Salomé Leclercq
email: salome.leclercq@univ-lille.fr

Sébastien Szaffarczyk

Pantelis Leptourgos

Pierre Yger

Alexandra Fahkri

Marielle Wathelet

Vincent Bouttier

Sophie Deneve

Renaud Jardri
email: renaud.jardri@univ-lille.fr

Alexandra Fakhri

Sophie Denève

Conspiracy beliefs and perceptual inference in times of political uncertainty

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Conspiracy theories (CTs) are appearing with increasing frequency in our modern societies, with criticism of mainstream knowledge and scientific evidence at center stage. CTs are commonly defined as beliefs assuming the existence of a secret group or organization operating maliciously and for its own benefit. Adherence to multiple unrelated CTs that contradict each other is common [START_REF] Goertzel | Belief in Conspiracy Theories[END_REF][START_REF] Swami | Conspiracist ideation in Britain and Austria: evidence of a monological belief system and associations between individual psychological differences and real-world and fictitious conspiracy theories[END_REF][START_REF] Drinkwater | Reality testing, conspiracy theories and paranormal beliefs[END_REF][START_REF] Wood | Dead and Alive: Beliefs in Contradictory Conspiracy Theories[END_REF] , suggesting common underlying mechanisms by which belief in CTs arises.

Interestingly, a first line of research revealed that these rigid beliefs often crystallize around highly polarizing societal or political events [START_REF] Van Prooijen | Conspiracy theories as part of history: The role of societal crisis situations[END_REF] and may serve as coping mechanisms for stress and loss of control when uncertainty increases sharply [START_REF] Whitson | Lacking Control Increases Illusory Pattern Perception[END_REF][START_REF] Sullivan | An existential function of enemyship: Evidence that people attribute influence to personal and political enemies to compensate for threats to control[END_REF][START_REF] Van Prooijen | The Influence of Control on Belief in Conspiracy Theories: Conceptual and Applied Extensions[END_REF][START_REF] Dow | Sense of control and conspiracy perceptions: Generative directions on a well-worn path[END_REF][START_REF] Farias | COVID-19 as an undesirable political issue: Conspiracy beliefs and intolerance of uncertainty predict adhesion to prevention measures[END_REF] . Although CTs can induce widespread misconceptions -as it has been observed during the COVID-19 pandemic -they also constitute intuitive explanations for complex issues (e.g., simple cause-effect relationships), that can meet the need to restore predictability [START_REF] Douglas | Understanding Conspiracy Theories[END_REF] at the cost of suboptimal reasoning.

A second line of research focused on the role of reasoning biases in CT emergence [START_REF] Wycha | It's a Conspiracy: Motivated Reasoning and Conspiracy Ideation in the Rejection of Climate Change[END_REF][START_REF] Brotherton | Intention seekers: conspiracist ideation and biased attributions of intentionality[END_REF][START_REF] Georgiou | Conspiracy theory beliefs, scientific reasoning and the analytical thinking paradox[END_REF] . According to this framework, it is thought that conspiracists bias the weight they attribute to certain stimuli to reduce uncertainty [START_REF] Pytlik | Thinking Preferences and Conspiracy Belief: Intuitive Thinking and the Jumping to Conclusions-Bias as a Basis for the Belief in Conspiracy Theories[END_REF][START_REF] Kabengele | Conspiracy Beliefs and Jumping to Conclusions[END_REF] , sometimes leading people to jump to conclusions (JTC) when probabilistic decisions must be made. Conspiracy ideations have also been associated with a more intuitive thinking style [START_REF] Drinkwater | Reality testing, conspiracy theories and paranormal beliefs[END_REF][START_REF] Georgiou | Conspiracy theory beliefs, scientific reasoning and the analytical thinking paradox[END_REF] than the common analytical approach. This tendency toward fast, preconscious and spontaneous processing could be based on specific reality-testing deficits in people endorsing CTs [START_REF] Lewandowsky | The Role of Conspiracist Ideation and Worldviews in Predicting Rejection of Science[END_REF] . These results have not always been replicated, leading some authors to wonder whether CTs could mainly be traced back to social constructs [START_REF] Raihani | An evolutionary perspective on paranoia[END_REF][START_REF] Müller | Linking paranormal and conspiracy beliefs to illusory pattern perception through signal detection theory[END_REF][START_REF] Hartmann | Illusory perception of visual patterns in pure noise is associated with COVID-19 conspiracy beliefs[END_REF] . However, others suggest that social learning depends on broader associative mechanisms responsible for the detection of predictive relationships in every natural domain [START_REF] Heyes | New thinking: the evolution of human cognition[END_REF] ; thus, Bayesian methods could be a complementary approach to addressing the existing link between CTs and uncertainty. This framework assumes that cognitive and perceptual factors are rooted in a common probabilistic mechanism [START_REF] Helmholtz | Concerning the perceptions in general, 1867[END_REF] . Surprisingly, only a few attempts have been made to investigate the potential links between perceptual inference and conspiracy ideations in a controlled experimental setting. Some results from the CT literature appear compatible with a probabilistic formalism. Dagnall and colleagues [START_REF] Dagnall | Conspiracy theory and cognitive style: a worldview[END_REF] explored the link between CTs and a wide range of cognitive-perceptual factors. They showed that such factors, including hallucination proneness, often conceptualized as false inferences [START_REF] Fletcher | Perceiving is believing: a Bayesian approach to explaining the positive symptoms of schizophrenia[END_REF] , were associated with CTs. Additionally, conspiracy ideations were found to be associated with illusory visual pattern detection [START_REF] Müller | Linking paranormal and conspiracy beliefs to illusory pattern perception through signal detection theory[END_REF][START_REF] Van Prooijen | Connecting the dots: Illusory pattern perception predicts belief in conspiracies and the supernatural[END_REF] , a phenomenon regularly explored through the prism of Bayesian theory [START_REF] Geisler | Illusions, perception and Bayes[END_REF] .

Very few papers have directly fitted computational models to behavioral data in nonclinical samples with some noticeable exceptions exploring paranoia and/or conspiracy ideations [START_REF] Suthaharan | Paranoia and belief updating during the COVID-19 crisis[END_REF][START_REF] Barnby | The computational relationship between reinforcement learning, social inference, and paranoia[END_REF] . Purely theoretical papers also confirmed that computational approaches could help to better understand the spreading of CTs on simulated or social-media data [START_REF] Cook | Rational Irrationality: Modeling Climate Change Belief Polarization Using Bayesian Networks[END_REF][START_REF] Madsen | Growing a Bayesian Conspiracy Theorist: An Agent-Based Model[END_REF] . Crucially, a more personalized computational lens [START_REF] Rigoli | Deconstructing the Conspiratorial Mind: the Computational Logic Behind Conspiracy Theories[END_REF] , and a study of CTs in their ecological environment [START_REF] Stojanov | Perceived lack of control and conspiracy theory beliefs in the wake of political strife and natural disaster[END_REF] seem required to decipher the respective contributions of sociopolitical factors and information weighting in CTs' emergence.

Thus, combining the strength of normative and ecological research during uncertain societal crises appears necessary to establish a bridge between CT and inference quantification. In the present paper, we referred to Circular inference (CI), a Bayesian framework that has proven effective in capturing both perceptual suboptimality in nonclinical populations [START_REF] Leptourgos | Circular inference in bistable perception[END_REF] and JTC in patients with psychosis [START_REF] Jardri | Experimental evidence for circular inference in schizophrenia[END_REF][START_REF] Simonsen | Taking others into account: combining directly experienced and indirect information in schizophrenia[END_REF] . We hypothesized that by fitting the CI model to a simple bistable task (which maximizes ambiguity at the perceptual level), we could benefit from an ideal setup to challenge the potential links between (i) the inferential mechanisms at play under conditions of extreme uncertainty, and (ii) the dynamics of conspiracy ideations in large populations exposed to natural sociopolitical stress.

RESULTS

Measuring multilevel inference before and after stressful political events FIGURE 1. A repeated-measures design framing stressful political events in 3 different countries.

Conspiracy ideations, political distress and perceptual stability were measured in the same participants (n = 623) via an online procedure, before and after the occurrence of a polarizing political event in three Western countries (M stands for month): the 2020 presidential election in the United States of America (n = 212, US), BREXIT implementation in the United Kingdom (n = 225, UK) and the 2022 presidential elections in France (n = 186, FR). We used T1 and T2 measures in the main model, while T3 was used in control analyses (see Supplementary Material section: Controlling for experimental design biases).

Because we assumed that periods of great sociopolitical uncertainty lead to significant increases in individual levels of distress and favor inferential biases such as conspiracy endorsements, we explored rigid beliefs and perceptual stability around polarizing political events in three independent Western countries (see Fig. 1): the United States of America (US, 2020 presidential elections), the United Kingdom (UK, 2021 BREXIT implementation) and France (FR, 2022 presidential elections). At each time point, healthy participants were instructed to rate their level of distress related to the ongoing event in their own country (later referred to as political distress, see Methods and Supplementary Material section: Self-reported measures).

Necker cube experiment

At each time point, the 623 enrolled participants performed an online bistable perception task based on the Necker cube (NC). The interpretation of the two-dimensional NC projected from a threedimensional space naturally alternates between two possible configurations: a seen from above (SFA), or a seen from below (SFB) cube (Fig. 2a). A perceptual stability score, ranging from 0 to 1, was estimated at the participant level. This score corresponds to the probability of switching from one interpretation to the other (0 means total instability, while 1 reflects a perceptive rigidity where the participant only sees one interpretation of the two, see the Methods section). Assuming a universal mechanism at the roots of belief formation, we merged the 3 samples after ensuring their comparability in terms of perceptual stability at baseline (Table .1, Fig. 2b-c; see also Supplementary Material section: Controlling for experimental design biases). Importantly, perceptual stability was tested for in lab/online within-subject reproducibility on a pilot independent sample before running the final online experiment (Fig. 2d-e). We also ensured that dynamic changes in stability between the different time points were not due to a simple training effect between the sessions (see Supplementary Material section: Controlling for experimental design biases). ). The x axis represents the average scores of the two methods. The y axis represents the mean difference between online and in-lab stability scores. The limits of agreements (LoA, pink dotted lines) are defined as the mean difference computed on all participants (pink line) ± 1.96 s.d., and each dot represents a participant. As all participants are included in the LoA, the methods are considered to be in agreement and may be used interchangeably.

Conspiracy adherence measures

The participants were instructed to self-rate their level of adherence to CTs, completing the Generic Conspiracist Beliefs Scale (GCB, see Methods section) at each time step. Replicating previous findings, we showed that conspiracy ideations were not normally distributed across the tested participants (W = .954 ; p = .440e - [START_REF] Wycha | It's a Conspiracy: Motivated Reasoning and Conspiracy Ideation in the Rejection of Climate Change[END_REF] , Fig. 3a, Fig. S1a), suggesting that only a subpart of the general population commonly endorses such beliefs. The distribution of the total GCB scores differed across the three samples (χ 2 = 31.5, p <.001, η 2 = .348e -07 ) despite a similar pattern across subscales (Fig. S1ab, Table S1), notably demonstrating a common preoccupation for information control.

Looking more precisely at the sociodemographic features associated with conspiracy endorsement, we replicated previous findings from the literature (see Supplementary Material section: Sociodemographic features of conspiracy theories), notably showing that despite an absence of a link with the sex of participants (Fig. 3b), GCB scores significantly differed as a function of age (F(2,620) = 3.10, p = .046, η 2 = .039, Fig. 3c) or education (F(2,620) = 13.5, p <.001, η 2 = .395e -05 , Fig. 3d). Thus, we retained those variables as covariates for later analyses.

Stress correlates at baseline

We assume that some participants might adopt information-processing strategies that can reduce the uncertainty induced by the framed political event. Notably, we expect that the search for stability would translate into high levels of confidence measurable at different levels of processing, from perception to conspiracy beliefs. Since belief in CTs has been proposed to be a coping strategy able to reduce the stress elicited by uncertainty, we also expect an association between great levels of confidence and low levels of distress. We first checked for associations between political distress at baseline (i.e., when uncertainty peaked) and: (i) perceptual stability on the one hand, and (ii) conspiracy endorsement on the other hand (Fig. 4a). Political distress was found to be negatively linked with both levels of inference (p = .028, ρ = -.120 and p = .007, ρ = -.094 respectively). We further confirmed these findings by splitting the sample into two subsamples according to stress: (i) a 'low stress' (LS) and (ii) a 'high stress' group (HS). Comparing these two groups at baseline, we confirmed a significant difference in both stability (U = 41385, p = .002, Cohen's d= .140) and GCB scores (U = 43411, p = .023, Cohen's d = .110), such as the LS group scored higher in both (Fig. 4d-e). We then looked at the influence of age, education, political distress and perceptual stability on GCB scores (F(4,618) = 11.1, p <.001, adjusted R 2 = .061). Again, we found that age (estimate = -.125, p = .009), and education (estimate = -1.88, p <.001) were significantly associated with CTs, further confirming that political distress (estimate = -.396, p = .009) had a significant impact on conspiracy endorsement, even after controlling for those sociodemographic factors.

Fitting the Circular Inference Model

Because we conceptualized perception as an inferential process, we also fitted the Circular Inference model to the Necker cube data (Fig. 4c). We have previously found that CI can explain NC data better than other pure Bayesian models [START_REF] Leptourgos | Circular inference in bistable perception[END_REF] . This approach allowed us to quantify four model parameters contributing to the perceptual decision: sensory weight, prior amplification, bias, and volatility. We checked whether these CI parameters could capture the effects of political distress and conspiracy adherence. Sensory weight was the only parameter positively associated with GCB scores at baseline (p = .030, ρ = .098, Fig. 4b), supporting the idea that participants more prone to CTs at baseline rely more on sensory evidence when asked to make a decision in a highly ambiguous environment. confirmed this GCB-sensory weight association (estimate = 1.20, p = .051) even after controlling for the effects of age, education and political distress (F(4,618) = 11.86, p <.001, adjusted R 2 = .065).

Measured changes after political event resolution

We then assessed changes in political distress, conspiracy ideations and perceptual stability over time (Table .2). We confirmed an overall stress reduction at T2 compared to that at baseline (W = 100834, p <.001, Cohen's d= -.250 ; Fig. 5a), despite some heterogeneity in the participants. Meanwhile, GCB scores significantly increased (W = 73048, p = .017, Cohen's d = .068), while stability scores decreased (W = 114427, p <.001, Cohen's d = -.139) -this tendency toward destabilization was observed in each national sample (see also Fig. S3). To account for the heterogeneity in stress evolution, we split the sample into two subgroups according to their trajectories: a first subsample with decreasing stress (Dec, n=330) and a second subsample with increased stress between T1 and T2 (Inc, n=227; Fig. 5b). Considering that the Dec group should have adopted the most efficient coping strategies, we checked how the CI parameters and degree of conspiracy ideations changed over the same period in these two subsamples (Table .

S3).

A delta measure for each CI parameter was computed (parameter value at retest minus value at baseline), such as a positive delta indicated a gain in the parameter value, while a negative delta reflected a decrease in this parameter. The Dec group showed increased reliance on prior information in the bistable task between T1 and T2 (mean ΔPrior=.0811, s.d=1.11), while the Inc group showed decreased use of priors in the same period (mean ΔPrior=-.121, s.d.=1.14). This difference was statistically significant (t(460.65) = 2.07, p = .039, Cohen's d = .180 ; Fig. 5c-left). We found no differences in the 3 other CI parameters (Fig. S3). We also computed a composite ΔGCB score corresponding to GCB at retest minus GCB at baseline, such that a positive delta corresponded to an increase in conspiracy adherence while a negative delta resulted in a decrease. We observed a trend for conspiracy strengthening in participants with decreased stress in comparison with that observed in the rest of the sample (t(429.52)=1.65, p = .099, Cohen's d = .145). Because conspiracy ideations were proposed to act as a coping mechanism when facing uncertainty, we next ran an oriented test to confront that hypothesis which reached significance (t(429.52)=1.65, p = .050, Cohen's d = 0.145 ; Fig. 5c-right). This finding supports a gain in the GCB score for the Dec group compared to the Inc group. To confirm the idea that the GCB score increase was directly associated with an increase in Prior in the Dec group, we compared ΔPrior and ΔGCB in this specific subsample; these measures were found to be positively associated (p = .035, ρ = .116, Fig. 5d).

DISCUSSION

A surge in CTs has been observed in recent years, and CTs have been proposed to act as coping strategies for the stress and perceived lack of control generated by global uncertainty [START_REF] Whitson | Lacking Control Increases Illusory Pattern Perception[END_REF][START_REF] Sullivan | An existential function of enemyship: Evidence that people attribute influence to personal and political enemies to compensate for threats to control[END_REF][START_REF] Van Prooijen | The Influence of Control on Belief in Conspiracy Theories: Conceptual and Applied Extensions[END_REF][START_REF] Dow | Sense of control and conspiracy perceptions: Generative directions on a well-worn path[END_REF][START_REF] Farias | COVID-19 as an undesirable political issue: Conspiracy beliefs and intolerance of uncertainty predict adhesion to prevention measures[END_REF] . CTs offer intuitive and easy-to-understand explanations to unsolved problems [START_REF] Van Prooijen | Why Education Predicts Decreased Belief in Conspiracy Theories[END_REF] . Links have already been established between conspiracy endorsement and some inference biases 3,12-14 . However, very few studies have primarily focused on low-level perceptual aspects of conspiracy [START_REF] Müller | Linking paranormal and conspiracy beliefs to illusory pattern perception through signal detection theory[END_REF][START_REF] Hartmann | Illusory perception of visual patterns in pure noise is associated with COVID-19 conspiracy beliefs[END_REF][START_REF] Dagnall | Conspiracy theory and cognitive style: a worldview[END_REF][START_REF] Van Prooijen | Connecting the dots: Illusory pattern perception predicts belief in conspiracies and the supernatural[END_REF] , and limited efforts have been made to delve into the potential mechanisms of information processing that may convey such associations.

To address these concerns, we combined online assessments of bistable perception in large international samples with Bayesian modeling. This approach allowed us to quantify perceptual inference mechanisms and to test their links with conspiracy ideations during periods of great sociopolitical uncertainty. We were able to capture the rigidification of conspiracy beliefs in nonclinical populations. Specifically, using the Circular Inference (CI) model, we highlighted a significant association between conspiracy endorsement and the overweighting of sensory information when ambiguity reaches a climax, later followed by a selective increase in prior reliance in those who subsequently decreased their stress levels.

Several attempts at modeling the features of conspiracy beliefs can be found in the literature. However, most of these models have either focused on the network scale [START_REF] Peruzzi | From confirmation bias to echo-chambers: a datadriven approach[END_REF] or remained purely theoretical, without experimental testing [START_REF] Rigoli | Deconstructing the Conspiratorial Mind: the Computational Logic Behind Conspiracy Theories[END_REF] . Recent findings highlighted the added value of a computational framework to account for the emergence of rigid beliefs during the COVID-19 pandemic [START_REF] Suthaharan | Paranoia and belief updating during the COVID-19 crisis[END_REF] and the protective aspect of CTs against distress in a social context [START_REF] Suthaharan | Assumed shared belief about conspiracy theories in social networks protects paranoid individuals against distress[END_REF] . These studies used high-level cognitive tasks and mainly focused on paranoia, a condition sharing some phenomenological features with CTs but also considered significantly different [START_REF] Greenburgh | Paranoia and conspiracy thinking[END_REF] , further justifying specific explorations. The quantitative approach proposed in the present work nicely completes these initiatives, adding the testing of low-level inference, together to measurements of conspiracy beliefs' emergence and rigidification.

Here, we provide the first evidence for an association between sensory information overweighting in ambiguous contexts and a high level of conspiracy endorsement. This finding suggests that when uncertainty peaks, a subpart of the population, more vulnerable to stress, is prone to embracing conspiracy explanations based on intuitive reasoning. Motivated by the need to cope with uncertainty, these participants first adopt an "exploration" strategy, seeking explanations in their direct environment to make their perceptual decisions. Interestingly, such a mechanism accounts for perceptual and inferential biases previously found to be associated with conspiracy ideations, such as illusory pattern detection [START_REF] Müller | Linking paranormal and conspiracy beliefs to illusory pattern perception through signal detection theory[END_REF][START_REF] Hartmann | Illusory perception of visual patterns in pure noise is associated with COVID-19 conspiracy beliefs[END_REF][START_REF] Van Prooijen | Connecting the dots: Illusory pattern perception predicts belief in conspiracies and the supernatural[END_REF] , intuitive thinking [START_REF] Swami | Analytic thinking reduces belief in conspiracy theories[END_REF][START_REF] Binnendyk | Intuition, reason, and conspiracy beliefs[END_REF] and the JTC phenomenon [START_REF] Pytlik | Thinking Preferences and Conspiracy Belief: Intuitive Thinking and the Jumping to Conclusions-Bias as a Basis for the Belief in Conspiracy Theories[END_REF][START_REF] Kabengele | Conspiracy Beliefs and Jumping to Conclusions[END_REF] .

We also explored the dynamic changes in model parameters after stress resolution by using a pre/post design surrounding the political events. We shed light on the association between prior knowledge amplification in perceptual decisions and the enhanced adherence to CTs in those who showed reduced stress level. This finding suggests that some participants coped with uncertainty by embracing conspiracy-oriented explanations, secondarily shifting to an "exploitation" strategy (Fig. S5), validating their newly established view and reinforcing their own beliefs. This second mode appears compatible with findings showing confirmation biases (Brotherton, 2015) and reality testing deficits [START_REF] Lewandowsky | The Role of Conspiracist Ideation and Worldviews in Predicting Rejection of Science[END_REF] in people with CTs, making these beliefs more resilient to counterevidence.

These results can also be compared with models of the emergence and maintenance of clinical beliefs, such as delusional ideations. Indeed, prior research conceptualized delusion formation as the result of impaired associative learning processes driven by excessive prediction error [START_REF] Corlett | Disrupted prediction-error signal in psychosis: evidence for an associative account of delusions[END_REF] , a framework that was later extended to account for delusion persistence as aberrant reinforcement of previously leaned associations [START_REF] Corlett | From drugs to deprivation: a Bayesian framework for understanding models of psychosis[END_REF] . Our results also add to previous work showing that parametric changes might mimic behaviors observed during the transition to psychosis [START_REF] Denève | Circular inference: mistaken belief, misplaced trust[END_REF] . It was shown using CI-based simulations that the seminal amplification of sensory information involved in the integration of aberrant causal relationships (during the transition to psychosis) subsequently constituted strong priors proposed as responsible for the stability of delusional contents from one psychotic episode to the next. Both approaches (predictive coding and Bayesian modeling) are congruent with (i) the idea that conspiracy endorsement is associated with the establishment of aberrant causal relations between random events [START_REF] Whitson | Lacking Control Increases Illusory Pattern Perception[END_REF] , and (ii) that conspiracy could be rooted in the self-reinforcement of previously integrated suboptimal beliefs.

While the endorsement of CTs may serve as an effective short-term coping strategy, it also appears to pave the way for the long-term rigidification of suboptimal beliefs (beliefs that would be computed through mechanisms deviating from Bayes' rule), making it maladaptive for stressregulation overall. The social implications of gaining a better understanding of this phenomenon are vast. Humankind has experienced repeated periods of heightened uncertainty throughout history, ranging from civilizational collapses or wars to economic crises. In extending the well-established association between political distress and the endorsement of CTs 5 , our model also explains the recent rise in extremism and populism observed since the beginning of the XXI st century in a global context of the pandemic, terror attacks and climate change.

We must acknowledge some limitations of this work. First, although significant, some results exhibit small effect-sizes (i.e., Cohen's d around 0.2). Of note, small effect-sizes were previously found to still have substantial significance when studies were conducted on large populations [START_REF] Mcneish | The effect of small sample size on two-level model estimates: A review and illustration[END_REF] . It is also important to remember that small effects were expected because we attempted to capture an association between a low-level inference process (bistable perception) and a more complex cognitive process (conspiracy). These findings still constitute an important proof-of-concept demonstration that the CI model can capture small variations in nonclinical populations' perceptual decisions, paving the way for promising advancements in deepening our understanding of the mechanisms underlying belief rigidification.

A second limitation is that we cannot rule out that some participants may have felt hesitant in honestly reporting their views about CTs, due to the controversy and potential stigma surrounding conspiracy thinking. However, we think that our experimental design offers two advantages in the valid assessment of conspiracy endorsement. First, its online nature ensured anonymity and encouraged freedom of speech, as frequently observed on the internet and digital social media. Second, the joint use of a low-level perceptual task, the NC, provided access to a proxy of inference processing that is rarely prone to social biases, such as interviewer compliance.

A third limitation is the representativity of the sample: we indeed chose to recruit participants from three Western educated countries, known for their high degree of polarization [START_REF] Fletcher | How Polarized Are Online and Offline News Audiences? A Comparative Analysis of Twelve Countries[END_REF] . Although our sample may not represent the world population, we argue that the phenomenon we are investigating follows some universal rules. First, links between sociopolitical uncertainty and the resurgence of conspiracy beliefs have already been observed at various times and locations, dating back to the Roman Empire [START_REF] Boddington | Whose Conspiracy?[END_REF] . Second, while the GCB total scores were distributed differently among our three samples (Fig. S1a), their qualitative distribution across GCB subscales followed the same pattern (Fig. S1b). Third, the main results and trends (i.e., sometimes not reaching significance due to the reduced statistical power) appear consistent across the 3 samples when tested separately (Fig. S2).

For the same reasons, we focused on the level of distress related to specific political events in the countries where the tests took place. Importantly, we did not consider other types of individual stress levels. Instead, we concentrated on the broader phenomenon of sociopolitical uncertainty.

Overall, this study highlights the potential of the Circular Inference model in examining subtle variations in inference processing associated with high-level cognitive beliefs. This model has already proven effective in accounting for the positive symptoms of schizophrenia [START_REF] Jardri | Experimental evidence for circular inference in schizophrenia[END_REF][START_REF] Simonsen | Taking others into account: combining directly experienced and indirect information in schizophrenia[END_REF][START_REF] Leptourgos | Can circular inference relate the neuropathological and behavioral aspects of schizophrenia?[END_REF] and schizotypal traits [START_REF] Derome | Functional connectivity and glutamate levels of the medial prefrontal cortex in schizotypy are related to sensory amplification in a probabilistic reasoning task[END_REF] ; however, this breakthrough opens up new avenues for applying quantitative approaches to dynamically explore subjective beliefs in nonclinical populations. By applying this computational framework, we delved deeper into the mechanisms underlying the emergence and maintenance of conspiracy beliefs, shedding light on their societal impact and providing insights that could be valuable for developing interventions aimed to counter the influence of CTs during highly uncertain periods.

of the participant's perceptual dynamics. A 10-second black screen display separated each block to minimize the influence of the previous block on later responses (Fig. 2a).

Participants were instructed to stare at the target located in the middle of the screen to neutralize the potential effects of eye movements. The two possible interpretations of the NC (SFA, SFB) were explicitly mentioned, and subjects were asked to look at the cube passively, without attempting to orient or force their perception. A short training session was performed beforehand to give participants the opportunity to become familiar with the stimulus and the task while ensuring that the instructions were well understood.

Judgment criterion

Various parameters can be used to understand and describe the phenomenon of bistable perception. We chose to focus on perceptual stability because we were interested in its dynamical dimension, i.e., how the system could stabilise and destabilise.

Perceptual stability is defined as the probability that a percept persists from one trial to the next. According to Markovian modeling, the current percept (one of the two interpretations SFA or SFB) depends on the previous percept and its updating by sensory observation. This implies a circularity in the integration of information where the percept at time t becomes the prior information at time t+1. A value was thus assigned to each trial "i": 0 if the response was different from the response to trial "i-1" and 1 if the response to trial "i" was identical to the response to trial "i-1". The average SP was thus calculated for all trials and separately for each interpretation (SP0 and SP1 for SFA and SFB, respectively). Overall, the SP was interpreted as the general probability that the system remains stable from one trial to the next, where 1 corresponds to a system with no perceptual change and 0 to a system governed by maximum instability.

A previously proposed way to assess perceptual stability is by computing stability curves representing SP as a function of different ISI values. Such a curve usually consists of an initial "destabilization" portion corresponding to a drastic drop in perceptual stability, and a "stabilization" portion reaching a "ceiling threshold", considered a good proxy of perceptual stability (Fig. 2b,d). This second portion of the curve was fitted to a reversed exponential function, and we considered the parameter corresponding to the last point of the curve as the stability score for each participant.

Self-reported measures

A sociodemographic form and some psychometric assessments were then conducted/collected on the Prolific © platform. Participants specified their age and educational attainment as defined in the International Standard Classification of Education (ISCED) [START_REF]I. for. International Standard Classification of Education (ISCED)[END_REF] . The participant demographics are shown in Table .1. When Likert or visual analogical scales were used, the cursor was coded to return to the center of the screen after each question to avoid the answer being biased by previous ones. Adherence to CTs was assessed using the 15-item Generic Conspiracist Beliefs Scale (GCB) [START_REF] Brotherton | Measuring Belief in Conspiracy Theories: The Generic Conspiracist Beliefs Scale[END_REF] and its French translation [START_REF] Lantian | Measuring Belief in Conspiracy Theories: Validation of a French and English Single-Item Scale[END_REF] . The GCB scores and subscores for each sample are shown in Table .1 and Table.S1. Participants were also asked to rate with a 10-point visual analogical scale how distressed they were regarding the target event in their country (political distress). The precise questions used are shown in the Supplementary Material section: Self-reported measures.

Data Analysis and Statistics

Characteristics of conspiracy adherence

The normality of the distributions was tested using the Shapiro-Wilk test. If nonnormally distributed, further analyses were run using nonparametric statistics. Notably, we compared GCB scores between the three US-UK-FR samples using a Kruskal-Wallis test. We compared GCB scores between males and females using a Mann-Whitney test, while GCB scores across ISCED levels of education and across different age levels were compared using Welch ANOVAs.

The correlates of stress at baseline

We conducted a series of model-free analyses to confirm the association between political distress, stability score, and GCB. Again, due to the nonnormal distribution of the GCB scores, we referred to Spearman rank correlations to explore linear associations, corrected for multiple comparisons based on the false discovery rate (FDR) method. These analyses were conducted on the whole sample, and on subsamples generated through a median split on the political distress score: the 'low stress' (LS, n=310) and 'high stress' (HS, n=313) subgroups. We used Mann-Whitney tests to assess the difference between these two subgroups regarding stability scores or GCB scores. We also used a linear regression model to confirm the association between political distress and GCB, adding age and education level as covariates to control for the effect of these sociodemographic factors.

Model fitting

To better understand the association between conspiracy theories and stress, we fitted a dynamical Circular Inference model (CI) to the data (for more details, see [START_REF] Leptourgos | Circular inference in bistable perception[END_REF] ). Applied to the NC task, CI describes the process through which participants combine prior expectations about the visual appearance of three-dimensional (3D) objects and (illusory) depth cues to compute a 3D interpretation of the two-dimensional (2D) NC : seen from above (SFA) or seen from below (SFB). Belief updating in CI can be formalized as follows: Function 𝛷() describes the (Markovian) dynamics of the system and is equivalent to a leak term. It captures the intuition that in the real-world, objects are not eternal and can appear, disappear or change abruptly. Markovian temporal statistics can be reduced to 2 parameters, 𝑟 𝑜𝑛 and 𝑟 𝑜𝑓𝑓 (corresponding to the probability of switching from SFB to SFA and from SFA to SFB respectively). This term pushes L toward its prior value (log ( 𝑟 𝑜𝑛 𝑟 𝑜𝑓𝑓 )). By making 𝑟 𝑜𝑛 greater than 𝑟 𝑜𝑓𝑓 , we can implement an implicit SFA bias.

The second term describes the auto-amplification of priors due to descending loops (parameter 𝑎). According to CI, prior information can be reverberated and counted several times [START_REF] Jardri | Circular inferences in schizophrenia[END_REF] . This overcounting of priors is akin to a positive feedback that strengthens and stabilizes currently held perceptual beliefs, resulting in bistable perception [START_REF] Leptourgos | Circular inference in bistable perception[END_REF] .

Finally, the third term describes the sensory noise that drives switches between the 2 interpretations. For simplicity, we assume that 𝑆 is sampled from a normal distribution with 0 mean and variance equal to 1. Furthermore, 𝑤 is a free parameter representing the overall sensory weight (sensory weight and climbing loops are mathematically indistinguishable, so they are both included in 𝑤).

In summary, this model of perceptual dynamics can be reduced to 4 free parameters: the overall gain of sensory inputs 𝑤 (sensory), the descending loops 𝑎 (prior), the transition rate 𝑟 𝑜𝑛 (volatility) and the bias (𝑟 𝑜𝑛 -𝑟 𝑜𝑓𝑓 ).

Changes after political event resolution

We assessed the evolution of political distress, stability scores and GCB scores over time using Wilcoxon signed-rank tests for repeated measures. We then split our sample into two groups: Dec and Inc comprising individuals who showed decreased or increased stress, respectively, between the two time points. We computed a delta measure for each parameter that corresponded to the parameter's value at retest minus that at baseline. A positive value indicated a gain in the parameter, while a negative value indicated a decrease. Due to the normal shape of distributions in these composite scores and our sample size, we referred to Welch tests for group comparisons.

The same procedure was used to compare the two groups regarding the gain in GCB (ΔGCB). We successively performed a two-tailed Welch's test, followed by Welch's test for the oriented hypothesis that the Dec subsample would significantly increase its GCB score compared with the Inc subsample. Finally, a Pearson correlation test was used to check for an association between ΔAlpha and ΔGCB in the Dec subgroup.

FIGURE 2 .

 2 FIGURE 2. The Necker cube (NC) task: procedure and validity. (a) The experimental procedure consisted of serial NC presentations. Each trial was decomposed into three steps (see Methods section). After a fixation cross of pseudorandomized duration (ISI) (1), the Necker cube was presented (2) until participants reported their interpretation of the stimulus: 'seen from above' (SFA) or 'seen from below' (SFB), using the right or left arrow of their keyboard, respectively (3). (b) Perceptual stability as a function of the interstimulus interval (ISI) for each national sample. US (United States of America, mean Stability Score= .587, s.d.= .172), UK (United Kingdom, mean Stability Score= .570, s.d.= .176) and FR (France, mean Stability Score= .565, s.d.= .1472). (c) Averaged stability scores at each time point for the three national samples. (d) Perceptual stability as a function of ISI, for online (mean stability score= .441, s.d.= .190) and in-lab methods (mean stability score= .500, s.d.= .140). (e) Bland-Altman plot of the agreement between online and in-lab methods comparing stability scores obtained in each condition for the same participants (n = 16). The x axis represents the average scores of the two methods. The y axis represents the mean difference between online and in-lab stability scores. The limits of agreements (LoA, pink dotted lines) are defined as the mean difference computed on all participants (pink line) ± 1.96 s.d., and each dot represents a participant. As all participants are included in the LoA, the methods are considered to be in agreement and may be used interchangeably.

FIGURE 3 .

 3 FIGURE 3. Sociodemographic features associated with conspiracy theories at baseline. (a) Left-skewed distribution of GCB scores across the entire international sample (N = 623). (b) Mean conspiracy scores in females (n = 310, mean = 33.8, s.d. = 13.5) and males (n = 312, mean = 33.8, s.d. =13.2). The between groups difference was not significant. (c) Mean conspiracy scores according to age level. Young participants (n = 310, age = [18;30]) displayed higher GCB scores (mean = 35, s.d. = 13.2) than the adults (n = 210, age = (30;45], mean = 33.1, s.d. = 13.2) and older adults (n = 103, age = (45;60], mean = 31.5, s.d. = 13.8). (d) Mean conspiracy scores according to educational attainment levels. The low education group (n = 86, ISCED = [0;3]) scored significantly higher on GCB (mean = 38.2, s.d. = 13.7) than the medium education (n = 363, ISCED = [3;6], mean = 34.7, s.d. = 13.5) and the high education groups (n = 179, ISCED = [6;9], mean = 30, s.d. = 12).

FIGURE 4 .

 4 FIGURE 4. Cognitive and perceptual inference correlates at baseline. (a) Heatmap depicting the strength of associations at baseline between political distress, conspiracy ideations measured with the Generic Conspiracist Beliefs Scale (GCB) and perceptual stability (Pearson's correlations, corrected for multiple comparisons using the false discovery rate method, FDR). Political distress was negatively associated with GCB (ρ = -.094) and perceptual stability scores (ρ = -.120). (b) Heatmap illustrating the strength of associations at baseline between GCB scores, political distress and Circular Inference parameters (sensory weight (𝑤), prior amplification (𝑎), bias (𝑟 𝑜𝑛 -𝑟 𝑜𝑓𝑓 ) and volatility (𝑟 𝑜𝑛 )). Pearson's correlations were corrected for multiple comparisons using FDR. GCB scores were significantly associated with sensory overweighting (ρ = .098). (c) The Circular Inference (CI) model relies on 4 parameters: the overall sensory gain (sensory, 𝑤), the descending loops (prior, 𝑎), the transition rate (volatility, 𝑟 𝑜𝑛 ) and the configuration preference (bias, (𝑟 𝑜𝑛 -𝑟 𝑜𝑓𝑓 )) (see Methods). (d) Mean conspiracy scores in the low stress (LS) group (n = 310, mean political distress = 2.10, s.d. = 2.06; mean conspiracy score = 35.3, s.d.= 14.4) and high stress (HS) group (n = 313, mean political distress = 8.01, s.d.= 1.58; mean conspiracy score = 32.3; s.d.= 12.0). GCB scores were significantly higher in the LS group (Cohen's d = .110). (e) Perceptual stability plotted as a function of inter-stimulus-interval (ISI) in LS (mean stability score = .597; s.d.= .176) and HS (stability score = .548; s.d.= .177) groups. Perceptual processing was found significantly more rigid in the LS group than in the HS group (Cohen's d = .140;). * stands for p<.05, ** for p<.01 and *** for p<.001.

FIGURE 5 .

 5 FIGURE 5. Computational and cognitive features associated with changes in political distress over time. (a) Illustration of political distress scores over testing sessions. Political distress decreased between baseline (T1, mean political distress = 5.35, s.d. = 3.33) and retest (T2, mean political distress = 4.54, s.d. = 3.16; Cohen's d = -.250). (b) Political distress over time according to stress trajectories. Some participants showed decreased political distress after uncertainty resolution (Dec, n = 330, mean Δpolitical distress = -2.78, s.d. = 2.45), while another subsample showed increased stress (Inc, n = 220, mean Δpolitical distress = 2.01, s.d. = 2.14). (c) Left: Illustration of the evolution of prior amplification (ΔPrior) over time for the 'Dec' (mean = .081, s.d. = 1.11) and 'Inc' (mean = -.121, s.d. = 1.14) groups. The intergroup difference was significant (Cohen's d = .180). Right: Evolution of conspiracy ideations (ΔConspiracy ideations) over time for the 'Dec' (mean = 1.29, s.d. = 10.3) and 'Inc' (mean = -.305, s.d. = 11.6) groups. The intergroup difference was significant (Cohen's d = .145) (d) Scatter plot showing the correlation between ΔPrior and ΔConspiracy ideations in the 'Dec' group (p = .035, ρ = .116; Spearman correlation, ρ = .116). * indicates p<.05, *** indicates p<.001.

  (𝐿) + 𝑎𝐿 + 𝑤𝑆This equation describes how the posterior belief about the ambiguous figure L changes over time (positive/negative L corresponds to SFA/SFB beliefs), under the influence of 3 driving "forces": dynamics (𝛷(𝐿)), descending loops (𝑎𝐿) and sensory noise (𝑤𝑆).

TABLE 1 . Description of the populations at baseline.

 1

		Sex (F /M) Age (y.o.) Education pol. distress	GCB	stability
	Whole Sample (n = 623)	310 / 311	33.0 ± 10.9 5.64 ± 1.41	5.07 ± 3.48	33.8 ± 13.3	.572 ± .178
	US (n = 212)	119 / 106	30.4 ± 9.83 5.22 ± 1.43	4.60 ± 3.88	37.7 ± 13.9	.585 ± .173
	UK (n = 225)	98 / 112	38.2 ± 10.7 5.52 ± 1.44	5.69 ± 3.32	33.3 ± 13.7	.566 ± .178
	FR (n = 186)	93 / 93	29.8 ± 9.90 6.25 ± 1.12	4.87 ± 3.07	30.0 ± 11	.565 ± .183
				The sex-ratio did not differ across samples (X²
	= 1.68, p = .431). UK participants were significantly older (F(2.408) = 44.255, p < 0.001, η² = 1.29e -19 ) and FR participants reached a
	higher educational attainment (F(2.411) = 35.458, p < 0.001, η² = 3.76e -13 ) than the other samples. UK participants demonstrated a
	higher level of political distress (F(2.408) = 5.8388, p = .00316, η² = 2.82e -3 ), while stability was consistent across samples (F(2.405) =
	0.81828, p = .442).

US: United States of America; UK: United Kingdom; FR: France; F/M: female or male; y.o., years old; Education levels are provided according to the International Standard Classification of Education (ISCED); pol. distress: political distress; GCB: Generic Conspiracist Beliefs Scale; stability: fitted stability score (see Methods section: Judgment criterion).

TABLE 2 . Population description at each time-step: scores, and CI parameters.

 2

		pol. distress	GCB	stability	sensory	prior	bias	volatility
	T1	5.35 ± 3.33	33.78 ± 13.33 .57 ± .18	1.70 ± .85	1.85 ± .91	.59 ± .07	-2.04 ± 1.50
	T2	4.54 ± 3.16	34.70 ± 13.56 .55 ± .18	1.65 ± .85	1.86 ± .93	.59 ± .06	-2.20 ± 1.40

Pol. distress: political distress; GCB: Generic Conspiracist Beliefs Scale; stability: estimated stability score (see also Methods section: Judgment criterion); Sensory: sensory overweighting (𝑤); Prior: prior amplification (𝑎).

METHODS

Participants

Three independent samples were recruited using the Prolific © web-platform: 212 US citizens, 225 British citizens and 186 French citizens. The same protocol was administered 1 month before and 1 month after a major stressful political event: the 2020 US presidential election, the 2021 UK BREXIT implementation and the 2022 French presidential election (Fig. 1). The targeted participants were aged between 18 and 60 and had normal-to-corrected vision. They were from the nationality of the country of interest for each sample and regularly used social media. Th exclusion criteria were a history of psychiatric or neurological disorder, strabismus, or eye surgery. From the initial sample (N = 755), 30 participants were excluded based on failed attentional checks (see Supplementary Material section: Controlling for experimental biases) or very-low reaction times (mean reaction time < 300ms), while 102 were lost longitudinally.

The Prolific © web-platform (https://www.prolific.co/) ensures data privacy following standards of the European and UK data protection law (i.e., General Data Protection Regulation (GDPR), transposed into UK law as the UK GDPR). Participants' sociodemographic characteristics were associated with their respective behavioral data through an anonymous ID randomly assigned at enrollment. The overall online survey complies with French regulations and ethics (Comité de Protection des Personnes Nord-Ouest IV).

Apparatus

The protocol was implemented in PsychoPy v.3, exported and hosted online on the Pavlovia.org platform. For the perceptual part of the experiment, participants were instructed to stand in total darkness, approximately 60 cm away from the screen and adjust it to be perpendicular to the floor with their eyes aligned to the fixation cross displayed at the center of the screen. The NC task and the self-reported assessment of beliefs were administered in a randomized order (see also

Supplementary Material section: Controlling for experimental biases).

The Necker Cube Task

Stimuli:

Visual stimuli representing Necker cubes (NC) were displayed in the center of a black screen. The stimulus size was standardized across the participants using a matching method based on a standard credit card displayed on the screen that the participant was required to adjust in size before starting the experiment (GitHub link to add).

Procedure:

The block-design of the task was inspired by Mamassian and Goutcher's 50 protocol. During each block, a NC was presented discontinuously. Referring to a forced-choice methodology, we asked participants to report their interpretation of the stimulus using their keyboard each time a new cube appeared on the screen. The cube disappeared after a pseudorandom duration (ISI ranging from 0.1 to 1.2 seconds). Each recorded response constituted a trial, and the experiment was divided into 10 blocks of 64 consecutive trials (i.e., 640 NC presentations per run), providing a discontinuous sample


Rawan Ghnemat
email: rawan.ghnemat@litislab.eu

Cyrille Bertelle
email: cyrille.bertelle@litislab.eu

Gérard H E Duchamp

Adaptive Spatial System Emergence from Community Swarm Optimization

This paper proposes a bottom-up approach to model spatial systems for urban dynamic and territorial management. Such approaches called micro-modelling have started with the use of cellular automata where local rules are defined and allow to implement some simulations, describing mainly dynamical diffusion processes over spatial grids. Our purpose is to deal with the detection of some emergent organizations during the simulation, using an implicit control, based on the use of genetic automata population. This automata population evolves by a swarm co-evolution, optimizing a fitness function which leads to an automatic emergence of dynamic communities.

SPATIAL SYSTEM EMERGENCE MODELLING

From Top-down to Bottom-up approches

. The first one is based on macromodelling corresponding to a top-down approach and the second one is based on micro-modelling corresponding to a bottom-up approach.

. Lowry's model of urban system, applied to the city of Pittsburg, proposed some "integrated" model, defining flow chart between the three main indicator classes: (i) the basic sector of industrial and business activities, (ii) the householder sector and (iii) the retail sector concerning the local population. This flow chart model already deals with a mile-square decomposition similar to spatial decomposition used later as an adaptation of cellular automata grid to geographical real space. The final output of the modeling process leads to a kind of socio-economic equilibrium state. This approach finds its limit because of its static description and dynamical models are essantial to understand the city evolution. Forrester proposed a dynamical modeling based on the application of industrial dynamics on urban dynamics. His model is based on non spatial stocks and flows models. Stocks are exchanged within a three income levels decomposition over housing, jobs and population. This model based on simple urban description was aimed to generate

simulation and Forrester claims the benefit of computer simulation to understand the city evolution and how we can predict its evolution by the modification of guiding policies within the system.

The stocks and flows models continue to be improved and to proposed more and more details, including transportation subsystem or land market, for example. One of the most complete, called Integrated Urban Model (IUM) was proposed by Bertuglia et al. [START_REF] Bertuglia | An integrated urban model[END_REF]. The computational complexity increase with the accuracity of the description and finally avoid to obtain reasonable estimates of the parameters. These models are more representational tools than simulation tools [START_REF] Benenson | Modeling Population Dynamics in the City: from a regional to a Multi-Agent Approach[END_REF].

To buid efficient simulation models, the idea was to simplify the description, using a more global one facilitating the analytical description. From the inspiration of dynamics of population theory, some researchers proposed to build urban dynamic models from ecological modeling. The paradigm of prey-predators systems is then used to give efficient simulation tools to investigate the main feature allowing to understand the global dynamics. For example, Dendrinos and Mulally [START_REF] Dendrinos | Urban Evolution: Studies in the Mathematical Ecology of the Cities[END_REF] use a prey-predator model, assuming that the increase of city population make decrease the economic status. The predators represent the urban population and the preys, the per capita income.

All the previous described models are based on top-down approachs to model the system dynamics. We first consider the whole phenomenon and we propose a way to split it in many sub-problems and then in stocks and flows or in different terms contituting the equational system. Another class of modeling is based on micromodeling and bottom-up representation of the city as a collection of individual-based descriptions, behavioral rules-based description and interaction systems. From this constructive approach, we want to obtain an emergent description of the whole system or of some sub-systems included in a hierarchical process. Two complementary methodologies can be used for that and we detail them in the following paragraphs.

The first methodology consists to generate a simulation where all the components, behaviors and rules, interact over a environment, perceiving and acting on it. The environment evolving is the support of emergent properties. The cellular automata modeling deals with this kind of simulations. The basic definition of cellular automata for urban or regional modeling, for exemple, consists in the decomposition of the city, region or any geographical area in a lattice of cells. Each cell is in some state which belongs to a finite set S. At each time step, the cells change its own state according to some transition rules based on its previous state and its neighbor cells. Many works based on cellular automata, have been developped for geographical systems and urban dynamics [START_REF] Batty | From cells to cities[END_REF][START_REF] Dubos-Paillard | Analyse de l'évolution urbaine par automate cellulaire. Le modèle SpaCelle[END_REF]. Extensions on environmental problems like water streaming are using these medels as efficient tools [START_REF] Langlois | RuiCells, automate cellulaire pour la simulation du ruissellement de surface[END_REF]. Cellular automata can be seen mainly as distributed tools to model diffusive penomena using rule-based systems. One of the first researchers in human sciences to propose models based on diffusive rule-based systems is T. Hägerstrand in very early period, during 50' [START_REF] Hagerstand | Innovation Diffusion as a spatial process[END_REF] but his work itself start to be diffused over the science community more than 15 years later, when the computer development become able to implement its model in realistic studies. One of the most famous cellular-based model for social modeling is du to T. Schelling [START_REF] Schelling | Dynamic Models of Segregation[END_REF], describing the segregation process. But with the implementation of this model, we face to an important extension to cellular automata where we need to represent moving individuals from a simple deliberative process. The mixing of spatial data and cellular automata with autonomous entities, like agents, is here needed [START_REF] Daude | Introduction à la modélisation multi-agents des systèmes complexes en géographie[END_REF].

The second methodology to deal with emergent description in micro-modelling, consists to complete the previous approach based on simulation, by introducing some computational processes which are able to detect emergent systems or organizations. The final goal of this method is then to be able to re-introduce these emergent systems or organizations inside the simulation and manage their evolutions and their interactions with the components of the system. The re-integration of the emergent systems, during the simulation, can be explicitely expressed like in the multiscale fluid flow simulation proposed by P. Tranouez [START_REF] Tranouez | Changing levels of description in a fluid flow simulation[END_REF] or it can be implicitly expressed using a self-controled process as we will describe in the following, using genetic algorithms.

The method proposed in this paper, called community swarm optimisation, is based on a swarm intelligence process which make coevolve a population of genetic automata, using a global control of the system, implemented by a fitness function leading to the emergence of communities, as defined in the next section.

A Specific Spatial System Context: the Community

We define here, the context of the community used in the swarm intelligence process proposed in the following.

Definition 1.1 (Community operational definition)

A community is a system or an organization which is characterized by a spatial property, a behavior property and the interaction between both.

Example 1.2 In ecology, a community is a group of plants or animals living in a specific region and interacting with one another.

Example 1.3

The spatial patterns generated by Schelling's segregation models [START_REF] Schelling | Dynamic Models of Segregation[END_REF] are some examples of communities and these spatial patterns are linked with some elementary behavioral rules implemented for each grid case. These rules describe, for each step, the movement of each individual according to its neighborhood.

In the Community Swarm Optimization(SCO) method which we propose and describe in the following, we need to represent an efficient way to describe the behavior of each entity and we use algebraic structures called automata with multiplicities [START_REF] Schützenberger | On the definition of a family of automata[END_REF]. The main advantage of these automata is to be associated with algebraic operators leading to automatic computation. With these operators, we can define behavioral distances for the entities modelled with these automata. The behavioral distance is one of the major keys of this new method. Section 2 presents a review of swarm optimization methods to introduce the original one we propose in this paper. Section 3 describes the algebraic basis for the automata management used in this method. In section 4, we describe the proposed method and in section 5, we discuss some applications which can be efficiently modelled by this method, according to their own complexity.

SWARM OPTIMIZATION METHODS

Decentralized algorithms have been implemented for many years for various purposes. In this algorithm category, multi-agent systems can be considered as generic methods [START_REF]Multiagent Systems[END_REF]. Agent-based programming deals with two main categories of agent concepts: cognitive agents and reactive agents. The first category concerns sophisticated entities able to integrate, for example, knowledge basis or communications systems. Generally, efficient computations, based on these cognitive architectures, implement few agents. The second category of agents, based on reactive architecture, is expected to be used inside numerous entity-based systems. The goals of programs using such architectures, is to deal with emergent organizations using specific algorithms called emergent computing algorithms. Swarm Intelligence is the terminology used to point out such reactive agent-based methods where each entity is built with the same basis of behavior, but reacts in autonoumous way. Swarm Optimization methods concern the problems of optimization where the computation of a function extramum is based on the concept of swarm intelligence.

Ant Colony Optimization (ACO) methods [START_REF] Bonabeau | Swam Intelligence, from natural to artificial systems[END_REF] is a bio-inpirated method family where the basic entities are virtual ants which cooperate to find the solution of graph-based problems, like network routing problems, for example. Using indirect communications, based on pheromon deposites over the environment (here a graph), the virtual ants react in elementary way by a probabilistic choice of path weighted with two coefficients, one comes from the problem heuristic and the other represent the pheromon rate deposit by all the ants until now. The feed-back process of the whole system over the entities is modelled by the pheromon action on the ants themselves.

Particule Swarm Optimization (PSO) is a metaheuristic method initially proposed by J. Kennedy and R. Ebenhart [START_REF] Kennedy | Particle Swarm Optimization In Proceedings of IEEE International Conference on Neural Networks[END_REF].This method is initialized with a virtual particle set which can move over the space of the solutions corresponding to a specific optimization problem. The method can be considered like an extension of a bird flocking model, like the BOIDS simulation from C.W. Reynolds [START_REF] Reynolds | Flocks, Herds and Schools: a distributed behavioral model In Computer Graphics[END_REF]. In PSO algorithm, each virtual particle moves according to its current velocity, its best previous position and the best position obtained from the particles of its neighborhood. The feed-back process of the whole system over the entities is modelled by the storage of this two best positions as the result of communications between the system entities.

Other swarm optimization methods have been developped like Artificial Immune Systems [START_REF] De Castro | Artificial immune system: a new computational approach[END_REF] which is based on the metaphor of immune system as a collective intelligence process. F. Schweitzer proposes also a generic method based on distributed agents, using approaches of statistical many-particle physics [START_REF] Schweitzer | Brownian Agents and Active Particles[END_REF]. The Communities Swarm Optimization method proposed in this paper, consists in the co-evolving of both the spatial coordinates and the behavior of each individual of a virtual population of automata. The feed-back process of the whole system over the entities is modelled by a genetic algorithm based on this co-evolving. The automata behaviors allow to define for each individual, a set of arbitrary complex transition rules. We develop the formalism needed to describe this method and the associated algorithm in the two next sections.

AUTOMATA-BASED AGENT MODELING AND COMPLEX SYSTEMS

Complex Systems Concepts

In this section, we give the basis of the conceptual tools which allow to extend the reactive and diffusive grid cases behavior to more sophisticated entities, using agent-based models. We propose to model the agent behavior with automata with multiplicities which are powerful algebraic stuctures.

According to General System Theory [21], a complex system is composed of entities in mutual interaction and interacting with the outside environment. A system has some characteristic properties which confer its structural aspects, as schematically described in part (a) of Figure 2: • The set elements or entities are in interactive dependance. The alteration of only one entity or one interaction reverberates on the whole system. • A global organization emerges from interacting constitutive elements. This organization can be identified and carries its own autonomous behavior while it is in relation and dependance with its environment. The emergent organization possesses new properties that its own constitutive entities do not have. • The global organization retro-acts over its constitutive components.

The interacting entities network as described in part (b) of Figure 2 leads each entity to perceive informations or actions from other entities or from the whole system and to act itself.

A well-adapted modeling consists of using an agent-based representation which is composed of the entity called agent as an entity which perceives and acts on an environment, using an autonomous behaviour as described in part (c) of Figure 2.

To compute a simulation composed of such entities, we need to describe the behaviour of each agent. This one can be schematically described using internal states and transition processes between these states, as described in part (d) of with multiplicities as described in the following section is welladapted for the agent behavior modelling.

Automata-based Modeling for Agent Behavior

An automaton with multiplicities is based on the fact that the output data of the automata with output belongs to a specific algebraic structure, a semiring, including real, complex, probabilistic, non commutative semantic outputs (transducers) [START_REF] Golan | Power algebras over semirings[END_REF][START_REF] Stanley | Enumerative combinatorics[END_REF]. In that way, we will be able to build effective operations on such automata, using the power of the algebraic structures of the output data. We are also able to describe automata by means of a matrix representation with all the power of the new (i.e. with semirings) linear algebra.

Definition 3.1 (Automaton with multiplicities)

An automaton with multiplicities over an alphabet A and a semiring K is the 5-uple (A, Q, I, T, F ) where

• Q = {S1, S2 • • • Sn} is the finite set of state; • I : Q → K is a
• λ ∈ K 1×Q which is a row-vector which coefficients are λi = I(Si), • γ ∈ K Q×1 is a column-vector which coefficients are γi = F (Si), • µ : A * → K Q×Q is a morphism of monoids (indeed K Q×Q is M(C) M(D)

Chromosome 1st component

(Automata-Based Agent Behavior)

We represent the agent behavior by automata with multiplicities (A, Q, I, T, F ) over a semiring K:

• The agent behavior is composed of a states set Q and of rulebased transitions between them. These transitions are represented by T ; I and F represent the initial and final transitions; • Alphabet A corresponds to the agent perceptions set;

• The semiring K is the set of agent actions, eventually associated to a probabilistic value which is the action realization probability (as defined in [START_REF] Ghnemat | Automata-Based Adaptive Behavior for Economic Modelling Using Game Theory[END_REF]).

Agent Behavior Metric Space

The main advantage of automata-based agent modelling is their efficient operators. We deal is this paragraph with an innovative way to define behavioral semi-distance as the essential key of selforganization processus proposed later.

Definition 3.4 (Evaluation function for automata-based behavior)

Let x an agent whom behavior is defined by A, an automaton with multiplicities over the semiring K, we define the evaluation function e(x) by: e(x) = V (A)

where V (A) stands for the vector of all coefficients of (λ, µ, γ), the linear representation of A, defined in remark 3.2.

Definition 3.5 (Behavioral semi-distance)

Let x and y two agents and e(x) and e(y) their respective evaluations as described in the previous definition 3. [START_REF] Bertelle | Genetic algorithms on automata with multiplicities for adaptive agent behaviour in emergent organizations In SCI[END_REF]. We define d(x, y) a semi-distance or pseudometrics 2 between the two agents x and y as d(x, y) = ||e(x)e(y)|| a vector norm of the difference of their evaluations.

Remark 3.6 In this paper, we propose a simple computation for the behavioral semi-distance. It is possible to define other behavioral semi-distances who can allow to introduce specific similarities on

specific path inside the automata and not the complete description of the automata as used here. This process consists in defining specific initial and final states and compute all the successfull paths between them [START_REF] Schützenberger | On the definition of a family of automata[END_REF] . We will not develop this extension in this paper.

2 see [START_REF] Bourbaki | Elements of Mathematics: General Topology[END_REF] ch IX

COMMUNITY SWARM OPTIMIZATION ALGORITHM

Spatial Automata-based Agent Definition (Spatial Automata-Based Agent)

A spatial automata-based agent is defined by its structural representation:

• An automaton with multiplicities corresponding to its behavior as a whole processus managing its perceptions and its actions over its environment. They include its communication capabilities and so its social behavior; • A spatial location defined on some specific metric space.

Genetic Operators on Automata Population

We consider in the following, a population of spatial automata-based agents, each of them is represented by a chromosome, following the genetic algorithm basis. We define the chromosome for each spatial automata-based agent as a couple of two sequences:

• the sequence of all the rows of the matrices of the linear representation of the automata. The matrices, associated to each letter from the alphabet of the agent perceptions, are linearly ordered by this alphabet and we order all the rows following these matrices order [START_REF] Bertelle | Genetic algorithms on automata with multiplicities for adaptive agent behaviour in emergent organizations In SCI[END_REF]. The figure 3 describes how this sequence is created from a linear representation of two matrices; • the sequence of all its spatial coordinates.

In the following, genetic algorithms are going to generate new automata containing possibly new transitions from the ones included in the initial automata.

The genetic algorithm over the population of automata with multiplicities follows a reproduction iteration broken up in two steps [START_REF] Goldberg | Genetic Algorithms[END_REF]:

• Reproduction (Duplication and Crossing-over): This operator is a combination of the standard duplication and crossing-over genetic operators. For each couple of spatial automata (called the parents), we generate two new spatial automata (called the children) as the result of the chromosome crossings and we keep, without change, the parent spatial automata. To operate for the crossing-over operation, we have to compute the automata of the behaviors of the two children. For this purpose, we consider a sequence of rows for each matrix of the linear representation of one of the two parents and we make a permutation on these chosen sequences of rows between the analogue matrix rows of the other parent. • Mutation: This operator deals only with the linear representation of the spatial automata-based agent. With a low probability, each matrix row from this linear representation is randomly chosen and a sequence of new values is given for this row (respecting some constraints if exist, like probabilistic values [START_REF] Bertelle | Genetic algorithms on automata with multiplicities for adaptive agent behaviour in emergent organizations In SCI[END_REF]).

Remark 4.2

The fitness is not defined at this level of abstract formulation, but it is defined corresponding to the context for which the automaton is a model, as we will do in the next section.

Adaptive Processus to Implement Community Detection

The community detection is based on a genetic algorithm over a population of spatial automata-based agents. The formation of the community is the result of the population evolution crossing by a selection process computed with the fitness function defined in the following.

For this computation, we deal with two distances defined on agent set. The first is the spatial distance associated to the agent spatial location and the second is the behavioral semi-distance defined in the definition 3.5.

Definition 4.3 Community clustering and detection fitness

Let Vx a neighbourhood of the agent x, relatively to its spatial location. We define f (x) the agent fitness of the agent x as :

f (x) =      card(Vx) y i ∈Vx d(x, yi) 2 if y i ∈Vx d(x, yi) 2 = 0
∞ otherwise where d(x, y) is the behavioral semi-distance between the two agents x and y.

On the figure 4, we represent an automata population where each automata is a colored chain representing its chromosom. Automata with similar colored chain must ne understanding as similar behavioral automata. In the left part of the figure, we focuss on one high fitness individual after computing its spatial neighbourhood and observing the behavioral similarity of all the automata included in this neighbourhood. In the right part of the figure, composed of the same population, we focuss on a low fitness individual, having not similar behavior with the other automata belonging to its neighbourhood.

The genetic evolution of the spatial automata-based agents leads to a self-organization which creates a clustering of the agents set in such way that each cluster contains agents of similar behavior. During the evaluation process, genetic algorithms can be turned such that individuals outside communities be attracted to them. The center of the clusters, the size of the clusters and the behavior of the agents in the center of each cluster are the result of the overall genetic processus which generates self-organization communities.

General Community Swarm Optimization Algorithm

Community Swarm Optimization (CSO) algorithm needs a initial step description which is the major issue of the modelling process. The way of going from the problem formulation to the initial spatial automata-based agents must be realized with accuracity. The formal description of the methodology to use, for this initial step, is described in Algorithm 1.

The core of the CSO algorithm is described by the iterative scheme defined in the Algorithm 2.

Conclusion and Perspective

In this paper, we describe Community Swarm Optimization (CSO) method which can be described as a swarm intelligence process. With the comparison of other methods from its category (Ant This method is expected to be used for adaptive spatial system emergence modelling. The swarm intelligence method proposed here manage artificial population leading to the emergent formation of communities. The genetic automata which compose this artificial population, allow the emergent communities to be self-controled using the fitness of the genetic process. This fitness express a global control over the community, using similarity evaluation on spatial neighborhood. This process mix individual representation and community system emergence: it is the formalization of a multi-scale description where the micro-macro interactions are implicit express by the genetic control of the emergent communities. The application expected to be model by this process, concern for exemple urban dynamic where quarters or city centers can emerge or evolve from the citizen behavior like in the gentrification problem.

Figure 1 .

 1 Figure 1. Support and feed-back comparison from Ant Colony Optilization (ACO), Particule Swarm Optimization (PSO) and Community Swarm Optimization (CSO)

Figure 2 .

 2 So Agent-based model for entity (d) automata-based model for agent behaviour

Figure 2 .

 2 Figure 2. Multi-scale complex system description: from global to individual models

Figure 3 .

 3 Figure 3. Chromosome first component building from the matrix rows of the linear representation of an automaton over the alphabet {C,D}

Figure 4 .

 4 Figure 4. Fitness evaluation for community detection

Algorithm 1 : 2 .Algorithm 2 :

 122 Methodology to model the initial step of CSO 1. Problem formulation by the definition of a set of transition rules ; Building of the behavioral automata based on the previous set of transition rules, describing the sequences and the context of their applications ; 3. Discretization of the spatial domain, according to its topological properties (Cellular automaton, network or graph, Geographical Information System) with the spatial location of the initial virtual population of spatial automata-based agents; Iteratice scheme of CSO Building the initial virtual population of the spatial automata-based agents (following the methodology of Algorithm (1)) ; repeat for Each couple of individuals in the population do Reproduction step generating 2 new children as described in the section (4.2) ; Mutation step as described in the section (4.2) ; Selection of the half population of the individuals corresponding to the highest values of the agent fitness described in section (4.3) ; until (the sum of the fitness values of the whole population reachs a threshold) or (the maximum iteration number is reached) ; Colony Optimization and Particule Swarm Optimization), CSO differs mainly on the modelling purpose. CSO deals with transition rules included in data structures (automata with multiplicities) for which algebraic operators allow to implement automatic computation for self-organizational phenomena.