Thomas Ricatte
email: thomas@ricatte.fr

Rémi Gilleron
email: remi.gilleron@univ-lille.fr

Marc Tommasi
email: marc.tommasi@univ-lille.fr

David Blei

Skill Rating for Multiplayer Games Introducing Hypernode Graphs and their Spectral Theory

Keywords: Hypergraphs, Graph Laplacians, Graph Kernels, Spectral Learning, Semisupervised Learning, Multiplayer Games, Skill Rating Algorithms

. As we leverage graph-based SSL algorithms and because games can be seen as relations between sets of players, we then generalize the approach. For this aim, we introduce a new finite model, called hypernode graph, defined to be a set of weighted binary relations between sets of nodes. We define Laplacians of hypernode graphs. Then, we show that the skill rating problem for multiplayer games can be formulated as arg min s s T ∆s where ∆ is the Laplacian of a hypernode graph constructed from a set of games. From a fundamental perspective, we show that hypernode graph Laplacians are symmetric positive semidefinite matrices with constant functions in their null space. We show that problems on hypernode graphs can not be solved with graph constructions and graph kernels. We relate hypernode graphs to signed graphs showing that positive relations between groups can lead to negative relations between individuals.

Introduction

We consider the skill rating problem for multiplayer games such as team sports and online games. We consider games between two teams where each team is composed of an arbitrary number of players. We assume, as in [START_REF] Herbrich | TrueSkill TM : A Bayesian Skill Rating System[END_REF], that the skill of a team is a weighted sum of the player skills. The problem is to infer individual player skills from a limited number of game outcomes. The skills can then be used for match-making, skill ranking or outcome prediction for new games.

Given a set of players and a set of games between teams of players, we model the player skills and the game outcomes by real-valued functions. We show that the skill rating problem is equivalent to solving a minimization problem arg min s s T ∆s where ∆ is a positive

G =    √ 2 2 √ 2 2 √ 2 - √ 2 - √ 2 0 1 0 0 -1 -1 1 0 0 √ 3 3 0 0 - √ 3 3   
Figure 1: Left: a hypernode graph h with three hyperedges h blue = {{1, 2, 3}, {4, 5}}, h black = {{4, 5}, {1, 6}} and h red = {{3}, {6}} where a hyperedge is represented by a rectangle and nodes in a hypernode are connected to the same side of the rectangle. Right : a gradient G of h which values are square root of the weights and are positive for one hypernode and negative for the other. The matrix G can be seen as an alternative definition of the hypernode graph h.

semidefinite matrix and some entries of s are constrained by the game outcomes. In order to solve this problem, we propose two algorithms. The first one is the semi-supervised learning (SSL) algorithm presented in Zhu et al. (2003) originally designed for graph-based SSL. The second one, also inspired from graph-based SSL, is to use a regression support vector machine using the Moore-Penrose pseudoinverse ∆ † of the matrix ∆ as a kernel.

In the experiments, we consider real data sets of multiplayer games (double tennis and online games) and apply our method in a batch setting. We get very competitive results with specialized algorithms such as Elo duelling (see [START_REF] Emrick | The Rating of Chess Players, Past and Present[END_REF] and TrueSkill (see [START_REF] Herbrich | TrueSkill TM : A Bayesian Skill Rating System[END_REF].

The above minimization problem can be solved by graph-based SSL algorithms because its formulation is similar to the case of graphs where ∆ is a graph Laplacian. Here, the positive semidefinite matrix ∆ expresses conditions on the players skills based on the game outcomes, where each game can be considered as a binary relation between two sets of players. Driven by these remarks, we introduce hypernode graphs as a finite model for relations between sets and we show that our matrix ∆ is the Laplacian of such an hypernode graph.

We define a hypernode graph to be a set of hyperedges. A hyperedge is a pair of disjoint hypernodes, where a hypernode is a set of nodes. Every node of a hypernode is given a non negative real-valued weight. Roughly speaking, a hypernode models a group, a hyperedge models a relation between two groups, and individual weights correspond to the contribution of each individual to the relation between the two groups. An example of hypernode graph is shown in Figure 1. Hypernode graphs generalize over graphs because there is a one to one correspondence between undirected graphs and hypernode graphs in which all hypernodes are singleton sets.

We consider real-valued node functions over hypernode graphs and we assume an additive model where the hypernode valuation is the linear combination of its node valuations. We assume that connected hypernodes tend to have similar values. For real-valued node functions over discrete structures, the gradient is a discrete derivative that allows to mea-sure such a similarity. Therefore, we define a gradient for hypernode graphs and we denote by G the gradient matrix of a hypernode graph h. An example is shown in Figure 1. We define the Laplacian ∆ of h to be ∆ = G T G. Then, the quantity f T ∆f measures the smoothness of a real-valued node function f on the hypernode graph h. In other words, f T ∆f is small when connected hypernodes have close values. Then, we show that a set of multiplayer games can be modeled by a hypernode graph and that our algorithm for the skill rating problem is finding a real-valued node function s minimizing Ω(s) = s T ∆s, where ∆ is the Laplacian of the constructed hypernode graph, under constraints on the s-values for game outcomes.

One question raised by the introduction of our new framework is whether problems on hypernode graphs can be solved using graphs. Indeed, for hypergraphs (a generalization of graphs where a hyperedge is a set of nodes), it has been proved in [START_REF] Agarwal | Higher Order Learning with Graphs[END_REF] that the hypergraph Laplacians introduced at this time (other hypergraph Laplacians have been defined since (see [START_REF] Hubert | Spectral properties of hypergraph laplacian and approximation algorithms[END_REF], and references within)), do not extend graph Laplacians because learning based on a hypergraph Laplacian can be proved equivalent to learning on a graph Laplacian using an adequate graph construction (such as clique expansion or star expansion). This is not the case for hypernode graphs. Indeed, we show that no graph construction allows to define a set of smooth functions over the graph equal to the set of smooth functions over a given hypernode graph. Thus, we state that hypernode graph Laplacians strictly generalize over graph Laplacians.

It can be noted that the class of hypernode graph Laplacians strictly contains the class of graph Laplacians and also contains the class of graph kernels (the Moore-Penrose pseudoinverse of a graph Laplacian). A convex linear combination of graph kernels (used in multiple graph kernel learning as in [START_REF] Argyriou | Combining Graph Laplacians for Semi-Supervised Learning[END_REF]) is not, in general, a graph kernel but we prove that it is a hypernode graph kernel (the Moore-Penrose pseudoinverse of a hypernode graph Laplacian). We conjecture that the class of hypernode graph Laplacians is the smallest class closed by convex linear combinations which contains all graph kernels. We also show that, for every hypernode graph, we can construct a signed graph with adjacency matrix W and degree matrix D such that D-W is the Laplacian of the hypernode graph. This shows that positive relations between groups can lead to negative relations between individuals. Let us recall that, for arbitrary signed graphs, the matrix D -W can be indefinite. From a spectral theory perspective, hypernode graphs correspond to the class of signed graphs such that D -W is positive semidefinite introduced by [START_REF] Koren | ACE: a fast multiscale eigenvectors computation for drawing huge graphs[END_REF] for the problem of drawing graphs. Last, we study how to define a distance between nodes in hypernode graphs and we leave open questions on connected components and random walks in hypernode graphs.

Skill Rating for Multiplayer Games

We consider competitive games between two teams where each team is composed of an arbitrary number of players. A first objective is to compute the skills of individual players from a batch of games with their outcomes. A second objective is to predict a game outcome for a new game. In this section, we show how to model a batch of games and to infer player skills that allow to predict game outcomes of new games.

Skill Rating as an Optimization Problem

Let us consider a set of players P = {1, . . . , n} and a set of games Γ = {1, . . . , m} together with their respective game outcome. Each game is between two teams of an arbitrary number of players. Let us also consider that a player i contributes to a game γ with a positive real value c γ (i). Let us assume that each player has a skill s(i), we suppose an additive model as in [START_REF] Herbrich | TrueSkill TM : A Bayesian Skill Rating System[END_REF] stating that the skill of a team is the weighted sum of the skills of the players in the team. More formally, given two teams of players A = {a 1 , a 2 , . . . , a } and B = {b 1 , b 2 , . . . , b k } playing game γ, then A is predicted to be the winner if

i=1 c γ (a i )s(a i ) > k i=1 c γ (b i )s(b i ) . (1)
Equivalently, one can rewrite this inequality by introducing a positive real number o γ on the right hand side such that

i=1 c γ (a i )s(a i ) = o γ + k i=1 c γ (b i )s(b i ) , (2)
where the real number o γ quantifies the game outcome. In the case of a draw, there is an equality between the team skills, thus the game outcome o γ is set to 0. Given a set of games with their respective outcome, the goal is to infer a skill rating function s ∈ R n that respects as much as possible Equation (2) for every game. We define the cost of a game γ with outcome o γ for a skill function s by

C γ (s) = i=1 c γ (a i )s(a i ) -o γ + k i=1 c γ (b i )s(b i ) 2 .
Consequently, given a set of games Γ and the corresponding game outcomes, the goal is to find a skill rating function s * that minimizes the sum of the different costs:

s * = arg min s γ∈Γ C γ (s) . (3)
We now reformulate this problem as a SSL problem over a positive semidefinite matrix. For this, let us define the matrix G ∈ R m× (n+m) , where n is the number of players and m the number of games, as follows. Let γ ∈ Γ be a game between teams A and B where A is supposed to be the winner, the γ-th row of G models game γ and is defined by: for

1 ≤ i ≤ n, G γ,i = c γ (a i ) if a i ∈ A, G γ,i = -c γ (a i ) if a i ∈ B,
and 0 otherwise (the player i is not involved in the game γ); and, for 1 ≤ γ ≤ m, G i,n+γ = -1 for γ = γ and 0 otherwise.

Then, it is easy to show that the skill rating problem (3) is equivalent to find a realvalued node function s ∈ R (n+m) solution of minimize s s T ∆s subject to for every game γ with outcome o γ , s(n

+ γ) = o γ (4)
where ∆ = G T G. We can note that the matrix ∆ is positive semidefinite.

Regularizing the Optimization Problem

When the number of games is small, player skills can be defined independently while satisfying the constraints and it will be irrelevant to compare them. In order to solve this issue, we introduce in Equation 4 a regularization term based on the standard deviation σ(s p ) of players skills where s p = (s(1), . . . , s(n)). This allows us to reduce the spread of the player skills and leads us to find a real valued node function s ∈ R (n+m) solution of minimize s s T ∆s + µσ(s p ) 2 subject to for every game γ with outcome o γ , s(n

+ γ) = o γ (5)
where µ is a regularization parameter. That is, we control the spread of s p avoiding extreme values for players involved in a small number of games. In order to apply SSL algorithms over a positive semidefinite matrix, we propose to rewrite Problem 5.

First, let us recall that if s is the mean of the player skills vector s p , then, for all q ∈ R,

we have σ(s p ) 2 = 1 n n i=1 (s(i) -s) 2 ≤ 1 n n i=1 (s(i) -q) 2 . Thus, Problem 5 is equivalent to minimize s∈R (n+m) ,q∈R s T ∆s + µ n n i=1 (s(i) -q) 2 subject to for every game γ with outcome o γ , s(n + γ) = o γ (6)
We now define the matrix n+m+1) . The first m rows of G µ are the rows of G with an additional null column. The last n rows of G µ are defined by, for every 1

G µ ∈ R (m+n)×(
≤ i ≤ n, G µ,(m+i,i) = -µ n , G µ,(m+i,n+m+1) = µ n
, and 0 otherwise. Note that the last n rows of G µ allow to compute µ n n i=1 (s(i) -q) 2 where s(n + m + 1) = q when computing s T G T µ G µ s. Thus Problem 6 is equivalent to finding a real-valued node function s ∈ R (n+m+1) solution of minimize

s s T ∆ µ s subject to for every game γ with outcome o γ , s(n + γ) = o γ (7)
where ∆ µ = G T µ G µ is positive semidefinite. Note that, assuming unitary contributions and a Gaussian distribution for skill ratings, we can prove that the value of µ/n should have the same order of magnitude than the average number of games played by a player. Indeed, this will allow us to obtain close expected values for the two terms s T ∆s and µσ(s p ) 2 . We follow this guideline in the experiments presented below.

Inferring Skill Ratings and Predicting Game Outcomes

Problem 7 can be viewed as a SSL problem because the question is to predict player skills (values s(i) for 1 ≤ i ≤ n) from known outcomes (values s(n + γ) for 1 ≤ γ ≤ m). We propose two algorithms: H-ZGL: as the matrix ∆ µ is positive semidefinite, we can use the SSL algorithm presented in Zhu et al. (2003). This algorithm was originally designed for graphs and solves exactly Problem 7 by putting hard constraints on the outcome values. It should be noted that the algorithm is parameter free.

H-SVR: another approach to solve the SSL problem is to use a regression algorithm. For this, we consider the Moore-Penrose pseudoinverse ∆ † µ of the matrix ∆ µ , we train a regression support vector machine and predict player skills. The main parameter is the soft margin parameter.

Given the player skills, we can deduce team skills, and finally predict game outcomes for new games. For this, we suppose that we are given a training set Γ l of games with known outcomes together with a test set Γ u of games for which game outcomes are hidden. The goal is to predict game outcomes for the test set Γ u . We follow the following protocol.

Algorithm 1 Predicting game outcomes Input: a training set Γ l of games and a test set Γ u of games 1: Build, as described in Sections 2.1 and 2.2, the matrix ∆ µ using game set Γ l 2: Compute an optimal skill rating s * using H-ZGL or H-SVR for Problem 7 3: Compute the mean skill s among players in Γ l 4: for each game in Γ u do 5:

Assign skills given by s * to players involved in Γ l , and s otherwise 6:

Compare the team skills and predict the winner according to Equation 17: end for

Experiments

In this section, we report experimental results for the inference of player skills and the prediction of game outcomes for different data sets in the batch setting described above. Note that other works have considered the online setting as in [START_REF] Herbrich | TrueSkill TM : A Bayesian Skill Rating System[END_REF] or Elo (1978).

Tennis Doubles

We consider a data set of tennis doubles collected between January 2009 and September 2011 from ATP tournaments (World Tour, Challengers and Futures). Tennis doubles are played by two teams of two players. Each game has a winner (no draw is allowed). A game is played in two or three winning sets. The final score corresponds to the number of sets won by each team during the game. The data set Γ consists in 10028 games with n = 1834 players.

Along the experimental process, given a proportion ρ, we draw randomly a training set Γ l of size ρ% of the number of games in Γ. The remaining games define the test set Γ u . We present in Figure 2 several statistics related to this process on the Tennis data set. First, it can be noticed that many players have played a small number of games. Second, when the number of games in the training set is small, half of the players are involved in games in the test set. Therefore, the skill rating problem and the game outcome prediction problem become far more difficult to solve when few games are used for learning.

The experimental setting is defined as follows. Let ρ be a proportion varying from 10% to 90% by 10%. For every value of ρ, we repeat ten times: draw a random training set Γ l of size ρ% of the number of games in Γ; let the remaining games define the test set Γ u ; infer player skills and predict game outcomes following Algorithm 1. We set all player The experiments presented in this section use µ/n = 16. We report in Figure 3 prediction results using: Algorithm 1 with H-ZGL, Algorithm 1 with H-SVR, Elo Duelling, and Trueskill. 1 It can be noted that Elo Duelling performs poorly but Elo was designed for one against one games. It should be noted that we have done similar experiments for tennis singles. The results are not reported here but they show that Elo Duelling and TrueSkill obtain similar results but are outperformed by H-ZGL and H-SVR. For the data set of tennis doubles, it must be noted that H-ZGL and H-SVR outperform TrueSkill for a small number of training games. Also, H-ZGL outperforms TrueSkill whatever is the number of training games. On this set of experiments H-ZGL outperforms H-SVR but it should be noted that H-ZGL is parameter free and that we use H-SVR with the default soft margin parameter value.

Xbox Title Halo2

The Halo2 data set was generated by Bungie Studio during the beta testing of the XBox title Halo2. It has been notably used in [START_REF] Herbrich | TrueSkill TM : A Bayesian Skill Rating System[END_REF] to evaluate the performance of the TrueSkill algorithm. We consider the Small Teams data set with n = 4992 players and 27536 games opposing up to 12 players in two teams which can have a different size. Each game can result in a draw or a win of one of the two teams. The proportion of draws is 22.8%. As reported in [START_REF] Herbrich | TrueSkill TM : A Bayesian Skill Rating System[END_REF], the prediction of draws is challenging and We consider the same experimental process. For the construction of the hypernode graph in Algorithm 1, we fix all players contributions in games to 1 and we again set the parameter value µ/n to 16. In the optimization problem (7), the game outcomes are set to 1 when the game has a winner and 0 otherwise because game scores vary depending on the type of game. The matrix ∆ µ has dimension n + 2 + 1 = 4995. We again compare the skill rating algorithms H-ZGL, H-SVR, Elo Duelling and TrueSkill. The number of prediction errors over game outcomes is computed assuming that a draw can be regarded as half a win, half a loss as in [START_REF] Lasek | The predictive power of ranking systems in association football[END_REF]. We present the experimental results in Figure 4. For a proportion of 10% of games in the training set, H-ZGL, H-SVR and TrueSkill give similar results while with larger training sets, our hypernode graph learning algorithms outperform TrueSkill. Contrary to the previous experiment, H-SVR (with default parameter value) outperforms H-ZGL.

Hypernode Graphs

In the previous section, we consider the skill rating problem as a SSL problem and we express it as a minimization problem using a positive semidefinite matrix ∆ which expresses conditions on the players skills based on the game outcomes. This allows us to use graphbased SSL algorithms. As games can be considered as relations between sets of players, there is a discrete structure underlying the skill-rating problem. The existing discrete structure and the similarity with graph-based SSL motivate the work presented in this section. Indeed, we now introduce hypernode graphs as a finite model for relations between sets, we show that our matrix ∆ is the Laplacian of such an hypernode graph, and we show that Problem 7 expresses the smoothness of a real-valued node function over the hypernode graph with some constraints on node values.

Introducing Hypernode Graphs

The following definition is our contribution to the modeling of binary relationships between sets of entities.

Definition 1 A hypernode graph h = (V, H) is a set of nodes V and a set of hyperedges H. Each hyperedge h ∈ H is an unordered pair {s h , t h } of two non empty and disjoint hypernodes (a hypernode is a subset of V ). Each hyperedge h ∈ H has a weight function w h mapping every node i in s h ∪ t h to a positive real number w h (i) (for i / ∈ s h ∪ t h , we define w h (i) = 0). Each weight function w h must satisfy the Equilibrium Condition defined by

i∈t h w h (i) = i∈s h w h (i) . (8)
We denote by n the number of nodes. We say that a node i belongs to a hyperedge h, that we denote by i ∈ h, if w h (i) = 0. We define the degree of a node i by deg(i) = h w h (i). The degree of a node is positive when it belongs to at least one hyperedge. We define the diagonal degree matrix by D = diag(deg(1), . . . , deg(n)).

An example of hypernode graph is shown in Figure 1. The hyperedge h blue links the sets {1, 2, 3} and {4, 5}. The weights of h blue satisfy the Equilibrium condition:

1/2 + 1/2 + √ 2 = √ 2 + √ 2.
The hyperedge h black links the sets {4, 5} and {1, 6} and satisfies the Equilibrium condition. The hyperedge h red links the two singletons {3} and {6} with equal weights, thus the hyperedge h red can be viewed as an edge with edge weight 1/3.

As noted in the previous example, when a hyperedge h is an unordered pair {{i}, {j}}, the Equilibrium Condition states that the weights w h (i) and w h (j) are equal. Therefore, every hypernode graph such that all hyperedges are unordered pairs of singleton nodes can be viewed as a graph with adjacency matrix W defined by W i,j = W j,i = w h (i) = w h (j) for every hyperedge {{i}, {j}}, and 0 otherwise. Conversely, every graph can be viewed as a hypernode graph where every hypernode is a singleton set.

Laplacians and Kernels of Hypernode Graphs

We are interested in evaluating real-valued functions on nodes and on hypernodes. We assume a weighted linear model such that any real-valued node function f can be extended to a hypernode u of a hyperedge h by f (u) = i∈u f (i) w h (i). Moreover, we will consider an homophilic assumption stating that connected hypernodes tend to have similar values. In order to implement this assumption, we introduce a gradient for hypernode graphs by Definition 2 Let h = (V, H) be a hypernode graph and f be a real-valued node function, the (hypernode graph) unnormalized gradient of h is a linear application, denoted by grad, that maps every real-valued node function f into a real-valued hyperedge function grad(f ) defined, for every h = {s h , t h } in H, by

grad(f )(h) = f (t h ) -f (s h ) = i∈t h f (i) w h (i) - i∈s h f (i) w h (i) , (9)
where an arbitrary orientation of the hyperedges has been chosen. We denote by G ∈ R p×n the matrix of grad. The square n × n real-valued matrix ∆ = G T G is defined to be the unnormalized Laplacian of the hypernode graph h.

As suggested by an anonymous reviewer, a hypernode graph could be defined by a gradient matrix as shown in Figure 1.

It should be noted that the Laplacian ∆ does not depend on the arbitrary orientation of the hyperedges chosen for defining the gradient. Also, by definition, the Laplacian ∆ is positive semidefinite. Moreover, because of the definition of the additive model and because of the Equilibrium Condition 8, the gradient of every constant node function is the zero-valued hyperedge function. Consequently, we have 1 ∈ Null(∆) where Null(∆) is the null space of ∆. Last, it could be noted that, when the hypernode graph is a graph (all hypernodes are singleton sets), then the hypernode graph Laplacian is equal to the unnormalized graph Laplacian.

The Laplacian allows to define the smoothness of a real-valued node function f over a hypernode graph h to be Ω(f ) = f T ∆f .

Last, we define the hypernode graph kernel of a hypernode graph h to be the Moore-Penrose pseudoinverse ∆ † of the hypernode graph Laplacian ∆.

Because a hypernode graph Laplacian is positive semidefinite, we can leverage the spectral learning algorithms defined in Von Luxburg (2007), [START_REF] Zhou | Learning from labeled and unlabeled data on a directed graph[END_REF]Zhu et al. (2003) from graphs to hypernode graphs.

Modeling the Skill Rating Problem with Hypernode Graphs

In this section, we show how to model a batch of games by a hypernode graph and how the problems ( 4) and ( 7) translate on the constructed hypernode graphs.

We consider a set of m games as in Section 2.1. The construction mimics the construction of the previous section with the additional trick to add a new node in order to satisfy the Equilibrium condition 8 for hyperedges. Indeed, we consider n + m + 1 nodes, n nodes for the players, m nodes for the outcomes and one lazy node. We can define, for every game γ a hyperedge h as follows.

1. The players of A define one of the two hypernodes of h, the weight of a player node i is defined to be c(i) 2 ;

2. do the same construction for the second team B;

3. add the outcome node associated with γ to the set of nodes of the losing team, its weight is set to 1;

4. add the lazy node to the set of nodes corresponding to the winning team, its weight is chosen in order to ensure the Equilibrium condition for the hyperedge h.

This defines the hypernode graph h = (V, H). Then, it is easy to show that Problem 4 is equivalent to minimizing s T ∆s where ∆ is the Laplacian of h with constraints for game outcomes and adding the additional constraint s(n + m + 1) = 0 for the lazy node.

For the regularization term, we define the hypernode graph h µ obtained from the hypernode graph h by adding a regularizer node and adding n hyperedges between every player node and the regularizer node R with (node) weights set to µ/n. Then, it is easy to show that Problem 7 is equivalent to minimizing s T ∆ µ s where ∆ µ is the Laplacian of h µ with constraints for game outcomes and adding the additional constraint s(n + m + 1) = 0 for the lazy node.

Then as said before, we can leverage graph-based SSL algorithms and consider, for instance, the algorithms H-ZGL and H-SVR of Section 2.3.

The Class of Hypernode Graphs Laplacians

Proposition 3 The class of hypernode graph Laplacians is the class of symmetric positive semidefinite real-valued matrices M such that 1 ∈ Null(M ) (the null space of M ).

Proof As a consequence of Definition 2, every hypernode graph Laplacian M is a symmetric positive semidefinite real-valued matrix such that 1 ∈ Null(M ). Conversely, let us consider the following algorithm Algorithm 2 Construction of a hypernode graph Input: M ∈ R n×n symmetric positive semidefinite such that 1 ∈ Null(M )

1: Set V to {1, . . . , n}; set H to ∅ 2: Compute a square root decomposition M = G T G 3: for each row r of G do 4: Create a new hyperedge h = {s h , t h } with s h = t h = ∅ 5:
for each node i ∈ V do 6:

Define w h (i) = r(i) 2 7: if r(i) < 0 then Add i to s h else if r(i) > 0 Add i to t h end if 8:
end for

9:

Add h to H 10: end for 11: return The hypernode graph h = (V, H) From a square root decomposition G T G of M , for each line of G, we define a hyperedge h = {s h , t h } where nodes in s h (respectively in t h ) have positive (respectively negative) values in the line, and node weights are chosen to be squares of values in the line. Then, the Equilibrium condition 8 is satisfied because 1 ∈ Null(M ). And it is easy to check that G is a gradient matrix of h and, consequently, that M = G T G is the Laplacian of h.

As a consequence of Proposition 3, the class of hypernode graph Laplacians

• is closed under convex linear combination,

• is closed under pseudoinverse,

• and strictly contains the class of graph Laplacians and the class of graph kernels (a graph kernel is the Moore-Penrose pseudoinverse of a graph Laplacian).

Linear combinations of graph kernels have been used for SSL (see for instance [START_REF] Argyriou | Combining Graph Laplacians for Semi-Supervised Learning[END_REF]. Another consequence of Proposition 3 is that any convex linear combination of graph kernels is a hypernode graph kernel (but not necessarily a graph kernel). To conclude this section, we propose

Conjecture:

The class of hypernode graph Laplacians is the smallest class closed by convex linear combination which contains all graph kernels.

The main point is to prove that every hypernode graph Laplacian is a convex linear combination of graph kernels which is difficult because a graph kernel has no simple analytic form.

Hypernode Graph Laplacians Strictly Generalize Graph Laplacians

As said above, the class of hypernode graph Laplacians strictly contains the class of graph Laplacians. But this does not allow us to claim that hypernode graph Laplacians provide a gain of expressiveness over graph Laplacians. Indeed, it could be the case that through a well chosen graph construction, the set of smooth functions f defined on hypernode graphs by f T ∆f = 0 could be made to coincide exactly with the set of smooth functions defined on graphs. The class of hypergraph Laplacians has been studied in that perspective of expressiveness. Indeed, it has been shown in [START_REF] Agarwal | Higher Order Learning with Graphs[END_REF] that hypergraph Laplacians such as the ∆ B from [START_REF] Bolla | Spectra, euclidean representations and clusterings of hypergraphs[END_REF], ∆ R from Rodríguez (2003) and ∆ ZHS from [START_REF] Zhou | Learning with hypergraphs: Clustering, classification, and embedding[END_REF] can be defined as (restrictions of) graph Laplacians using a graph construction such as the clique expansion (where each hyperedge is replaced by a clique graph with uniform weights) or the star expansion (where, for every hyperedge, a new node is added and is linked with all the nodes in the hyperedge).

While one can think of similar constructions for the case of hypernode graphs, we prove that there does not exist a (finite) graph expansion of a hypernode graph which defines the same set of smooth functions over the original set of nodes. The proof is based on the very simple hypernode graph h shown in Figure 5. Note that a function is smooth over h if and only if it satisfies (f (1

) + f (2) -f (3) -f (4)) 2 = 0 . (C)

We show that

Proposition 4 There do not exist a finite graph g whose node set contains {1, 2, 3, 4} and that satisfies the conditions:

1. All the smooth functions on g satisfy (C).

2. Any function that satisfy (C) can be extended to a smooth function on g.

Proof Let us define S = {1, 3} and denote by ∆ the Laplacian matrix of h. The indicator vector 1 S is in Null(∆) and, thus, define a smooth function on h. Let us consider a graph g e = (N e , E e ) whose nodeset N e contains the nodeset N = {1, 2, 3, 4} and that satisfies the two conditions presented in Proposition 4. We denote by ∆ e the Laplacian matrix of g e . Because of the second condition, the function 1 S can be extended to a smooth function f e on g e .

Since g e is a graph, we know that Null(∆ e ) is spanned by the indicator vectors of the connected components of g e (see for instance Von Luxburg, 2007). Since f e (1) = 1 S (1) = f e (2) = 1 S (2), 1 and 2 must be in different components in g e . Note that the same holds with f e (1) = f e (4), f e (3) = f e (2) and f e (3) = f e (4). By following a similar reasoning with S = {1, 4}, we eventually deduce that the four original nodes 1, 2, 3 and 4 must be in distinct components in g e .

Let us now consider the components C of g e that contains the node 1. As stated above, we have necessarily f e (2) = f e (3) = f e (4) = 0. The indicator function f e of C is smooth on g e but we have

(f e (1) + f e (2) -f e (3) -f e (4)) 2 = (1 + 0 -0 -0) 2 = 0 .
Consequently, f e does not satisfy (C), which violates the first condition and concludes the proof.

The first condition in Proposition 4 ensures that smooth functions on g are related to smooth functions over the original hypernode graph h. The second condition in Proposition 4 ensures that smooth functions on the original hypernode graph can be defined from smooth functions over expanded finite graphs. The combination of these two conditions ensure that the problem of finding smooth functions on h can be rephrased as a graph problem (finding the smooth functions on g and compute their restrictions to {1, 2, 3, 4}). The fact that no graph g can satisfy both conditions at the same time for a simple hypernode graph h shows that smoothness properties over hypernode graphs can not be defined over undirected expanded graphs.

1 2 3 4 1 1 1 1 ∆ =     1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1    
G = 1 1 -1 -1 0 0 -1 1 1 2 3 4 0.5 0.5 2 0.5 0.5 2 G = - √ 2 2 - √ 2 2 √ 2 0 - √ 2 2 - √ 2 2 0 √ 2
Figure 6: two L-equivalent hypernode graphs with their respective gradient which are square root of the Laplacian ∆ shown in Figure 7.

∆ =     1 1 -1 -1 1 1 -1 -1 -1 -1 2 0 -1 -1 0 2     ; W =     0 -1 1 1 -1 0 1 1 1 1 0 0 1 1 0 0     ; D =     1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 2     .
Figure 7: Laplacian matrix, pairwise weight matrix, and degree matrix for the two Lequivalent hypernode graphs shown in Figure 6.

root decomposition is not unique, there are several hypernode graphs with the same Laplacian that we called L-equivalent. Examples of L-equivalent hypernode graphs are given in Figure 6. In order to study the L-equivalent relation and to decide whether a hypernode graph is L-equivalent to a graph, we first show that hypernode graphs are related to signed graphs.

It is known that the Laplacian matrix ∆ of a graph can be written D -W where W is the adjacency matrix of the graph and D is the corresponding degree matrix. Let us consider a hypernode graph h with Laplacian ∆, we define the pairwise weight matrix W of h by W i,j = -∆ i,j if i = j, and 0 otherwise. Note that the pairwise weight matrix coincides with the classic adjacency matrix when the hypernode graph is a graph. Let us define the degree of a node i to be deg(i) = j∈V W i,j and let us denote by D the diagonal matrix of all deg(i). Then, because of the property 1 ∈ Null(M ) in Proposition 3, it is immediate that, for every i, ∆ i,i = j∈V W i,j . As a consequence, we have Proposition 5 Let h = (V, H) be a hypernode graph, let W be the pairwise weight matrix of h, and let D be the diagonal degree matrix of h. Then, the Laplacian of h is ∆ = D -W .

An example is shown in Figure 7. As a consequence of the above proposition, we can leverage the pairwise weight matrix to characterize L-equivalent hypernode graphs by Proposition 6 Two hypernode graphs are L-equivalent if and only if they have the same pairwise weight matrix. Two L-equivalent hypernode graphs have the same degree matrix.

The pairwise weight matrix W contains in general negative weights and can be thus interpreted as the adjacency matrix of a signed graph. Therefore, we define the reduced signed graph of a hypernode graph h to be the signed graph g with adjacency matrix W . An example is shown in Figure 8. Then, we can show that

• the reduced signed graph of a graph g (viewed as a hypernode graph) is the graph g,

• a hypernode graph h is L-equivalent to a graph if and only if its reduced signed graph is a graph,

• a signed graph with adjacency matrix W is the reduced signed graph of a hypernode graph if and only if the matrix D -W is positive semidefinite.

The definition of the pairwise weight matrix may seem ad hoc to mimic the definition of the Laplacian in the graph case. But, we can show that the pairwise weight matrix can be defined directly from the hypernode graph using the following formula ∀i = j, W i,j = h∈H P (h, i, j) w h (i) w h (j) ,

where P (h, i, j) = 1 if i and j belongs to two different ends of h, P (h, i, j) = -1 if i and j belongs to the same end of h, and 0 otherwise. The pairwise weight W i,j can be interpreted as a flow measure between node i and j in the hypergraph. It is computed as a sum over all the hyperedges involving nodes i and j. For every term in the sum, the quantity w h (i) can be viewed as the cost of entering the hyperedge h at node i, the quantity w h (j) as the cost of exiting the hyperedge h at node j, and P (h, i, j) as a sign depending whether i and j are in the same end or in different ends of the hyperedge.

Defining a distance in Hypernode Graphs

We now study whether a distance can be defined between nodes of a hypernode graph. Let us consider a hypernode graph h = (V, H) with Laplacian ∆. Let ∆ † be the Moore-Penrose pseudo-inverse of ∆, let us define d by, for every i, j in V ,

d(i, j) = ∆ † i,i + ∆ † j,j -2∆ † i,j . (11)
Because ∆ † is symmetric positive semidefinite, we have Proposition 7 d defines a pseudometric on h: it is positive, symmetric and satisfies the triangle inequality (for all i, j, k the inequality d(i, j) ≤ d(i, k) + d(k, j) holds).

For the pseudometric d to be a distance, d should satisfy also the coincidence axiom: d(i, j) = 0 ⇒ i = j. This is not true in general as d(1, 2) = 0 for the hypernode graph in Figure 5. The intuition is that the nodes 1 and 2 can not be distinguished because the smoothness condition is on the sum f (1) + f (2). Nevertheless we can show that Proposition 8 When Null(∆) = Span(1), d defines a metric (or distance) on h.

Let us recall that, in the graph case, Proposition 7 holds and that Proposition 8 holds when the graph is connected. By analogy with the case of graphs, the condition Null(∆) = Span(1) can be viewed as an algebraic definition of a connected hypergraph. So far, we have not found an alternative algorithmic definition of connected components in hypernode graphs.

Finally, let us note that, in the graph case, for connected graphs, both d and d 2 are metrics while, for hypernode graphs, d 2 does not satisfy the triangle inequality even if Null(∆) = Span [START_REF] Lee | Parameters of Elo and TrueSkill are the default parameters of Hamilton[END_REF]. We provide in [START_REF] Ricatte | Hypernode Graphs for Learning from Binary Relations between Groups in Networks[END_REF] a formulation of d 2 in terms of diffusion potentials but do not find an interpretation of d 2 using random walks in hypernode graphs.

Conclusion

We have introduced a new model and new algorithms to learn from binary relations between groups. We showed that our method obtained state-of-the-art results for skill rating in multiplayer games. We are convinced that our method could be successfully applied to other games where players have different roles (such as team sports like basketball, american football or baseball) by modeling roles with weights. From a machine learning perspective, we believe that this model will open the way to solving new learning problems in networks when relations between sets of nodes are meaningful. From a fundamental perspective, our model is a strict generalization of graphs. Many theoretical studies remain to be done on the spectral theory of hypernode graphs. Note that we leave open some questions such as finding a characterization of the class of hypernode graphs, finding an algorithmic definition of connected components, and how to interpret random walks and distances on hypernode graphs. Also the study of directed hypernode graphs remains to be done from a machine learning perspective.

  Figure 2: [left] Distribution of the number of players against the number of played games;[right] Average percentage of players in Γ u which are involved in some game in Γ l

Figure 3 :

 3 Figure 3: Predictive error for Double Tennis data set

Figure 4 :

 4 Figure 4: Predictive error for Halo2 data set

Figure 5 :

 5 Figure 5: A hypernode graph h and its Laplacian ∆

Figure 8 :

 8 Figure 8: The reduced signed graph with adjacency matrix W of Figure 7 of the two Lequivalent hypernode graphs shown in Figure 6.

Acknowledgments

The authors are very grateful to the referees for their constructive comments and to the Editor David Blei for his unwavering support.

Xiaojin Zhu, Zoubin Ghahramani, John Lafferty, et al. Semi-supervised learning using gaussian fields and harmonic functions. In Proceedings of the 20th International conference on Machine learning (ICML-03), pages 912-919, 2003.


Adrien Toutant

General and exact pressure evolution equation

A crucial issue in fluid dynamics is related to the knowledge of the fluid pressure. A new general pressure equation is derived from compressible Navier-Stokes equation. It is argued that this new pressure equation allows unifying compressible, low-Mach and incompressible approaches. Moreover, this equation should be able to replace the Poisson equation in isothermal incompressible fluids. For computational fluid dynamics, it can be seen as an alternative to Lattice Boltzmann methods and as the physical justification of artificial compressibility.

Introduction

Incompressible Navier-Stokes equations (INS) describes a fluid characterized by infinite sound speed. It is valid in the case of fluid flows in isothermal configuration and at low Mach numbers (Mach number, M a = U /c is the ratio of the characteristic flow speed U and the speed of sound c defined at some reference temperature T and density ρ ). INS equations correspond to a mixture of hyperbolic and elliptic partial differential equations. They can be written

∂ t u i + u j ∂ j u i + ∂ i P = 1 Re ∂ j ∂ j u i ∂ i u i = 0 ( 1
)
where u is the fluid velocity, P is the pressure and Re the Reynolds number, which represents the ratio between inertial and viscous forces [1]. The pressure in (1) is not an independent thermodynamic variable. It can be seen as a Lagrangian multiplier of the incompressibility constraint. It is determined by the Laplace or Poisson equation:

∂ i ∂ i P = -(∂ j u i )(∂ i u j ) (2)
In very anisothermal flow, the low Mach number hypothesis conducts to a similar system [START_REF] Paolucci | On the filtering of sound from the Navier Stokes equations[END_REF][START_REF] Toutant | Turbulence statistics in a fully developed channel flow submitted to a high temperature gradient[END_REF]. Considering that only density ρ depends on temperature, the low Email address: adrien.toutant@univ-perp.fr (Adrien Toutant)

Preprint submitted to Physics letter A May 15, 2017 Mach number equations can be written

∂ t u i + u j ∂ j u i + 1 ρ ∂ i P = 1 ρRe ∂ j ∂ j u i + 1 3 ∂ i S ∂ i u i = S (3)
where ρ depends on temperature and S is a source term linked to conductive heat transfer (S depends on temperature). Again, the pressure is determined by a Poisson equation. It can be given by:

∂ i 1 ρ ∂ i P = 1 Re 4 3ρ ∂ j ∂ j S + 1 Re ∂ j 1 ρ ∂ i ∂ i u j + 1 3 ∂ i S -(∂ j u i )(∂ i u j ) -u j ∂ j S -∂ t S (4)
The physical meaning of ( 2) and ( 4) is that in a system with infinitely fast sound propagation, any pressure disturbance induced by the flow is instantaneously propagated into the whole domain. This elliptic problem is a crucial issue for fluid dynamics. Indeed, the INS equations are difficult to study analytically and numerically. This difficulty has motivated the search of alternative numerical approaches to determine pressure without solving the Poisson equation. Two different ways have been found. The first is the so-called artificial compressibility method where a pressure evolution equation is postulated [START_REF] Chorin | A numerical method for solving incompressible viscous flow problems[END_REF]. The second way is the Lattice Boltzmann method (LBM) which uses a velocity-space truncation of the Boltzmann equation from the kinetic theory of gases [START_REF] Benzi | The lattice boltzmann equation: theory and applications[END_REF].

In this paper, we determine a general and exact pressure evolution equation. It gives the physical bases of artificial compressibility method and it allows the study of very anisothermal flow contrary to LBM. The goal is similar to the reduced compressible Navier-Stokes equations (RCNS) derived by Ansumali et al. [START_REF] Ansumali | Thermodynamic theory of incompressible hydrodynamics[END_REF] and the proposed pressure equation is very similar to the grand potential equation derived by Karlin et al. [START_REF] Karlin | Kinetically reduced local navier-stokes equations: An alternative approach to hydrodynamics[END_REF]. However we will argue that the use of pressure instead of the grand potential simplifies the equation of compressible hydrodynamics. Moreover, because the proposed pressure equation is valid without any assumptions concerning the flow or the fluid, it builds bridge between compressible, low-Mach and incompressible approaches.

In section 2, we will determine the general and exact pressure evolution equation (without any assumptions). This equation generalizes the one used by Zang et al. in the particular case of an ideal gas [START_REF] Zang | Direct and largeeddy simulations of threedimensional compressible navierstokes turbulence[END_REF]. In section 3, we will simplify this equation in the low Mach number limit. Finally, in section 4, we will reduce the equation for low Mach number and isothermal flow.

General pressure evolution equation

The total energy E conservation is given by

∂ t (ρE) + ∂ j ((ρE + p)u j ) = ∂ j (σ ij u i ) -∂ j q j ( 5
)
with σ ij the shear-stress tensor for a Newtonian fluid

σ ij = 2µS ij - 2 3 µδ ij S kk S ij = 1 2 (∂ i u j + ∂ j u i ) (6)
and q i the conductive heat flux

q i = -κ∂ i T (7)
Introducing internal energy U = E -uiui 2 and enthalpy H = U + P ρ , one gets

ρD t H = D t P -∂ i q i + Φ Φ = σ ij ∂ i u j (8)
with D t the material derivative (total derivative [1]):

D t φ = ∂ t φ + u i ∂ i φ (9)
Using the relations of heat capacity at constant pressure c p and of the isobaric thermal expansion coefficient α

∂H ∂T P = c p ∂H ∂P T = 1 ρ (1 -T α) with α = - 1 ρ ∂ρ ∂T P (10)
an alternative formulation can be derived for temperature

ρc p D t T = T αD t P -∂ i q i + Φ (11)
We propose to derive a new pressure equation from the temperature formulation [START_REF] Serra | Thermal large eddy simulation in a very simplified geometry of a solar receiver[END_REF]. We introduce the isothermal compressibility coefficient

χ T = 1 ρ ∂ρ ∂P T
and we consider temperature as a function of density and pressure

D t T = ∂T ∂ρ P D t ρ + ∂T ∂P ρ D t P .
Using mass conservation and ( 11), the recomputation poses no difficulties and we here write the result:

(ρc p χ T α -αT )D t P = -∂ i q i - ρc p α ∂ i u i + Φ (12)
In order to simplify this expression, one introduces isochoric heat capacity c v = ∂U ∂T ρ , heat capacity ratio γ = cp cv and the Mayer relation

α 2 T = ρc v χ T (γ -1) (13)
One obtains

D t P + γ χ T ∂ i u i = α ρc v χ T (Φ -∂ i q i ) (14)
Sound velocity c and the isentropic compressibility coefficient χ S are given by

c 2 = ∂P ∂ρ S χ S = 1 ρ ∂ρ ∂P S (15)
Using the Reech relation γ = cp cv = χ T χ S , one obtains the general and exact pressure evolution equation

D t P + ρc 2 ∂ i u i = α ρc v χ T (Φ -∂ i q i ) (16)
In the particular case of an ideal gas α = 1 T , χ T = 1 P and c 2 = γrT with r the specific gas constant, this equation is equivalent to the one used by Zang et al. [START_REF] Zang | Direct and largeeddy simulations of threedimensional compressible navierstokes turbulence[END_REF]9]. It is worth noting that equation ( 16) can be used for any fluid (gas or liquid) without restriction on Mach number or temperature gradient. It gives the physical bases of artificial compressibility methods that postulate the pressure equation. The pressure equation ( 16) can be seen as an energy equation: to complete the system, one has to consider in addition mass conservation, momentum conservation and an equation of state.

Pressure equation for low Mach number flow

At this step, the pressure evolution equation ( 16) depends on the total derivative. At low Mach number, the first simplification consists in assuming that viscous dissipation is negligible. We now show that, at low Mach number, advection can be neglected. One defines the following nondimensionalized quantities:

ρ X = ρ ρ u X i = u i c P X = γP ρ (c ) 2 t X P = 1 M a 2 tU x (17)
It is worth noting that in the classical low Mach number assumption, nondimensionalized time is defined by t X U = tU x . The factor 1 M a 2 is justified by the fact that pressure time evolution is much faster than velocity time evolution (subscripts U and P indicate that the nondimensionalized time corresponds to velocity or pressure respectively). One defines moreover, the Reynolds number Re, the Prandtl number P r and the Peclet number P e:

Re = ρU x µ P r = ν a T P e = P rRe (18)
One uses the asymptotic expansion of pressure, temperature and velocity

P X = P 0 + M a 2 P 1 (19)
T X = T 0 + M a 2 T 1 (20) u X i = M a(u i0 + M a 2 u i1 ) (21)
where P X is the nondimensionalized pressure, P 0 the zero-order pressure that is constant in space and P 1 the second-order pressure (see [START_REF] Paolucci | On the filtering of sound from the Navier Stokes equations[END_REF][START_REF] Majda | The derivation and numerical solution of the equations for zero mach number combustion[END_REF]). It follows that the orders of magnitude of the different terms of equation ( 16) are given by:

∂ t P = ρ x γ (c ) 3 1 M a ∂ t (P 0 + M a 2 P 1 ) (22)
u i ∂ i P = ρ x γ (c ) 3 M a 3 u i0 ∂ i P 1 ( 23
)
ρc 2 ∂ i u i = ρ x (c ) 3 M a∂ i u i0 (24) -α ρc v χ T ∂ i q i = ρ x (c ) 3 M a ReP r ∂ i ∂ i T 0 (25)
Consequently, the leading order gives:

∂ t P 0 = 0 (26)
The zero-order pressure P 0 is the reference pressure that is constant in time. It is worth noting that in the classical low Mach number assumption the zero-order pressure P 0 is not constant in time. This difference between the classical low Mach number assumption and the new low Mach number assumption proposed in this paper is due to the fact that asymptotic expansion depends of the chosen nondimensionalized quantities. More precisely, the zero-order pressure of this new low Mach number assumption is constant in time because of the choice of characteristic time for pressure variation. This choice is justified by the fact that pressure has very fast time variation compared to velocity for small Mach number. The second-order pressure P 1 corresponds to the thermodynamic pressure and the pressure of the momentum equation. The time evolution of the second-order pressure is given by:

∂ t (P 1 ) + γ∂ i u i0 = γ ReP r ∂ i ∂ i T 0 (27)
The realized asymptotic expansion conducts to neglect advection for small Mach number. This hypothesis is further confirmed using the results of large eddy simulations and direct numerical simulations of biperiodic turbulent channel flow at different Reynolds numbers and different temperature ratios. The simulations are realized with the ideal gas assumption. The temperature ratio is defined as the ratio of wall imposed temperature at the hot wall to the wall imposed temperature at the cold wall (temperatures are in Kelvin). The studied temperature ratios are 2 and 5. The studied turbulent Reynolds numbers (based on the friction velocity) are 180 and 395. More details about the simulations can be find in [START_REF] Serra | Thermal large eddy simulation in a very simplified geometry of a solar receiver[END_REF][START_REF] Serra | Turbulent kinetic energy spectrum in very anisothermal flows[END_REF][START_REF] Serra | High-temperature gradient effect on a turbulent channel flow using thermal large-eddy simulation in physical and spectral spaces[END_REF][START_REF] Toutant | Turbulence statistics in a fully developed channel flow submitted to a high temperature gradient[END_REF][START_REF] Sanchez | Large eddy simulations of thermal boundary layer spatial development in a turbulent channel flow[END_REF][START_REF] Aulery | Spectral analysis of turbulence in anisothermal channel flows[END_REF]. In all these cases, at a given distance from the wall, the statistic average of the term ρc 2 ∂ i u i = γP ∂ i u i is a thousand times bigger than the statistic average of u i ∂ i P . Written with dimensionalized variables, the pressure evolution equation becomes for low Mach number (viscous dissipation is assumed to be negligible):

∂ t P + ρc 2 ∂ i u i = -α ρc v χ T ∂ i q i (28)
In the particular case of an ideal gas, one finds the equation used for P 0 in the low Mach number approximation [START_REF] Paolucci | On the filtering of sound from the Navier Stokes equations[END_REF][START_REF] Majda | The derivation and numerical solution of the equations for zero mach number combustion[END_REF]. We show here that equation ( 28) can be used for the pressure of momentum equation P 1 and for all fluids (gas and liquid). It is a local nonadvected equation for the scalar thermodynamic field. Combined with mass conservation, momentum conservation and an equation of state, it constitutes a system very similar to Lattice Boltzmann method (LBM): low Mach number equations avoiding the nonlocality of pressure (there is no Poisson equation and so no need for elliptic solver). Compared to LBM, the system is valid without the ideal gas assumption and the small temperature gradient restriction. Indeed, it is well known that LBM has stability problems in very anisothermal flows. Compared to fully compressible Navier-Stokes, the energy equation is expressed in a pressure form without the ideal gas assumption and without the advection term.

Pressure equation for isothermal low Mach number flow

At this stage, the pressure evolution equation (28) depends on temperature. We now consider the isothermal limit of this equation. Considering temperature as a function of pressure and density, one gets:

∂ i ∂ i T = ∂ 2 T ∂P 2 ρ (∂ i P )(∂ i P ) + ∂ 2 T ∂ρ 2 P (∂ i ρ)(∂ i ρ) + ∂T ∂P ρ ∂ i ∂ i P + ∂T ∂ρ P ∂ i ∂ i ρ (29)
For sufficiently large time-space scales (see the discussion in [START_REF] Ansumali | Thermodynamic theory of incompressible hydrodynamics[END_REF]), one can neglect density variation (ρ, χ T and α are supposed contant and evaluated at equilibrium) and temperature variation (around a globally uniform equilibrium temperature) becomes a function of pressure:

∂ i ∂ i T ≈ ∂T ∂P ρ ∂ i ∂ i P = χ T α ∂ i ∂ i P (30)
It is worth noting that the pressure in this equation corresponds to P 1 the second-order pressure. This pressure corresponds to the pressure of the momentum equation (P 0 the zero-order pressure is constant in space). In the isothermal limit, thermal conductivity is assumed to be constant: -∂ i q i = κ∂ i ∂ i T . The isothermal pressure evolution (IPE) equation is finally given by:

∂ t P + ρc 2 ∂ i u i = κ ρc v ∂ i ∂ i P (31)
Note that all "material parameters" appearing in (31) (κ, ρ and c v ) are evaluated at a constant equilibrium temperature. A very interesting point of this equation is the presence of a diffusion term. By analogy with temperature diffusivity a T = κ ρcp , one can define the pressure diffusivity a P = κ ρcv . The ratio of pressure/temperature diffusivities corresponds to the heat capacity ratio γ = ap a T . This diffusion term is a crucial difference with artificial compressibility method [START_REF] Chorin | A numerical method for solving incompressible viscous flow problems[END_REF]. This term allows to stabilize the simulation. It guarantees the numerical applicability of the method. Indeed, Ohwada and Asinari [START_REF] Ohwada | Artificial compressibility method revisited: Asymptotic numerical method for incompressible navierstokes equations[END_REF] proposed to introduce a dissipation term in order to improve the quality of numerical solution obtained with artificial compressibility method. Moreover, the kinetically reduced local Navier-Stokes (KRLNS) equations [START_REF] Karlin | Kinetically reduced local navier-stokes equations: An alternative approach to hydrodynamics[END_REF] contains exactly the same diffusion term applied to grand potential instead of pressure. In fact, the KRLNS grand potential equation is exactly the same than the proposed IPE equation ( 31) if one replaces grand potential G by pressure P . This observation leads to two conclusions. In one hand, it is expected that numerical simulations using IPE equation associated with momentum equation capture the correct transient behavior of complex flows as KRLNS does [START_REF] Hashimoto | Higher order numerical simulation of unsteady viscous incompressible flows using kinetically reduced local navierstokes equations on a {GPU}[END_REF]. On the other hand, because KRLNS grand potential equation and IPE equation are the same, kinetic energy k = P -G can be neglected in KRLNS. Indeed, the substraction of the KRNLS grand potential equation from the proposed IPE equation (31) gives

∂ t k = κ ρc v ∂ i ∂ i k (32)
In the case of a Prandtl number different from heat capacity ratio (P r = γ), this last equality will not be accurate unless kinetic energy and its space/time variations are very small. Finally, it is interesting to write the IPE equation (31) in a nondimensional form. Omitting the exponent X for the sake of simplicity, the nondimensionalized form of (31) is

∂ t P + γ∂ i u i = γ M a 2 ReP r ∂ i ∂ i P (33)
It shows that "pressure diffusion" cannot be neglected for pressure time evolution because its order of magnitude can be similar to the one of pressure time derivative due to the Peclet number. Again, it is a fundamental difference with artificial compressibility method. It allows to stabilize the simulation. Furtheremore, considering a Reynolds number around one and a Prandtl number similar to the Mach number squared (very small Prandtl number), pressure diffusion has the same order of magnitude of the divergence term. Consequently, it is anticipated that isothermal low Mach number flows may not be incompressible.

To the best of our knowledge, experimental evidence of compressible (∂ i u i = 0) isothermal low Mach number flows does not exist. It would be very interesting to investigate. Considering numeric interest of IPE, because the Prandtl number has no effect on isothermal incompressible flow, it is expected that it can be used in the IPE equation (31) as a numerical parameter to stabilize the simulation. Obviously, the Peclet number has to be much bigger to the Mach number if one wants to approximate the incompressible limit.

Conclusion

This work establishes the general and exact pressure equation evolution. The pressure equation has been obtained

• without any assumptions,

• in the low Mach number limit and • in the isothermal limit.

Without any assumptions, the obtained equation generalizes the ideal gas pressure equation. In the low Mach number limit, we propose a new asymptotic analysis that conducts to neglect advection. The proposed low Mach number assumption conducts to the same equation than the classical one. However, the new equation corresponds to the pressure of momentum equation. It means that the coupling between the energy equation and the momentum equation is increased. Consequently, the new assumption is less restrictive than the classical one. It is expected that it better corresponds to the physics of turbulent anisothermal flows where turbulent time scales can be of the same order of magnitude as pressure time scales. In the isothermal limit, the pressure equation gives the physical bases of ACM. The obtained equation is different from ACM. Indeed, a new diffusion term is added. This new term is coherent with the fact that both viscosity and thermal conductivity are involved in acoustic wave attenuation.

In a fundamental point of view, this equation provides a thermodynamic theory of incompressible hydrodynamics. More precisely, it gives a thermodynamic derivation of incompressibility without the isentropic flow assumption. Such a thermodynamic derivation goes far beyond academic interest. Indeed, the general pressure evolution equation paves the way for new technique or methodology for computational fluid dynamics. In future works, we will use this equation as an alternative numerical approach to determine pressure without solving the Poisson equation as ACM or LBM.