summed up with Figure3.

Fundamental result Spectral convergence ofTˆnunder Markovian dynamic, see Section2.1

⇓

Guarantee for the recovery of: Algorithm (a)envelopep, see (5) ↔ SCCHEi (b)latent distancesri, see (9) ↔ HEiC2019 Figure 3.Presentation of our method to recover the envelope and the latitude functions.

Define λ(Tn) := (λ1, . . . , λn) and resp. λ( ˆTn) :=

(ˆλ₁, . . . ,ˆλ_n) the spectrum of T_n and resp. Tˆ_n, see (1).

Building clusters fromλ( ˆTn), Algorithm1(SCCHEi) esti- mates the spectrum of envelopepwhile Algorithm3(Araya and De Castro,2019) (HEiC, see SectionEin Appendix) extractsdeigenvectors ofTˆnto uncover the Gram matrix of the latent positions. Both can then be used to estimate the unknown functions of our model (see Figure2).

Previous works Non-parametric estimation of RGGs on Euclidean sphere has been investigated in (Castro et al., 2020) with i.i.d. latent points. Estimation of latent point relative distances with HEiC Algorithm has been introduced in (Araya and De Castro,2019) under i.i.d. latent points assumption. Phase transitions on the detection of geome- try in RGGs (against Erd¨os R´enyi alternatives) has been investigated in (Bubeck et al.,2016).

For the first time, we introduce latitude function and non- parametric estimations of envelope and latitude using new results on kernel matrices concentration with dependent variables (see Appendix).

Outline Sections2and3present the estimation method with new theoretical results under Markovian dynamic.

These new results are random matrices operator norm con- trol and resp. U-statistics control under Markovian dynamic,

shown in the Appendix at SectionGand resp. SectionF. The envelopeadaptiveestimate is built from a size constrained clustering (Algorithm1) tuned by slope heuristic (6), and the latitude function estimate (see Section3.1) is derived from estimates of latent distancesri. Sections5and6inves- tigate synthetic data experiments¹. We propose heuristics to solve link prediction problems and to test for a Marko- vian dynamic. Our method can handle random graphs with logarithmic growth node degree (i.e., new comer at time T =nconnects toO(logn)previous nodes), referred to as relatively sparsemodels, see Section4.

Notations Consider a dimensiond≥3. Denote byk · k2

(resp.h·,·i) the Euclidean norm (resp. inner product) onR^d. Consider thed-dimensional sphereS^d−1 := {x ∈ R^d : kxk₂ = 1}and denote by σthe uniform distribution on S^d−1. For two real valued sequences(un)_n∈Nand(vn)_n∈N, denoteun =

n→∞O(vn)if there existk1 >0andn0 ∈N such that∀n > n₀,|u_n| ≤k₁|v_n|.

Given two sequences x, y of reals–completing finite se- quences by zeros–such thatP

ix²_i +y_i² <∞, we define the`2rearrangement distanceδ2(x, y)as

δ₂²(x, y) := inf

π∈S

X

i

(x_i−y_π(i))²,

whereSis the set of permutations with finite support. This distance is useful to compare two spectra.

2. Nonparametric estimation of the envelope function

One can associate withW(x, y) = p(hx, yi)the integral operatorTW : L²(S^d−1) → L²(S^d−1)such that for any g∈L²(S^d−1),

∀x∈S^d−1, (T^Wg)(x) = Z

S^d−1

g(y)p(hx, yi)dσ(y), wheredσis the Lebesgue measure onS^d−1. The operator TW is Hilbert-Schmidt and it has a countable number of bounded eigenvaluesλ^∗_k with zero as only accumulation point. The eigenfunctions ofT^W have the remarkable prop- erty that they do not depend onp(see (Dai,2013) Lemma 1.2.3): they are given by the real Spherical Harmonics. We denoteHlthe space of real Spherical Harmonics of degree lwith dimensiondland with orthonormal basis(Yl,j)_j∈[dl] where

dl:= dim(Hl) =







1 ifl= 0

d ifl= 1

l+d−1 l

− ^l+d−3_l−2

otherwise.

1Our code is available at https://github.com/quentin- duchemin/Markovian-random-geometric-graph.git

We define also for allR ∈N,R˜ :=PR

l=0d_l. We end up with the following spectral decomposition

p(hx, yi) =X

l≥0

p^∗_l X

1≤j≤dl

Yl,j(x)Yl,j(y) =X

k≥0

p^∗_kckG^β_k(hx, yi), (4) where λ(T^W) = {p^∗₀, p^∗₁, . . . , p^∗₁, . . . , p^∗_l, . . . , p^∗_l, . . .} meaning that each eigenvaluep^∗_l has multiplicitydl; and G^β_k is the Gegenbauer polynomial of degree k with pa- rameterβ := ^d−2₂ andck := ^2k+d−2_d−2 (see AppendixC).

Sincepis bounded, one hasp∈L²((−1,1), wβ)where the weight functionwβ is defined bywβ(t) := (1−t²)^β−12, and it can be decomposed asp ≡ P

k≥p^∗_kckG^β_k and the Gegenbauer polynomialsG^β_k are an orthogonal basis of wβ(t) := (1−t²)^β−12.

Weighted Sobolev space The spaceZ_w^s

β((−1,1))with regularity s > 0 is defined as the set of functionsg = P

k≥0g_k^∗c_kG^β_k ∈L²((−1,1), w_β)such that kgk^∗_Zs

wβ((−1,1)) :=

"_∞ X

l=0

dl|g_l^∗|²(1 + (l(l+ 2β))^s)

#^1/2

<∞.

2.1. Integral operator spectrum estimation with dependent variables

One key result is a new control ofU-statistics with latent Markov variables (see Section F) and it makes use of a Talagrand’s concentration inequality for Markov chains (see (Adamczak,2007)). This article follows the hypothesis made on the Markov chain(Xi)i≥1by (Adamczak,2007).

Namely, we referred to as “mild conditions” the assumption that the latitude function fL is bounded away from zero withkfLk∞<∞; and the first and the second regeneration times associated with thesplitchain to have sub-exponential tail. Those assumptions are fully described in sectionF.

Theorem1is a theoretical guarantee for a random matrix approximation of the spectrum of integral operator with dependentlatent variables. Theorem3in AppendixGgives explicitly the constants hidden in the big O below which depend on the spectral gap of the Markov chain(Xi)_i≥1. Theorem 1 Assume mild conditions on the Markov chain (Xi)_i≥1and assume the envelopephas regularitys >0.

Then, it holds Eh

δ₂²(λ(T^W), λ(Tn))∨δ²₂(λ(T^W), λ^Ropt( ˆTn))i

= O

n log²(n)

−_2s+d−1^2s ! ,

with λ^Ropt( ˆTn) = (ˆλ1, . . . ,λˆR˜_opt,0,0, . . .)and Ropt = b n/log²(n)2s+d−1¹

cwhereˆλ₁, . . . ,λˆ_n are the eigenval- ues ofTˆ_nsorted in decreasing order of magnitude.

Remark In Theorem1and Theorem2, note that we re- cover, up to alogfactor, theminimax rate of non-parametric estimationofs-regular functions on a space of (Riemannian) dimensiond−1. Even with i.i.d. latent variables, it is still an open question to know if this rate is the minimax rate of non-parametric estimation of RGGs.

Eq.(4) shows that one could use an approximation of (p^∗_k)k≥1to estimate the envelopepand Theorem1states we can recover(p^∗_k)k≥1up to a permutation. The problem of finding such a permutation is NP-hard and we introduce in the next section an efficient algorithm to fix this issue.

2.2. Size Constrained Clustering Algorithm

Note the spectrum ofT^W is given by (p^∗_l)_l≥0 where p^∗_l has multiplicity dl. In order to recover envelope p, we build clusters from eigenvalues ofTˆnwhile respecting the dimensiondlof each eigen-space ofT^W. In (Castro et al., 2020), an algorithm is proposed testing all permutations of {0, . . . , R}for a given maximal resolutionR. To bypass the high computational cost of such approach, we propose an efficient method based on the tree built fromHierarchical Agglomerative Clustering(HAC).

In the following, for anyν₁, . . . , ν_n ∈ R, we denote by HAC({ν1, , . . . , ν_n}, dc)the tree built by a HAC on the real valuesν₁, . . . , ν_n using the complete linkage functiond_c defined by∀A, B⊂R,d_c(A, B) = max_a∈Amax_b∈Bka−

bk₂. Algorithm1describes our approach.

Algorithm 1Size Constrained Clustering for Harmonic Eigenvalues (SCCHEi).

Data: ResolutionR, matrixTˆn=_n¹A, dimensions(dk)^R_k=0. 1: Letλˆ1, . . . ,λˆnbe the eigenvalues ofTˆnsorted in decreasing

order of magnitude.

2: SetP:={ˆλ1, . . . ,ˆλR˜}anddims= [d0, d1, . . . , dR].

3: whileAll eigenvalues inPare not clustereddo 4: tree←HAC(nonclustered eigenvalues inP,dc) 5: for d∈dims do

6: Search for a cluster of sizedintreeas close as possible to the root.

7: if such a cluster Cd exists then

Update(dims, tree,Cd, d).

8: end for

9: ford∈dimsdo

10: Search for the groupCintreewith a size larger thand and as close as possible tod.

11: if such a group exists then Update(dims, tree,C, d) elseGo to line3.

12: end for 13: end while

Return:Cd₀, . . . ,Cd_R,{λˆR+1˜ , . . . ,λˆn}

Algorithm 2Update(dims, tree,C, d).

1: Save the subsetC_dconsisting of thedeigenvalues inC with the largest absolute values.

2: Delete fromtreeall occurrences to eigenvalues inC_d and deletedfromdims.

2.3. Adaptation: Slope heuristic as model selection of Resolution

A data-driven choice of model sizeRcan be done byslope heuristic, see (Arlot,2019) for a nice review. One main idea of slope heuristic is to penalize the empirical risk by κpen( ˜R) and to calibrate κ > 0. If the sequence (pen( ˜R))R˜ is equivalent to the sequence of variances of the population risk of empirical risk minimizer (ERM) as model sizeR˜grows, then, penalizing the empirical risk (as done in (6)), one may ultimately uncover an empirical ver- sion of theU-shaped curve of the population risk. Hence, minimizing it, one builds a model sizeRˆthat balances be- tween bias (under-fittingregime) and variance (over-fitting regime). First, note that empirical risk is given by the intra- class variance below.

Definition 1 For any output(Cd0, . . . ,CdR,Λ) of the Al- gorithm SCCHEi, the thresholded intra-class variance is defined by

I_R:= 1 n







R

X

k=0

X

λ∈C_dk



λ− 1 d_k

X

λ⁰∈C_dk

λ⁰





2

+X

λ∈Λ

λ²





, and the estimations(ˆpk)k≥0of the eigenvalues(p^∗_k)k≥0is given by

∀k∈N, pˆ_k = ( ₁

dk

P

λ∈C_dkλ ifk∈ {0, . . . ,R}ˆ

0 otherwise.

(5) Second, as underlined in the proof of Theorem1(see The- orem3in the Appendix), the estimator’s variance of our estimator scales linearly inR.˜

Hence, we apply Algorithm SCCHEi forRvarying from 0 toRmax := max{R ≥ 0 : R˜ ≤ n}to compute the thresholded intra-class varianceIR(see Definition1) and given someκ >0, we select

R(κ)∈ arg min

R∈{0,...,Rmax}

nI_R+κ R˜ n o

. (6)

The hyper-parameterκcontrolling the bias-variance trade- off is set to2κ₀whereκ₀is the value ofκ >0leading to the “largest jump” of the functionκ7→R(κ). OnceRˆ:=

R(2κ₀)has been computed, we approximate the envelope functionpusing (5) (see (19) in Appendix for the closed

form). In AppendixD, we describe this slope heuristic on a concrete example and our results can be reproduced using the notebook in the Supplementary Material.

3. Nonparametric estimation of the latitude function

3.1. Our approach to estimate the latitude function in a nutshell

In Theorem 2(see below), we show that we are able to estimate consistently the pairwise distances encoded by the Gram matrixG^∗where

G^∗:= 1

n(hXi, Xji)_i,j∈[n].

Taking the diagonal just above the main diagonal (referred to assuperdiagonal) ofGˆ- an estimate of the matrixGto be specified - we get estimates of the i.i.d. random vari- ables(hX_i, X_i−1i)_2≤i≤n = (r_i)_2≤i≤n sampled from f_L. Using (ˆr_i)_2≤i≤n the superdiagonal ofnG, we can buildˆ a kernel density estimator of the latitude functionf_L. In the following, we describe the algorithm used to build our estimatorGˆwith theoretical guarantees.

3.2. Spectral gap condition and Gram matrix estimation

The Gegenbauer polynomial of degree one is defined by G^β₁(t) = 2βt, ∀t ∈ [−1,1]. As a consequence, us- ing theaddition theorem(see (Dai,2013, Lem.1.2.3 and Thm.1.2.6)), the Gram matrixG^∗is related to the Gegen- bauer polynomial of degree one. More precisely, for any i, j∈[n]it holds

G^∗_i,j = 1

2βnG^β₁(hXi, Xji) = 1 nd

d

X

k=1

Y1,k(Xi)Y1,k(Xj).

(7) Denoting V^∗ ∈ R^n×d the matrix with columnsv_k^∗ :=

√1

n(Y1,k(X1), . . . , Y1,k(Xn))fork∈[d], (7) becomes G^∗:=1

dV^∗(V^∗)^>.

We will prove that fornlarge enough there exists a matrix Vˆ ∈ R^n×d where each column is an eigenvector ofTˆ_n, such thatGˆ:= ¹_dVˆVˆ^>approximatesG^∗well, in the sense that the normkG^∗−Gkˆ Fconverges to0. To choose thed eigenvectors of the matrixTˆnthat we will use to build the matrixVˆ, we need the following spectral gap condition

∆^∗:= min

k∈N, k6=1|p^∗₁−p^∗_k|>0. (8) This condition will allow us to apply Davis-Kahan type inequalities.

Now, thanks to Theorem1, we know that the spectrum of the matrixTˆnconverges towards the spectrum of the inte- gral operatorT^W. Then, based on (7), one can naturally think that extracting thedeigenvectors of the matrixTˆn

related with the eigenvalues that converge towardsp^∗₁, we can approximate the Gram matrixG^∗of the latent positions.

Theorem2proves that the latter intuition is true with high probability under the spectral gap condition (8). The algo- rithm HEiC (Araya and De Castro,2019) (See SectionEfor a presentation) aims at identifying the above mentionedd eigenvectors of the matrixTˆ_nto build our estimate of the Gram matrixG^∗.

Theorem 2 Assume mild conditions on the Markov chain (Xi)_i≥1, assume∆^∗>0, and assume that graphonW has regularitys >0. We denoteVˆ ∈R^n×dthedeigenvectors of the matrixTˆnassociated with the eigenvalues returned by the algorithm HEiC and we defineGˆ := ¹_dVˆVˆ^>. Then fornlarge enough and for some constantD >0, it holds with probability at least1−5/n²,

kG^∗−Gkˆ F ≤D n

log²(n)

_2s+d−1^−s

. (9)

Based on Theorem2, we propose a kernel density approach to estimate the latitude function fL based on the super- diagonal of the matrixG, namelyˆ

ˆ

ri:=Gbi−1,i

i∈{2,...,n}. In the following, we denotefˆ_Lthis estimator.

4. Relatively Sparse Regime

Although this paper deals with the so-calleddenseregime (i.e. when the expected number of neighbors of each node scales linearly withn), our results may be generalized to the relatively sparsemodel connecting nodesiandjwith prob- abilityW(Xi, Xj) =ζnp(hXi, Xji)whereζn∈(0,1]sat- isfieslim inf

n ζnn/logn≥Z for some universal constant Z >0.

In the relatively sparse model, one can show following the proof of Theorem1that the resolution should be cho- sen as Rˆ = _nζ

n

1+ζnlog²n

_2s+d−1¹

. Specifying that λ^∗ = (p^∗₀, p^∗₁, . . . , p^∗₁, p^∗₂, . . .) and Tˆ_n = A/n, Theorem1 be-

comes for a graphon with regularitys >0

E

"

δ²₂ λ^∗,λ( ˆTn) ζn

!#

=O

nζn

1 +ζnlog²n

_2s+d−1^−2s ! .

Figure4illustrates the estimation of the latitude and the envelope functions in some relatively sparse regimes.

Figure 4.Results of our algorithms for graph of size2,000with functions of Eq.(10) and sparsity parameterζn= log^kn/n,k∈ {2,3,4}.

5. Experiments

We test our method usingd= 3and considering the follow- ing functions

p:x7→1x≥0, and fL:x7→

₁

2g(1−x; 2,2) ifx≥0

1

2g(1 +x; 2,2) otherwise , (10) whereg(·; 2,2)is the density of the beta distribution with pa- rameters(2,2). Figure5.(a)presents theδ₂-error between the spectra of the true envelope (resp. latitude) function and the estimated one when the size of the graph is increasing.

In Figure5.(b), we propose a visualization of the cluster- ing performed by SCCHEi withn = 1000and R = 4.

Blue crosses represent theR˜ eigenvalues ofTˆn with the largest magnitude, which are used to form clusters corre- sponding to the five-first spherical harmonic spaces. The red circles are the estimated eigenvalues(ˆpk)_0≤k≤4(plot- ted with multiplicity) defined from the clustering given by our algorithm SCCHEi (see (5)). Those results show that SCCHEi achieves a relevant clustering of the eigenvalues ofTˆnwhich allows us to recover the envelope function (see Figure2.(a)).

(a)δ2error on envelope and lat- itude functions.

(b) Clustering adjacency eigen- values (R= 4).

Figure 5.Non-parametric estimation of envelope and latitude us- ing algorithms described in Sections2and3. In (a), bars represent standard deviation ofδ2errors between true and estimated func- tions.

6. Applications

In this section, we apply the MRGG model to link prediction and hypothesis testing in order to demonstrate the usefulness

of our approach as well as the estimation procedure.

6.1. Markovian Dynamic Testing

As a first application of our model, we propose a hypothesis test to statistically distinguish between an independent sam- pling the latent positions and a Markovian dynamic. The null is then set toH0:nodes are independent and uniformly distributed on the sphere(i.e.,no Markovian dynamic). Our test is based on estimatefˆL of latitude and thus the null can be rephrased asH₀: f_L =f_L⁰wheref_L⁰is the latitude of uniform law, dynamic is then i.i.d. dynamic. Figure6 shows the power of a hypothesis test with level5%(Type I error). One can use anyblack-box goodness-of-fit testcom- paringfˆ_Ltof_L⁰, and we chooseχ²-test discretizing(−1,1) in70regular intervals. Rejection region is calibrated (i.e.,

Figure 6.Hypothesis testing.

threshold of theχ²-test here) byMonte Carlo simulations under the null. It allows us to control Type I error as de- picted by dotted blue line. We choose alternative given by Heaviside envelope and latitudef_Lof (10). We run our al- gorithm to estimate latitude from which we sample a batch to compute theχ²-test statistic. We see that for graphs of size larger than1,000, the rejection rate is almost1under the alternative (Type II error is almost zero), the test is very powerful.

6.2. Link Prediction

Suppose that we observe a graph withnnodes. Link pre- diction is the task that consists in estimating the probability of connection between a given node of the graph and the upcoming node.

6.2.1. BAYESLINKPREDICTION

We propose to show the usefulness of our model solving a link prediction problem. Let us recall that we do not es- timate the latent positions but only thepairwise distances (embedding task is not necessary for our purpose). De-

noting proj_X⊥

n(·) the orthogonal projection onto the or- thogonal complement ofSpan(Xn), the decomposition of hXi, Xn+1idefined by

hX_i, X_nihX_n, X_n+1i+p

1− hX_n, X_n+1i²

×p

1− hX_i, X_ni²h proj_X⊥ n(Xi) kproj_X⊥

n(Xi)k2

, Y_n+1i, (11) shows that latent distances are enough for link prediction.

Indeed, it can be achieved using a forward step on our Markovian dynamic, giving the posterior probability (see Definition2)ηi(D1:n)defined by

Z

[−1,1]²

p

hXi, Xnir+p

1−r²p

1− hXi, Xni²u

fL(r)drdu 2 . (12)

Definition 2 (Posterior probability function)

The posterior probability function η is defined for any latent pairwise distances D_1:n = (hXi, X_ji)1≤i,j≤n ∈ [−1,1]^n×nby

∀i∈[n], ηi(D1:n) =P(Ai,n+1= 1|D1:n), whereAi,n+1 ∼ B(p(hXi, Xn+1i)is a random variable that equals1if there is an edge between nodesiandn+ 1, and is zero otherwise.

We consider a classifierg(see Definition3) and an algorithm that, given some latent pairwise distancesD1:n, estimates Ai,n+1by putting an edge between nodesXiandXn+1if gi(D1:n)is1.

Definition 3 A classifier is a function which associates to any pairwise distancesD1:n= (hXi, Xji)1≤i,j≤n, a label (gi(D1:n))_i∈[n] ∈ {0,1}ⁿ.

The risk of this algorithm is as inbinary classification, R(g,D1:n) := 1

n

n

X

i=1

P(gi(D1:n)6=Ai,n+1|D1:n)

= 1 n

n

X

i=1

(1−ηi(D1:n))1g_i(D_1:n)=1+ηi(D1:n)1g_i(D_1:n)=0 ,

(13) where we used the independence between A_i,n+1 and g_i(D_1:n)conditionally onσ(D_1:n). Pushing further this analogy, we can define the classification error of some clas- sifier gbyL(g) = E[R(g,D_1:n)]. Proposition 1shows that the Bayes estimator - introduced in Definition4- is optimal for the risk defined in (13).

Definition 4 (Bayes estimator)

We keep the notations of Definition2. The Bayes estimator g^∗of(Ai,n+1)_1≤i≤nis defined by

∀i∈[n], g_i^∗(D_1:n) =

1 ifη_i(D_1:n)≥ ¹₂ 0 otherwise.

Proposition 1 (Optimality of the Bayes classifier for the riskR)

We keep the notations of Definitions2and4. For any classi- fierg, it holds for alli∈[n],

P(g_i(D_1:n)6=A_i,n+1|D_1:n)

−P(g^∗_i(D1:n)6=Ai,n+1|D1:n)

= 2

ηi(D1:n)−1 2

×E

1^gi(D_1:n)6=g_i^∗(D_1:n)|D1:n , which immediately implies that

R(g,D1:n)≥ R(g^∗,D1:n)and thereforeL(g)≥L(g^∗).

6.2.2. HEURISTIC FORLINKPREDICTION

One natural method to approximate the Bayes classifier from the previous section is to use theplug-in approach. This leads to the MRGG classifier introduced in Definition5.

Definition 5 (The MRGG classifier)

For anynand anyi∈[n], we defineηˆ_i(D_1:n)as Z

ˆ p

bri,nr+p

1−r²q

1−rb²_i,nu

fˆL(r)drdu

2 , (14)

wherepˆandfˆ_Ldenote respectively the estimate of the enve- lope function and the latitude function with our method and wherebr:=nG. The MRGG classifier is defined byb

∀i∈[n], g_i^{M RGG}(D_1:n) =

1 ifηˆ_i(D_1:n)≥ ¹₂ 0 otherwise.

To illustrate our approach we work with a graph of2,000 nodes with d = 3, Heaviside envelope and fL(r) =

1

2f(5,1)(^r+1₂ )wheref(5,1)is the pdf of the Beta distribution with parameter(5,1). Figure7shows that we are able to recover the probabilities of connection of the nodes already present in the graph with the coming nodeX_n+1. Using the decomposition ofhX_i, X_n+1igiven by (11), orange crosses are computed using (12). Green plus are computed similarly replacingpandf_Lby their estimationspˆandfˆ_Lfollowing (14). Blue stars are computed using (12) by replacingf_Lby

w_β

kwβk1 = ¹₂ (sinced= 3) which amounts to consider that the points are sampled uniformly on the sphere.

In Figure8, we compare the risk of therandomclassifier - whose guessgi(D1:n)is a Bernoulli random variable with parameter given by the ratio of edges compared to complete graph - with the risk of the MRGG classifier (see Defini- tion5). Figure8shows that the risk of the MRGG classifier is significantly smaller than the risk of random classifier, it is clearly better than random. Moreover, Figure8shows that the MRGG classifier gives similar results compared to the optimal Bayes classifier.

Figure 7.Link predictions between the future nodeXn+1and the 10first nodesX1, . . . , X10.

Figure 8.Comparison between the risk (defined in (13)) of the MRGG classifier, the random classifier and the risk of the optimal Bayes classifier.

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Supplementary Material

Markov Random Geometric Graph (MRGG):

A Growth Model for Temporal Dynamic Networks

Guidelines for the supplementary material SectionsAtoC: Basic definitions and Complements

In SectionAwe recall basic definitions on Markov chains which are required for SectionBwhere we describe some properties verified by the Markov chain(Xi)i≥1. SectionCprovides complementary results on the Harmonic Analysis on S^d−1which will be useful for our proofs.

SectionsDtoE: Algorithms and Experiments²

SectionDdescribes precisely the slope heuristic used to perform the adaptive selection of the model dimensionR. Section˜ E provides a complete description of the HEiC alogorithm used to extractd-eigenvectors of the adjacency matrix that will be used to estimate the Gram matrix of the latent positions.

SectionsFtoH: Proofs of theoretical results

Thereafter, we dig into the most theoretical part of the supplementary material. In SectionF, we provide a full description of the assumptions we made on the Markov chain(Xi)_i≥1and which we have been referring to so far bymild conditions.

SectionFis also dedicated to the presentation of a concentration result for a particular U-statistic of the Markov chain (Xi)i≥1 that is an essential element of the proof of Theorem1which is provided in SectionG. Finally, the proof of

Theorem2can be found in SectionH.

A. Definitions for general Markov chains

We consider a state spaceEand a sigma-algebraΣonEwhich is a standard Borel space. We denote by(Xi)i≥1a time homogeneous Markov chain on the state space(E,Σ)with transition kernelP.

A.1. Ergodic and reversible Markov chains

Definition 6 (Roberts and Rosenthal,2004, section 3.2) (φ-irreducible Markov chains)

The Markov chain(Xi)_i≥1is saidφ-irreducible if there exists a non-zeroσ-finite measureφonEsuch that for allA∈Σ withφ(A)>0, and for allx∈E, there exists a positive integern=n(x, A)such thatPⁿ(x, A)>0(wherePⁿ(x,·) denotes the distribution ofXn+1conditioned onX1=x).

Definition 7 (Roberts and Rosenthal,2004, section 3.2) (Aperiodic Markov chains)

The Markov chain (X_i)_i≥1 with invariant distributionπis aperiodic if there do not existm ≥ 2 and disjoint subsets A₁, . . . , A_m ⊂EwithP(x, A_i+1) = 1for allx∈A_i (1 ≤i≤m−1), andP(x, A₁) = 1for allx∈A_m, such that π(A₁)>0(and henceπ(A_i)>0for alli).

Definition 8 (Roberts and Rosenthal,2004, section 3.4) (Geometric ergodicity)

The Markov chain (Xi)_i≥1 is said geometrically ergodic if there exists an invariant distribution π, ρ ∈ (0,1) and C:E→[1,∞)such that

kPⁿ(x,·)−πkT V ≤C(x)ρⁿ, ∀n≥0, π−a.ex∈E, wherekµk_{T V} := sup_A∈Σ|µ(A)|.

Definition 9 (Roberts and Rosenthal,2004, section 3.3) (Uniform ergodicity)

The Markov chain(Xi)_i≥1is said uniformly ergodic if there exists an invariant distributionπand constants0< ρ <1and

2Our code is available athttps://github.com/quentin-duchemin/Markovian-random-geometric-graph.git

L >0such that

kPⁿ(x,·)−πkT V ≤Lρⁿ, ∀n≥0, π−a.ex∈E, wherekµkT V := sup_A∈Σ|µ(A)|.

Remark:A Markov chain geometrically or uniformly ergodic admits a unique invariant distribution and is aperiodic.

Definition 10 A Markov chain is said reversible if there exists a distributionπsatisfying π(dx)P(x, dy) =π(dy)P(y, dx).

A.2. Spectral gap

This section is largely inspired from (Fan et al.,2018). Let us consider that the Markov chain(X_i)_i≥1admits a unique invariant distributionπonS^d−1.

For any real-valued,Σ-measurable functionh:E→R, we defineπ(h) :=R

h(x)π(dx). The set L2(E,Σ, π) :={h:π(h²)<∞}

is a Hilbert space endowed with the inner product hh1, h2iπ=

Z

h1(x)h2(x)π(dx), ∀h1, h2∈ L²(E,Σ, π).

The map

k · kπ:h∈ L2(E,Σ, π)7→ khkπ=p hh, hiπ,

is a norm onL2(E,Σ, π).k · kπnaturally allows to define the norm of a linear operatorTonL2(E,Σ, π)as Nπ(T) = sup{kT hkπ :khkπ= 1}.

To each transition probability kernelP(x, B)withx∈EandB ∈Σinvariant with respect toπ, we can associate a bounded linear operatorh7→R

h(y)P(·, dy)onL2(E,Σ, π). Denoting this operatorP, we get P h(x) =

Z

h(y)P(x, dy), ∀x∈E, ∀h∈ L2(E,Σ, π).

LetL⁰₂(π) :={h∈ L2(E,Σ, π) : π(h) = 0}. We define the absolute spectral gap of a Markov operator.

Definition 11 (Spectral gap) A Markov operatorP reversible admits a spectral gap1−λif λ:= sup

kP hk_π khkπ

: h∈ L⁰₂(π), h6= 0

<1.

The next result provides a connection between spectral gap and geometric ergodicity for reversible Markov chains.

Proposition 2 (Ferr´e et al.,2012, section 2.3)

A uniformly ergodic Markov chain admits a spectral gap.

B. Properties of the Markov chain

Let P be the Markov operator of the Markov chain(X_i)_i≥1. By abuse of notation, we will also denoteP(x,·) the density of the measure P(x, dz)with respect todσ, the uniform measure on S^d−1. For any x, z ∈ S^d−1, we denote R^z_x∈R^d×da rotation matrix sendingxtoz(i.e.R^z_xx=z) and keepingSpan(x, z)^⊥fixed. In the following, we denote ed:= (0,0, . . . ,0,1)∈R^d.

B.1. Invariant distribution and reversibility for the Markov chain Reversibility of the Markov chain(Xi)i≥1.

Lemma 1 For allx, z∈S^d−1,P(x, z) =P(z, x) =P(e_d, R_z^edx).

Proof of Lemma1.

Using our model described in section2, we getX2=rX1+√

1−r²Y where conditionally onX1,Y is uniformly sampled onS(X1) := {q ∈ S^d−1 : hq, X1i = 0},and whererhas densityfL on[−1,1]. Let us consider a gaussian vector W ∼ N(0, Id). Using the Cochran’s theorem and Lemma2, we know that conditionally onX1, the random variable

W−hW,X1iX1

kW−hW,X1iX1k2 is distributed uniformly onS(X1).

Lemma 2 LetW ∼ N(0, Id). Then, _kW^W_k

2 is distributed uniformly on the sphereS^d−1. In the following, we denote=^L the equality in distribution sense. We have conditionally onX1

R^e_X^d

1

W − hW, X1iX1

kW− hW, X₁iX₁k₂ =

Wˆ − hW , eˆ died

kWˆ − hW , eˆ diedk2

,

whereWˆ =R^e_X^d

1W ∼ N(0, I_d). Using Cochran’s theorem, we know thatWˆ − hW , eˆ _die_dis a centered normal vector with covariance matrix the orthographic projection matrix onto the spaceSpan(e_d)^⊥, leading to

Wˆ − hW , eˆ died

=L

Y 0

,

whereY ∼ N(0, I_d−1). Using Lemma2, we conclude that conditionally onX1, the random variable_kW^W_−hW,X^−hW,X1iX1

1iX₁k₂ is distributed uniformly onS(X₁)(because the distribution ofY is invariant by rotation).

We deduce that X2

=L rX1+p

1−r² W− hW, X1iX1

kW − hW, X₁iX₁k₂

=L rX1+p

1−r² R^X_X¹

2W⁰− hR^X_X¹

2W⁰, X₁iX₁ kR_X^X1

2W⁰− hR^X_X¹

2W⁰, X₁iX₁k₂, whereW⁰:=R^X_X²

1W. Note thatW⁰∈R^dis also a standard centered gaussian vector because this distribution is invariant by rotation. SincehR^X_X¹

2W⁰, X1i=hW⁰, X2iandkR^X_X¹

2qk2=kqk2, ∀q∈S^d−1, we deduce that X₂−rX₁=^L R^X_X¹

2

p

1−r² W⁰− hW⁰, X₂iX₂ kW⁰− hW⁰, X2iX2k2

. (15)

R^X_X²

1 is the rotation that sendsX1toX2keeping the other dimensions fixed. Let us denotea1:=X1,a2:= _kX^X2−rX1

2−rX1k2 and complete the linearly independent family(a1, a2)in an orthonormal basis ofR^dgiven bya:= (a1, a2, . . . , ad). Then, the matrix ofR_X^X2

1 in the basisais





r −√

1−r² 0^>_d−2

√1−r² r 0^>_d−2 0_d−2 0_d−2 I_d−2



.

We deduce that

R^X_X¹

2

−1

(X2−rX1) =R^X_X²

1(X2−rX1)

=kX₂−rX₁k₂R^X_X²

1

X₂−rX₁ kX2−rX1k2

=kX₂−rX₁k₂R^X_X²

1a₂

=kX₂−rX₁k₂h

−p

1−r²a₁+ra₂i

=−p

1−r²kX2−rX1k2X1+rX2−r²X1

=−(1−r²)X1+rX2−r²X1

=−X1+rX2. Going back to (15), we deduce that

X1

=L rX2+p 1−r²

W˜ − hW , X˜ 2iX2

kW˜ − hW , X˜ 2iX2k2

, (16)

whereW˜ =−W⁰is also a standard centered gaussian vector inR^d. Thus, we proved the first equality of Lemma1. Based on (16) we have,

R^e_X^d

2X₁=^L rR^e_X^d

2X₂+p

1−r²R_X^ed

2

W˜ − hW , X˜ ₂iR^e_X^d

2X₂ kW˜ − hW , X˜ ₂iX₂k₂

=re_d+p

1−r² R_X^ed

2

W˜ − hR_X^ed

2

W , e˜ died

kR^e_X^d

2

W˜ − hR^e_X^d

2

W , e˜ diedk2

, which proves thatP(ed, R^e_x^d₂x1) =P(x2, x1)for anyx1, x1∈S^d−1becauseR^e_X^d

2

W˜ is again a standard centered gaussian vector inR^d.

Invariant distribution of the Markov chain.

Proposition 3 The uniform distribution on the sphereS^d−1is an invariant distribution of the Marokv chain(X_i)_i≥1. Proof of Proposition3.

Let us considerz∈S^d−1. We denotedσ≡dσ_dthe Lebesgue measure onS^d−1anddσ_d−1the Lebesgue measure onS^d−2. Using (Dai,2013, Section 1.1), it holdsb_d:=R

S^d−1dσ= _Γ(d/2)^2πd/2. We have Z

x∈S^d−1

P(x, z)dσ(x) bd

= Z

x∈S^d−1

P(e_d, R^e_z^dx)dσ(x) bd

(Using Lemma1)

= Z

x∈S^d−1

P(ed, x)dσ(x)

b_d (Using the change of variablex7→R_z^edx)

= Z π

0

Z

S^d−2

P

ed, ξsinθ

cosθ

(sinθ)^d−2dθdσ_d−1(ξ)

b_d (Using (Dai,2013, Eq.(1.5.4) Section 1.5))

= Z 1

−1

Z

S^d−2

P

ed, ξ√

1−r² r

(1−r²)^d−32 drdσd−1(ξ) bd

= Z 1

−1

f_L(r)dr× Z

S^d−2

1 b_d−1

dσ_d−1 b_d = 1

b_d,

which proves that the uniform distribution on the sphere is an invariant distribution of the Markov chain.