Lilian Loubet
email: lilian.loubet@univ-lehavre.fr

Géraldine Del Mondo
email: geraldine.del_mondo@insa-rouen.fr

Julius Millano

Eric Sanlaville
email: eric.sanlaville@univ-lehavre.fr

Pierrick Tranouez
email: pierrick.tranouez@univ-rouen.fr

Modélisation de la gouvernance des villes portuaires européennes de taille moyenne

Keywords: gouvernance portuaire, gouvernance, parties prenantes, modélisation, théorie des graphes Classification JEL : R49 -R58 -O20 -C63 governance, port governance, stakeholder, modelling, graph theory

Ce travail présente un modèle d'analyse de la gouvernance portuaire considérant que la capacité des acteurs à adopter des comportements coopératifs constitue l'élément moteur du développement. Il se concentre sur les ports moyens européens ; quinze cas ont été étudiés (plus de 80 entretiens réalisés). Toutefois, si ce matériau riche permet de comparer la gouvernance de deux, voire trois ports, la finesse des relations mises en lumière complexifie considérablement la comparaison au sein d'un ensemble plus important. Aussi cet article propose un traitement semiautomatique participant à pallier cette difficulté au moyen d'un modèle informatique basé sur la théorie des graphes. Il s'agit de modéliser un système en se basant sur les relations entre les entités du système. Ainsi, les relations entre acteurs ont été analysées afin de créer des typologies et d'envisager à terme des modèles types de gouvernance. Afin de territorialiser le propos, six cas représentatifs ont été retenus sur les quinze : Le Havre, Nantes-Saint Nazaire, Dunkerque (France), Gdynia (Pologne), Klaipėda (Lituanie), Hamina-Kotka (Finlande). Tous ces environnements portuaires modélisés selon un même format (i.e. un graphe) donnent lieu à l'application d'un certain nombre de métriques permettant de les comparer. Deux principales métriques ont été présentées à titre illustratif dans le cadre de cet article : la "Densité" et la "S_metric". Elles ont été couplées à d'autres indicateurs (distribution des degrés et nombre de hubs par port) qui ont permis de mesurer l'intensité des relations, la répartition de cette intensité entre les acteurs, d'identifier les acteurs majeurs ou peu influents.

Introduction

There is a strong correlation between port and urban development. Many cities have grown out of ports (VIGARIE, 1999) and there is often a link between the size of a port and the size of the conurbation in which it is located (RODRIGUE et alii, 2017). Changes in international trade and containerization, however, have led to a transformation in ports, forcing them to move further away from cities. The result is a physical disassociation without any functional associations actually being eliminated [START_REF] Ducruet | Structures et dynamiques spatiales des villes portuaires: du local au mondial[END_REF]. This distancing has led to a reshaping of territories which impacts local governance. Medium-sized port-cities are particularly illustrative on the matter. Although they are not often the focus of maritime geography studies, which prefer to concentrate on large ports, as their port-city relationship is particularly strong. The reciprocal impact of the territorial and port dynamics is more evident when comparing it to larger portcities. The disconnection of port and city, the peri-urbanization, with the perimeters of the port, industry and housing converging and overspreading, have forced local stakeholders to evolve in an environment which has to be constantly renegotiated. Beyond a simple coexistence, therefore, the driving force in the development of these port territories resides in the creation of synergies and stakeholders' capacity to adopt cooperative behaviours [START_REF] Loubet | Logiques d'acteurs et développement territorial dans les villes portuaires moyennes. Le cas du Havre et de Klaipeda[END_REF].

Our analysis is built upon a research programme1 based, amongst others, on a European benchmark integrating fifteen medium-sized European port-cities (bloc "Qualitative Analysis, actors interviews", cf. Fig. 1). Each port was the subject of a qualitative survey and analysis of actors' discourse (over 80 anonymous interviews conducted with the principal stakeholders). This wealth of material makes it possible to compare the governance of two or even three ports. However, taking into account the delicate nature of the relationships brought to light, it is made more difficult to compare on a larger scale. As a consequence, this article will present a computer model analysing relationships between stakeholders, their capacity to influence a decision, so as to facilitate the comparison between a large number of ports, create typologies, and even, in the longer term, develop standard models.

The first phase of the work consists in defining the model (bloc "Graph modelling", cf. Fig. 1) by basing it on the characteristics of what we understand by "port governance". The use of the concept of governance in the port domain has been the subject of much research (BROOKS et alii, 2017;CAHOON et alii, 2013;DEBRIE et alii, 2013;NOTTEBOOM et alii, 2013;etc.). This work has helped to clarify the territorial, political, economic, organizational and institutional dynamics at work. Yet a considerable extent of work still remains to be done, especially in identifying the most propitious models as regards efficiency and the development of a port and the territory in which it is located: "One example of this scholarly effort to examine governance structures (i.e., to not only document the experiences of ports in fourteen different countries but to also evaluate the initial effectiveness of the reform programs) culminated in the publication by BROOKS and CULLINANE (2006). They concluded that there was considerably more work to do to fully understand what models are in place, and which of them are the most effective in delivering the objectives sought". (BROOKS and PALLIS, 2012, p.491-492).

In the framework of this paper, we will focus our attention on the organizational and territorial dimension of port governance by considering in particular the relationships between stakeholders and cooperative dynamics. Beyond the port cluster concept, that names action coordination and resource sharing between interrelated firms inside a given port region [START_REF] Brett | The potential for clustering of the maritime transport sector in the Grater Dublin Region[END_REF][START_REF] Haezendonck | Essays on Strategy Analysis for Seaports[END_REF]LAMBROU et alii, 2008;[START_REF] De Langen | The performance of port clusters; a framework to analyze cluster performance and an application to the port clusters of Durban, Rotterdam and the Lower Mississippi[END_REF][START_REF] Musso | Ports and Regional Economies[END_REF]ROH et alii, 2007), we shall mobilize the french concept of "place portuaire", which is imperfectly translated as "port community". Indeed, the concept of "place portuaire" broadens the scope to local institutions, town, urban area, region, chamber of commerce and professional organizations, thus getting closer to the hanseatic model of gouvernance for the town-port couple. Last, as they are located in coastal areas with high environmental sensitivity, the coexistence of specific environments, industrial, portuary, urban, logically questions the governance of these complex spaces (LÉVÊQUE, 2016, p. 263). As "place portuaire" and "port community" do not perfectly match, we shall simply use the word "port" inside this article framework.

A conventional way of modelling these interactions between stakeholders in computing science is to use a model based on the graph theory [START_REF] West | Introduction to graph theory[END_REF] in which the stakeholders are represented by vertices and interactions between stakeholders by edges. This mathematical representation standardizes the definition of every port, making the application of comparison operators possible (blocs "Comparison operators definition", and "Analysis Comparison operators", cf. Fig. 1). The challenge is to produce a graph model which would be sufficiently close to the semantics one wishes to convey, so that the comparison can provide pertinent results.

The main question will be to specify the types of relationships between stakeholders, essential information for the understanding of port governance. However, the model should also produce a representation which is abstract enough for the automated calculations of comparison to be feasible. The idea is not to imagine getting rid of experts when analysing the results but to enable them to reason simultaneously on a greater area of terrains, through an appropriate visualization of the findings of pertinent metrics (blocs "Visualization" and "Analysis -comparison operators", cf. Fig. 1). Thus the main contribution of this paper is the synergy between a governance problematic and a graph based model. If the model itself is a new one (according to the semantic of the governance application), it draws upon known concepts like hypergraphs and classical graph metrics.

The general workflow of the work developed in this paper is illustrated in Fig. 1. The first part of this paper will present the different methodological stages which led to the conception of the graph model. In the second part, the model will be defined and we shall put forward the application of comparison metrics between the graphs representing the governance of the different ports. All the metrics whose result we shall present have been calculated using an application that we developed. Lastly, the results obtained will be discussed and the research perspectives of the work presented, focusing in particular on a reinforcement of the use of semantics present in the graphs in order to define new comparison metrics. The sampling method of the medium-sized port-cities will be set out beforehand, as well as an outline of the qualitative survey and elements structuring the computer model.

As was touched on above, this work is done in the context of an ongoing research project.

Intermediate results of the qualitative survey have already been published (XX, 2020;XX, 2020;XX, 2019;XX, 2022;etc.). They will not be presented in detail but will feed into the discussion in the second part. We shall, however, offer a succinct description of the method developed in this survey as it conditions the modelling, the subject of our paper. These interviews have been conducted during 2017-2019. Since then, territorial and port contexts may have been the subject of deep recompositions. The strengthening of HAROPA 2 in the Havre case illustrates this phenomenon. Thus, the issue is not to give an account of the actual situation (2022) but to study the logics, representations and relationships between stakeholders at the time the speeches were collected (hence avoiding anachronisms).

1.1 Defining the medium-sized European port-city "Unidentified real objects" (BRUNET, 1997, p.188), describes medium-sized cities as being difficult to understand. In Europe, a medium-sized city might have a population of between 100 000 and 500 000, according to certain studies (GIFFINGER et alii, 2007). But this scale varies depending on the author and national contexts. For example, Frédéric SANTAMARIA (2000) and Frédéric TESSON (2012), from case studies in the UK, France, Portugal and Spain, propose definitions that emphasize different dimensions. The first focuses on the demographic variable, the second on the domination of the metropolis over medium-sized towns. The ORATE/ESPON batch 1.4.1 programme ( 2006) also develops a functional approach through various case studies in Austria, France, Germany, Italy, Poland, Spain and Sweden. Similarly, François TAULELLE (2010) proposes going beyond the demographic criterion alone and defining the "medium-sized city" according to the extent of its urban functions.

In the port domain, the diversities of configurations and the many exploitable criteria in the definition of medium-sized ports [START_REF] Bird | Seaport and Seaport Terminals[END_REF]) also obscure this intermediary dimension. Indeed, several indicators can be used: "an institutional criterion can be used, since large ports are often under national supervision and smaller ones are administered at local level. Such a definition, however, ignores spatial or national disparities, such as the presence of private ports or sometimes even the major role of municipal jurisdiction. The tonnage is certainly the most widely used instrument. The European Association of Maritime Ports defines medium-sized ports by traffic of between 10 and 50 million tonnes [START_REF] Verhoeven | European Port Governance: Report of an Inquiry into the Current Governance of European Seaports (The ESPO 'Fact-Finding Report[END_REF]. But this procedure also has its limits, like consideration not being given to the value added to the goods. Above all, it is worth putting the notion of size into perspective depending on the diversity of seaboards: a small Chinese port would be likened to a large South American harbour. Consequently, using tonnage to differentiate between large seaports and medium-sized ones is specifically regional (COMTOIS et alii, 1993)" [START_REF] Loubet | Logiques d'acteurs et développement territorial dans les villes portuaires moyennes. Le cas du Havre et de Klaipeda[END_REF]. Associating these two objects with irregular outlines, therefore, adds complexity to the analysis all the more so as, according to C. [START_REF] Ducruet | Les villes-ports laboratoires de la mondialisation[END_REF], the port-city itself (of whatever size) is not subject to a consensual definition. As a consequence, in a context where urban-port configurations vary greatly from one country to another, it is difficult to define what "a mediumsized port-city" is. There are many port-city typologies. The approach taken by J. MARCADON (1997) combines size, function and relationships with the urban environment; M. HESSE (2010) cross-references the dynamics of port activity (decline, stability, growth, etc.), organizational strategies, the impact on the hinterland and the regional economy.

In the context of this article, we shall take another look at the criteria of the research project XX. "Medium-sized port-cities" have been defined according to a dual component (urban and port) in accordance with conventional quantitative indicators (taking account, however, of national specificities). The medium-sized port-cities included subscribe henceforth to the following model (cf. Fig. 2):

• cities of between 100 000 and 250 000 inhabitants with maritime traffic exceeding 10 million tonnes; • and cities of over 20 000 inhabitants and fewer than 500 000 inhabitants with port traffic of between 10 and 50 million tonnes. The concept of governance becomes during the eighties a precious tool for the analysis of collective action. The port field does not escape this dynamic and also makes use of this concept: "More and more port reforms in the international arena are analyzed under the light of port governance. Meanwhile, some theoretical or base studies are emerging with focuses on the basic issues associated with port governance, such as the port governance model (BROOKS and CULLINANE, 2006), the relationship between port governance model and port performance [START_REF] Brooks | Assessing port governance models: process and performance components[END_REF], the interactions between institutions, port governance reforms and port authority routines (NOTTEBOOM et alii, 2013) and the common characteristics of the port governance reform processes (BROOKS et alii, 2017) (1996), BAIRD (1995BAIRD ( , 2000)), SUYKENS (1995); NOTTEBOOM (many references); HARALAMBIDES (2017,2018); (KNATZ, 2020); EVERETT (many references); HEAVER (1995) Privatisation, competitive pricing, productivity and performance of port governance Economy (Source: based on work by [START_REF] Daudet | Approche par le compromis des pratiques de gouvernance portuaire : pistes prospectives à partir de l'étude de cas des ports du Havre et d'Abidjan[END_REF]; translation by the author).

However, despite this lack of consensus and the protean nature of the concept, "governance" is a central element in the analysis of a port's development process: "Governance principles are most important in the case of ports. Ports are critical infrastructure for an economy, contributing to the realization of trade and movement. At the same time, port management, operations, and development are capital intensive, consume (public) scarce land, generate externalities (noise, emissions), and involve many decision-makers and stakeholders such as the port authority, terminal operators, rail operators, trucking companies, logistics providers, and port-cities " (PALLIS, 2022, p.281).

So we share Jason Monios' observation, although a different conceptual apparatus is mobilized in our work.The author considers that current static models of port governance place too much value on hierarchy and internal formal structures (mostly the government, the port authority and the port terminal operator), to the detriment of explicitly including informal organisations (e.g. citizens and environmental groups) and external stakeholders (e.g. carriers, shipping regulators) in a more dynamic conceptualization (MONIOS, 2019, p.26).

1.3 The qualitative analysis as preliminary element to modelling.

Our work is based on the study of fifteen ports. In this paper, we shall deal succinctly with six typical cases chosen in view of their closeness similarities in terms of governance or conversely their major differences: Le Havre, Nantes-Saint Nazaire, Dunkirk (France), Gdynia (Poland), Klaipėda (Lituania), Hamina-Kotka (Finland). This phase of the survey is based on individual, anonymous interviews with leading stakeholders. We believe that leaders (NEGRIER, 2001;[START_REF] Faure | Dynamique intercommunales, leadership et territoire[END_REF]BARAIZE, 2001) shape governance, influencing decision-making and cooperative dynamics over and above other stakeholders. In fact, the "territorial configuration" (BARAIZE, 2001) [here urban-port] is strongly determined by the personality, methods and aims of the leaders in charge. Territorial construction, actor learning, decision-making and leadership structuring all seem to be correlated. (LOUBET, 2019). In order to define the level of leadership and the role of port stakeholders we shall transpose the method used to other configurations: the analysis of inter-municipal governance of the conurbations of Toulouse and Le Havre (XX, 2012 and 2019), respectively ranked 4 th and 15 th French cities in terms of numbers of inhabitants.

BROOKS [START_REF] Brooks | Devolution, Port Governance and port performance[END_REF], 2017); BROOKS & PALLIS (2012); PALLIS (many references); DE LANGEN (2007LANGEN ( , 2017)); TALLEY (1998TALLEY ( , 1994)); LAM (2016); VERHOEVEN (2012VERHOEVEN ( , 2017) ) Divestment, deregulation, corporatisation and stakeholder management in port governance

Management

As a consequence, in the port context, the level of leadership materializes the position of actors according to their institutional standing, but above all the role played in the port decision-making process. Their presence in the chain of command in the different bodies has been identified as well as their levels of participation in (leadership) decisions. Here, the informal rationale and stakeholder representations impose themselves on official organizational charts and our assessment is essentially the fruit of an analysis of the contents of the interviews [START_REF] Drisko | Content analysis[END_REF] and the mixture of responses from the actors as a whole to the questions below:

• position occupied by the actor questioned (What is your job, your role in the organization? / Your degree of investment / The status of your organization in the port?...); • decision-making process and governance (How are decisions made within the port? And by whom? / How could the decision-making mechanism be improved?...); • What is the nature of the forces present? (Who are the stakeholders or groups of leading players / The other groups of actors? / The stakeholders at the centre of the organization? / The actors at the margin distancing themselves from the official objectives of the port project? / Difficulties in finding agreements? Which? Why? Who is opposed to whom?) / What is your position in this process? / According to you, how are you identified by the other actors? / What influences do the different groups of actors have on the port dynamics?...); • Relationships with the other territories, other territorial scales, management of the portcity interface; • impacts of municipal policies on port activities;

• conversely, we questioned the elected representatives on the impacts of port activity on the development of the territory for which they are responsible; • etc.

In total, following the creation of this interview guide, 33 interviews were conducted with leading stakeholders from the six ports. These interviews were then encoded and a textual analysis performed on the stakeholders' discourse so as to identify the nature of their relationships, their games and strategies. During this stage, one textual analysis software package ("NVivo")3 was employed. The interviews were processed by means of contents analysis followed by a comparative analysis highlighting the convergences, divergences and strong trends.

As regards the following stage, it was based on the creation of a computer model. It aimed to align the representation of each of the port governances so as to be able to compare them. In this context, the use of a model based on the theory of graphs enabled us to easily model the actors and their interactions. For example, the graph in Fig. 3 represents four stakeholders (municipal scale, port authority, state, unions) and two relationships ("mediation", "control"; the nature of the relationship appears here as a result of the prior qualitative analysis). This made it possible to summarize the essential information obtained from the interviews as well as to standardize it for each of the ports. The model expressivity should be able to represent all the characteristics of the interviews deemed necessary for the comparison of governances of the different ports.

Authors, 2023

Work was done to define an adequate graph model. This is a directed graph because some of the stakeholders' relationships have a direction. They are "unilateral" relationships (versus "bilateral": cf. Table 3): in Fig. 3, for instance, the stakeholder state controls the stakeholder port authority, whereas the reverse has not been proven. Reciprocity in relationships is not necessarily the rule. Moreover, hyper-graphs must be used (AUSIELLO and LAURA, 2017; DEVEZAS and NUNES, 2020) since relationships are not solely relationships between two stakeholders. They can, for instance, be relationships between stakeholder A and two stakeholders B and C (e.g. Fig. 3 the "mediation" relationships, where the stakeholder at the municipal scale carries out a "mediation" between port authority stakeholders and port workers' unions). All these types of relationships were retained, and their descriptions are available Table 3. For each of the relationships, the degree of intensity is specified (as for the nature of the relationship, the degree of intensity of the relationship appears as a result of the previous content analysis). In fact, the graph is labelled by the type of relationships (arcs) and the category of stakeholder (vertices), and weighted, on the one hand by the intensity of relationships (arcs) and on the other by the level of leadership on the part of the stakeholders (vertices).

The formalization of this model is presented in the following part (1.4). It is clear that the use of directed hyper-graphs will have a significant impact on the definition of the metrics thereafter (AUSIELLO and LAURA, 2017; DEVEZAS and NUNES, 2020), but it is necessary in order to capture as well as possible the semantics expressed by the interviews. (2016), where the use of graphs to model maritime networks provide new insights into the history, spatial distribution, vulnerability… of such networks. Semantics can also be associated with graphs quite naturally. In [START_REF] Pinson | Representation of interdependencies between urban networks by a multi-layer graph[END_REF] an example of the use of semantic graphs for urban networks can be found, it models technical networks (electricity, water, railway etc.) at different levels of granularity (component, infrastructures, urban system) and the semantic interaction between these elements (inter-dependencies) potentially spatialized. The model based on a graph (G) proposed in this paper takes account of all the actors identified in the framework of the qualitative analysis and represents them through the definition of its vertices.

Let V port be all the stakeholders for a given port.

Each stakeholder belongs to a certain category, let C a be all these available categories.

A relationship between stakeholders is represented by an arc between the corresponding vertices, let E port be all the relationships identified between the stakeholders of a given port. Each arc is labelled by the appropriate type of relationship, let T r be all the available types of relationships. A level of leadership is potentially associated with each stakeholder (N 1 = {1, ..., n}, n ⋲ ℕ) and to each arc a level of relational intensity (Ni = {high, moderate, low, easy, difficult}).

Our qualitative analysis (section 1. Therefore, for a given port, the formal directed weighted hyper-graph: G port = (V port , E port , α, γ)

with:

E port = {(T, HE) ∶ HE ⊂ V port T ⊂ V port }
T is called tail (source) and HE is called head (target) of the hyper-arc (T, HE), and in our case, T ≠ HE. (DEVEZAS and NUNES, 2020)

α : V port ⟶ C a , N l γ : E port ⟶ T r , N i
If one instantiates this model on the example of the port of Le Havre one obtains the Fig. 4 representation (the graphs representing the six ports studied are available in the annex). For reasons of legibility the decision-making scenes and the relationships associated are not represented. The vertices 'H' are not stakeholders, they are only introduced to help with the visualization of a hyper-arc and linked to the use of an image library 6 . Here, for instance, the municipal scale stakeholder carries out a "mediation" ("difficult", solid line) between the port workers' union and the port authority. The municipal scale stakeholder conducts another mediation ("easy") between the port authority and the private operators sector. These two mediations are presented alone in Fig. 5 to reinforce the explanation of the representation of the hyperlinks in the application. Let us add that this article is a general presentation. Its format does not allow us to present all the possibilities of the model. We have therefore chosen to present and discuss only certain indicators (cf. 2.2). In the context of this first paper, we chose to momentarily evade the institutional dimension by not mentioning the decision-making scene (both formal and informal).

Yet the analysis of port governance might gain from a set of metrics relative to this dimension (cf. conclusion).

Results and discussion: a model for comparing port governance

Our work aims to identify over time the processes of the most suitable governance for encouraging cooperative dynamics and, as a result, the development of ports and their territory.

The first sub-part (2.1) will highlight the link between governance and development concisely. A lengthy development of a correlation at the basis of local development and territorial development, as well as economical geography, and generally speaking of regional sciences, is out of scope of this paper. The second (2.2) will attempt to demonstrate the benefit of computer modelling making it possible to qualify the governance of the ports studied.

Governance and development

This part analyses the way relationships between stakeholders mould the ports and impact territorial development. Indeed, territorial development is based here on two main engines: governance and production [START_REF] Torre | Les moteurs du développement territorial[END_REF]. According to the author, relationships between stakeholder are at the core of these two dimensions. They materialize a territorialized mechanic producing decisions, that defines governance. They structure the production according to two central decisions: technical tinnovation, and the systematic character of local relationships. Therefore the stakeholders capacity to adopt cooperative behaviors at the heart of the territories appears as a critical element.

As mentioned above, the approach proposed is in keeping with the philosophy of "territorial development" which associates territorial governance and development: "Work in the field of territorial development focuses more on the role of the mechanisms of governance on territorial dynamics. What the many definitions of territorial governance share is putting the accent on forms of partnership between public and private actors, on the interactions between levels and the role of networks, but also on the modalities of implementing participation in a context of growing partnership heterogeneity, leading to a "mosaic of stakeholders" (TORRE, 2011)" (REY-VALETTE and MATHÉ, 2012).

In this we take a particular interest in the first (organizational and territorial) dimension of the concept of governance suggested below by M.R. BROOKS and K. CULLINANE ( 2007), inside their works on port research. Indeed, these authors point out four main components which structure the concept and direct the work in this field: "In essence, therefore, the systems, structures and processes that organize groups of individuals to a common purpose can be perceived as constituting the governance structure of the group, society or voluntary organization. At the other end of the continuum, the legislation and regulations that the government imposes on a business or not-for-profit entity also shape the governance structure within which such organizations operate. The structures and processes put in place by national laws, such as the requirements for open procurement processes, do form part of the governance of government. Most frequently, however, the term governance is used to refer more specifically to corporate governance, an area that is widely researched" (BROOKS and CULLINANE, 2007, p.10-11).

In addition, we shall make use of the term "good governance". At the end of the 1980s, the expression "good" governance entered the vocabulary of the international community (BANERJI, 2015). Very soon there evolved a correlation between "governance" and "development". K. ANNAN, former Secretary General of the UN, stated 7 that good governance was without doubt the most important factor in the promotion of development. Developing good governance was to become a qualifying principle for certain countries, especially developing ones, for them to obtain subsidies.

In the context of this paper, however, we shall not retain all the criteria that structure the notion of "good governance" according to the institutions, most of which are international (World Bank Institute, UN, OECD, etc.). When applied to the port, we consider that "good governance" is a relational and decision-making system that encourages cooperation between port actors and hinders conflicts. As mentioned above, we consider that "good governance" is favourable to port and territorial development. In this we promote the organizational dimension of the concept and maintain the very essence of this notion in its territorial dimension: the necessary cooperation between actors for the purpose of development.

Thus, with regard to the relationships we have identified in Table 3 and Fig. 4, it seems that there are relationships that are more or less favourable to the emergence of "good governance. Table 4 presents these links observed in the different port terrains. Our objective in this paper is to demonstrate the interest of computer modelling in this applicative framework, through the application of metrics, on the graphs representing ports. In the first instance, the metrics we propose are relatively simple but make it possible to validate the approach. Over time, metrics will be proposed that will be based more on the semantics of the relationships. They will make it possible to tend towards the final objective of characterizing "good" or "bad" governance (more precisely "governance unfavourable to territorial development"). We shall give some pointers in the conclusion to this paper.

Selection of two metrics: density and S-metric

The number of stakeholders and the relationships between the different stakeholders vary significantly depending on the port. This variation gives us information on the intensity of governance, in other words, stakeholder participation in decisions which commit port development in absolute values (the degree of openness to a multiplicity of stakeholders) and in relative values (to what extent stakeholders communicate and interact with one another). The idea is to be able to categorize the ports and be able to tell if two ports have a similar intensity of governance or conversely, a very different one. The metric we have chosen to do this is density.

The general definition of density on a simple graph G = (V, E) is expressed as follows:

d = (2 * |E|) / (|V| * |V -1|)
Note that, if | | expresses cardinality, |V| *(|V|-1)/2 is the maximum number of edges in a graph. As a consequence, the density is a rational number of between 0 and 1. As our data are represented by hypergraphs, we made the choice for these first results to transform the data for the calculation of this metric. A hyperedge is replaced by as many simple edges as necessary, e.g. the hyperedge (A, {B,C}) will be replaced by (A, B) and (A, C). This transformation was already suggested in AUSIELLO and LAURA (2017) in particular.

The results are presented in Fig. 6. We note that there are ports for which links are more developed. In the light of Fig. 6, and especially Fig. 7, Table 5 and the S_metric, which will be presented later, we shall demonstrate that a segmentation in two groups can be implemented. A "group 1" (Dunkirk, Hamina-Kotka and Gdynia) characterized by more developed links; a "group 2" (Le Havre, Klaipeda, Nantes-Saint Nazaire) not as well equipped in the matter.

The density, however, provides only one piece of information on the intensity of relationships generally and none on the distribution of this intensity. Another metric may complete this first indicator -the S_metric [START_REF] Li | Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)[END_REF]. Using this metric can identify the prominent stakeholders in a governance. This makes it possible to identify if the decision is in the hands of a small group of stakeholders (low S-metric) or, on the contrary, if the governance is more participatory, associating a wider circle of stakeholders (high S-metric). The S_metric is calculated as follows:

s = ∑ (i,j) ∈ E d i * d j
with d i and d j the degrees of the corresponding vertices, in other words, the number of respective connections of these vertices. The results are presented in Fig. 7. In order to verify the results relative to this metric we calculated the number of hubs contained in these graphs for every port, that is to say the number of prominent stakeholders. The definition of a hub has not really been normalized so we chose to apply the following definition:

Let avg(d g ) be the average of the degrees calculated on a graph, let avg Gn (avg(d g )) be the average of the avg(d g ) on the graphs processed as a whole. A vertex is considered as a hub if its degree is above this threshold avg Gn (avg(d g )).

The results are presented in table 5.

Stakeholders acting as hubs in the ports (> Average of the Average Degrees, avg Gn (avg(d g )) = 4.33162) The analysis of the S_metric (Fig. 7) confirms the segmentation set out beforehand in view of the density indicator. In effect, the S_metrics of Dunkirk ( 2136) and Hamina Kotka (3187) are very high compared to group 2 (723 on average for this group). Consequently, 8 and 6 hubs respectively are identified (Table 5) in these two ports. The connectivity is therefore high and shows a greater distribution.

At Dunkirk, the overall intensity of relationships is therefore strong (high density) and the port governance is exercised by means of relationships between a large number of stakeholders (high S_metric). The capacity to influence the port dynamics is therefore shared, although a chain of command can be seen among its leaders. Consequently, the institutional stakeholders are in a rather conventional position of power in the leader rankings (cf. Fig. 12 in the annex) in the different ports observed (the state ranked first, the port authority second, and finally the local scales, the municipal scale or inter-municipal scale in 3 rd or 4 th place). However, the analysis of the S_metric and degrees enable us to go further than the analysis based solely on stakeholder leadership. It highlights finer dynamics; the power play around stakeholders not necessarily occupying the highest positions in the ranking. Consequently, the local port group (Norlink) (degree 12, cf. Table 5), the private operators in the port sector (degree 11) constitute hubs around which gravitate a large number of stakeholders. Interestingly enough, among the six ports studied, the port of Dunkirk is the only one where the local port group stakeholder constitutes a hub which, more significantly, is of great importance (cf. Table 5 and Fig. 8).

Fig. 8 Actors' degree compared by ports

The group 2 ports (Le Havre, Klaipedia and Nantes-Saint Nazaire) are of the same order of magnitude on these metrics. They show relatively low S_metrics compared with the ports in group 1 (cf. Table 5). Furthermore, there is a similar number of hubs (4) in these three ports, which signifies that connectivity is not very uniform, as well as being poor (see the result in terms of density (cf. Fig. 6). These elements corroborate the results of our qualitative (XX, 2022;XX, 2021;XX, 2020;XX, 2019;etc.).

To finish our analysis and so as to territorialize further the subject in question we shall propose a concise illustration of the use of the S_metric with a short study of the cases in group 2.

At Le Havre9 , governance is structured around some stakeholders who concentrate powers and struggle to exchange words outside a limited circle (S-metric: 767, confirmed by a number of limited hubs ( 4)). The state appears to be the most influential stakeholder, notably through the management board of the port authority (degree 10, cf. Table 5) which it controls. The Prime Minister (Édouard Philippe at the time of processing the data) was formerly the mayor of Le Havre and this reinforces the particular interest shown in the development of this port of national strategic interest10 . The municipal scale (Le Havre) (degree 9, cf. Table 5) comes in second position. A large number of informal relationships between the president of the port authority of the Major Seaport (Grand Port Maritime) of Le Havre11 and the mayor (also president of the inter-municipal scale of greater Le Havre (le "Grand Havre") enables the latter to weigh heavily on strategic directions involving the port. In this context, the "quadripartite" (a powerful decision-making body, informal, but known to all) materialises the cooperative relationship between the representatives of the State (in particular the Port Director) and the Mayor:

"The Quadripartite, a meeting which takes place three or four times a year and brings together the mayor of Le Havre, the president of the CODAH12 , who is in fact one and the same, the president of the CCI and the president of the port board of directors […]. This is a powerful, yet totally informal, decision-making body […]. On sensitive questions, where a choice must be made between several scenarios, where arbitration is absolutely essential, that's what it's there for. Sometimes there's no need to be four -in fact there are three -and sometimes it's just city and port". A councillor of Le Havre, 2017. "That's where [at the quadripartite] discussions take place and then, depending on the decisions, well, decisions are taken at that level". A Grand Port Maritime of Le Havre representative, 2017.

This dynamic is strengthened in view of the close ties between the current mayor and his predecessor, prime minister at the time the data were processed and returned once again as mayor since then. In second place come the port workers' unions (degree 7) after the state and the municipal scale which are particularly influential. According to the stakeholders interviewed, their leadership is very important in view of their capacity to carry out national blockades and the effects that strikes have on the port dynamics. This tendency is especially identifiable in the discourse of the private operators in the port sector representatives (degree 7):

« If the unions don't like the State's decision, you've got a month of strike action, a month without work and a few million euros lost... so I would have said that, in terms of the forces at work: effectively, the unions come first.... ». UMEP 13 representative, 2017.

In the case of the port of Nantes-Saint-Nazaire 14 it reflects similar dynamics in terms of density and a concentration of relationships to just a few stakeholders (S-metric 618, number of hubs 4). This French port subscribes to the GPM model, with at its head the port authority's management board heavily controlled by the state. Yet given the hubs calculated and the degrees of connections by the stakeholders associated, the importance of the state or its relay the port authority does not emerge (cf. Table 5, degree 4 versus 10 for Le Havre). In effect, contrary to the case of Le Havre, the model of governance at Nantes-Saint-Nazaire was developed more in line with "horizontal cooperation". An example of the dynamics is illustrated in the Be My Port cluster (cf. Fig. 10 in the annex) associating local economic stakeholders in particular.

The recent creation of this port promotion cluster seems to have had an impact on the decisionmaking process. This local initiative brings together public and private players 15 in an effort to promote and unite the port community. However, in addition to the promotional dimension, informal exchanges and monthly meetings foster proximity between stakeholders and the process of collective reflection on port strategy (XX, 2022). " We had the impression that everyone was working in their own corner: the CCI on one side, the Port on one side, us on the other, and the local authorities on the other. It's true that if we can get everyone working together, it would be fantastic! It's not going to be easy, because it's not easy to coordinate all that, but if we set ourselves a few major strategic priorities and we all agree on them, and then put in place the actions we need to take in relation to the strategic priorities we've set, we'll have taken a big step forward ". Lastly, the institutional management of the port of Klaipeda (S-metric: 786, number of hubs 4) has certain similarities with that of French ports. A greater presence of central government can be noted. The state (degree 5), therefore, represented by the port authority17 (degree 7) enjoys very strong leadership in the port.

"We are a staterun business and were set up by the ministry of transport. We are like a subsidiary of the ministry of transport". A manager of Klaipeda's port authority, 2017.

In this context, the municipal scale (degree 9) has experienced difficulty in guiding the development of the territory in a direction that conforms to municipal policy.

"If I said that we have no impact on the port, this would almost be the truth. Lithuania only has one port. It's a staterun port and the municipality has no rights over it; other than the fact that we are convened to two advisory councils. The port council in which we don't have the right to vote. There's also the council for development in which we have four seats out of 23 and in which nobody pays any attention to us". The deputy mayor of Klaipeda, 2017.

Here, the qualitative dimension of our analysis should be developed so as to explain certain dynamics observed, which elude the sole breakdown in terms of degree, which might appear paradoxical. In effect, as in numerous territories, local and central powers can be opposed on different issues concerning the port (land tenure, conflicts of use). Therefore, if we return to the terminology in Table 3, the relationships here have been qualified as "are in stiff competition".

The difficulties in local governance can also be observed in the relationships between the municipal scale and the Organizations for the defence of business interests (degree 8) (relationships qualifying as "difficult negotiations", cf. Table 3). These (associations, lobbies) take an active part in the management of port territories. They negotiate and cooperate with the government (cf. Fig. 14 in the annex where relationships between Organizations for the defence of business interests and the state are qualified as "easy negotiations", "strong influence"). With central government, these organizations constitute coalitions which have the effect of marginalizing the municipal scale concerning the challenges relating to port development. In this context, "the lack of concertation, the imbalance in the capacity to influence between the port authority and the municipality encourage the municipality to adopt a defensive stance (perhaps to the detriment of port development)" (LOUBET and SERRY, 2020, p.648). As a consequence, this case illustrates the need to take into consideration "the relationships" and the "level of intensity" (cf. Table 3) in the analysis of the connections between stakeholders. In effect, the municipal scale seems particularly linked to other stakeholders. Yet these relationships, qualified as "unfavourable to good governance" (cf. Table 4), testify to a relational system where the organization is in a weak position faced with the other leaders in coalition.

Conclusion

Considering that cooperative dynamics and local development go hand in hand, we have analysed the organizational and territorial dimension of the governance of six European ports. By means of a comparative approach, we qualified and measured the intensity of relationships between stakeholders. Our previous work provided us with rich material, enabling us to compare the governance of two or three ports. However, as we stated in the introduction, given the delicate nature of the relationships, comparison on a larger scale (multiplying the cases) seemed difficult. As a consequence, it was appropriate to resort to computer modelling.

In the framework of this paper, we proposed a model based on graph theory. This made it possible to take into account the essential characteristics of port governance, but above all to compare them semi-automatically. The first piece of work (still being developed) presents a certain number of metrics. In order to illustrate the model, we chose two principal metrics which validate the general philosophy of the approach. In fact, we managed to establish a classification of the ports in two distinct groups according to their capacity to develop a more or less participatory governance in absolute as well as relative values (cf. section 2.2 metrics Density and S_metric). The analysis of the intensity of relationships was completed by a study of the number of connections between stakeholders (distribution of degrees) and the number of hubs per port. The use of the S_metric coupled with these other quantitative indicators made it possible to identify the extent of active participation in decision-making, by the major stakeholders and those with little influence. If the density calculation allowed us to establish a classification of the general intensity of relationships between stakeholders, the S_metric put the calculation into perspective in terms of distribution. In effect, if a strong S_metric provides information about the strong distribution of the intensity of relationships, a weak S_metric implies that the intensity of the relationships is concentrated on a limited number of stakeholders. We confirmed this metric by calculating the number of hubs, which confirms the prominence of certain actors. This part of the analysis via these metrics enabled us to carry out an initial semiautomatic comparison of ports. The model also facilitates a comparative reading (in terms of density, S-metric, and semantic metrics, etc.) of the role of a specific stakeholder (the state, the region, the local port group, etc.) in the governance of the different ports studied.

Lastly, a more territorialized approach aimed to show the relevance of the model by reexamining our previous results (XX, 2022;XX, 2021;XX, 2020;XX, 2019;etc.) against the first metrics selected. This short piece of work shed light on numerous convergences. Paradoxical elements also appeared, however, which forced us to refine and nuance our previous conclusions (which did not benefit from the inputs of the model presented in this paper). Stakeholders we had judged to be secondary in our previous analyses appear to play a significant role in local governance when one looks at it against the indicators in question. The case of Dunkirk is particularly illustrative. The local port group, initially identified as a very subsidiary actor, appears as a central vertex of governance notably multiplying the "mediation" relationships between stakeholders. Besides the inputs of the two metrics studied, this example shows how important it is also to take account of the nature of the relationship and therefore the semantic aspect. The example of Klaipeda is also part of this observation. As a consequence, the model could be upgraded subsequently by the introduction and processing of semantic metrics, that is to say metrics taking account of the semantics of the arcs, in other words the types of relationships involved. It would therefore propose to the expert a semi-automatic categorization of the typologies of governance, finer, more in accordance with the criteria. Identifying the greater or less favourable relationships for the emergence of "good governance", as presented in Table 4, constitutes a pre-requisite to this approach. Similarly, as pointed out in section 1.4, in the context of this first paper, we chose to momentarily evade the institutional dimension by not mentioning the decision-making scene (both formal and informal). Yet the analysis of port governance might gain from a set of metrics relative to this dimension. "The number of decision-making scenes", "the "formal" or "informal" character of decision-making scenes", "the number of participants taking part in these decision-making scenes", "the level of leadership of the participants", "the presence of "decision-making scenes governing relationships", etc. provide multiple possibilities for cross-referencing and interpreting the results.

APPENDICES

Fig. 1

 1 Fig. 1 General workflow

Fig

  Fig. 2 Medium-sized port-cities in Europe

Fig. 3

 3 Fig. 3 An example of interactions between stakeholders.

  3) enabled us to identify C a = {State, Port Authority, Municipal Scale, Inter-municipal Scale, Port Workers' Unions, Private operators in the port sector, Organizations for the defence of business interests, Shipowners, Intermediary territorial level, Local port groups, Decision-making scene for regulating relationships, Strategic council, Economic and Social Council, Local bodies, Industrial firms} and T r = {Control, Defends the interests, Is in competition, Is included, Influence, Mediation, Same person, Same organization, Negotiations, Partnership}. 4 5

Fig. 4 Fig. 5

 45 Fig. 4 Graph of the port governance of Le Havre.

Fig. 6 .

 6 Fig. 6. Density of the graphs relative to each port, taking account of all the vertices

Fig. 7

 7 Fig. 7 Calculation of the S_metric for each port

Fig. 9 2023 Fig. 10 2023 Fig. 13 2023 Fig. 14

 9202310202313202314 Fig. 9 Graph of the port governance of 'Hamina Kotka

Table 1

 1 Summary of the main multidisciplinary fields of study in the scientific literature on port governance

Main academic authors Fields of study Main discipline

	BRAUDEL (many references); MARNOT (2012); DUCRUET (many references); LEROY-LADURIE (1978); BUCHET (1998)	Accumulation and transmission of port governance knowledge	History
	VIGARIÉ (many references); BIRD (1971); CHARLIER (1985);
	SLACK ET COMTOIS (many references); WANG (2017); DEBRIE (2017); LACOSTE (2008); RODRIGUE (2017, 2018);	Spatial scales, decentralisation and territories of port governance	Geography
	SERRY (2019); LOUBET & SERRY (2020, 2022a, 2022b)
	THORRANCE (2014, 2019); GROSIDIDIER DE MATONS (1999); THE WORLD BANK (many references); JUHEL (2001); BAIRD (2004); CHAPON (1993)	Regulatory engineering and financial optimisation of port governance	Finance
	GROSDIDIER DE MATONS (1971, 2012); REZENTHEL (many references); SUPIOT (2008); DUPUIS (1978); LICHERE ET SALAS (2020) MARTOR (2007); SANCHEZ (2020); SANCHEZ & PEREZ-	Legal construction and organisation governance of public-private partnerships in port	Law
	CASS

Table 2

 2 Survey methodology

		Studied ports
	Period	(bold : ports studied in this	Methods and tools	Type of actors surveyed
		paper)
			For the 6 ports : 33 interviews
			For the15 ports : 80 semi-directive	representatives of port authorities
	2017-		interv iewss
	2019	Le Havre
		Dunkirk	These interviews, under the cover of anonimity, have been recorded then	port worker union representative
			integrally transcribed.

Table 3

 3 Definition of the different relationships between stakeholders Modelling of port governance using a semantic graph Modelling port governance means being capable of capturing the relationship between different stakeholders that it comprises. Graphs, as well as hyper-graphs, come from discrete mathematics, and have established themselves as an unrivalled tool in the modelling of interactions[START_REF] West | Introduction to graph theory[END_REF][START_REF] Ausiello | Directed hypergraphs: Introduction and fundamental algorithms[END_REF]. Examples are countless in the literature. For instance in the field of maritime transport, one may cite the multidisciplinary book edited by DUCRUET

	Nature of the	Relationship	Type of	Intensity level	Definition
	relationship		relationship
			(observation)
	Powers	"Control"	Unilateral	Low, Moderate,	The organization (most often
				High	represented by its leader) dominates
					another organization. It can compel it to
					do what it wishes.
		"Influence"	Unilateral	Low, Moderate,	The organization acts on the decisions of
				High	another and in part determines the
					choices thereof.
	Powers &	"Mediation"	Unilateral	Easy, difficult	The organization acts upon the
	Cooperative				relationships fostered between other
	Dynamics				organizations in order to limit conflicts
					and encourage cooperative dynamics.
	Cooperative	"Partnership"	bilateral**	Low, Moderate,	Relationship based on (or aiming to
	dynamics			High	establish) a close collaboration between
					organizations associated within a system
		"Defends the	unilateral	Low, Moderate,	The organization preserves and
		interests"		High	strengthens the position of another
					organization including vying with
					competing organizations (or antagonistic
					interests).
		"Negotiations"	Bilateral	Easy, difficult	The organizations have discussions in
					order to handle a business deal, a
					project, negotiate the terms of a contract,
					etc. The objective being to reach an
					agreement.
		"Is in	bilateral	Low, Moderate,	Several organizations target the same
		competition"		High	object or pursue the same goal
					(including on the symbolic market:
					"powers")
	Participation	"Is included"	bilateral		An organization (embodied by its
	in decision-				representatives) sits or participates in a
	making				(formal or informal) decision-making
	scenes				scene
	*"unilateral": non-reciprocal
	** "bilateral": reciprocal.
				Authors, 2023
	1.4

Table 4

 4 Assessment of the quality of governance according to the types of relationships between organizations (vertices)

	Relationships

favourable to "good governance" Varies according to the contexts and depending on the original stakeholder Relationships which are unfavourable to "good governance"

	"Partnership"	"Control"	Difficult "negotiations"
	"Mediation"	"Influence"	"Is in stiff competition"
	Easy "negotiations"	Is in low, moderate "competition"
	"Defends the interests"
		Authors, 2023

Table 5

 5 Synthesis of the hubsnumber and contents per port

	Port	# Hubs Stakeholder	Degree	Average Degree of Hubs
	Le Havre	4	Port Authority	10	8.25
			Municipal Scale	9
			Port workers' union	7
			Private operators in the port sector	7
	Klaipeda	4	State	5	7.25
			Port Authority	7
			Municipal Scale	9
			Organization for the defence of business interests	8
	Dunkirk	8	Port Authority	11	8.5
			Inter-municipal Scale	7
			Port workers' union	6
			Private operators in the port sector	11
			Organization for the defence of business interests	7
			Local port group	12
			Industrial firms	6
			Organization for the defence of business interests 2	8
	Nantes St.	4	Private operators in the port sector	7	6.25
	Nazaire		Organization for the defence of business interests	6
			Local associations	6
			Industrial firms	6
	Hamina Kotka	6	Port Authority	19	9.67
			Municipal Scale	7
			Private operators in the port sector	7
			Shipowners	7
			Industrial firms	11
			Municipal Scale 2	7
	Gdynia	4	Port Authority	14	11.0
			Municipal Scale	15
			Port workers' union	7
			Private operators in the port sector	8
			Authors, 2023

  13 Union Maritime et Portuaire du HAVRE: is a maritime and port federation which federate professional organizations of the Port of LE HAVRE14 In 1966 the ports of Nantes and Saint-Nazaire merged and became "Port Autonome" (self-governing port) before obtaining the status of GPM in 2008. It is the fifth French port in terms of overall traffic, third in terms of liquid bulk and possesses the largest methane terminal in Europe (Montoir-de-Bretagne). 15 The cluster comprises GPM management board, the Union Maritime Nantes Ports (UMNP), the Chamber of Commerce and Industry (CCI), the territorial authorities and their organizations (inter-municipal scale of greater St Nazaire region, inter-municipal scale of greater Nantes, Département and Region), and the PASCA association (Plateforme régionale d'innovation des Pays de la Loire who develops the competitiveness of businesses and territories by optimizing logistics and purchasing strategies).

  The manager of a port service company, 2019. " Territorial promotion means working with port stakeholders [...] it's one of the current issues, between the Union Maritime, the CCI... It's clear that everyone is working on economic development issues, there's the Regional Agency... So in the end, we're hunting in packs! The term is perhaps a little aggressive, but it's there.". A manager within the GPM, 2019.Therefore, if institutional organization of the GPM generally finds it difficult to integrate local businesses and industries in major decision-making scenes, it is different at Nantes-Saint Nazaire. As opposed to Le Havre, industrial companies (degree 6) and the private operators port sector (degree 7) have partly overcome this deficit by a clever use of representation provided by organizations for the defence of business interests (degree 6), as the cluster Be My Port 16 (see Fig.10in appendix). Similarly, in a territory that has experienced strong mobilization (in light of the domestic downturn) against the building of the Notre-Dame-des-Landes airport project, local associations (degree 6) are capable of bearing influence on urban-port governance, namely where environmental issues are concerned.

Research project XX

The organization HAROPA aims at merging, inside a large sea river port the three ports of the Seine axis, Paris, Rouen, Le Havre.

"NVivo", a qualitative data analysis (QDA) computer software package produced by QSR International.

This list of stakeholders is the result of the questionnaire survey (section 1.3). It enabled us to identify the key stakeholders and rank them according to their ability to influence port decisions (leadership). Among other things, we asked each actor interviewed to rank the leading stakeholder in their port, from the most influential to the least influential. We then cross-checked their results to obtain an overall ranking for each port.

Each port was the subject of a content analysis. The aim of this article is to formalize the analysis. We therefore identified the types of relationships observed in all the ports studied (as part of the XX research project and in our various publications) and categorized them in order to create the model.

Statement made by the Secretary General, Mr Kofi Annan, delivered on 28 July 1997 at the opening meeting of the International Conference on Governance for Sustainable Growth and Equity at the UN Headquarters On line: https://www.un.org/press/en/1997/19970728.SGSM6291.html

"it is because he defines the new referential that a stakeholder takes on the leadership of the sector by asserting his hegemony but, at the same time, it is because this stakeholder asserts his hegemony that his world view progressively becomes the norm"(MULLER, 1995, p.165). Therefore, through the active promotion of the referential [which could be understood according to the following terms in our case: it is necessary for the actors as a whole to cooperate in order to develop the port community] the mediator "acquires force and exerts influence by virtue of its capacity to specify the roles of various actors in a system of relationships, conferring on them not only meaningful identities, but also influence, since this system invariably embodies relationships of power"(HALL, 2015, p.240).

The GPM (Grand Port Maritime -Major Seaport) of Le Havre was the first port to be the subject of analysis of its governance in the framework of our study. During interviews with the leading stakeholders (2018), HAROPA (the organization aiming to merge the three ports of the Seine axis: Paris, Rouen and Le Havre, within a large fluviomaritime port) was never cited by the stakeholders who had an impact on the governance of the port of Le Havre. A second phase of interviews will update the results and certainly give more weight to HAROPA.

 10 The port of Le Havre is the first container port of France, second in terms of overall

traffic. 11 The 2008 law for port reform constituted an institutional change which impacted French ports in a differentiated way. The three self-governing French ports analysed in this paper became "grands ports maritimes" (GPM) or major seaports in English. They are among the seven GPM, state-owned establishments, which assume sovereign functions as well as development of the port domain,

which they own. 12 "Communauté d'agglomération du Havre": inter-municipal scale of greater Le Havre

However, this type of local cooperation remains delicate, and this cluster disappeared in 2022.

Under the direct trusteeship of the state through the Lithuanian transport ministry.

Authors, 2023

The local port group emerges as a central vertex in the governance of this port, notably multiplying the "mediation" relationships between the different stakeholders (cf. Fig. 12 in the annex). In effect, according to the work on "the cognitive analysis of public policies" the process of "mediation" (MULLER, 1995, p.153) is accompanied by a take-over of power by the "mediators" 8 [START_REF] Hall | Cognitive Approaches: A French Touch: Three Anglo-American Perspectives on French Policy Analysis[END_REF][START_REF] Muller | Les politiques publiques comme construction d'un rapport au monde[END_REF]. As for the port authority (degré 11), it is once again positioned in the top three. As a consequence, simply cross-analysing, in terms of "leadership" and "S_metric", may prove to be highly instructive. There remains the port of Gdynia which has a strong S_metric (1965) but few hubs (4). This means that in this port there is a strong connectivity (also confirmed by the density value Fig. 6) which is not widely distributed. The analysis of these three metrics combined (i.e. density, number of hubs, S_metric) enables us to infer that the four stakeholders concerned are prominent ones for this port (in particular the port authority and municipal scale).


domain: info chim

text:
Alexis Front

Christine Mottet

Composition-dependent chemical ordering predicted in Pt-Ag nanoalloys

Pt

-Ag nanoalloys display an astonishing chemical organization depending on size and composition. It has been shown recently a reversed size-dependent stabilization of ordered nanophases [Pirart et al., Nature Communications, 2019, 10, 1982] around equiconcentration. We extend this study by a theoretical investigation on the whole range of composition showing a significant composition-dependent chemical ordering in Pt-Ag nanoalloys. At low silver content, the surface exhibits a strong Ag segregation coupled to a (2x1) superstructure on the (100) facets. By increasing silver concentration, the system displays an L1 1 ordered phase in the core, interrupted in a narrow range of concentration by a concentric multishell structure characterized by an alternation of Ag-pure/Pt-pure concentric layers starting from the surface shell to the core. Experimentally, the L1 1 ordered phase has been observed whereas the concentric multishell structure is lacking due to the difficulty of the experimental characterization.

Introduction

Nanoalloys is a term to designate nanoparticles of metallic alloys where both size and composition induce attractive properties in catalysis, optics or magnetism. [START_REF]Nanoalloys: From Theory to Application[END_REF][START_REF] Ferrando | [END_REF][START_REF] Nanoalloys | Synthesis, Structure and Properties[END_REF][START_REF] Ferrando | Structure and Properties of Nanoalloys[END_REF] For example in electrocatalysis, Pt-based nanocatalyst are vastly investigated in order to keep the high catalytic activity of platinum while in the same time reducing platinum content and therefore the global cost. [START_REF] Debe | [END_REF] Intermetallic Ag-Pt surfaces are highly active for electrochemical oxidation of formic acid 6,[START_REF] Peng | Advanced Functional Materials[END_REF] or in selective methanol oxidation to methyl formate. [START_REF] Wisniewska | [END_REF] Under the shape of nanoparticles, Ag-Pt stellated bimetallic nanoparticles 9 , core-shell nanoparticles 10 or nanocages 11 activity for methanol oxidation and oxygen reduction reactions. Concerning the direct synthesis of H 2 O 2 from dihydrogen and dioxygen, Ag-Pt octahedra have shown to increase by a factor ten the selectivity compared to pure Pt octahedra. 12,13 A crucial point to be used as electrocatalysts in a PEM fuel cell is their stability and durability in reaction conditions. 11,14,15 An other interesting aspect of Pt-Ag nanoboxes synthesized by galvanic replacement reactions are their optical properties and the possibility to tune the surface plasmon resonnance peak across the entire visible spectrum. 16 To better understand the relation between structure and properties we present here a theoretical study of the structure and chemical ordering of Pt-Ag nanoalloys as it has been done with other systems (Ag-Cu, Ag-Co, Ag-Ni, and Au-Ni nanoalloys). 17 The originality of Pt-Ag system resides in its hybrid chemical tendency. According to the bulk phase diagram [18][19][20] , Pt-Ag system displays a wide immiscible domain as in the case of Ag-Cu, Ag-Co or Ag-Ni systems but, in addition, it presents an intermetallic compound around equiconcentration with the L1 1 ordered structure. In general, face centered cubic metallic alloys at equiconcentration display the L1 0 ordered phase with an alternation of pure atomic plans in the (100) direction, as in the well known Cu-Au or Co-Pt systems. 21,[START_REF] Ducastelle | Order and Phase Stability in Alloys[END_REF] In the case of Ag-Pt, the L1 1 ordered phase forms an alternation of pure atomic plans in the (111) direction, as for Cu-Pt alloys, the only known one. [START_REF] Lang | [END_REF] To characterize the Pt-Ag nanoalloys, we have determined a realistic interatomic potential for the Pt-Ag system 24,25 able to reproduce the main chemical tendencies of the bulk phase diagram. As mentioned before, this system presents an immiscible gap rather extended in the Pt-rich side, and the L1 1 ordered phase at equiconcentration. This leads, in bulk alloys for intermediate compositions, to a phase coexistence between this ordered intermetallic phase and an almost Pt-pure or Ag-pure phase. At the nanoscale, the phase separation tendency is a driving force to surface segregation, and, in the same time, the L1 1 ordered phase could coexist depending on the concentration and the temperature. This raises to the main interrogation of this study: which chemical configurations are stabilized in nanoalloys ? A theoretical study at high temperature near the melting transition has been performed with the same interatomic potential on Pt-Ag nanoalloys for the three intermediate compositions AgPt 3 , AgPt and Ag 3 Pt showing the influence of Ag surface segregation on premelting. 26 Concerning the core, optical properties and chemical ordering of Pt-Ag nanoalloys have been addressed 27 and compared to other similar systems mixing coinage metals (Ag, Au) with (Pt, Pt) transition metals. 28 It has been shown that the remarkable L1 1 ordering was only favoured in small AgPt particles but the chemical ordering in the core does not induce specific optical signature.

Theoretical model and numerical simulations

The energetic model to represent the interactions between the atoms is a many-body potential based on a tightbinding framework within the second moment approximation (TB-SMA) of the density of states of d-band, well fitted for transition and noble metals. 29 The originality of this potential, in the case of Pt-Ag system, is to reproduce bulk phase diagram properties, notably the quite rare ordered L1 1 phase at equiconcentration formed by an alternation of pure (111) atomic layers [18][19][20] as in the better known Cu-Pt system. [START_REF] Lang | [END_REF][START_REF] Hultgren | Selected Values of the Thermodynamic Properties of Binary Alloys[END_REF] The TB-SMA interatomic potential has been modified to stabilize the specific L1 1 phase at equiconcentration by adding an attractive contribution with a gaussian shape. This Gaussian function is localized at the second neighbour distance where the mixed bonds are responsible of the stabilization of the L1 1 phase as compared to the L1 0 one. This original potential is written as follows with an attractive term (bonding origin) including a classical square root dependence with the neighboring atoms and a gaussian contribution centered on the second neighbors distance:

E Bond i = - ∑ j,r i j <r 3 αβ ξ 2 αβ e -2q αβ r i j r 0 αβ -1 -∑ j,r i j <r 3 αβ G αβ e - (r i j -r 2 ) 2 2σ 2 αβ (1)
and an empirically repulsive term of the Born-Mayer type:

E Rep i = ∑ j,r i j <r 3 αβ A αβ e -p αβ r i j r 0 αβ -1 (2)
where (α, β ) is the nature of the metal atom. In these expressions, r i j is the distance between the atom at site i and their neighbors at site j, r 3 αβ is a cutoff distance for the third neighbor distance and r 0 αβ is the first neighbor distance depending on the nature of the atoms.

The homo-atomic (α = β ) parameters are fitted to the cohesive energies, lattice parameters and elastic constants calculated by density functional theory (DFT). For the hetero-atomic interactions (α = β ), the parameters p αβ and q αβ are the average of the pure metal ones. Finally, A αβ , ξ αβ , G αβ and σ αβ are fitted to the formation enthalpies of the L1 0 and the L1 1 phases as well as the solution energies of one impurity in the matrix of the other metal. G αβ =0.0440 eV and σ αβ =0.1393 Angströms. More details on the fitting procedure can be found in Ref. 25 .

Monte Carlo simulations are performed in the semigrand canonical ensemble using the Metropolis algorithm [START_REF] Metropolis | [END_REF] which means that the total number of atoms in the binary system, the temperature and the pressure are fixed whereas the number of atoms per species can change. As a consequence, the concentration at equilibrium is a result of the Monte Carlo process. In the Metropolis algorithm, starting with the system in the "i" state, with energy E(i), one performs one atomic trial, which can be either moving an atom or permuting the chemical nature of an atom, we calculate the new energy E( j). If E( j) is lower than E(i), the new configuration is accepted and the simulation goes on starting from it, else, the " j" state is accepted according to the following probability:

exp - E( j) -E(i) + dn(µ B -µ A ) + 3 2 ln( m B m A ) k B T ( 3
)
where dn is the permutation sign equal to 1 if we permute an atom of type A into an atom of type B or -1 if we permute an atom of type B into an atom of type A. m A and m B are the atomic mass of the elements A and B.

In a standard Monte Carlo simulation with N atoms we run five to ten thousands macrosteps after leaving the system being equilibrated during two to five thousands steps. One macrostep consists in proposing randomly either displacements of atoms or permutations of chemical species, N-times. We usually start with a Pt-pure or Ag-pure system and increase, respectively decrease the chemical potential difference µ B -µ A in order to scan the entire concentra- tion range and we start the new Monte Carlo run for each chemical potential from the last configuration.

Finally we calculated the atomic local stress on each atomic site of the nanoalloys after optimizing the atomic structure at 0 K. The local atomic pressure formula is given by Kelires and Tersoff 32 :

p i = - dE i dlnV i (4)
where E i is the energy on site i and V i the atomic volume proportional to r 3 i j , j designating the neighbor sites of i. This leads to the following expression of the local atomic pressure on site i, expressed in GPa, when the energy E i is in eV/at.: [33][34][35]

p i = -400 q e (a α ) 3 ∑ j,r i j <r 3 αβ dE i 3dr i j r i j (5)
where a α is the lattice parameter of the metal of type α (Pt or Ag), in Angströms, and q e = 1.60219 10 -19 C is the electron charge to convert the eV/at. in GPa, knowing that the atomic volume in the face centered cubic structure is equal to the cube of dimension a α divided by 4 since there is four atoms per mesh. The local atomic pressure represents the trace of the stress tensor σ :

p i = - 1 3 tr( σ ) = - 1 3 ∑ i=1,3 σ ii (6)
It is well to recall that in elasticity the stress is positive for positive strain (extension) and negative for negative strain (contraction) whereas it is the opposite for a pressure since by convention a positive pressure is associated to a compression (i.e. contraction) whereas a negative pressure is associated with a tension (i.e. extension).

Results and discussion

We considered truncated octahedra (TOh) of 1289 and 2951 atoms which represent Wulff equilibrium shape on the fcc structure (the one of the bulk Ag and Pt metals) and because experiments have shown essentially fcc structures with that shape for PtAg nanoparticles. 24 We performed semi-grand canonical Monte Carlo simulations to determine isotherms by sweeping the whole range of composition of the alloy. The Monte Carlo simulations give an equilibrium chemical configuration including atomic relaxations for any compositions, at a given temperature, leading to what we call "isotherms" (Fig. 1). Thus we characterized the equilibrium chemical configuration of the Pt-Ag nanoalloy by distinguishing the silver concentration on different group of sites of the surface and in the core. Typically we distinguish vertex, edge, (100) and (111) facet sites on the surface (Fig. 1-top graph). In the core, we distinguish the concentric shells starting from the surface up to the center of the nanoparticle (Fig. 1-middle graph) and the core sublattices (Fig. 1-bottom graph), specially determined to characterize the L1 1 ordered phase. In fact we chose two sublattices in a (111) atomic plane and the other two in the next (111) plane, in a way to characterize the fcc atomic mesh but in order to distinguish the L1 1 phase. These two core site analysis are complementary to give a precise characterization of the chemical ordering in the core of the Pt-Ag TOh. All these specific concentrations are plotted as a function of the global silver concentration of the nanoparticle. These equilibrium concentrations, on well chosen characteristic sites of the nanoalloys, allow to give a precise idea of the chemical configuration of the nanoparticle at any global composition of Pt and Ag. Some of them are illustrated on Fig. 2. At low Ag composition, silver atoms occupy first the verteces, followed by the edges and (100) sites simultaneously. The silver edge concentration in Fig. 1 concerns two different kind of edges: the one between two (111) facets and the one between (100) and (111) facets. They are occupied differently. At low Ag concentration, only the one between (100) and (111) facets are occupied by silver, as shown in Fig. 2-(a). This is correlated to the occupation of the (100) facets. The (100) facets are characterized by an alternation of Ag-pure and Pt-pure atomic rows corresponding to the (2x1) surface reconstruction. This superstructure displays a plateau over a large range of composition (from 7 to 55% of Ag). In the meantime, the edges between two adjacent (111) facets remain Ptpure (Fig. 2-(a)). It is worth noticing that for this cluster size, the square (100) facets are made of an odd number of atomic rows which allows to form the (2x1) surface reconstruction and to fill the edges with Ag on the border of the facets. In case of an even number of atomic row on the facet, it would lead to a frustration between the (2x1) surface reconstruction and the filling of the edges. The origin of this peculiar chemical ordering on (100) facets comes both from the chemical ordering tendency linked to the L1 1 ordered phase and also from the misfit between the two elements bringing elastic strains. However such surface reconstruction depends on temperature and disappears at higher temperature as shown in Fig. 1 at 700 K where the plateau is attenuated. At 10 % of Ag, the edges curve exhibits an inflexion which concerns the edges between two (111) facets and which is correlated to the filling of the (111) facets by silver. Meanwhile (111) facets begin to be occupied by silver atoms. The Ag concentration on (111) facets is strictly linear up to 70 % corresponding to a Langmuir mode with random occupation sites, weakly temperature-dependent (see Fig. 1). We notice an inversion of the usual hierarchy of segregation from the lower coordinated sites to the higher coordinated sites, i.e. between the (100) and ( 111) facets, because the (111) facets are Ag-pure before the (100) ones. This effect persists at 700 K. This has to be compared with the surprising anisotropy of surface segregation in CuPt alloys surfaces 36 where the Cu surface enrichment is stronger on the close-packed (111) surface than for the more open (100) one.

At 34 % of Ag, (111) facets display a dimming of the Ag enrichment characterized by an inflexion of the curve. This phenomenon is correlated to the filling of the remaining Pt sites located at the outline of the (111) facets, i.e. in contact with the Ag edges (Fig. 2-(b)). The (2x1) superstructure on the (100) facets remains up to 55 % of Ag (Fig. 1-top graph) whereas all other surface sites are fully occupied by silver atoms.

All these particular chemical configurations on the surface shell of the nanoparticles may have an high impact on catalytic properties with different surface sites occupied either by Ag or by Pt atoms and surrounded respectively by Pt or Ag atoms.

Chemical ordering in the core is characterized by shell concentration (Fig. 1-middle graph) and core sublattice occupations (Fig. 1-bottom graph). At 34 % of Ag, the core displays alternation of some Ag-pure/Pt-pure (111) atomic planes corresponding to the beginning of the formation of a L1 1 ordered phase. By increasing the silver concentration, all the core is ordered according to the L1 1 phase up to 54 % of Ag (Fig. 2-(c)). On the bottom graph of the Fig. 1, the L1 1 phase is characterized by two sublattices occupied by Pt (α and δ ) and two occupied by Ag (β and γ).

Then, for a concentration slightly higher, around 60 % of Ag (Fig. 2-(d)), the nanoparticle is composed of an almost Ag-pure shell (only a few Pt atoms isolated on (100) facets remain) and a core displaying an alternation of Ag and Pt concentric layers. This multishell structure is characterized by an alternation of Ag-rich (grey) and Pt-rich (blue) concentric layers starting from a silver surface (n), then a Pt subsurface (n-1) and so on, as illustrated in the Fig. 1 (middle graph). Increasing the Ag composition to 65 %, one recovers the L1 1 phase in the core (Fig. 2-(e)) but with an alternation over three concentric shells at the vicinity of the surface, remaining from the last structure. To go deeper in the analysis, a decomposition of the core into concentric shells is illustrated in Fig. 3. It is worth noticing that the (100) facets of the shell under the surface (namely the n-1 shell) are filled by silver atoms whereas all the other sites are Pt-pure. On the following shell, namely the n-2 shell, (100) facets, which consist in only one atom, are filled by platinum, whereas this shell is Ag-pure. The other shells do not present (100) facets, they are octahedral. The origine of such chemical arrangement may be linked to the peculiar (100) facet reconstruction with a mixed Ag-Pt concentration to accommodate elastic strain. This leads to have pure concentric shells except on the (100) facets of each shell.

Comparing to a higher temperature (700 K), the L1 1 ordered phase in the core does no more exist as displayed by the core sublattices occupation on Fig. 1 which are almost equaly filled. In the same time, the 3-concentric shells at the vicinity of the surface remain because this phenomenon is driven by surface segregation, whereas the ordered phase is only stabilized beneath a critical order/disorder temperature as in bulk alloys.

The main result which consists in a core, either ordered according to the L1 1 phase, or with a concentric multishells structure, depending on the concentration, is general and appears at larger sizes. As an illustration, we have extended the previous investigation to a truncated octahedron of 2951 atoms (around 5 nm). We first compare the energies of the L1 1 phase and the multishell structures because the Monte Carlo simulations for large sizes is not able to switch from one structure to the other in the same simulation. In fact, we start with an initial configuration, either the perfect L1 1 ordered cluster or the perfect alternation of Pt/Ag concentric shells. For each configuration, we both increase or decrease the chemical potential to cover the entire concentration range. Finally, we keep the curve that minimizes the energy as shown in Fig. 4. From 54 % to 60 % of Ag, the concentric multishells structure is stable. Above and below these thresholds, the L1 1 phase with a 3-concentric shells is stable. Above 70 % of Ag, all the core sublattices gather corresponding to a solid solution in the Ag-rich side, as in the bulk phase diagram. This is the first time that such a chemical configuration transition has been predicted on nanoalloys. The concentric multishells structure itself is unusual in binary nanoalloys and has only been observed on fivefold symmetry and L1 0 ordered domains in decahedral Co-Pt nanoparticules. 37 The L1 1 chemical ordered phase has been experimentally observed recently by high resolution scanning transmission electron microcopy with high angle annular dark field mode (STEM-HAADF). 24 If the experimental studies concluded in the L1 1 ordered phase on small nanoparticles (up to 2.5 nm) with a silver surfaceshell, it can not fully exclude Pt subsurface segregation. Theoretical investigation showed that Ag surface segregation induces a very strong stress in compression inside the nanoparticle that can be released by putting smaller atoms i.e. Pt atoms on subsurface. 24 In the present study, we show that the alternation of pure atomic concentric layers which can be seen as an L1 1 nanophase on each (111) orientation of the truncated octahedron is in competition with the perfect L1 1 phase in the core. By increasing the Ag composition, one of the (111) orientation in the cluster is chosen to display only one L1 1 nanophase in the core. It is important to notice that when only one L1 1 nanophase exists in the nanoparticle, it can be characterized by STEM-HAADF but the alternated concentric layers not. This can explain why the authors did not observe experimentally this phase. Moreover, the formation of the L1 1 induces a rhombohedral deformation along the (111) orientation, as in the CuPt system, [START_REF] Lang | [END_REF] which enters in competition with the silver surface segregation. To release the stress, the system forms an alternation of concentric layers near the surface which accommodates the elastic deformation induced by the L1 1 phase. To understand this chemical ordering transition, we have calculated the local atomic pressure represented in the Fig. 5. The stress is better redistributed on each type of element (silver or platinum atoms) in the structures with concentric shells: either the full multishells structure or the partial concentric shells near the surface (multishells/L1 1 ). In the perfect L1 1 structure with an Ag surface shell, the majority of the stress in compression with high values (around 10 GPa) is supported by silver atoms in the core. In overall, the two structures with concentric shells are less constraints than the perfect L1 1 phase with an Ag shell at the surface.

Conclusion

To conclude we have shown that in Pt-Ag nanoalloys there can be an alternative chemical ordered phase compared to the L1 1 phase observed experimentally 24 which consists in an alternation of pure atomic planes in the (111) direction (as expected from the bulk phase diagram). We have shown that beyond the silver surface segregation, there is also a platinum subsurface segregation coupled to a silver subshell forming a 3-concentric shells around the L1 1 ordered phase in the core on a quite large range of composition (between 40% and 70% of silver). What is really surprising is the apparition, in a narrow range of concentration (between 54% and 60% of Ag), of a full concentric shell structure over the whole nanoparticle, which is more stable than the L1 1 phase. We have shown that the concentric shells allow to release the stress in the nanoparticle as compared to the perfect L1 1 phase with a silver shell. Such very peculiar chemical ordering, never reported before, is established for two different sizes: 3 and 5 nm. We think it may be general when increasing the size because of the necessity to release the stress that increases with the size. We have also shown a (2x1) reconstruction on the (100) facets stable on a large range of silver concentration (from very low to equiconcetration) which can have some interesting properties in catalysis. Finally our predictions could motivate new experimental studies on Pt-Ag nanoalloys on a wider range of composition and should have strong implications in catalysis and optical properties.

  demonstrated enhanced electrochemical a Aix-Marseille University / CNRS, CINaM UMR 7325, Campus de Luminy, 13288 Marseille, France. E-mail: christine.mottet@univ-amu.fr b Present address:Department of Chemistry and Materials Science, Aalto University, 02150 Espoo, Finland; Department of Applied Physics, Aalto University, FI-00076 Espoo, Finland.

Fig. 1

 1 Fig. 1 Isotherms at 200 K and 700 K of the Pt-Ag truncated octahedra of 1289 atoms (around 3 nm): surface site concentration (top), concentric shells concentration (middle), core sublattices concentration (bottom) as a function of the global Ag concentration of the nanoparticle. The concentric shells start from the surface (n) to the center of the nanoparticle (n-5). The core sublattices are considered in the central part of the nanoparticle by removing the three external layers (n, n-1, n-2). The structure corresponding to the compositions indicated by (a-e) are presented in Fig.2.

Fig. 2

 2 Fig. 2 Snapshots of the TOh 1289 at 200 K corresponding to the compositions pointed in Fig.1: (a) 7% of Ag, (b) 34% of Ag, (c) 54% of Ag, (d) 61% of Ag and (e) 65% of Ag. The top row represents the external surface whereas the bottom row displays a cutting view. Ag atoms are in grey ant Pt atoms in blue

Fig. 3

 3 Fig. 3 Decomposition in concentric shells of TOh 1289 at 200 K corresponding to the compositions (d) multishell structure and (e) L1 1 phase surrounded by 3-concentric shells, pointed in Fig.1 and illustrated in Fig.2. Ag atoms are in grey ant Pt atoms in blue.

Fig. 4

 4 Fig. 4 Isotherms at 300 K of the Pt-Ag truncated octahedra of 2951 atoms (around 5 nm): total energies of the concentric multishells phase and the L1 1 phase surrounded by 3-concentric shells (top), concentric shells concentration (middle), core sublattices concentration (bottom) as a function of global Ag concentration. The concentric shells start from the surface (n) to the center of the nanoparticle (n-5). Core sublattices are considered in the central part of the nanoparticle by removing the three external layers (n, n-1, n-2), as in Fig.1. The right column shows cutting view of the two structures. Ag atoms are in grey and Pt atoms in blue.

Fig. 5

 5 Fig. 5 Cutting view of the 2951 atoms TOh with in top row the nature of the atoms (Ag in grey, Pt in blue) and in bottom row the stress map with color code corresponding to sites in compression in red (up to 10 GPa) and sites in tension in blue (up to -5 GPa).

Table 1

 1 TB-SMA interatomic potential parameters. r 3 αβ represents a cutoff distance around the 3 rd neighbors where the effective TB-SMA potential is connected to a polynomial form of the potential between third and fourth neighbors distance and r[START_REF] Ferrando | Structure and Properties of Nanoalloys[END_REF] αβ represents the cutoff distance around the 4 th neighbors where the potential is zero.

	α -β	p αβ	q αβ	A αβ	ξ αβ	r 3 αβ	r 4 αβ
	Pt-Pt	10.7960 3.1976 0.1993 2.2318 4.8745 5.6286
	Ag-Ag 11.7240 2.8040 0.0748 1.0064 5.0949 5.8831
	Pt-Ag	11.2600 3.0008 0.1456 1.5920 5.0949 5.6286

| 1-7

| 1-7

Acknowledgements

The authors thank Riccardo Ferrando for fruitful discussion, and acknowledge support from the International Research Network-IRN "Nanoalloys" of CNRS. This work was granted access to the HPC ressources of IDRIS under the allocation 2017-096829 mage by GENCI.

Notes and references

Author Contributions

All authors contributed equally.

Conflicts of interest

There are no conflicts to declare.