Carole Adam
email: carole.adam@imag.fr

Julie Dugdale
email: julie.dugdale@imag.fr

Catherine Garbay
email: catherine.garbay@imag.fr

Multi-factor model and simulation of social cohesion and its effect on evacuation

In this paper we propose an analysis of social cohesion in terms of 3 factors: emotions, social norms, and mutual knowledge. These factors have previously been analysed separately in terms of Beliefs, Desires, Intentions (BDI) logics, but have not been merged into a unified model. The goal of this paper is to provide a unified agent-based model and describe its implementation in the GAMA simulation platform. The simulator is applied to an evacuation case study and different scenarios are run to evaluate the impact of the 3 factors on cohesion, as well as the effect of cohesion on evacuation. This paper describes first results and concludes with interesting future prospects.

Introduction

Understanding social cohesion and its dynamics is becoming increasingly important in the current political and social situation. The aim is to support its positive aspects, such as solidarity, and prevent its negative aspects, such as divided communities. Crisis situations are particularly appropriate for observing social cohesion, its dynamics and its impact. In such situations new and unexpected forms of cohesion can emerge, such as altruism and solidarity, or racism and a lack of communication and openness between groups. Studying the dynamic behaviour of social cohesion in crisis situations cannot easily or safely be undertaken in real life. Therefore computer simulation offers a viable and powerful alternative. There exist various approaches to computer simulation, among which the agent-based approach is particularly suitable for modelling human behaviour at the micro level [START_REF] Macal | Tutorial on agent-based modelling and simulation[END_REF]. Agents are autonomous entities, capable of reactive and proactive behaviour depending on the context, and that can communicate with others. In these respects they are therefore well adapted to represent humans. As Bonabeau notes [START_REF] Bonabeau | Agent-based modeling: methods and techniques for simulating human systems[END_REF], agent based systems are able to capture emergent phenomena that are representative of complex adaptive systems, and provide a natural, realistic description of these systems. Finally, they are flexible, allowing us to study social systems at different levels of abstraction by varying the complexity of our agents or by aggregating agents into subgroups.

Modelling and simulating social cohesion has received very little attention in the literature. In this paper we introduce a first approach to this issue. We argue that there are three main factors that can influence cohesion and its dynamics: emotions, mutual knowledge, and norms (institutional rules or social conventions). Our contribution is to focus on the interplay between these three factors, how they evolve through environmental and social dynamics, and how they mediate an entity's individual beliefs, desires and intentions. Our analysis is grounded on a set of properties (autonomy vs dependence, altruism vs obedience, justice vs injustice) that show these complex relationships. A key result is to understand how these factors influence the efficacy and speed of evacuation.

The structure of the paper is as follows. Section 2 discusses the three main factors contributing to social cohesion, from a theoretical perspective and with a particular focus on their interaction. Section 3 then describes our agent based model of human behaviour including these factors. The presentation is anchored in an evacuation case study. This model is implemented in the GAMA platform and applied to practical scenarios, which are described in Section 4 along with the results of our experiments. Section 5 concludes this article with a discussion on the relevance of the findings to social cohesion, and ideas for future work.

Factors of cohesion 2.1 Emotions

The influence of emotions on human reasoning is now well accepted and many computer models of emotional agents exist. These models are generally based on psychological theories of cognitive appraisal [START_REF] Lazarus | Stress, appraisal, and coping[END_REF][START_REF] Ortony | The cognitive structure of emotions[END_REF], that describe emotions as resulting from an individual's evaluation of the situation. This consists of two distinct processes: appraisal and coping.

1

Appraisal triggers emotions in response to the individual assessment of the situation. This explains why different individuals may exhibit different emotions in the same situation, since the situation is evaluated differently by different people. Coping is an attempt to handle these emotions. This may be trying to suppress negative emotions via various strategies. Rational strategies, such as planning or action, may focus on the problem if it is controllable. Alternatively, strategies may focus on the emotion. These strategies may be more or less irrational and aim to modify the individual's mental attitudes with respect to the problem. Examples may be finding the silver lining, denying that the problem exists and ignoring clues, or taking one's mind off the problem.

Emotions seen as reasoning heuristics are even more relevant to study in crisis situations where decisions must be made quickly and with incomplete information. Research has shown that reasoning in crises is influenced by cognitive biases and affective heuristics [START_REF] Kinateder | A Review of Risk Perception in Building Fire Evacuation[END_REF]; these biases have been highlighted in natural disasters such as bushfires [START_REF] Arnaud | The role of cognitive biases in reactions to bushfires[END_REF].

Other works show the importance of social attachment in evacuation after disasters such as earthquakes [START_REF] Bangate | A review on the influence of social attachment on human mobility during crises[END_REF]. Attachment to a target (in a wide sense: object, place, person, group, etc) [START_REF] Mawson | Mass panic and social attachment: the dynamics of human behavior[END_REF] is the source of emotions, for example, worry about one's family. Attachment is also the result of shared emotions, for example attachment to a familiar place that is linked with good memories, or to a person who has shared strong emotional experiences. Thus, strong cohesion can be observed in groups such as a fire brigade where members share the same emotions, understand and can empathise with each other. Research has shown that emotions rather than cognition determines team work in stressful situations [START_REF] Nair | The role of emotions in multiagent teamwork[END_REF].

Conversely, some emotions such as fear can hinder cohesion. Crisis managers are trained to suppress and not express these emotions in order not to break group cohesion. For instance firemen often use humour as a coping strategy, to regulate emotions, and as a bonding mechanism [START_REF] Alexander | When the bells go down: Resilience and vulnerability in firefighters[END_REF][START_REF] Moran | Positive reactions following emergency and disaster responses[END_REF][START_REF] Moran | Stress and emergency work experience: a non-linear relationship[END_REF][START_REF] Moran | Recruits predictions of positive reactions in diasaster and emergency work[END_REF].

Social norms

Cohesion is discussed in this section from an institutional and collaborative perspective, i.e an activity involving several organizations whose roles and missions are governed by different rules. It is therefore important to distinguish between the cohesion of individual activity, which depends on the individual's adherence to the rules governing its organization, and the cohesion of collective activity, which depends on the quality of coordination between actors.

In a crisis situation, the institutional rules differ according to the involved actors and the context. They are valid only in specific contexts, and are inherently incomplete because it is impossible to list all crisis situations and all possible responses. Their appropriation and mastery by the actors can be partial. Their application can be a source of error, because of the difficulties for the actors to perceive the global situation and its dynamics. In addition, at the inter-organizational level, one cannot assume that the set of rules is coherent and non-contradictory.

It is also important to distinguish between the prescribed activity and the real activity. The prescribed activity is described by a set of norms and whose cohesion depends on the coherence and comprehensiveness of this set. Whereas the real activity is that which is actually carried out by human actors and may not be in line with the norms, either through ignorance of the norms, or when the situation requires applying commonsense norms instead. Simulation plays a participative role and provides an evaluative feedback on the consequences of the modelled norms. It may lead to (i) an improvement in the completeness and coherence of the intra and inter-organizational sets of norms, and (ii) an improvement in the cohesion of the actors' behaviour towards these norms, i.e. to improve the appropriation and mutual awareness of these norms [START_REF] Thevin | Evaluating plans and human response using a normative multi-agent system[END_REF].

Mutual knowledge

Mutual knowledge is knowledge that interacting agents share and know they share [START_REF] Krauss | Intellectual Teamwork: Social and Technical Bases of Collaborative Work[END_REF]. It is also called 'common ground' because it is also an integral part of coordination actions and collaborative decision-making [START_REF] Cramton | The mutual knowledge problem and its consequences for dispersed collaboration[END_REF].

This notion can be analysed from the point of view of both victims and crisis managers. Previous work has shown that the first action of people in a crisis situation is not to search for shelter but to seek information [START_REF] Averill | Occupant behavior, egress, and emergency communications, federal building and fire safety investigation of the world trade center disaster[END_REF], via traditional media (e.g. radio, TV), social media (Facebook, Twitter, SMS), or direct interactions with other individuals in the same situation. Information seeking aims at building situation awareness and understanding by the individual in order to prepare a coordinated response (e.g. collective flight decision). Seeking and sharing information through (physical or digital) social interactions allows the individual to improve their mutual knowledge of the situation, and strenghtens social links and cohesion.

From a theoretical point of view, the emergence of mutual awareness in a group can be explained as a percolation phenomenon [START_REF] Dugdale | Coupling agent based simulation with dynamic networks analysis to study the emergence of mutual knowledge as a percolation phenomenon[END_REF] and is necessary for efficient cooperation and coordination in a collective task. However, the temporality of gaining mutual mutual knowlegde differs dramtically between everyday situations, where it can be gained over a long period of time, versus crisis situations, where victims typically only have brief interactions that strongly constrain the acquisition of mutual knowledge and make it imperfect.

Mutual knowledge can be established via several mechanisms [START_REF] Krauss | Intellectual Teamwork: Social and Technical Bases of Collaborative Work[END_REF]: direct knowledge created during shared experiences where individuals can obtain an informed judgement about what others know; interactional dynamics, when individuals meet the discussion is first focused on what is considered to be mutual knowledge [START_REF] Stasser | Effects of information load and percentage of shared information on the dissemination of unshared information during group discussion[END_REF]; and belonging to a category, which facilitates mutual knowledge. In crisis situations the usual social borders are broken and a new "victim" category emerges that reinforces social cohesion. Contrarily to non critical situations, where individuals are not likely to share their information [START_REF] Cramton | The mutual knowledge problem and its consequences for dispersed collaboration[END_REF], in crisis situations the need to understand the situation pushes them to share what they know. It has been shown that information sharing facilitates the emergence of mutual knowledge and can result in exceptional system robustness [START_REF] Pavard | Underlying concepts in robustness and resilience and their use in designing socio-technical systems[END_REF], which is a characteristic that emergency managers wish to obtain.

Interactions between these 3 factors

Our previous work leads us to look at the interaction between the gravity of crisis situations, emotions, mutual knowledge and norms. The gravity of a situation leads to the emergence of strong emotions and the obligation to obey specific norms. However, because of the tight time constraints it can also result in more effective communication since there is a conscious and concerted effort to inform others.

Emotions are linked to institutional norms in many ways. Norms can trigger certain emotions; they specify which emotions may or may not be expressed according to the context; they also provide a behavioral guide when the level of stress reduces the ability to make decisions. The moral or social emotions (e.g. shame and guilt, pride, admiration, etc.) felt towards a group come from respecting or violating the moral and cultural values shared by the group. These emotions can influence group cohesion by pushing to respect the norms or trying to repair the violations (repair actions, excuses) so that individuals can continue to be accepted in the group. The need to respect the norms can be inconsistent with individual goals and lead to dilemmas, such as choosing between a moral action or a selfish one. According to von Scheve [START_REF] Scheve | My agents love to conform: Norms and emotion in the micro-macro link[END_REF], the relationship between emotions and social norms, more precisely the place of emotions in normative multi-agent systems, have not yet been intensively studied, despite some pioneering work. Nevertheless it has been possible to show how emotion is essential in maintaining social norms, and in particular how sanctions may be imposed when there is non-compliance with norms. Furthermore, social norms, which are sources of emotion, can determine the processes of coping and regulating emotions.

Cairo [START_REF] Caire | A normative multi-agent systems approach to the use of conviviality for digital cities[END_REF] focuses on the articulation between institutional (rather formal) norms and "friendliness" (rather implicit) norms and on the impact of violating these two types of norms. He notes that friendliness appears rather as a "positive" norm, but also has a negative side when it intervenes to ground relations of power. He emphasizes the importance of friendliness to improve communication, increase the effectiveness of cooperation by reducing conflicts, and strengthen social cohesion by highlighting its values. Friendliness is brought into play through the expression of feelings, such as praising and encouraging people who conform to norms, and expressing resentment and anger towards those who do not. This is both a condition of social interaction and a valuable instrument for the internal regulation of social systems.

Emotions are linked to mutual knowledge in several ways. Beliefs about others can trigger positive emotions (e.g. empathy, compassion) or negative ones (e.g. resentment, jealousy, envy, etc.). Also through a process of reverse appraisal, the emotions expressed by others can be analyzed to build knowledge about the mental attitudes that may have caused that emotion. Finally, mutual knowledge favors the understanding of the emotions of the other agents and thus the cohesion of the group. The term emotional knowledge has been introduced by Shaver et al. [START_REF] Shaver | Emotion knowledge: further exploration of a prototype approach[END_REF] to describe sharing and describing emotional reactions to past or present events. This mutual emotional knowledge advantageously increases the efficiency of the social group [START_REF] Rhee | Shared emotions and group effectiveness: the role of broadening-and-building interactions[END_REF]. In crisis situations, this can lead to groups of victims coordinating their response or evacuation actions.

Knowledge and understanding of dynamic situations are vital for deciding how to act and which normative rules to apply. Artefacts, such as social media, also play a role in supporting mutual awareness since they allow people to understand other peoples actions, possibly recognise their intentions, and to comprehend the emotions evoked by the situation. Social norms are in fact mutual knowledege that is shared by the collective. Mutual awareness of norms is an important pre-requisite, but is not limited to the sharing of formal knowledge, involving the sharing of beliefs and intentions over the situations encountered. Rauwolf et al. [START_REF] Rauwolf | Modelling normative awareness: First considerations[END_REF] discussed the importance of the notion of "normative awareness", emphasizing the need to distinguish it from the more common notion of "situation awareness". They propose three levels: perception, comprehension, and ability to anticipate the impact of norms of others. According to them it is the third level, that is to say the awareness that other agents are guided by their own beliefs and intentions, which is the most important to consider. The absence of studies and models on sharing intentionality leads to the simplifying assumption, which is present in most works, of a general awareness of all norms.

Emotions, norms, and mutual knowledge have already been studied independently in the scope of group behaviours, but they have not been combined in order to explain the dynamics of social cohesion. The goal of this work is to combine these factors in a unified agent model that shows their impact on the dynamics of beliefs and decisions of the agent at the micro scale, and on the effect of group cohesion. The following section describes the proposed unified model.

Agent-based model of cohesion 3.1 Case study

The case study concerns a situation of people evacuating from a large room in a building, for example a shopping mall. In order to practically demonstrate how the three factors identified above contribute to the emergence of social cohesion, the case study has been manipulated so that people can belong to one of two different social groups. Furthermore the room has a number of exits, some allowing only agents from one or the other social group to pass through. The number of exits can be parameterised, so that injust conditions can be created and studied. Not all of the agents know the location of exits, so they do not have mutual knowledge. However, as the simulation proceeds they can share their knowledge.

The environment is a large square room (in future it can be made more realistic by adding walls and furniture, etc, but it was not considered necessary at this point). The room contains an alarm system (located at the center of the room) that can be switched on or off, and a number of exits of two different types (square and circle). It is populated with a number of agents, representing humans, that belong to two different social groups (denoted by circles and squares). Their model is detailed below,

Human agents attributes

The human agents in the model have the following main psychological attributes: level of obedience to norms, i.e. how likely they are to respect the norms they are subject to; level of altruism, i.e. how likely they are to help other agents around them, for instance by providing information or guidance; level of leadership, i.e. how likely they are to explore for exits proactively versus just following other agents; risk aversion; and emotional sensitivity. Human agents also have various physical attributes such as a level of energy (which is depleted as they move around, and can be refilled when someone helps them), a list of known exits, and a radius of perception (how far they can see around them).

Agents BDI model

The BDI architecture based on formal BDI logics [START_REF] Rao | Modeling rational agents within a BDI-architecture[END_REF] is widely used in agent-oriented programming [START_REF] Padgham | Developing intelligent agent systems: a practical guide[END_REF] and is starting to gain interest in social simulation in order to design more realistic agents [START_REF] Adam | BDI Agents in social simulations: a survey[END_REF], in particular when extended with emotions [START_REF] Adam | A logical formalization of the OCC theory of emotions[END_REF][START_REF] Bourgais | Emotion modeling in social simulation: a survey[END_REF]. Agents with this architecture have different types of mental attitudes: beliefs (what they know about the world); desires (how they would like the world to be, the desires may possibly be inconsistent or infeasible); intentions (these are selected feasible, consistent desires that the agent is committed to achieve with some degree of persistence). They might also have norms or ideals (which are externally imposed descriptions of how things should be). Finally they have a library of plans (recipes to reach their intentions).

The reasoning of BDI agents is organised in the following phases: perception of their environment to update their mental attitudes, and infer new mental attitudes from the existing ones based on a set of rules; decision: selecting an intention from the agents desires (usually the highest priority feasible desire is selected), and selecting a plan to achieve this intention in the current context; action: performing the first action of the current plan. Below we detail the specific features of the BDI agents in our model.

Predicates

Logical predicates describe the agents world and represent mental attitudes. The necessary predicates, a sort of ontology of the domain of application, for our case study is shown in Table

Perceptions, new beliefs

The human agents can perceive the world around them, including other agents. Their observations result in new beliefs, desires, social relationships, and emotions.

• Perception of the alarm being switched on adds a belief that there is danger, and removes the desire to keep shopping.

• Perceiving an exit within the agent's perception radius adds beliefs that the agent knows an exit, and that the agent is located at an exit.

• When the agent knows of more exits for the other group than for its own, it adds a belief that the situation is injust (this belief can be mistaken since it is based on the agent's incomplete perception of its environment: there could be more exits that it does not know about yet).

• Perceiving an exit at the immediate proximity adds a desire to exit, whose strength depends on the type of exit: if it is the correct exit, the priority is maximal; if it is an incorrect exit, the priority is moderated by the level of subjective danger (the higher the danger, the more chance to exit) and the level of obedience (the more obedient, the less chance of using the wrong exit)

• Perceiving another agent in perception radius creates a social link with them: if they are heading to an exit then trust towards them increases; if they are helping someone, then liking towards them increases; if they are in the same social group, solidarity increases.

• Perceiving an agent who is stopped, lost, out of energy, or waiting for a leader, adds a belief that they need help and a desire to help, with priority based on the agent's level of altruism.

• Perceiving an agent helping another agent (whether they provide energy or information about exits) creates a new belief about that helping action, storing who helped who.

Concerning the above perceptions, we assume that agents can access the other agents' current plan. This is a slight simplification, but it can be justified by the fact that observable actions usually suggest what people are doing: people heading straight to a known exit move faster and more confidently than people who are exploring around; people who are helping others usually slow down or stop next to them; etc.

Triggering emotions

The emotions triggered in our agents are based on their definitions by Ortony, Clore and Collins [START_REF] Ortony | The cognitive structure of emotions[END_REF] and on their logical formalisation by Adam [START_REF] Adam | A logical formalization of the OCC theory of emotions[END_REF].

• Hearing the alarm triggers fear of the potential danger, with an intensity based on the agent's risk aversion level.

Fear intensity then increases with time as long as the alarm keeps ringing, making the potential danger more imminent.

• When the alarm stops, fear is suppressed immediately.

• Perceiving an agent needing help, needing information, or waiting for a leader, triggers compassion for that agent, with an intensity based on the agent's altruism and liking for the target agent.

• Perceiving an agent from a group that has more (known) exits than the other group triggers jealousy in the agents from that other group.

• Perceiving an agent helping someone else triggers admiration; if they are helping oneself it triggers gratitude.

Rules, new desires

Rules describe logical links allowing the agents to infer new mental attitudes from the ones they already have and the ones they deduce from their observations above. In particular, these rules can activate contextual desires that appear only under some conditions. For instance we have the rules below:

• When the alarm is ringing and the agent knows of at least one exit, it infers the desire to head to the exit, with a priority based on its subjective evaluation of danger.

• When the alarm is ringing but the agent does not know any exits, it infers the desire to find exits, with a priority proportional to its level of leadership

• When the alarm is on and the agents knows no exit but knows potential leaders, it infers the desire to follow, with a strength inversely proportional to its leadership

• When the alarm is on, the agents knows no exit and no potential leader, it infers the desire to find a leader, with a strength inversely proportional to its leadership.

These situations are not mutually exclusive, so each agent might infer a number of different and possibly contradictory desires. For instance an agent knowing no exit might desire to find one itself (priority proportional to its leadership) and at the same time desire finding and following a leader towards an exit (priority inversely proportional to its leadership). The agents therefore need to rank their desires in order of priority, and select the one with the highest priority to make it their intention. In the previous example, an agent with high leadership will favor exploration for exits, while an agent with low leadership will prefer to find someone to follow. Contrary to desires, intentions have to be feasible, consistent, and persistent. Once selected, the agent commits to pursue its current intention until it is either reached, or deemed unfeasible (due to a change of context). Intentions lead to action because of the plan library.

Plans

Our agents have a library of plans available to achieve their intentions in different contextual conditions.

• Intention to go shopping: there is only one plan that consists of wandering around in the shopping mall;

• Intention to be at the exit: there is only one plan, which is applicable when an exit is known, and which consists in heading straight to that exit;

• Intention to know an exit: two plans are applicable when an exit is not yet known. The first one is an organised exploration where the agent aims for a wall and then follows the wall until it finds an exit. The second one is a more disorganised exploration where the agent moves around randomly until it finds an exit by chance; it is selected if the first one fails for any reason;

• Intention to follow towards an exit: three plans are applicable. If a known leader is within reach, the first plan deals with following them. If no known leader is visible, but other agents are around, the second plan chooses the best one (either the closest, or fastest, or with highest trust / liking / altruism) and follows them. If no potential leader is within reach, the third plan allows the agent to follow the general direction of others around it;

• Intention to find a leader: there is a single plan that lets the agent wait until a leader comes past (its perception will trigger the belief that the agent now has a leader, which will finish the intention and plan to wait);

• Intention to exit: two plans are available when the agent is within reach of a door. The first plan is applicable if the exit is the correct one (i.e. allowed by the norms), and lets the agent use that door. The second plan is applicable if the exit is wrong but the agent believes that it is an emergency, or is not very obedient. If neither of these 2 plans is applicable (e.g. the agent is at the wrong door and exiting is not yet that urgent), the intention is unfeasible, dropped, and another desire comes forth (e.g. exploring to find more exits).

At each time step, the agent either continues executing its current plan, or selects a new one, or selects a new intention. Indeed, if a plan fails for some reason the agent will try another plan available for the same intention; if no plan is feasible for the current intention, it is deemed unfeasible, dropped, and another one is selected. Having backup plans for each intention ensures robust behaviour of the agents, making their behaviours more like real humans.

In the next section we describe how this model was implemented, and the different experiments that were conducted to evaluate cohesion, its links with the 3 investigated factors (emotions, norms, mutual knowledge), and its impact on evacuation.

Implementation and experiments 4.1 Implementation

GAMA platform

The model has been implemented and simulation have been run using the GAMA platform.

GAMA is an open-source agent-based modelling and simulation platform, offering easy integration of GIS data into the models. It has recently been extended with a BDI architecture for the agents [START_REF] Taillandier | A situated BDI agent architecture for the gama modelling and simulation platform[END_REF] as well as social and emotional features [START_REF] Bourgais | An Agent Architecture Coupling Cognition and Emotions for Simulation of Complex Systems[END_REF]. Agents with this architecture have their behaviour modelled in terms of beliefs, desires, intentions, norms. Emotions are triggered based on the agent's mental attitudes, they decay over time and are contagious. Agents can also create and maintain social relationships with other agents. GAMA is therefore well adapted to implement our BDI model of the three factors of cohesion.

User interface In the interface, the user can set the values of a number of parameters of the simulation, such as the density of people in the room (betwen 20 and 200 people), or the initial percentage of people that know an exit at the start of the simulation. The user can choose the minimum and maximum values of the different psychological attributes (altruism, leadership, and obedience to norms) in order to constrain the profile of the population. They can also use some actions to modify the simulation, in particular start and stop the alarm. They can also obtain feedback as the simulation is running though a console. In particular at each cycle information is provided about each agent, their current intention, plan, and level of energy. The main displays also show agents with a different colour based on their current plan. Other displays show emotions and their targets, and leader-follower groups (see Figure 1). When the simulation stops, statistics are provided about the total duration, percentage of agents evacuated, and if the norms were respected. By manipulating these parameters, we explored various scenarios to analyse the separated and combined influence of the 3 factors of social cohesion, and the impact of cohesion on evacuation.

Measuring cohesion

There is no consensus on the definition and measurement of cohesion. Based on our theoretical analysis above, we have devised several possible measures or indicators of cohesion, detailed here.

• Geographical measure: we used a DBscan algorithm [START_REF] Ester | A density-based algorithm for discovering clusters in large spatial databases with noise[END_REF] to create clusters of agents based on geographic proximity, which reflects groups of agents following the same leader (see Figure 2). DBscan is a popular density-based clustering algorithm that when given a set of points in an environment it groups together points that are closely packed together (close neighbours). It also marks remote points that are alone in low-density areas (these are points whose nearest neighbours are too far away). We measured various properties of the clusters as a first indication of cohesion: number of groups, average size of groups, average lifetime of groups, and average time spent alone by the lonely agents.

• Emotional measure: we implemented a display in the simulator to show the agents, their social group (circle or square shape) and their emotions in the form of coloured labeled arrows (green for positive emotions, orange for negative ones) pointing at the target of the emotion. This display visualises the emotions shared inside of a group versus the emotions towards agents outside of the group.

• Normative measure: at the end of the simulation (when all agents have evacuated or stopped moving), the number of agents who have used the right exit, wrong exit, or no exit at all is measured. This gives an idea of cohesion in terms of adherence to norms: if all agents respect the norms, or all agents violate the norm, the population as a whole has more cohesion w.r.t. this norm than if behaviour is more varied.

• Informational measure: during the simulation, we measure the number of agents knowing of zero, one, or more exits. This provides an idea of cohesion in terms of information sharing, since at first only some agents know the location of exits.

Scenario 1: autonomy vs dependence

In a first scenario, we compared two extreme situations with respect to the initial knowledge of the agents, on a population of 20 agents:

• Situation 1: all agents know the location of the exit and have a high leadership. In this case, all agents are autonomous and immediately head to their known exit. No agent needs help, and no cohesion emerges.

• Situation 2: no agent knows the location of the exit. In that case, they explore the room to find an exit, some agents follow others. Since they lack information, they depend on others who might have more information than themselves to provide information or guidance. In this case cohesion emerges by need.

Table 2 shows the results in terms of geographical cohesion (number of groups and average size) and impact on evacuation (time and completeness). This suggests that with complete knowledge (which in our case study is limited to knowledge of the exits), the agents do not need each other and cohesion does not emerge. Table 2: Results of experiment 1: evacuation time and percentage, average size and number of groups, in both situations

Scenario 2: obedience vs altruism

In a second scenario we manipulated the parameters of obedience and altruism of the agents:

• Situation 1: the agents are very altruistic. They help each other and form more groups, guiding others to the exits. Since they do not consider social groups, they might guide someone to the wrong exit. As a result, many agents use the wrong exit, but very few agents, if any, are left behind.

• Situation 2: the agents are not altruistic but very obedient to norms. They do not stop to guide others. Agents who find a wrong exit first will refuse to use it due to their high obedience, so they will keep exporing to find a correct exit. As a result, the evacuation might be slower, with agents left behind, but most agents use the right exit (unless they feel it is too dangerous to obey the norm).

Table 3 shows the results of this experiment with populations of 20 agents. This scenario suggests that not all forms of cohesion are the same. On the one hand, altruism creates cohesion around leaders who guide others to exits and share information about the exits, via the triggering of positive emotions of compassion, gratitude or admiration. This is an inclusive form of cohesion, regardless of the agents' norms or social groups: no one is excluded, and bigger groups emerge. On the other hand, obedience creates cohesion in terms of respecting the same norms. form of cohesion which depends on having the same norms or belonging to the same social group: insiders are cohesive but outsiders are rejected. Further scenarios could investigate with "altruistic norms", for instance a norm that one should help others, or at least some types of others (children, elderly, women, etc). This could show differences in helping behaviours induced by altruism versus the same behaviours induced by enforcing norms. We suspect that norm-enforced helping behaviours would work well to counteract egoistic behaviours in normal situations, but would be less resistant to panic in case of an emergency, where individual survival will preempt if the person is not truly altruistic.

Scenario 3: positive vs negative emotions

The simulator allows triggering and visualising different types of emotions in the agents (see Figure 3).

• Positive emotions: altruistic agents feel compassion towards agents perceived as needing help. Agents receiving help feel gratitude towards those who help them, and are then more likely to help in return. They also feel admiration towards agents who are witnessed helping others, and are then more likely to choose them as leaders.

• Negative emotions: agents feel reproach towards agents violating a norm (not helping others or using the wrong door), and anger towards agents who refuse to help themselves. Subsequently, they are less likely to follow or help these disliked agents.

Positive emotions therefore play a role in cementing cohesion, making groups of agents emerge and stick together. On the contrary, negative emotions separate different groups, by excluding the agents who are the target of these emotions. For instance lost or weak agents will feel gratitude or admiration towards an altruistic leader who helped them by providing information or guidance, and they will follow them as a result. On the other hand egoistic agents might be rejected by agents who feel reproach towards them.

Scenario 4: justice vs injustice

In the future we want to investigate scenarios where we create an unfair distribution of exits, with either more exits for one of the groups, or exits that are easier to locate. When perceiving such an unfair situation (less exits for one's own group), our agents feel jealousy towards any member of the social group having more exits, even though these agents are not directly responsible for this situation. This is often observed in the real world where quick categorisations and generalisations are made to judge people based on the group they belong to, in particular after crisis events. In such scenarios we hope to show the emergence of stronger cohesion in the discriminated group, by joining forces against a common enemy. We expect this to be a form of exclusive cohesion where foreign agents will be rejected for not sharing the same values.

Impact of cohesion on evacuation

Further experiments are needed to assess the impact of cohesion on evacuation in more detail. However, we can already see that cohesion helps some people evacuate that would not have been able to if left alone (see Figure 4). Indeed, some people are slower, shorter sighted, or ignore the location of exits. Having other agents who stop to help and guide them allows them reach the exit. On the contrary if they are unlucky and no one finds them (which can be the case with a lower density of agents in the room), they might panic or freeze and never evacuate. This suggests that evacuation might be slower but more complete with cohesive groups. Therefore, it is crucial that the alarm alerts the population well in advance so that there is more time to evacuate everybody.

Conclusion

In this article we have shown that the dynamics of social cohesion result from the intertwining of several factors, namely emotions, mutual knowledge, and norms. We proposed a BDI agent model to integrate these 3 factors and implemented them in the GAMA platform. The interest of the BDI architecture is to offer a range of options to the agents where they can choose an intention among all their desires, and then choose one of the available plans to reach their intention. This allows us to simulate heterogeneous, non-predictable behaviour, where each agent has different priorities leading to different behaviours. The experiments showed that cohesion may emerge in many different forms, with various bases and dynamics. Positive emotions, with behaviours driven towards the welfare of others have been shown to play a cementing role for an inclusive form of cohesion, relying on sharing. Conversely, unfair situations, with behaviours driven towards the obedience to shared norms, have been shown to create the emergence of exclusive forms of cohesion, relying on discrimination. The need for cohesion may also rely on the need for others (lack of information, lack of resources): no cohesion can emerge when the agents are fully autonomous.

How cohesion evolves and is analysed according to its positive or negative effects remains an open research field. Important work remains to be done to gain a deeper insight into the dynamics of cohesion, to account for the mutual dynamics of emotions, knowledge and norms, and to possibly integrate other factors of cohesion. Another important field of study would be to investigate how the search for and feeling of cohesion might bias individual behaviour, emotional response and obedience to norms. In future work we will develop a serious game simulating a crisis situation to test the influence of different aspects (quality of management plans and regulations, coordination of managers) on the emergence of cohesion.

Figure 1 :

 1 Figure 1: Display of the leader-follower groups emerging during evacuation.

Figure 2 :

 2 Figure 2: DBScan, 3 clusters (pink, green and red groups), 5 lonely agents (in grey).

Figure 3 :

 3 Figure 3: Displaying positive emotions inside 3 groups of agents.

Figure 4 :

 4 Figure 4: Evolution of distribution of plans in the agents. We can observe alarm starting at step 2, and first agent evacuating at step 15

Table 1 :

 1 1. Domain ontology: predicates, values, meaning

	shopping	None	the agent is currently busy shopping in the mall
	danger	Starting cycle	there is a danger (signalled by the alarm)
	injustice	Ratio of circle/square exits	the situation is injust (more exits for the others than self)
	at exit	Which exit	the agent is at an exit
	know exits	Which exits	the agent knows an exit (initial or after exploring)
	circle exit	None	the agent can use circle exits
	square exit	None	the agent can use square exits
	has leader	Name of leader	the agent has a leader to follow
	following	Name of leader	the agent is following another agent
	exiting	Which door	the agent is exiting through a door
	helping	Names of helper and helped	an agent is helping another agent
	need help	Agent name	an agent needs help (low energy)
	need info	Agent name	an agent needs exit info
	need leader	Agent name	an agents needs a leader to follow (low autonomy)

Table 3 :

 3 It creates reproach and other negative emotions towards those who have different norms and values, those who disobey and are not punished, or those who are favoured by unfair norms. This is an exclusive Results of experiment 2: agents exiting through correct door, incorrect door, not exiting at all, and total time.

	S Correct Wrong Not exited Time
	S1	15	2	3	127
	S2	10	9	1	79


Carole Adam
email: carole.adam@imag.fr

Julie Dugdale
email: julie.dugdale@imag.fr

Agent-based analysis of the spread of awareness in the population in the prodromal phase of bushfires

Keywords: crisis communication, agent-based modelling and simulation, bushfires, human behaviour modelling, BDI architecture, phone tree

Efficient communication is essential in disasters in order to coordinate a response and assure effective evacuation. This paper focuses on the case study of the Melbourne bushfires in 2009. We first analysed some interviews of the population to know who the population communicates with (neighbours, family, authorities, etc), and using what channel (radio, phone, internet, etc). We then developed and implemented communicative actions in a Belief-Desire-Intention model of the population's behaviour. Finally, we ran experiments in order to compare the speed at which the population becomes aware of the fires in different scenarios with different types of communication (more or less organised). Our first results show that more organised modes of communication would provide significant benefits in terms of propagation of awareness in the population.

Introduction

In order to provide an effective and unified response to crisis situations it is crucial that emergency managers and responders communicate efficiently both at the intra-and inter-organisational level [START_REF] Wenger | Disaster analysis: emergency management offices and arrangements[END_REF][START_REF] Lindell | Organizing response to disasters with the incident command system/incident management system (ics/ims)[END_REF]. However, a successful crisis response also requires communication with citizens and citizen groups to ensure safe evacuation or management of the situation [START_REF] Laajalahti | Crisis communication competence in co-producing safety with citizen groups[END_REF]. At the ground level, citizens need to interact and share opinions with other citizens in order to inform each other and prepare for a response [START_REF] Sutton | Backchannels on the front lines: Emergent uses of social media in the 2007 southern california wildfires[END_REF][START_REF] Haataja | Citizens' communication habits and use of icts during crises and emergencies[END_REF]. Indeed one of the first actions immediately following a disaster is for people to search for information in order to understand and make sense of the situation. This information may come: from news agencies, broadcasted by radio, TV or Internet; from emergency response organisations, such as the Red Cross or FEMA, communicated via social media or given to the news agencies; or more immediately from fellow citizens, through direct communication, telephone calls, and SMS. In cases where there is warning of an impending crisis, such as with bushfires or hurricanes, citizens will monitor local news sources, contact each other to find more information, warn others, or simply try to be close to family or friends.

There are several types of communication that are relevant: from the authorities to the population (usually unidirectional broadcast on TV, radio or Internet); from the population to the authorities (requesting information or help, and increasingly also providing field information via social medias); between and inside the different disaster management agencies; and within the population (exchanging information, reassuring one's family, etc).

In previous work we modelled various communication strategies that the authorities could use to influence the population's decision to evacuate the population during a bushfire. In this paper we focus on the interactions of the population with various interlocutors (their family and friends, their neighbours, and the authorities). We intend to show that these interactions could and should be more organised to improve effectiveness in spreading awareness in the population. A significant proportion of the population of Victoria live in small communities in bushfire prone areas, some of which are quite remote. Unlike in other parts of the world where the responsibility for rescue and defence of property lies with the fire services, in Australia it is often the case that the local population is responible for defending their own property or evacuating themselves to safe areas. Many people died needlessly in bushfires because they were isolated from others and became aware of the danger too late to safely react. On the contrary, other less isolated people might be warned several times. We implemented and tested several ways of organising intra-population interactions to avoid such situations, and compared how fast awareness of the danger spreads in the population with each organisation mode.

1

This paper is structured as follows: Section 2 first discusses related works in crisis communication. Section 3 then introduces the data available along with our methodology and the results of our analysis. Section 4 then presents our model and simulation of the population's behaviour based on this analysis. Section 5 describes the experiments and discusses results obtained from our simulations. Section 6 concludes the paper.

Literature about crisis communication

During an incident, information is as critically important to people as food or water. Not only can accurate information mean the difference between life and death, it can provide reassurance that response and recovery are truly underway [START_REF]Effective communication (student manual) -lesson 3: communicating in an emergency[END_REF].

Yet, unfortunately there are very few works that use simulation to test the effect of communication strategies on a realistic simulated population, taking into account the psychological aspects of message acceptance and impact on behaviour change. One exception is the simulation we cited earlier [START_REF] Bailly | An interactive simulation for testing communication strategies in bushfires[END_REF], but it focuses on the interactions from the authorities to the population, and not on how the information then spreads inside the population.

Information spread was addressed at a theoretical level in [START_REF] Dugdale | Coupling agent based simulation with dynamic networks analysis to study the emergence of mutual knowledge as a percolation phenomenon[END_REF] by combining a multi-agent system and social network analysis to study communication within groups of people. They found that the emergence of mutual knowledge may be explained as a percolation phenomenon originating from the physics domain.

Nevertheless these simulation works are often focused on small groups of people. For instance [START_REF] Dugdale | A pragmatic development of a computer simulation of an emergency call centre[END_REF] provided an agentbased model and simulation of different modes of communication in an emergency control room, showing that telephonic communications are often more efficient than other forms of communication even when people are closely co-located.

A notable exception in the simulation domain that looks at communications in disasters for a wider group of people is [START_REF] Clerveaux | Information simulation model: Effective risk communication and disaster management in a mixed cultural society[END_REF], who used a GIS to analyse different modes of communication during a tsnumani. They found that a correlation exists between the time that information was received and the number of casualties.

Outside of the simulation domain, other works are interested in crisis communication. For example, [START_REF] Chauhan | Providing online crisis information: An analysis of official sources during the 2014 carlton complex wildfire[END_REF] compared the timeliness and relevance of 4 different sources of official online information during crises. Their results provide an insight about how official sources can better manage their online communication, and how the public can find timely and relevant information online (websites and social media).

In this paper we have purposely ignored social media, as they are not mentioned in our data source. However, Twitter has indeed been used during the bushfires, and some works have already exploited Twitter. For instance, Power et al. [START_REF] Power | Emergency situation awareness: Twitter case studies[END_REF] propose the Emergency Situation Awareness (ESA) system, that monitors Twitter to provide emergency managers with relevant information, generating alerts when detecting an unusual increase in the frequency of certain words.

The use of social medias has also been investigated by [START_REF] Moore | The tweet before the storm: Assessing risk communicator social media engagement during the prodromal phase -a work in progress[END_REF], who studied the engagement of risk communicators with social media in the prodromal phase of hurricane Sandy. The focus on the prodromal phase is a shared originality with our work here. Of course this prodromal phase, before the disaster actually starts but when it is already expected, is only relevant for slow-onset, predictable disasters such as bushfires or storms. Such work is not applicable to earthquakes or terrorist attacks for instance.

Finally, another originality of our approach is to design our model based on citizens' interviews where they describe their behaviour, rather than using quantitative data such as statistics. Such qualitative data is harder to analyse but provides valuable insight into the psychological determinants of behaviour, that cannot be quantified.

Data analysis

This section introduces our case study of the Melbourne "Black Saturday" bushfires in 2009, the data available concerning that disaster, and how it was analysed.

Context and data

In February 2009, severe bushfires hit the state of Victoria in South-East Australia, killing 173 people and injuring another 414. A research commission was formed and produced a report [START_REF] Teague | Victorian Bushfires Royal Commission[END_REF] that showed that communication to the population must be improved in order to avoid such high fatality and injury rates in the future.

In particular the report contains the interviews of 100 people that were involved in the bushfires: residents of the fire-affected areas, firemen who fought the fires, or relatives of deceased people. In these interviews, people recount in detail what they did before, during and after the bushfires. These interviews are quite heterogeneous in terms of their author and length. Since the interviews are in narrative form they are hard to analyse automatically.

Nevertheless, these interviews provide very rich information about the population's behaviour in the bushfires. Consequently, they have been used to extract different stereotypical behaviour profiles [START_REF] Services | Review of the community response in recent bushfires[END_REF]: can-do defenders, livelihood [START_REF] Adam | Modelling human behaviours in disasters from interviews: application to melbourne bushfires[END_REF][START_REF]Modelling human behaviours in disasters from interviews: application to melbourne bushfires[END_REF]. This model uses a finite-state-machine architecture that provides a rather simplistic and rigid description of behaviour. In addition, it completely ignores communication between the individuals.

Methodology and results

In this paper, interviews have been exploited to show the importance of communication in the population's reactions to the bushfires, and to subsequently update a model to include communicative behaviour. In the absence of existing tools or methodologies for extracting a human behaviour model from narratives, we adopted the following methodology.

After reading the interviews, we empirically selected various relevant communication keywords, that we sorted into different categories of communication channels (TV, radio, Internet, social media, etc) and different categories of interlocutors (family, police, fire agencies, neighbours). We implemented a parsing tool to count occurrences of these keywords in each interview and generated statistics.

Results

The results are shown in Figure 2 for the communication channels, and Figure 1 for the interlocutors. As we can see, most people mention communicating, all of them used the phone, and most of them also used the radio. All of them also mention their family, and most of them mention their neighbours, showing the importance of social links inside these communities. Communication is important and happens between all types of actors involved in the crisis, with also many mention of the police and fire agencies.

Discussion

A more detailed semi-automated analysis of the interviews is still ongoing, trying to exhaustively list all the interactions described, with the people involved; the channel used (phone, Internet, face to face, etc); the type and content of the interaction (information, recommendation); its timing (before, during or after the fires); its trigger (why did the person decide to communicate at that time); and its effect on behaviour (did the receiver change their behaviour as a result). In the future we also intend to compare these results with communication in other types of crises (various natural disasters, terrorist attacks, etc.) and in other countries and cultures.

However this first analysis already provided interesting insights. Two points in particular are worth discussing further. First, we can see that surprisingly, there was no mention of social medias (Facebook, Twitter) in any of the interviews. It may be likely that these communication media were used more by people outside of the area. Another explanation is that social media was not as frequently used in 2009 as it is today. As a result we decided to ignore social media in the model presented here.

Second, several people mentioned in their interview that they wished that a phone tree had been organised. A phone tree [18] is a list of phone numbers where each person is asked to contact the next person in the list, etc. in order to efficiently share information in a community. There are different types of phone trees: simple (an unordered list of numbers); linear (a sequence of numbers with a defined start and end); circular (a sequence that can start anywhere in the circle); tree (the starter rings two people to trigger parallel branches of the list); or combinations. The choice of the appropriate organisation usually depends on the size of the group. In our model we decided to implement various organisations of communication in order to compare their efficiency.

Model of communicative behaviour

Based on this analysis, we intend to develop a model of the communicative behaviour of the population before and during the bushfires. We focus on the communication inside the population, ignoring the other levels of communication (population with the authorities, response agencies with each other). Messages from the authorities to the population have been investigated elsewhere, for example [START_REF] Bailly | An interactive simulation for testing communication strategies in bushfires[END_REF].

Modelling choices

Selected human agent architecture

In agent-based modelling and simulation, there are various levels of complexity for the underlying agent model. A wellknown agent architecture in artificial intelligence is the BDI model [START_REF] Rao | Modeling rational agents within a BDI-architecture[END_REF], which describes behaviour in terms of mental attitudes: Beliefs, Desires and Intentions. Beliefs describe how the agent sees the world, and can be incomplete or incorrect; Desires describe how the agent wants the world to be, and can be inconsistent or unrealistic; Intentions finally describe what the agent commits to do, and must be consistent and persistent.

Other works have discussed the interest of BDI models in social simulations in general [START_REF] Adam | BDI Agents in social simulations: a survey[END_REF], and shown the interest of using a BDI model (over more simplistic reactive models) specifically to simulate the population in bushfires [START_REF] Adam | Comparing agent architectures in social simulation: Bdi agents versus finite-state machines[END_REF]. In particular BDI models are easier to develop and extend, and the code and results are easier to understand. This is very important in crisis management, where models are likely to be used by field experts who are not computer scientists. The level of abstraction of a BDI model also matches the vocabulary used by people to describe their behaviour, making it easier to abstract a model from the interviews, and to explain it to field experts. This is essential for models aimed at raising awareness or supporting decision making, which are much more likely to reach their goal if they are well understood. Finally, BDI offers more flexibility and realism for modelling human behaviour.

The BDI model is also more adapted to describe communication. The semantics of messages can be expressed in terms of mental attitudes, explaining why a message is sent (e.g. because the sender wanted to let someone know about something, because they believed it was important, etc). Also, adding a new state in a Finite-State Machine architecture as the one proposed by [START_REF]Modelling human behaviours in disasters from interviews: application to melbourne bushfires[END_REF] would have required specifying the transitions between that new state and all the other states in the FSM. Adding communication is much more straightforward in the BDI architecture, where one just needs to write new plans that can be selected by the agent when they want to communicate. As a result, we chose to use a BDI architecture for the human agents in our model.

Selected simulation platform

GAMA [START_REF] Grignard | Gama 1.6: Advancing the art of complex agent-based modeling and simulation[END_REF] is an open-source platform for agent-based modelling and simulation, offering an integrated programming language and development framework to develop elaborated models with up to several million agents. GAMA is supported by an active and growing community and offers many benefits for our simulation:

• The GAMA Modelling Language (GAML) is a high level agent-based language based on Java, specifically designed to be easy to use even for non-computer scientists, allowing domain experts to create and maintain their own models.

• GAMA also provides native management of GIS (Geographical Information Systems) data allowing the integration of geographical data files into simulations.

• GAMA offers interactive functions (user commands) enabling the use of participatory dynamics and the involvement of the user in interactive simulations, which is useful when aiming at raising awareness.

• GAMA has both a GUI and a batch simulation mode: the GUI mode allows visualising one simulation as it unfolds, while the batch mode allows the repetition of several simulations with various parameter values and drawing statistics and graphs.

• GAMA now provides a BDI plugin [START_REF] Taillandier | A situated bdi agent architecture for the gama modelling and simulation platform[END_REF] to implement human agents with a BDI architecture.

We therefore chose to implement our model and simulation in the GAMA platform.

Our general model

There are different types of agents in our model: elements of the environment (land, rivers, roads, buildings, shelters); humans (civilians and fire-fighters); and exchanged messages.

Environment

The environment is built from OpenStreetMap1 (OSM) data for the town of Marysville, which greatly suffered from the 2009 bushfires. GAMA provides native functions to handle GIS data, that we used to implement a tool to create the agents in our environment (land, rivers, roads, buildings) from an OSM file. This provides us with a more realistic environment.

Firemen

This environment is populated with humans agents with two possible roles: civilians and fire-fighters. We are not yet interested here in the decision process of the fire-fighters nor their interactions together and with the population. In order to optimise the simulation, we therefore modelled them with a very simple reflex architecture: their only available behaviour is to head towards and fight fires. In particular, they cannot yet inform the population or give them recommendations, but this is to be added in future work.

Communication model

Messages exchanged by the human agents are represented by agents with the following attributes, based on communication theory [START_REF] Shannon | A mathematical theory of communication[END_REF]: sender, list of receivers, content (a predicate), channel. For now the possible messages are only informative, and although the possible channels include phone, radio and TV, our model only uses phone. In future work we intend to add face-to-face communication between neighbours and with local fire-fighters and policemen, as well as recommendation messages in addition to information.

Civilian model

In this paragraph, we detail the BDI model of civilians, which is the main focus of this paper.

Attributes and initialisation

Civilian agents have the following attributes:

• Physical attributes: ability, perception radius, speed, health, location, home address, work address, known shelters, known fires

• Psychological attributes: minimum home resistance required, sociability (size of social network), risk aversion, home attachment, altruism, obedience, inaction threshold (minimal motivation necessary to stop being passive)

• Communication attributes: focus on the different channels, list of neighbours, list of friends, list of calls made.

These attributes are initialised randomly for each agent at the start of the simulation, creating a heterogeneous population in terms of ability, willingness to communicate, etc. Initially the agents have no beliefs or desires.

Perception and reasoning

Civilians then observe their environment and can perceive other humans, fires, buildings, and messages sent to them. Their beliefs are updated following various rules, in particular:

• When the resistance of their home exceeds their required minimum, they adopt the belief that their house is ready (which is a necessary condition to defend it).

• When they detect or are informed about a fire, they adopt the belief that there are fires (they become aware).

• When they detect or are informed about a fire closer than 20m to their house, they adopt the belief that the fires are close (this is the trigger for some behaviours).

These beliefs in turn activate new desires. For instance becoming aware of the fires triggers the following desires:

• Desire to prepare themselves and their home.

• Desire to inform others about these fires, with a priority influenced by their level of altruism.

• If they are part of an organised phone tree: desire to inform others in the phone tree about the fires, with a priority influenced by their level of obedience to norms.

• If ready: desire to stay safe, with a priority based on risk aversion.

• If ready and home ready: desire to protect their home, with a priority influenced by home attachment.

When the fires get very close to their home, they change their desires again, dropping the desire to get ready (since there is no time left) and immediately adopting the two contradictory desires of escaping to safety (depending on risk aversion) and of protecting their home (depending on home attachment).

The reasoning engine then prioritises the desires depending on their respective intensity and on the context (distance to the fires, etc), in order to select an appropriate intention to pursue. For instance if home attachment is greater than risk aversion, the civilian will adopt the intention to protect their home. If no desire has a priority above the inaction threshold, the civilian does not adopt any intention and stays passive.

Plans and action

Civilians have the following plans to try to achieve their intentions:

• Prepare themselves: improve their resistance (this simulates wearing appropriate gear, etc), and make phone calls if needed;

• Prepare their home: improve the resistance of the building (this simulates turning on sprinklers, etc);

• Escape to a shelter: follow the roads towards the closest known shelter;

• Fight the fires: decrease the intensity of fires at a 20m distance around them, depending on their physical ability;

• Communicate with their family (not currently used)

• Propagate information about the fires: choose a receiver, try to call them (which can fail if the line is busy or the receiver is not sufficiently focused on the phone), and if they answer the phone, inform them about the fires.

Psychology and communication

The civilian agents therefore have several psychological attributes that influence the communication:

• Level of altruism: influences their willingness to share fire information with others (friends and neighbours), i.e. the priority of their desire to propagate fire information with the social and geographical strategies.

• Level of obedience to norms: influences their willingness to follow the phone tree, and to keep trying if calls fail, i.e. the priority of their desire to propagate fire information with the phone tree strategy.

• Focus on the phone channel: influences the probability that they answer the phone when someone calls them.

Communication modes

We have implemented four modes of organising communication in this simulated population: no communication at all, communication based on social relations (call one's friends), communication based on geographical proximity (call one's neighbours), and communication organised along a phone tree. The communication mode can be set by the user in the parameters of the simulation. Concretely, the communication mode modifies the selection of receivers in the communication plan of the civilian agents:

• No communication: delete receivers, no message is sent.

• Social communication: potential receivers are the agent's friends (the number of friends depends on the sociability attribute) who were not yet contacted. If an attempt fails, the agent might retry later but new calls have priority over re-calls.

• Geographical communication: potential receivers are the agent's neighbours at a distance of 200m. In future work this distance will be made a parameter to study the influence of the size of a neighbourhood.

• Phone tree: the potential receivers are the residents of the next home to call in the list. If the call fails several times, there is a backup strategy to skip that home and call the next one in the list. We implemented a circular phone tree where the agent will loop through the entire list if needed.

Experiments and results

Our simulation aims at comparing the modes of organising the communication within the population described above. The simulator is intended as a tool to raise awareness by providing interesting visualisations of information propagation, as well as a tool to investigate new communication modes.

Indicators

In order to evaluate each communication mode, we defined the following indicators:

• Number of communication failures: an agent tried to call but failed to reach the intended receiver (line busy, receiver not focused);

• Number of messages successfully sent: the phone call reached the receiver;

• Number of useful messages: the receiver did not already know the information received;

• Percentage of agents aware of the fires after 200 cycles, and/or number of simulation cycles before all agents are aware of the fires.

Experiment modes

We have implemented two different experiments described below: GUI mode (visualisation of information propagation in a population using one particular communication mode); batch mode (comparison of average results with four communication modes).

GUI mode

In GUI mode, the user interface has the following elements:

• Parameters: global level of altruism, strength of the fire, communication mode

• User commands: send official warning to one agent, start fires

• Map with civilian agents to observe behaviour (see Figure 3)

• Other dislpays: phone network, messages exchanged

• Graphs (see Figure 4) Figure 3 shows the user interface with the parameter selection window, the console, and the city map display. The agents change colour when they become aware of the fires, and an arrow is shown between the sender of a message and the receiver. This allows the user to visualise the propagation of information live. The user can interact with the simulation by sending the official warning (which will make one random initial agent aware of the fires) and/or starting fires (which can be perceived directly by some agents, who will then propagate their awareness).

The interface also offers some graphs to summarise interesting information. Figure 4 illustrates this display. At the top, a time series graph shows the evolution over time of the number of agents that are alive (green line) and aware (blue line). At the bottom left, a pie chart shows the percentage of agents that are communicating at a particular moment. At the bottom right, a histogram counts the number of failed interactions, successful interactions (phone call reached recipient), and useful interactions (information not already known).

GAMA batch experiments

In batch mode, we run 50 repeats of the simulation with each of the four communication modes, and compute average values of our indicators. The 50 repeats allow smoothing out the randomness of each simulation and obtaining more statistically significant values. Figure 5 illustrates the results.

These results show some interesting insights about the different modes of communication:

• No communication: only the agent receiving the official warning is aware, since the fires have not started yet to allow direct perception.

• Social communication is quick and motivated by people's altruism rather than obedience to norms. There is however a high redundancy (few useful messages) as the same people might be called several times by different friends. On the other hand some people might never be called if they are socially isolated, and the percentage of people aware reaches a maximum at about two thirds of the total population.

• Geographical communication is also influenced by altruism and solidarity. The information is local and therefore more relevant and useful. However the information does not spread out of the local neighbourhood, requiring warnings to be issued to each community. It is not the case in our model, so only a small percentage of the population is aware at the end of the simulation.

• Communication along a phone tree is the most efficient strategy because of the backup (if a call fails, the agent will call the next number, until someone is reached). However this backup means that an agent might call a number of an agent that was already informed before, reducing the percentage of useful messages. Without this backup, the tree might quickly stop when one agent does not answer, leaving the sequel of the list uninformed. This needs to balance precision (percentage of useful calls, to avoid loss of time and loss of interest or cognitive overload) with

Discussion and limitations

These first results are interesting but we would like to discuss some limitations. First, we have limited ourselves to interactions via phone, while in the interviews many residents mention having had direct interactions with their neighbours or with firemen or policemen going door-to-door to inform the population. Second, we have limited the content of our messages to information about fires, while other contents were mentioned: information about shelters, about safety status, about the fire plan; recommendations from the authorities about what to do; negotiation about what to do inside a family, etc.

Finally, for the sake of experimentation, we have only explored very "artificial" situations: official warning sent to a single agent, no direct perception of fires (focus on the prodromal phase when initially there is no fire), use of one single strategy (no combination). In fact, the residents can receive information about fires through both direct observation and indirect communication, and strategies can be combined: using a phone tree does not prevent calling one's friends or neighbours as well.

Many things also remain to explore in future work. In the short-term, we will investigate other types of phone trees (with or without backup, circular vs linear, etc). We will also investigate less artificial situations, and observe how communication evolves when the fire starts. We will also experiment with using more parameters: size of neighbourhood, best strategy depending on various levels of alruism vs obedience, etc.

In the medium-term, we will also enrich our communication model in several ways: add face-to-face interactions; endow the fire-fighter agents with more complex behaviour, in particular interactions with the population and with each other; implement different types and contents of messages; and link communication with more psychological parameters such as trust and emotions.

Conclusion

In this paper we have provided an agent-based model and simulation of the propagation of information in the population in the prodromal phase of a bushfire. The advantages of our approach is to use a BDI model of the population, providing a high-level description of their behaviour. The implemented simulator offers a visualisation of the propagation of awareness in the population before the fires actually starts, as well as measuring some interesting indicators. Despite the limitations discussed above, this work already provides interesting results about the benefits of a more organised communication, which was not in place during the 2009 bushfires.

This work opens many future avenues of work: Natural Language Processing techniques could be used to perform a deeper analysis of the interviews; more complex interactions should be investigated (not only information about the fires, but also about safety status, and about the decision to stay or evacuate within a family); and the influence of more parameters could also be studied, such as trust in the different sources of information, or the influence of emotions. In the longer-term we would also like to compare communication in different types of crises in order to generalise our model.

Figure 1 :Figure 2 :

 12 Figure 1: Percentage of interviews mentioning each type of interlocutor

Figure 3 :

 3 Figure 3: Interface of the simulator: parameters (top left), console (bottom left), city map (right)

Figure 4 :Figure 5 :

 45 Figure 4: Graphs obtained with geographical communication

https://www.openstreetmap.org/


Maxence Lepers

Hui Li

Jean-François Wyart

Goulven Quéméner

Olivier Dulieu

Ultracold rare-earth magnetic atoms with an electric dipole moment

We propose a new method to produce an electric and magnetic dipolar gas of ultracold dysprosium atoms. The pair of nearly degenerate energy levels of opposite parity, at 17513.33 cm -1 with electronic angular momentum J = 10, and at 17514.50 cm -1 with J = 9, can be mixed with an external electric field, thus inducing an electric dipole moment in the laboratory frame. For field amplitudes relevant to current-day experiments, i.e. an electric field of 5 kV/cm, we predict a large magnetic dipole moment up to 13 Bohr magnetons, and sizeable electric dipole moment up to 0.22 debye. When a magnetic field is present, we show that the induced electric dipole moment is strongly dependent on the angle between the fields. The lifetime of the field-mixed levels is found in the millisecond range, thus allowing for suitable experimental detection and manipulation.

Introduction. In a classical neutral charge distribution, a dipole moment appears with a separation between the barycenter of positive and negative charges [START_REF] Jackson | Classical Electrodynamics[END_REF]. An obvious example is provided by an heteronuclear diatomic molecule, which possesses a permanent dipole moment along its interatomic axis. It will manifest in the laboratory frame when such a molecule is placed in an external electric field, acquiring a preferred orientation along the direction of the field. Moreover, a neutral atom placed in an external electric field acquires a small dipole moment, as the spherical symmetry of space is broken. This effect is spectacularly maximized in Rydberg atoms, where the maximal induced dipole moment scales as n 2 , where n is the principal quantum number of the considered Rydberg state [START_REF] Gallagher | Rydberg atoms[END_REF].

At the single-particle scale, the external electric field mixes even and odd-parity levels of the energy spectrum: rotational levels for a diatomic molecule (see e.g. [START_REF] Brieger | [END_REF]), or levels with different orbital angular momenta for Rydberg atoms (see e.g. [4]). In both cases, this leads to a pronounced linear Stark shift on the energy levels, revealing the existence of a permanent dipole moment in the laboratory frame. More suprisingly, it has been observed that a homonuclear diatomic molecule formed by a Rydberg atom bound to a ground state atom located inside the Rydberg electron orbital, can exhibit a permanent electric dipole moment in the laboratory frame [5,6].

The search for such dipolar systems, involving especially lanthanide atoms, is currently very active in the context of ultracold dilute gases [7][8][9][10][11][12][13][14][15][16][17][18]. Indeed, the particles of the gas interact through a highly anisotropic long-range potential energy varying as the inverse cubic power of their spatial separation [START_REF] Stone | The Theory of Intermolecular Forces[END_REF][START_REF] Lepers | Cold Chemistry: Molecular Scattering and Reactivity Near Absolute Zero[END_REF]. Prospects related to many-body physics, quantum simulation and ultracold chemistry are nowadays within reach experimentally [START_REF] Baranov | [END_REF][22][23]. A particular attention is paid to gases with an electric and a magnetic dipole moment, which up to now consist of paramagnetic polar diatomics [24][25][26][27][28][29][30][31][32][33].

In dysprosium (Dy), there exists a pair of quasidegenerate opposite-parity levels at 19797.96 cm -1 , split by less than 0.01 cm -1 , possessing the same electronic angular momentum J = 10, and which have been employed for fundamental measurements [34][35][36][37][38][39][40]. However, the reduced transition dipole moment (TDM) coupling those two levels, equal to 0.038 D [35] (1 D = 0.393456 ea 0 , e being the unsigned electron charge and a 0 the Bohr radius), is not sufficient to observe dipolar effects. On the contrary, we demonstrate in the present work, that the odd-parity level |a at E a = 17513.33 cm -1 with J a = 10 and main configuration [Xe]4f 10 6s6p ([Xe] being the xenon core), and the even-parity level |b at E b = 17514.50 cm -1 with J b = 9 and main configuration [Xe]4f 10 5d6s [41], which present a reduced TDM of 8.16 D, are promising candidates to produce an electric and magnetic dipolar gas of ultracold atoms.

In this Letter, we set up the effective Hamiltonian for the two opposite-parity excited levels |a and |b in the presence of electric and magnetic fields with arbitrary relative orientation. We use energies and g-factors from Ref. [START_REF] Kramida | NIST Atomic Spectra Database (ver. 5.3)[END_REF], and TDMs and radiative lifetimes based on our own well-established semi-empirical electronic-structure calculations. For an experimentally accessible electric field of 5 kV/cm and magnetic field of 100 G, we predict a field-mixed level with a permanent magnetic dipole moment (PMDM) of µ * = 13 µ B (with µ B the Bohr magneton), to our knowledge the largest value observed in ultracold experiments, and an induced electric dipole moment (IEDM) of d * = 0.22 D, which is similar to the value obtained in the pioneering experiment on KRb [START_REF] Ni | [END_REF]. We demonstrate a strong control of the IEDM, which ranges from 0 to d * as a function of the angle between the fields. We also calculate the atomic radiative lifetime as functions of the fields parameters, and obtain a few milliseconds for the level characterized by µ * and d * .

Finally, we show that our method is applicable for all bosonic and fermionic isotopes of Dy.

Model. We consider an atom in two energy levels |a and |b , of energies E i and total angular momentum J i (i = a, b). Here E a = 17513.33 cm -1 , J a = 10 and E b = 17514.50 cm -1 , J b = 9 [START_REF] Kramida | NIST Atomic Spectra Database (ver. 5.3)[END_REF]. Firstly, we consider bosonic isotopes which have no nuclear spin, I = 0. In absence of field, each level |i is (2J i + 1)-time degenerate, and the corresponding Zeeman subslevels are labeled with their magnetic quantum number M i . The atom is submitted both to a magnetic field B = Be z , with e z the unit vector in the z direction, taken as quantization axis, and to electric field E = Eu, with u a unit vector in the direction given by the polar angles θ and φ = 0. In the basis

{|M a = -J a , ..., | + J a , |M b = -J b , ..., | + J b } spanned
E i Ji Mi=-Ji |M i M i | + ŴZ + ŴS . (1)
The Zeeman Hamiltonian ŴZ only contains diagonal terms equal to M i g i µ B B, with g a = 1.30 and g b = 1.32 the Landé g-factors [START_REF] Kramida | NIST Atomic Spectra Database (ver. 5.3)[END_REF]. The last term of Eq. ( 1) is the Stark Hamiltonian, which couples sublevels |M a with sublevels |M b as

M a | ŴS |M b = - 4π 3(2J a + 1) a d b E × Y * 1,Ma-M b (θ, 0) C JaMa J b M b ,1,Ma-M b , (2)
where a d b is the reduced TDM, Y kq (θ, φ) a spherical harmonics and C cγ aαbβ a Clebsch-Gordan coefficient [START_REF] Varshalovich | Quantum theory of angular momentum[END_REF]. For given values of E, B and θ, we calculate the eigenvalues E n and eigenvectors

|Ψ n = i=a,b Ji Mi=-Ji c n,Mi |M i (3)
of the Hamiltonian in Eq. ( 1). The reduced TDM a d b and the natural line widths γ i are calculated using the method developed in our previous works [START_REF] Lepers | [END_REF][46][47][48], based on the least-square fitting procedure of energies present in the Cowan codes [START_REF] Cowan | The theory of atomic structure and spectra[END_REF], and extended by us to Einstein coefficients. Firstly, oddlevel energies are taken from Ref. [47] which include the electronic configurations [Xe]4f 10 6s6p and [Xe]4f 9 5d6s 2 . Even-level energies are calculated with the configurations [Xe]4f 10 6s 2 , [Xe]4f 10 5d6s and [Xe]4f 9 6s 2 6p [START_REF] Wyart | [END_REF]. Secondly, following Ref. [47], we adjust the monoelectronic TDMs by multiplying their ab initio values by appropriate scaling factors [46], equal to 0.794 for 6s|r|6p , 0.923 for 4f |r|5d and 0.80 for 5d|r|6p [START_REF]This value is arbitrary[END_REF]. From the resulting Einstein coefficients, we can extract a d b = 3.21 a.u., as well as the natural gauss, the two Zeeman manifolds overlap; but because the magnetic field conserves parity, the sublevels of |a and |b are not mixed. Provoking that mixing is the role of the electric field.

On figure 1(b), we plot the 21 lowest eigenvalues of Eq. ( 1) as functions of the electric field for B = 100 gauss and θ = 0 • . We focus on the eigenstates converging to the sublevels of |a when E → 0. In the range of field amplitudes chosen in Figs. 1(a) and (b), which corresponds to current experimental possibilities, the influence of E is much weaker than the influence of B. On Fig. 1(b), the energies decrease quadratically with the electric field, because the sublevels of |a are repelled by the sublevels of |b . Since θ = 0 • , the z component of the total angular momentum is conserved, and so, the sublevels for which M a = M b are coupled in pairs. In consequence, the sublevels |M a = ±10 are insensitive to the electric field, as they have no counterparts among the sublevels of |b (recalling that J b = 9).

The only way to couple the |M a = ±10 sublevels to the other ones is to rotate, say, the electric field, and thus break the cylindrical symmetry around the z axis. On figure 1(c), the 21 lowest eigenvalues of Eq. ( 1) are now shown as function of the angle θ, for fixed field amplitudes, E = 5 kV/cm and B = 100 gauss. Even if the corresponding eigenvectors are not associated with a single sublevel |M i (unlike Figs. 1(a) and (b)), they can conveniently be labeled |M i after their field-free or θ = 0 counterparts. For a given eigenstate, the θ-dependence of energy is weak. However for |M a = ±10 , the energy decrease reveals the repulsion with sublevels of |b , which is maximum for θ = 90 • .

Permanent magnetic and induced electric dipole moments. The z component of the PMDM associated with the eigenvector |Ψ n is equal to

µ n = -µ B i=a,b g i Ji Mi=-Ji |c n,Mi | 2 M i . (4)
Since the eigenvectors are mostly determined by their field-free counterparts, µ n does not change significantly in our range of field amplitudes; it is approximately equal to µ n ≈ -M a g a µ B for n ∈

d n = - 1 E Ma,M b c * n,Ma c n,M b M a | ŴS |M b + c.c. , (5)
where the matrix element of ŴS is given in Eq. ( 2). graph is symmetric about the y axis. All the curves increase linearly with E; all, except the lowest and highest ones, correspond to two eigenstates. In agreement with Fig. 1(b), the curve d n = 0 is associated with |M a = ±10 (n = 1 and 21). The lowest and highest curves belong to M a,b = 0, for which by contrast, the PMDM vanishes.

As shows figure 2(b), the IEDMs change dramatically as functions of the angle θ. In particular, the IEDM of the eigenstate |M a = -10 ranges continuously from 0 to a maximum d * = 0.224 D for θ = 90 • . The eigenstate |M a = 10 follows a similar evolution, except that its curve is sharper around its maximum. In contrast, the IEDM of the eigenstate |M a = 0 , which is the largest for θ = 0 • , becomes the smallest for 90 • . Compared to the eigenstates |M a , the curves corresponding to the eigenstates |M b exhibit an approximate reflection symmetry around the y axis. Finally, it is important to mention that the influence of the magnetic field on the IEDMs is weak in the amplitude range of Fig. 1(a).

Radiative lifetimes. The radiative lifetime τ n = 1/γ n associated with eigenvector |Ψ n is such that γ n is an arithmetic average of the natural line widths of |a and |b , Figure 3 displays the lifetimes of all eigenstates of Eq. ( 1) as functions of the angle θ for E = 5 kV/cm and B = 100 G. Because the natural line widths γ a and γ b differ by 6 orders of magnitude, the lifetimes τ n are also spread over a similar range. At the field amplitudes of Fig. 3, the eigenvectors |M a are composed at least of 90 % of sublevels of |a , and similarly for eigenstates |M b . Therefore, the lifetimes of eigenstates |M b (not shown on Fig. 3) are approximately 1/γ b , and they weakly depend on θ. As for the eigenstates |M a , their small |b components, say ε, induces lifetimes roughly equal to ≈ τ b /ε 2 .

τ n = γ -1 n =   i=a,b γ i Ji Mi=-Ji |c n,Mi | 2   -1 . (6)
For |M a = ±10 , the lifetime ranges from τ * = 4.22 ms for θ = 90 • to τ a = 28.1 s for θ = 0 • . Again, this illustrates that the coupling with the sublevels of |b is maximum for perpendicular fields and absent for colinear ones.

The inset of figure 3 shows the lifetime of the eigenstate |M a = -10 as function of E. In this range of field amplitude, τ 1 scales as E -2 . So, a large amplitude E can strongly affect the lifetime of the atoms; but on the other hand, E needs to be sufficient to generate a sizeable IEDM. So, there is a compromise to find between IEDM and lifetime, by tuning the electric-field amplitude and the angle between the fields.

Fermionic isotopes. There are two fermionic isotopes of dysprosium, 161 Dy and 163 Dy, both with a nuclear spin I = 5/2. A given hyperfine sublevel is characterized by the total (electronic+nuclear) angular momentum F i and its z-projection M Fi , where |J i -I| ≤ F i ≤ J i + I, and -F i ≤ M Fi ≤ F i . Namely, F a ranges from 15/2 to 25/2, and F b ranges from 13/2 to 23/2. The hyperfine sublevels are constructed by angular-momentum addition of J i and I, i.e.

|F i M Fi = MiM I C FiM F i JiMiIM I |J i M i |IM I .
Compared to Eq. ( 1), the Hamiltonian Ĥ is modified as [54] and minimal lifetime τ * obtained for different isotopes of dysprosium for an electric field E = 5 kV/cm, a magnetic field B = 100 G and an angle θ = 90 2) and text above). After diagonalizing Eq. ( 7), one obtains 240 eigenstates (compared to 40 in the bosonic case). Despite their large number of curves, the plots of energies, IEDMs and lifetimes show similar features to figures 1-3. The eignestates |Ψ n can be labeled |F i M Fi after their field-free counterparts |F i M Fi . Moreover, the "stretched" eigenstates |F a M Fa = |25/2, ±25/2 are not sensitive to the electric field for θ = 0 • , and maximally coupled for θ = 90 • ; and so, their IEDMs range from 0 up to d * and their lifetimes from τ a down to τ * . As shows Table I, for the same field characteristics, the values of d * and τ * are very similar from one isotope to another.

Ĥ = i=a,b Fi E Fi M F i |F i M Fi F i M Fi | + ŴZ + ŴS (7)
Table I also contains the so-called dipolar length a d = m(d * / ) 2 [54]. It characterizes the length at and beyond which the dipole-dipole interaction between two particles is dominant. For the 161 Dy isotope, one can reach a dipolar length of a d = 2299 a 0 . To compare with, at E = 5 kV/cm and an IEDM of 0.22 D [START_REF] Ni | [END_REF], 40 K 87 Rb has a length of a d = 1734 a 0 . Similarly, a length of a d = 1150 a 0 was reached [11] for magnetic dipolar Feshbach molecules of 168 Er 2 . Therefore one can obtain comparable and even stronger dipolar character with excited Dy atoms than with certain diatomics. However, the collisions are likely to affect the sample lifetime, unless the atoms are loaded in an optical lattice.

Conclusion. We have demonstrated the possibility to induce a sizeable electric dipole moment on atomic dysprosium, in addition to its large magnetic dipole moment. To do so, the atoms should be prepared in a superposition of nearly degenerate excited levels using an electric and a magnetic field of arbitrary orientations. We show a remarkable control of the induced electric dipole mo-ment and radiative lifetime by tuning the angle between the fields. Being metastable, those two levels are not accessible by one-photon transition from the ground level. Instead, one could perform a two-photon transition to a high-lying even-parity level and then rely on spontaneous emission towards |a [35], or a Raman transition between the ground level |g (J g = 8) and the level |b (J b = 9), through the upper levels at 23736.61, 23832.06 or 23877.74 cm -1 , whose [Xe]4f 10 6s6p character insures significant transition strengths with |g and |b . In the spectrum of other lanthanides, there exist pairs of quasi-degenerate levels accessible from the ground state, for instance the levels at 24357.90 and 24660.80 cm -1 in holmium, but in turn their radiative lifetime is much shorter [55].

  by the Zeeman sublevels of |a and |b , the Hamiltonian can be written Ĥ = i=a,b

  FIG. 1. Eigenvalues of the atom-field Hamiltonian (1) as functions of: (a) the magnetic field B for vanishing electric field and angle E = θ = 0; (b) the electric field E for B = 100 G and θ = 0 • ; (c) the angle θ for B = 100 G and E = 5 kV/cm. In panels (b) and (c), the origin of energies is taken at (Ea + E b )/2 = 17513.92 cm -1 . The blue curve with crosses corresponds to the eigenstate converging towards |Ma = -10 when θ → 0.

  Figure 1(a) shows the eigenvalues of the Hamiltonian (1) as functions of the magnetic field for E = θ = 0. The field splits levels |a and |b into 21 and 19 sublevels respectively, each one associated with a given M a or M b ; from now on, we emphasize the lowest sublevel |M a = -10 . Due to the close Landé g-factors, the two Zeeman manifolds look very similar, i.e. the branches characterized by the same values M a = M b are almost parallel. For B ≥ 1000

[ 1 ;

 1 21] and µ n ≈ -M b g b µ B for n ∈ [22; 40]. For instance, the state |M a = -10 has the maximal value µ * = 13.0 × µ B . The mean IEDM d n = Ψ n |d • u|Ψ n associated with the eigenvector |Ψ n in the direction u of the electric field is

FIG. 2 .

 2 FIG. 2. Induced electric dipole moments, see Eq. (5), associated with the eigenstates of the atom-field Hamiltonian (1) as functions of: (a) the electric field E for B = 100 G and θ = 0 • ; (b) the angle θ for B = 100 G and E = 5 kV/cm. The blue curve with crosses corresponds to the eigenstate |M a = -10 .

FIG. 3 .

 3 FIG. 3. Radiative lifetimes, see Eq. (6), associated with the eigenstates |M a as functions of the angle θ for B = 100 G and E = 5 kV/cm. The blue curve with crosses corresponds to the eigenstate |M a = -10 . The inset shows the lifetime of this eigenstate as a function of E for B = 100 G and θ = 90 • .

TABLE I .

 I Maximal induced electric dipole moment d * , dipolar length a d

  • . The results of 162 Dy are also valid for the other bosonic isotopes 156 Dy, 158 Dy, 160 Dy and 164 Dy. Fi is the hyperfine energy depending on the magnetic-dipole and electric-quadrupole constants A i and B i . For 163 Dy, they have been calculated in Ref.[START_REF] Dzuba | [END_REF]:A a =225 MHz, B a = 2434 MHz, A b = 237 MHz and B b = 706 MHz. For 161 Dy, we apply the relations ( 161A i ) = -0.714 × ( 163 A i ) and ( 161 B i ) = 0.947 × ( 163 B i )given in Ref.[53]. The matrix elements of the Zeeman ŴZ and Stark Hamiltonians ŴS are calculated by assuming that they do not act on the nuclear quantum number M I , and by using the formulas without hyperfine structure (see Eq. (

		d * (D)	a d (a0)	τ * (ms)
	161 Dy	0.225	2299	4.18
	162 Dy	0.224	2293	4.22
	163 Dy	0.222	2266	4.23
	where E

Acknowledgments. We acknowledge support from "DIM Nano-K" under the project "InterDy", and from "Agence Nationale de la Recherche" (ANR) under the project "COPOMOL" (contract ANR-13-IS04-0004-01). We also acknowledge the use of the computing center "MésoLUM" of the LUMAT research federation (FR LUMAT 2764).