Abstract
 We propose an opinion dynamics model that combines processes of vanity and opinion propagation. The interactions take place between randomly chosen pairs. During an interaction, the agents propagate their opinions about themselves and about other people they know. Moreover, each individual is subject to vanity: if her interlocutor seems to value her highly, then she increases her opinion about this interlocutor. On the contrary she tends to decrease her opinion about those who seem to undervalue her. The combination of these dynamics with the hypothesis that the opinion propagation is more efficient when coming from highly valued individuals, leads to different patterns when varying the parameters. For instance, for some parameters the positive opinion links between individuals generate a small world network. In one of the patterns, absolute dominance of one agent alternates with a state of generalised distrust, where all agents have a very low opinion of all the others (including themselves). We provide some explanations of the mechanisms behind these emergent behaviors and finally propose a discussion about their interest.
 Keywords:
 Opinion Dynamics, Vanity, Leviathan
Introduction

"Again, men have no pleasure (but on the contrary a great deal of grief) in keeping company where there is no power able to overawe them all. For every man looketh that his companion should value him at the same rate he sets upon himself, and upon all signs of contempt or undervaluing naturally endeavours, as far as he dares (…), to extort a greater value from his contemners, by damage; and from others, by the example." T. Hobbes, Leviathan, chapter 13.
 1.1
 In this paper, our goal is to revisit Hobbes thesis using opinion dynamics models and computer simulations. We propose a simple model including a vanity process, inspired from the above citation, in which agents measure themselves in the eyes of the others and retaliate against this judgement. Then, we observe the collective patterns that emerge for different values of the parameters.
 1.2
 Our approach is in the line of many others in the field of social simulation or in sociophysics. It consists in making a few simple assumptions about the rules of interactions between agents and then studying the obtained emerging behaviors. In some of such models, the agents have binary opinions (SznadjWeron 2005; Galam 2008), while in others the opinions are continuous (Gargiulo & Huet 2010; Deffuant et al. 2000; Huet & Deffuant 2010; Fortunato 2004; Urbig et al. 2008; Huet et al. 2008; Deffuant 2006; Lorenz 2007) (see Castellano et al. 2009 for a review). Our model is closer to recent ones which include a set of affinities between agents, leading to emerging networks (Bagnoli et al. 2007; Carletti et al. 2011).
 1.3
 In the proposed model we assume that each agent can have a continuous opinion about every other agent and we truncate it to remain between 1 and +1. In the initial state, we suppose that the agents don't have any opinion about the others. The agents interact by randomly chosen pairs and two different processes apply. The first one supposes that during any interaction, each agent propagates her opinions about herself, about her interlocutor and about several randomly chosen other agents that she knows. In this propagation, highly valued agents are more influential. The second process represents a vanity effect: an agent likes to be highly valued by the others, thus she increases her opinion on those who value her well. On the contrary she decreases her opinion about those who undervalue on her. These assumptions are inspired by Hobbes, but also by more recent experiments and observations from socialpsychologists (Stephan & Maiano 2007; Wood & Forest 2011; Buckley et al. 2004; Srivastava & Beer 2005; Leary et al. 2006; Fein 1997). Moreover, we suppose that the access to the opinion of the others is not perfect: people may not express exactly what they think and the listener may misinterpret these expressions. To take this into account in the model, the propagated opinions are distorted by some noise.
 1.4
 Despite its simplicity, the model shows a surprising variety of patterns when changing the parameters. We identified the following five major ones:
 Equality. Each agent has a positive opinion about herself, she is connected by strong positive mutual opinions with a small set of agents and has very negative opinions about all the others. All agents have a similar number of positive (and negative) links. For some parameters, the network of positive links shows the characteristics of small world networks.
 Elite. The pattern shows two categories of agents: the elite and second category agents. The elite agents have a positive selfopinion and are strongly supported by a friend, but they have a very negative opinion of all the other elite agents and of all the second category agents. The second category agents have a very negative selfopinion, they have a very negative opinion of all the other second category agents and their opinion about the elite agents is moderate.
 Hierarchy. All agents share a similar opinion about every other agent (called reputation) and the reputations are widely spread between 1 and +1. There are more agents of low reputation than of high reputation: this gives the image of a classical hierarchy with a wide basis and progressively shrinking when going up to the top.
 Dominance. As in the hierarchy pattern, all agents share a similar opinion about every other agent, but a single agent has a high reputation while most of the other agents have a very low reputation.
 Crisis. Each agent has a very negative opinion of all the others and of herself.
 1.5
 Moreover, we observe also mixtures of these patterns in a same run, in particular crisis and dominance patterns alternate in some simulations. These mixtures of patterns add significantly to the richness of the model. We believe that this richness cannot be explored in details in a single paper. Therefore, our goal is rather to draw a global view of the main features of this model and to derive the main theoretical explanations of some of them. Doing so, we identify several issues that would require more detailed studies.
 1.6
 The paper is organised as follows. We firstly describe in details the dynamics of the model. Then we describe the main patterns obtained and some maps of their presence in the parameter space. We then propose some theoretical or qualitative explanations of these patterns. Finally, we propose a discussion about the potential use of the model in philosophical or sociological debates.
The dynamics of the model
 2.1
 This section describes the model in details and particularly its evolution rules representing opinion propagation and vanity.
 2.2
 We consider a set of N agents, each agent i is characterised by her list of opinions about the other agents and about herself: (a_{i,j})_{1 ≤ i,j ≤ N} .
 2.3
 We assume a_{i,j} lies between 1 and +1 , or it is equal to nil if the agent i never met j and nobody talked to i about j yet. At the initialisation, we suppose that the agents never met, therefore all their opinions are set to nil.
 2.4
 The individuals interact by uniformly randomly drawn pairs (i, j) and at each encounter, we apply both processes: opinion propagation and vanity.
 2.5
 Let us remark that we always keep the opinions between 1 and +1 , by truncating them to 1 if their value is below 1 after the interaction, or to +1 if their value is above +1 . Moreover, in the following, we consider that one iteration, i.e. one time step t →t + 1, is N/2 random pair interactions (each individual interacts N times on average during one iteration).
Opinion propagation
 2.6
 Let us assume that agents i and j have been drawn. During an encounter, we suppose that agent j propagates to i her opinions about herself (j), about i, and about k agents randomly chosen among her acquaintances. Moreover, we suppose that if i has a high opinion of j, then j is more influential.
 2.7
 This hypothesis is implemented by introducing a propagation coefficient, denoted p_{i,j}, which is based on the difference between the opinion of i about j (a_{i,j}) and the opinion of i about herself (a_{i,i}). It uses the logistic function with parameter σ . If a_{i,j} = nil (j is unknown to i), we assume that i has a neutral opinion about j and we set a_{i,j} ← 0. Let us also observe that, at the initialisation an agent has no opinion about herself (a_{i,i} = nil), before she takes part of a first encounter, thus we also set a_{i,i } ← 0. Then we compute the propagation coefficient p_{i,j}, which rules the intensity of the opinion propagation from j to i :
(1)  2.8
 Figure 1 represents the value of p_{i,j} when the difference a_{i,j } a_{i,i } varies (between 2 and +2), for three different values of parameter σ . One can observe that p_{i,j} tends to 1 when a_{i,j } a_{i,i} is close to 2 (i values j higher than herself), and tends to 0 when it is close to 2 (i values j lower than herself). The influence of j on i is then expressed as follows ( ρ is a parameter of the model ruling the importance of opinion propagation):
(2) and
(3) Where we denoted by Random ( δ , δ ) a uniformly distributed random number between  δ and + δ , that can be seen as a noise that distorts the perception that i has about j 's opinions. The parameter δ rules the amplitude of this noise.
Figure 1. Examples of variations of the propagation coefficient p_{i,j} when a_{i,j}  a_{i,i} varies, and for three values of parameter σ . Moreover, j propagates her opinion about (at most) k of her acquaintances. More precisely, let n_{j} be the number of acquaintances of j different from i (number of individuals q such that a_{j,q} is not nil and j ≠ i )^{[1]}. We choose at random with reinsertion min(k, n_{j}) agents among j 's acquaintances (i.e. an acquaintance of j can be selected several times, while others are not selected).
 2.9
 The propagation to i of j 's opinion about q is expressed thus by:
(4) and this will be repeated for every q selected at random.
 2.10
 In the interaction, we apply the influence of j on i and then the reciprocal one of i on j. Indeed, we suppose that the interactions take place in both ways (from i to j and from j to i), the order being randomly chosen.
Vanity dynamics
 2.11
 The second ingredient of the model is the dynamics representing agent's vanity. This dynamics expresses that agents tend to reward the agents that value them more positively than they value themselves and to punish the ones that value them more negatively than they value themselves.
 2.12
 We assume that the modification of i's opinion about j is simply linear with the difference between the opinion of i about herself and the opinion of j about i (slightly modified randomly):
(5)  2.13
 If i has a lower (resp. higher) selfopinion than her perception of the opinion j has about her (i), then a_{i,j} is increased (resp. decreased). Parameter ω rules the importance of the vanity dynamics. We also assume that the intensity of the vanity effect is independent from the opinions. Indeed, it is possible that one forgives more easily a disappointment coming from a loved person because precisely of this affection. But on the other hand, it is well known that hatred can also easily come from disappointed love. Since there is no clear intuition in one way or the other, we made the simplest assumption that the vanity effect is linear with the difference of opinions.
Summary
 2.14
 The Algorithm 1 describes one iteration: N/2 random pairs of individuals are drawn, with reinsertion, and we suppose that each individual influences the other during the encounter. Algorithm 2 codes the interaction with the two aspects of the dynamics: opinion propagation and vanity.
Algorithm 1: Iteration
for N/2 times do ;
i ← Random(1,N) ;
j ← Random(1,N, ≠i) ;
Interaction(i,j) ;
Interaction(j,i) ;
endAlgorithm 2: Interaction(j,i)
if a_{i,i} = nil then a_{i,i} ← 0 ;
if a_{i,j} = nil then a_{i,j} ← 0 ;
if a_{j,i} = nil then a_{j,i} ← 0 ;
if a_{j,j} = nil then a_{j,j} ← 0 ;
ρ(i,j) ← 1/(1+exp((a_{i,j} a_{i,i})/σ) ) ; // computing propagation coefficient
a_{i,i } ← a_{i,i} + ρ p_{i,j}(a_{j,i}a_{i,i} + Random(δ,δ)) ; // j propagates a_{j,i}
a_{i,j} ← a_{i,j } + ρ p_{i,j}(a_{j,j}a_{i,j} + Random(δ,δ)) ; // j propagates a_{j,j}
for min(k,n_{j}) times do ; // n_{j }number of individuals known by j
q ← Random(KnownAgents(j), ≠i, ≠j); // q random known by j
if a_{i,q} = nil then a_{i,q} ← 0;
a_{i,q } ← a_{i,q }+ ρ p_{i,j}(a_{j,q}a_{i,q} + Random(δ,δ)) ; // j propagates a_{j,q}
end
a_{i,j } ← a_{i,j}+ ω (a_{j,i}a_{i,i} + Random(δ,δ)) ; // vanity dynamics, i modifies a_{i,j}
 2.15
 Summarizing the parameters of the model are thus:
 N, number of agents;
 ρ , ruling the intensity of the opinion influence;
 ω , ruling the intensity of the vanity;
 k , number of acquaintances about which the pair of agents discuss in the opinion influence;
 δ , intensity of noise perturbing the evaluation of other's opinions;
 σ , ruling the slope of the logistic function determining the propagation coefficients.
Representations of the population state
 2.16
 We use two different representations of the population state:
 Matrix representation: the opinion list of each agent is represented as the row of a N×N square matrix. The element a_{i,j} from line i and column j is the opinion of agent i about agent j. We use colours to code for the opinions: blue for negative and red for positive opinions with light colours meaning that the absolute value is close to 0. This representation provides all the information about the state of the population at a given time step, but it is sometimes difficult to interpret.
 Network representation: we represent the agents as nodes of a network, in a 2D space and we draw links between two agents, only when one of the agents has a positive opinion about the other. The position in the 2D space of the nodes representing the agents is obtained by a dynamical algorithm, which at each interaction moves the nodes in order to get a distribution where the distances between the nodes are as close as possible to the values of a simple function depending on the sum of the relative agents' opinions. The function yields a large value when two agents have negative opinions about each other, and reciprocally, a small value when their opinions about each other are positive. When the sum of the opinions is close to 0, the function yields a medium value. The colour of the links is yellow when the link is close to 0 and it gets close to red when the value of the link is close to 1. This representation does not show all the information about the state, because we do not represent the negative links nor the asymmetry of the links. However, some features of the emergent structures appear more easily with this representation.
The observed patterns and their sensitivity to parameters
 3.1
 This section describes the five main prototypical patterns that we use to describe the behaviour of the model. Some patterns of the model are mixtures of these main ones; they are described in a following section.
The main patterns
Equality
 3.2
 Figure 2 shows, using both matrix and network representations, three states of the model for N = 40, ω = 0.3, ρ = 0.01, δ = 0.2, σ = 0.35, k = 5. One can observe two phases before reaching a stable state:
 In a few hundred iterations, the model reaches a state in which the opinion matrix is symmetric, with only extreme values (1 or +1). Each agent separates the population between two groups: the ones she hates and the ones she loves. The first group is larger than the second (more blue squares in the matrix representation at t = 1000). Another feature is striking: the opinions on the diagonal are significantly positive: the agents have a good opinion of themselves (though not as good as their opinion about their loved ones).
 During a few thousand iterations, many positive connections are progressively destroyed before reaching a stable state (we tested this stability during several hundred thousand iterations). Looking at the network representation of the stationary state (iteration 50000), we observe the presence of small connected sets made by 3 or 4 individuals. The sets tend to be located far from each other (on a circle), indicating that each connected set has a negative opinion about the others (as shown in the matrix representation).
 3.3
 When doing other runs with the same parameters, we always get a final stable state with a similar number of friends in the networks. This pattern is typically obtained when the vanity is dominating over the opinion propagation (we describe the positions of the patterns in the parameter space later). For some parameters, the network of friends shows the properties of small world networks (this is the case of the one shown on Figure 2). We call this pattern "'equality"' because all the agents are in a similar situation.
Figure 2. Example of equality pattern. Parameters are: N = 40, ω = 0.3, ρ = 0.01, σ = 0.35, k = 5, δ = 0.2. The final average degree of the network of positive links is 3.4, the clustering coefficient is 0.25, whereas in a random network with the same number of nodes and average degree, the clustering coefficient is 0.08 (average over 10 replicas). The mean shortest path is 3.05 which is smaller than the mean shortest path of the equivalent random network (5.02 average on 10 replicas). See Section 2.4 for general explanation about the representations. Elite
 3.4
 In this pattern, shown on Figure 3, we note that after a while, there are two categories of agents:
 elite agents have a very positive selfopinion, they have a good friend who is also in the elite, but all the other members of the elite are their foes. These appear as red nodes in the centre of the network representation for t = 5000 and t = 50000 . In the matrix representation, they correspond to the red squares of the diagonal.
 second category agents have a very negative selfopinion, they have a very negative opinion of the other second category agents, and they have a moderate opinion of the members of the elite (which can be moderately negative or moderately positive according to the parameters). These agents are represented as blue nodes on a circle in the network representation of Figure 3. In this case, the opinions of the second category agents about the elite are moderately positive (yellow links). These links appear as partial light columns in the matrix representation. These columns are cut with blue squares which correspond to the negative opinions that second category agents have about each other.
 3.5
 We notice that the evolution from the initial state is also very different from the one of the equality pattern. Indeed, we see that after 1000 iterations, the matrix is less symmetric than in the equality pattern after the same number of iterations. Then we observe that the structure with the two categories of agents is already reached at t = 5000. This structure is rather stable, it has not changed much at t = 50000. However, one can obtain configurations where the elite agents often become second category and vice versa, with different values of the parameters.
Figure 3. Example of elite pattern. N = 60, ω = 0.3 and ρ = 0.1, δ = 0.2, σ = 0.3, k = 5. Two types of agents appear, the elite agents (in the middle of the network representation) have a positive selfopinion. The second category agents have a negative selfopinion. See section 2.4 for general explanation about the representations. Hierarchy
 3.6
 When the opinion propagation gets stronger than in the previous examples, all agents tend to have the same opinion about each other agent. This is shown for instance on Figure 4 obtained using N = 40, ω = 0.2 and ρ = 0.5, δ = 0.2, σ = 0.3, k = 10. In the matrix representations, this is visualized by cells with similar colours in the same column. As explained more formally later, this agreement of opinions takes place because of the strength of the averaging dynamics in the opinion propagation. We call the reputation r_{i} of agent i the average of the opinions about this agent. The random noise added to the opinion propagation introduces some random fluctuations of the opinions around the average value of the reputation and also some fluctuations of this average value.
 3.7
 Moreover, if we observe the distribution of the reputations, it appears that the number of agents is decreasing with the reputation. This gives an image of a hierarchy of reputations with a large basis and a progressively shrinking number of agents while the reputation is increasing. The average opinion is significantly negative, reflecting the higher number of low reputation agents in the population. It can be observed on the matrix representation that agents with the highest reputation are continuously changing: they can drop after a while to the basis of the hierarchy and conversely an agent of the lowest reputation can reach the top of the hierarchy.
 3.8
 Moreover, the distribution itself is not fully stable. From time to time, we observe a few agents that have a reputation close to 1, whereas all the other reputations are close to 1. The highest reputation agents generally rapidly lose their high reputation which decreases to medium levels (0.5 or even a bit lower), and then the pattern with high reputations appears again. For instance, the highest reputation at t = 20000 is higher than at t = 50000.
Figure 4. Example of hierarchy pattern. N = 40, ω = 0.2 and ρ = 0.5, δ = 0.2, σ = 0.3, k = 10. See section 2.4 for general explanation about the representations. a is the average opinion at the considered iteration. We observe that it is significantly negative. Dominance
 3.9
 For some values of the parameters, we observe one agent who is highly valued by all the others (the one corresponding to the red column in the matrix representation and the red node at the centre of the network representation in Figure 5 for t = 10000 and t = 50000 ), whereas most of the other agents are homogeneously viewed negatively. Hence one can consider that the agent with the highest reputation strongly dominates the group. This dominating agent (called afterwards the leader for short) is metastable; after a while the leadership is taken by another agent. For instance, during the simulation illustrated by Figure 5, the leader changed 3 times in 50000 iterations. The transition between the leaders takes place when a second agent gets a reputation as high as the one of the current leader, then there is a competition between these two agents and only one remains the leader. In this example a leader maintains her dominance for typically more than 10000 iterations. There is no transition phase between the leader and her successor, as we can observe in another pattern (mixed pattern described later).
Figure 5. Example of dominance pattern. ω = 0.4, ρ = 0.8, N = 40, δ = 0.2, k = 2, σ = 0.3. See section 2.4 for general explanation about the representations. a is the average opinion Crisis
 3.10
 For some values of the parameters, after a while, all the opinions remain negative (often below 0.5). This state corresponds to a situation where each agent is the enemy of all the others and even of herself. In the matrix representation, all the squares are blue, and in the network representation, the nodes are not connected and they tend to be located on a circle because the algorithm tends to maximise the distance each agent has with all the others. In some cases, this state of general distrust is not stable and can lead to other patterns such as absolute dominance or elite (the mixed patterns are described in more details later).
Figure 6. Example of crisis pattern. ω = 0.4, ρ = 0.35, N = 40, δ = 0.2, k = 2, σ = 0.5. See section 2.4 for general explanation about the representations. a is the average opinion which is very negative for this pattern. Maps of patterns in the parameter space
Principle of the experiments
 3.11
 The model includes 6 parameters and it is difficult to make an exhaustive study in the complete parameter space. We decided to study with more attention the influence of parameters ρ and ω because they rule the respective influence of vanity and opinion propagation which are the main components of the model. We decided to fix δ = 0.2 in this first analysis of the model, because it seems to us that the influence of this noise parameter is the easiest to interpret. Therefore, in the following, we propose maps in the space ρ and ω , both varying from 0.05 to 1 with a step of 0.05 , leading to 400 pairs of values, for a small number of selected values of N , k and σ . For each couple of parameter values ρ and ω , we run the model for 210 000 iterations (one iteration corresponding to N/2 random pair interactions), and we repeat this for 30 replicas. From iteration 10 000 to 210 000 every 100 iteration, we classify the state of the population into one of the 5 possible patterns: equality, elite, hierarchy, dominance, crisis. We do not consider the first 10 000 iterations because they are a transitory state that is not meaningful for the pattern. The rules for this classification are as follows:
 equality: One opinion of the individual with the highest reputation is higher than 0.5, and one opinion about this individual is lower than 0.5 and one opinion about the individual with the lowest reputation is higher than 0 (i_{M} such that r_{iM} = max(r_{i}), i_{m} such that r_{im}= min(r_{i}) , ∃ j such that a_{iM,j} > 0.5 and ∃ k such that a_{k,iM} <  0.5 and ∃ l such that a_{l,im} > 0);
 elite: One opinion of the individual with the highest reputation is higher than 0.5, and one opinion about this individual is lower than 0.5 and all the opinions about the lowest reputation individual are negative ( ∃ j such that a_{iM,j} > 0.5 ∃ k such thata_{k, iM} <  0.5 and ∀ l,a_{l,im} < 0);
 hierarchy: The highest reputation is above 0, and there are more than one reputation between the highest reputation and the highest reputation minus 0.5 (r_{iM} > 0 and #{i such that r _{i} > r_{iM}  0.5} > 1);
 dominance: The highest reputation is above 0, and there is at most one reputation which is between the highest reputation and the highest reputation minus 0.5 ( r_{iM} > 0, and #{i such that r_{ i } > R  0.5} ≤ 1);
 crisis: All opinions are below 0.5, and all reputations below 0 (a _{ i, j } ≤  0.5, for 1 ≤ i, j ≤ N and r _{i} ≤ 0 for 1 ≤ i, j ≤ N ).
 3.12
 Then, for the 30 replicas, we compute the percentages of presence of each pattern during the runs. These percentages are used for computing the maps that are presented in the following sections: We show the sign corresponding to one pattern as soon as this pattern appears in more than 20% of considered iterations.
Effect of k and σ for N = 40
 3.13
 Figure 7 shows the maps of these patterns for the following values:
 N = 40 , k = 10 , σ = 0.3 ;
 N = 40 , k = 10 , σ = 0.5 ;
 N = 40 , k = 2 , σ = 0.3 ;
 N = 40 , k = 2 , σ = 0.5 ;
 3.14
 This first set of experiments shows the respective influence of parameters k and σ for a given value of N. The figure can be conveniently interpreted by considering a radius, starting on the ρ axis and turning around the point ( ρ = 0, ω = 0) until it reaches the ω axis. Following this rotation, the patterns are found in the same order in the four maps (hierarchy, dominance, crisis, elite and equality) with important overlaps of two and sometimes of three patterns (we study some of these mixed patterns in the next sections). The hierarchy occurs therefore when there is a low vanity and a high opinion propagation, whereas the equality pattern occurs for a high vanity and a low opinion propagation. The other patterns appear for stronger coupling between the two processes.
 3.15
 Moreover, we note that the area covered by each pattern vary significantly when k and σ vary. More precisely, increasing k, the number of acquaintances talked about, increases significantly the sectors of hierarchy and dominance, and decreases the sectors of equality and elite. The effect on the crisis pattern is not clear from these experiments. From the dynamics of the model, we know that increasing k increases the opinion propagation which has similar effects to increasing ρ , which is confirmed by the observation that the limits between the sectors of different patterns tend to turn upward when k is growing. We also observe that increasing σ increases the sector of equality and hierarchy and decreases the sector of elite.
Figure 7. Maps of patterns for N = 40, δ = 0.2, different values of k (number of acquaintances talked about in the opinion propagation) and σ (parameter ruling the slope of the propagation coefficient function). A pattern is visualised on the map when it appears in more than 20% of the considered iterations in the 30 replicas, whenever two or three patterns are present in more than 20% of replicas, two or three symbols are superposed (see text for more details). Maps of patterns for N = 10 and N = 100
 3.16
 In this section, we try to get first observations on the effect of the size of the population on the sectors of patterns in the parameter space.
 N = 10 , k = 2 , σ = 0.3 ;
 N = 10 , k = 2 , σ = 0.5 ;
 N = 100 , k = 10 , σ = 0.3 ;
 N = 100 , k = 10 , σ = 0.5 ;
 3.17
 This second set of experiments shows that the maps are also organised in sectors of patterns in the same order for different values of the number of agents (see figure 8). We observe that the crisis pattern does not appear for N = 10 whereas it is more present for N = 100 than for N = 40. It seems that when the population reaches a crisis state, it takes more time to get out of it when the population is large. For N = 10 the dominance pattern is covering a larger sector than for the other values of N. This is probably due to the very simple criterion to distinguish between dominance and hierarchy, which should take the size of the population into account in a more refined version.
Figure 8. Maps of patterns for δ = 0.2, N = 10, k = 2 and N = 100, k = 10, for σ = 0.3 or σ = 0.5. A pattern is visualised when it appears in more than 20% of the considered iterations in the 30 replicas for N = 10 or 10 replicas for N = 100, whenever two or three patterns are present in more than 20% of replicas, two or three symbols are superposed (see text for more details). Stability of the patterns over the replicas
 3.18
 We do not think that it is very useful to provide the maps of variations of the patterns over the replicas. Globally, we observe that the main patterns for which vanity dominates (equality) or for which opinion propagation dominates (hierarchy) are very stable. The mixed patterns are the less stable, particularly around the diagonal. We suspect that this variability, at least in some cases, could nevertheless be misleading. Indeed, for instance in the case of the mixed crisisdominance pattern, the length of episodes of dominance or of crises can vary significantly with time. An interesting property to test for these patterns is the ergodicity of the system: running it during a very long time would be equivalent to make many replicas. If this ergodicity was verified, we would have to evaluate the necessary time to run the model in order to get a good view of the distribution of events. We consider that this work is out of the scope of this first paper. We now present two examples of mixed patterns.
First example of mixed pattern: equality and elite
 3.19
 Figure 9 illustrates the evolution of the opinions in the case of a mixed pattern between equality and elite where ω = 0.8, ρ = 0.15, in the case N = 40, k = 2, σ = 0.3. We observe that the matrix is not symmetric; a few horizontal lines appear with a majority of slightly negative values, the value of the diagonal for these lines being negative. Moreover, these lines are not stable, their number and positions vary in time. When looking at the network representation, we observe several blue nodes in addition to the red nodes which are connected together. These correspond to the second category agents in the elite pattern. The second category agents tend to establish connections between them and then, they tend to increase their selfopinion, and they may come back to the elite (not shown on the figure). Meanwhile, other elite agents can get a negative selfopinion and then her opinion about all the others become less negative, then lose all their friends and become second category. Globally the number of agents of each category fluctuates around a constant average value over the time. This pattern can be considered as transitory between equality and elite. The model satisfies the rules of classification for elite when one agent of the second category is disconnected from all the others. This rule may appear too simple because in this case, there are more elite than second category agents, which is not the idea we have in general of an elite. This could be refined in further studies of the model.
Figure 9. Example of mixed pattern between equality and elite. N = 40 , ω = 0.8 and ρ = 0.15 , k = 2 , σ = 0.3 , δ = 0.2 . Several individuals, with a low selfopinion appear, and the matrix is no more symmetric with respect to the diagonal (horizontal lines appear). See section 2.4 for general explanation about the representations. Second example of mixed pattern: dominance and crisis
 3.20
 With this pattern (for instance for ω = 0.5 and ρ = 0.3, for N = 40, k = 10, σ = 0.3), we observe periods of strong dominance by one or two agents, followed by periods of crisis, all the agents having very negative opinions of all the others and also about themselves. Figure 10 shows the matrix and network representations of six states around the first peak of dominance. From iteration 6500 to iteration 7500, two individuals, appearing as red columns in the matrix representation whereas all the other columns are blue and in red nodes located in the centre of the network representations, are valued highly positively (red links) by all the others which are colored in blue and located far from the circle center. After the dominance period, there are no more positive links, and all the nodes are blue (negative self value).
 3.21
 Other dominance episodes start with only one dominant individual. In this case, the dominance tends to last longer and it can lead to progressive hierarchy that are similar to the ones observed in the hierarchy pattern. The dominance ends when a second dominant individual emerges and reaches the same level of reputation as the first dominant one. This situation is very instable and it leads rapidly to back to the crisis.
Figure 10. Example of mixed pattern between dominance and crisis. Study around the first peak of the dominance indicator for ω = 0.5, ρ = 0.3, N = 40, k = 10, σ = 0.3, between iterations 5000 and 9000. In a first phase, two agents reinforce each other to dominate the whole population and in the second phase they compete and destroy each other's reputation. See section 2.4 for general explanation about the representations.
Analytical study

When vanity dominates
Case of ρ = 0
 4.1
 In order to better understand the patterns when ρ is small compared with ω (i.e. when vanity dominates), we simulate the model with the vanity dynamics alone, i.e. ρ = 0, starting with all agents not knowing each other, hence with all the opinions set to nil. We get the evolution represented on Figure 11. After relatively few time steps, all the opinions become either 1 or +1, except the selfopinions which change from nil to 0 at the first interactions, and then remain at 0 because no change occurs on the selfopinions in the vanity dynamics (the diagonal of the matrix representation remains white). Each agent separates the population into two groups of almost equal size: the ones she hates and the ones she loves. We also note that the diagonal is a symmetry axis of the matrix: if I love you, you love me, or if I hate you, you hate me. Note finally that the average of all opinions over the whole population is close to 0.
Figure 11. Vanity only. ρ = 0, ω = 0.4. The matrix remains symmetric and each agent splits the population between friends and foes. See section 2.4 for general explanation about the representations  4.3
 We now propose some analytical explanations of the above result. By definition of the vanity process, and because a_{i,i} = a_{j,j} = 0, when agents i and j meet, the opinions evolve as follows:
(6) From these equations, we can derive the following points:
 If we have, for a couple (i, j) with i ≠j , at a given iteration t, a_{i,j}(t) > δ and a_{j,i}(t) > δ, then a_{j,i}(t) + Random(δ,δ) > 0 and a_{i,j}(t) + Random(δ,δ ) > 0 . Hence, in this case, after iteration t, both a_{i,j} and a_{j,i} keep increasing until they are truncated to 1.
 Similarly, if a_{i,j}(t) <  δ and a_{j,i}(t) <  δ, then both a_{i,j} and a_{j,i} keep decreasing afterwards until they are truncated to 1 .
 If a_{i,j}(t) <  δ and a_{j,i}(t) > δ, then at the next steps a_{i,j}(t) will increase and a_{j,i}(t) will decrease. Of course we get the same when inverting i and j.
 4.4
 This allows us to describe the evolution of the reciprocal opinions of each couple (i, j) in two phases:
 In a first phase, when the absolute values of a_{i,j} and a_{j,i} are smaller than δ, the effect of randomness is dominating, and the opinions tend to be close to random walks. When both opinions get close to δ, these random walks are biased towards common growth, when they both get close to δ, they are biased towards common decrease, and if their signs are different, the opinions tend both to go to 0.
 As a result of these random processes each opinion couple (a_{i,j}, a_{j,i}) necessarily ends up either by being both higher than δ or both lower than δ, with an equal probability. This leads to the opinions couples being both at +1 or 1 in the end.
Analysis of the equality pattern
 4.5
 This analytical study helps formulate qualitative explanations of the equality pattern. First, we make the hypothesis that the dynamics of opinion propagation is negligible except on the diagonal of the matrix, because out of the diagonal the vanity dynamics are dominating. The evolution of the opinions outside the diagonal is mostly driven by the vanity equations:
(7)  4.6
 If all the values of the diagonal are positive, we see from these equations that this tends to decrease on average the values of a_{i,j} compared with the case of ρ = 0, which explains why these values are more frequently negative. It remains to be explained why the values of the diagonal are positive. Suppose agent j propagates her opinion about i to i herself, we have:
(8)  4.7
 We note that p_{i,j} is higher than 0.5 when a_{j,i}  a_{i,i} is positive. On the contrary, p_{i,j} is lower than 0.5 when a_{j,i}  a_{i,i} is negative. Therefore, the propagation of opinions favours the positive contributions to the diagonal, explaining why this diagonal is positive.
 4.8
 Qualitatively, the agents love those who flatter their ego and believe more those they love, hence they believe those who flatter them. As a result, the agents tend to have a high selfopinion. The positive selfopinions tend to shift the other opinions towards negative values. Indeed, having a high selfopinion, the agents feel often undervalued by the others and they decrease their opinion about them by vanity. In return, these agents do the same. After a while, the selfopinions fluctuate around an equilibrium value that can be approximated by considering that the sum of the modifications by the interactions with all the other agents is 0:
(9)  4.9
 Neglecting the random term, this can be rewritten as:
(10)  4.10
 If this selfopinion of equilibrium is such that a_{i,i} > 1  δ, then the random fluctuations can decrease a_{i,j} , even if a_{i,j} = 1 , because we can have a_{j,i} + Random(δ , δ ) < a_{i,i} . In this case, the couples (a_{i,j}, a_{j,i}) that have converged to +1, can enter in negative retaliation loops and finally converge to 1. This leads a_{i,i} to a smaller equilibrium value. The process continues until a_{i,i} < 1  δ , for all i. This explains the two phases that we observe in the simulations. In the first phase, the selfopinions converge to a value which is too high, with a too large number of friends. Then progressively this number of friends decreases, each time one of these friends has her opinion going below the considered selfopinion. This process stops when the selfopinion reaches an equilibrium value for an adequate number of friends that keeps its fluctuations always below 1  δ, preventing any underevaluation by friends. When such a value is reached, the number of friends cannot change any more and the configuration is stable. Therefore, at the stable state, we must have, for each agent i:
(11)  4.11
 Putting a_{i,i} at its maximum value in the computation of p_{i,j}, and a_{i,j} ≈ 1 for friends, a_{i,j} ≈ 1 for foes we get, for friends:
(12)  4.12
 Similarly, for foes:
(13)  4.13
 If we call f the number of friends, at stability, we can compute the function s(f):
(14)  4.14
 An estimate of the number of friends at stability is given by the maximum value of f such that s(f ) < 1  δ . For the case of the equality pattern shown on Figure 2 (N = 40 and σ = 0.35), we have: for f = 3, s(3) = 0.79, f = 4, s(4) = 0.85 and 1  δ = 0.8. Therefore, an evaluation of the number of friends at stability is f = 3, corresponding to the number of friends observed on average (3.4).
 4.15
 This provides an evaluation of the number of friends that each agent has. The tests shown on Figure 12 confirm that this approximation is correct. However, it should remain in a range where ρ is small compared with ω , and increasing k increases also the effect of the opinion propagation, therefore it can also modify the validity of the approximation.
Figure 12. Variations of the average number of friends of each agent in the equality pattern when σ varies, compared with the theoretical prediction for N = 40 and N = 20. The measured values are averaged on 10 replicas. The error bars correspond to the minimum and maximum over the 10 replicas. They show that the average number of friends does not vary significantly with the replicas. The other parameter values are: ω = 0.3, ρ = 0.01, δ = 0.2, k = 5.  4.16
 This analysis allows us to predict the effect of the main parameters on this pattern. For instance:
 Increasing δ (noise parameter) decreases the final equilibrium value of the selfopinion, and hence the final number of friends of each agent;
 Increasing σ (ruling the slope of the sigmoid function of the influence parameter) will decrease the difference of weights between friends and foes in equation 10, and therefore decrease the equilibrium value for a given number of friends. This will thus result in a larger number of friends at the stable state;
 Increasing the number of agents in the population decreases the value of the selfopinions for the same number of friends (see eq. 10). Therefore, it will result in a larger number of friends at the stable state.
 Increasing ρ increases the fluctuations around the equilibrium selfopinion and hence will decrease the number of friends at the stable state.
First analysis of the small world properties in the equality pattern
 4.17
 In the presentation of the equality pattern, we mentioned that the network of friends shows in some cases the properties of small worlds. We did not perform a systematic study to identify the limits of these properties in the parameter space. From our first investigations however, it seems that the small world properties require very small values of parameter ρ compared with parameter ω .
 4.18
 When ρ is small, we are in the domain of validity of the approximation made in the previous paragraph, and the general scheme applies: a fast establishment of a too large network of friends, leading to too high selfopinions, leading to the progressive elimination of some friends until reaching a sustainable equilibrium with more moderate selfopinions. A reasonable hypothesis is that during this phase of progressive elimination of friends, the friends who have positive links between them tend to be more protected than friends who have negative links between them. As an example, let's consider agent i, who has two friends j_{1} and j_{2}. If j_{1} and j_{2} are friends, they will tend to support each other when they discuss with i. Therefore, if there a vanity struggle between i and j_{1} starts, j_{2} will tend to influence i positively. The influence of j_{2} on i is important because j_{2} is a friend of i. On the contrary, if j_{1} and j_{2} are foes, each one tends to influence negatively i about the other. This influence favours in general a higher clustering coefficient than in random networks.
 4.19
 This rapid analysis explains why the smallworld properties can take place but not systematically, because:
 they require an adequate level of opinion propagation: if it is too high, the phase where the friends are too numerous does not take place, if it is too small, it is not sufficient to influence the elimination of friends during the elimination phase.
 the parameter k (number of acquaintances talked about) should not be too small compared with N (size of the population), because otherwise the influence of the friends is very limited.
 4.20
 Table 1 shows the results of a few experiments that confirm this analysis: In these experiments, k = 10, favouring the propagation of opinion, gives a higher clustering coefficient than k = 2. We also observe that, in these examples, decreasing the value of ρ tends to decrease the clustering coefficient, but also the mean shortest path. Of course, more systematic explorations would be necessary to come to confirm and refine these conclusions.
Table 1: Comparison of clustering coefficient and shortest path between simulations of the model and random networks of same number of nodes and average degree. Each result is the average of 10 replicas. Note that in these experiments when k = 10, the network shows the small world properties, this is not the case for k = 2. Parameters Average
degreeClustering
coefficientRandom
network
clustering
coefficientMean
shortest
pathRandom
network
mean
shortest pathω ρ σ k 0.3 0.01 0.35 2 3.83 0.09 0.08 2.88 3.36 10 3.02 0.23 0.07 4.82 7.08 0.6 0.05 0.5 2 7.19 0.18 0.19 2.03 2.04 10 3.51 0.25 0.09 3.43 4.58 Analysis of the Elite pattern
 4.21
 The elite pattern occurs when the opinion propagation is stronger than in the previous case which leads to the situation where the number of friends is less than one on average. In this case, with the random fluctuations from the initialisation, the model converges to a situation where there are two types of agents:
 the elite agents who have one friend supporting them and who have a positive selfopinion, and they have a very negative opinion of all the others (except themselves and their friend);
 the second category agents who did not manage to have a friend and have a negative selfopinion, and a very negative opinion of all the other second category agents.
 4.22
 The second category agents are sensitive to the opinion propagation from the elite agents. Indeed, the second category agents having a strongly negative selfopinion, they are influenced even by agents for whom they have a negative opinion (for an opinion which is the same as the selfopinion, the propagation coefficient is 0.5, see Figure 1). Let us suppose that there are m elite agents, each having a single friend with an opinion of 1 about them, and having a selfopinion of about 1  δ, and m2 other elite agents having an opinion 1 about them. At a stationary state, the opinion a_{s,e} of the second category agents about the elite agents would respect an equilibrium between the influences of the different elite agents:
(15)  4.23
 In equation 15, the influence of the selfopinion is multiplied by 1, because the considered elite agent always influences the second category agents about herself, whereas the influence of the others' opinion takes place when they talk about her, having a probability k / N2 to be talked about (because the random drawing excludes the discussing pair of agents). Moreover, there is one elite agent having an opinion +1, and m2 having opinion 1. Finally, we neglect the influence of the second category agents because they all have a similar opinion about the elite agents. We get:
(16)  4.24
 We verified on two examples with N = 40, k = 2, δ = 0.2, σ = 0.3 that this approximation gives the right order of magnitude:
 For ω = 0.3 and ρ = 0.15, the number of members of the elite is 24. The measured average opinion of the second category agents about the elite agent is 0.15 whereas the approximation yields 0.14.
 For ω = 0.5 and ρ = 0.4, the number of members of the elite is 4. The measured average opinion of the second category agents about the elite agent is 0.61 whereas the approximation yields 0.64.
 4.25
 This result seems satisfactory, given the level of approximation made in the reasoning. We can deduce from this that the higher the number of the elite, the lower will be the opinion of the second category for the elite. But when the opinion of the second category agents is high, this tends to increase the selfopinion of the elite agents, and this increases the risk of instability. This corresponds to the observations we made.
 4.26
 The second category agents are more stable if the opinion propagation coefficient ρ is relatively high because in this case, the influence of the very negative opinion of elite agents tends to prevent them to grow with possible positive loops of vanity among second category agents. This rise of couples of second category agents allowing them to come back the elite is possible though, especially in the mixed pattern equality and elite.
Analysis of the mixed pattern between equality and elite
 4.27
 In the mixed pattern between equality and elite (see Figure 9) we have some elite agents that become temporarily second category and then become elite again. We are also in a case where the number of friends per agent is around 1, and with the fluctuations the agents have chances to lose their only friend. Once the agent becomes isolated, she progressively loses her high selfopinion because all the other agents are sending her messages of negative value. This explains why these agents form a second ring in the network representation. It can happen that the agent loses her good selfopinion first and then loses her friend. This can be explained with the random fluctuations of the model, where they can interact for a long time only with their foes, and their selfopinion can reach a threshold below which their foes become more influent than her single friend (for which the propagation coefficient does not increase much). With this negative selfopinion, the agent propagates a negative opinion of herself, which progressively influences her friend.
 4.28
 When her selfopinion becomes very negative, she becomes more influenced by the agents of the elite, who have a high selfopinion. This increases the opinion of the second category agents about the elite and, this opinion cannot increase too much because each agent of the elite is viewed very negatively by all the other elite agents except her single friend (see the reasoning made in the previous paragraph).
 4.29
 The main difference with the previous pattern is that, the vanity being stronger, there is a high chance that the second category agents enter into positive loops of friendship reinforcement, despite the opinion propagation of the elite. When this takes place, the second category agents reinforce each other, they restore their positive selfopinion and come back to the elite. This explains why there can be fluctuations between equality and elite patterns.
When opinion propagation dominates
Study of the case ω = 0
 4.30
 Studying the model with the opinion influence only (ω = 0) can give us some clues for understanding the model when ω is small. Figure 13 shows the corresponding pattern, that suggests the following observations:
 Like in the hierarchy and dominance patterns, all the agents have similar opinions about each agent, leading to matrix representations with columns of almost homogeneous colours (the reputation r_{i} of agent i).
 Unlike in the hierarchy pattern, after a while, the values of reputations r_{i} are almost uniformly distributed on the segment [ 1, + 1]. The values r_{i} are not stable over time, they can take any value in the segment [ 1, + 1] if the simulation is long enough.
Figure 13. Evolution of the opinions with the dynamics of opinion propagation only (N = 40, ω = 0, ρ = 0.3).  4.31
 In the case ω = 0, it is easier to show analytically why the columns of the matrix tend to be homogeneous.
 4.32
 Let us assume i and j meet, the propagation of opinion process implies:
(17)  4.33
 Let us introduce the variables D_{i,j}(t) = a_{i,j}(t)  a_{j,j}(t), hence
(18) Where Δ = ρp_{i,j}δ_{j,j}(t)  ρp_{j,i}δ_{i,j}(t) is the stochastic contribution. By definition p_{i,j} + p_{j,i} ∈ [0, 2] and it can be equal to zero only a_{i,i}  a_{i,j} > > 1 and a_{j,j}  a_{j,i} > > 1 . So neglecting for a while the stochastic term and defining θ = 1  ρ(p_{i,j }+ p_{j,i}), we have:
(19)  4.34
 Therefore, if we suppose ρ < 0.5, we have θ < 1 and then, as t increases D_{i,j}(t + 1) goes to zero. This is the required result.
 4.35
 Now, with the addition of the stochastic part, the dynamics of the reputations seems to be a random walk between 1 and +1. However, when computing the distribution of the reputations over 50000 iterations in a sliding window, we observe that this distribution is not perfectly uniform. Figure 14 on the left shows the distribution of the selfopinions (which are close to the reputations) and we observe that the average value tends to be slightly negative, and the distribution has two maxima close to its extremes (around 0.7 and around +0.8), with a minimum between these maxima around 0.3.
 4.36
 Moreover, we observe on Figure 14 on the right that the average selfopinion tends to be slightly higher than the reputation, except when the selfopinion of the agents is close to the extremes. The values of the extremes are due to side effects: the reputation tends to fluctuate more slowly than the selfopinion. The average positive bias for the selfopinion is more interesting. It is due to the propagation coefficient which tends to be higher when the selfopinion gets higher. Indeed, because of this difference, when an agent selfopinion is higher than her reputation, the others have less influence on the selfopinion than when the selfopinion is lower than the reputation (everything else being equal). However, the effect of this average difference between the selfopinion and the reputation depends on the value of the agent's reputation:
 If the agent's reputation (and hence her selfopinion) is among the lowest, then she is very sensitive to the influence of most of the others that she values more than herself (they have thus a high propagation coefficient). Therefore, when the agent selfopinion is higher than her reputation (which statistically takes place more often than the opposite), the agent's selfopinion tends to follow her reputation (the average opinion of the others about herself), thus it decreases. This explains a tendency towards negative values.
 If the agent's reputation (and hence her selfopinion) is among the highest, then on average she tends to impose her selfopinion to the others, because her propagation coefficient is high in the exchanges. Thus, on the contrary, when her selfopinion is higher than her reputation (statistically the most frequent), she tends to increase her reputation towards her selfopinion.
 4.37
 To summarise, the highly valued individuals tend to lead the other's opinions and, with the statistical bias for a selfopinion higher than the reputation, they tend to increase their reputation. This is the contrary for the badly valued individuals who tend to naturally decrease their selfopinion, only by the effect of the propagation coefficient. Nevertheless, these are statistical tendencies, which take place stochastically and there are random movements in the opposite directions. This explains the shape of the reputation distribution. Below a threshold of the selfopinion, the reputations and selfopinions tend to be biased towards negative values, and above towards positive values. The value of this threshold depends on parameter σ determining the propagation coefficient and also on k the number of agents about whom the opinions are propagated during the encounters. Indeed, this number has an impact on how the agents propagate their opinion about the others.
Figure 14. Graphs for ω = 0, ρ = 0.3. On the left: the density of selfopinion averaged on 50000 iterations. On the right: the average difference between the selfopinion and the reputation (a_{i,i}  r_{i}). Analysis of the hierarchy pattern
 4.38
 These observations are useful to understand the processes behind the emergence of hierarchy. First note that the vanity process enhances the tendency of selfopinions to be higher than the reputations. Indeed, the small statistical positive bias for selfopinion that is due to the opinion propagation leads, on average, the agents to consider themselves as (slightly) undervalued by the others, thus they devalue them by vanity. This is very similar to the process that we observed in the equality pattern, but it is slower because of the averaging effect of the opinion propagation. This explains the tendency to get negative opinions in hierarchy pattern.
Analysis of mixed pattern between crisis and dominance
 4.39
 The crisis pattern appears as an extreme result of the tendency for a positive self bias that is amplified by increasing the vanity. But in the considered mixed pattern, the state of crisis is not stable, it generates episodes of dominance that we now try to explain qualitatively:
 There are fluctuations of the reputations during the crisis. When an agent's reputation (and thus selfopinion) becomes higher enough than the others, it becomes less sensitive to the opinion propagation from the others. In this case, the opinions of this leading agent become more strongly driven by the vanity than by the opinion propagation. In particular, this agent can establish loops of positive reinforcements with a few other agents that are not averaged immediately by opinion propagation. This increases the selfopinion of the leader who propagates this good opinion with a strong influence. Moreover, the leader propagates her good opinion about the agents that have a good opinion of her, which reinforces these agents in the population and their propagation of their good opinion about the leader. In summary, when an agent reaches some level of reputation (with all the others having very low reputations) she tends to lead the opinions, reinforcing herself and those who propagate a good opinion of her.
 When this leader's reputation becomes close to 1, it cannot grow anymore. Then a struggle with her potential rivals takes place. Indeed, in general, her allies, with the fluctuations, end up by having a lower opinion of the leader than the leader's selfopinion. In this case, the leader begins to decrease her opinion about this challenger. The leader is followed by the rest of the population with limited damage for her reputation if the challenger is not too strong. On the contrary, when the challenger has reached a reputation which is as strong as the one of the leader, then the fight is fatal for both of them and leads to come back to the generalised distrust.
 The rival that reaches the level of the leader and causes her loss can appear more or less rapidly. In some cases, the leader and her rival grow together right at the beginning and reach the top reputation almost at the same time. In this case, the dominance episode is short (first peak in the graph of dominance around iteration 7000). Moreover, the rapid growth of the two leaders selfopinions leads them to decrease their opinions about all the rest of the population, by vanity. Figure 15 illustrates the rise and fall of a couple of leaders.
 On the contrary, when the leader manages to reach the top reputation with a significant difference with her followers, she may keep her dominance for more than 10 thousand time steps, with several attempts from challengers that she manages to stop soon enough. In these dominance episodes, there are periods where a progressive hierarchy is established with several agents having intermediate reputations. Figure 16 illustrates the rise of a single leader and figure 17 her successful struggle with a challenger.
Figure 15. Crisisdominance pattern: Rise (first line of matrices) and fall (second line of matrices) of double leadership. At t = 6050, we observe that two agents (i = 11 and i = 16) support strongly each other (their reciprocal opinions are represented by the two red squares symmetric with respect to the diagonal), and both have positive self opinions. Moreover both tend to have very negative opinions of the rest of the population. At t =6 200, 6300, we observe that this process reinforces the reputations of both agents and decrease all the other opinions that become close to 1. At t = 7400, we observe that the two leaders do not support each other anymore, on the contrary, their opinion about each other (lighter squares) are lower than their respective reputations. Hence they enter in a negative loop of vanity in which they lead the rest of the population (t = 7600, 7800). The crisis pattern will take place after a few hundred iterations. Figure 16. Crisisdominance pattern: Rising of a single leader. At t = 200100, we see that an agent (column with light pink cells at i = 7) has a higher reputation than the others, and she supports 3 other agents more than the average (white squares at j = 1, 12, 17). At t = 200300; we observe that her reputation significantly increased, that she supports other agents (particularly the pink square at j = 6). At t = 200500, the dominance of agent 7 is complete, and she supports strongly several agents that have their reputation rising (columns with several white or pink squares). Figure 17. Crisisdominance pattern: the leader manages to get through the competition with a challenger. At t = 204500, the opinion of the challenger (i = 25) for the leader (i = 7) and of the leader for the challenger are significantly lower than their respective selfopinions. They enter in a negative loop of vanity that ends up at t = 205700 with the challenger returning into the majority of very low reputations, and the leader having still some limited support from one other agent. The leader's reputation decreased much, but it is high enough to rise again (t = 206000, 206300) and a new episode of strong dominance will take place. Conclusions of the theoretical analysis
 4.40
 Globally, our analysis points out the main drivers of the model behaviour when vanity or opinion propagation dominates. In both cases, the relation between the selfopinion and the reputation is important. It is responsible for the general tendency of the model to yield negative opinions. We propose some quantitative approximations correctly predicting some measures of the model behaviour in the equality and elite patterns. For the patterns where opinion propagation dominates, our explanations are more qualitative. Nevertheless, this analytical work is far from complete and many questions remain, like for instance explaining the stability of the dominance pattern, the quantification of the dominance periods in the crisisdominance pattern, etc...
Discussion  conclusion
 5.1
 In the last decades, the researches in the field of complexity showed that the collective behaviour of a population of interacting agents can be very difficult to predict because of cascade effects or global structure emergence from local interactions. Computer simulations give the possibility to simulate and study these effects. We argue that this approach can be a source of new arguments in philosophical debates making hypotheses about individual interactions and we believe that this paper illustrates this point. We translated Hobbes hypothesis about the role of vanity in human relations into a computer model and explored by simulation the consequences of this hypothesis on the emergence of patterns. Our work can thus be seen as using computer simulation as a tool for philosophical reflection: it allows one to make thought experiments with idealised populations of interacting agents.
 5.2
 We think that our results can shed a new light on Hobbes theses. Indeed, it is striking that our model leads to two situations on which Hobbes focuses particularly: the general distrust (crisis) where men are all enemies of each other (and of themselves), and the absolute dominance of one agent (the Leviathan). In this respect, the model confirms the main intuitions of Hobbes. Moreover, in the crisisdominance pattern, these two situations alternate dynamically, as if they were the two sides of the same coin. The main point is that they take place spontaneously as an effect of the individual interactions. Moreover, we show that the same hypotheses can lead to very different patterns that were not considered by Hobbes (equality, elite).
 5.3
 The model can also be related with more recent general theories of social interactions, in particular the ones of René Girard (1972; 1982). Indeed, with his thesis about mimetism as being the main driver of social interactions, Girard puts forward a mixture of the ingredients that are present in our model: imitation (opinion propagation) and rivalry (vanity). Girard considers that these ingredients can lead to a state of generalised crisis, where all hierarchies are abolished, and also to the absolute dominance of one agent. Moreover, he insists on the intrinsic instability of the leadership, the book of Job in the Bible being for him a prototypical illustration of such instability. Our model is in accordance with these general views: the same mechanisms can lead to the general crisis and to the dominance, and the dominance is always unstable. However, Girard assumes also that the same mechanisms lead to scapegoat structures where an agent is universally despised. We have not observed this last pattern with our model. If this is confirmed that the model cannot generate such a pattern, it would mean that our model does not implement correctly Girard's hypotheses about individual interactions, or that Girard is wrong about such interactions leading to scape goat patterns.
 5.4
 Another philosophical debate in which our model could contribute is about how to define a human subject. Indeed, in the model, the agents have a perception of themselves which is always mediated by the others. Their selfopinion derives from the opinion that the others express about them. This is an important option which differs strongly from the Cartesian subject who is able to found herself. The subject who builds herself in the interaction with the others corresponds better to the more recent philosophical traditions which give a particular importance to language (after the "linguistic turn").
 5.5
 These remarks show that this model could help contribute to some philosophical debates. The question is then: can it be related to more concrete social observations?
 5.6
 One could answer positively to this question, arguing that the patterns emerging from the model show major common features with familiar real social situations. For instance:
 When vanity dominates strongly, the agents are in priority careful to reward a good image of themselves or to punish a bad one, and otherwise they are not influenced much by other opinions, then the model predicts that agents do not establish a hierarchy and establish strong links with a small set of friends, all the others being enemies. In some cases, the structure of this friendship network shows the properties of small worlds. This equality pattern could be related to the rejection of hierarchies and the tendency to privilege privatized lives in modern societies, turning exclusively to the small circle of family and close friends. It can be argued that the corresponding dynamics of the model (strong vanity, low opinion propagation) are coherent with the tendencies of the modern individual. The fact that the network has a small world property for some parameter values is an interesting emerging feature, coming from the dynamics of interactions where people talk about the people they know and value their friends. Some more detailed properties of the model behaviour related to the equality pattern can also be interesting to interpret in a sociological perspective:
 The number of friends is related to the selfopinion of the agent. If this selfopinion is too high, then there is a high probability of being offended by a friend and entering a loop of negative vanity retaliations, ending up by losing this friend.
 The number of friends tends to decrease when the value of parameter σ increases. This parameter can be interpreted as ruling the importance given to differences of values between individuals. In a society where the differences count much (small σ) the number of friends tends to be low because the individuals do not change their selfopinion when criticised by agents they value lower than themselves. As a consequence, they get easily a high opinion of themselves which increases their probability to loose friends. On the contrary, when σ is high, the difference of values counts less and the agents tend decrease their selfopinion with the critical opinions of the agents they value lower than themselves. This lower selfopinion is more compatible with a higher number of friends.
 Finally, we observe that in a larger population, the agents tend to have a larger number of friends, because these friends compensate the hostility of a larger number of agents.
 When opinion propagation is more important and vanity dominates less strongly, there is a transition towards the elite pattern, where two types of agents appear: the elite and the second category agents. The process leading to this differentiation can also be interpreted in sociopsychological terms:
 The second category agents are those who have lost all their friends, and had their selfopinion dramatically decreased consequently. This situation can be related to depression due to isolation.
 The elite agents have only one friend and they hate the other elite agents as well as the second category agents. Being in the elite is a constant competition where each has a low number of friends and many foes.
 The second category agents have a moderately good opinion of the elite agents, and a strongly negative opinion of the other second category agents. The elite agents can count on some support from the second category agents.
 When the elite agents are many, they tend to be less valued by the second category agents and more stable than when they are a low number.
 The second family of patterns takes place when the opinion propagation dominates. This situation can be interpreted as a larger influence of the population opinion on each individual, corresponding to a traditional society where the group dominates the individual. In this case, the first striking feature is that the population has almost always a consensus on its opinion about each individual. Therefore, there is a very small difference between the selfopinions and the reputation of the individuals (whereas this difference is very strong in the vanity dominating patterns). Hence, this is curiously when the agents care less about their image (low vanity) that the reputation becomes meaningful and shared by the whole population. This family of patterns suggests the following remarks related to psychosociological interpretations:
 The hierarchy pattern presents a pyramidal figure of the group, with a number of agents increasing when the reputation is decreasing. It could be related to traditional, hierarchical societies (as opposed to modern individualist societies), and it emerges from hypotheses corresponding to a more traditional view of the influence of the group.
 A positive bias for the agent selfopinion is responsible for a global tendency to adopt negative opinions about the others, when some vanity is introduced. This positive self bias is generally observed in social psychology (Hoorens 1993). The studies of links between selfesteem, leadership and groups (see e.g. Hogg & van Knippenberg 2003) could be related to the mechanisms we analysed in our model.
 In the crisisdominance pattern, we observe that the leadership is given to agents who tend to have a more positive view of the others than average and who are able to resist to the general opinion. This corresponds to the charismatic leader as defined in (van Knippenberg et al. 2004). In our model, the dominant agents are able to make the others change strongly their opinions, which also corresponds to a recognised feature of leadership in the literature (Mary & UhlBien 2006; Hogg & van Knippenberg 2003).
 The emergence of absolute leadership in totalitarian regimes has been interpreted as an attempt to restore a traditional order in a modern society by Gauchet (2010). This can be related with our results where this absolute leadership takes place when mixing the vanity (that can be related to the modern individualism) and opinion propagation (that can be related to the traditional power of the society on the individual).
 When vanity dominates strongly, the agents are in priority careful to reward a good image of themselves or to punish a bad one, and otherwise they are not influenced much by other opinions, then the model predicts that agents do not establish a hierarchy and establish strong links with a small set of friends, all the others being enemies. In some cases, the structure of this friendship network shows the properties of small worlds. This equality pattern could be related to the rejection of hierarchies and the tendency to privilege privatized lives in modern societies, turning exclusively to the small circle of family and close friends. It can be argued that the corresponding dynamics of the model (strong vanity, low opinion propagation) are coherent with the tendencies of the modern individual. The fact that the network has a small world property for some parameter values is an interesting emerging feature, coming from the dynamics of interactions where people talk about the people they know and value their friends. Some more detailed properties of the model behaviour related to the equality pattern can also be interesting to interpret in a sociological perspective:
 5.7
 However, a closer analysis would lead to remain careful about some of these interpretations:
 Experiments show that a low selfesteem is very rarely expressed by subjects, whereas the majority of agents have a low selfopinion in several patterns of the model (hierarchy, dominance and crisis). A possible interpretation is that the usual subjects submitted to experiments are more likely to be closer to the equality pattern where everybody has a high selfesteem. Moreover, as underlined previously, our model yields the usually observed positive self bias, compared with the other's opinions, hence this failure to match psychosociology observations about selfesteem should probably be discussed further.
 A major problem is that interpretations considering large societies are certainly too demanding to the model. In its present version, it is indeed limited to small populations, because it is supposed that everybody can meet with everybody and know everybody. Therefore, it is closer to group dynamics.
 5.8
 In order to extend the study to larger sets of agents (thousands to millions for instance), it will be necessary to make new assumptions: to limit the number of agents that an agent can have in mind and discuss about, to limit the agents she can discuss with through different assumptions of a priori networks of interactions (possibly evolving with the opinions). Moreover, in the present setting, we considered identical agents at the beginning. It could be interesting to introduce some diversity, for instance in the propagation coefficient. However, we think that there is still a lot of work to complete the experimental and theoretical study of this first version of the model, particularly for the patterns of dominance, hierarchy and crisis where our analysis remains qualitative.
Acknowledgements
 The work of T. Carletti has been partially supported by the FNRS grant Mission Scientifique 20102011 allowing him to visit the Irstea  LISC during the fall 2010.
Notes
 ^{1}We also tested without checking that the acquaintance is different from i, and did not notice any major change in the emergent patterns of the model.
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