Laurent Beauguitte

César Ducruet

SCALE-FREE AND SMALL-WORLD NETWORKS IN GEOGRAPHICAL RESEARCH: A CRITICAL EXAMINATION 1

Keywords: Scale-free network, small-world network, geography, regional science, social network analysis

A rapid surge of interest for networks in the late 1990s throughout natural and social sciences has witnessed the emergence and the diffusion of new concepts and measures. In this paper, we wish to examine how two recent models of networks (i.e. scale-free and small-world) have been integrated in the works of geographers, what have been the benefits, and whether such concepts are likely to increase their influence in further works on networks. First, we propose a critical review of the "scalefree" and "small-world" concepts, notably based on a review of the physics literature. The second section examines the spatial dimension in networks studies, and the third one evaluates how geographers have used these measures and concepts in their works. In conclusion, we question the benefits of these two models of networks to geography compared with other approaches such as the ones developed in sociology.

INTRODUCTION

A rapid surge of interest for networks in the late 1990s throughout natural and social sciences has witnessed the emergence and diffusion of new concepts and measures. However, little has been done evaluating the impact and outcome of the latter on geography, with reference to the recent reviews by [START_REF] Borgatti | Network analysis in the social sciences[END_REF] and [START_REF] Crossley | Review article: The new social physics and the science of small world networks[END_REF][START_REF] Crossley | Small-world networks, complex systems and sociology[END_REF] about sociology. In this paper, we wish to examine how two recent models of networks (i.e. scale-free and small-world) have been integrated in the works of geographers, what have been the benefits, and whether such concepts are likely to increase their influence in further works on networks. In a first attempt to evaluate the benefits of these approaches in geography, [START_REF] Rozenblat | A small world perspective on urban systems[END_REF] noticed that "this type of empirical approach combining a conceptual approach of "small world theory" and dedicated tools has not been developed in geography": is the situation different four years later?

Since the quantitative revolution of the 1960s, the status of network analysis in geography has remained rather simple. This tool is mainly used by transport geographers focusing primarily on graph theory with applications to planar and technical networks (e.g., roads and railways, see [START_REF] Kansky | Structure of Transportation Networks: Relationship Between Network Geometry and Regional Characteristics[END_REF][START_REF] Haggett | Network Analysis in Geography[END_REF][START_REF] Dupuy | Réseaux territoriaux[END_REF][START_REF] Mathis | Graphes et réseaux. Modélisation multi-niveaux[END_REF]. Relations with Social Network Analysis (SNA) remain limited (Freeman, 2004:85) although sociologists developed their tools from the 1920s while focusing exclusively on non-planar graphs [START_REF] Wasserman | Social Network Analysis: Methods and Applications[END_REF]. Network analysis in geography has thus relatively stagnated notwithstanding advances in Geographical Information Systems (GIS), while regional scientists continued developing their own models on spatial analysis and flow optimisation [START_REF] Waters | Network and nodal indices. Measures of complexity and redundancy: A review[END_REF].

The sudden interest by physicists in network analysis in the late 1990s principally provided models of networks based on two main dimensions: the small-world network, based on average distance path and density of neighbourhoods (Watts and Strogatz, 1998;[START_REF] Watts | Six degrees. The Science of a Connected Age[END_REF], and the scale-free network based on the hierarchy of hubs [START_REF] Barabási | Emergence of scaling in random networks[END_REF]. The first concept originates from the famous experience of Stanley [START_REF] Milgram | The small world problem[END_REF] which had very limited diffusion in geography except from a paper by [START_REF] Stoneham | The small-world problem in a spatial context[END_REF]. While such concepts have quickly spread across various disciplines and scientific fields [START_REF] Gastner | Optimal design of spatial distribution networks[END_REF], their integration in geography still appears rather limited to a few studies in transport, urban and economic geography. Conversely, physicists have increasingly integrated the spatial dimension in their works (see the extensive review of [START_REF] Barthélemy | Spatial networks[END_REF].

The structure of the paper is as follows. First, we propose a critical review of the "scale-free" and "small-world" concepts based on a review of the physics literature. The second section examines the space dimension in networks studies using these two models, and the third one studies how geographers have used these measures and concepts in their works. In conclusion, we question the benefits of these measures to geography compared with other approaches such as the ones developed in SNA, which have been far less integrated by geographers despite their a priori relevance at meso (regional) levels of analysis [START_REF] Grabher | Trading routes, bypasses, and risky intersections: Mapping the travels of `networks" economic sociology and economic geography[END_REF]. This research also provides floor for further discussing how geography integrates (or not) scientific innovations.

SCALE-FREE AND SMALL-WORLDS NETWORKS

Most researches in graph theory and network analysis in general have long been focused primarily on regular and random networks. The concepts of "small-world network" (hereafter SWN) and "scale-free network" (SFN) were first proposed by the respective works of Watts and Strogatz (1998) and [START_REF] Barabási | Emergence of scaling in random networks[END_REF]. Their goal was to define models of network organisation differing from regular and random networks [START_REF] Erdös | On random graphs[END_REF]. Random networks are characterised by a normal degree distribution, whereas SFNs contain few large degree nodes and a majority of small degree nodes, resulting in a strong hierarchical structure. SWNs exhibit a small average path length between pairs of nodes and a high local clustering coefficient (also called transitivity: probability for nodes that their adjacent neighbours are linked). SWNs and SFNs have in common several features [START_REF] Gastner | Optimal design of spatial distribution networks[END_REF]:

-They are more efficient in terms of ease of circulation within the network, which can be measured by the average shortest path length, as those networks allow limiting the number of stops between two nodes on average; -They include many hubs (bridge nodes) and isthmuses (crucial links) between densely and tightly connected communities or "clusters", based on the idea of cliques.

But such networks also have some important differences:

-SFNs are less "clustered" than SWNs due to the stronger influence of large degree nodes, which lowers the clustering coefficient (cf. hub-and-spoke configurations in air transport). SFNs seem more efficient as the presence of hubs provides optimal circulation and less friction;

-SWNs are denser than SFNs because removing few hubs would result in the identification of communities. Yet, large degree nodes tend to form cliques in SFNs and this can be measured by the rich-club coefficient.

Usually, the structure of SFNs is described by plotting node frequency over degree distribution in a log-log diagram. The slope (exponent) of the power-law line gives an indication whether the network is scale-free, i.e. when values of the exponent are over 1 or even 2. Another way testing the existence of scale-free properties is dynamic. As SFNs are evolving by preferential attachment, when new nodes are added to the network, they primarily connect the already large nodes, which function as magnets towards new entrants, thereby reinforcing the structure. Such dynamical properties inherent to SWNs and SFNs were in fact already expressed in a number of related concepts and growth models, as underlined by the recent review provided by [START_REF] Zaidi | Analysis, Structure and Organization of Complex Networks[END_REF] on complex networks:

-Scale-free: the power-law structure and the preferential attachment process were already described by the Yule (1925) process, by the Gibrat (1931) law (growth is proportional to size), by Jackson (1935) with the "rich get richer" idea, by Zipf (1949) on the rank-size rule, by Price (1965Price ( , 1976) ) about the cumulative advantage process observed in citation networks, and by Merton (1968) about inequality in credit attribution among researchers;

-Small-world: the works of [START_REF] Milgram | The small world problem[END_REF] as well as Travers and Milgram (1969) were in fact the first to demonstrate the topological proximity among distant individuals and to label it as "small-world phenomenon". Earlier, Simmel and Wolff (1950) first proposed the concept of triad to depict mutual acquaintances in a social network, and how they are likely to evolve.

In fact, the merit of recent works by Watts and Strogatz (1998) and [START_REF] Barabási | Emergence of scaling in random networks[END_REF] has been to reincorporate such ideas into clearly defined models of network structure and evolution, together with associated measures and methods directly usable for empirical research. Several measures have been proposed to highlight the properties of networks and the following list does not pretend to be exhaustive:

-Nodal hierarchy: fitness of degree distribution with power-law trend, that is similar to the rank-size rule proposed by Zipf (1949) and widely applied in urban studies for decades; -Transitivity: probability for nodes to have their adjacent neighbours connected with each other, measured by the "global clustering coefficient". Although it is not explicitly addressed, this measure actually refers to the fraction of closed triplets proposed by sociologists decades ago (transitivity). This indicator is also measured at the node level by the "local clustering coefficient"; -Efficiency: average shortest path length is the average number of links needed to connect all node pairs in the graph;

-Assortativity: refers to the global measure of "degree correlation" depicting the level of homogeneity of the network"s structure. It is the result of the Pearson correlation between degree scores of nodes for all links; -Density, connectivity: the "rich-club coefficient" proposed by physicists is a derivative of the γ index proposed by [START_REF] Kansky | Structure of Transportation Networks: Relationship Between Network Geometry and Regional Characteristics[END_REF], also called density by social network analysis (SNA) researchers , applied to links between nodes over a certain degree threshold.

In the end, physicists have mostly relied on existing measures, but they have also modified and improved them in order to take the weights into account, which has long remained a drawback of graph theory (Opsalh and Panzarasa, 2009). Weighting, for instance, the clustering coefficient and the rich-club coefficient provide very useful answers to the question whether larger nodes are more strongly interconnected with each other than with smaller nodes, i.e. whether hubs form cliques in the graph. If weight is taken into account, directionality is barely considered by physicists, even when links are obviously directed (cf. Barábasi"s studies of the Internet structure), which is less the case in SNA where main methods are suited for directed graphs (blockmodeling, equivalence, Siena model, etc.).

INTEGRATING SPACE IN NETWORK RESEARCH

A number of research papers seeking small-world and scale-free properties in networks of all kinds has been produced in the last two decades at exponential pace, thus making it difficult reviewing the field exhaustively. Given the similarity of methods and measures from one case study to the other, we review a number of works principally focusing on the spatial dimension of SFNs and SWNs.

Natural scientists have themselves criticized mainstream research on networks due to the noninclusion of the spatial, social or economic dimension in general measures and models [START_REF] Watts | Six degrees. The Science of a Connected Age[END_REF]. The material and geographical embedding of some networks appears only implicitly in the smallworld model, while it is absent of the scale-free model2 . In addition, Watts (1999) found that preferential attachment often occurs over shorter distances. A distinct category of "spatial networks"3 thus emerged in an explicit way in the physics literature, which actually applies to a majority of realworld networks in contrast with theoretical models of networks (Gastner and Newman, 2004). Yet, many works on spatial networks do not specifically measure their spatial dimension, as they only look at the topological dimension as in the case of maritime networks [START_REF] Hu | Empirical analysis of the worldwide maritime transportation network[END_REF] and commuter flows (De [START_REF] Montis | Modeling commuting systems through a complex network analysis: A study of the Italian islands of Sardinia and Sicily[END_REF] where distance parameters are not included. Spatial networks are defined by physicists [START_REF] Barthélemy | Spatial networks[END_REF] through some fundamental properties:

-Physical embedding: nodes and/or links are grounded in a physical (Euclidian) space, which in turn constrains the multiplication of links and orientates the layout of the network, with the crucial importance of borders; -Interaction range: distance metrics (and related costs) play a central role in the emergence, distribution, and weight of links, since spatial proximity is one dominant factor favouring short-range versus long-range interaction.

Several scholars have thus explored the influence of spatial structure on network topology in static and dynamics ways [START_REF] Waters | Network and nodal indices. Measures of complexity and redundancy: A review[END_REF]. Among the earliest attempts to validate this idea, [START_REF] Barthélemy | Crossover from scale-free to spatial networks[END_REF]2010) showed that spatial networks in general exhibit higher clustering coefficients than nonspatial networks due to the importance of proximity in node connectivity. Further results were provided by [START_REF] Barrat | The effects of spatial constraints on the evolution of weighted complex networks[END_REF] based on the case of air transport highlighting several other properties of spatial networks such as: fewer global hubs and more regional hubs, higher disassortativity as the network grows, higher correlation between degree and betweenness, increased influence of the barycentre on betweenness values, and increased cliquishness. However, the fundamental difference between planar and non-planar networks is not always considered by physicists. For instance, planar spatial networks are more physically constrained and thus are more assortative, with a higher probability to contain a giant component (i.e. connected subgraph including a majority of the nodes), while non-planar spatial networks are more likely to exhibit scale-free properties [START_REF] Bullock | Spatial embedding and the structure of complex networks[END_REF]. Some recent interest toward planar graphs can be seen among physicists, and the class of "Apollonian networks", being simultaneously planar, scale-free and small-world, illustrates it [START_REF] Andrade | Apollonian networks: simultaneously scale-free, small world, euclidian, space filling, and with matching graphs[END_REF]. It seems too soon however to check the empirical relevance of this new network model. Methods used for testing the influence of spatial embedding range from the inclusion of simple distance parameters to the simulation and modelling of complex networks embedded in twodimensional space (for an early review see Boccaletti et al., 2006, pp. 205-212), based on the hypothesis that connectivity is a function of distance [START_REF] Barnett | Spatially embedded random networks[END_REF]). Notably, Crucitti et al. (2006) take into account physical distance in their calculation of node centrality, arguing that their results may be more useful to urban and design. In the same vein, Cardillo et al. (2010) include metric distance in their analysis of urban streets, notably comparing observed and optimal efficiency. Other works adopted a node redundancy approach to study the influence of spatial structure on cascading failures (Huang et al., 2006), arguing that stronger geographical constraints foster the "reservoir effect" of hubs (i.e. redistribution of traffic from smaller nodes to larger nodes situated in close proximity), while such spatial networks face higher risks of becoming disconnected due to their higher density. When it comes to the simultaneous analysis of several spatial networks, [START_REF] Parshani | Inter-similarity between coupled networks[END_REF] notably demonstrated that location matters in the inter-similarity of networks, based on ports and airports" geographical coordinates.

SCALE-FREE AND SMALL-WORLD NETWORKS IN GEOGRAPHY AND REGIONAL SCIENCE

While the quantitative revolution of the 1960s favoured a rapid diffusion of graph theoretical concepts and methods, network analysis in geography stagnated during the following decades. The wider paradigm shift from structural to behavioural approaches is seen by [START_REF] Waters | Network and nodal indices. Measures of complexity and redundancy: A review[END_REF] as a main cause for the declining interest in spatial analysis as a whole, where network analysis "remained somewhat of a backwater" despite improvements provided by Geographical Information Systems (GIS) since the 1980s. Since the early 1990s, transport geographers observed the emergence of hub-and-spokes networks in various industries (e.g. airlines) without any reference to SFNs. In such context, geographers remained isolated from research on networks in other disciplines, relying on dated measures of network structure most of them coming from classic graph theory. This may also be explained by the absence of collaborations with sociology where network analysis has been at centre stage since the 1920s. Social geographers have often given privilege to qualitative methods rather than quantitative, since social geography emerged in opposition to the quantitative revolution.

Such picture has changed in the 1990s when geographers started to represent and analyse non-planar networks, thereby stepping out of "classic" graph theory and thus needing more advanced tools to represent and analyse such networks. Studies of the European urban system progressively integrated a network dimension with the works of [START_REF] Cattan | Attractivity and internationalisation of major European cities: The example of air traffic[END_REF] on airlines, and [START_REF] Rozenblat | The location of multinational firms in the European urban system[END_REF] on multinational firms. Yet, these works innovate both by the methods used for analysing geographical networks and by the paths they opened towards new ways considering systems of cities. Anyway, according to Gorman et al. (2004), few scholars have improved existing connectivity indices, although such progress remained hindered by a lack of knowledge about the history and development of network analysis in geography and in other disciplines [START_REF] Waters | Network and nodal indices. Measures of complexity and redundancy: A review[END_REF]. Although Gorman and Kulkarni (2004) consider [START_REF] Batty | Editorial; Cities as small worlds[END_REF] as the main propeller of complex network research within urban geography and economics, the latter is not always cited in the pioneering attempts by regional scientists to integrate a network perspective in their works [START_REF] Banister | European Transport and Communications Networks: Policy Evolution and Change[END_REF]. Instead, regional scientists directly refer to physicians, as in the case of Jiang and Claramunt's work on urban streets as SWNs (2004).

A review of recent applications of scale-free and small-world networks by geographers and regional scientists provides a somewhat balanced picture of the varied outcomes. Several works done by geographers remain direct applications of complex network measures on various datasets, as in the case of airline networks, notwithstanding some innovation through the refinement of clustering methods [START_REF] Amiel | Réseaux multi-niveaux : l'exemple des échanges aériens mondiaux de passagers[END_REF], the comparison of centrality measures with local socio-economic data [START_REF] Wang | Exploring the network structure and nodal centrality of China's air transport network: A complex network approach[END_REF], or the combination with maritime networks (Ducruet et al., 2011). Distance or territorial aspects themselves are therefore barely introduced by geographers in their study of networks. It was found, however, that seaports with larger degree connect over longer kilometric distances on average based on worldwide maritime links [START_REF] Ducruet | Maritime constellations: A complex network approach to shipping and ports, Maritime Policy and Management[END_REF]. When studying the Indianapolis road network, [START_REF] Gleyze | Making allowances for spatial and network effects when assessing indicators on infrastructure network nodes[END_REF] proposed to distinguish among networks effects and spatial effects in the measurement of betweenness centrality and eccentricity. Innovating thus takes place in the discipline without modifying or even discussing the benefits and limits of such methods and models. Other works, however, clearly use complex networks as a complement to their own existing methods, such as [START_REF] Patuelli | Network analysis of commuting flows: A comparative static approach to German data[END_REF] combining complex networks and Spatial Interaction Models (SIM), Gorman and Kulkarni (2004) referring SNA (e.g. structural equivalence) and complex methods in their analysis of Internet backbone networks in the US. In the same vein, Ducruet et al. (2011) apply successively three sets of methods on the combined air-sea global network: scalefree dimension of the network on different levels of node aggregation, Multiple Regression Quadratic Assignment Procedure (QAP) to reveal the correlation between network topologies, and the more classic "nodal regions" algorithm 4 for highlighting relevant sub-trees in the network. For [START_REF] Vinciguerra | The geography of internet infrastructure: An evolutionary simulation approach based on preferential attachment[END_REF], the goal is to better understand the evolution of the European Internet backbone network through the testing of the Barabási-Albert model of preferential attachment. Other works use complex networks as a tool to perform their analysis, such as Rozenblat (2010) works on the location logics of multinational firms in a multi-level perspective within and among cities, or Comin (2009) using available centrality measures (degree, closeness, betweenness) to depict the situation of European cities in partnership networks and their dynamics. The diachronic dimension is indeed much taken into account by geographers when it comes to actual data rather than simulation models.

CONCLUSION

SFNs and SWNs became quite quickly a popular hobby for physicists, and some included spatial issues in their research. Geographers remained mostly apart from this trend, despite some recent exceptions. While cross-disciplinary interactions between geography and other fields remained higher with physics (and computer sciences) than with sociology, most geographers kept on considering networks as planar and technical graphs.

There may be various causes to this state of affairs. On the one hand, the shift away from structural approaches in the 1970s can be seen as one of them. Yet, this cannot explain why quantitative geographers have not been faster in adopting SFNs and SWNs, as they did in the 1960s when integrating graph theory. On the other hand, one likely reason is the limited innovation brought by complex networks research to geography, but again, no geographer has clearly expressed such critique in a formal review of the field as it was done by the aforementioned sociologists.

Many tools exist nowadays (e.g. GIS, R packages) which can provide a valuable help to test measures and methods on networks. Geographers focusing on migration, economy, political geography, without mentioning transport geography, could surely examine further the relevance of these innovations.

Questioning their relevance cannot be done without testing these measures. In addition, geographers have the opportunity to improve the integration of space (and time) in network research, which has many concrete applications and is being adopted by decision-makers as a relevant approach to their problems.

The works of physicists indeed still has some weaknesses despite their plethoric (and sometimes redundant) production: absence of critical discussion on data quality and relevance, limited knowledge on the specific study field, and shortage of results" interpretation and implications. Such drawbacks, combined with the trend to re-discover well-known measures by giving them a new name (the most obvious being the transitivity renamed global clustering coefficient), still cannot fully erode the potential benefits of SFN and SWN models. Further research in geography may insist on the necessity to analyze networks as elements of wider territorial structures. Can we identify socio-economic invariants in the hierarchy of places and in the emergence of dense communities in the network? Are there determinants of network evolution beyond the sole role of costs and Euclidian distances?

Paper presented at the 17th European Colloquium on Theoretical and Quantitative Geography (ECTQG), Athens, Greece,

2-5 September 2011.

Barabási (2002) recognized that cities connecting thousands of highways simply do not exist, thereby recognizing implicitly physical constraints to network growth.

Spatial networks are also coined geographical networks, technological networks, infrastructure networks, adhoc networks, or physical networks in the literature.


Jarmo Hietarinta
email: jarmo.hietarinta@utu.fi

Claude Viallet
email: claude.viallet@upmc.fr

Integrable lattice equations with vertex and bond variables

We present integrable lattice equations on a two dimensional square lattice with coupled vertex and bond variables. In some of the models the vertex dynamics is independent of the evolution of the bond variables, and one can write the equations as non-autonomous "Yang-Baxter maps". We also present a model in which the vertex and bond variables are fully coupled. Integrability is tested with algebraic entropy as well as multidimensional consistency.

Introduction

Along with the classification results for integrable lattice maps on quad-graphs with one component fields on vertices [START_REF] Adler | Classification of Integrable Equations on Quad-Graphs. The Consistency Approach[END_REF], various more general integrable lattice equations have been proposed recently. This includes the "Yang-Baxter maps" (solutions to the functional Yang-Baxter equations [START_REF] Drinfeld | On some unsolved problems in quantum group theory[END_REF]) with variables on bonds [START_REF] Veselov | Yang-Baxter maps and integrable dynamics[END_REF][START_REF] Adler | Geometry of YangBaxter Maps: pencils of conics and quadrirational mappings[END_REF][START_REF] Papageorgiou | Yang-Baxter maps and multi-field integrable lattice equations[END_REF], and number of higher order as well as multi-component cases [START_REF] Hietarinta | Boussinesq-like multi-component lattice equations and multi-dimensional consistency[END_REF][START_REF] Spicer | Higher Analogues of the Discrete-Time Toda Equation and the Quotient-Difference Algorithm[END_REF][START_REF] Nijhoff | A higher-rank version of the Q3 equation[END_REF].

We describe here integrable lattice models which have both vertex and bond variables. 1 These models were found in the analysis of a coarse graining process of known integrable models on a square lattice [START_REF] Hietarinta | Weak Lax pairs for lattice equations[END_REF]. We first describe the basic procedure we used, and the way to test integrability via algebraic entropy (section 2). We then list various models obtained in this way, giving explicitly the evolution they define and their entropy. We also show that the evolution may be determined from a set of algebraic relations which have special rationality properties (section 3). We finally examine the three dimensional consistency of these models, and show that some of them can be interpreted as solutions of non-autonomous functional Yang-Baxter equations (section 4).

A road map 2.1 The setting

The starting point is a regular planar square lattice. Its vertices are labeled by two integers (n, m). We introduce two kinds of variables: the vertex variables, denoted w n,m , and the bond variables denoted X n,m for the horizontal bonds, and Y n,m for the vertical bonds. The bond variables are indexed by their base vertex, which is to the left (respectively down) along the bond, as shown in Figure 1(a). As usual we use the shorthand notation where only shifts are indicated, e.g., X 2 = X n,m+1 , w 12 = w n+1,m+1 , etc... The models are given by three relations between the eight variables associated to each square cell, in such a way that they define an evolution: from the three defining relations, it is possible to calculate X 2 , Y 1 , w 12 (open circles in Figure 1(a)) from w, X, Y, w 1 , w 2 (black disks). To fully define an evolution, we have to give a suitable set of initial data. Such data may be given on a diagonal staircase, after which the vertex and bond variables may be evaluated on the entire lattice, using the local relations on each cell.

We will be interested in models where the evolution is rational, i.e., X 2 , Y 1 , w 12 are rationally expressed in terms of w, X, Y, w 1 , w 2 . For the models we will consider, one may actually permute the role of the four corners, and in particular define a backward evolution calculating w, X, Y in terms of Y 1 , X 2 , w 1 , w 2 , w 12 . This is a form of rational invertibility, similar to birationality for maps. For the models we describe, it is actually possible to define rational evolution in all four directions on the lattice.

A construction procedure

There is one very simple way to generate such integrable maps from known one-component lattice maps.

Suppose we are given an integrable quad map. As shown in Figure 1(b), consider {x, x 11 , x 22 , x 1122 } as vertex variables of a coarser lattice (renaming them {w, w 1 , w 2 , w 12 }, the rule being w n,m = x 2n,2m ), and {x 2 , x 112 , x 1 , x 122 } as bond variables (and renaming them {Y, Y 1 , X, X 2 }, i.e., X n,m = x 2n+1,2m , Y n,m = x 2n,2m+1 ). The quad map determines x 12 in terms of x, x 1 , x 2 , and x 122 in terms of x 2 , x 22 , x 12 , and so on, after which one may forget the central x 12 . One signature of this construction is that X 2 does not depend on w 1 , and Y 1 does not depend on w 2 .

The purpose of this paper is to present a number of models which cannot be obtained

t ❞ ❞ ❞ t t t t w 2 w 1 w 12 w Y (a) Y 1 X X 2 t ❞ ❞ ❞ t t t t x 22 x 11 x 1122 x x 2 (b) x 112 x 1 x 122 x 12
In order to construct such maps we start from quad maps having a 2 ×2 Lax pair [START_REF] Fw Nijhoff | Lax pair for the Adler (lattice Krichever-Novikov) system[END_REF][START_REF] Bobenko | Integrable systems on quad-graphs[END_REF]. We use this Lax pair to write the zero curvature condition on the larger square of Figure 1(b). From the size of the Lax matrices, we see that this zero curvature condition can give at most three scalar conditions. We know that one rational solution can be obtained in the way described above, we call this solution "regular". What is remarkable is that for certain quad maps, there exists another rational solution, which we called "exotic" [START_REF] Hietarinta | Weak Lax pairs for lattice equations[END_REF].

In other words, we use the fact that the zero curvature condition written on a coarser lattice is ambiguous. In most cases there is only one solution giving rational evolution, it is the regular one. There are usually other solutions that are not rational, but we discard them. Only in some exceptional cases do we find rational solutions satisfying (1). 2

Testing integrability

When an exotic solution exists, there is no guarantee a priori that it leads to an integrable evolution, and we have to check its integrability.

The setting is appropriate to use the vanishing algebraic entropy criterion [START_REF] Bellon | Algebraic Entropy[END_REF][START_REF]Viallet Algebraic entropy for lattice equations[END_REF]. Given initial data on a basic diagonal staircase, one evaluates the w's, X's and Y 's away from the initial diagonal in terms of these data. The entropy can then be extracted from the sequences {d n } of the successive degrees respectively for the w's, X's and Y 's. The entropy is the limit

η = lim n→∞ 1 n log(d n ).
This limit always exists. The characteristic of integrability is the vanishing of η. It is equivalent to polynomial growth of the sequences {d n }. Non-vanishing of the entropy, that is to say exponential growth of a sequence {d n }, means non-integrability.

Remark : The simplest way to find the exact value of the entropy is by fitting, if possible, the generating function ζ(s) = n d n s n of the sequence of degrees by a rational fraction.

The entropy is then determined by the position of the poles of that fraction [START_REF] Bellon | Algebraic Entropy[END_REF][START_REF]Viallet Algebraic entropy for lattice equations[END_REF].

Simplified defining equations

We get the exotic solutions from a set of three algebraic conditions equivalent to a zero curvature condition, having more than one solution. It is possible, for all the models we describe below, to write three simpler equations, which have, as a unique solution, the exotic model.

These equations define a rational variety of dimension 5 in an 8 dimensional space. They have special (multi)-rationality properties: for any choice of a corner in Figure 1(a), the three corresponding variables (one vertex variable and the two adjacent bond variables) can be expressed rationally in terms of the five others. This special rationality property allows one to define rational evolutions in all four directions of the square lattice. This is a generalized form of the notion of quadrirationality introduced in [START_REF] Adler | Geometry of YangBaxter Maps: pencils of conics and quadrirational mappings[END_REF].

Equivalently, choosing two adjacent bonds, the relations define 2 → 2 birational maps between the variables attached to these bonds and the two remaining ones. These maps and their inverses actually happen to have the same form, but they are not involutions a priori.

3 The models

dpKdV (H1)

The simplest case is obtained from the lattice potential KdV (H1 in [START_REF] Adler | Classification of Integrable Equations on Quad-Graphs. The Consistency Approach[END_REF]). Computing the zero curvature condition on the coarser lattice we found two rational solutions [START_REF] Hietarinta | Weak Lax pairs for lattice equations[END_REF]. We give here for reference the regular one, which does not fulfill condition (1):

X 2 = Y + (q -p)(Y -X) (Y -X)(w -w 2 ) -(q -p) , (2a)
Y 1 = X + (p -q)(X -Y ) (X -Y )(w -w 1 ) -(p -q) , (2b)
w 12 = w + (p -q) (p -q)(w 1 + w 2 -2w) + 2(w -w 1 )(w -w 2 )(X -Y ) (p -q) 2 -(w -w 1 )(w -w 2 )(X -Y ) 2 . ( 2c
)
The exotic solution is:

w 12 = w 1 + w 2 -w, (3a) X 2 = Y + P, (3b) Y 1 = X + P, (3c)
P = - (X -Y )[(p -r)(w -w 2 ) + (q -r)(w -w 1 )] (p -r)(w -w 2 ) -(q -r)(w -w 1 ) -(w -w 1 )(w -w 2 )(X -Y ) . ( 3d
)
The parameters p, q, r appearing in the solutions come from the construction of the models: they are present in the Lax pairs we used.

The sequence of degrees for the bond variables is

{d n } b = 1, 4,
The generating function of the sequence ( 4) is

ζ(s) = 1 + 4 s 2 + s (1 -s) 3
The sequence has quadratic growth and the entropy vanishes.

Simplified form: By taking suitable linear combinations of equations ( 3) we can also write the equations in the form

w 12 -w 1 -w 2 + w = 0, (5a) X + X 2 -Y -Y 1 = 0, (5b)
X X 2 -Y Y 1 -(X -X 2 ) q -r w -w 2 + (Y -Y 1 ) p -r w -w 1 = 0. ( 5c
)
The equations ( 5) have the form typical for most of our results: an independent linear or linearizable w equation, a linear or linearizable equation for X, Y, X 2 , Y 1 , and a coupling equation. It is then easy to see that the evolution is rational in any direction: solve the w of any corner from the first equation, then its adjacent X, Y variables can be solved rationally from the remaining two.

Remark: The parameters appearing in the results (here and below) follow from the Lax set-up [START_REF] Hietarinta | Weak Lax pairs for lattice equations[END_REF], p, q, r corresponding to the three coordinate directions of the consistency cube. In the end the regular solution typically depends only on p -q, while the exotic solution depends on p -r and q -r. Clearly any finite r can be absorbed into p, q. The limit r → ∞ is also possible, but in the present case it produces a model that is linear in X, Y .

H1ǫ

This quad map was introduced in [START_REF] Adler | Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases[END_REF]. There in fact are two kinds of integrable models of type H1ǫ, related by inversion.

(x -x 12 )(x 1 -x 2 ) -(p -q)(1 + ǫxx 12 ) = 0 (6) (x -x 12 )(x 1 -x 2 ) -(p -q)(1 + ǫx 1 x 2 ) = 0, (7)
When extended to the whole lattice they must alternate thereby forming a black-white (checkerboard) lattice [START_REF] Xenitidis | Symmetries and integrability of discrete equations defined on a black-white lattice[END_REF][START_REF] Hietarinta | Weak Lax pairs for lattice equations[END_REF].

We may consider two different configurations for the 2 × 2 basic cell [START_REF] Spicer | Higher Analogues of the Discrete-Time Toda Equation and the Quotient-Difference Algorithm[END_REF] α :

β :

(6) (7) (7) (6)
To each of these patterns corresponds a zero curvature condition. Each of these again have two rational solutions, the regular one and an exotic one. The exotic solutions are as follows:

H1ǫα

For configuration α, we have:

Original form:

X 2 =Y + P/Q, Y 1 = X + P/Q, (8a) P =(X -Y )[(p -r)(w -w 2 )(1 + ǫww 1 ) + (q -r)(w -w 1 )(1 + ǫww 2 )] + ǫ(p + q -2r)(p -q)(w -w 1 )(w -w 2 ) (8b) Q =(w -w 1 )(w -w 2 )(X -Y ) -(p -r)(w -w 2 )(1 + ǫww 1 ) + (q -r)(w -w 1 )(1 + ǫww 2 ) ( 8c
)
w 12 + w = (w 1 + w 2 )(1 + ǫw 2 ) 1 + ǫ(ww 1 + ww 2 -w 1 w 2 ) . ( 8d
)
Entropy: The sequence of degrees are respectively:

{d n } v = 1,
for the vertex variables, and

{d n } b = 1,
for the bond variables. The sequence (9) has linear growth, signaling the linearizability of the evolution of the vertex variables (see below). The sequence [START_REF] Hietarinta | Weak Lax pairs for lattice equations[END_REF] has the generating function

ζ(s) = n d n s n = 1 + 5 s 2 + 2 s (1 -s) 3 (11)
which means quadratic growth of the degrees, i.e. integrability.

Simplified form: Transforming the vertex variables by the Moebius transformation w → (w -1)/[κ(w + 1)] with ǫ = -κ 2 etc, takes (8d) into

w w 12 = w 1 w 2 . ( 12
)
Then we can write the equations in the form

w w 12 -w 1 w 2 = 0, (13a) X 2 + X -Y 1 -Y = 0, (13b) X 2 X -Y 1 Y -(X -X 2 )(q -r)κ w + w 2 w -w 2 + (Y -Y 1 )(p -r)κ w + w 1 w -w 1 +κ 2 [(p -r) 2 -(q -r) 2 ] = 0. (13c)
In comparison with [START_REF] Papageorgiou | Yang-Baxter maps and multi-field integrable lattice equations[END_REF], which was linear in w, this is multiplicative.

H1ǫβ

For the configuration β we have the exotic solution:

Original form:

X 2 = Y + P 2 /Q 2 , Y 1 = X + P 1 /Q 1 (14a) P 1 =(X -Y )(1 + ǫX 2 )[(p -r)(w -w 2 ) + (q -r)(w -w 1 )], (14b) Q 1 =(X -Y )(w -w 1 )(w -w 2 ) -(p -r)(w -w 2 )(1 + ǫX 2 ) + (q -r)(w -w 1 )(1 -ǫX(X -2Y )), (14c) P 2 =(X -Y )(1 + ǫY 2 )[(p -r)(w -w 2 ) + (q -r)(w -w 1 )], (14d) Q 2 =(X -Y )(w -w 1 )(w -w 2 ) -(p -r)(w -w 2 )(1 -ǫY (Y -2X)) + (q -r)(w -w 1 )(1 + ǫY 2 ), (14e)
w 12 =w 1 + w 2 -w. (14f)
Entropy: The entropy calculation leads to the same conclusion as for the configuration α: the vertex evolution is linear, and independent of the bonds. The degree sequence for the bonds is the same as [START_REF] Hietarinta | Weak Lax pairs for lattice equations[END_REF].

Simplified form: Starting with ( 14) and using the Moebius transformation X → (X -1)/[κ(X + 1)], etc. with ǫ = -κ 2 , we get:

w 12 -w 1 -w 2 + w = 0, (15a) X + X 2 -Y -Y 1 + 2κ q -r w -w 2 (X -X 2 ) -2κ p -r w -w 1 (Y -Y 1 ) = 0, ( 15b
)
XX 2 -Y Y 1 = 0. ( 15c
)
Remarks: H1ǫα and H1ǫβ are deformations that in the limit ǫ → 0 reduce back to H1. As a consequence the original forms ( 8) and ( 14) reduce to (3), as can be readily verified. However, the simplified forms ( 13) and ( 15) are obtained with a transformation that is singular in ǫ. In these equations ǫ appears through κ and the κ → 0 limit trivializes them.

The models again depend on p -r and q -r, so r can be eliminated, except that r → ∞ is a possible limit. In the H1ǫβ case the r → ∞ limit leads to a nonlinear model, which may be interesting on its own.

dmKdV (H3 δ=0 )

If we start from the discrete modified KdV equation (H3 with δ = 0 in [START_REF] Adler | Classification of Integrable Equations on Quad-Graphs. The Consistency Approach[END_REF]) we obtain the following:

Original form:

w 12 w = w 1 w 2 , (16a) X 2 X = Y 1 Y, (16b) X 2 Y = Y 1 X = (q 2 w 2 -r 2 w)(w -w 1 )pX -(p 2 w 1 -r 2 w)(w -w 2 )qY (r 2 w 2 -q 2 w)(w 1 -w)pY -(r 2 w 1 -p 2 w)(w 2 -w)qX (16c)
Entropy: The sequence of degrees we get are respectively

{d n } v = 1,
for the bond variables, actually the same as for H1. The vertex evolution is linearizable, and the bond evolution is integrable.

Simplified form: By taking suitable linear combinations of the original equations we can write the result also as

w w 12 -w 1 w 2 = 0, (19a) (X + X 2 )p(q 2 + r 2 ) -(X -X 2 )p(q 2 -r 2 ) w + w 2 w -w 2 -(Y + Y 1 )q(p 2 + r 2 ) + (Y -Y 1 )q(p 2 -r 2 ) w + w 1 w -w 1 = 0, (19b) X X 2 -Y Y 1 = 0. ( 19c
)
Remark: For H3 the parameter dependence is through p/r and q/r. Therefore we may put r = 1, but the special limits r = 0 and r → ∞ also produce interesting sub-cases.

dSKdV (Q1 δ=0 )

This model was obtained from the discrete Schwarzian KdV equation [START_REF] Nijhoff | Direct linearization of nonlinear difference-difference equations[END_REF] (Q 1 with δ = 0 in [START_REF] Adler | Classification of Integrable Equations on Quad-Graphs. The Consistency Approach[END_REF]). For the elementary square we used the Lax matrices

L(x 1 , x; p) = px 1 + r(x -x 1 ) -pxx 1 p -px + r(x -x 1 ) , ( 20
)
where L(x 1 , x; p) describes the parallel transport from x to x 1 , and L(x 2 , x; q) from x to x 2 . When constructing the Lax pair for the bigger 2 × 2 lattices, we get expressions depending on the corner variables x, x 11 , x 22 , x 1122 which will be relabeled as w, w 1 , w 2 , w 12 , respectively, while x 1 , x 2 , x 112 , x 122 will become bond variables X, Y, Y 1 , X 2 , respectively.

The results also depend on the three parameters (p, q, r) in an homogeneous way. The "true" parameters are thus p/r and q/r. If r is not vanishing we do get a rational model, but that model is not integrable, as we will see. (We will not give here the explicit solution for r = 0, as it may be extracted from the simplified form of the defining equations given at the end of this section.) In the r → 0 limit, we have the following defining relations:

Original form:

w 12 = A B with (21)
A = -(X -Y ) (2 w -X -Y ) w 1 2 w 2 2 + -2 Y X 2 -4 Y w 2 + XY 2 + 4 wY X + w 3 w 1 2 w 2 -2 w (X -Y ) w 2 -Y X w 1 w 2 --2 Y 2 X -4 w 2 X + 4 wY X + wX 2 + w 3 w 1 w 2 2 +Y 2 (w -X) 2 w 1 2 -X 2 (w -Y ) 2 w 2 2 -wY 2 (w -X) 2 w 1 + wX 2 (w -Y ) 2 w 2 , B = -2 w 3 X -Y 2 X 2 + 4 wX Y 2 -4 w 2 X Y -w 2 Y 2 w 1 -w (w -X) 2 w 2 2 + -4 w 2 X Y + 2 Y w 3 -Y 2 X 2 + 4 wX 2 Y -X 2 w 2 w 2 + w (w -Y ) 2 w 1 2 +2 (X -Y ) w 2 -Y X w 1 w 2 + (w -X) 2 w 1 w 2 2 -(w -Y ) 2 w 1 2 w 2 +w 2 (X -Y ) (wX + Y w -2 Y X) ,
and

X 2 = C D with (22)
C = X 2 (w -Y ) 2 w 2 -Y 2 (w -X) 2 w 1 + Y (w -X) 2 w 1 2 -2 Y 2 X -Y X 2 -Y w 2 -2 wY X + 2 w 2 X w 1 w 2 + (X -Y ) (-X + 2 w -Y ) w 1 2 w 2 , D = -2 Y 2 X -Y X 2 -Y w 2 -2 wY X + 2 w 2 X w 1 +Y (w -X) 2 w 2 + (w -Y ) 2 w 1 2 -(w -X) 2 w 1 w 2 + w (X -Y ) (-2 Y X + wX + wY ) .
By symmetry, Y 1 is obtained from X 2 by the exchange (X ↔ Y, w 1 ↔ w 2 )

Entropy: For r = 0 the sequence of degrees for the vertex variables is

{d n } v = 1,
fitted by the generating function

ζ v (s) = 1 + 7 s + 9 s 3 -s 4 (1 + s) (1 -s) 4 (24)
For the bond variables we get the sequence of degrees

{d n } b = 1,
fitted by

ζ b (s) = (1 + s) (1 + 3 s + 4 s 2 ) (1 -s) 4 (26)
The growth of the degrees is cubic, showing integrability. It is interesting to notice that this example does not have the quadratic growth so commonly observed. The vertex and bond variables are moreover intertwined in a non-trivial way.

The behaviour when r = 0 has to be contrasted with the generic r = 0 behaviour, where the degree calculation yields for the vertex variables

{d n } v = 1,
ζ v (s) = 1 + 6 s -s 2 + 6 s 3 + s 4 (1 -s) (1 + s + s 2 ) (1 -4 s + s 2 ) , ζ b (s) = 1 + 14 s 2 + 13 s 3 + 8 s 4 + 4 s (1 -s) (1 + s + s 2 ) (1 -4 s + s 2 ) ,
indicating non-integrability (the rate of growth of the degrees is given by the roots of s 2 -4s + 1, i.e. 2 + √ 3).

Simplified form: The transformation

X → Xw -w 1 X -1 , Y → Y w -w 2 Y -1 , X 2 → X 2 w 2 -w 12 X 2 -1 , Y 1 → Y 1 w 1 -w 12 Y 1 -1 ,
simplifies the equations to

X 2 Y -Y 1 X = 0, (27a) Y 1 Y (w -w 1 ) + w 12 -w 2 = 0, (27b) X 2 X(w -w 2 ) + w 12 -w 1 = 0, (27c)
Here is also, for reference, the (nonintegrable) form before taking the r → 0 limit:

X 2 Y -Y 1 X = r(AX 2 + BY 1 ), Y 1 Y (w -w 1 ) + w 12 -w 2 = 0, X 2 X(w -w 2 ) + w 12 -w 1 = 0, where A = (w -w 1 )(w -w 2 ) q XY -((w -w 2 )X -w 1 + w 2 ) 2 p p q (w -w 1 )(w 1 -w 2 ) , B = (w -w 1 )(w -w 2 ) p XY -((w -w 1 )Y -w 1 + w 2 ) 2 q p q (w -w 2 )(w 1 -w 2 ) .
4 Non-autonomous functional Yang-Baxter equations, three dimensional consistency

One interesting property of the four models proposed in Sec. 3.1-3.3 is that they provide non-autonomous solutions to the functional form of the Yang-Baxter equations [START_REF] Drinfeld | On some unsolved problems in quantum group theory[END_REF], alias Yang-Baxter maps. This may be seen by examining the 3 dimensional consistency of the models on the configuration shown in Figure 2. The vertices introduce position dependence, and the maps induced on the bonds verify a modified functional Yang-Baxter equations if the model is 3D consistent. Suppose that w, w 1 , w 2 , w 3 , X, Y, Z are given. Then the relation on each face allows one to evaluate X 2 , X 3 , Y 1 , Y 2 , Z 1 , Z 2 , w 12 , w 13 , w 23 in a unique way, X 23 , Y 13 , Z 12 in two different ways, and w 123 may be computed in three different ways. Consistency means they all have to give the same results.We have checked that all the integrable models described here verify this 3 dimensional consistency.

w 2 w w 1 w 12 Y 3 X 3 Z 2 Y X 2 Y 1 X Z 1 Z Z 12
It should be noted that for the model presented in section 3.4, the consistency is not verified when r = 0, precisely in the case where it is not integrable, but is verified for r = 0 (that is to say the limit p = q = s = ∞), where it is integrable, as ascertained by the vanishing of the algebraic entropy.

The first four sets of simplified equations have similar structure. The vertex equation is independent of the bonds, and the values w might be considered as parameters. They introduce non-autonomy in the bond evolution, which turns out to be the functional equivalent of the quantum Yang-Baxter equation introduced in [START_REF] Gervais | Novel Triangle Relation and Absence of Tachyons in Liouville String Field Theory[END_REF]. The vertex equations are either linear (additive case), or linearizable (multiplicative case), and can be solved explicitly. In the additive case w + w 12 -w 1 -w 2 = 0 the general solution is w n,m = F (n) + G(m), and in the multiplicative case w w 12 -w 1 w 2 = 0, we have w n,m = F (n) G(m), with F and G arbitrary functions. The equations determining X and Y contain one linear part, and a coupling equation which introduces non-linearity.

dpKdV and H1ǫ, configuration α

We can solve dpKdV (5) and H1ǫ, configuration α, (13) together. The vertex evolution is additive for (5a) and multiplicative for (13a) and is solved as described above. Equations (5c) and (13c) can then be cast in the common form:

X X 2 -Y Y 1 -g(m) (X -X 2 ) + f (n) (Y -Y 1 ) + ω = 0, (28)
where f and g are readily expressible in terms of the aforementioned F (n) and G(m) and ω is a constant.

We can solve (28) together with the constraint X+X 2 = Y +Y 1 for arbitrary f (n), g(m), ω, obtaining

X 2 = Y + R, Y 1 = X + R, with R = f (n) 2 -g(m) 2 -ω Y -X + f (n) -g(m)
.

Redefining the X and Y by the shifts

X = X ′ + f (n) + σ(n) + ρ(m), Y = Y ′ + g(m) + σ(n) + ρ(m),
where σ, ρ are determined from σ(n + 1) -σ(n) = 2 f (n) and ρ(m + 1) -ρ(m) = 2 g(m), we finally get

X ′ 2 = Y ′ + P, Y ′ 1 = X ′ + P, with P = f (n) 2 -g(m) 2 -ω X ′ -Y ′ . ( 29
)
This is a non-autonomous version of the Adler map [START_REF] Adler | Recuttings of polygons[END_REF] (F V in the classification of [START_REF] Adler | Geometry of YangBaxter Maps: pencils of conics and quadrirational mappings[END_REF]).

4.2

H1ǫ, configuration β and dmKdV (H3 δ=0 )

Equations ( 15) and ( 19) can also be solved together. In both cases the coupling equations (15b) and (19b) can be written as

µ (X + X 2 ) -ν (Y + Y 1 ) -g(m) (X -X 2 ) + f (n) (Y -Y 1 ) = 0, (30)
where f and g are, as above, some functions related to F (n), G(m), and µ and ν are constants. Equation (30) together with the constraint X X 2 -Y Y 1 = 0 is solved in the generic case by

X 2 = Y Q, Y 1 = X Q, Q = (µ -g(m)) X -(ν -f (n)) Y (ν + f (n)) X -(µ + g(m)) Y , (31)
Scaling X and Y by

X = X ′ σ(n)ρ(m)/(f (n) + ν), Y = Y ′ σ(n)ρ(m)/(g(m) + µ), (32)

Conclusion and perspectives

We have presented lattice models with values given at both the vertices and the bonds of a 2D square lattice. They were constructed using the ambiguity of the zero curvature condition on a coarse grained lattice. The integrability of these models is guaranteed by the vanishing of the algebraic entropy and by the Consistency-Around-the-Cube property.

Four out of five models provide non autonomous generalizations of known Yang-Baxter maps, the fifth case (27) (related to dSKdV) being seemingly different. They all share some remarkable algebraic properties, in particular their multi-rationality, which might be the frame for further examples and an eventual classification.

Many more aspects will have to be examined (especially for (27)), like the existence of Bäcklund transforms, proper Lax pairs, symmetries, reductions (periodic reductions as well as similarity reductions) which should produce interesting integrable maps, including discrete Painlevé equations. Their continuous limits should also be considered. All these go beyond the scope of this paper, and require further studies.

Figure 1 :

 1 Figure 1: (a): the basic lattice square, and (b): its precursor, a 2 × 2 sublattice. in the simple way described above, and to analyze their properties. They will in particular satisfy ∂X n,m+1 ∂w n+1,m • ∂Y n+1,m ∂w n,m+1 = 0, ∀n, m.(1)

13 Figure 2 :

 132 Figure 2: The consistency cube

  13, 28, 49, 76, 109, 148, 193, 244, 301 . . .

  3, 5, 7, 9, 11, 13, 15, 17, 19, 21, . . .

  5, 17, 37, 65, 101, 145, 197, 257, 325, 401, 485, 577, . . .

  8, 33, 92, 201, 376, 633, 988, 1457, 2056, 2801, 3708, 4793, . . .

  10, 38, 149, 565, 2110, 7882, 29425, 109817, 409850, 1529590, . . . and for the bond variables {d n } b = 1, 8, 45, 186, 711, 2672, 9991, 37304, 139239, 519666, 1939437, . . .

	fitted respectively by the generating functions

In some respects this is similar to using conservation laws to derive dynamics, which sometimes allows multiple solutions[START_REF] Quispel | Duality for discrete integrable systems[END_REF].

Acknowledgments

One of us (JH) was partially supported by the Ville de Paris in the "Research in Paris" program. Some of the computations were done using REDUCE [23]. We would like to thank F. Nijhoff and J. Perk for additional references.

where σ, ρ now solve

we get the equations in the form

with

Equation ( 34) is a non-autonomous version of F III of [START_REF] Adler | Geometry of YangBaxter Maps: pencils of conics and quadrirational mappings[END_REF].

It should be noticed that the non-autonomous nature of the above equations parallels the results of [START_REF] Sahadevan | Hydon Integrability conditions for nonautonomous quad-graph equations[END_REF].

dSKdV (Q1 δ=0 )

In the case of equation ( 27) the w evolution is not linearizable, as indicated by the algebraic entropy analysis. If we consider (27) as three equations for two variables X, Y then we can derive from their compatibility an equation for w, which can be written as

where T, H are 3-point Schwarzian-like derivatives of n, m, respectively:

Equation ( 35) connects the points of a 3 × 3 sublattice, except the center point (in the above w(n + 1, m + 1)). The algebraic entropy analysis indicates the rare cubic growth. Equations defined on a 3 × 3 sublattice are typical for Boussinesq-type lattice equations [START_REF] Nijhoff | Discrete Painlevé Equations and Symmetry Reduction on the Lattice[END_REF], but they usually have quadratic growth.

The Schwarzian KdV equation is given by

From (35) one can now derive

which can be solved and we find that w solves the non-autonomous SKdV equation

Thus when the model ( 27) is considered on larger cells made of four adjacent squares, we find an independent non-trivial evolution of the vertex variables (35), driving the evolution of the bond variables according to (27b,c).


MD, MPH Valérie Bourdès

MD Jean Ferrières

MD Jacques Amar

MD Elisabeth Amelineau

PhD Stéphane Bonnevay

Ph.D Maryse Berlion

MD Nicolas Danchin

Dr Bourdes Valérie
email: valerie.bourdes@themis-rd.fr

Dr Spaan Jos

Dr Dear

Spaan

MD, MPH Valérie Bourdes

Prediction of persistence of combined evidence-based cardiovascular medications in patients with acute coronary syndrome after hospital discharge using neural networks

Keywords: Acute coronary syndrome, evidence-based medical therapy, long-term persistence, artificial neural networks Acute coronary syndrome, evidence-based medical therapy, long-term persistence, artificial neural networks

Prediction of persistence of combined evidence-based

cardiovascular medications in patients with acute coronary syndrome after hospital discharge using neural networks.

Introduction

Acute coronary syndrome (ACS) includes life-threatening clinical conditions ranging from unstable angina to non-ST-elevation myocardial infarction (NSTEMI) and ST-elevation myocardial infarction (STEMI) that are a major cause of emergency medical care and hospitalization in industrialized countries [START_REF] Grech | Acute coronary syndrome: unstable angina and non-ST segment elevation myocardial infarction[END_REF][START_REF]Myocardial infarction redefined--a consensus document of The Joint European Society of Cardiology/American College of Cardiology Committee for the redefinition of myocardial infarction[END_REF]. Patients with ACS after the initial phase carry a high risk of recurrence of ischemic events. Therefore, active secondary prevention is an essential element of long-term management. European clinical practice guidelines (European Society of Cardiology) support the use of four major classes of drugs, i.e., angiotensinconverting enzyme inhibitors (ACEI) and/or angiotensin II receptor blockers (ARB), beta blockers, antithrombotic agents and statins for long-term treatment of patients after an acute coronary event [START_REF] Bassand | Guidelines for the diagnosis and treatment of non-ST-segment elevation acute coronary syndromes. ESC Guidelines[END_REF][START_REF] Silber | Task Force for Percutaneous Coronary Interventions of the European Society of Cardiology (2005) Guidelines for percutaneous coronary interventions. ESC Guidelines[END_REF][START_REF] Van De Werf | Task Force on the Management of Acute Myocardial Infarction of the European Society of Cardiology (2003) Management of acute myocardial infarction in patients presenting with ST-segment elevation. ESC Guidelines[END_REF]. Unfortunately, there is evidence that these therapies are neither consistently prescribed when appropriate nor adhered to by patients in daily clinical practice [START_REF] Lee | Use of secondary prevention drug therapy in patients with acute coronary syndrome after hospital discharge[END_REF]. In a recent analysis of real-world use of secondary prevention therapies over [2003][2004] within the 90 days following an hospitalization for ACS, Lee et al. [START_REF] Lee | Use of secondary prevention drug therapy in patients with acute coronary syndrome after hospital discharge[END_REF] found exposure rates of 52% for ACEIs or ARBs, 64% for beta-blockers and 63% for statins in a cohort of 1135 patients. Similar results were obtained in three European observational studies [START_REF] Carruthers | Contemporary management of acute coronary syndromes: does the practice match the evidence? The global registry of acute coronary events (GRACE)[END_REF][START_REF]Lifestyle and risk factor management and use of drug therapies in coronary patients from 15 countries; principal results from EUROASPIRE II Euro Heart Survey Programme[END_REF][START_REF] Hasdai | A prospective survey of the characteristics, treatments and outcomes of patients with acute coronary syndromes in Europe and the Mediterranean basin; the Euro Heart Survey of Acute Coronary Syndromes (Euro Heart Survey ACS)[END_REF].

Long-term persistence of evidence-based cardiovascular medications (EBCM) remains a key concern to achieve optimal outcomes. In a survey of 1326 patients with coronary artery disease assessing the adherence to EBCM by comparing baseline prescription at hospital discharge to self-reported medical regimen at 1 year, Kulkarni et al. [START_REF] Kulkarni | Long-term adherence with cardiovascular drug regimens[END_REF] reported that only 54% of patients were adherent to all of their initial medications. In 1700 patients hospitalized for ACS in 2006 in France, the four-drug combination therapy had been prescribed in 46% of patients at hospital discharge, 80% of them were still taking the combination 14 months after hospital discharge [START_REF] Amar | Persistence of combination of evidence-based medical therapy in patients with acute coronary syndromes[END_REF].

Defining predictors of long-term EBCM persistence has implications for the disease management approach of individual patient. Traditional statistical approaches such as logistic or Cox regressions, commonly used for prediction analyses in cardiology, tend to select only variables which have a high level of linear correlation with the outcome variable. Complex nonlinear relationships between independent and dependent variables can be accurately detected by artificial neural networks (NNs) [START_REF] Cross | Introduction to neural networks[END_REF]. Artificial NNs are biologically inspired by computer programs designed to simulate the way in which the human brain processes information [START_REF] Rosenblatt | The perceptron: a probabilistic model for information storage and organization in the brain[END_REF].

NN is formed from hundreds of single units, artificial neurons or processing elements, connected with coefficients (weights), which constitute the neural structure and are organized in layers [START_REF] Agatonovic-Kustrin | Basic concepts of artificial neural network (ANN) modeling and its application in pharmaceutical research[END_REF]. NNs present the main advantage not to be based on "a priori" assumptions and to allow detection of links between factors that conventional statistical techniques may not be able to detect [START_REF] Agatonovic-Kustrin | Basic concepts of artificial neural network (ANN) modeling and its application in pharmaceutical research[END_REF][START_REF] Bourdès | Comparison of artificial neural network with logistic regression as classification models for variable selection for prediction of breast cancer patient outcomes[END_REF][START_REF] Tu | Advantages and disadvantages of using artificial neural networks versus logistic regression for predicting medical outcomes[END_REF]. In studies carried out in the cardiovascular field, NNs provided a better predictive accuracy than did traditional statistical techniques [START_REF] Baldassarre | Recognition of patients with cardiovascular disease by artificial neural networks[END_REF][START_REF] Bigi | Artificial neural networks and robust Bayesian classifiers for risk stratification following uncomplicated myocardial infarction[END_REF][START_REF] Voss | Prediction of risk of coronary events in middle-aged men in the Prospective Cardiovascular Munster Study (PROCAM) using neural networks[END_REF].

In the present French, cross-sectional, observational PREVENIR-5 study, NNs (Multi-Layer Perceptron) have been applied to a large sample of patients with recent ACS to identify persistence factors of EBCM. The prediction accuracy of NN was compared with that of logistic regression (LR) using classification performance indices, i.e., Area Under Receiver-Operating Characteristics (AUROC), sensitivity, specificity, accuracy and positive predictive value (PPV).

Methods

Study design

Male or female adult patients hospitalized for a first episode of ACS (NSTEMI, STEMI or unstable angina) during 2004 or 2005, and followed routinely by a general practitioner (GP), were included in this study. No exclusion criteria were applied except for current participation in a clinical trial. The physicians were randomly selected from a comprehensive national database of French GPs. The sample was regionally stratified to ensure the representativeness of the study with regard to French medical practice. The study was proposed by mail to 39665 GPs and 3122 (7.9%) gave their consent to participate and signed the financial agreement. Each participating GP was asked to include the first three consecutive patients, who fulfilled the eligibility criteria over a three-month period. The following data were prospectively collected by the GP at the inclusion visit (day of consultation): patients' and ACS characteristics, medications at hospital discharge and at inclusion visit, cardiovascular risk factors, last biological measurements, concomitant cardiovascular and other diseases, and concomitant treatments. The study met the quality standard described in the declaration of Helsinki and the Good Epidemiological Practices [START_REF]IEA guidelines for proper conduct in epidemiologic research (2007) Good Epidemiological Practice (GEP)[END_REF] and was approved by local legal authorities ("French Information Technology and Liberties Commission" and "French Medical Committee").

Participation in the study was voluntary. This observational study did not modify the clinical practices of the physicians.

Clinical endpoints

The endpoints were to identify key persistence factors of EBCM in ACS patients using NN and LR and to compare the two models prediction performances. Persistence was defined as current use of EBCM prescribed at hospital discharge and non persistence as the discontinuation of at least one of the drugs the day of consultation. The analysis was performed on the subgroup of patients who received the four-drug combination at hospital discharge (N=2132), i.e., any antithrombotic agent + beta-blocker + statin + ACEI and/or ARB (EBCM).

Variable selection with neural networks

Neural network constructions were carried out using the Statistica Neural Networks software release 7.1. LR was performed with SAS Software 8.2. Two types of NN models were built, one using the LR variable selection (NN-varLR) and the other using the NN variable selection (NN-varNN). Therefore, three models were compared (NN-varLR, NN-varNN and LR). Three different methods were used to select the most significant variables for each NN model: forward and backward stepwise feature selections and a genetic input selection algorithm. Forward selection consists in choosing the most predictable variable then checks for a second variable that, added to the first, most improves the model; this process is repeated until either all variables have been selected or no further improvement is made. Backward stepwise feature selection is the reverse process: it starts with all the variables and then removes a variable at each stage which less degrades the model. Genetic algorithm [START_REF] Goldberg | Genetic algorithms in search, optimization and machine learning[END_REF] selection is a heuristic seeking the optimal set of input variables. This heuristic builds a model by a succession of artificial transformations (mutation, crossover, and selection) from an initial population of variables sets. Each of our genetic selections was made from a population of 100 individuals (one individual corresponds to one set of variables) on 100 generations. Each set of variables corresponds to a binary string where a 0 indicates that the variable is not in the set of variables, and a 1 indicates that the variable is in this set. This set is tested with the help of a neural network, and the objective function is the error of this neural network on a training set. With our choice (100 individuals and 100 generations), it performs 10,000 evaluations of sets of variables. For example, the selection of one set of variables is about 60 times longer with the genetic algorithm than the backward selection, but genetic algorithms are well suited for feature selection as there is a large number of possible variables. Because of their differences and complementarities, we decided to combine these three methods to select the inputs of our NN models.

With a view to improving generalization capability of networks and to decreasing the network size and execution size, a penalty can be used to penalize the large sets of variables. In this way, a penalty parameter is multiplied by the number of selected variables and added to the error level. Following different analyses, we finally used a small penalty equal to 0.0001 with half of selection algorithms and no penalty with the other half selection algorithms. Each method calculates the value of the set of covariates selected at each step while building the NN. We have chosen to perform each method 40 times: 20 without penalty and 20 with a penalty equal to 0.0001. Indeed, 120 (3×40) selections were performed. Covariates were kept as an input for the final model if it was selected in at least 80% of the 120 individual models; this value of 80% leads to the inclusion of a reasonable number of variables regarding the complexity of the NN models. Table 1 displays the selected variables for LR and NN for EBCM subgroup.

Neural networks modelling

The Multi Layer Perceptron (MLP) NN was applied as a classifier, with a single-layer of hidden nodes using hyperbolic activation functions [START_REF] Hornik | Multilayer feedforward neural networks are universal approximators[END_REF]. The number of neurons on the hidden layer was determined according to the number and the nature of variables at the entry. Weights and bias of NN were determined by training with a two-phase procedure. The first phase is a quite short burst of backpropagation, with a moderate training rate. The second phase is a longer run of conjugated gradient descent, a much more powerful algorithm, which is less likely to encounter convergence problems than otherwise due to the use of backpropagation first. During this learning process, the weights in a MLP are adjusted using least squares fitting together with the training two-phase procedure to minimize a root mean square error function. In order to interpret the network outputs as probabilities and to make them comparable to the results of logistic regression, we used a cross entropy error function to adjust weights. This cross entropy function is specially designed for classification problems where it is used in combination with hyperbolic activation function.

A continuous input value is prescaled to a range between 0 and 1; a two-state nominal variable, which corresponds to one entry of the neural network, is represented by transformation into a numeric value (e.g., "Gender" = 0 or 1); a many-state nominal variable is recoded into as many binary entries as modalities (e.g., "Age" is a 4-state nominal variable which correspond to 4 inputs). The NN-varLR was composed of: (a) one input layer corresponding to the 11 covariates (given by the Logistic Regression). Seven covariates with two modalities correspond to a binary entry. The other covariates are recoded into as many entries as modalities (one covariate with 3 modalities, one with 4 modalities and two covariates with 5 modalities). The network built that way has 24 binary entries; (b) one hidden layer composed of 6 neurons with hyperbolic activation function and (c) one output layer composed of the one neuron with logistic activation function. The NN-varNN was composed of (a) one input layer corresponding to the 8 covariates (given by the neural selection). Three covariates correspond to a binary entry and the other covariates are recoded into as many entries as modalities (2 covariates with 3 modalities, one with 4 modalities and two with 5 modalities). The network built that way has 23 binary entries;

(b) one hidden layer composed of 8 neurons with hyperbolic activation function and (c) one output layer composed of the one neuron with logistic activation function.

In order to avoid over-fitting and to obtain networks with a strong capacity of generalization, we divided data randomly in two datasets by cross validation: a learning set to build the models and a testing set for the evaluation (not used for construction). The learning set was composed of 70% of the total random population (testing set with 30% of the remaining population).

Comparison between logistic regression and neural networks

The predictive accuracy of the three models,

Rule extraction methodology

To identify patients profiling, the Orthogonal Search-based Rule Extraction (OSRE) methodology was used to extract rules from NN [START_REF] Etchells | Orthogonal search-based rule extraction (OSRE) for trained neural networks: a practical and efficient approach[END_REF][START_REF] Etchells | Orthogonal search-based rule extraction for modelling the decision to transfuse[END_REF]. OSRE is a rule extraction algorithm that efficiently extracts comprehensive rules from smooth models such as those created by NNs. This method, which is a refinement of RULENEG algorithm, constructed a hierarchical list of rules from the best to the less good: decreasing order of sensitivity and PPV. This construction was performed on the learning set then validated on the remaining data.

Results

From October 2006 to April 2007, 5111 patients were enrolled by 1838 active GPs. A total of 261 patients (5.1%) were excluded from the analyzed population due to non respect of inclusion criteria (n=9), patients hospitalized out of the 2004-2005 period (n=77), patients seen in consultation before the beginning of the survey (n=175). Overall, 4850 patients were enrolled by 1811 GPs with a mean time ( SD) of occurrence of first ACS of 24  7 months (range, 10.8 to 39.3 months).

Patients' characteristics

Patients' characteristics (N=4850) are described in Table 2. The majority was male (80.1%) and the mean age was 64 years. ACS included STEMI (43.7%), unstable angina (33.1%), NSTEMI (18.4%) or unknown (4.8%). The majority of patients were regularly followed by their GP with quarterly, monthly or bimonthly visit rates for 40.7%, 36.5% and 19.1% of patients, respectively. Moreover, 96.1% of the patients were also followed by a specialist, mainly a private (65.0%) or hospital (34.6%) cardiologist.

EBCM persistence

At hospital discharge, EBCM was prescribed to 2132 patients (44.0%). At inclusion visit, EBCM was persistent in 1849 patients (86.7%). Persistence rates of antithrombotic agents, ACEI and/or ARB, beta-blockers, statins were 99.4%, 94.3%, 92.9% and 98.1%, respectively (Table 3). Non persistence was mostly due to the discontinuation of beta-blockers (7.1%) and ACEI and/or ARB (5.7%). The main reason for treatment discontinuation reported by the practitioner was intolerance particularly for statins (48.8%), ACEI and/or ARB (36.1%) and beta-blockers (38.4%) (Table 4).

Persistence profile

The rule extracted from the OSRE algorithm was:  25< body mass index-BMI <30 kg/m²) and  low educational level, and  hypercholesterolemia, and  no coronary by-pass grafting (CABG) for the first ACS.

Rules generated indicated that patients with the association of overweight, hypercholesterolemia, no CABG for their first ACS, and low educational level represented a relevant persistence profile with a PPV of 0.958.

Comparison between NN and LR

The AUROC curves of persistence showed better predictive accuracy for NN models compared with LR model for EBCM, whatever the variables selection with AUC of 0.78, 0.80 and 0.69for NN-varLR, NN-varNN and LR, respectively (Figure 1).

Discussion

This large-scale, epidemiological and cross-sectional study reflects real-life general medical practice regarding the long-term therapeutic management of ACS in France. NN models successfully identified persistence profile of patients with a high PPV, confirming their relevance as predictive tools. The NN model showed a higher prediction performance than LR and should be considered as a helpful tool in medical decision support by identifying specific subgroups of patients.

The MLP neural network is the most widely applied to real world problems in medical diagnosis and prediction [START_REF] Hornik | Multilayer feedforward neural networks are universal approximators[END_REF]. One of the criticisms towards NN is that their process inside is unknown and some authors consider them as "black boxes". To prevent this criticism and to enhance the NN selection process, we decided to use several variables selection techniques perfectly coded and to use one penalty for some selection methods. Using these three different techniques for the variables selection may be criticized as a time-consuming process. However, it was a guarantee for avoiding over-fitting, which is the main limit of the NN [START_REF] Agatonovic-Kustrin | Basic concepts of artificial neural network (ANN) modeling and its application in pharmaceutical research[END_REF] and leading to a gain in selection improvement.

The PREVENIR-5 survey confirms the low prescription of combined EBCM at hospital discharge in patients with ACS, despite the well documented efficacy of aspirin, beta-blockers, statins and ACEIs as secondary prevention of coronary artery disease [START_REF] Ramanath | Evidence-based medical therapy of patients with acute coronary syndromes[END_REF]. Only 44% of patients received treatment with an antithrombotic, a beta-blocker, a statin and an ACEI and/or ARB. This prescription rate was consistent with that observed in the recent PREVENIR-4 observational study carried out in 2006, in which 46% of patients received the combination therapy (-blocker, antithrombotic, statin and ACEI) [START_REF] Amar | Persistence of combination of evidence-based medical therapy in patients with acute coronary syndromes[END_REF]. However, it seems to be better than results reported by Danchin et al. [START_REF] Danchin | Impact of combined secondary prevention therapy after myocardial infarction: data from a nationwide French registry[END_REF] in French patients admitted to intensive care units in 2000 for acute myocardial infarction; only 27% of these patients received the quadritherapy at hospital discharge. Yan et al. [START_REF] Yan | Optimal medical therapy at discharge in patients with acute coronary syndromes: temporal changes, characteristics, and 1-year outcome[END_REF] reported a significant increase in the discharge use of all Page 13 / 26 therapy in the Canadian ACS I (September 1999-June 2001) and ACS II (October 2002-December 2003) registries, respectively. Some factors associated with lower rate prescription have been identified: elderly patients, female gender, missing of low density lipoproteincholesterol measurement, history of peripheral artery disease or stroke, and finally the difficulty of observance [START_REF] Philippe | Changes in the prescription of cardiovascular prevention drugs in France between 1995 and 2003: factors influencing the gap between evidence base medicine and clinical practice[END_REF].

The rates of persistence for EBCM as well as for each compound separately were high in PREVENIR-5 patients as already observed in the most recent studies. In PREVENIR-4 survey carried out in 2006, 80% of coronary patients were still taking the quadritherapy 14 months on average after hospital discharge [START_REF] Amar | Persistence of combination of evidence-based medical therapy in patients with acute coronary syndromes[END_REF]. When analyzed separately, beta-blockers, antithrombotics, statins and ACEIs were persistent over the 14-month period for 92%, 97%, 96% and 86% of patients, respectively [START_REF] Amar | Persistence of combination of evidence-based medical therapy in patients with acute coronary syndromes[END_REF]. In 13830 patients from the GRACE registry, therapy was maintained 6 months after hospital discharge for myocardial infarction or unstable angina for 92% of patients taking aspirin on discharge, 88% of those taking beta-blockers, 80% of those taking ACEIs and 87% of those taking statins [START_REF] Eagle | Adherence to evidence-based therapies after discharge for acute coronary syndromes: an ongoing prospective, observational study[END_REF]. The high persistence rates for PREVENIR-5 patients can be explained by the fact that almost all the patients were regularly followed by both GPs and cardiologists. A regular follow-up by the attending physician and/or specialist suggests effective physician-patient interactions and efficient communication is also thought to be an important determinant of adherence to treatment [START_REF] Eagle | Adherence to evidence-based therapies after discharge for acute coronary syndromes: an ongoing prospective, observational study[END_REF]. A patient who clearly understands the usefulness of a drug and the potential consequences of not taking it is more likely to continue the treatment. Moreover, half of the patients were less than 65 years old and few patients presented concomitant disease. Patients were started on medication at the time of hospital discharge and adherence is generally superior in this setting to when therapies are initiated in the outpatient setting [START_REF] Feely | The influence of hospital-based prescribers on prescribing in general practice[END_REF]. These rates of persistence were higher than those reported in the early to mid 1990s suggesting an improvement in adherence to treatment over the past decade [START_REF] Harder | Prescription of cardiovascular drugs in outpatient care: a survey of outpatients in a German university hospital[END_REF][START_REF] Jones | Discontinuation of and changes in treatment after start of new courses of antihypertensive drugs: a study of a United Kingdom population[END_REF][START_REF] Krumholz | Determinants of appropriate use of angiotensin-converting enzyme inhibitors after acute myocardial infarction in persons > or = 65 years of age[END_REF][START_REF] Roe | Compliance with and dosing of angiotensin-converting-enzyme inhibitors before and after hospitalization[END_REF]. This gradual improvement may be explained by the diffusion of the findings of major randomized trials into medical practice and a good follow-up of the clinical guidelines.

Using NNs, the associated predictive factors correlated with a persistence profile were absence of CABG, overweight, hypercholesterolemia and low educational level for patients receiving combined treatment at hospital discharge. The cardiovascular risk factors particularly hypercholesterolemia were frequently shown to be associated with treatment persistence, especially for statin therapy [START_REF] Jackevicius | Adherence with statin therapy in elderly patients with and without acute coronary syndromes[END_REF]. Conversely to our results, adherent patients were significantly more likely to have a higher educational status in an American observational study [START_REF] Kulkarni | Long-term adherence with cardiovascular drug regimens[END_REF]. Low educational status may be a marker of limited financial resources to afford medications in United States but not in France where patients benefit from a different health care system. In France, cardiovascular risk factors such as overweight or hypercholesterolemia are frequently associated with low educational status. Although prior studies have suggested age-influenced adherence [START_REF] Ali | Age and persistent use of cardiovascular medication after acute coronary syndrome: results from medication applied and sustained over time[END_REF], our study did not find an association between patient age and persistence whatever the model used.

Our study had several limitations, the most significant being inherent to the cross-sectional design of the study. As a result it was not possible to examine the time relationship between patient factors and persistence. However, no impact on NN or LR models was expected.

In conclusion, long-term persistence of EBCM prescribed after hospital discharge for a first ACS was high. The neural approach and rule extraction allowed the identification of patient profiling for persistence of EBCM, i.e., absence of CABG, overweight, hypercholesterolemia and low educational level. This large epidemiological French study shows that NN should be recognized as powerful predictive tools, to be considered routinely alongside standard LR.

  LR, NN with inputs of LR selection (NN-varLR) and NN with inputs of NN selection (NN-varNN), was assessed using AUROC. The AUROC is a good measure of the overall predictive accuracy of an analytic tool. It represents a plot of sensitivity versus (1 -specificity). Sensitivity measures the fraction of positive cases classified as positive and specificity measures the fraction of negative cases classified as negative. The positive predictive value (PPV) corresponds to the number of true positives divided by the sum of true positives and false positives.

Table 2

 2 Patients and disease characteristics

	Overall patients (N=4850)
	Mean  SD Number of patients	%

Abbreviations: ACS, acute coronary syndrome; BMI, body mass index; CABG, coronary artery bypass grafting; NSTEMI, non-ST-elevation myocardial infarction; PCI, percutaneous coronary intervention; SD, standard deviation; STEMI, ST-elevation myocardial infarction.

Table 3

 3 Persistence of treatments at consultation visitTreatment at consultation visit, n (%) (at least one treatment or combination)

	EBCM subgroup at hospital
	discharge (n=2132)

Abbreviations: ACEI, angiotensin-converting enzyme inhibitor; ARB, angiotensin II receptor blocker; EBCM, evidence-based cardiovascular medications.

Table 4

 4 Reasons for treatment discontinuation in EBCM subgroupAbbreviations: ACEI, angiotensin-converting enzyme inhibitor; ARB, angiotensin II receptor blocker; EBCM, evidence-based cardiovascular medications.

	Reason of non persistence -EBCM	ACEI and/or ARB	Beta-blocker	Statin	Aspirin
		(n=122)	(n=151)	(n=41)	(n=13)
	Bad tolerance	44 (36.1%)	58 (38.4%)	20 (48.8%) 3 (23.1%)
	Replacement by another class	9 (7.4%)	22 (14.6%)	5 (12.2%)	3 (23.1%)
	No indication	33 (27.0%)	17 (11.3%)	1 (2.4%)	1 (7.7%)
	Benefit/risk ratio judged as insufficient	13 (10.7%)	4 (2.6%)	1 (2.4%)	1 (7.7%)
	Patient's refusal	2 (1.6%)	6 (4.0%)	4 (9.8%)	0
	Contra-indication	5 (4.1%)	17 (11.2%)	2 (4.9%)	2 (15.4%)
	Other	15 (12.3%)	25 (16.6%)	6 (14.6%)	2 (15.4%)
	Missing data	1 (0.8%)	2 (1.3%)	2 (4.9%)	1 (7.7%)

Acknowledgments

This work was financially supported by Bristol Myers Squibb, France. The authors thank Prof.

Paulo Lisboa and his team (Liverpool John Moore University, United Kingdom) for the rule extraction methodology and the data management and statistics department of ICTA-PM (Dijon, France).

Authors: The aim of this process is not to build the NN, but only to select the best set of variables. At each step of this process (each set of variables and each generation: 100*100=10000 step) a NN is build and an error is calculated. Then, with the best set of variables, a NN is build according to a classical learning process with test and training sets.

*

Methods / Variable Selection [Line 19]: The authors use 100 population over 100 generations. Can the authors justify these numbers? Did the population sufficiently converge over 100 generations? We carefully read the last comments of the reviewers and we revised the manuscript taking into account all the changes requested by the reviewers.

Tables

The replies to the reviewers, comment by comment, are given below and we submit our revised manuscript.

Yours Authors: There's no convergence reason; these choices were only guided by the time-consuming process.

 Methods / Variable Selection: Can the authors give the cross-over and mutation coefficient values?

Authors:

The single-point crossover rate was set to 0.8 and the mutation rate was set to 0.1.

Reviewer #2:

From my point of the view the authors have accomplished the reviewers´ suggestions and have incorporated all necessary changes.