TSE‐652

“Howtocalculatethebarycenterofaweightedgraph”

SébastienGadat,IoanaGavraandLaurentRisser

May2016

How to calculate the barycenter of a weighted graph

Sebastien GadatToulouse School of Economics, Universite Toulouse 1 Capitole sebastien.gadat@math.univ-toulouse.fr,

http://perso.math.univ-toulouse.fr/gadat/

Ioana GavraInstitut de Mathematiques de Toulouse, Universite Toulouse 3 Paul Sabatier Ioana.Gavra@math.univ-toulouse.fr,

Laurent RisserInstitut de Mathematiques de Toulouse, CNRS lrisser@math.univ-toulouse.fr, http://laurent.risser.free.fr/

Discrete structures like graphs make it possible to naturally and flexibly model complex phenomena. Sincegraphs that represent various types of information are increasingly available today, their analysis has becomea popular subject of research. The graphs studied in the field of data science at this time generally havea large number of nodes that are not fairly weighted and connected to each other, translating a structuralspecification of the data. Yet, even an algorithm for locating the average position in graphs is lackingalthough this knowledge would be of primary interest for statistical or representation problems. In this work,we develop a stochastic algorithm for finding the Frechet mean of weighted undirected metric graphs. Thismethod relies on a noisy simulated annealing algorithm dealt with using homogenization. We then illustrateour algorithm with two examples (subgraphs of a social network and of a collaboration and citation network).

Key words : metric graphs; Markov process; simulated annealing; homogeneizationMSC2000 subject classification : 60J20,60J60,90B15OR/MS subject classification : Primary: Probability - Markov processes; secondary: Networks/graphs -

Traveling salesmanHistory :

1. Introduction.Numerous open questions in a very wide variety of scientific domains involve complex discretestructures that are easily modeled by graphs. The nature of these graphs may be weighted or not,directed or not, observed online or by using batch processing, each time implying new problemsand sometimes leading to difficult mathematical or numerical questions. Graphs are the subject ofperhaps one of the most impressive growing bodies of literature dealing with potential applicationsin statistical or quantum physics (see, e.g., [21]) economics (dynamics in economy structured asnetworks), biology (regulatory networks of genes, neural networks), informatics (Web understand-ing and representation), social sciences (dynamics in social networks, analysis of citation graphs).We refer to [37] and [43] for recent communications on the theoretical aspects of random graphmodels, questions in the field of statistics and graphical models, and related numerical algorithms.In [34], the authors have developed an overview of numerous possible applications using graphsand networks in the fields of industrial organization and economics. Additional applications, detailsand references in the field of machine learning may also be found in [28].

In the meantime, the nature of the mathematical questions raised by the models that involvenetworks is very extensive and may concern geometry, statistics, algorithms or dynamical evolutionover the network, to name a few. For example, we can be interested in the definition of suitablerandom graph models that make it possible to detect specific shape phenomena observed at dif-ferent scales and frameworks (the small world networks of [47], the existence of a giant connectedcomponent for a specific range of parameters in the Erdos-Renyi random graph model [20]). A com-plete survey may be found in [39]. Another important field of investigation is dedicated to graphvisualization (see some popular methods in [45] and [35], for example). In statistics, a popular topic

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deals with the estimation of a natural clustering when the graph follows a specific random graphmodel (see, among others, the recent contribution [36] that proposes an optimal estimator for thestochastic block model and smooth graphons). Other approaches rely on efficient algorithms thatanalyze the spectral properties of adjacency matrices representing the networks (see, e.g, [6]). Afinal important field of interest deals with the evolution of a dynamical system defined througha discrete graph structure: this is, for instance, the question raised by gossip models that maydescribe belief evolution over a social network (see [1] and the references therein).

We address a problem here that may be considered as very simple at first glance: we aim todefine and estimate the barycenter of weighted graphs. Hence, this problem involves questionsthat straddle the area of statistics and the geometry of graphs. Surprisingly, as far as we know,this question has received very little attention, although a good assessment of the location of aweighted graph barycenter could be used for fair representation issues or for understanding thegraph structure from a statistical point of view. In particular, it could be used and extended toproduce “second order” moment analyses of graphs. We could therefore generalize a PrincipalComponent Analysis by extending this framework. This intermediary step was used, for example,by [13] to extend the definition of PCA on the space of probability measures on R, and by [14]to develop a suitable geometric PCA of a set of images. A popular strategy to define momentsin complex metric spaces is to use the variational interpretation of means (or barycenters), whichleads to the introduction of Frechet (or Karcher) means (see Section 2.2 for an accurate definition).This approach has been introduced in the seminal contribution [23] that makes it possible to definep-means over any metric probability space.

The use of Frechet means has met with great interest, especially in the field of bio-statistics andsignal processing, although mathematical and statistical derivations around this notion constitutea growing field of interest.• In continuous domains, many authors have recently proposed limit theorems on the empirical

Frechet sample mean (only a sample of size n of the probability law is observed) towards itspopulation counterpart. These works were mostly guided by applications to continuous manifoldsthat describe shape spaces introduced in [19]. For example, [38] establishes the consistency of thepopulation Frechet mean and derives applications in the Kendall space. The study of [9] establishesthe consistency of Frechet empirical means and derives its asymptotic distribution when n−→+∞,whereas the observations live in more general Riemannian manifolds. Finer results can be obtainedin some non-parametric restrictive situations (see, e.g., [11, 12, 15] dedicated to the so-called shapeinvariant model). Many applications in various domains involving signal processing can also befound: ECG curve analysis [10] and image analysis [44, 2], to name a few.• Recent works treat Frechet means in a discrete setting, especially when dealing with phylo-

genetic trees that have an important hierarchical structure property. In particular, [8] proposeda central limit theorem in this discrete case, whereas [41] used an idea of Sturm for spaces withnon-positive curvature to define an algorithm for the computation of the population Frechet mean.Other works deal with the averaging of discrete structure sequences such as diagrams using theWasserstein metric (see, e.g., [42]) or graphs [27].Our work deviates from the above-mentioned point since we build an algorithm that recovers theFrechet mean of a weighted graph while observing an infinite sequence of nodes of the graph,instead of finding the population Frechet mean of a set of discrete structures, as proposed in[8, 41, 42, 27]. Our algorithm uses recent contributions on simulated annealing ([3, 4]). It relies ona continuous-time noisy simulated annealing Markov process on graphs, as well as a second processthat accelerates and homogenizes the updates of the noisy transitions in the simulated annealingprocedure.

The paper is organized as follows: Section 2 highlights the different difficulties raised by thecomputation of the Frechet mean of weighted graphs and describes the algorithm we propose. In

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Section 3.1, theoretical backgrounds to understand the behavior of our method are developed onthe basis of this algorithm, and Section 3.2 states our main results. Simulations and numericalinsights are then given in Section 4. The convergence of the algorithm is theoretically establishedin Section 5, whereas Section 6 describes functional inequalities in quantum graphs that will beintroduced below.

Acknowledgments.The authors gratefully acknowledge Laurent Miclo for his stimulating discussions and helpful com-ments throughout the development of this work, and Nathalie Villa-Vialaneix for her interest andadvice concerning simulations. The authors are also indebted to zbMATH for making their databaseavailable to produce numerical simulations.

2. Stochastic algorithm on quantum graphsWe consider G= (V,E) a finite connected and undirected graph with no loop, where V = {1, . . . ,N}refers to the N vertices (also called nodes) of G, and E the set of edges that connect some couplesof vertices in G.

2.1. Undirected weighted graphsThe structure of G may be described by the adjacency matrix W that gives a non-negative weightto each edge E (pair of connected vertices), so that W = (wi,j)1≤i,j≤N while wi,j = +∞ if there isno direct link between node i and node j. W indicates the length of each direct link in E: a smallpositive value of wi,j represents a small length of the edge {i, j}. We assume that G is undirected(so that the adjacency matrix W is symmetric) and connected: for any couple of nodes in V , wecan always find a path that connects these two nodes. Finally, we assume that G has no self loop.Hence, the matrix W satisfies:

∀i 6= j wi,j =wj,i and ∀i∈ V wi,i = 0

We define d(x, y) as the geodesic distance between two points (x, y) ∈ V 2, which is the length ofthe shortest path between them. The length of this path is given by the addition of the length oftraversed edges:

∀(i, j)∈ V 2 d(i, j) = mini=i1→i2→...→ik+1=j

k−1∑`=1

wi`,i`+1.

When the length of the edges is constant and equal to 1, it simply corresponds to the number oftraversed edges. Since the graph is connected and finite, we introduce the definition of the diameterof G:

DG := sup(x,y)∈V 2

d(x, y).

To define a barycenter of a graph, it is necessary to introduce a discrete probability distributionν over the set of vertices V . This probability distribution is used to measure the influence of eachnode on the graph.Example 1. Let us consider a simple scientometric example illustrated in Figure 1 and consider

a “toy” co-authorship relation that could be obtained in a subgraph of a collaboration network likezbMATH 1. If two authors A and B share kA,B joint papers, it is a reasonable choice to use a weightwA,B = φ(kA,B), where φ is a convex function satisfying φ(0) = +∞, φ(1) = 1 and φ(+∞) = 0. Thismeans that no joint paper between A and B leads to the absence of a direct link between A and B

1 https://zbmath.org/

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ν(1) = 0.5

ν(2) = 0.1 ν(3) = 0.1

ν(4) = 0.2

ν(5) = 0.11/5

1/10

1

1/10

1/20

1/30

Figure 1 Example of a weighted graph between five authors, where the probability mass is {0.5,0.1,0.1,0.2,0.1}.In this graph, Author 1 shares five publications with Author 2 and ten publications with Author 5, andso on.

on the graph. On the contrary, the more papers there are between A and B, the closer A and B willbe on the graph. Of course, this graph may be embedded in a probability space with the additionaldefinition of a probability distribution over the authors that can be naturally proportional to thenumber of citations of each author. This is a generalization of the Erdos graph. Note that this typeof example can also be encountered when dealing with movies and actors, leading, for example, tothe Bacon number and graph (see the website: www.oracleofbacon.org/).

2.2. Frechet mean of an undirected weighted graphFollowing the simple remark that the p-mean of any distribution ν of Rd is the point that minimizes

x 7−→EZ∼ν [|x−Z|p],

it is legitimate to be interested in the Frechet mean of a graph (G,ν) where ν refers to theprobability distribution over each node. The Frechet mean is introduced in [23] to generalize thisvariational approach to any metric space. Although we have chosen to restrict our work to thecase of p= 2, which corresponds to the Frechet mean definition, we believe that our work could begeneralized to any value of p≥ 1. If d denotes the geodesic distance w.r.t. G, we are interested insolving the following minimization problem:

Mν := arg minx∈E

Uν(x) where Uν(x) =1

2

∫G

d2(x, y)ν(y). (1)

Hence, the Frechet mean of (G,ν), denoted Mν , is the set of all possible minimizers of Uν . Notethat this set is not necessarily a singleton and this uniqueness property generally requires someadditional topological assumptions (see [3], for example). At this point, we want to make threeimportant remarks about the difficulty of this optimization problem:• The problem of finding Mν involves the minimization of Uν , which is a non-convex function

with the possibility of numerous local traps. To our knowledge, this problem cannot be efficientlysolved using either a relaxed solution or using a greedy/dynamic programing algorithm (in thespirit of the Dijkstra method that makes it possible to compute geodesic paths [18]).

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• It is thus natural to think about the use of a global minimization procedure, and, in particular,the simulated annealing (S.A. for short) method. S.A. is a standard strategy to minimize a functionover discrete spaces and its computational cost is generally high. It relies on an inhomogeneousMarkov process that evolves on the graph with a transition kernel depending on the energy esti-mates Uν . In other words, it requires the computation of Uν that depends on an integral w.r.t. ν.Since we plan to handle large graphs, this last dependency can be a very strong limitation.• In some cases, even the global knowledge of ν may not be realistic, and the importance of

each node can only be revealed through i.i.d. sequential arrivals of new observations in E thatare distributed according to ν. This may be the case, for example, if we consider a probabilitydistribution over E that is cropped while gathering interactive forms on a website.

2.3. Outline of Simulated Annealing (S.A.)The optimization with S.A. introduces a Markov random dynamical system that evolves either incontinuous time (generally for continuous spaces) or in discrete time (for discrete spaces). Whendealing with a discrete setting, S.A. is based on a Markov kernel proposition L(., .) : V × V −→[0,1] related to the 1-neighborhoods of the Markov chain, as introduced in [30]. It is based onan inhomogeneous Metropolis-Hastings scheme, which is recalled in Algorithm 1. We can deriveasymptotic guarantees of the convergence towards a minimum of U as soon as the cooling scheduleis well chosen.Algorithm 1: M.-H. Simulated Annealing

Data: Function U . Decreasing temperature sequence (Tk)k≥0

1Initialization: X0 ∈ V ;2for k= 0 . . .+∞ do

3 Draw x′ ∼L(Xk, .) and compute pk = 1∧{eT−1k

[U(Xk)−U(x′)] L(x′,Xk)

L(Xk,x′)

}4 Update Xk+1 according to Xk+1 =

{x′ with probability pk

Xk with probability 1− pk5end6Output: limk−→+∞Xk

When dealing with a continuous setting, S.A. uses a drifted diffusion with a vanishing variance(εt)t≥0 over V , or an increasing drift coefficient (βt)t≥0. We refer to [32, 40] for details and we recallits Langevin formulation in Algorithm 2.

Algorithm 2: Langevin Simulated Annealing

Data: Function U . Increasing inverse temperature (βt)t≥0

1 Initialization: X0 ∈ V ;2 ∀t≥ 0 dXt =−βt∇U(Xt)dt+ dBt3Output: limt−→+∞Xt

In both cases, we can see that S.A. with U = Uν given by (1) involves the computation of thevalue of U in Line 4 of Algorithm 1, or the computation of ∇U in Line 2 of Algorithm 2. These twocomputations are problematic for our Frechet mean problem: the integration over ν is intractablein the situation of large graphs and we are naturally driven to consider a noisy version of S.A. Apossible alternative method for this problem is to use a homogenization technique: replacing Uν inthe definition of pk by Uy(·) = d2(·, y), where y is a value from an i.i.d. sequence (Yn)n≥0 distributedaccording to ν . Such methods have been developed in [29] as a modification of Algorithm 1 withan additional Monte-Carlo step in Line 4, when UY (x) follows a Gaussian distribution centeredaround the true value of U(x) = EY∼νUY (x). This approach is still problematic in our case sincethe Gaussian assumption on the random variable d2(x,Y ) is unrealistic here. Another limitationof this MC step relies on the fact that it requires a batch average of several (U(x,Yj))1≤j≤nk where

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nk is the number of observations involved at iteration k, although we also plan to develop analgorithm that may be adapted to on-line arrivals of the observation (Yn)n≥0. Lastly, it is importantto observe that the non-linearity of the exponential prevents the use of only one observation Yk inthe acceptation/reject ratio involved in the S.A. since it does not lead to an unbiased evaluationof the true transition:

EY∼ν[eT−1k

[UY (Xk)−UY (x′)]]6= eT

−1k

EY∼ν [UY (Xk)−UY (x′)] .

This difficulty does not arise in the homogenization of the simulated annealing algorithm in thecontinuous case since the exponential is replaced by a gradient, i.e., the process we use is a Markovprocess of the form:

dXt =−βt∇UYt(Xt)dt+ dBt,

where Bt is a “Brownian motion” on the graph G, β−1t refers to the inverse of the temperature, and

Yt is a continuous time Markov process obtained from the sequence (Yn)n∈N. This point motivatesthe introduction of the quantum graph induced by the initial graph. Of course, dealing with acontinuous diffusion over a quantum graph G deserves special theoretical attention, which will begiven in Section 3.1.

2.4. Homogenized S.A. algorithm on a quantum graphWe now present the proposed algorithm for estimating Frechet means. To do so, we first introducethe quantum graph ΓG derived from G= (V,E) that corresponds to the set of points living insidethe edges e∈E of the initial graph. Once an orientation is arbitrarily fixed for each edge of E, thelocation of a point in ΓG depends on the choice of an edge e ∈E and on a coordinate xe ∈ [0,Le]where Le is the length of edge e on the initial graph. The coordinate 0 then refers to the initialpoint of e and Le refers to the other extremity.

While considering the quantum graph ΓG, it is still possible to define the geodesic distancebetween any point x ∈ ΓG and any node y ∈G. In particular, when x ∈ V , we use the initial defi-nition of the geodesic distance over the discrete graph, although when x∈ e∈E with a coordinatexe ∈ [0,Le], the geodesic distance between x and y is:

d(x, y) = {xe + d(e(0), y)}∧ {Le−xe + d(e(Le), y)} .

This definition can be naturally generalized to any two points of ΓG, enabling us to consider themetric space (ΓG, d).

Consider a positive, continuous and increasing function t 7−→ αt such that:

limt 7−→+∞

αt = +∞ and ∀t≥ 0 βt = o(αt).

We introduce (Nαt )t≥0, an inhomogeneous Poisson process over R+ with intensity α. It is standard

to represent Nα through a homogeneous Poisson process H of intensity 1 using the relationship:

∀t≥ 0 Nαt =Hh(t), where h(t) =

∫ t

0

αsds.

Using this accelerated process Nα, our optimization algorithm over ΓG is based on the Markovprocess that solves the following stochastic differential equation over ΓG:{

X0 ∈ ΓG

dXt =−βt∇UYNαt (Xt)dt+ dBt.(2)

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Using the definition of Nt and the basic properties of a Poisson Process, it can be observed thatfor all ε > 0 and all t≥ 0:

E[Nt+ε−Nt

ε

]=

1

ε

∫ t+ε

t

αsds

Hence, α should be understood as the speed of new arrivals in the sequence (Yn)n∈N.Our definition (2) is slightly inaccurate: for any y ∈ V (G), the function Uy is C1 except at a finite

set of points of ΓG:• This can be the case inside an edge x∈ e with xe ∈ (0,Le) when at least two different geodesic

paths from x to y start in opposite directions. Then, we define ∇Uy(x) = 0.• This can also be the case at a node x ∈ V when several geodesic paths start from x to y. In

that case, we once again arbitrarily impose a null value for the “gradient” of Uy at node x. Thisprior choice will not have any influence on the behavior of the algorithm.

We will first detail below the theoretical objects involved in (2), then we will describe an efficientdiscretization of (2) that makes it possible to derive our practical optimization algorithm.

Algorithm 3: Homogenized Simulated Annealing over a quantum graph

Data: Function U . Increasing inverse temperature (βt)t≥0. Intensity (αt)t≥0

1Initialization: Pick X0 ∈ ΓG. ;2T0 = 0 ;3for k= 0 . . .+∞ do4 while Nα

t = k do5 Xt evolves as a Brownian motion, relatively to the structure of ΓG initialized at X−Tk .

6 end7 Tk+1 := inf{t :Nα

t = k+ 1};8 At time t= Tk+1, draw Yk+1 = YNαt according to ν.9 The process Xt jumps from X−t towards Yk+1:

Xt =X−t +βtα−1t

−−−−−−−→XtYNαt , (3)

where−−−−→XtYNαt represents the shortest (geodesic) path from Xt to YNαt in ΓG.

10end11Output: limt−→+∞Xt

Algorithm 3 could be studied following the road map of [5]. Nevertheless, this implies seriousregularity difficulties on the densities and the Markov semi-group involved. Hence, we have chosento consider Algorithm 3 as a natural Euler explicit discretization of our Markov evolution (2): fora large value of k, the average time needed to travel from Tk to Tk+1 is approximately α−1

Tk−→ 0 as

k−→+∞. On this short time interval, the drift term in (2) is the gradient of the squared geodesic

distance between XTk and YNαTk

, which is approximated by our vector−−−→XtYk, multiplied by βTk ,

leading to (3). Xt now evolves as a Brownian motion over ΓG between two jump times and thisevolution can be simulated with a Gaussian random variable using a (symmetric) random walkwhen the algorithm hits a node of ΓG. Figure 2 proposes a schematic evolution of (Xt)t≥0 over asimple graph ΓG with five nodes. We will prove the following result.

Theorem 1. A constant c?(Uν) exists such that if αt = 1 + t and βt = b log(1 + t) with b >{c?(Uν)}−1, then (Xt)t≥0 defined in (2) converges a.s. to Mν defined in (1), when t goes to infinity.

A more rigorous form of this result and more details about the process defined in (2) are presentedin Section 3.2. The proof of Theorem 3 is deferred to Section 5.

3. Inhomogeneous Markov process over ΓGThis section presents the theoretical background needed to define the Markov evolution (2).

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V1

V2 V3

V4

V5

+X−T1+XT1

YT1

Jump of size {βT1α−1T1}× d(X−

T1, YT1

)

+X−T2

Brownian displacement YT2

+XT2

Jump of size {βT2α−1T2}× d(X−

T2, YT2

)

Figure 2 Schematic evolution of the Homogenized S.A. described in Algorithm 3 over the quantum graph. Wefirst observe a jump at time T1 towards YT1 = V1 and a Brownian motion on ΓG during T2 − T1. Asecond node is then sampled according to ν: here, YT2 = V5 and a jump towards YT2 occurs at time T2.

3.1. Diffusion processes on quantum graphsWe adopt here the convention introduced in [24] and fix for any edge e ∈ E of length Le anorientation (and parametrization se). This means that se(0) is one of the extremities of e andse(Le) is the other one. By doing so, we have determined an orientation for ΓG.

Dynamical system inside one edge Following the parametrization of each edge, we can define thesecond order elliptic operator Le as ∀(x, y, t)∈ e×V (G)×R+:

Lef(x, y, t) =−βt∇xUy(x) +1

2∆xf(x, y) +αt

∫G

[f(x, y′)− f(x, y)]dν(y′), (4)

which is associated with (2) when x ∈ e: the x-component follows a standard diffusion drifted by∇xUy(.) inside the edge e, although the y-component jumps over the nodes of the initial graph Gwith a jump distribution ν and a rate αt.

Since the drift term ∇Uy(.) is measurable w.r.t. the Lebesgue measure over e and the second-order part of the operator is uniformly elliptic, Le uniquely defines (in the weak sense) a diffusionprocess up to the first time the process hits one of the extremities of e (see, e.g., [33]), which leadsto a Feller Markov semi-group.

Dynamical system near one node We adopt the notation of [26] and write e∼ v when a vertexv ∈ V is an extremity of an edge e∈E. For any function f on ΓG, at any point v ∈ V , we can definethe directional derivative of f with respect to an edge e∼ v according to the parametrization ofe. If nv denotes the number of edges e such that e∼ v, we then obtain nv directional derivativesdesignated as (def(v)):

def(v) =

lim

h−→0+

f(se(h))− f(se(0))

hif se(0) = v

limh−→0−

f(se(Le +h))− f(se(Le))

hif se(Le) = v.

It is shown in [26] that general dynamics over quantum graphs depend on a set of positivecoefficients:

A :=

{(av, (ae,v)e∼v)∈R1+nv

+ s.t.av +∑e∼v

ae,v > 0 : v ∈ V

}.

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There then exists a one-to-one correspondence between A and the set of all possible continuousMarkov Feller processes on ΓG. More precisely, if the global generator Lt is defined as

∀(x, y)∈ ΓG×V Ltf(x, y) =Le,t(f)(x, y) when x∈ e,

while f belongs to the domain:

D(L) :=

{∀y ∈G f(., y)∈ C∞b (ΓG) : ∀v ∈ V

∑e∼v

aedef(v, y) = 0

}, (5)

then the martingale problem is well-posed (see [26, 22]). In our setting, the gluing conditions aredefined through the following set of coefficients A:

∀v ∈ V av = 0 and ∀e∼ v ae,v =1

nv.

The gluing conditions defined in (5) induce the following dynamics: when the x component of(Xt, Yt)t≥0 hits an extremity v ∈ V of an edge e, it is instantaneously reflected in one of the nv edgesconnected to v (with a uniform probability distribution over the connected edges) while spendingno time on v. Using the uniform ellipticity of Le and the measurability of the drift term, Theorem2.1 of [26] can be adapted, providing the well-posedness of the Martingale problem associated with(L,D(L)), and the next preliminary result can then be obtained.

Theorem 2. The operator L associated with the gluing conditions A generates a Feller Markovprocess on ΓG×V ×R+, with continuous sample paths on the x component. This process is weaklyunique and follows the S.D.E. (2) on each e∈E.

For simplicity, ∆x will refer to the Laplacian operator with respect to the x-coordinate on thequantum graph ΓG, using our gluing conditions (5) and the formalism introduced in [26].

3.2. Convergence of the homogenized S.A. over ΓG

As mentioned earlier, we use a homogenization technique that involves an auxiliary sequence ofrandom variables (Yn)n≥1, which are distributed according to ν. More specifically, the stochasticprocess (Xt, Yt)y≥0 described above is depicted by its inhomogeneous Markov generator, which canbe split into three parts:

Ltf(x, y) =L1,tf(x, y) +L2,tf(x, y).

In the equality above, L1,t is the part of the generator that acts on Yt:

L1,tf(x, y) = αt

∫[f(x, y′)− f(x, y)]ν(dy′), (6)

describing the arrival of a new observation Y ∼ ν with a rate αt at time t. Concerning the actionon the x component, the generator is:

L2,tf(x, y) =1

24xf(x, y)−βt <OxUy,Oxf(x, y)> . (7)

Since the couple (Xt, Yt)t≥0 is Markov with a renewal of Y with ν, it can be immediately observedthat the y component is distributed at any time according to ν. We introduce the notation mt

to refer to the distribution of the couple (Xt, Yt) at time t, and we define nt as the marginaldistribution of Xt. In the following, we will also need to deal with the conditional distribution of

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Yt given the position Xt in ΓG. We will refer to this probability distribution as mt(y|x). To sumup, we have:

L (Xt, Yt) =mt with nt(x)dx=

∫V

mt(x, y)dy and mt(y |x) := P[Y = y |Xt = x]. (8)

A traditional method for establishing the convergence of S.A. towards the minimum of a functionUν consists in studying the evolution of the law of (Xt)t≥0 and, in particular, its close relationshipwith the Gibbs field µβt with energy Uν and inverse temperature βt:

µβt =1

Zβtexp(−βtUν(x)), (9)

where Zβt is the normalization factor, i.e., Zβt =

∫e−βtUν(x)dx.

Using a slowly decreasing temperature scheme t 7−→ β−1t , it is expected that the process (Xt)t≥0

evolves sufficiently fast (over the state space) in the ergodic sense so that its law remains close to µβt .In addition, the Laplace method on the sequence (µβt)t≥0 ensures that the measure is concentratednear the global minimum of Uν (see, for example, the large deviation principle associated with(µβ)β→+∞ in [25]).

Hence, a natural consequence of the convergence “L (Xt)−→ µβt” and of the weak asymptoticconcentration of (µβt)t≥0 around Mν would be the almost sure convergence of the algorithm towardsthe Frechet mean:

limt−→+∞

P (Xt ∈Mν) = 1.

We refer to [32] for further details. In particular, a strong requirement for this convergence can beconsidered through the relative entropy of nt (the law of Xt) with respect to µβt :

Jt :=KL(nt||µβt) =

∫ΓG

log

[nt(x)

µβt(x)

]dnt(x) (10)

The function βt will be chosen as a C1 function of R+, and since µβ is a strictly positive measure overΓG, it implies that t 7−→ µβt(x)−1 is C1(R+×ΓG). Moreover, (t, x) 7−→ nt(x) follows the backwardKolmogorov equation, which induces a C1(R+ × ΓG) function. Since the semi-group is uniformlyelliptic on the x-component, we have:

∀t > 0 ∀x∈ ΓG nt(x)> 0.

On the basis of these arguments, we can deduce:

Proposition 1. Assume that t 7−→ βt is C1, then (t, x) 7−→ nt(x) defines a positive C1(R+×ΓG)function and t 7−→ Jt is differentiable for any t > 0.

If we define:αt = λ(t+ 1) and βt = b log(t+ 1),

with b a constant smaller than c?(Uν)−1, where c?(Uν) is the maximal depth of a well containing a

local but not global minimum of Uν , defined in (34), then our main result can be stated as follows:

Theorem 3. For any constant λ > 0 such that αt = λ(t + 1) and βt = b log(1 + t) with b >{c?(Uν)−1}−1, then:

limJt = 0 as t−→+∞.

Author: On the calculation of a graph barycenter00(0), pp. 000–000, c© 0000 INFORMS 11

This ensures that the process Xt will almost surely converge towards Mν and, therefore, towardsa global minimum of Uν .

The idea of the proof is to obtain a differential inequality for Jt, which implies its convergencetowards 0. It is well known that the Gibbs measure µβt is the unique invariant distribution of thestochastic process that evolves only on the x component, whose Markov generator is given by:

L2,t(f)(x) =1

2∆xf −βt〈∇xf,∇xUν〉. (11)

Therefore, a natural step of the proof will be to control the difference between L2,t and L2,t andto use this difference to study the evolution of mt and nt. It can be seen that L2,t may be writtenas an average action of the operator thanks to the linearity of the gradient operator:

L2,t(f) =1

2∆xf −βtEy∼ν〈∇xf,∇xUy〉.

When Xt = x, we know that Yt is distributed according to the distribution mt(y|x). Consequently,the average action of L2,t on the x component is:

L2,t(f) =1

2∆xf −βtEy∼mt(.|x)〈∇xf,∇xUy〉=

1

2∆xf −βt

∫V

〈∇xf,∇xUy〉mt(y|x)dy, (12)

whose expression may be close to that of L2,t if mt(.|x) is close to ν.Thus, another important step is to choose appropriate values for αt and βt, i.e., to find the

balance between the increasing intensity of the Poisson process and the decreasing temperatureschedule, in order to quantify the distance between ν and mt(.|x). The main core of the proofbrings together these two aspects and is detailed in Section 5.

Another important step will be the use of functional inequalities (Poincare and log-Sobolevinequalities) over ΓG for the measure µβ when β = 0 and β −→+∞. The proof of these technicalresults are given in Section 6.3.

Corollary 1. Assume that βt = b log(t+ 1) with b < c?(Uν)−1 and αt = λ(t+ 1)γ with γ ≥ 1,

then for any neighborhood N of Mν:

limt−→+∞

P[Xt ∈N ] = 1.

Proof: The argument follows from Theorem 3. Consider any neighborhood N of Mν . The conti-nuity of Uν shows that:

∃δ > 0 N c ⊂ {x∈ ΓG :U(x)>minU + δ}.

Hence,

P[Xt ∈N c] ≤ P[U(Xt)>minU + δ]

=

∫ΓG

1U(x)>minU+δdnt(x)

=

∫ΓG

1U(x)>minU+δdµβt(x) +

∫ΓG

1U(x)>minU+δ[nt(x)−µβt(x)]dx

≤ µβt {U >minU + δ}+ 2dTV (nt, µβt)

≤ µβt {U >minU + δ}+√

2Jt,

where we used the variational formulation of the total variation distance and the Csiszar-Kullback inequality. As soon as limt−→+∞ Jt = 0, we can conclude the proof observing thatµβt {U >minU + δ} −→ 0 as βt −→+∞. �

Author: On the calculation of a graph barycenter12 00(0), pp. 000–000, c© 0000 INFORMS

It can actually be proven (not shown here) that limt−→+∞P[Xt ∈ N ] = 1 if and only if theconstant b is chosen to be lower than c?(Uν)

−1. This means that this algorithm does not allowfaster cooling schedules than the classical S.A. algorithm. Nevertheless, this is a positive resultsince this homogenized S.A. can be numerically computed quickly and easily on large graphs.Finally, we should consider this result to be theoretical. However, in practice, the simulation of thishomogenized S.A. is performed during a finite horizon time and efficient implementations certainlydeserve a specific theoretical study following the works of [17] and [46].

4. Numerical simulation resultsWe now present different practical aspects of our strategy. After giving numerical details abouthow to make it usable on large graphs, plus different insights into its calibration, we will presentresults obtained on social network subgraphs of Facebook2 and a citation subgraph of zbMATH.

4.1. Algorithmic complexityIt is widely recognized that the algorithmic complexity of numerical strategies dealing with graphscan be an issue. The number of vertices and edges of real-life graphs can indeed be quite large.For instance, the zbMATH subgraph of Section 4.3.2 has 13000 nodes and approximately 48000edges. In this context, it is worth justifying that the motion of (Xt)t≥0 on the graph ΓG across theiterations of Algorithm 3 is reasonably demanding in terms of computational resources. We recallthat this motion is driven by a Brownian motion (lines 4 to 6 of Algorithm 3) and an attractiontowards the vertex YNαt (line 9 of Algorithm 3) at random times (Tk)k≥0. The number of verticesand undirected edges with non-null weights in ΓG is also N and |E|, respectively. Note finally thatN − 1≤ |E| ≤N(N − 1)/2 if ΓG has a unique connected component.

Neighborhood structure A first computational issue that can arise when moving Xt is to find allpossible neighbors of a specific vertex v. This is indeed performed every time Xt moves from oneedge to another and directly depends on how ΓG is encoded in the memory. Encoding ΓG in a listof edges with non-null weights is common practice. In that case, the computer checks the vertexpairs linked by all edges to find those containing v, so the algorithmic cost is 2|E|. A second classicstrategy is to encode the graph in a connectivity matrix. In this case, the computer has to gothrough all the N indexes of the columns representing v to find the non-null weights. We insteadsparsely encode the graph in a list of lists: the main list has a size N and each of its elements liststhe neighbors of a specific vertex v. If the graph only contains edges with strictly positive weights,this strategy has a computational cost N , and in all of the other cases, the cost is <N . On average,the computational cost is 2|E|/N so that this strategy is particularly advantageous for sparsegraphs, where |E|<<N(N − 1)/2, which are common for the targeted applications. For instance,|E|= 4 ∗ 103 and N(N − 1)/2 = 1.24 ∗ 105 on the smallest Facebook subgraph of Section 4.3.1, and|E|= 4.8 ∗ 104 and N(N − 1)/2 = 8.44 ∗ 107 on the zbMATH subgraph of Section 4.3.2.

Geodesic paths Another potential issue with our strategy is that it seeks at least one optimalpath between the vertices Xt and YNαt at each jump time (line 9 of Algorithm 3). This may beefficiently done using a fast marching propagation algorithm (see, e.g., [18] for details), where:

(i) the distance to Xt is iteratively propagated on the whole graph ΓG until no more optimaldistance to reach Xt is updated, and

(ii) considering the shortest path between Xt and YNαt .In this case, step (i) is particularly time-consuming and cannot be reasonably performed at eachstep of Algorithm 3. Fortunately, the algorithmic cost to compute the distance between all pairs ofvertices is equivalent to that of computing step (i). We then compute these distances once and for

2 Subgraphs obtained on the Stanford Large Network Dataset Collection: https://snap.stanford.edu/data/

Author: On the calculation of a graph barycenter00(0), pp. 000–000, c© 0000 INFORMS 13

all at the beginning of the computations and store the result in a N ×N matrix in the RAM of

the computer 3. We can therefore very quickly use these results at each iteration of the algorithm

and, in particular, deduce the useful part of the geodesic paths involved to travel from Xt to YNαt .

The only limitation of this strategy is that the distance matrix can be memory-consuming. It can

therefore be used on small to large graphs but not on huge graphs (typically when N > 105) on

current desktops. An extension to deal with this scalability issue is a current subject of research,

and we will therefore not describe applications of Algorithm 3 on huge graphs in this work.

4.2. Parameter tuning

Several parameters influence the behavior of the simulated process Xt. Some of them are directly

introduced in the theoretical construction of the algorithm, i.e., the intensity of the Poisson process

or the temperature schedule. Other ones come from the practical implementation of the algorithm,

i.e., the maximal time up to which we generate Xt. The theoretical result given in Theorem 3 gives

an upper bound for the probability of Xt to be not too distant from the set of global minima. This

bound depends on βt and αt as well as on different characteristics of the graph such as its diameter

and number of nodes. We then propose an empirical strategy for parameter tuning, which we will

use later in Section 4.3.

For a graph G with N nodes and a diameter DG, we define:

T ∗max = 100 + 0.1N and β∗t = 2 log(t+ 1)/DG.

We then choose the intensity of the Poisson process so that it will generate a reasonably large

amount of Yn at the end of the algorithm, depending on the discrete probability distribution ν.

More specifically, let S∗ be the average number of jumping times between T ∗max− 1 and T ∗max. We

then set S∗ = 1000, which can be obtained by choosing:

α∗t = λ(2t+ 2) with λ=S∗

2T ∗max + 1.

4.3. Results

This section presents results obtained on graphs of different sizes and using different parameters.

We used the empirical method of Section 4.2 to define default parameters and altered them to

quantify the sensitivity of our algorithm to parameter variations.

Since the process Xt lives on a quantum graph, its location at time t is between two vertices, on

an edge of ΓG. However, our primary interest is to study the properties of the initial discrete graph

and therefore, the output of the algorithm will be the vertex considered as the graph barycenter.

To achieve this, we associate a frequency to each vertex. For a vertex v, this frequency is the

portion of time during which v was the closest vertex to the simulated process Xt. In our results,

we consider frequencies computed over the last 10% of the iterations of the algorithm.

4.3.1. Facebook subgraphs

3 In our experiments, we used the all-shortest-paths function of the Python library NetworkX.

Author: On the calculation of a graph barycenter14 00(0), pp. 000–000, c© 0000 INFORMS

Experimental protocol We first tested our algorithm on three subgraphs of Facebook from the

Stanford Large Network Dataset Collection: (FB500) has 500 nodes and 4337 edges and contains

two obvious clusters; (FB2000) has 2000 nodes and 37645 edges and fully contains (FB500);

and (FB4000) has 4039 nodes and 88234 edges and fully contains (FB2000). For each of these

subgraphs, we considered the probability measure ν as the uniform distribution over the graph’s

vertices and used a length of 1 for all edges. We also explicitly computed the barycenter of these

graphs using an exhaustive search procedure. We are able to do that because we considered a

simplified case where the distribution over the nodes is uniform. For example, this exhaustive search

procedure required approximately 6 hours for the (FB4000) subgraph. Nevertheless, it allowed us

to compare our strategy with ground-truth results.

We used the strategy of Section 4.2 to define default parameters adapted to each subgraph. We

also tested different values for parameters β, S and Tmax in order to quantify their influence. We

repeated our algorithm 100 times for each parameter set to evaluate the algorithm stability.

In the tables representing quantitative results, Error represents the number of times, out of

100, that the algorithm converged to a node different from the ground-truth barycenter. It is a

rough indicator of the ability of the algorithm to locate the barycenter of the graph, that could be

replaced by a measure of the average distance between the last iterations of the algorithm and the

ground-truth barycenter (not shown in this work). For each subgraph and parameter set, column

Av. time contains the average times in seconds for the barycenter estimation, keeping in mind that

the Dijkstra algorithm was performed once for all before the 100 estimations. This preliminary

computation requires approximately 1, 30 and 80 seconds on a standard laptop.

Effect of β From a theoretical point of view, (βt)t≥0 should be chosen in relation to the constant

c?(Uν), which is unknown in practice. Therefore, the practical choice of (βt)t≥0 is a real issue to

obtain a good behavior of the algorithm. Table 1 gives the results obtained on our algorithm with

different values of βt.

FB500 FB2000 FB4000β Error Med. Freq. Av. time Error Med. Freq. Av. time Error Med. Freq. Av. time14β∗ 15 % 0.6042 0.95 s 28 % 0.3519 10.12 s 27 % 0.3314 12.41 s

12β∗ 2 % 0.4184 1.38 s 1 % 0.7268 4.25 s 5 % 0.6534 14.11 sβ∗ 0 % 0.8008 1.48 s 0 % 0.9418 12.36 s 1 % 0.8913 13.55 s2β∗ 0 % 0.8321 11.31 s 0 % 0.9892 8.52 s 0 % 0.9647 16.79 s4β∗ 0 % 0.8233 13.19 s 0 % 0.9930 15.58 s 0 % 0.9824 31.43 s8β∗ 0 % 0.7717 2.36 s 0 % 0.9750 23.25 s 0 % 0.9445 44.46 s

Table 1 Experiments on the three Facebook subgraphs, while varying the values of (βt)t≥0.

We can observe that when (βt)t≥0 is too small, then the behavior of the algorithm is deteriorated,

revealing the tendancy of the process (Xt)0≤t≤T∗maxto have an excessively slow convergence rate

towards its local attractor in the graph. Roughly speaking, in such a situation, the process does

not learn fast enough. When the value of β is chosen in the range [β∗; 2β∗], we can observe a really

good behavior of the algorithm: it almost always locates the good barycenter in a quite reasonable

time of computation (less then 20 seconds for the largest graph). Finally, we can observe in the

column, Med. Freq., that in most of the last iterations of the algorithm (more than 80%), the

process evolves around its estimated barycenter, so that the decision to produce an estimator is

quite easy when looking at an execution of the algorithm.

Author: On the calculation of a graph barycenter00(0), pp. 000–000, c© 0000 INFORMS 15

Effect of S and Tmax Table 2 gives the results obtained with our algorithm while using differentvalues of S and Tmax. As expected, we observe that increasing the ending time of simulation alwaysimproves the convergence rate (column Error in Table 2) of the algorithm towards the right node.The behavior of the algorithm is also improved by increasing the value of S, which quantifies thenumber of arrivals of nodes observed along the averaging procedure. Of course, the counterpart ofincreasing both S and Tmax is an increasing cost of simulation (see column Av. time).

FB500 FB2000 FB4000S Tmax Error Med. Freq. Av. time Error Med. Freq. Av. time Error Med. Freq. Av. time12S∗ T ∗max 2 % 0.7667 0.59 s 0 % 0.9344 3.22 s 0 % 0.8614 6.70 sS∗ 2T ∗max 0 % 0.8049 2.79 s 0 % 0.9610 9.30 s 0 % 0.8970 24.42 sS∗ 4T ∗max 0 % 0.8101 5.60 s 0 % 0.9677 22.21 s 0 % 0.9222 54.73 s2S∗ T ∗max 1 % 0.8345 2.26 s 0 % 0.9512 8.39 s 0 % 0.9062 20.95 s2S∗ 2T ∗max 0 % 0.8361 4.57 s 0 % 0.9586 23.30 s 0 % 0.9121 23.30 s2S∗ 4T ∗max 0 % 0.8423 11.16 s 0 % 0.9735 11.16 s 0 % 0.9366 96.84 s

Table 2 Experiments on the three Facebook subgraphs, while varying the values of S and Tmax.

As an illustration of the (small) complexity of the Facebook sub-graphs used for benchmarkingour algorithm, we provide a representation of the FB500 graph in Figure 3. This representationhas been obtained with the help of Cytoscape software and is not a result of our own algorithm.In Figure 3, the red node is the estimated barycenter, which is also the ground-truth barycenterlocated by a direct exhaustive computation. The blue nodes are the “second rank” nodes visitedby our method.

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Figure 3 Region of interest presenting the results obtained on the FB500 graph.

4.3.2. zbMATH subgraph The zbMATH subgraph built from zbMATH4 has been obtainedby an iterative exploration of the co-authorship relationship in an alphabetical order. We thus nat-urally obtained a connected graph. This graph has been weighted by the complete number of cita-tions obtained by each author, leading to the distribution probability ν ∝ ](number of citations).Finally, all the edges in the graph are fixed to have a length of 1. This exploration was initialized onthe entry of the first author’s name (i.e., S. Gadat) and we stopped the process when we obtained13000 nodes (authors) on the graph. This stopping criterion in the exploration of the zbMATHdatabase corresponds to a technical limitation of 40 GB memory required by the distance matrix

4 https://zbmath.org/authors/

Author: On the calculation of a graph barycenter16 00(0), pp. 000–000, c© 0000 INFORMS

obtained with the Dijsktra algorithm. In particular, this limitation and the starting point of theexploration induce an important bias in the community of authors used to build the subgraph fromthe zbMATH dataset: the researchers obtained in the subgraph are generally French and appliedmathematicians. The graph is more or less focused on the following research themes: probabil-ity, statistics and P.D.E. Consequently, the results provided below should be understood as anillustration of our algorithm and not as a bibliometric study!

Our experiments rely on the same choice of parameters (β∗t )t≥0, S∗, T ∗max and α∗t as above,

indicated at the beginning of Section 4.2. Again, our algorithm produces the same outputs but ofcourse, in this situation, we do not know the ground-truth barycenter of the zbMath subgraph.In Figure 4, we present a representation of the subgraph obtained with Cytoscape software, and azoom on a region of interest (ROI for short) in Figure 4.

Again, the main nodes visited by our algorithm are represented with a red square and the sizeof the used square is larger when the node is frequently visited. According to the results obtainedon the Facebook subgraphs, we then assume that the larger red square is the barycenter of thezbMath subgraph.

ROI

Szulga

Bonichon

Marckert

Mallows

vanDijk

Demichel

Raugi

Simatos

Horowitz

Yamazaki

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Geniet

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Yuan

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Chalendar

Kleinberg

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Peleg

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Zhang

Liang

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Merhav

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Celani

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RobinsonMolla

Pietracaprina

Augustine

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Pisanti

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Robert

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Antunes

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It is also possible to assert the robustness of our method with respect to several Monte Carloruns of our algorithm. We have produced some boxplots for each of the main authors located in thesubgraph, according to the occupation measure of the process over the last 10% of the iterationswith 10 Monte-Carlo replications. These “violin” plots are represented in Figure 5. Each executionof the algorithm requires approximately 3 hours of computations. The algorithm seems to producereliable conclusions concerning the top nodes visited all along the ending iterations. Nevertheless,it appears to be necessary to extend our investigations in order to obtain a scalable method forhandling larger graphs.

5. Proof of the main result (Theorem 3)We establish a differential inequality that will imply the convergence of Jt. The computationsactually lead us to a system of two differential inequalities. We therefore introduce another quantity

Author: On the calculation of a graph barycenter00(0), pp. 000–000, c© 0000 INFORMS 17

Yor

Cohen

Str

oock

Ledoux

Jaff

ard

Bre

zis

Zeit

ouni

Lions

Nuala

rt

1

2

3

4

5

6

7

8

Frequency

(%

)

Figure 5 “Violin” plot of the occupation measure of the algorithm on the zbMATH subgraph for 10 MC repli-cations. The average frequency is located in the middle of the “violin” plot, although the minimal andmaximal values are shown in the extremity of the representation.

It that measures the average closeness (w.r.t. x) of the conditional law of y given x at time t to ν,defined as:

It :=

∫KL(mt(.|x)||ν)dnt(x) =

∫ [∫log

mt(y|x)

ν(y)mt(y|x)dy

].dnt(x) (13)

The next proposition links the evolution of Jt (in terms of an upper bound of ∂tJt) with thespectral gap of µβt over ΓG, the diameter of the graph DG and the divergence It.

5.1. Study of ∂tJt

Proposition 2. Let c?(Uν) be the term defined in Equation (34) and CΓG be the constant givenin Proposition 5. We then have:

∂tJt ≤D2Gβ′t + 8D2

Gβ2t It−

e−c?(U)βt

CΓG(1 +βt)Jt.

Proof: We compute the derivative of Jt and separately study each of the three terms:

∂tJt =

∫∂t{log(nt(x))}dnt(x)︸ ︷︷ ︸

J1,t

−∫∂t{log(µβt(x))}dnt(x)︸ ︷︷ ︸

J2,t

+

∫log

[nt(x)

µβt

]∂tnt(x)dx︸ ︷︷ ︸

J3,t

(14)

Study of J1,t: This term is easy to deal with:

J1,t =

∫ΓG

∂t{log(nt(x))}nt(x)

Author: On the calculation of a graph barycenter18 00(0), pp. 000–000, c© 0000 INFORMS

=

∫ΓG

∂t{nt(x)}nt(x)

nt(x) =

∫ΓG

∂t{nt(x)}dx= ∂t

{∫ΓG

nt(x)dx

}= ∂t{1}= 0. (15)

Study of J2,t: Using the definition of µβt given in Equation (9), we obtain for the second term:

J2,t =−∫

ΓG

∂t{log(µβt(x))}dnt(x) =−∫

ΓG

∂t{−βtUν(x)− logZβt}dnt(x)

= β′t

∫ΓG

Uν(x)dnt(x) +∂t{Zβt}Zβt

.

According to the definition of Zβt , we have:

∂t{Zβt}Zβt

=Z−1βt∂t

{∫ΓG

e−βtUν(x)dx

}=−β′t

∫ΓG

Uν(x)e−βtUν(x)

Zβtdx=−∂t{βt}

∫ΓG

Uν(x)µβt(x)dx.

Hence, we obtain

J2,t = β′t

∫ΓG

Uν(x)[nt(x)−µβt(x)]dx

The graph has a finite diameter DG. We therefore have:

∫ΓG

Uν(x)dnt(x)≤D2G. The same inequality

holds using the measure µβt so that:|J2,t| ≤ D2

Gβ′t. (16)

Study of J3,t: The last term J3,t involves the backward Kolmogorov equation. First, since nt is

the marginal law of Xt, we have: nt(x) =

∫mt(x, y)dy.

Using the backward Kolmogorov equation for the Markov process (Xt, Yt)t≥0 and the Fubinitheorem, we have, for any smooth enough function ft : x∈ ΓG 7−→R :

∫ΓG

ft(x)∂t{nt(x)}dx=

∫ΓG

ft(x)∂t

{∫V

mt(x, y)dy

}dx=

∫ΓG

∫V

ft(x)∂t{mt(x, y)}dxdy

=

∫ΓG

∫V

Lt(ft)(x)mt(x, y)dxdy=

∫ΓG

∫V

[L1,t +L2,t](ft)(x)mt(x, y)dxdy,

where L1,t and L2,t are defined in Equations (6) and (7). Since the function ft is independent of y,we have L1,t(ft) = 0. For the part corresponding to L2,t, we have:

∫ΓG

∫V

L2,t(ft)(x)mt(x, y)dxdy

=

∫ΓG

∫V

[1

2∆xft(x)−βt∇xUy(x)∇xft(x)

]mt(x, y)dxdy

=

∫ΓG

1

2∆xft(x)nt(x)dx−βt

∫ΓG

∫V

∇xUy(x)∇xft(x)nt(x)mt(y|x)dxdy.

because nt is the marginal distribution of Xt and mt(y|x)×nt(x) =mt(x, y).

Thus, for any smooth enough function ft(x), using the operator introduced in (12), we have:∫ΓG

ft(x)∂tnt(x)dx=

∫ΓG

L2,t(ft)(x)nt(x)dx.

Author: On the calculation of a graph barycenter00(0), pp. 000–000, c© 0000 INFORMS 19

Replacing ft(x) by lognt(x)

µβt(x), we obtain:

J3,t =

∫ΓG

lognt(x)

µβt∂tnt(x) =

∫ΓG

L2,t

[log

nt(x)

µβt(x)

]nt(dx).

Since µβt is the invariant distribution of L2,t (see Equation (11)), it is natural to insert L2,t:

J3,t =

∫ΓG

L2,t

[log

nt(x)

µβt(x)

]nt(dx)

=

∫ΓG

L2,t

[log

nt(x)

µβt(x)

]nt(dx)−

∫ΓG

(L2,t−L2,t

)[log

nt(x)

µβt(x)

]nt(dx)︸ ︷︷ ︸

:=κt

. (17)

Since L2,t is a diffusion operator and µβt is its invariant measure, it is well known (see, e.g., [7])that the action of L2,t on the entropy is closely linked to the Dirichlet form in the following way:

∫ΓG

L2,t

[log

nt(x)

µβt(x)

]nt(dx) =−2

∫ΓG

(∇x

{√nt(x)

µβt(x)

})2

µβt(dx), (18)

and therefore translates a mean reversion towards µβt in the first term of (17).

We now study the size of the difference between L2,t and L2,t and introduce the “approximation”term of ∇xUν(x) at time t:

Rt(x) =

∫V

∇x(Uy)(x)mt(y|x)dy.

The relationship ∇x log(f) = 2∇x√f√

f, the Cauchy-Schwarz inequality and 2ab≤ a2 + b2 yield:

|κt| = βt

∣∣∣∣∫ΓG

(Rt(x)−∇xUν(x)

)∇x{

lognt(x)

µβt(x)

}nt(dx)

∣∣∣∣= 2βt

∣∣∣∣∣∫

ΓG

(Rt(x)−∇xUν(x)

)∇x

{√nt(x)

µβt(x)

}√µβt(x)

nt(x)nt(dx)

∣∣∣∣∣≤ 2βt

√∫ΓG

(Rt(x)−∇xUν(x)

)2

nt(dx) ·

√√√√∫ΓG

∇x

(√nt(x)

µβt(x)

)2

µβt(x)dx

≤ β2t

∫ (Rt(x)−∇xUν(x)

)2

nt(x)dx+

∫ΓG

(∇x

{√nt(x)

µβt(x)

})2

µβt(dx)

If dTV denotes the total variation distance, the first term of the right hand side leads to:∣∣∣Rt(x)−∇xUν(x)∣∣∣= ∣∣∣∣∫

V

∇xd2(x, y)[mt(y|x)− ν(y)]dy

∣∣∣∣≤ ‖∇xd2(x, y)‖∞

∣∣∣∣∫V

[mt(y|x)− ν(y)]dy

∣∣∣∣≤ 2‖∇xd2(x, y)‖∞dTV (mt(.|x), ν)

≤√

2‖∇xd2(x, y)‖∞

√∫V

logmt(y|x)

ν(y)mt(y|x)dy

Author: On the calculation of a graph barycenter20 00(0), pp. 000–000, c© 0000 INFORMS

where the last line comes from the Csiszar-Kullback inequality. Since d2(·, y) is differentiable a.e.and its derivative is bounded for all y ∈ V by 2DG, we can use It defined in Equation (13) to obtain:∫

ΓG

(Rt− Rt

)2

(x)nt(x)dx≤ 8D2GIt

Consequently, we obtain:

|κt| ≤ 8D2Gβ

2t It +

∫ΓG

(∇x

{√nt(x)

µβt(x)

})2

µβt(dx) (19)

Taking the inequalities (18) and (19), we now obtain in (17):

J3,t ≤ 8D2Gβ

2t It−

∫ΓG

(∇x

{√nt(x)

µβt(x)

})2

µβt(dx)

We denote ft =√

ntµβt

. Since ‖f‖22,µβt = 1, one can easily see that Entµβt (f2t ) = Jt. Now, the Loga-

rithmic Sobolev inequality on the (quantum) graph ΓG for the measure µβt stated in Proposition5 shows that: ∫

ΓG

(∇x

{√nt(x)

µβt(x)

})2

µβt(dx)≥ e−c?(Uν)βt

CΓG(1 +βt)Jt,

where c?(Uν) is defined in Equation (34) and is related to the maximal depth of a well containinga local but not global minimum. We thus obtain:

J3,t =

∫L2,t

[log

nt(x)

µβt(x)

]nt(dx)≤ 8D2

Gβ2t It−

e−c?(Uν)βt

CΓG(1 +βt)Jt.

The proof is concluded by regrouping the three terms. �

5.2. Study of ∂tItProposition 3. Assume that DG ≥ 1 and βt is an increasing inverse temperature with βt ≥ 1,

then:∂tIt ≤−αtIt− ∂tJt +D2

G[β′t + 6β2t ]. (20)

Proof: We compute the derivative of It. Observing that mt(x, y) = nt(x)mt(y|x), we have:

∂tIt = ∂t

{∫ΓG

nt(x)

(∫V

logmt(y|x)

ν(y)mt(y|x)dy

)dx

}= ∂t

{∫ΓG

∫V

logmt(y|x)

ν(y)mt(x, y)dxdy

}=

∫ΓG

∫V

∂t{logmt(y|x)}mt(x, y)dxdy︸ ︷︷ ︸:=I1,t

−∫

ΓG

∫V

∂t{log ν(y)}mt(x, y)dxdy︸ ︷︷ ︸:=I2,t

+

∫log

mt(y|x)

ν(y)∂tmt(x, y)dxdy︸ ︷︷ ︸

:=I3,t

The computation of the first two terms is straightforward. For the first one we have:

I1,t =

∫ΓG

∫V

∂t{logmt(y|x)}mt(x, y) =

∫ΓG

∫V

∂tmt(y|x)

mt(y|x)mt(x, y)dxdy

=

∫ΓG

∫V

∂tmt(y|x)nt(x)dxdy=

∫ΓG

nt(x)

(∫V

∂tmt(y|x)dy

)dx

Author: On the calculation of a graph barycenter00(0), pp. 000–000, c© 0000 INFORMS 21

=

∫ΓG

nt(x)∂t

∫V

mt(y|x)dy︸ ︷︷ ︸:=1

dx= 0

The computation of I2,t is easy since ν does not depend on t, implying that I2,t = 0.

For the third term, we use the backward Kolmogorov equation and obtain:

I3,t =

∫ΓG

∫V

logmt(y|x)

ν(y)∂tmt(x, y) =

∫ΓG

∫V

Lt(

logmt(y|x)

ν(y)

)mt(x, y)dx dy

=

∫ΓG

∫V

L1,t

(log

mt(y|x)

ν(y)

)mt(x, y)dx dy︸ ︷︷ ︸

:=I13,t

+

∫ΓG

∫V

L2,t

(log

mt(y|x)

ν(y)

)mt(x, y)dx dy︸ ︷︷ ︸

:=I23,t

The jump part L1,t exhibits a mean reversion on the entropy on the conditional law: applying theJensen inequality for the logarithmic function and the measure ν, we obtain:

I13,t = αt

∫ΓG

∫V

[∫V

(log

mt(y′|x)

ν(y′)− log

mt(y|x)

ν(y)

)ν(y′)dy′

]mt(x, y)dxdy

= αt

∫ΓG

∫V

[∫V

(log

mt(y′|x)

ν(y′)

)ν(y′)dy′

]mt(x, y)dxdy−αtIt

≤ αt∫

ΓG

∫V

log

(∫V

mt(y′|x)

ν(y′)ν(y′)dy′

)mt(x, y)dxdy−αtIt

≤ αt∫

ΓG

∫V

log

(∫V

mt(y′|x)dy′

)mt(x, y)dxdy−αtIt

≤ αt∫

ΓG

∫V

log 1 ·mt(x, y)dxdy−αtIt ≤−αtIt (21)

If we consider the action of L2,t on the entropy of the conditional law, using mt(x, y) =mt(y|x)nt(x)yields:

I23,t =

∫ΓG

∫V

L2,t

(log

mt(y|x)

ν(y)

)mt(x, y)dx dy

=

∫ΓG

∫V

L2,t

(log

mt(x, y)

ν(y) ·nt(x)

)mt(x, y)dx dy

=

∫ΓG

∫V

L2,t log (mt(x, y))mt(x, y)dxdy−∫

ΓG

∫V

L2,t log (nt(x, y))mt(x, y)dxdy, (22)

because L2,t(ν)(y) = 0 (L2,t only involves the x component). We now study the first term of (22):

∫ΓG

∫V

L2,t (logmt(x, y))dmt(x, y) =1

2

∫ΓG

∫V

∆x [logmt(x, y)] dmt(x, y)

−βt∫

ΓG

∫V

<∇xUy,∇x logmt(x, y)> dmt(x, y)

=1

2

∫ΓG

∫V

∂2x {logmt(x, y)}dmt(x, y)

−βt∫∂xUy∂x logmt(x, y)dmt(x, y)

Author: On the calculation of a graph barycenter22 00(0), pp. 000–000, c© 0000 INFORMS

The first term of the right-hand side deserves special attention:∫ΓG

∫V

∂2x {logmt(x, y)}dmt(x, y) =

∫ΓG

∫V

∂x

{∂x{mt(x, y)}mt(x, y)

}dmt(x, y)

=

∫ΓG

∫V

∂2x{mt(x, y)}mt(x, y)

dmt(x, y)−∫

ΓG

∫V

[∂x{mt(x, y)}]2

mt(x, y)dxdy

=

∫ΓG

∫V

∂2x{mt(x, y)}dxdy− 4

∫ΓG

∫V

(∂x√mt(x, y)

)2

dxdy

=∑e∈E

∫V

[∂xmt(e(Le), y)]− ∂xmt(e(0), y)dy

− 4

∫ΓG

∫V

(∂x√mt(x, y)

)2

dxdy,

where {e(s),0≤ s≤ Le} is the parametrization of the edge e ∈ E introduced in Section 3.1. Thegluing conditions (5) yield:∑

e∈E

∫V

∂xmt(e(Le), y)− ∂xmt(e(0), y) =

∫V

∑v∈V

[∑e∼v

−demt(v, y)

]= 0.

As a consequence, we obtain:∫ΓG

∫V

L2,t (logmt(x, y))dmt(x, y) =−2

∫ΓG

∫V

(∂x√mt(x, y)

)2

dxdy

−βt∫

ΓG

∫V

∂xUy∂x logmt(x, y)dmt(x, y) (23)

The Cauchy-Schwarz inequality applied on the second term leads to:∣∣∣∣βt ∫ΓG

∫V

∂xUy∂x logmt(x, y)dmt(x, y)

∣∣∣∣≤ βt‖∂xd2(., .)‖2,mt

√∫ΓG

∫V

(∂x logmt(x, y))2dmt(x, y)

≤ βt||∂xd2(x, y)||∞

√∫ΓG

∫V

(∂x logmt(x, y))2mt(x, y)dxdy

≤ 4βtDG

√∫ΓG

∫V

(∂x√mt(x, y)

)2

dxdy

≤ 2D2Gβ

2t + 2

∫ΓG

∫V

(∂x√mt(x, y)

)2

dxdy

Inserting this in Equation (23) leads to:∫ΓG

∫V

L2,t logmt(x, y)dmt(x, y)≤ 2D2Gβ

2t . (24)

We study the second term of (22) and use the proof of Proposition 2: the decomposition (14) withEquations (15) and (16) yield:

∂tJt ≤ β′tD2G +

∫L2,t

(log

nt(x)

µβt(x)

)mt(dx,dy).

This implies:

−∫

ΓG

∫V

L2,t (lognt(x))mt(dx,dy)≤−∂tJt +β′tD2G−

∫L2,t (logµβt(x))mt(dx,dy). (25)

Author: On the calculation of a graph barycenter00(0), pp. 000–000, c© 0000 INFORMS 23

Using the definition of µβt , we obtain:

−∫

ΓG

∫V

L2,t loge−βtUν(x)

Zβt=

∫ΓG

∫V

[L2,t (βtUν(x)) +L2,t (logZβt)]mt(dx,dy)

= βt

∫ΓG

∫V

L2,t (Uν(x))mt(dx,dy)

= βt

∫ΓG

∫V

1

2∆xUν(x)−βt∇xUy(x)∇xUν(x)mt(dx,dy)

≤ βt2

+ 4β2tD2

G

This inequality used in (25) yields:

−∫L2,t (lognt(x))mt(x, y) =−∂tJt +β′tD2

G +βt2

+ 4β2tD2

G (26)

We now use (24) and (26) in (22) and our assumptions βt ≥ 1 and DG ≥ 1 to obtain:

I23,t ≤−∂tJt +D2

G[β′t + 6β2t ] (27)

Combining (21) and (27) leads to the desired inequality given by Equation (20). �

5.3. Convergence of the entropyThe use of Propositions 2 and 3 makes it possible to obtain a system of coupled differential inequal-ities.

Proof of Theorem 3: If we denote a=D2G, we can write:{

J ′t ≤− e−c?(Uν )βt

CΓG(1+βt)

Jt + aβ′t + 8aβ2t It

I ′t ≤−αtIt−J ′t + a(β′t + 6β2t )

We introduce an auxiliary function Kt = Jt+ktIt, where kt is a smooth positive decreasing functionfor t large enough, so that lim

t→∞kt = 0. We use the system above to deduce that:

K ′t = J ′t + k′tIt + ktI′t

≤ J ′t + ktI′t

≤ J ′t + kt(−αtIt−J ′t) + akt(β′t + 6β2

t )≤ (1− kt)J ′t − ktαtIt + akt(β

′t + 6β2

t ).

In the first inequality, we use the fact that k′t ≤ 0 and It is positive. The second inequality is givenby (20). The upper bound of Proposition 2 now leads to:

K ′t ≤−εt(1− kt)Jt− ktαtIt + 8a(1− kt)β2t It + aβ′t + 6aktβ

2t ,

where we denoted εt = e−c?(Uν )βt

CΓG(1+βt)

. We choose the function kt to obtain a mean reversion on Kt:

kt :=8aβ2

t

αt + 8aβ2t − εt/2

. (28)

Note that this function is decreasing for sufficiently large t as soon as βt = o(αt), which is thecase according to the choices described in Theorem 3. Moreover, a straightforward consequence islimt−→+∞ kt = 0. This ensures that a positive T0 exists such that:

∀t≥ T0 0≤ kt ≤1

2.

Author: On the calculation of a graph barycenter24 00(0), pp. 000–000, c© 0000 INFORMS

Consequently, we deduce that:

∀t≥ T0 K ′t ≤−εt(1− kt)Jt− ktεt2It + aβ′t + 6aβ2

t kt

≤−εt2Jt− kt

εt2It + aβ′t + 6aβ2

t kt ≤−εt2Kt + aβ′t + 6aβ2

t kt︸ ︷︷ ︸:=ηt

The next bound is an easy consequence of the Gronwall Lemma:

∀t≥ T0 Kt ≤KT0e−

∫ t

T0

εs2

ds+

∫ T

T0

ηse−

∫ t

s

εu2

duds,

which in turn implies that Kt −→ 0 as t−→+∞ as soon as ηt = ot∼+∞(εt) with∫∞

0εudu= +∞.

We are now looking for a suitable choice for αt and βt. Let us assume that αt ∼ λtγ for any γ > 0.This choice leads to:

kt ∼8ab2 log(t+ 1)2

λtγ,

so that:

ηt ∼ab

t+

48a2b4 log(t+ 1)4

λtγ=O(t−(1∧γ) log(t)4).

At the same time, we can check that εt ∼ t−c?(Uν )b

bCΓGlog t

. Now, our conditions on ηt and εt imply that:∫ ∞0

εudu= +∞ ⇐⇒ b c?(Uν)≤ 1.

At the same time:

ηt = o(εt) ⇐⇒ 1∧ γ > bc?(Uν).

The optimal calibration of our parameters γ and b (minimal value of γ, maximal value of b) inducesthe choice γ = 1 and b < c?(Uν)

−1. Up to the choices αt ∼ λt and βt ∼ b log(t), we deduce thatKt −→ 0 when t goes to infinity. Since It, kt and Jt are positive, this also implies that Jt −→ 0when t goes to infinity. �

6. Functional inequalitiesThis section is devoted to the proof of the Log-Sobolev inequality for the measure (µβ)β>0. We dealtwith the Dirichlet form using this inequality (see Equation (18)) in Proposition 2. In particular, weneed to obtain an accurate estimate when β −→+∞. For this purpose, we introduce the genericnotation for Dirichlet forms (see, i.e., [7] for a more in-depth description):

∀β ∈R∗+ ∀f ∈W 1,2(µβ) Eβ(f, f) = ‖Of‖2µβ =

∫ΓG

|Of(x)|2dµβ(x).

If we denote < f >µβ= µβ(f) =

∫ΓG

fdµβ, we are interested in showing the Poincare inequality:

‖f−< f >µβ ‖22,µβ

=

∫ΓG

(f−< f >µβ )2dµβ ≤ λ(β)Eβ(f, f) (29)

Author: On the calculation of a graph barycenter00(0), pp. 000–000, c© 0000 INFORMS 25

and the Log-Sobolev Inequality (referred to as LSI below):∫ΓG

f2 log

(f

‖f‖2,µβ

)2

dµβ ≤C(β)Eβ(f, f) (30)

The specific feature of this functional inequality deals with the quantum graph settings anddeserves careful adaptation of the pioneering work of [31]. This technical section is split into twoparts. The first one establishes a preliminary estimate when β = 0 (i.e., when dealing with theuniform measure on ΓG). The second one then uses this estimate to derive the asymptotic behaviorof the LSI when β −→+∞.

6.1. Preliminary control for µ0 on ΓGWe consider µ0 the normalized Lebesgue measure and use the standard notation for any measureµ on ΓG:

‖f‖W1,p(µ) := ‖f‖p,µ + ‖∇f‖p,µLet us establish the next elementary result:

Lemma 1. The ordinary Sobolev inequality holds on ΓG, i.e., for all measurable functions f wehave:

‖f−< f >0 ‖2p,µ0≤ e2/eL2

[DGL

2+ 1

]E0(f, f), (31)

where DG is the diameter of the graph and L its perimeter defined by:

L=∑e∈E

Le.

Proof: First, let us remind the reader that for a given interval I in dimension 1 equipped with theLebesgue measure λI , the Sobolev space W 1,p(λI) is continuously embedded in L∞(I) (compactinjection when I is bounded). In particular, it can be shown (see, e.g., [16]) that while integratingw.r.t. the unnormalized Lebesgue measure:

∀p≥ 1 ‖f‖L∞(I) ≤ e1/e‖f‖W1,p(λI ).

We now consider f ∈W 1,p(µ0). Since G has a finite number of edges we can write: ΓG =⋃n

i=1 ei,where each ei can be seen as an interval of length `i. We have seen that for each edge ei:

‖f‖L∞(ei) ≤ e1/e‖f‖W1,p(λei )

.

We can use a union bound since ΓG represents the union of edges, and deduce that:

‖f‖L∞(ΓG) = max1≤i≤n

‖f‖L∞(ei) ≤ e1/e max

1≤i≤n‖f‖W1,p(λei )

≤ e1/e‖f‖W1,p(λΓG),

where the inequality above holds w.r.t. the unnormalized Lebesgue measure. If L denotes the sumof the lengths of all edges in ΓG, we obtain:

‖f‖L∞(ΓG) ≤ e1/eL‖f‖W1,p(µ0). (32)

Second, we establish a simple Poincare inequality for µ0 on ΓG. For any function f ∈W 1,2(µ0),we use the equality:

‖f−< f >0 ‖22,µ0=

1

2

∫ΓG

∫ΓG

[f(x)−f(y)]2dµ0(x)dµ0(y) =1

2

∫ΓG

∫ΓG

[∫γx,y

f ′(s)ds

]2

dµ0(x)dµ0(y).

Author: On the calculation of a graph barycenter26 00(0), pp. 000–000, c© 0000 INFORMS

where γx,y is the shortest path that connects x to y, parametrized with speed 1 and f ′(s) refers tothe derivative of f w.r.t. this parametrization at time s. It should be noted that such a path existsbecause the graph ΓG is connected. The Cauchy-Schwarz inequality yields:

‖f−< f >0 ‖22,µ0≤ 1

2

∫ΓG

∫ΓG

[∫γx,y

|∇f(s)|2ds

]|γx,y|dµ0(x)dµ0(y)≤ DGL

2‖∇f‖22,µ0

. (33)

The Sobolev inequality is now an obvious consequence of the previous inequality: consider f ∈W 1,p(µ0) and note that since µ0(ΓG) = 1, then (32) applied with p= 2 leads to:

∀q≥ 1 ‖f−< f >0 ‖2q,µ0=

(∫ΓG

|f−< f >0 |qdµ0

)2/q

≤ ‖f−< f >0 ‖2L∞(ΓG)µ0(ΓG)

≤ e2/eL2‖f−< f >0 ‖2W1,2(µ0).

We can use the Poincare inequality established for µ0 on ΓG in Equation (33) and obtain:

∀q≥ 1 ‖f−< f >0 ‖2q,µ0≤ e2/eL2

[DGL

2+ 1

]‖∇f‖22,µ0

= e2/eL2

[DGL

2+ 1

]E0(f, f),

which concludes the proof. �Remark 1. The constants obtained in the proof of Lemma 1 above could certainly be improved.

Nevertheless, such an improvement would have little importance for the final estimates obtainedin Proposition 5.

6.2. Poincare Inequality on µβIn the following, we show a Poincare Inequality for the measure µβ for large values of β. Thispreliminary estimate will be useful for deriving LSI on µβ. This functional inequality is stronglyrelated to the classical minimal elevation of the energy function Uν for joining any state x to anystate y.

We first introduce some useful notations. For any couple of vertices (x, y) of ΓG, and for anypath γx,y that connects them, we define h(γx,y) as the highest value of Uν on γx,y:

h(γx,y) = maxs∈γx,y

Uν(s).

We define H(x, y) as the smallest value of h(γx,y) obtained for all possible paths from x to y:

H(x, y) = minγ:x→y

h(γ)

Now, for any pair of vertices x and y, the notation γx,y will be reserved for the path that attainsthe minimum in the definition of H(x, y). Such a path exists for any x, y because ΓG is connectedand possesses a finite number of paths that connect any two given vertices.

Finally, we introduce the quantity that will mainly determine the size of the spectral gap involvedin the Poincare inequality and the constant in the LSI (see the seminal works of [25] for a LDPprobabilistic interpretation and [32] for a functional analysis point of view):

c?(Uν) := max(x,y)∈Γ2

G

[H(x, y)−Uν(x)−Uν(y)] + minx∈ΓG

Uν(x) (34)

In the following, every time we write minUν we refer to minx∈ΓG

Uν(x).

Author: On the calculation of a graph barycenter00(0), pp. 000–000, c© 0000 INFORMS 27

Theorem 4 (Poincare inequality for µβ). For all measurable functions g defined on ΓG:

V arµβ (g)≤ |E|maxe∈E Lee2DG

2eβc

?(Uν)Eµβ (g, g).

Proof: Since the graph ΓG is connected, for any two points x and y, we can find a minimal pathγx,y that links them and that minimizes H. We denote x0 = x,x1, · · ·xl, xl+1 = y where the sequencex1, · · · , xl refers to the nodes included in the path γx,y.

V arµβ (g) =1

2

∫ΓG

∫ΓG

(g(y)− g(x))2dµβ(x)dµβ(y)

=1

2

∫ΓG

∫ΓG

(l∑i=0

(g(xi+1)− g(xi)

)2

dµβ(x)dµβ(y)

=1

2

∫ΓG

∫ΓG

(l∑i=0

∫ xi+1

xi

g′(s)ds

)2

dµβ(x)dµβ(y)

=1

2

∫ΓG

∫ΓG

(∫γx,y

g′(s)ds

)2

dµβ(x)dµβ(y)

≤ 1

2

∫ΓG

∫ΓG

∫γx,y

|∇g(s)|2ds|γx,y|dµβ(x)dµβ(y),

The last line is implied by the Cauchy-Schwarz inequality.

We can organize the terms involved in the above upper bound in the following way: for any edgeof the graph e∈E, we denote Ve the set of points (x, y) such that γx,y ∩ e 6= ∅. We therefore have:

V arµβ (g)≤ 1

2

∑e∈E

∫e

|∇g(s)|2ds

∫Ve

|γx,y|dµβ(x)dµβ(y)

≤ 1

2

∑e∈E

∫e

|∇g(s)|2µβ(s)

∫Ve

|γx,y|1

µβ(s)dµβ(x)dµβ(y)ds

≤ 1

2

∑e∈E

[∫e

|∇g(s)|2µβ(s)ds

][sups∈e

1

µβ(s)

∫Ve

|γx,y|dµβ(x)dµβ(y)

]≤ 1

2

∫ΓG

|∇g(s)|2µβ(s)ds sups∈ΓG

1

µβ(s)

∫Ves

|γx,y|dµβ(x)dµβ(y)

In this case, es denotes the edge of the graph that contains the point s and the set Ves is stillthe set of couples (x, y) defined above, associated with each edge es. We introduce the quantity Adefined as:

A= sups∈ΓG

1

µβ(s)

∫Ves

|γx,y|dµβ(x)dµβ(y).

Using this notation, we have obtained that for all functions g, we have the Poincare inequality:

V arµβ (g)≤ A2Eµβ (g, g). (35)

All that remains to be done is to obtain an upper bound of A.

A= sups∈ΓG

1

µβ(s)

∫Ves

|γx,y|dµβ(x)dµβ(y)

Author: On the calculation of a graph barycenter28 00(0), pp. 000–000, c© 0000 INFORMS

= sups∈ΓG

ZβeβUν(s)

∫Ves

e−β(Uν(x)+Uν(y))

Z2β

|γx,y|dxdy

=1

Zβsups∈ΓG

∫Ves

eβ(Uν(s)−Uν(x)−Uν(y))|γx,y|dxdy

Since γx,y is the minimal path for H(x, y), we have H(x, y) = maxs∈γx,y

Uν(s). Therefore:

A≤ 1

Zβsups∈ΓG

∫Ves

eβ((H(x,y)−Uν(x)−Uν(y))|γx,y|dxdy

≤ 1

Zβsups∈ΓG

∫Ves

eβ(c?(Uν)−min(Uν))|γx,y|dxdy

≤ eβ(c?(Uν)−min(Uν))

Zβsups∈ΓG

∫Ves

|γx,y|dxdy

≤ eβ(c?(Uν)−min(Uν))

Zβ|E|max

e∈ELe.

Using the definition of Zβ, we have:

A≤ eβc?(Uν)|E|maxe∈E Le∫

ΓG

e−β(Uν(x)−min(Uν))dx

. (36)

If x? ∈Mν is a Frechet mean that minimizes Uν , we designate B(x?,

1

β

), as the ball of center x?

and radius1

β, for the geodesic distance d on the graph ΓG. It is easy to check that:

|Uν(x)−Uν(x?)|=∣∣EY∼ν [d2(x,Y )− d2(x?, Y )

]∣∣≤EY∼ν ∣∣d2(x,Y )− d2(x?, Y )∣∣≤ 2DG× d(x,x?).

We can then deduce a lower bound on the denominator involved in (36):∫ΓG

e−β(Uν(x)−min(Uν))dx≥∫B

x?, 1β

e−β(Uν(x)−min(Uν))dx

≥∫B

x?, 1β

e−2DGβd(x,x?)dx

≥ e−2DG

∫B

x?, 1β

dx.

The Lebesgue measure of B(x?,

1

β

)may be lower bounded by β−1 since there is, at the least, one

path in ΓG passing by the point x?. Inserting this inequality in (36) gives:

A≤ |E|maxe∈E

Lee2DG βeβc

?(Uν).

Using this upper bound in (35) leads to the desired Poincare inequality. �

Author: On the calculation of a graph barycenter00(0), pp. 000–000, c© 0000 INFORMS 29

6.3. Sobolev Inequalities on µβ6.3.1. Preliminary control on Dirichlet forms The next result will be useful to derive

a LSI for µβ from a Poincare inequality on µβ (given in Theorem 4). It generalizes the Poincareinequality for p norms with p > 2 while using the Sobolev inequality given in Lemma 1.

First, we introduce the maximal elevation of Uν as:

M = supx∈ΓG

Uν(x)− infx∈ΓG

Uν(x). (37)

Note that in our case, M may be upper bounded by D2G.

Proposition 4. For any p > 2 and all measurable functions f , we have:

∀β ≥ 0 ‖f−< f >β ‖2p,µβ ≤ 4L2

(DGL

2+ 1

)e2/e+MβEβ(f, f). (38)

Proof: The Jensen inequality applied to the convex function x 7−→ xp yields:

‖f−< f >β ‖pp,µβ =

∫ΓG

|f−< f > |pβdµβ

=

∫ΓG

∣∣∣∣f(x)−∫

ΓG

f(y)dµβ(y)

∣∣∣∣p dµβ(x)

=

∫ΓG

∣∣∣∣∫ΓG

f(x)− f(y)dµβ(y)

∣∣∣∣p dµβ(x)

≤∫

ΓG

∫ΓG

|f(x)− f(y)|p dµβ(y)dµβ(x)

≤∫

ΓG

(|f(x)−< f >0 |+ |f(y)−< f >0 |)p dµβ(y)dµβ(x)

Again, the Jensen inequality |a+ b|p ≤ 2p−1[|a|p + |b|p] implies that:

‖f−< f >β ‖pp,β ≤ 2p−1

∫ΓG

|f(x)−< f >0 |p + |f(y)−< f >0 |pdµβ(y)dµβ(x)

≤ 2p∫

ΓG

|f(x)−< f >0 |pdµβ(x) = 2p‖f−< f >0 ‖pp,µβ

We conclude that:‖f−< f >β ‖2p,µβ ≤ 4‖f−< f >0 ‖2p,µβ . (39)

Using the fact that:

µβ(x) =e−βminUν

Zβ≤ e−βminUν

Zβ×Z0µ0(x),

we also have: ‖f−< f >0 ‖pp,µβ ≤Le−βminUν

Zβ‖f−< f >0 ‖pp,µ0

, since Z0 is the perimeter L of the

graph ΓG. Consequently, we obtain:

‖f−< f >0 ‖2p,µβ =(‖f−< f >0 ‖pp,µβ

)2/p

≤(Le−βminUν

Zβ

)2/p

‖f−< f >0 ‖2p,µ0

Using inequality (39), the fact that Zβ ≤Le−βminUν and the assumption 2/p < 1, we conclude that:

‖f−< f >β ‖2p,µβ ≤4Le−βminUν

Zβ‖f−< f >0 ‖2p,µ0

.

Author: On the calculation of a graph barycenter30 00(0), pp. 000–000, c© 0000 INFORMS

The Sobolev inequality (1) implies that:

‖f−< f >β ‖2p,µβ ≤4L3 (DGL/2 + 1)e2/e−βminUν

ZβE0(f, f). (40)

We now find an upper bound for the Dirichlet form E0(f, f) that involves Eβ(f, f):

E0(f, f) =

∫ΓG

<Of,Of > dµ0

=Zβ

∫ΓG

<Of,Of > eβUν(x) e−βUν(x)

Zβdµ0

≤Zβ exp

(β supx∈ΓG

Uν(x)

)Z0

∫ΓG

<Of,Of > dµβ

≤Zβ exp

(β supx∈ΓG

Uν(x)

)L

Eβ(f, f) (41)

Putting (40) and (41) together concludes the proof. �

6.3.2. Log-Sobolev Inequality on µβ For all probability measures µ and all measurablefunctions f , we denote:

Entµ(f2) =

∫ΓG

f2 log

(f2

‖f‖22,µ

)dµ

Proposition 5. The Log-Sobolev Inequality holds on ΓG. A constant CΓG exists such that forall β ≥ 0 and all µβ-measurable functions f , we have:

Entµβ (f2)≤CΓG [1 +β]ec?(Uν)βEβ(f, f)

Proof: We consider β ≥ 0 and a µβ measurable function f . We apply the Jensen inequality for thelogarithmic function and the measure f2/‖f‖22,µβdµβ to obtain:

∫ΓG

(f

‖f‖2,µβ

)2

log

(f2

‖f‖22,µβ

)dµβ =

2

p− 2

∫ΓG

(f

‖f‖2,µβ

)2

log

(|f |p−2

‖f‖p−22,µβ

)dµβ

≤ 2

p− 2log

(∫ΓG

|f |p−2

‖f‖p−22,,µβ

f2

‖f‖22,,µβdµβ

)

≤ 2

p− 2log

(‖f‖pp,µβ‖f‖p2,µβ

)=

p

p− 2log

(‖f‖2p,µβ‖f‖22,µβ

).

Observing that for all x, δ > 0 we have log(xδ)≤ δx, we obtain log(x)≤ xδ+ log(δ−1). Therefore:

∀δ > 0

∫ΓG

f2 log

(f

‖f‖2,µβ

)2

dµβ = ‖f‖22,µβ

∫ΓG

(f

‖f‖2,µβ

)2

log

(f2

‖f‖22,µβ

)dµβ

≤p‖f‖22,µβp− 2

log

(‖f‖2p,µβ‖f‖22,µβ

)

≤p‖f‖22,µβp− 2

[δ‖f‖2p,µβ‖f‖22,µβ

+ log1

δ

].

Author: On the calculation of a graph barycenter00(0), pp. 000–000, c© 0000 INFORMS 31

Replacing f with f−< f >β and choosing δ= e−Mβ leads to:∫ΓG

(f−< f >β)2 log

(f−< f >β

‖f−< f >β ‖2,β

)2

dµβ

≤ p

p− 2

[e−βM‖f−< f >β ‖2p,µβ +Mβ‖f−< f >β ‖22,µβ

].

Proposition 4 and the Poincare inequality established in Theorem 4 yields:

Entµβ[(f−< f >β)2

]=

∫ΓG

(f−< f >β)2 log

(f−< f >β

‖f−< f >β ‖2,µβ

)2

dµβ

≤ p

p− 2

(4L2DGL+ 2

2e2/e +Mβ|E|max

e∈ELee2DG

2βec

?(Uν)β

)Eβ(f, f) (42)

It remains to use Rothau’s Lemma (see Lemma 5.1.4 of [7]) that states that for any measure µ andany constant a:

Entµ(g+ a)2 ≤Entµg2 + 2

∫ΓG

g2dµ.

Let p= 4 in Equation (42). Putting this together with Rothau’s Lemma and Theorem 4, we obtainthe following:

Entµβ (f2) =

∫ΓG

f2 log

(f2

‖f‖22,µβ

)dµβ

≤Entµβ[(f−< f >β)2

]+ 2V arµβ (f)

≤Eβ(f, f)[β|E|max

e∈ELee

2DG(1 +Mβ)ec?(Uν)β + 4L2(2 +DGL)e2/e

]≤CΓG [1 +β]ec

?(Uν)βEβ(f, f),

where CΓG is a large enough constant (independent of β) that could be made explicit in terms ofconstants DG, |E| and L since we trivially have M ≤D2

G. �

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