Ren É Van Den Brink

Agnieszka Rusinowska
email: agnieszka.rusinowska@univ-paris1.fr

Frank Steffen
email: steffen2@liverpool.ac.uk

Marlies Ahlert

Yener Altunbas

Matthew Braham

Shanti Chakravarty

David Hojman

Manfred Holler

Christian Klamler

Hart- Mut Kliemt

Andreas Nohn

Hannu Nurmi

Georg Von

Measuring Power and Satisfaction in Societies with Opinion Leaders: Dictator and Opinion Leader Properties

Keywords: JEL Classification: C7, D7 Bipartite digraph, influence, inclination, collective choice, opinion leader, follower, satisfaction, power, dictator properties, opinion leader properties

A well known and established model in communication policy in sociology and marketing is that of opinion leadership. Opinion leaders are actors in a society who are able to affect the behavior of other members of the society called followers. Hence, opinion leaders might have a considerable impact on the behavior of markets and other social agglomerations being made up of individual actors choosing among a number of alternatives. For marketing or policy purposes it appears to be interesting to investigate the effect of different opinion leader-follower structures in markets or any other collective decision-making situations in a society.

We study a two-action model in which the members of a society are to choose one action, for instance, to buy or not to buy a certain joint product, or to vote yes or no on a specific proposal. Each of the actors has an inclination to choose one of the actions. By definition opinion leaders have some power over their followers, and they exercise this power by influencing the behavior of their followers, i.e. their choice of action. After all actors have chosen their actions, a decision-making mechanism determines the collective choice resulting out of the individual choices. Making use of bipartite digraphs we introduce novel satisfaction and power scores which allow us to analyze the actors' satisfaction and power with respect to the collective choice for societies with different opinion leader-follower structures. Moreover, we study common dictator and opinion leader properties of the above scores and illustrate our findings for a society with five members.

Introduction

The concept of opinion leadership received considerable attention in sociology and marketing. It rose out of the two-step flow of communication theory introduced by the 'Lazarsfeld group' (see e.g. [START_REF] Katz | Personal Influence: The Part Played by People in the Flow of Mass Communication[END_REF][START_REF] Katz | Personal Influence: The Part Played by People in the Flow of Mass Communication[END_REF]Lazarsfeld et al., 1968, [29, 31]). In its most rudimentary form it claims that 'ideas often flow from radio and print to the opinion leaders and from them to the less active sections of the population' (Lazarsfeld et al., 1968, [31]). For instance, Lazarsfeld et al. (1968, [31]) investigated the influence of mass communication on the 1940 presidential election campaign in the US. They found that the voters' choices were more influenced by actors which they called opinion leaders than by mass communication and concluded that the communication process is not a one-but a two-step process. According to this model information distributed by mass media first reaches the so-called opinion leaders. These are actors who are specified as highly self-confident with strong opinions. In Lazarsfeld et al. (1968, [31]) they act as intermediaries between the mass media and the recipients. In general, the latter actors are called followers. They feel attracted by the opinion leaders holding them in high esteem and are prepared to accept their opinion for their own behavior. Hence, a major characteristic of opinion leaders is their exercised power over their followers. After critiques of the model by the 'Lazarsfeld group' (see e.g. Bostian, 1970, [6]), Troldahl (1966, [41]) introduced a modified version of their model called the two-cycle flow of communication model which corresponded to other results in the field (see e.g. Deutschmann and Danielson, 1960, [START_REF] Deutschmann | Diffusion of knowledge of the major new story[END_REF]). Troldahl's model distinguishes between two phases in the communication process. Phase one is a flow of information from the mass media to the members of the society which is assumed to be a one-step process, i.e. the information goes directly to all members of the society. Phase two is the flow of influence on beliefs and behavior which is assumed to be a two-step process. In a first step opinion leaders form their own opinion based on additional information provided by experts, such as academics, while in a subsequent second step they try to influence the behavior of their followers. Since Troldahl's contribution the literature on opinion leadership has provided a strong body of knowledge of how and why opinion leaders influence followers choices (see Hoyer and Stockburger-Sauer, 2007, [27]).

Opinion leaders form an attractive group for marketing and policy purposes (see e.g. Hoyer and Stockburger-Sauer, 2007, [START_REF] Hoyer | A comparison of antecedents and consequences of market mavens and opinion leaders[END_REF]) as the existence (or non-existence) of opinion leaders in a society and their relations to their followers may have a considerable impact on market behavior (such as consumer or financial markets), and other social agglomerations being made up of individual actors choosing among a number of alternatives (open to them at a given time). Hence, it appears to be interesting to investigate the effect of different opinion leader-follower structures in markets or other collective decision-making situations in a society. This includes questions such as whether it would be worthwhile to establish a new opinion leader in a society or whether a change in an existing opinion leader-follower structure can be expected to make a difference to the society. However, to our best knowledge there exists no study which addresses such issues on bare theoretical grounds. In this paper we lay the foundation to fill out this lacunae by introducing novel power and satisfaction scores for societies with opinion leaders. 1 The former informs us about the power distribution among the members of the society with respect to their ability to affect the state of the society concerning a specific outcome, while the latter tells us to which degree members of the society can be expected to end up with an outcome that they like. Moreover, in this paper we study common dictator and opinion leader properties of the above scores and illustrate our findings for a society with five members.

For our analysis we consider the example of binary choice as it can also be found in Sinha and Raghavendra (2006, [37]) who study the effect of opinion leaders on market outcomes. It is assumed that an actor can choose among two alternatives. For instance, this can be a market in which the actors have to decide whether they should buy or not buy a joint product, or a voting situation in which the members of the society have to choose to vote either yes or no on a specific proposal. From now onwards we will refer to a voting situation only. However, all results presented in this paper also apply to markets. We assume that the actors in a society have to decide whether they would like to remain with the status quo or whether a specific exogenous proposal leading to a new state of the society should be adopted. We assume that the proposal has been distributed among all actors. Each actor has to form its own opinion on the proposal, i.e. without being influenced by any other actor. We will call this the actor's inclination. The society is partitioned into opinion leaders, followers, and independent actors. Followers are free to choose their own opinion leaders. Then, in line with the inherent idea of opinion leadership we suppose that via informal discussions of the proposal the inclinations of the opinion leaders are becoming public information prior to the real decision. Only after these discussions all actors will (secretly) choose their action which coincide with the actors inclination if it is an opinion leader or independent actor. Concerning the followers we assume that for their choice of action they -independently of their own inclination -adopt the inclination of a certain fraction of its opinion leaders if these have the same inclination. Finally, based on the individual choices of all actors a decisionmaking mechanism determines the collective choice, i.e. whether the proposal is adopted or not.

In the literature we can find several scores and measures being introduced for analyzing collective decision-making situations with a possible influence between the actors. 2 For instance, some measures for arbitrary digraphs have been studied in van den Brink and Borm (2002, [START_REF] Brink Van Den | Digraph competitions and cooperative games[END_REF]) and van den Brink and Gilles (2000, [START_REF] Brink Van Den | Measuring domination in directed networks[END_REF]). Since in the present paper we consider an opinion leader-follower structure which can be represented by a bipartite digraph, our model is related to the studies done in [START_REF] Brink Van Den | Digraph competitions and cooperative games[END_REF][START_REF] Brink Van Den | Measuring domination in directed networks[END_REF]. Coming from a slightly different direction are the works presented in the voting power literature. One of the traditional measures is the Rae index (Rae, 1969, [35]) which measures the success of an actor in a voting situation. An actor is said to be successful if its vote coincides with the voting outcome. Such a successful actor can be additionally powerful. For the calculation of the voting power of an actor a number of measures have been suggested. The most prominent measures are the Banzhaf measure [START_REF] Banzhaf | Weighted voting doesn't work: a mathematical analysis[END_REF], [START_REF] Banzhaf | Weighted voting doesn't work: a mathematical analysis[END_REF], see also [START_REF] Dubey | Mathematical properties of the Banzhaf power index[END_REF][START_REF] Dubey | Mathematical properties of the Banzhaf power index[END_REF]Owen, 1975, [14, 34]), and the Shapley-Shubik index (Shapley and Shubik, 1954, [36]). They ascribe power to an actor if its vote coincides with the voting outcome, but this outcome would have been different if the actor changed its vote. 3 As we are also concerned with measuring satisfaction and power distributions, our research is also related to the work on measurement of voting power. In the present paper we analyze satisfaction in a digraph by the number of times the collective choice is the same as the inclination of an actor. Power of an actor in a bipartite digraph is measured by the number of times the actor has a swing, where by a swing we mean that an actor by changing its inclination, given the inclinations of the others, enforces a change in the collective choice. We show that the power and satisfaction scores that we introduce satisfy some common dictator and opinion leader properties.

The paper is structured as follows. In Section 2 we describe the model, and in Section 3 we introduce novel satisfaction and power scores for actors in societies with opinion leaders. In Section 4, we suggest some desirable properties of satisfaction and power for such societies and prove that they are satisfied by our scores. In Section 5, we illustrate the introduced scores and their properties for a society represented by five-actors digraphs. In Section 6, we draw some conclusions, discuss some possible extensions of the model and a future research agenda. Proofs are presented in the Appendix.

The model

We consider the following model. There is a specific exogenous proposal on which n actors have to decide upon either by choosing the yes-or no-action. Let 1 and 0 stand for the choice of the yes and no-action, respectively, and let N = {1, ..., n} denote the set of all actors forming the society which is partitioned into opinion leaders, followers, and independent actors. We assume that the opinion leader-follower relations are given, i.e. it is given which actors might influence the choice of action of certain other actors by exercising some power over them. 4 Furthermore, we suppose that each actor has already formed an inclination to choose either the yes-or no-action. These are given by an inclination vector I = (I 1 , ..., I n ) ∈ {1, 0} n . This is a vector which k th component, I k , is 1 if actor k has the inclination to choose the yes-action, and 0 if it is inclined to choose the no-action. It is assumed that via informal discussions of the proposal the inclinations of the opinion leaders have already become public information, i.e. each follower is aware of the inclination of its opinion leader(s). While we allow that these discussions change the inclinations of some actors, we suppose that when choosing their action and after that inclinations do not alter. Moreover, we suppose that the choice of action is done secretly which implies that we consider a simultaneous decision-making situation.

The structure of such 'opinion leader-follower' relations is represented by a bipartite directed graph (or bipartite digraph) (N, D) with set of nodes N representing the actors and D ⊂ N × N a binary relation on N . Since we take the set of actors N fixed, we represent a digraph (N, D) just by its binary relation D. We denote the collection of all bipartite digraphs on N , represented by their binary relation, by D N .

Let S D (k) and P D (k) denote the set of successors and predecessors of actor k in digraph D, respectively, i.e., for each k ∈ N ,

S D (k) = {j ∈ N : (k, j) ∈ D} P D (k) = {j ∈ N : (j, k) ∈ D}.
As we assume that each actor is either an opinion leader, follower or independent actor, we consider digraphs D such that

|S D (k)| • |P D (k)| = 0 for each k ∈ N (1)
where |X| denotes the cardinality of set X. Let OL(D), FOL(D), and IND(D) denote the sets of all opinion leaders, followers, and independent actors in digraph D, respectively, i.e.

OL(D) = {k ∈ N : S D (k) = ∅} FOL(D) = {k ∈ N : P D (k) = ∅} IND(D) = N \ (OL(D) ∪ FOL(D)).
Therefore, by assumption (1) we have that

OL(D) ∩ FOL(D) = ∅,
and thus the sets OL(D), FOL(D) and IND(D) form a partition of the set N . We refer to a pair (I, D) with I ∈ {0, 1} n and a bipartite digraph D ∈ D N as described above as an opinion leader-follower collective choice situation. We assume that the actors in OL(D) ∪ IND(D) make their simultaneous choice of action according to their own inclinations. If a follower has just one opinion leader, it will choose according to the inclination of its opinion leader. If a follower has more than one opinion leader, then it will not follow its own inclination if and only if a certain fraction of its opinion leaders have the same inclination and this differs from its own inclination.

Let V = V (I, D) ∈ {0, 1} n denote the choice vector, that is, a vector which k th component, V k , is 1 if actor k has chosen the yes-action, and 0 if k has chosen the noaction. We assume that for every follower it is specified what fraction (more than one half) of its predecessors it will follow. Assuming this fraction to be uniform over the actors, there exists q ∈ [ 1 2 , 1) such that the choice vector V = V (I, D) ∈ {0, 1} n is recursively given by:

V k = I k if k ∈ OL(D) ∪ IND(D),
and for k ∈ FOL(D) :

V k = x if |{j ∈ P D (k) : I k = x}| > [q • |P D (k)|] I k otherwise, (2)
where x ∈ {0, 1} and [Z] denotes the largest integer not greater than Z. According to (2), given (I, D), if more than the fraction q of the opinion leaders of follower k has the same inclination, then k will follow these opinion leaders, otherwise k will decide according to its own inclination. In particular, if q = 1 2 , and half of the opinion leaders of k are inclined to choose the yes-action and half to choose the no-action, the follower k will follow its own inclination. For sufficiently large q, actor k follows its opinion leaders independently of its own inclination only if they are all unanimous in their inclinations.

After all actors have chosen their actions, a collective choice is resulting according to the decision-making mechanism in use. The decision-making mechanism is given by the collective decision function C : {0, 1} n × D N → {0, 1} which assigns an outcome to every pair (I, D) ∈ {0, 1} n × D N , that is, the value 0 if the collective decision is no, and the value 1 if the collective decision is yes. Usually one only considers collective decision functions C that are neutral5 and anonymous 6 . In this paper, we consider the collective decision function by simple majority voting. Let for an action x ∈ {0, 1} and

V = V (I, D) ∈ {0, 1} n n x (V (I, D)) = |{k ∈ N : V k = x}|
be the number of actors choosing the action x. We restrict our analysis to situations in which the number of actors is odd. The collective decision function is defined, for each (I, D), as follows:

C(I, D) = 1 if n 1 (V (I, D)) > n 0 (V (I, D)) 0 if n 0 (V (I, D)) > n 1 (V (I, D)). (3)
3 Measuring satisfaction and power

In this section we define satisfaction and power scores for bipartite digraphs which represent a collective decision-making situation as described above. In general, a score or measure for a bipartite digraph is a function f : D N → R n which assigns an n-dimensional real vector to every bipartite digraph on N . Success of an actor in a voting game means that the actor's vote coincides with the voting outcome. Since in our model actors have their inclinations to choose either the yes-or no-action before they actually choose, we propose to measure the satisfaction of an actor which is related to how often an actor's inclination prior to its actual choice coincides with the collective choice. 7 First, we define a score of satisfaction of an actor under the given inclination vector, i.e., for each (I,

D) ∈ {0, 1} n × D N and k ∈ N SAT k (I, D) = 1 if C(I, D) = I k 0 otherwise.
Next, based on satisfaction of an actor under each inclination vector, we define a score of satisfaction of actor k in a bipartite digraph, SAT :

D N → R n , by SAT k (D) = I∈{0,1} n SAT k (I, D) for each k ∈ N. (4)
Even more frequently than measuring success, the voting literature studies scores and measures of power of an actor. Roughly speaking, we ascribe power to a successful actor in a voting game if given the votes of the others, by changing its vote the actor changes the voting outcome, i.e. the actor has a swing. Since we consider situations where the actors have their inclinations before they choose their action, and the power of an actor is related to its ability to alter the collective choice, we can relate the power of an actor in this model to its inclination. Consequently, we ascribe power to an actor if the actor by changing its inclination alters the collective choice.

Actor k ∈ N has a swing in (I, D) according to collective decision function C if C(I, D) = C(I ′ , D) with I ′ k = I k and I ′ j = I j for all j ∈ N \ {k}. We measure power under the given inclination vector by the number of swings:

P OW k (I, D) = 1 if k has a swing in (I, D) 0 otherwise.
Then, the power score P OW : D N → R n is given by

P OW k (D) = I∈{0,1} n P OW k (I, D) for each k ∈ N. (5)
Note that both the satisfaction and power scores depend on the quota q, and hence they should be formally denoted by SAT q k (D) and P OW q k (D), respectively. Nevertheless, for notational convenience we will always write SAT k (D) and P OW k (D) as denoted in ( 4) and ( 5). In the next section we will study common dictator and opinion leader properties of the satisfaction and power scores given by ( 4) and ( 5).

Properties for satisfaction and power scores based on following a qualified majority of opinion leaders

Let us consider the scores SAT and P OW defined in (4) and ( 5), respectively. Each follower is assumed to follow a qualified majority of its opinion leaders, i.e. the choice vector V is as defined in [START_REF] Bolger | Power indices for multicandidate voting games[END_REF]. We consider the neutral and anonymous collective decision function C as defined in [START_REF] Bolger | A value for games with n players and r alternatives[END_REF]. First, we state some properties for a score (or measure) f : D N → R n . We start with the property which says that the score for actors with a symmetric position in the bipartite digraph is the same.

Property 1 (Symmetry) If S D (k) = S D (j) and P D (k) = P D (j) then f k (D) = f j (D).
Next, we consider two dictator properties. Clearly, a dictator, i.e. a unique opinion leader who is followed by all other actors, has the power to change the outcome for any voting profile by changing its own vote and, if the dictator votes according to its inclination then the outcome will be the inclination of the dictator. Therefore the dictator property states that, if there is a dictator, then the score of the dictator is equal to the total number of possible inclination vectors. Note, that since we assume that no actor can be at the same time a follower and an opinion leader, the dictator as defined above cannot be a follower.

Property 2 (Dictator property)

If D ∈ D N and h ∈ N is such that S D (h) = N \{h}, then f h (D) = 2 n .
Secondly, since a follower who has only one opinion leader has always to follow this opinion leader, dictated independence states that the score of a follower with one opinion leader does not change as long as this follower is dictated by a sole opinion leader.

Property 3 (Dictated independence) If D, D ′ ∈ D N and k ∈ N are such that |P D (k)| = |P D ′ (k)| = 1, then f k (D) = f k (D ′ ).
Next we present the three opinion leader properties saying something about the changes in the score for different actors when the opinion leader-follower structure changes, in particular when an actor gets a new opinion leader. The properties that we consider are inspired by similar properties for solutions in cooperative game theory, where studying these kinds of properties has a longer history. The most widely studied and applied property in this field is fairness introduced by Myerson (1977, [START_REF] Myerson | Graphs and cooperation in games[END_REF]) in the context of cooperative games in which the players belong to some binary communication structure, stating that deleting a communication link between two players changes their individual payoffs by the same amount. In van den Brink (1997, [START_REF] Brink Van Den | An axiomatization of the disjunctive permission value for games with a permission structure[END_REF]) this type of equal gain/loss property is stated in terms of games in which the players belong to some hierarchical structure, the so-called games with a permission structure, where it is stated that deleting a domination arc between a successor and a predecessor changes the payoffs of these two players by the same amount. In this paper we consider such kind of property for the opinion leader-follower collective choice situation. Suppose that a follower gets one more opinion leader. The equal absolute change property states that the changes in scores of this follower and of its new opinion leader are either the same or are opposite, but the same in absolute values.

D ′ = D ∪ {(h, j)}. Then either f h (D ′ ) -f h (D) = f j (D ′ ) -f j (D) or f h (D ′ ) -f h (D) = f j (D) -f j (D ′ ).
Although it is not obvious whether the scores of a follower and new opinion leader should change in the same or in opposite direction, it seems reasonable that these changes are in opposite direction in case the follower was an independent actor before the change. Taking this into account, the opposite gain property states that, if an actor becomes a sole opinion leader of another actor who was previously independent, then the sum of the scores of these two actors does not change. This implies that in case the opinion leader gains then this goes fully at the cost of the follower.

′ = D ∪ {(h, j)}. Then f h (D ′ ) -f h (D) = f j (D) -f j (D ′ ).
Finally, in the context of cooperative TU-games, Lehrer (1988, [32]) and Haller (1994, [START_REF] Haller | Collusion properties of values[END_REF]) introduced properties that consider collusion of players. In particular, Haller (1994, [START_REF] Haller | Collusion properties of values[END_REF]) considers different types of collusion neutrality properties requiring that the sum of the payoffs of two colluding players does not change. In van den Brink (2010, [START_REF] Brink Van Den | Axiomatizations of Banzhaf permission values for games with a permission structure[END_REF]), these properties are applied to the above mentioned games with a hierarchical permission structure. In that model, deleting a domination link between a successor and a predecessor is interpreted as a collusion between this predecessor and another predecessor with respect to the influence over the successor, leading to a power neutrality property stating that the sum of payoffs of the two colluding predecessors should not change. A similar reasoning in our underlying opinion leader-follower collective choice situation is reflected in power neutrality for two opinion leaders which states that, if a follower with one opinion leader gets a second opinion leader, then the sum of scores of the 'old' and 'new' opinion leaders does not change. In other words, the change for the new (second) opinion leader is opposite but in absolute value equal to the change for the old (first) opinion leader.

D ′ = D ∪ {(h, j)}. Then f h (D ′ ) + f g (D ′ ) = f h (D) + f g (D).
Next, we state the first main result which shows that the satisfaction score satisfies all the six properties mentioned above, if the choice vector is defined by (2).

Theorem 1 Let the choice vector V be defined by [START_REF] Bolger | Power indices for multicandidate voting games[END_REF]. The satisfaction score SAT : D N → R n defined in (4) satisfies symmetry, the dictator property, dictated independence, the equal absolute change property, the opposite gain property, and power neutrality for two opinion leaders.

The proof of this theorem can be found in the appendix. We note that from the proof that SAT satisfies the equal absolute change property we obtain as a corollary that whenever the satisfaction of two actors changes in the same direction after one becomes opinion leader of the other, then in the new situation the follower chooses its action according to its own inclination. It turns out that the power score POW also satisfies all six axioms, if the choice vector is defined by (2).

Theorem 2 Let the choice vector V be defined by [START_REF] Bolger | Power indices for multicandidate voting games[END_REF]. The power score P OW : D N → R n defined in (5) satisfies symmetry, the dictator property, dictated independence, the equal absolute change property, the opposite gain property, and power neutrality for two opinion leaders.

The proof of this theorem also can be found in the appendix.

Examples

In order to illustrate the developed scores and obtained results we analyze a society consisting of five actors, i.e. N = {1, 2, 3, 4, 5}. We assume eight potential opinion leaderfollower structures for this society, each represented by a digraph. For the followers we assume three different levels of the qualified majorities of the opinion leaders required to adapt their own behavior,

q = 1 2 , q ′ = 2 3 , q ′′ = 3 4 .
We analyze the following digraphs (see Figures 1a -1h): 1a, no opinion leaders) 1b, actor 2 is the sole opinion leader of actor 1) [START_REF] Banzhaf | Weighted voting doesn't work: a mathematical analysis[END_REF][START_REF] Bolger | A value for games with n players and r alternatives[END_REF], [START_REF] Banzhaf | Weighted voting doesn't work: a mathematical analysis[END_REF][START_REF] Bolger | A consistent value for games with n players and r alternatives[END_REF], (1, 5)} (Figure 1f, actor 1 is a dictator) 1g, actor 2 follows 1, actor 4 follows 3) [START_REF] Bolger | A value for games with n players and r alternatives[END_REF][START_REF] Bolger | A consistent value for games with n players and r alternatives[END_REF], (5, 4)} (Figure 1h, actor 2 follows 1, actors 3 and 5 are the opinion leaders of 4).

D 0 = ∅ (Figure
D 1 = {(2, 1)} (Figure
D 2 = {(2,
D 5 = {(1, 2),
D 6 = {(1, 2), (3, 4)} (Figure
D 7 = {(1, 2),
For this setup we show how the number of opinion leaders affects the relations between actors' satisfaction and power scores.

FIGURE 1 ABOUT HERE

Note that in digraph D 3 , in which actor 1 has three opinion leaders, when following the qualified majority of its opinion leaders, under q = 1 2 , actor 1 will always decide according to the inclination of at least two of its opinion leaders with the same inclination. Under q ′ = 2 3 and q ′′ = 3 4 , it will follow its opinion leaders independently of its own inclination only when they are unanimous. In digraph D 4 , in which actor 1 has four opinion leaders, under q and q ′ actor 1 will follow at least three unanimous opinion leaders, but it will decide according to its own inclination if two of its opinion leaders have the positive inclination, and two have the negative inclination. Under q ′′ agent 1 follows its opinion leaders 2, 3, 4, 5, independently of its own inclination only if they are unanimous in their inclinations. When an actor has two opinion leaders, like in D 2 and D 7 , following the qualified majority of the opinion leaders means, of course, following the opinion leaders independently of their own inclination only when they are unanimous.

Tables 1 and2 present the chosen individual actions and the collective choices based on the collective decision function as given in (3) for digraphs D 0 to D 3 , and D 4 to D 7 , respectively. The sign "-" means that for a given I, the result (either the chosen action or the collective choice) is the same as in D 0 .

TABLE 1 ABOUT HERE TABLE 2 ABOUT HERE

Table 3 shows the satisfaction and power scores for the actors in the situations represented by digraphs D 0 to D 7 .

TABLE 3 ABOUT HERE

Note that for digraph D 3 with actor 1 as follower and actors 2, 3, and 4 as its opinion leaders, the satisfaction and power score of the follower (the opinion leaders, respectively) increases (decreases, respectively) when the quota increases from q to q ′ , and each of these two scores remains the same when the quota increases from q ′ to q ′′ . On the other hand, the scores of each actor in digraph D 4 are the same for q, q ′ , and q ′′ . Using the scores calculated in Table 3, we can illustrate the properties presented in Section 4. Symmetry is obviously satisfied by both scores in all digraphs. The scores of satisfaction and power of actors in digraph D 5 with actor 1 as a dictator, satisfy the dictator property. Dictated independence can be illustrated with the scores in question of actor 1 in D 1 , actors 2 and 4 in D 6 , and actor 2 in D 7 , with satisfaction always equal to 16, and power always equal to 0.

In order to illustrate the equal absolute change property, let us first consider digraphs D 2 and D 3 with q = 1 2 , where the follower 1, having already two opinion leaders in D 2 (actors 2 and 3), gets his new opinion leader in D 3 , actor 4. The following holds:

SAT 4 (D 3 ) -SAT 4 (D 2 ) = 24 -20 = 20 -16 = SAT 1 (D 2 ) -SAT 1 (D 3 ) and P OW 4 (D 3 ) -P OW 4 (D 2 ) = 16 -8 = 8 -0 = P OW 1 (D 2 ) -P OW 1 (D 3 ).
On the other hand, when considering D 3 and D 4 with q = 1 2 , where actor 5 becomes a new opinion leader of actor 1, we have:

SAT 5 (D 4 ) -SAT 5 (D 3 ) = 22 -16 = SAT 1 (D 4 ) -SAT 1 (D 3 ) and P OW 5 (D 4 ) -P OW 5 (D 3 ) = 12 -0 = P OW 1 (D 4 ) -P OW 1 (D 3 ).
The opposite gain property can be seen when comparing D 0 with D 1 , and D 1 with D 6 . Indeed, for the latter case, we have: We want to wind up this section with some observations concerning satisfaction, power and resulting outcome effects of different opinion leader-follower structures. Comparing D 0 with D 3 for q ′ and q ′′ and D 4 for q, q ′ and q ′′ from Tables 1 and2, we can find that for all actors' satisfaction and power is always identical, i.e. it has no effect whether an actor is an opinion leader, follower, or independent actor or whether the society is structured in one way or the other. Moreover, if actors become opinion leaders or followers their effect on the outcome does not change. Hence, for these structures it does not matter whether there exists no or three or four opinion leaders if there is only up to one follower. However, to establish an opinion leader in a society without any opinion leader (moving from D 0 to D 1 ) has some effect on the outcome of the collective choice: it changes in 18.75 percent of the cases, but adding another opinion leader to an existing opinion leader (moving from D 1 to D 2 ) decreases the number of cases where the outcome changes. In contrast to the case without an opinion leader, the outcome alters only in 6.25 percent of the cases. Hence, the existence of only one opinion leader has a stronger effect to society as having a second one if there exists only one follower.

Future research

The existence of opinion leaders and their influence over other actors can be seen in every day life situations: in small as well as in large societies be it in politics or business. Both satisfaction and power are the very natural measures of actors' strength or status in such situations. Although, as mentioned in the introduction, there exist several related theoretical studies in the literature on voting models and on networks, the approach which we use in the paper, i.e. the analysis of opinion leader-follower structures and the properties of the scores in question has brought up several innovative elements and can also be regarded to contribute to knowledge, in particular, in marketing.

However, there are several improvements one could bring to this framework in future research. First of all, after stating some initial properties of satisfaction and power scores for opinion leader-follower structures, a next step is to deliver full axiomatic characterizations of the satisfaction and power scores, which would show a difference between the scores from the axiomatic point of view. In van den Brink, Rusinowska and Steffen (2010, [START_REF] Brink Van Den | Measuring power and satisfaction in societies with opinion leaders: An axiomatization[END_REF]) properties presented in this paper are used to axiomatize these scores in the special case of unanimity, i.e. independently from its own inclination a follower will choose the action which corresponds to the inclination of its opinion leader(s), if all of them have the same inclination, but will choose an action which corresponds to its own inclination, if the inclinations of its opinion leader(s) are not all the same. In this axiomatic system the satisfaction and power scores differ only with respect to some normalization axiom.

In this paper we assumed that an actor cannot be at the same time an opinion leader for some actor(s) and a follower of some other actor(s). In a future research on this topic, one could try to relax this assumption and to consider a more general digraph, allowing the sets of predecessors and successors of a given actor to be both nonempty. In terms of Troldahl's (1966, [41]) two-cycle flow of communication model this would allow us to include the experts as an additional group of actors into our analysis acting as opinion leaders of the opinion leaders.

Furthermore, we could apply the same approach to some related models on organizational hierarchies based on subordinates and their superiors (see, e.g., Hammond and Thomas, 1990, [26]), where an organizational choice is to be made. Although such topics are naturally related to our present work, we expect that results on the properties of the scores in question in the superior-subordinate structures will be quite different from the ones obtained in the present model.

Since we consider the two-action model, a natural and useful generalization of the framework will be to enlarge the set of possible actions, i.e., to follow some works on abstention (see, e.g. Braham and Steffen, 2002[START_REF] Machover | Ternary voting games[END_REF][START_REF] Machover | The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes[END_REF][START_REF] Machover | Models and reality: the curious case of the absent abstention[END_REF], Tchantcho et al., 2008, [7, 16-18, 40]), and on multi-choice games (see e.g. Grabisch and Rusinowska, 2010, and Hsiao and Raghavan, 1993, [START_REF] Grabisch | A model of influence with an ordered set of possible actions[END_REF][START_REF] Hsiao | Shapley value for multichoice cooperative games[END_REF]). Related models are also games with r alternatives, where the alternatives are not ordered; see [START_REF] Bolger | Power indices for multicandidate voting games[END_REF][START_REF] Bolger | A value for games with n players and r alternatives[END_REF][START_REF] Bolger | A consistent value for games with n players and r alternatives[END_REF]Bolger ( , 2002, [2-5], [2-5]). Also in Freixas (2005aFreixas ( , 2005b, [19, 20], [19, 20]) and Freixas and Zwicker (2003, [21]), the authors consider decision-making situations, i.e. voting systems, with several levels of approval in the input and output, where those levels are qualitatively ordered. They introduce (j, k) simple games, in which each actor expresses one of j possible levels of input support, and the output consists of one of k possible levels of collective support.

Finally, another issue of future research agenda would be to apply the opinion leaderfollower collective choice situation to concrete situations of real life.

Then, SAT h (I, D ′ ) -SAT h (I, D) = SAT g (I, D) -SAT g (I, D ′ ) = 1. Thus, with (4) we have SAT h (D ′ ) -SAT h (D) = SAT g (D) -SAT g (D ′ ), showing that SAT satisfies power neutrality for two opinion leaders.

Proof of Theorem 2

It is straightforward that P OW satisfies symmetry.

Since a dictator has a swing in every inclination vector, P OW satisfies the dictator property. Since an actor with a unique opinion leader never has a swing, P OW satisfies dictated independence. In a similar way we can consider the case when [q • |P D (j)|] + 1 opinion leaders of j have one inclination, and [q • |P D (j)|] + 1 opinion leaders have the other inclination. Thus, with (5), we get P OW h (D ′ ) -P OW h (D) = P OW j (D) -P OW j (D ′ ). Case II: Suppose that [q

• |P D (j)|] < q • |P D (j)|, and [q • |P D (j)|] + 1 = [q • (|P D (j)| + 1)]. We distinguish the following cases. (i) If I ∈ {0, 1} n is such that [q • |P D (j)|] + 2

Property 4 (

 4 Equal absolute change property) Let D, D ′ ∈ D N , j ∈ FOL(D), h ∈ OL(D) ∪ IND(D), and

Property 5 (

 5 Opposite gain property) Let D, D ′ ∈ D N , j ∈ IND(D), h ∈ OL(D) ∪ IND(D), and D

Property 6 (

 6 Power neutrality for two opinion leaders) Let D, D ′ ∈ D N , j ∈ FOL(D) and g ∈ OL(D) such that P D (j) = {g}, h ∈ OL(D) ∪ IND(D), and

Corollary 1

 1 For every D, D ′ ∈ D N and h, j ∈ N such that P D (j) = ∅, h ∈ (OL(D) ∪ IND(D))\P D (j) and D ′ = D∪{(h, j)}, whenever SAT h (I, D ′ )-SAT h (I, D) = SAT j (I, D ′ )-SAT j (I, D) = 1, we have V j (I, D ′ ) = I j .

  1), (3, 1)} (Figure 1c, actors 2 and 3 are the opinion leaders of actor 1) D 3 = {(2, 1), (3, 1), (4, 1)} (Figure 1d, actors 2, 3 and 4 are the opinion leaders of 1) D 4 = {(2, 1), (3, 1), (4, 1), (5, 1)} (Figure 1e, actors in N \ {1} are the opinion leaders of 1)

SAT 4 (

 4 D 1 ) + SAT 3 (D 1 ) = 20 + 20 = 16 + 24 = SAT 4 (D 6 ) + SAT 3 (D 6 ) and P OW 4 (D 1 ) + P OW 3 (D 1 ) = 8 + 8 = 0 + 16 = P OW 4 (D 6 ) + P OW 3 (D 6 ). Power neutrality for two opinion leaders can be shown when comparing D 1 with D 2 , and D 6 with D 7 . Indeed, for the follower 1 in D 1 and D 2 , and actors 2 and 3 as actor 1's opinion leaders, we get as follows: SAT 2 (D 1 ) + SAT 3 (D 1 ) = 28 + 20 = 24 + 24 = SAT 2 (D 2 ) + SAT 3 (D 2 ) and P OW 2 (D 1 ) + P OW 3 (D 1 ) = 24 + 8 = 16 + 16 = P OW 2 (D 2 ) + P OW 3 (D 2 ).

  To show the equal absolute change property, let D, D ′ ∈ D N , j ∈ FOL(D), h ∈ OL(D) ∪ IND(D) and D ′ = D ∪ {(h, j)}. Note that P OW h (I, D) = 1 implies that P OW h (I, D ′ ) = 1. Case I: Suppose that [q • |P D (j)|] = q • |P D (j)|. We distinguish the following subcases: (i) If I ∈ {0, 1} n is such that [q • |P D (j)|] + 1 opinion leaders of j have the same inclination, then P OW j (I, D) = P OW j (I, D ′ ) = 0 and P OW h (I, D) = P OW h (I, D ′ ). (ii) If I is such that less than [q • |P D (j)|] opinion leaders of j have the inclination x ∈ {0, 1} and less than [q • |P D (j)|] opinion leaders have the inclination y ∈ {0, 1} \ {x}, then P OW j (I, D) = P OW j (I, D ′ ) and P OW h (I, D) = P OW h (I, D ′ ). (iii) Suppose that I ∈ {0, 1} n is such that [q • |P D (j)|] opinion leaders have the same inclination, say x ∈ {0, 1}, and less than [q • |P D (j)|] have the different inclination y ∈ {0, 1} \ {x}. (a) If I j = x = y = I h , then P OW j (I, D) = P OW j (I, D ′ ) and P OW h (I, D) = P OW h (I, D ′ ). (b) Let x = I h = y = I j . If P OW j (I, D) = 1, then P OW j (I, D) -P OW j (I, D ′ ) = 1 and P OW h (I, D ′ ) -P OW h (I, D) = 1. Moreover, for I ′ ∈ {0, 1} n such that x = I ′ j = I ′ h = y, I ′ k = I k for k ∈ {j, h}, P OW j (I ′ , D) -P OW j (I ′ , D ′ ) = 1 and P OW h (I ′ , D ′ ) -P OW h (I ′ , D) = 0, but for I ′′ ∈ {0, 1} n such that x = I ′′ j = I ′′ h = y, I ′′ k = I k for k ∈ {j, h}, P OW j (I ′′ , D) -P OW j (I ′′ , D ′ ) = 0 and P OW h (I ′′ , D ′ ) -P OW h (I ′′ , D) = 1.

Figure 1 :

 1 Figure 1: Digraphs D 0 -D 7

Note that this research is in some respects also related to work on opinion leaders and the Condorcet Jury Theorem (see, e.g.,Estlund, 1994, [15]), threshold models of collective behavior (see, e.g.,[START_REF] Granovetter | Threshold models of collective behavior[END_REF][START_REF] Granovetter | Threshold models of collective behavior[END_REF] Soong, 1986, [23,

[START_REF] Granovetter | Threshold models of interpersonal effects in consumer demand[END_REF]) and, in more general terms to the literature on network externalities.

For the distinction between scores, measures, and indices, see Felsenthal and Machover (1998,[START_REF] Machover | The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes[END_REF]).

Both measures can also be derived from a probabilistic framework. See, e.g.,[START_REF] Straffin | Homogeneity, independence, and power indices[END_REF]Straffin ( , 1978, [38, 39], [38, 39]) and, more recently,Laruelle and Valenciano (2005, [30]).

As a result of this influence, the ability of the followers to determine the outcome of the collective choice, i.e. their power to do something (with respect to the outcome of the collective choice) might be affected.

A collective decision function C is neutral if [C(I, D) = 1 if and only if C(I c , D) = 0], where I c k = 1 if and only if I k = 0.

A collective decision functions C is anonymous if for every permutation π : N → N , C(I, D) = C(π(I), π(D)) with π(I) k = I π(k) and (π(k), π(j)) ∈ π(D) if and only if (k, j) ∈ D.

Note, that this implies the general belief that (at least from the followers' perspective) the existence of opinion leaders is beneficial for a society. As already said above followers are prepared to accept the views of their chosen opinion leaders even if this implies to choose an action against their own inclination. This is comparable with the behavior of voters in democracy. They usually also do not question the existence of democracy if the outcome of the collective choice is not in line with their view. Another example are fashion goods. Here followers usually choose opinion leaders from which they assume that they correctly predict what will be popular in the future. Followers often adopt their view even if they do not like the style they declare to be trendy in the future. Hence, we can distinguish between two types of inclinations. The above type of inclinations given by I being related to the outcome of the collective choice, and inclinations related to the procedure, i.e. that under certain conditions followers have the inclination to align their individual choice with the view of their chosen opinion leaders even if this is against their own inclinations. This also implies the existence of two notions of satisfaction: one with respect to the outcome and one with respect to the procedure. In this paper we suggest a score for the former one.

Appendix

In this appendix we provide all proofs of the results in this paper.

Proof of Theorem 1

It is straightforward that SAT satisfies symmetry.

The dictator property follows straightforward since a dictator is followed in all 2 n inclination vectors in {0, 1} n , i.e., if S D (h) = N \{h}, then C(I, D) = I h for all I ∈ {0, 1} n .

To show dictated independence, note that actor k always chooses an action according to j's inclination if P D (k) = {j}. That means that the collective choice is independent of actor k's inclination, i.

Then it must hold that actor j had to deviate from its inclination and follow h, and this must result in a change of collective choice from I j to I h , with

showing that SAT satisfies the opposite gain property.

To show power neutrality for two opinion leaders, let

Then it must hold that actor j initially had to choose an action against its inclination and now can choose an action according to its inclination because its new opinion leader h has the same inclination. So, for g ∈ P D (j) we have C(I, D) = I g = I j = I h and C(I, D ′ ) = I j = I h = I g .

with q with q with q ′ , q ′′ with q ′ , q ′′ (0,0,0,0,0)

,0,0,0,1) 0 (0,0,0,0,1) -(0,0,0,0,1) -(0,0,0,0,1) -(0,0,0,0,1)

1 (0,0,1,0,1) 0 --(0,0,1,0,1) 0 --(1,0,0,1,1) 1 (0,0,0,1,1) 0 (0,0,0,1,1) 0 (0,0,0,1,1) 0

Table 1. Individual and collective choices for digraphs D0 -D3 I V (I, D4) C(I, V ) V (I, D4) C(I, V ) V (I, D5) C(I, V ) V (I, D6) C(I, V ) V (I, D7) C(I, V ) with q, q ′ with q, q ′ with q ′′ with q ′′ (0,0,0,0,0) ----(0,0,0,0,0) -----(1,0,0,0,0) (0,0,0,0,0) -(0,0,0,0,0) -(

,0,0,0) -(0,1,0,0,0) ----(0,0,0,0,0) -(0,0,0,0,0) -(0,0,0,0,0) -(0,0,1,0,0) ----(0,0,0,0,0) -(0,0,1,1,0) -(0,0,1,0,0) -(0,0,0,1,0) ----(0,0,0,0,0) -(0,0,0,0,0) -(0,0,0,0,0) -(0,0,0,0,1)

,0,0,1,0) (0,0,0,1,0) ---

,0,0,0,1) (0,0,0,0,

(1,1,0,0,1) 1 (0,1,1,0,0) ----(0,0,0,0,0) -(0,0,1,1,0) -(0,0,1,0,0) -(0,1,0,1,0) ----(0,0,0,0,0) -(0,0,0,0,0) -(0,0,0,0,0) -(0,1,0,0,1) ----(0,0,0,0,0) -(0,0,0,0,1) -(0,0,0,0,1) -(0,0,1,1,0) ----(0,0,0,0,0) -----(0,0,1,0,1) ----(0,0,0,0,0) -(0,0,1,1,1) 1 (0,0,1,1,1) 1 (0,0,0,1,1) ----(0,0,0,0,0) -(0,0,0,0,1)


M.D. C Christoph Schultz

MD Igor Nenadic

MD Kathrin Koch

PhD Gerd Wagner

PhD; Martin Roebel

MD Claudia Schachtzabel

Thomas W Mühleisen

Markus Nothen

MD Sven Cichon

PhD; Thomas Deufel

PhD Markus M Nöthen

MD; Michael Kiehntopf

MD Marcella Rietschel

MD Jürgen R Reichenbach

PhD Heinrich Sauer

MD Ralf G M Schlösser

Reduced cortical thickness is associated with the glutamatergic regulatory gene risk variant DAOA Arg30Lys in schizophrenia

Keywords: rs2391191, G72, schizophrenia, cortical thickness, susceptibility gene, DAOA

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Risk carriers with schizophrenia exhibit a significantly thinner cortex in two almost inversely arranged clusters on the left and right hemisphere comprising middle temporal, inferior parietal and lateral occipital cortical areas. The clusters encompass an area of 1174 mm² (left) and 1156 mm² (right). No significant effect was observed in healthy controls.

The finding of our study that the Arg30Lys risk variant is associated with a distinct cortical thinning provides new evidence for the pathophysiological impact of DAOA in schizophrenia. The affected areas are mostly confined to cortical regions with a crucial role in the theory of mind network and visual processing, which both can be influenced via glutamatergic modulation. Our finding thus underlines the importance of DAOA and related glutamatergic processes as a putative target for therapeutic interventions in schizophrenia.

Introduction

In light of current etiological concepts of schizophrenia the glutamatergic system plays an essential role for the pathophysiology of the disorder offering multiple options for new treatment strategies [START_REF] Javitt | Glutamate and schizophrenia: phencyclidine, N-methyl-D-aspartate receptors, and dopamine-glutamate interactions[END_REF]. Genetic findings of the recent years fundamentally broadened our knowledge of schizophrenia [START_REF] O`donovan | Genetics of psychosis; insights from views across the genome[END_REF] and particularly in view of the glutamatergic system as several identified genes are implicated to interact with the glutamatergic neurotransmission [START_REF] Coyle | Glutamate and schizophrenia: beyond the dopamine hypothesis[END_REF].

However, as a further translational step the identification of the exact pathophysiological mechanisms is mandatory for the development of new treatment strategies. A new emerging approach, imaging genetics, intends to identify the pathophysiological role of genetic variations by analyzing the impact on brain function and structure. In particular, schizophrenia susceptibility genes offering potential targets for new treatment strategies via their biochemical products are of special interest. The G72 gene is one of the identified susceptibility genes in schizophrenia.

Its biochemical products play an important role in the glutamatergic neurotransmission through the interaction with the D-amino acid oxidase (DAAO) and was therefore identified as D-amino oxidase activator (DAOA) [START_REF] Chumakov | Genetic and physiological data implicating the new human gene G72 and the gene for D-amino acid oxidase in schizophrenia[END_REF]. DAOA and related glutamatergic mechanism are suggested to constitute potential targets for treatment strategies in schizophrenia [START_REF] Javitt | Glutamate and schizophrenia: phencyclidine, N-methyl-D-aspartate receptors, and dopamine-glutamate interactions[END_REF][START_REF] Williams | Commentary: genome-based CNS drug discovery: D-amino acid oxidase (DAAO) as a novel target for antipsychotic medications: progress and challenges[END_REF]. Genetic variation in this gene was first found to be associated with schizophrenia by [START_REF] Chumakov | Genetic and physiological data implicating the new human gene G72 and the gene for D-amino acid oxidase in schizophrenia[END_REF] and [START_REF] Schumacher | Examination of G72 and D-amino-acid oxidase as genetic risk factors for schizophrenia and bipolar affective disorder[END_REF]. A recent meta-analysis of all published association studies supported G72 as a risk gene for schizophrenia (Detera-Wadleigh and McMahon, 2006b). Furthermore, DAOA variants predicted the progression from prodromal to the first episode in schizophrenia [START_REF] Mossner | DAOA/G72 predicts the progression of prodromal syndromes to first episode psychosis[END_REF]. The importance of DAOA gene variants has additionally been demonstrated using functional magnetic resonance imaging (fMRI) [START_REF] Jansen | A putative high risk diplotype of the G72 gene is in healthy individuals associated with better performance in working memory functions and altered brain activity in the medial temporal lobe[END_REF][START_REF] Krug | Genetic variation in G72 correlates with brain activation in the right middle temporal gyrus in a verbal fluency task in healthy individuals[END_REF]. A recent study of [START_REF] Hartz | G72 influences longitudinal change in frontal lobe volume in schizophrenia[END_REF] The DAOA rs2391191 variant (M15) SNP showed one of the strongest associations with schizophrenia in a previous meta-analysis (Detera-Wadleigh and McMahon, 2006b). It causes an arginine-to-lysine substitution at codon 30 (Arg30Lys) and might therefore be of putative functional relevance [START_REF] Donohoe | DAOA ARG30LYS and verbal memory function in schizophrenia[END_REF]. Recently, it has been shown that the DAOA SNP rs2391191 variant is associated with a significantly lower visual memory capacity in patients with a psychotic disorder [START_REF] Soronen | Association of a nonsynonymous variant of DAOA with visuospatial ability in a bipolar family sample[END_REF]. Additionally, the rs2391191 SNP variant is associated with an impaired verbal memory in schizophrenia [START_REF] Donohoe | DAOA ARG30LYS and verbal memory function in schizophrenia[END_REF]. However, the pathophysiological basis for the functional deficits impacted by the rs2391191 SNP remains unclear.

Surface based MRI studies investigate cortical shape parameters in terms of cortical thickness and curvature. Surface based MRI studies showed that cortical thickness is reduced in chronic [START_REF] Kuperberg | Regionally localized thinning of the cerebral cortex in schizophrenia[END_REF][START_REF] Nesvag | Regional thinning of the cerebral cortex in schizophrenia: effects of diagnosis, age and antipsychotic medication[END_REF]Schultz et al, 2010a) and first episode schizophrenia [START_REF] Narr | Mapping cortical thickness and gray matter concentration in first episode schizophrenia[END_REF]Schultz et al, 2010b) in mainly fronto-temporal, but also parietal regions. ROI analyses demonstrated cortical thinning in key regions of the medial surface including the anterior cingulum [START_REF] Fornito | Surfacebased morphometry of the anterior cingulate cortex in first episode schizophrenia[END_REF] and the entorhinal cortex (Schultz et al, 2010c). As cortical thickness is assumed to reflect neurodevelopmental mechanisms [START_REF] Hilgetag | Role of mechanical factors in the morphology of the primate cerebral cortex[END_REF], cortical thickness might be a suitable parameter for the investigation of a potential association of genetic variants and brain morphology. Thus, cortical thinning might constitute one underlying pathomechanism of the DAOA Arg30Lys risk variant.

However, the association of the DAOA Arg30Lys coding variant and cortical thickness has to our best knowledge not been explored.

Objectives and Hypothesis

Based on these considerations, we compared whole brain cortical thickness computed with a surface based MRI method between the different groups in patients and healthy controls according to the genotype status, to identify a potential association of G72 SNP rs2391191 and cortical thickness in patients with schizophrenia and healthy controls. According to the meta -analysis of Detera-Wadleigh (2006a) the A allele represents the risk allele in schizophrenia. Therefore, we hypothesized carriers of the A allele (AA/AG) exhibiting a thinner cortex both in patients with schizophrenia and healthy controls in comparison to homozygous G carriers (GG) reflecting a potential pathomechanism of this DAOA risk variant.

Subjects and methods

Participants

We studied 52 patients with schizophrenia and 42 healthy controls. All participants were right-handed [START_REF] Annett | The binomial distribution of right, mixed and left handedness[END_REF]. Diagnoses were established by a clinical psychiatrist (M. R.) based on the Structured Clinical Interview for DSM-IV and were confirmed by 2 independent psychiatrists (R. S. and C. Chr. S.). All patients met DSM-IV criteria for schizophrenia and had no second psychiatric diagnosis. They were on stable medication, mostly with second-generation antipsychotics.

Healthy volunteers were screened for major medical, neurological and psychiatric history. None of the healthy subjects had a current or history of a psychiatric disorder and first-degree relatives with a psychiatric disorder according to DSM-IV. Exclusion criteria for all participants were neurological disease or damage, medical disorders potentially influencing neurocognitive function. All participants were caucasian. All participants gave written informed consent to the study approved by the Ethics Committee of the Friedrich-Schiller University. Sociodemographic and psychopathological data are given in Table 1.

<Insert table 1>

MRI-Acquisition

We acquired high-resolution anatomical T1-weighted scans in a 1.5 T Siemens Magnetom Vision whole-body system using a three-dimensional spoiled gradient echo sequence: 1 mm sagittal slices with TR=15 ms, TE=5 ms, flip angle 30°, FOV=256, matrix=256 x 256, number of sagittal slices=192.

All scans were inspected for motion artefacts and a neuroradiologist confirmed absence of gross pathological findings.

MR scan processing and calculation of cortical thickness

We used the FreeSurfer software package (version 4.0.5, http://surfer.nmr.mgh.harvard.edu) for processing of images [START_REF] Dale | Cortical surface-based analysis. I. Segmentation and surface reconstruction[END_REF][START_REF] Fischl | Cortical surface-based analysis. II: Inflation, flattening, and a surface-based coordinate system[END_REF]. The implemented processing stream comprises the removal of non-brain tissue, transformation to Talairach-like space, and segmentation of gray/white matter tissue. White and gray matter boundary is tessellated and topological defects are automatically corrected. After intensity normalization, transition of gray/white matter and pial boundary is identified by detecting the greatest shift in intensity through surface deformation. The entire cortex of each subject was then visually inspected and any inaccuracies in segmentation were manually edited. After creation of the cortical representations the cerebral cortex is automatically parcellated into anatomical structures [START_REF] Fischl | Automatically parcellating the human cerebral cortex[END_REF]. Cortical thickness is computed by finding the shortest distance between a given point on the estimated pial surface and the gray/white matter boundary and vice versa and averaging these two values [START_REF] Fischl | Measuring the thickness of the human cerebral cortex from magnetic resonance images[END_REF]. As the generated cortical maps are not restricted to the voxel resolution of the original data, they are capable of detecting sub-millimeter differences between diagnostic groups. This automated approach of cortical thickness measurement have been validated against manual measurements in schizophrenia [START_REF] Kuperberg | Regionally localized thinning of the cerebral cortex in schizophrenia[END_REF].

Genetic analyses and identifying of subgroups

Fifty-two patients with schizophrenia and 42 healthy controls were genotyped for the G72 SNP (M15 = rs2391191). Genotyping was performed as part of a large-scale multi-center study using high throughput technology including 768 SNPs (GoldenGate assay; Illumina, San Diego) [START_REF] Treutlein | Dissection of phenotype reveals possible association between schizophrenia and Glutamate Receptor Delta 1 (GRID1) gene promoter[END_REF]. Primer sequences and PCR conditions can be obtained from the authors upon request. The raw genotypes were quality controlled by visual inspection of the genotype cluster plots.

The 94 participants recruited in the study center Jena underwent structural MRI and were analyzed in the present study. To estimate the replicate error rate for quality comparison purposes we genotyped a subset of the sample twice. Two out of 94 DNA samples were randomly chosen for this purpose. For the SNPs genotyped in the framework of the multi-center study, all genotypes between duplicates were consistent (0% replicate error rate). By a standard 1 df chi-square test, there were no significant deviations from Hardy-Weinberg equilibrium for the genotype distributions of the studied sample. Based on the considerations given in the introduction section we examined only the DAOA Arg30Lys SNP with regard to a potential association with cortical thickness. Depending on the M15 genotype, patients with schizophrenia and healthy controls were divided in subgroups resulting in two genotype subgroups per diagnostic group, i.e. 2 x A/A and A/G; 2 x G/G.

Statistical analysis

Statistical cortical maps

Each cortical thickness measurement of each vertex of the subjects' surface was mapped on a common spherical coordinate system using a spherical transformation.

Maps were smoothed using a Gaussian kernel of 10 mm. A general linear model controlling for the effect of age was used to estimate differences in cortical thickness between the diagnostic groups according to the genotype in patients and healthy controls separately at each vertex of the surface.

Right and left hemispheres were tested separately.

Monte carlo simulation and clustering

As a conservative procedure for the correction of multiple comparisons a Monte Carlo simulation with 10000 iterations and cluster analysis was then performed to identify contiguous clusters of significant cortical thickness differences between the genotype groups in patients and healthy controls (p < 0.05).

The simulation and clustering approach is implemented in FreeSurfer and based on the AlphaSim algorithm [START_REF] Ward | Simultaneous Inference for FMRI Data[END_REF]. The significance level is obtained through a combination of a probability threshold and cluster size threshold. The Monte Carlo simulation computes the distribution of the maximum cluster size. The p values of the resulting clusters of the original data expressed as cluster wise probability (CWP)

represent the probability to identify a maximum cluster of that size or larger during the simulation step. Hence, the cluster wise probability resulting from the simulation and clustering is equivalent to the overall alpha significance level.

Results

Schizophrenia patients with the AA/AG genotype in comparison to patients with the GG genotype exhibit a significantly (corrected for multiple comparisons) thinner cortex in two almost inversely arranged clusters on the left and right hemisphere. The clusters comprised parts of the middle temporal, inferior parietal and lateral occipital cortex of both hemispheres (s. figure 1). Table 2 summarizes the characteristics of the two clusters.

<Insert figure 1> <Insert table 2> In healthy controls the AA/AG carriers demonstrated no significant differences in cortical thickness in comparison to healthy GG carriers.

We additionally performed a genotype x diagnosis interaction analysis with the same four groups as defined for the comparisons of cortical thickness between the genotype groups (HC: AA/AG; HC: GG; SZ: AA/AG; SZ: GG) and applying the same statistical conditions (Monte Carlo simulation, automated clustering). We did not detect a significant difference between the diagnostic groups regarding the effect of the genotypes.

Discussion

In order to identify a potential pathomechanism of the glutamatergic regulatory gene variant DAOA Arg30Lys, the present study is the first to specifically investigate the association of the DAOA Arg30Lys variant on cortical morphology in schizophrenia and healthy controls. Our data demonstrate that this variant has a significant impact on cortical thickness in schizophrenia, but not in healthy controls. In accordance to our hypotheses based on the results of the meta-analysis of the G72 gene (Detera-

Wadleigh and McMahon, 2006b

) A carriers with schizophrenia demonstrated a thinner cortex in an area covering the temporo parietal junction and lateral occipital regions on both hemispheres in comparison to GG carriers.

Although the recent meta -analysis from SchizophreniaGene Database of Schizophrenia Research Forum (http://www.schizophreniaforum.org/res/sczgene), which did not include family-based association data, did not point into direction of a significant association of the Arg30Lys SNP and schizophrenia, our data implicate that the rs2391191 variant has a distinct impact on cerebral morphology in schizophrenia and might influence pathophysiological processes relevant for the disorder.

Reduced cortical thickness in risk carriers and potential functional implications: the theory of mind network and visual processing deficits in schizophrenia

Beyond other cortical regions comprising the medial prefrontal [START_REF] Vollm | Neuronal correlates of theory of mind and empathy: a functional magnetic resonance imaging study in a nonverbal task[END_REF] and the posterior cingulate cortex [START_REF] Gobbini | Two takes on the social brain: a comparison of theory of mind tasks[END_REF] the temporo-parietal junction has been indicated to be an anatomical node of the ToM network [START_REF] Andreasen | Theory of mind and schizophrenia: a positron emission tomography study of medication-free patients[END_REF][START_REF] Brune | An fMRI study of theory of mind in schizophrenic patients with "passivity" symptoms[END_REF]. Impairments of ToM network are confirmed by two meta-analyses [START_REF] Bora | Theory of mind impairment in schizophrenia: metaanalysis[END_REF][START_REF] Sprong | Theory of mind in schizophrenia: meta-analysis[END_REF] and can be regarded as a trait marker for schizophrenia. In the present study cortical thinning of the temporo-parietal junction has been demonstrated in G72 risk allele carriers. Thus, this subtype of schizophrenia patients might be predominantly vulnerable to develop ToM deficits.

This assumption would additionally strengthen the view that the phenotypical heterogeneity of schizophrenia might be reduced by defining endophenotypical and genetically subgroups. It should be emphasized, however, that this assumption is very speculative and further, particularly multimodal, studies are needed to elucidate this issue.

In addition to the observed reduced cortical thickness of the temporo-parietal junction the clusters of cortical thinning in G72 risk carriers comprise inferior parietal and lateral occipital areas. These cortical regions are mostly confined to parts of the "dorsal stream" of visual processing in humans [START_REF] Ettlinger | Object vision" and "spatial vision": the neuropsychological evidence for the distinction[END_REF]. Visual processing deficits are core parts of the pathophysiology in schizophrenia [START_REF] Coleman | Schizophrenia patients show deficits in shifts of attention to different levels of globallocal stimuli: evidence for magnocellular dysfunction[END_REF][START_REF] Kantrowitz | Seeing the world dimly: the impact of early visual deficits on visual experience in schizophrenia[END_REF]. FMRI studies revealed that lateral occipital areas, the inferior parietal cortex and the human motion selective cortex (hMT) demonstrate a disturbed activation during visual processing [START_REF] Green | Regional brain activity associated with visual backward masking[END_REF]. Studies in unaffected siblings of schizophrenia patients illustrated that visual processing deficits are related to genetic liability [START_REF] Green | Forward and backward visual masking in unaffected siblings of schizophrenic patients[END_REF]. Interestingly, it has also been demonstrated that theory of mind impairments in schizophrenia are associated with deficits of motion perception as an upward consequence of altered visual processing [START_REF] Kelemen | Theory of mind and motion perception in schizophrenia[END_REF]. Visual processing has been shown to be modulated via the glutamatergic system [START_REF] Javitt | When doors of perception close: bottom-up models of disrupted cognition in schizophrenia[END_REF]. In accordance, a study of [START_REF] Uhlhaas | Perceptual organization in ketamine users: preliminary evidence of deficits on night of drug use but not 3 days later[END_REF] demonstrated visual perception deficits in ketamine users similar to those deficits found in schizophrenia. The G72 gene investigated in the present study is strongly connected to the glutamatergic system.

4.2

Functional role of G72 and its impact as a potential target for therapeutic

interventions in schizophrenia

The G72 gene is directly involved in the cerebral glutamatergic neurotransmission.

The product of G72 (DAOA) interacts with D -amino acid oxidase (DAAO), which degrades D-amino acids in the human brain. The activity of DAAO is increased in schizophrenia [START_REF] Habl | Increased D-amino acid oxidase expression in the bilateral hippocampal CA4 of schizophrenic patients: a post-mortem study[END_REF][START_REF] Madeira | Increased brain D-amino acid oxidase (DAAO) activity in schizophrenia[END_REF]. These findings would imply a relative cerebral deficit of D-amino acids such as D-Serine, which acts as modulator at the N-methyl-D-aspartate (NMDA) ionotropic glutamate receptor. Indeed, decreased D-Serine serum levels have been shown in schizophrenia [START_REF] Hashimoto | Decreased serum levels of D-serine in patients with schizophrenia: evidence in support of the N-methyl-D-aspartate receptor hypofunction hypothesis of schizophrenia[END_REF]. The D-amino acid oxidase activator (DAOA) might act as negative effector for the DAAO [START_REF] Sacchi | pLG72 modulates intracellular D-serine levels through its interaction with D-amino acid oxidase: effect on schizophrenia susceptibility[END_REF]. Thus, a genetically impaired and inactive DAOA would lead to hyperactivity of DAAO. Subsequently, hyperactivity of DAAO may result in decreased D-Serine levels. Decreased D-Serine levels induce a hypofunction of the glutamatergic system via understimulation of NMDA receptors. Dysfunction of NMDA receptors is a key pathophysiology of schizophrenia [START_REF] Coyle | NMDA receptor function, neuroplasticity, and the pathophysiology of schizophrenia[END_REF][START_REF] Coyle | Converging evidence of NMDA receptor hypofunction in the pathophysiology of schizophrenia[END_REF][START_REF] Javitt | Glycine modulators in schizophrenia[END_REF]. Accordingly, clinical trials demonstrated that the addon treatment with D-Serine has a significant beneficial effect on negative and cognitive symptoms in schizophrenia [START_REF] Heresco-Levy | D-serine efficacy as add-on pharmacotherapy to risperidone and olanzapine for treatmentrefractory schizophrenia[END_REF][START_REF] Tsai | D-serine added to antipsychotics for the treatment of schizophrenia[END_REF].

In accordance to current neurodevelopmental models of schizophrenia [START_REF] Du Bois | Early brain development disruption from NMDA receptor hypofunction: relevance to schizophrenia[END_REF] the present study underlines the relevance of a glutamatergic regulatory gene variant for cortical thickness in regions relevant for schizophrenia.

This corroborates the assumption, that glutamatergic dysregulation is associated with cortical thinning processes as an underlying neuropathological feature of schizophrenia.

Potential cerebral mechanism compensating for the effect of the G 72 risk variant in healthy controls

The biological effect of the amino acid change caused by the Arg30Lys variant can be assumed to be present in both healthy subjects and patients. Hence, we hypothesized finding cortical thickness differences between the genotype groups in both patients with schizophrenia and healthy controls. In the present study, we did not find an impact of the risk genotype on cortical thickness in healthy controls.

Various cerebral mechanisms might explain this finding. Cortical thinning originated from different processes of brain development and at differing time points. In early stages of brain development cortical thickness is impacted by neuronal migration, forming of cortical layers and building of cortico-cortical connections [START_REF] Hilgetag | Developmental mechanics of the primate cerebral cortex[END_REF]. Processes of net synaptic pruning occurring in the adolescence period [START_REF] Huttenlocher | Regional differences in synaptogenesis in human cerebral cortex[END_REF]) also have an influence on cortical thickness. Later in life age related processes additionally lead to cortical thickness reduction, which might be accelerated in schizophrenia [START_REF] Pol | Volume changes in gray matter in patients with schizophrenia[END_REF]. At which stage a distinct G72 risk genotype impacts these cortical processes is speculative. However, it gets obvious that at every stage compensatory mechanism (i.e. lower synaptic pruning, preservation of accelerated age related processes) might lead to unaltered cortical morphology in healthy risk carriers so that the critical point of vulnerability to manifest the disorder is not reached.

4.4

Potential limitations of the study 1. 5 T MRI was used in the present study to explore the association of the Arg30Lys polymorphism and cortical thickness in schizophrenia and healthy controls demonstrating a significant association of the Arg30Lys risk variant and cortical thinning in schizophrenia. As higher magnetic field strength might be more sensitive for the detection of structural alterations [START_REF] Perri | Signal abnormalities on 1.5 and 3 Tesla brain MRI in multiple sclerosis patients and healthy controls. A morphological and spatial quantitative comparison study[END_REF] we cannot definitely exclude that a significant association of the Arg30Lys variant and cortical thickness in healthy controls might get unveiled with the use of higher scanner field strength.

Thus, further studies are needed to shed more light on this issue.

Concluding remarks

The principal finding of our study that the glutamatergic regulatory gene variant DAOA Arg30Lys is associated with a distinct cortical thinning demonstrates that this variant has an impact on brain morphology in functional relevant areas in

Titles and legends to figures

Cortical statistical maps displaying cortical thickness differences in schizophrenia patients between carriers of AA/AG and GG. Displayed is the lateral surface of both hemispheres, pial and inflated view, p-values corrected for multiple comparison

  examined the association of a four locus haplotype of DAOA including the rs2391191 SNP with longitudinal cerebral lobe volume change in schizophrenia. This association was tested between the volumes of the segregated cerebral lobes, total cerebral volume and the resulting three allele representations of the DAOA haplotype in a longitudinal design in schizophrenia. A significant association of total frontal lobe volume decrease after a period of on average three years of illness for one specific allele representation of the DAOA haplotype was detected indicating that the DAOA risk variants might be associated with progressive cerebral changes in schizophrenia.

  schizophrenia. The affected areas are mostly confined to cortical regions constituting central nodes in the theory of mind network and higher visual processing. Both processes might be influenced via glutamatergic modulation. Our finding thus adds new evidence for the pathophysiological relevance of DAOA in schizophrenia and might open new perspectives for DAOA and related glutamatergic mechanisms as a putative target for therapeutic interventions.

Figure 1

 1 Figure 1

Table 2

 2 Significant clusters for the left and right hemisphere, cluster size in mm², p-values from the Monte carlo simulation and clustering as cluster wise probability (CWP), resulting from the vertex wise comparison of cortical thickness between the two patients groups

	Cortex area	Size in mm²	Talx	Taly	Talz	CWP
	Left middle temporal -
	inferior parietal -lateral	1174.9	-38.9	-61.5	30.7	0.0108
	occipital
	Right middle temporal -
	inferior parietal -lateral	1156.1	37.8	-58.3	15.3	0.0179
	occipital

Acknowledgments

This work was supported by the Bundesministerium für Bildung und Forschung, BMBF Grant 01GW0740 and the Interdisciplinary Center for Clinical Research of the University of Jena (IZKF). The sponsor served no role in the study design; in the collection, analysis and interpretation of data; in the writing of the report; or in the decision to submit the paper for publication.

Disclosure Statement

The authors declare that HS received compensation from Otsuka and Lilly in 2008.

All of the authors reported no financial, personal or other relationships with other people or organizations that could inappropriately influence, or be perceived to influence, their work.

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