Antoine Billot

How to shake the Invisible Hand (when Robinson meets Friday)

Keywords: A13, D71, J15 preferences, social interaction, invisible hand JEL Classi…cation: A13-D71-J15 Preferences, social interaction, invisible hand

We propose to de…ne the invisible hand by (i) modelling the mechanism itself (not to just assume its existence) and (ii) making explicit the limit conditions for its working. For that purpose, we simply assimilate the working of the invisible hand mechanism to the existence of a social preference such that individual and social optimalities are consistent. In introducing the possibility of interaction among individuals, we then suggest that the standard 'Robinson case' or social atomicity is just a degenerate feature of a more general requirement that we call the Global Network Agreement. Our main result is that the invisible hand mechanism does keep on working when there is an interaction between Robinson and Friday if the former (resp. the latter) is sensitive to the latter (resp. the former) in such a way that they exhibit some agreement in preferences. Hence, the 'Robinson case' naturally satis…es this property since nor Robinson neither Friday can disagree with himself. But more cooperative situations are also allowed in order to extent the invisible hand mechanism to cases with interactions. cependant, d'autres cas, beaucoup plus coopératifs sont eux aussi aussi compatibles avec le mécanisme de la main invisible.

Nous proposons de dé…nir la Main Invisible sous la forme d'un mécanisme formel explicite. Pour cela, nous assimilons ce mécanisme à l'existence d'une relation de préférence sociale telle que les optimalités individuelle et collective soient compatibles. Puis en introduisant une possibilité d'interaction entre agents, nous suggérons que le cas standard (Adam Smith) d''atomicité sociale'constitue le cas limite d'une classe plus générale de situations satisfaisant toutes une certaine contrainte appelée le "Global Network Agreement"(ou Accord Global de Réseau). Notre résultat principal est alors que le mécanisme de la main invisible s'applique même en présence d'interactions dès lors que les agents qui interagissent sont sensibles en préférence les uns aux autres d'une manière telle qu'ils manifestent entre eux un consensus minimal. Aussi, le cas standard (chaque agent est un Robinson) satisfait naturellement cette propriété, I thank Maurice Salles, Jacques-François Thisse, James Friedman, Claudio Mezzetti, the participants of the doctoral seminars at Venice International University and two referees for comments and suggestions. Furthermore, I specially thank Tzachi Gilboa whose remarks clearly improve the proof and even the meaning of the second theorem. Usual disclaimer applies. For correspondence : Université Panthéon-Assas, Paris 2, 92 rue d'Assas, 75006, Paris, France, e-mail address : billot@u-paris2.fr

Introduction

The idea that vice or sel…shness leads to virtue or social good appears in 1723 with La Fable des Abeilles or Les Vices Privés Font le Bien Public proposed by Bernard Mandeville. 1 According to Mandeville, a hive is analogous to a human society. And what do we observe in a hive? Two things. First, the bees seem to behave in a coordinated and cooperative way while in reality they are just sel…sh maximizers of their own (private) welfare. Second, such decentralized actions of the individual bees promote the hive welfare better than could be achieved by a 'central planner bee'. How is it possible? In his famous book on The Wealth of Nations written in 1776, Adam Smith answered: because of the so-called 'invisible hand'. Two centuries later, the invisible hand metaphor still works: the liberal heirs of the classical economists can a¢ rm that markets would spontaneously produce socially optimal results if only government refrains from interference. Yet, such a use of the Mandeville-Smith's intuition is purely ideological: it is a defence for laissez-faire capitalism. (The interpretation of the concept of the invisible hand is de…nitely subtle and consequently ambiguous: see [START_REF] Nozick | Philosophical Explanations[END_REF][START_REF] Rothschild | Adam Smith and the Invisible Hand[END_REF], for further discussions). The existence and the proper working of the invisible hand mechanism are taken for granted, without specifying the limit conditions for it to work. In addition, how should one precisely characterize an invisible hand mechanism? First, the outcome to be explained must be produced as a result of a decentralized process: there are no explicit agreements among the individuals. Second, the invisible hand mechanism must be non-intentional: the individuals do not intend to produce the outcome, that is social welfare; a by-product of their actions is the social desirability of the …nal outcome. Actually, since the process should work even if the individuals have no knowledge of the process, it is called invisible. 2In this paper, we propose to de…ne the invisible hand mechanism in a more systematic methodological way. Speci…cally, we implicitly consider that it is contingent if the mechanism is actually working. What is interesting is rather (i) to model the mechanism itself, not to just assume its existence and (ii) to make explicit the limit conditions for it to work.

For that purpose, recall that an individual optimal decision is a maximal element of an individual preference order and a social optimal decision is a maximal element of a social preference order. Then, we suggest that an invisible hand mechanism naturally works when there is no con ‡ict between each individual optimal decision and the social optimal one. Consequently, we consider the working of an invisible hand mechanism as the existence of a social preference for which individual and social optimalities are consistent. The formal conditions of existence of such a social preference characterize the environment for the invisible hand mechanism to work.

Traditionally, one thinks that the invisible hand mechanism needs social atomicity, namely the fact that no individual can in ‡uence the social decision by means of his only preferences. However, in introducing interactions among individuals (à la [START_REF] Akerlof | Social Distance and Social Decisions[END_REF], we suggest that social atomicity, that is the 'Robinson case', is a particular feature of a more general requirement that we call Global Network Agreement: the invisible hand mechanism requires that individuals agree in some sense in preferences when they interact. In other words, each Robinson naturally satis…es this property since no one can disagree with himself. But other situations, de…nitely more cooperative, are allowed as well.

The organization of the paper is as follows: In Section 2, we present the model of interaction and the utility representation of individual preferences translating interactions (Theorem 1). In Section 3, we model the invisible hand mechanism and prove the limit condition of its working, namely the presence of a local network disagreement (Theorem 2). Finally, in Section 4, we illustrate our result with two examples where the invisible hand mechanism works: …rst, the Robinson case (no interaction) and second, an Interaction case with Global Network Agreement.

Individuals

Individual Preferences

Consider a given and …nite set S of individuals i (i : 1; :::; n). Call coalition any nonempty subset of S.

Consider an arbitrary given set C R of (monetary) consequences expressed by real numbers and X a set of acts, that is functions x from the set S of individuals to that C of consequences. Any act x of X is such that x (i) corresponds to the consequence of the act x for individual i.

For any coalition A S, x (A) denotes the tuple [x (i)] i2A . Furthermore, given a coalition A and two acts x and y, an act denoted

(x A ; y A ) is such that (x A ; y A ) (i) = x (i) for all individuals i 2 A and (x A ; y A ) (j) = y (j) for all other individuals j = 2 A. By convention, we write (i) x y if x (i) y (i) for all individuals i 2 S, (ii) x > y if x y and x (i) > y (i) for some individuals i 2 S and (iii) x y if x (i) > y (i) for all individuals i 2 S. Each individual i 2 S is characterized by a preference relation < i ,
that is a binary relation de…ned over X X such that x < i y means that for individual i, x is at least as good as y. Two acts x; y are considered indi¤erent, which is denoted x i y, when we simultaneously have x < i y and y < i x.

Besides, all preference relations are supposed to be (i) nontrivial: for all i 2 S, (x i ; z i ) i (y i ; z i ) for some x; y; z 2 X and (ii) monotone: for all i 2 S, if x (i) > y (i), then (x i ; z i ) i (y i ; z i ) for all z 2 X.

We also assume the following standard axioms.

Axiom 1 (Complete Weak Ordering) : 8i 2 S, the preference relation < i is a complete weak-order on X 2 .

Axiom 2 (Continuity) : 8i 2 S, 8x; y; z 2 X such that x < i y < i z, there exists 2 [0; 1] such that x + (1 ) z i y.

Axiom 3 (Independence) : 8i 2 S, 8x; y; z 2 X such that x < i y, 8 2 ]0; 1[:

x + (1 ) z < i y + (1 ) z.

Interaction and vNM Expected Utility Representation

Several economic situations involve basic interactions sometimes called 'network externalities' such as urban congestion, fashion e¤ect, conformism or altruism and so on... The common idea of all these approaches is that individuals are assumed to be interdependent whenever their preferences may be a¤ected by the social neighborhood (see e.g., [START_REF] Postlewaite | The Social Basis of Interdependent Preferences[END_REF][START_REF] Billot | How Liberalism Kills Democracy: Sen's Theorem Revisited[END_REF]. Here, if i interacts with j then i may be 'su¢ ciently sensitive to j' to broaden his own set of preferences determinants in order to encapsulate the consequence for j of all acts i faces. 3 For instance, [START_REF] Luttmer | Group Loyalty and the Taste for Redistribution[END_REF] shows that numerous young black lawyers issued from the poor suburb of Chicago …nally militate in favor of a …scal policy that proposes to largely redistribute towards the poor social classes (even if, for that purpose, they will individually pay more taxes) rather than choosing to support a liberal program which maximizes their sel…sh interests.

Interaction

Consider that there exists a binary relation I de…ned on S 2 such that 'iIj'means that 'i interacts with j'and assume that this relation I is re ‡exive: 8i 2 S, iIi. Consider now, for any i 2 S, a probability distribution p i : 2 S ! [0; 1] with P j2S p i (j) = 1 where p i (j) is the probability for i to be sensitive to j. Assume that if i has no interaction with j, then p i (j) = 0, i.e. i's sensitivity to j is null. On contrary, each time iIj, p i (j) > 0 and i is sensitive to what happens to j. (Intuitively, the more i interacts with j, the more he is sensitive to him.) Note that since we suppose I to be re ‡exive, then p i (i) > 0. Finally, the support of the distribution p i naturally corresponds to i's individual network, that is the set of all individuals with whom he has an interaction. We denote it S i :

S i = fj 2 S : iIjg = def fj 2 S : p i (j) > 0g .
(1)

Note that the re ‡exivity of I can also be written as

[8i 2 S, i 2 S i ]. 4
Two cases occur:

(1) (Ic as Interaction case). There exists at least one individual whose network is nonsingleton:

[9i; j 2 S, i 6 = j, s.t. iIj] () [9i 2 S s.t. p i (i) < 1] () [9i 2 S s.t. S i 6 = fig] .
For convenience, we say in that case that p i is a nonsingleton distribution.

(2) (Rc as Robinson case). All individual networks are singleton:

[8i 2 S, / 9j 2 S, i 6 = j, s.t. iIj] () [8i 2 S, p i (i) = 1] () [8i 2 S, S i = fig] .
For convenience, we say in that case that any p i is a singleton distribution.

The standard case, namely that which is commonly used in general microeconomics, is obviously given by Rc.

Remark 1 : Transitivity of the interaction I -i.e. 8i; j; k 2 S, [iIj and jIk] )

[iIk] -, can also be written as [j 2 S i ) S j S i ].5

vNM Expected Utility Representation

Following Morris intuition in (1997Morris intuition in ( , 2000)), we then propose to translate the behavior of an individual who is sensitive to other individuals through the probability distribution which a¤ects his utilities in a setting à la von Neumann-Morgenstern (vNM) where u i (x (i)) translates i's utility when facing the consequence of x for him, namely x (i), while u i (x (j)) corresponds to i's utility when considering the consequence of x for j, namely x (j).

Theorem 1 (vNM) : Under A1-A3, for each individual i 2 S, there exists a strictly increasing and continuous utility function u i : X ! R such that:

x < i y , X j2S p i (j) u i (x (j)) X j2S p i (j) u i (y (j)) , (2)
where p i is a given probability distribution, p i : 2 S ! [0; 1] with P j2S p i (j) = 1, translating i's interactions in S. Moreover, u i is unique up to a linear transformation.

The proof is strictly equivalent to that of Proposition 5.4 in Kreps (1988, Ch. 5., 46-50).

Therefore each individual preference < i satisfying A1-A3 has a vNM representation as de…ned in (2). This way, if iIj, i takes into account -by empathy -the consequence x (j) of x weighted by the degree p i (j) of his sensitivity to j. On the contrary, if i does not interact with j, the in ‡uence of the consequence x (j) in i's evaluation of x is null.

In the Robinson case, (2) naturally becomes:

x < i y , u i (x (i)) u i (y (i)) (3)
since S i = fig (see also [START_REF] Samet | Between Liberalism and Democracy[END_REF], for the interpretation of such a constraint as the quali…cation rule of liberalism).

Throughout the sequel, we consider that each individual preferences has a vNM representation as de…ned by (2).

Invisible Hand

Consider a …nite subset of acts, X, and call it a problem. A problem corresponds to a particular decision which must be made by the society S.

For each problem X and each individual i 2 S, it can be de…ned a set of i's optimal solutions, that is the acts dominating in preferences all other acts that are candidates to solve the problem: i = fx 2 : x < i y, for all y 2 g . ( 4)

Proposition 1 : Under A1, each individual optimal solution set i associated to any problem is nonempty.

Proof : Straightforward by …niteness of and weak-ordering completeness of < i .

De…ne now a decision rule as a function : S ! and say to be optimal when 8i 2 S, (i) 2 i .

(5)

Remark 2 : A decision rule does take into account the individuals' social network since each i is based upon i's preference and each i's preference is related to S i by its vNM representation as de…ned in (2).

When such an optimal decision rule is followed, (i) determines the acts which dominate -for i -all other possible solutions of the problem. Such a rule generates an abstract new act ! ' , not necessarily in , such that ' (i) = i (i), 8i 2 S (where i (i) is the i's consequence of the act (i)).The class of acts generated by optimal decision rules is denoted and de…ned as follows:

= fx 2 X : x (i) = i (i) , 8i 2 S, for some optimal g . ( 6)

Proposition 2 : Under A1-A3 the class of issues generated by optimal decision rules is nonempty.

Proof : This follows from Proposition 1 and (6).

Each element x of the nonempty class of acts generated by optimal decision rules is designed such that the individual consequence x (i) is < i -maximal by de…nition of i and (5-6). Then, if there is no social preference such that each act of is weakly preferred to any other act of the problem , we know that a con ‡ict arises between the society S as a whole and some of its members: hence the invisible hand does not seem to work. A contrario, if there is no contradiction between the society S as a whole and any of its members in the sense where the best decision for the former is consistent with each best decision of the others, the invisible hand seems to work.

Introduce one supplementary condition on the individual preferences. Given individual preferences (< i ) i2S and a coalition A S, say that x < A y if x < i y for all i 2 A, and x A y if x < A y and x i y for some i 2 A.

Global Network Agreement : 8i 2 S, x < i y ) x < S i y for all x; y 2 X.

Global Network Agreement means that within each given individual network, no con ‡ict arises between the individual and its own network. In return a local (network) disagreement corresponds to the fact that there exists an individual, say i, such that x < i y ; x < S i y for all x; y 2 X, i.e. 9x 0 ; y 0 2 X such that x 0 < i y 0 while y 0 S i x 0 .

Lemma 1 : In the Robinson case, there is a Global Network Agreement.

Straigthforward from (Rc).

Note, indeed, that a Global Network Agreement does not imply that S i = fig, 8i 2 S. Thus, let us study other situations, especially situations where individual networks are nonsingleton.

Lemma 2 : There is a Global Network Agreement i¤ the interaction I is transitive.

Proof : ()) Transitivity of I can also be written as [j 2 S i ) S j S i ] (see Remark 1). It means that, 8i; j; k 2 S, if i 2 S j and j 2 S k then i 2 S k . Suppose now, on the contrary, that there exist three individuals 1; 2; 3 2 S such that 1 2 S 2 , 2 2 S 3 but 1 = 2 S 3 . Let x (1) = 1 and x (i) = 0 for all i 6 = 1. By monotonicity of the preferences, x 1 0 n (since 1 2 S 1 by re ‡exivity of I), x 2 0 n (since 1 2 S 2 ), x 3 0 n (since 1 = 2 S 3 ) and …nally x < i 0 n for all i 2 S, where 0 n = (0; :::; 0). Thus, x S3 0 n since 2 2 S 3 . But, x 3 0 n while x S3 0 n means that there is a local disagreement.

(() Suppose that there is a local disagreement. It means that, for some i 2 S, x Si 0 n while x i 0 n . Hence, there exists j 2 S i such that x j 0 n . Let x (k) = 1 and x (i) = 0 for all i 6 = k. If x j 0 n , this means that k 2 S j , k 6 = j, i.e. fj; kg S j . Since x i 0 n , then k = 2 S i . Consequently j 2 S i 6 ) S j S i and I is not transitive.

De…ne a social preference < as a preference relation which satis…es (A1-A3) and a Strong Monotonicity condition: 8i 2 S, 8x; y 2 X, (x i ; z i ) < (y i ; z i ) , x (i) y (i) .

(SM)

This condition translates the fact that, ceteris paribus, the society is better o¤ when allocating more than less to any given individual. The modeling of an invisible hand mechanism that we propose is then that acts in (which may or may not be included in the problem ) must be at least as good for the social preference < as acts in , that is x < y for all acts x 2 and y 2 .

Invisible Hand (IH) : There exists a social preference < such that for each …nite problem X there exists an act x 2 such that x < y, for all y 2 . The Global Network Agreement condition implies that the sensitivity of an individual to another one, when it exists (i.e. in the Interaction case), can be translated through its individual network. Nevertheless, this result does not clarify the implication relation between IH and a Global Network Agreement. Does IH implies Global Agreement or the other way round? If it is straightforward that IH is consistent with the Robinson case, that is when all the individual networks are singleton, we must know if it can be consistent as well with nonsingleton individual networks.

Theorem 2 : If IH holds, then there is a Global Network Agreement.

Lemma 1 shows that there is a Global Network Agreement i¤ the interaction I is transitive. Hence we have now to precise the relationship between IH and the properties of the interaction I: Lemma 3 : If IH holds, then the interaction I is transitive.

Proof : Suppose that I does not satisfy transitivity, i.e. [j 2 S i 6 ) S j S i ] for some i; j; k 2 S. This means for instance that there exist three individuals 1; 2; 3 2 S such that 2 2 S 1 , 1 2 S 3 but 2 = 2 S 3 . Now, by A2, it is possible to de…ne a problem = fx; 0g where the act x is de…ned as (" 1 ; 1; " 3 ; 0; :::; 0) (with x (1) = " 1 , x (2) = 1, x (3) = " 3 and x (i) = 0 for all i 6 = 1; 2; 3 with " 1 , " 3 positive and very small) such that the act 0 n 1 x (since f1; 2g S 1 and " 1 very small), 0 n 2 x (since 2 2 S 2 while 1 = 2 S 2 and 3 = 2 S 2 ) but x 3 0 n .6 Hence, if y 2 , this implies y (3) = " 3 and y (i) = 0 for all i 6 = 3, i.e. = f0; 0; y(3); 0; :::; 0g. Thus, 0 n y for all y 2 . This is a contradiction with IH.

Proof of Theorem 2: It is straightforward from Lemmas 1 and 2.

If a con ‡ict arises between the society as a whole and some of its members, the invisible hand does not work. This is clearly consistent with a society based on individuals who behave like Robinson. However, Theorem 2 shows that it is possible for the invisible hand mechanism to be relevant whenever 'Robinson meets Friday who meets...' if the interaction between the individuals is transitive such that there is no local disagreement. In that sense, if individuals throught networks are su¢ ciently connected in preferences, the standard invisible hand mechanism can be extented to subsets of individuals.

Examples

As an illustration of situations where IH holds, consider the two following examples:

Example 1 (Robinson Case) : Suppose three individuals f1; 2; 3g whose individual distributions are singleton, i.e. S 1 = f1g, S 2 = f2g and S 3 = f3g, and utilities are given by u i (x) = x for i : 1; 2; 3. Then, for two programs x; y: 8 < :

x < 1 y x < 2 y x < 3 y , , ,

x (1) y (1) , x (2) y (2) , x (3) y (3) .

Consider now the following problem: = fx; 0g, where the act x = "; 1; " 2 , i.e. x (1) = ", x (2) = 1 and x (3) = " 2 . Now, it is obvious that the act 0 n=3 = (0; 0; 0) is optimal for individuals 2 and 3 and the act x is optimal for 1. Thus, = f(x (1) ; 0; 0)g, that is ("; 0; 0). Now, by SM, the act ("; 0; 0) (0; 0; 0) for each social preference and then IH holds.

Example 2 (Interaction Case with Global Agreement) : Suppose three individuals f1; 2; 3g such that 1 and 3 have nonsingleton individual distributions, i.e.

p 1 (1) = 1 2 = p 1 (2), p 3 (2) = 1 2 = p 3 (3) while 2's individual distribution is p 2 (2) = 1. This corresponds to S 1 = f1; 2g, S 2 = f2g and S 3 = f2; 3g. Suppose that all utilities are given by u i (x) = x for i : 1; 2; 3. Then, for two acts x; y, we have: 8 < :

x < 1 y x < 2 y x < 3 y , , ,

Actually, the …rst version of this text is a long poem published in 1705 under the title: The Grumbling Hive or Knaves Turned Honest.

However, the invisibility is not an essential requirement for a mechanism to be considered as an invisible hand mechanism. The result in question needs not be unknown to the individuals participating in its production. Modern discussion about the invisible hand allows the individuals to know the result of their actions. It is nevertheless required that they do not intend to produce that result. The result should be an unintended consequence of the action, that is a by-product of intentions to do something else.

This sort of interaction does not suppose any strategic behavior nor coordination: for instance, an individual i can be sensitive to what happens to another individual j without j being sensitive to i.

In the language of epistemic logics à la[START_REF] Kripke | Semantical Analysis of Modal Logic[END_REF] (see for instance[START_REF] Hughes | An Introduction to Modal Logic[END_REF] Cresswell, 1989, or Billot and[START_REF] Billot | Epistemic Properties of Knowledge Hierarchy[END_REF], the individual network S i is called a syntactical operator and the distribution p i is its valued semantical counterpart while the society S plays the role of the set of possible worlds.

See Billot and Walliser (1999, §2.4, 190-192) for details about the way semantical properties can be equivalently interpreted in syntax.

Take for instance, p 3 (1) = p 3 (3) = :5 with " 1 > " 3 > 0.


Dimitri Papadimitriou

Pedro Pedroso

Piet Demeester

Performance Analysis of a Dynamic Compact Multicast Routing Scheme

Keywords: routage compact, point-à-multipoint, dynamique, performance

Les algorithmes de routage compact ont pour objectif de trouver le meilleur compromis entre le nombre de bit mémoire nécessaire pour stocker les entrées des tables de routage et le coût des chemins de routage que ces algorithmes produisent. Dans cet article, nous présentons un algorithme de routage compact permettant la construction dynamique de chemins de routage point-à-multipoint pour la distribution de données depuis une source quelconque vers un ensemble quelconque de destinations. L'arbre déterminé par ce chemin et servant à la distribution du trafic est communément appelé arbre de distribution de trafic multicast. Au moyen de cet algorithme, cet arbre est capable d'évoluer dynamiquement en fonction de l'arrivée des demandes de souscription à la source de trafic associée. Nous étudions les compromis entre l'étirement produit par l'algorithme de routage, la taille et le nombre d'entrées résultantes dans les tables de routage ainsi que le coût de la communication.

Introduction

Compact unicast routing aims to find the best tradeoff between the memory-space required to store the routing table (RT) entries at each node and the stretch factor increase on the routing paths it produces. Such routing schemes have been extensively studied following the model developed in [START_REF] Peleg | A Trade-off between Space and Efficiency for Routing Tables[END_REF]. Since then, in accordance to the distinction between labeled (nodes are named by polylogarithmic size labels encoding topological information) and name-independent (node names are named by topologically independent) compact routing schemes have been designed, notably in [START_REF] Thorup | Compact routing schemes[END_REF] and [START_REF] Abraham | Compact name-independent routing with minimum stretch[END_REF]. Recently introduced in [START_REF] Abraham | Compact Multicast Routing[END_REF], dynamic compact multicast routing algorithms construct point-to-multipoint routing paths from any source to any set of destinations (or leaves). These paths define multicast traffic distribution tree (MDT) since the algorithm instantiates local routing state so that each MDT node can derive the entries to forward the multicast traffic received from a given source to its leaves.

In this paper, we propose in Section 2 a dynamic compact multicast routing (CMR) algorithm that enables the construction of point-to-multipoint routing paths for the distribution of multicast traffic from any source to any set of leaf nodes. Compared to [START_REF] Abraham | Compact Multicast Routing[END_REF], the proposed name-independent CMR algorithm is i) leaf-initiated: the join/leave requests are initiated by the leaf nodes, ii) distributed: transit nodes process the join/leave requests and compute the routing table entries (no centralized processing by the root of the MDT), iii) dynamic: join/leave requests are processed on-line as they arrive without re-computing and/or re-building the MDT from scratch, and iv) protocol independent of any underlying unicast routing topology: the local knowledge of the cost to direct neighbor nodes is sufficient for the proposed algorithm to properly operate. In Section 3, we evaluate the CMR performance by measuring the stretch of the routing paths it produces, the memory space needed to store the resulting RT entries as well as its communication cost, i.e., the number of message exchanged to build the MDT. For this purpose, we simulate the CMR algorithm on synthetic power law graphs comprising 10k nodes that are representative of the Internet topology. Two reference schemes, the Shortest Path Tree (SPT) and the Steiner Tree (ST) algorithm are used to compare the CMR performance over the same topologies. We conclude this paper in Section 4 by analyzing our results against possible improvements left as future work.

CMRA Algorithm

The objective of the proposed algorithm is to minimize the routing table (RT) size at each node v ∈ V, |V| = n, at the expense of i) routing multicast packets on point-to-multipoint paths with relative small deviation compared to the optimal stretch obtained with the Steiner Tree (ST), and ii) higher communication cost compared to the shortest path tree (SPT). For this purpose, the CMR algorithm reduces the storage of routing information by maintaining during the MDT construction the direct neighbor-related entries per node v, i.e., only the local routing information (degree(v) entries) instead of global routing information (|n-1| entries). The information needed to reach a given multicast source s is acquired by means of a search mechanism that returns the upstream neighbor node along the least cost branching path to the MDT sourced at s. Such mechanism is triggered whenever a node decides to join a given multicast source s as part of a multicast group g. After a node becomes member of the MDT, a multicast routing entry is dynamically created and stored in the so-called tree information base (TIB) from which a multicast forwarding entries is derived. The reduction in memory space consumed by the RT of each node results however in higher communication cost compared to the SPT and ST reference algorithms. In order to keep the communication cost as low as possible, we introduce a maximum (dissemination) path budget in each message so as to discard messages with too long and unneeded range.

Preliminaries

Consider a network topology modeled by an undirected graph G = (V,E,c) where the set V, |V| = n, represents the finite set of nodes or vertices (all being multicast capable), the set E, |E| = m, represents the finite set of links or edges, and c a non-negative cost function c: E → Z + that associates a cost c(u,v) to each link (u,v) ∈ E. For u, v ∈ V, let c(u,v) denote the cost of the path p(u,v) from u to v in G, where the cost of a path is defined as the sum of the costs along its edges. Let S be the finite set of source nodes, S ⊂ V, and let D be the finite set of all possible destination nodes that can join a multicast group g, D ⊆ V\{S}. Let s ∈ S, and d ∈ D denote a source node and a destination (or leaf) node, respectively. A multicast distribution tree T s,M = (V T , E T ) is defined as a connected sub-graph of G without cycles, i.e., a tree rooted at s ∈ S with leaf node set M, M ⊆ D. The set M corresponds to the current set of nodes at a given construction step of the MDT identified by the tuple multicast source s-multicast group g (<s,g>).

Description

The multicast distribution tree T s,D is constructed iteratively. At each step ω (ω = 1,2,..,|D|) of the leaf-initiated construction, a randomly selected node u joins T s,M , M ⊆ D. If node u is already part of T s,M (u ∈ V T ) then it is either a transit or a branching node of the MDT. Otherwise, node u is not part of T s,M (u ∈ D\{V T }) and it must search for the least cost branching path from node u to node v ∈ T s,M . Among the set P u,v of possible paths p(u,v) from node u ∉ T s,M to node v ∈ T s,M , the least cost branching path p(u,v)* = min{c(u,v) | p(u,v) ∈ P u,v }. The cost c(u,v) of the path p(u,v) is defined as the sum of the cost c(u,w) of the edge (u,w), w being the upstream neighbor node of u (or succ(u)) and the cost c(w,v) of the path p(w,v). Two types of messages are involved at each step of the construction, namely the request (type-R) messages flowing in the upstream direction towards the multicast source s, and the response (type-A) messages sent in the downstream direction towards the joining leaf node u. Type-R messages comprise the following information i) a sequence number {u id , r id } to prevent duplication of messages, where u id identifies the leaf node u and r id identifies its request to join/leave the multicast source/group pair <s,g>, ii) the leaf node u's timer value τ(u) that sets the waiting time at intermediates nodes before answering back to the downstream neighbor node, and iii) a path budget π, starting from π(u) = π max , set at leaf node u. The π max value is bound by the graph diameter (the length of the longest shortest path) for which approximation algorithms exist, as well as method for computing a lower and upper bounds [START_REF] Magnien | Fast computation of empirically tight bounds for the diameter of massive graphs[END_REF]. Starting from node u, the path budget π is decremented at each node w according to the travelled edge cost. If π(w) reaches 0 at node w, the latter does not further propagate the type-R message in order to keep the communication cost as low as possible. Type-A messages sent in response to type-R messages comprise i) the cost c(w,v)* of the locally selected least cost path p(w,v)* from the local node w to v ∈ T s,M .

Type-R message

When leaf node u decides to join the multicast source-group pair <s,g>, it sends a type-R message to all its direct neighbor nodes (succ(u)) to find the least cost branching path p(u,v)* to a node v ∈ T s,M (v ∈ V T ). The algorithm applied at node w to process and to further propagate type-R messages is described in Frame.I. The processing of type-A message sent in response by node w to its downstream neighbor nodes pred(w) is described in Frame.II. Observe that node w maintains no additional routing information besides the degree(v) entries required at each step of the execution. At waiting timer τ(u) expiration, if the set of type-A message received by node u is empty or the cost c(succ(u),v) in all received type-A message is set to infinite, node u declares the multicast source s unreachable. Otherwise, leaf node u selects among the received type-A messages the upstream node succ(u) along the least-cost branching path p(u,v)*, v ∈ T s,M . Node u then further proceeds by sending a message to succ(u) to join T s,M . At the end of this process, the routing table of each node v belonging to T s,M (v ∈ V T ) includes: i) one RT entry (stored in the multicast routing information base or MRIB) that indicates the upstream neighbor node to which any Join/Prune message is to be sent for that MDT and ii) one multicast traffic routing entry (stored in the TIB) for the tuple <s,g> and the total number of RT table entries.

Performance Analysis

To analyze its performance, the CMR algorithm is executed on a large-scale topology (10k nodes) generated by means of GLP [START_REF] Bu | On distinguishing between Internet power law topology generators[END_REF] that produces power-law graphs representative of the Internet topology. The execution scenarios consider the construction of point-to-multipoint routing paths for multicast groups of increasing size from 500 to 2000 nodes (selected randomly) with increment of 500 nodes. We consider the Shortest-Path tree (SPT) and the Steiner tree (ST) algorithms to compare the performance of the CMR algorithm. The SPT provides the communication cost reference whereas the ST gives the reference in terms of stretch. The SPT is constructed from a loop-avoidance path-vector routing algorithm carrying the multicast source identifier and the routing path to reach that source. Each node keeps an RT entry per neighbor node (to exchange routing message with its neighbors) and a RT per path to the multicast source s. In order to obtain the near optimal solution for the ST, we consider the ST-Integer Linear Programming formulation. For this purpose, we have adapted the formulation provided in [START_REF]Sage's Graph Library, free open-source mathematics software system[END_REF] for bi-directional graphs. The communication cost for the ST is based on the dissemination from current MDT nodes (at each step of its construction) of the minimal information for remote nodes not (yet) belonging to the MDT to join it. By this mean, each node knows how to reach the closest node of the MDT. Thus, although the ST is computed centrally, the communication cost accounts for the total number of messages exchanged during the MDT construction as a dynamic scenario would perform.

Stretch

Fig. 1 shows the results obtained for multiplicative stretch ratio between the CRM and the ST algorithm and between the SPT and the ST algorithm. From this figure, we can observe that the CMR algorithm leads to a multiplicative stretch slightly higher than 1 (from 1,09 to 1,06 for multicast group sizes ranging from 500 to 2000 nodes) and provides a gain of at least 5% compared to the SPT (for a multicast group size of 500 nodes) that remains approximately constant with increasing group size.

Communication cost

As shown in Fig. 2, the communication cost for the CMR is relatively high compared to the SPT even if twice lower than the communication cost implied by the ST when the multicast group size reaches 2000. This observation can be explained by the presence of high degree nodes (nodes that have a degree of the order to 100 or even higher) in the power law graph. These results suggest that further improvements on the communication process are desirable to apply the CMR on power law graphs comprising of the order of 10k nodes. Following the results obtained in [START_REF] Pedroso | A name-independent compact multicast routing algorithm, available as[END_REF], enforcing type-R message propagation by parts shows an improvement of at least one order of magnitude on the communication cost at the expense of slightly deteriorating stretch. Using this method, the joining node u searches locally for a node v ∈ T s,M in its vicinity B(u) ⊂ V where |B(u)| ~ n 0.5 /log(n); if T s,M is unreachable in B(u), node u then initiates a global search on the remaining topology V \ {B(u)}) .

Routing Table Size

Routing tables (RT) include the MRIB, the TIB as well as the unicast RIB entries (for the SPT scheme that relies on the underlying unicast routing topology). As shown in Fig. 3, the gain in terms of number of RT entries obtained for the ST compared to the CMR algorithm decreases from 8,8 to 3,4 when the multicast group size increases from 500 to 2000. The same trend is observed when comparing the SPT to the CMR algorithm. As shown in Fig. 4, the gain in terms of memory space consumption obtained for the ST compared to the CMR algorithm varies from 10,1 to 3,8 when the multicast group size increases from 500 to 2000. This gain decreases from about 8,8 to 3,4 when comparing the SPT to the CMRA. Even if this gain decreases with increasing multicast group size, both figures show significant benefit when the group size remains relatively small.

Conclusion

This paper proposes a leaf-initiated name-independent compact multicast routing algorithm with distributed computation of the RT entries. Our performance analysis considers the multiplicative stretch, the memory space consumption by the RT entries produced by the algorithm and the communication cost required to build point-tomultipoint routing paths. For this purpose, the proposed CMR algorithm has been executed over power-law graphs of 10k nodes and its performance compared to the SPT and ST algorithms ran over the same topologies.

The results obtained by simulation show that the CMR algorithm provides a basis for balanced stretch and memory space consumption. Compared to the SPT, the gain in memory space results from the elimination of the underlying unicast RT entries whereas, compared to the ST, this gain is mainly due to the elimination of the RT entries required at each step of the routing path construction. Our initial results show that the communication cost, inherent to the CMRA search phase, can be significantly reduced by segmenting this process. Work is currently conducted to confirm if this technique can effectively decrease the communication cost.

  If port(pred(w),w) in state open then Set port(pred(w),w) to state close π(w) ← π(pred(w)) -c((pred(w),w)) If π(w) > 0 then If w ∈ T s,M (u ≠ w = v) then Set c(w,v) ← 0 Set port(pred(w),w) to state reach Send type-A message to pred(w) Else If π(w) -c(w,succ(w)) > 0 then If port(w,succ(w)) in state open then Set port(w,succ(w)) to state wait Send type-R message to succ(w) Wait type-A message from succ(w) Else Wait type-A message from succ(w) Else (π(w) -c(w,succ(w)) ≤ 0) c(w,v) ← inf Set port(pred(w),w) to state unreach Send type-A message to pred(w) Else (π(w) ≤ 0) c(w,v) ← inf Set port(pred(w),w) to state unreach Send type-A message to pred(w) Type-A message While port(w,succ(w)) in state wait or τ(w) > 0 If c(succ(w),v) finite then c(w,v) ← c(w,succ(w)) + c(succ(w),v) Set c(w,v)* = min{c(w,v) | p(w,v) ∈ P w,v } Set port(w,succ(w*)) to state reach Set port(w,succ(w)) to state unreach Else (c(succ(w),v) inf) Set port(w,succ(w)) to state unreach If c(w,v)* finite then If port(pred(w),w) in state close then Set port(pred(w),w) to state reach Send type-A message to pred(w) Else If port(pred(w),w) in state close then Set port(pred(w),w) to state unreach Send type-A message to pred(w)

Fig. 3 : 4 :

 34 Fig.3: RT entries in function of the multicast group size Fig.4: RT memory in function of the multicast group size

: Stretch in function of the multicast group size Fig.2: Communication cost in function of the multicast group size

		Stretch				Communication Cost
	1,20			1600
	1,18
	1,16			1350
	1,14			1100
	1,12
	1,10			850
	1,08
				600
	1,06
	1,04		CMR/ST	350			CMR/SPT
	1,02		SPT/ST				ST/SPT
	1,00			100
	500	1000	1500	2000	500	1000	1500	2000
		Number of nodes of the mcast group			Number of nodes of the mcast group
	Fig.1Routing Table (number of entries)			Routing Table (memory)
	9,00			12,00
	8,00		SPT/CMR				SPT/CMR
	7,00		ST/CMR	10,00		ST/CMR
	6,00			8,00
	5,00
				6,00
	4,00
	3,00			4,00
	2,00
				2,00
	1,00
	0,00			0,00
	500	1000	1500	2000	500	1000	1500	2000
		Number of nodes of the mcast group			Number of nodes of the mcast group


PhD Tasnime N Akbaraly
email: tasnime.akbaraly@inserm.fr

PhD Marie-Laure Ancelin

Isabelle Jaussent Msc

MD Craig Ritchie

MD Pascale Barberger-Gateau

PhD Carole Dufouil

PhD Mika Kivimaki

MD Claudine Berr

PhD Karen Ritchie

Metabolic syndrome and onset of depressive symptoms in elderly: Finding from the Three-City Study Running title: Metabolic Syndrome and depression in elderly

Keywords: Depressive symptoms, metabolic syndrome, elderly, prospective study

Objective: Given the increasing prevalence of both metabolic syndrome (MetS) and depressive symptoms during old age, we aimed to examine prospectively the association between MetS and onset of depressive symptoms according to different age-groups in a large general elderly population.

Research Design and Methods: Prospective cohort study of 4446 men and women aged 65 to 91 and free of depression or depressive symptoms at baseline (the Three-City study, France). MetS was defined using the NCEP-ATP III criteria. New onset of depressive symptoms (the Center for Epidemiologic Studies Depression Scale (CES-D) score≥16 and use of antidepressant treatment) was assessed at 2-and 4-year follow-ups.

Results:

After adjusting for a large range of potential confounders, we observed MetS to be associated with a 1.73-fold (95% CI: 1.02-2.95) odds for new-onset depressive symptoms in the youngest age group (65 to 70 at baseline), independently of cardiovascular diseases. No such association was seen in older age groups.

Conclusion:

Our findings suggest that the link between MetS and depressive symptoms evidenced until now in middle-aged can be extended to older adults but not to the oldest ones.

Further research is needed to examine if a better management of MetS prevents depressive symptoms in people aged 65 to 70.

While depression has long shown to be associated with the development of metabolic syndrome (MetS) [START_REF] Raikkonen | Depressive symptoms and stressful life events predict metabolic syndrome among middle-aged women: a comparison of World Health Organization, Adult Treatment Panel III, and International Diabetes Foundation definitions[END_REF], more recent evidence of a "bi-directional" association has been reported in young [START_REF] Pulkki-Raback | Depressive symptoms and the metabolic syndrome in childhood and adulthood: a prospective cohort study[END_REF] and middle-aged women [START_REF] Koponen | Metabolic syndrome predisposes to depressive symptoms: a population-based 7-year followup study[END_REF][START_REF] Raikkonen | The relationship between psychological risk attributes and the metabolic syndrome in healthy women: antecedent or consequence?[END_REF] and older men [START_REF] Almeida | Obesity and metabolic syndrome increase the risk of incident depression in older men: the health in men study[END_REF] suggesting a more complex etiological pathway than has previously been considered. We have recently confirmed the association between MetS and depressive symptoms in a larger middle-aged population [START_REF] Akbaraly | Association between metabolic syndrome and depressive symptoms in middle-aged adults: results from the Whitehall II study[END_REF],

suggesting that a better management of MetS might reduce the incidence of depressive symptoms in the 40-60 years age range. The adverse consequences of depression on quality of life in older adults [START_REF] Blazer | Depression in late life: review and commentary[END_REF] combined to the increasing prevalence of MetS [START_REF] Eckel | The metabolic syndrome[END_REF] during old age, raised the interest in establishing whether MetS continues to increase the risk of depressive symptoms in older persons. Any potential association benefits from a biologically plausible hypothesis; research suggests that the etiology of late-onset depression is linked to vascular causes, such as diseases of the blood vessels and circulation [START_REF] Alexopoulos | Vascular depression' hypothesis[END_REF]. There are also clear associations between diabetes and neurodegenerative disease and within these diseases, depression and other mood disorders can be an early and observable feature. Thus, our objective was to assess prospectively the association between MetS and onset of depressive symptoms according to different age-groups in the elderly, by using data from a large general population cohort.

RESEARCH DESIGN AND METHODS

Study population

The Three-City Study (3C) is an ongoing multisite cohort study of communitydwelling persons aged 65 years or older and recruited from electoral rolls of three French cities (Bordeaux, Montpellier and Dijon) from 1999 to 2001 (N=9294) [START_REF] Study | Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF]. Participants were interviewed by trained staff and underwent a number of clinical examinations at baseline, two and four years. The study protocol was approved by the ethics committee of the University-Hospital of Bicêtre, France, and written informed consent was obtained from each participant.

Participants with dementia (n=500), those who presented depressive symptoms (n=2066) or who reported past or current major depressive disorders episodes (n=540) at baseline were excluded. Of these 6188 participants, the present analyses were carried out on the 4446 participants who had complete data on MetS, and all covariates measured at baseline and with at least one assessment of depressive symptoms available at 4-year follow-up.

Data Collection

Assessment of metabolic syndrome

The MetS was defined at baseline, using the National Cholesterol Education program Adult Treatment Panel III (NCEP-ATP III) criteria [START_REF] Day | Metabolic syndrome, or What you will: definitions and epidemiology[END_REF] based on the presence of 3 or more of the following; waist circumference: men >102 cm, women >88 cm; serum triglycerides:≥ 1.7 mmol/L; HDL cholesterol: men <1.04 mmol/L, women < 1.29 mmol/L; systolic blood pressure:≥130 mmHg, diastolic blood pressure 85 mmHg systolic ; fasting glucose:≥6.1 mmol/L or presence of type 2 diabetes. As data on established diagnosis of type 2 diabetes by a practitioner were not available, the use of anti-diabetic treatment was considered as a proxy.

Details of procedures regarding waist circumference, blood pressure, fasting blood glucose, HDL-cholesterol, and triglycerides measurements have been described previously [START_REF] Akbaraly | Metabolic syndrome, its components, and mortality in the elderly[END_REF].

Assessment of depression and depressive symptoms at baseline and over the follow-up.

At each wave, depressive symptomatology was assessed using the Center for Epidemiologic Studies Depression Scale CES-D scale which has been validated in general population [START_REF] Radloff | The CES-D scale: A self-report depression scale for research in the general population[END_REF].

Depressive symptomatology refers to any symptoms of depression reported by subjects on the CES-D, with the higher number of symptoms being considered to represent greater severity.

A cut-off point of 16 on the CES-D provides a categorical division of symptoms with higher scores corresponding to clinically significant levels of depressive symptomatology warranting clinical intervention [START_REF] Radloff | The CES-D scale: A self-report depression scale for research in the general population[END_REF]. Subjects with low levels of depressive symptomatology are referred to as a low symptom group. In longitudinal analyses, incident depressive symptoms were identified from this low symptom group not treated by antidepressants at baseline but who subsequently had incident depressive symptoms or began an antidepressant treatment during the follow-up. To strengthen the assumption that participants included in the present analyses were free of depressive symptoms, we also excluded subjects with lifetime major depressive episodes according to DSM-IV criteria by using the Mini International Neuropsychiatry Interview -a standardized psychiatric examination validated in the general population [START_REF] Ryan | Late-life depression and mortality: influence of gender and antidepressant use[END_REF].

Assessment of covariates at baseline.

Socio-demographic variables consisted of sex, age, study center, marital status (living alone /married or cohabited), educational level (no formal education or primary school / lower secondary education /higher secondary education or university degree), Health behaviors considered were smoking status (non /former /current smoker), alcohol consumption (no drinker /moderate defined by 3 (2) glasses or less per day for men (women) /high: > 3 [START_REF] Pulkki-Raback | Depressive symptoms and the metabolic syndrome in childhood and adulthood: a prospective cohort study[END_REF] glasses per day for men (women)). Health status was ascertained BMI categories (normal: BMI <25 kg/m² /overweight: 25BMI<30 kg/m 2 / obesity: BMI≥30 kg/m²), cognitive impairment defined as having score below 24 on the Mini-Mental State Examination (MMSE), disability (evaluated using the instrumental activities of daily living (IADL) scale, (score>0)), and MetS treatment (lipid lowering drugs and treatment of hypertension including hypotensor, angiotensor, beta-blocker and diuretics, lipid lowering drugs). An inventory of all drugs (prescription and over-the-counter drugs) used during the preceding month was included in a standardized interview. Medical prescriptions and, where feasible, the medications themselves were checked by the interviewer. In addition, self-report history of cardio-or cerebrovascular disease (CVD) (angina pectoris, myocardial infarction, coronary balloon dilation or artery bypass, stroke, or peripheral artery disease surgery) at baseline and incident CVD (including stroke) over the follow-up was also considered.

Statistical analyses

Student's t test (for continuous variables) and the χ 2 test (for categorical variables) were used to compare characteristics of participants according to 1) the presence of new onset of CES-D depression 2) MetS status at baseline. Logistic regression models were performed to assess the association between MetS, its components and onset of CES-D depression. The statistically significant interaction found between age and MetS on new-onset depressive symptoms (Wald test p=0.03) led us to conduct these analyses separately within each agequartile group. The mean age (SD) for each group was: age-group1: 67.9 (1.04); age-group2: 71.1 (1.0); age-group3: 74.7 (1.1); age-group4: 79.9 (2.5). These analyses were adjusted for age at baseline (by year), sex, study center, education, marital status, smoking habits, alcohol consumption, BMI categories, cognitive impairment, disability, MetS treatment, self-report history of CVD at baseline. Furthermore, to explore whether any association between MetS and CES-D depression could be driven by CVD, we repeated the analyses after excluding participants who self-reported history of CVD at baseline or who developed incident CVD over the follow-up. All analyses were conducted using SAS software, version 9.2 (SAS Institute).

RESULTS

Over the 4-year follow-up, 827 (18.6%) new cases of CES-D depression were observed. Characteristics of the population as a function of incident CES-D depressive symptoms over the 4-year follow-up are detailed in Table 1. Overall, the prevalence of the MetS was 12.9% (n=574) at baseline. Compared to MetS-free participants, those with MetS were more likely to be men (p=0.002), former smokers (p<10 -4 ), in higher BMI categories (p<10 -4 ), to have less education (p=0.002), more disability (p=0.02) and more cognitive deficit (p=0.002) and to report history of CVD (p=0.0003) and to develop CVD during the study (p=0.009) (Appendix-Table-A).

Results of table 1 show that incidence of depressive symptoms increased significantly across age groups (p < 0.0001). Table 2 shows the association of MetS with incidence of depressive symptoms by age-groups categorized according to quartile distribution. In age-group1 (first quartile), participants with MetS were more likely to develop depressive symptoms (OR=1.73; 95% CI=1.02; 2.95) compared to participants without MetS at baseline, after adjusting for a large range of socio-demographic characteristics (sex, study center, educational attainment, marital status), health behaviors (smoking, alcohol consumption) and health status factors (MetS treatment, cognitive deficit, disability, BMI, self-report history of CVD at baseline). In older age groups, there was no statistically significant association between MetS and new-onset of depressive symptoms.

To assess the robustness of this finding, we performed analyses in which age-groups were categorized a priori: (n=424). These analyses confirmed the main analysis by showing that MetS was associated with a 1.8-fold odds ratio for new-onset depressive symptoms in elderly aged 65 to 70 (OR=1.82; 95% CI=1.12, 2.95), but no association was found for older participants

(Appendix-Table-B).
The MetS is defined as a clustering of five metabolic disorders including elevated abdominal obesity, low HDL-cholesterol, high triglycerides, high blood pressure and high fasting glucose/ type 2 diabetes. We performed an analysis to examine which specific MetS components were associated with new-onset of depressive symptoms over the 4-year followup, especially in age-group1. Of the five MetS components, only low HDL-cholesterol component was significantly associated with new onset of depressive symptoms in that age group.

The evidence of an association between CVD and depression [START_REF] Krantz | Effects of mental stress in patients with coronary artery disease: evidence and clinical implications[END_REF] combined to the fact that MetS is an establish risk factor for CVD [START_REF] Eckel | The metabolic syndrome[END_REF], led us to explore whether the association between the MetS and new onset depression would be driven by CVD. Analyses were re-run after excluding participants firstly who reported a history of CVD at baseline (n=666) and then those who developed CVD during the 4-year follow-up (n=201). The agedependent association was largely replicated with MetS retaining its observed association with a higher odds of new-onset depression in age-group1, after excluding participants with self-reported history of CVD (OR=1.85;95% CI=1.04, 3.27) and after further excluding participants with incident cases of CVD during the follow-up (OR=2.06; 95% CI=1.15, 3.67).

CONCLUSION

To our knowledge, this is the first study to report the association between MetS and the onset of depressive symptoms across different age-groups in a prospective, multicentric, elderly general population cohort. MetS was associated with an almost doubling of the odds for new-onset depressive symptoms in age-group 65 to 70, even in subjects without apparent cardio-or cerebro-vascular pathologies, while for older age groups the association was not significant.

Several cross-sectional studies have shown an association between the MetS and depression in young adults [START_REF] Kinder | Depression and the metabolic syndrome in young adults: findings from the Third National Health and Nutrition Examination Survey[END_REF] and middle aged populations [START_REF] Gil | Relationship between the prevalence of depressive symptoms and metabolic syndrome. Results of the SOPKARD Project[END_REF][START_REF] Heiskanen | Metabolic syndrome and depression: a cross-sectional analysis[END_REF][START_REF] Skilton | Associations between anxiety, depression, and the metabolic syndrome[END_REF]. Although in several studies the assumption has been that depression predicts the MetS, depression could also be a consequence of the MetS. To date, this direction of the association has been prospectively investigated in middle-aged population [START_REF] Koponen | Metabolic syndrome predisposes to depressive symptoms: a population-based 7-year followup study[END_REF][START_REF] Raikkonen | The relationship between psychological risk attributes and the metabolic syndrome in healthy women: antecedent or consequence?[END_REF][START_REF] Akbaraly | Association between metabolic syndrome and depressive symptoms in middle-aged adults: results from the Whitehall II study[END_REF] and a "two-way street" between depression and the MetS is now evidenced. Our observation of an elevated risk of developing depressive symptoms in both pre-elderly men and women with MetS is consistent with a previous report carried out in a large middle aged British population including men and women [START_REF] Akbaraly | Association between metabolic syndrome and depressive symptoms in middle-aged adults: results from the Whitehall II study[END_REF]. Another prospective study carried in middle-aged population has suggested a sex-specific association by finding an association in women but not in men [START_REF] Koponen | Metabolic syndrome predisposes to depressive symptoms: a population-based 7-year followup study[END_REF] and was in accordance with a previous study reporting an association between MetS and depressive symptoms in a cohort of middleage women [START_REF] Raikkonen | The relationship between psychological risk attributes and the metabolic syndrome in healthy women: antecedent or consequence?[END_REF]. The absence of relationship in men may however, result from the lack of statistical power due to lower prevalence of MetS combined to the lower incidence of depressive symptoms in men compared to women. In our study no evidence of an interaction with sex was found in the MetS-depressive symptoms relationship. Our results are in line with those reported by Almeida et al. carried out on a large cohort of Australian older men [START_REF] Akbaraly | Metabolic syndrome, its components, and mortality in the elderly[END_REF]216 men aged 65-84 years) indicating that MetS was associated with an increase in the risk of incident depression [START_REF] Almeida | Obesity and metabolic syndrome increase the risk of incident depression in older men: the health in men study[END_REF].

MetS has gained clinical currency as a robust predictor of cardiovascular morbidity [START_REF] Eckel | The metabolic syndrome[END_REF] which may consequently raise the prevalence of "vascular depression" [START_REF] Alexopoulos | Vascular depression' hypothesis[END_REF]. It is therefore crucial to determine whether the association between the MetS and depressive symptoms is not driven by depressive symptoms generated by manifest CVD. To our knowledge, very few studies has investigated this question [START_REF] Gans | The metabolic syndrome, depression, and cardiovascular disease: interrelated conditions that share pathophysiologic mechanisms[END_REF]. The present data provide evidence that MetSdepression association observed may be independent of past and current and incident cardioand cerebro-vascular diseases and thus constitutes a novel finding.

Regarding the specific components of MetS, our data showed that low HDL cholesterol component was associated with increased odds of new onset depressive symptoms in those aged 65-70 Corresponding associations in that age group were not observed for other MetS components. This data is in agreement with recent findings on the importance of dyslipidemia in the etiology of late-life depression [START_REF] Ancelin | Gender and genotype modulation of the association between lipid levels and depressive symptomatology in community-dwelling elderly (the ESPRIT study)[END_REF] and may also imply that the association observed between MetS and depressive symptoms may be partially driven by HDL cholesterol in the pre-elderly participants. Further work is needed to assess whether a better management of HDL-cholesterol would reduce incidence of depressive symptoms in this age-group.

The reason why MetS would predict new onset of depressive symptoms until 70 years old but not after remains unclear. MetS, as an entity, is an empirical concept [START_REF] Eckel | The metabolic syndrome[END_REF] and it is possible that its clinical utility is less relevant in late elderly than in young elders. Another explanation would be that the MetS-depression relationship in older aged is masked by the higher disability, morbidity and mortality rate associated with older age, as poor health is associated with both the MetS [START_REF] Akbaraly | Metabolic syndrome, its components, and mortality in the elderly[END_REF] and late onset of depression [START_REF] Ryan | Late-life depression and mortality: influence of gender and antidepressant use[END_REF] in our previous work carried on the Three-City Study and other cohorts. Finally, our results may also reflect the fact that late onset of late-life depression (after 70 years old) did not share the same etiologyand therefore the same risk factors -as onset of depressive symptoms in middle aged, pre-and "young"elderly [START_REF] Blazer | Depression in late life: review and commentary[END_REF]. Further work is needed to further investigating whether MetS-depressive symptoms relationship is different before and after 70 years old.

The longitudinal and multicentric design and the large sample size including more than 4000 elderly subjects from the general population constitute the major strengths of this investigation. Furthermore to strengthen the assumption that participants included in the present analyses are free of depressive symptoms, we made the choice to also exclude all participants who were diagnosed at baseline as having current or past major depressive episodes based on the MINI psychiatric interview which was administered in the entire Threecity cohort only at baseline. The limitations of the present report included, first, the classification of clinical level of depression using CES-D. This instrument has however been validated in elderly general population, a cut-off point of 16 corresponding to clinically significant levels of depressive symptomatology warranting clinical intervention [START_REF] Radloff | The CES-D scale: A self-report depression scale for research in the general population[END_REF]. A second drawback involves reliance on the NCEP-ATP III in defining MetS while other definitions also exist [START_REF] Eckel | The metabolic syndrome[END_REF]. However, the NCEP-ATP III is the most widely used definition, thus allowing comparison of our results with other studies. Furthermore as the diagnosis of type 2 diabetes by a general practitioner was not assessed in the Three-city Study at baseline, we considered participants who reported the use of anti-diabetic drugs as having type 2 diabetes to compute MetS according to NCEP-ATP III criteria. Third, the design of our study -an observational epidemiological study-, with MetS only assessed at baseline, does not permit us to conclude that there is a causal link between MetS and depressive symptoms.

Furthermore the description of factors associated with MetS condition suggests that participants with MetS at baseline constitute a vulnerable population and thus the possibility that unmeasured confounders may partly explain the observed associations remains. However, our results were robust to adjustments for a large range of socio-demographic, health behavior and health status factors making less probable that they were attributable to confounding or obtained by chance. Further investigation is needed to establish MetS as an etiological factor for depression in pre-elderly; especially it remains to be shown whether a better management of MetS or its reversion is associated with lower incidence of depressive symptoms. Despite these limitations, by exploring the MetS-depressive symptoms across different age groups in the elderly, our findings suggest that the MetS -depressive symptoms link evidenced until now in middle-aged can probably be extended to young-elderly populations but not to the oldest ones. At this stage it is too early to present MetS as a predictor of depression, but under the hypothesis that prevention and treatment of MetS may be important for the prevention of depressive symptoms in middle-age [START_REF] Akbaraly | Association between metabolic syndrome and depressive symptoms in middle-aged adults: results from the Whitehall II study[END_REF], results of our study would suggest that such intervention studies would also be justified in order to prevent depressive symptoms in populations between 65 to 70 years of age.

Author contribution

TNA conducted the statistical analyses and co-wrote the initial and final drafts, and is guarantor; MLA, PBG, CD, CB and KR contributed to the design of the 3C study; MLA, IJ, CR, PBG, CD, MK, CB and KR co-wrote the final draft.

TNA had full access to all of the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis.

TABLES

  [65-70[, (n=1300); [70-75[, (n=1558); [75-80[, (n=1164); [80-91],

Table 1 :

 1 Characteristics of the 4446 of the Three-City Study participants according to the onset of new CES-D depression cases over the 4-year of follow-up.

	Cumulative onset of depressive
		symptoms *
	No (%)	Yes (%)	p
	(N=3619)	(N=827)

Table 2 :

 2 Association of metabolic syndrome and each of its components with onset of depressive symptoms over the 4-year follow-up by age quartile group.Odds ratio adjusted for age at baseline (by year), sex, study center, MetS treatment (use of lipid lowering drugs, anti-hypertensive drugs), educational attainment, marital status, smoking, alcohol consumption, cognitive deficit (MMSE<24), disability (assessed by IADL scale), BMI, self-report history of CVD (incl. stroke) at baseline. The reference (OR=1) corresponded to the subjects without MetS (or MetS components), for each age group and conditions. † BMI categories were not included in this model

		Without MetS or	With MetS or
		components	components
	Age group	(n, % depressed)	(n, % depressed)	OR (95% CI)*	p
	Overall MetS
	[65.0-69.4]	989 (14.8%)	123 (21.1%)	1.73 (1.02;2.95)	0.04
	[69.4-72.8]	963 (16.4%)	148 (16.9%)	1.18 (0.70; 1.97)	0.58
	[72.8-76.8]	960 (19.8%)	152 (20.4%)	0.94 (0.57;1.52)	0.81
	[76.8-91.1]	960 (22.9%)	151 (20.5%)	0.89 (0.56;1.42)	0.63
	Central obesity component †
	[65.0-69.4]	863 (14.6)	249 (18.5)	1.18 (0.80;1.75) † 0.40
	[69.4-72.8]	821 (16.0)	290 (17.9)	1.00 (0.68;1.46) † 0.99
	[72.8-76.8]	801 (19.5)	311 (20.9)	0.98 (0.69;1.38) † 0.91
	[76.8-91.1]	770 (22.5)	341 (22.9)	0.86 (0.62;1.19) † 0.37
	High TG component
	[65.0-69.4]	933 (15.4)	179 (15.6)	1.14 (0.71;1.84)	0.58
	[69.4-72.8]	916 (16.5)	195 (16.4)	1.11 (0.71;1.74)	0.63
	[72.8-76.8]	947 (19.4)	165 (22.4)	1.18 (0.77;1.80)	0.45
	[76.8-91.1]	943 (22.7)	168 (22.0)	0.99 (0.65;1.50)	0.95
	Low HDL-cholesterol component
	[65.0-69.4]	1025 (14.8)	87 (23.0)	1.81 (1.04;3.17)	0.03
	[69.4-72.8]	998 (16.5)	113 (15.9)	1.03 (0.59;1.80)	0.91
	[72.8-76.8]	1009 (19.4)	103 (24.3)	1.23 (0.74;2.03)	0.42
	[76.8-91.1]	1021 (22.1)	90 (27.8)	1.37 (0.82;2.29)	0.22
	Hypertension component

*

ACKNOWLEDGEMENTS

Financial support: The 3C Study is conducted under a partnership agreement between the Institut National de la Santé et de la Recherche Médicale (INSERM), Victor-Segalen Bordeaux-2 University, and Sanofi-Aventis. The Fondation pour la Recherche Médicale funded the preparation and initiation of the study. The 3C Study is also supported by the Caisse Nationale Maladie des Travailleurs Salariés, Direction Générale de la Santé, MGEN, Institut de la Longévité, Conseils Régionaux d"Aquitaine et Bourgogne, Fondation de France and Ministry of Research -INSERM Programme "Cohortes et collections de données biologiques". Part of this project is financed by grant from the Agence Nationale de la Recherche (project ANR-2006-PNRA -005; ANR2007 LVIE 003;ANR 2007-LVIE-004).Tasnime Akbaraly is supported by the BUPA Foundation, UK. Mika Kivimäki is supported by the Academy of Finland, Finland, the BUPA Foundation, the UK, and the National Heart, Lung, and Blood Institute (R01HL036310-20A2) and the National Institute on Aging (R01AG034454-01), NIH, US.

The funders had no role in the design and conduct of the study, collection, management, analysis, and interpretation of the data; and preparation, review, or approval of the manuscript.

Authors have no conflicts of interest to declare