Yann Balouin
email: y.balouin@brgm.fr

François Bourrin

Florian Meslard

Eric Palvadeau

Nicolas Robin

Assessing the Role of Storm Waves and River Discharge on Sediment Bypassing Mechanisms at the Têt River Mouth in the Mediterranean (Southeast France)

River mouths along sandy coastlines are influenced by alongshore transport of littoral sands and the interaction of hydrodynamics processes at the river mouth. These interaction are responsible for opening/closure of river outlets and subsequent episodes of sand bypassing by spitbreaching. Here, we study these mechanisms of alongshore sediment bypassing at a river mouth along the French Mediterranean coast. Over the last 6 years, bypassing processes were found to be associated with spit elongation and spit breaching. Hydrodynamics measurements were implemented in order to assess the respective role of river jet and wave conditions on these opening/breaching episodes. Results tends to point on one major mechanism: spit overtoping that induces breaching. For the floods events analysed, the river jet has apparently only a very low contribution to this opening.

INTRODUCTION

Intermittently open/closed estuaries (IOCE, McSweeney et al., 2017) are dynamic coastal areas where morphology and by-passing mechanisms are driven mainly by waves and fluvial processes.

As for tidal inlets, several bypassing processes can be observed (spit breaching, bar bypassing, channel rotation, [START_REF] Fitzgerald | Natural mechanisms of sediment bypassing at tidal inlets[END_REF][START_REF] Fitzgerald | New England tidal inlets with special reference to riverine-associated inlet systems[END_REF][START_REF] Balouin | Sediment transport pattern at the Barra Nova inlet, South Portugal: A conceptual model[END_REF][START_REF] Nienhuis | Breaking wave driven sediment bypassing of river mouth: mechanisms and effect on delta evolution[END_REF]. In wave dominated river outlets, interaction between the river discharge and the coastal dynamics generates complex hydrodynamic processes interacting with morphological features. Spit growing can partialy or fully close the river mouth during the dry season but also generates abrupt opening during floods, storm surge or concimittance of these events. In these environments, such behaviours are key parameters in the definition of coastal hazards.

Indeed, the spit disturbes the river jet inducing an increase of the water level on the river side, while limiting the impact of waves and marine surge in the river. When conditions are met to breach the spit, it allows river jet flushing and penetration of waves in the outlet. This breaching is also the main mechanism for sand bypassing the oulet. Here, we study the mechanisms by which the interaction between the flash flood and storm surge controls the openings of the river entrance and sediment bypassing at the Têt river mouth in SE France.

Fieldsite

Têt river's mouth is located in the Roussillon coast, along the French Mediterranean coast. This is a small mountaineous river draining a catchment around 1400 km². Hydrological regime is typical of Mediterranean area. The rainfall pattern is characterized by long dry periods interrupted by short mediterranean events leading to flash floods.

The mean water discharge at Perpignan station is around 11m 3 /s but can reach 1800 m 3 /s during major floods [START_REF] Bourrin | Sediment dispersal from a typical Mediterranean flood: the Têt River, Gulf of Lions[END_REF]. The river mouth tend to a seasonal closure during the summer period and re-opening during the winter period.

The Roussillon coast is a microtidal, wave dominated sandy coast. The tidal range is very low (< 0.30 m at mean spring tides). Nevertheless, large variations in water level can occur in response to wind forcing and atmospheric pressure fluctuations. Maxium water level can exceed 1 to 1.5 m near the shore [START_REF] Certain | Morphodynamics of a microtidal barred sandy coast : The Gulf of Lions[END_REF] under the combined action of storm surge and waves. These events are able to overtop sand spit that has an elevation ranging from 1.2 to 2 m above mean sea level.At the coast, significant wave heights (H s ) are generally low (H s < 0.3 m for 75% of the time and H s < 1.5 m for 94%) but can reach more 3 m during each winter storms. The residual longshore drift is northward and is estimated to be around 200 000 m3/y [START_REF] Kulling | Deformation du rivage et dérive littorale sur les plages du Golfe du Lion[END_REF].

Two wind orientations prevail in the study area: NW offshore winds (60% of the time with velocities pouvant atteindre 100 km/h) inducing short-period waves directed southwards and E to SE onshore winds (30% of the time) during marine storm event.

METHODS

Morphology Monitoring

The Roussillon coast is monitored since summer 2013 within the Coastal Observatory of the sandy catalan coast (http://obscat. fr). A particular attention was paid to river mouths [START_REF] Balouin | Sediment bypassing mechanisms at river mouths: example in the Mediterranean[END_REF] where topo-bathymetry, UAV photogrammetric surveys are done by ®Ecoceanodrone every 6 months before and after the most energetic winter period. Both Aerial and marine UAV permit to obtain an accurate topography survey (grid 2*2 cm, Dz~5 cm) of the river mouth.

Additional information is provided by Sentinel-2 satellite images obtained using the Coastsat algorithm (Vos et al., 2019(Vos et al., , 2019b)). After filtering images having a too large cloud cover, the spacing between 2 images varies between 5 and 15 days ( 235images from 2016 to 2019). These images were used to describe opening events of the river mouth, as well as spit morphology and outlet alongshore migration.

Hydrodynamics

Offshore waves are recorded by the regional authority (DREAL Occitanie) at the Leucate wave buoy (around 10 km northward from the site), water levels at Port-La-Nouvelle (Refmar) and water discharge at Perpignan station (Banque Hydro) (Figure 1).

To better assess the interactions between river jet and waves, two stations (Figure 2) were implemented since June 2018 within the Dem'Eau project: one offshore of the site (10 m depth) with a ©Nortek AWAC current profiler coupled with an Optical backscatter sensor (OBS), the other one in the river mouth with a ©Nortek Aquadopp profiler and an OBS. Datasets used in this study include water level offshore and in the river mouth (every 10 min), current profiles (0.1 m cells in the river mouth and 1 m cells offshore) and wave measurement every 30 min (20 min bursts at 2 Hz).

RESULTS

Hydrodynamic and Wind Conditions

During the monitoring period (2013-2019), offshore wave climate (Figure 3) was typical from the area with at least two storm events per year over 4 m in Hs.

Morphological Response during Marine Storm/Flash Flood Events

The role of spit growing under the prevailing northwards longshore drift is evident, but the mechanisms involved in spit breaching and subsequent sand bypassing remain unclear. The available dataset indicates several key processes: regular breaching, dominant effect of concomitant marine storm/flash flood events, spit growth closing the entrance under prevailing SE waves or NW offshore winds. To document these mechanisms, 4 opening events (14-17/10/2018, 17-20/11/2018, 21-24/04/2019 and 22-24/10/2019) were analysed. As shown on Figure 5, all these events were characterized by an important river discharge, except for the one of april 2019.

During the first event (Oct 14-17 th 2018, Figure 6a), storm waves and high river discharge were recorded. At the beginning of the event, the spit was elongated northward and the river outlet mostly closed. An increase in wave high and river water level was firt observed until Oct 15 th . Then, an important increase of the sea water level (storm surge) was observed. In the river mouth, current velocities were very low. When water level offshore reaches its maximum, spit breahing occurred, probably resulting from wave overtopping the narrow spit. A rapid decrease of river mouth water level and an increase of the river jet was measured, confirming the opening of the river mouth that allowed the river jet to flush seaward. Water temperature decrease also indicates the rapid mixing of river and sea water masses.

During the second event (Figure 6b), the same mechanisms was observed, but the river current velocities remained low and the opening was observed north from the main axis of the river. During this event, wave highs were mostly the same (around 4 m in Hs), but the storm surge was very low.

The fourth event presents the same characteristics than the first one, the opening, based on the sudden increase in flushing current velocity, being observed at the storm peak (in wave and surge elevation).

The third event is quite different. There was no flash flood during this event, and thus only marine conditions are supposed to be involved in the opening. The river mouth presented a large meander southward, but a very narrow sand spit. Before the storm peak, a narrow opening was artificially done in the northern part of the river entrance. At the pic of the storm, the opening in the southern extremity of the river mouth was observed, yielding a decrease in the river mouth water level. Given the very low discharge during this event, the river water leverl is supposed to be mostly due to wave beaking and wave setup at the river mouth.

The artificial opening in the northern part does not seem to have had any effect as this narrow channel was closed during the event, and the main chanel opened more than 150 m southward.

Bypassing Mechanisms

The outer part of the Tet river mouth is not stabilized and the most important morphological evolution is the spit development during the summer period and spit breaching when concomittant marine storm/flash flood events exceeding 100 m3/s water discharge are observed. The outer river delta is very limited (only a few meters offshore) with low or inexistent bar by-pass on the small shoals. More offshore bar by-pass can be deduced from the bathymetry surveys, but data indicates that this sediment volume stays in the nearshore and does not reach the downdrift coast, at least not in the first hundred meters.

As shown on Figure 7, the position of the river mouth can be more than 200 m from the river axis in both northward and southward direction according to the wave climate (it can even exceed 700 m northwards during the longer monitoring period). This indicates that opening of the river outlet during storm not always indicate the prevailing northwards sand bypassing. After long northern conditions (dec 2018 as an example), a very long southward elongation can be observed and in this situation, opening will not induce any sand bypassing to the downdrift north coast.

DISCUSSION

Concomittance between marine storm and flash floods appears to be the main factor influencing the river entrance opening by spit breaching. These events can generate an important amount of sediment from the spit to the downdrift coast. As an example, more than 30 000 m 3 were transferred to the northern coast during the event of winter 2014-2015 when a new entrance was opened 400 m southward (Figure 7). The residual longshore drift is estimated to be around 200 000 m 3 /y [START_REF] Kulling | Deformation du rivage et dérive littorale sur les plages du Golfe du Lion[END_REF], suggesting than most of the sediment transfer from the updrift to the downdrift coast is done while the river entrance is closed.

Despite a very dynamic evolution (development and closure during fair weather conditions, abrupt opening and relocation during more energetic conditions), the river entrance is quite stable at a pluri-annual timescale. This is obvisouly due to the fact that the internal river is stabilized, but also derives from the high frequency of opening/closure of the outlet that prevent spit growth and formation of dunes.

CONCLUSIONS

Sand bypassing at the Tet river mouth has been monitored since 2013. The main processes observed are associated with spit elongation under the prevailing northward longshore drift, and spit breaching during the most energetic events. To analyse these energetic events yielding to the opening of the river outlet, hydrodynamic measurements were performed on the nearshore and in the river mouth. Most of the time, flash floods are associated with storm waves, and the main mechanisms explaining the breaching remained unclear. Our results suggest that the river jet itself has a very low contribution to the opening of the river outlet and the subsequent sand bypassing. Four energetic events were analysed and each time, the breaching occurs when the significant wave high exceed 3 m in the nearshore, and when the water level offshore is increased by the storm surge and wave setup. Despite no visual observation are available during these events, it is evidenced that breaching is dur to wave overtopping the spit, putting in relation sea and river waters masses. The river jet is then fushed out of the outlet with a strong increase of current velocities observed.

Monitoring os going and further instrumentation (video monitoring) should be deployed in the next months to better document the timing and kinetics of these breaching episodes.

Figure 1 .

 1 Figure 1. Localisation of the fieldsite along the French Mediterranean coast.

Figure 2 .

 2 Figure 2. Measurement of hydrodynamics offshore and in the Têt river mouth.

Figure 3 .

 3 Figure 3. Hydrodynamic conditions during the monitoring period: from top to bottom: Hs (m), Longshore wave power (kW/m), water level (m); strong winds (km/h) and daily-averaged discharge (m3/s). Vertical blue lines indicate the field surveys.

Figure 4 .

 4 Figure 4. UAV aerial photos and DTM undertaken at the Tet river mouth between 2014 and 2018.

  Winter 2016-2017 was more energetic with 6 successive events. The most important storm waves were March 2013, Nov 2014, Oct 2016 and March 2018. These events were associated with high water levels due to an important storm surge. Half of the storm events are associated with flash floods. Main flash floods occurred during the winter periods and are always combined with SE storm waves (except 30/04/2013). The most important river discharges were recorded in March 2013 and Nov 2014, with daily-averaged discharges reaching more than 600 m 3 /s. Important NW offshore winds were observed, particularly at the beginning of 2014 and 2015. They can play an important role on sediment transport inducing a reverse drift along the Roussillon coast. Mid-term Morphological Evolution of the River Mouth Figure 4 present the UAV aerial photos of the Têt river mouth between 2014 and 2018. Morphology evolution is characterized by a cycle involving spit elongation and spit breaching. At the beginning of the survey (2013-2014), the narrow spit was elongated northwards more than 500 m, inducing a strong meandering of the river mouth. The marine storm/flash flood event of Nov 2014 opened a new outlet to the south, that immediately began to migrate northwards. Successive marine storm/flash flood events of Nov 2015 and Feb 2017 opened a new river outlet, and the closure of the previous one. In March 2018, a major marine storm was observed, that almost closed the river mouth.. In such conditions, the marine storm/flash flood event of April 2018 induced the only northwards relocation observed during the survey of the outlet. Impact of strong NW offshore winds is observed (e.g., April 2015, Oct 2016 or sept 2017) and induced a local reverse drift southwards, able to generate small southwards spits. Nevertheless this outlet behavior appears to be limited even if temporary favouring temporairement spit growth and its vertical accretion.

Figure 5 .

 5 Figure 5. Têt river discharge (m3/s) at Perpignan station.

Figure 6 .

 6 Figure6. Hydrodynamics during the most important events: for each event: top: wave and sea water level (m), middle: river level and temperature; bottom current velocity at mean water column (m/s); right: sentinel-2 images before and after the event.Urban monitoring system. The shadow area represents the time of breaching estimated to be correlated with the sudden increase in flushing current velocity.

Figure 7 .

 7 Figure 7. Migration of the river entrance: North-South distance of the entrance from the main river axis. Top: position of the river mouth since 2013; bottom: zoom on the position from mid 2018 to 2019.

ACKNOWLEDGEMENTS

This work was funded by BRGM, European Fund for Regional Development and Perpignan Mediterranée Métropole as part of the regional project Dem'Eau. Topobathymetry and UAV surveys were acquired within the Coastal Observatory OBSCAT, and offshore hydrodynamics provided by the DREAL Occitanie throught the CANDHIS network and DATA-SHOM).


Nikolaj Horsevad
email: nikolajhorsevad@gmail.com

David Mateo

Robert E Kooij

Alain Barrat

Roland Bouffanais
email: roland.bouffanais@uottawa.ca

Transition from simple to

published or not. The documents may come

Introduction

Complex systems, be them natural or artificial, operate on the basis of a particular connectivity between constituting elements, which orchestrates the execution of specific dynamical processes. Such a high-level abstraction encompasses wildly different systems giving rise to a range of emergent behaviors, such as fish schooling [START_REF] Swain | Coordinated speed oscillations in schooling killifish enrich social communication[END_REF][START_REF] Rosenthal | Revealing the hidden networks of interaction in mobile animal groups allows prediction of complex behavioral contagion[END_REF][START_REF] Sosna | Individual and collective encoding of risk in animal groups[END_REF] , social opinion formation [START_REF] Centola | How Behavior Spreads: The Science of Complex Contagions[END_REF] , disease spreading [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] , cascading failures in power grids [START_REF] Kinney | Modeling cascading failures in the North American power grid[END_REF][START_REF] Koç | The impact of the topology on cascading failures in a power grid model[END_REF] , target tracking by swarm robotic systems [START_REF] Ren | Distributed Consensus in Multi-Vehicle Cooperative Control[END_REF][START_REF] Kwa | Optimal swarm strategy for dynamic target search and tracking[END_REF] , etc. Such collective dynamics have been found to be crucially dependent on the underlying network topology [START_REF] Rosenthal | Revealing the hidden networks of interaction in mobile animal groups allows prediction of complex behavioral contagion[END_REF][START_REF] Sosna | Individual and collective encoding of risk in animal groups[END_REF][START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF][START_REF] Barrat | Dynamical Processes on Complex Networks[END_REF][START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF][START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF][START_REF] Centola | Complex contagions and the weakness of long ties[END_REF][START_REF] Olfati-Saber | Ultrafast consensus in small-world networks[END_REF][START_REF] Santos | Scale-free networks provide a unifying framework for the emergence of cooperation[END_REF] , which conditions the efficient transmission of behavioral change.

The propagation of state changes within a social system-or an engineered networked one-has been acknowledged to be akin to a contagion process, which can be either 'simple' or 'complex' [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF][START_REF] Centola | Cascade dynamics of complex propagation[END_REF][START_REF] Watts | A simple model of global cascades on random networks[END_REF][START_REF] Firth | Considering complexity Animal social networks and behavioural contagions[END_REF][START_REF] Romero | Differences in the mechanics of information diffusion across topics: idioms, political hashtags, and complex contagion on twitter[END_REF] . With a simple contagion, the behavior propagates through a single exposure or interaction. On the other hand, if social reinforcement is required-following Centola and Macy's definition [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF] : "if its transmission requires an individual to have contact with two or more sources of activation"-the contagion is said to be complex.

Numerous models of behavioral propagation in networked systems (including social ones) have been considered over the years. Threshold models have been the predominant modeling framework used to characterize a wide range of complex contagion processes, such as the adoption of technological innovations [START_REF] Centola | How Behavior Spreads: The Science of Complex Contagions[END_REF] or preventative health measures [START_REF] Centola | The spread of behavior in an online social network experiment[END_REF] , and the spread of misinformation on social media [START_REF] Romero | Differences in the mechanics of information diffusion across topics: idioms, political hashtags, and complex contagion on twitter[END_REF][START_REF] Törnberg | Echo chambers and viral misinformation: Modeling fake news as complex contagion[END_REF] (Note that complex-like contagions are also observed with stochastic models of epidemic-like processes [START_REF] Karsai | Complex contagion process in spreading of online innovation[END_REF][START_REF] Dodds | Universal behavior in a generalized model of contagion[END_REF][START_REF] Böttcher | Failure and recovery in dynamical networks[END_REF][START_REF] Böttcher | Critical behaviors in contagion dynamics[END_REF] ). The growing interest in threshold models can be traced to their mathematical simplicity, their paradigmatic nature and their success in modeling the spread of behaviors in various social settings [START_REF] Centola | How Behavior Spreads: The Science of Complex Contagions[END_REF][START_REF] Centola | Complex contagions and the weakness of long ties[END_REF][START_REF] Centola | Cascade dynamics of complex propagation[END_REF][START_REF] Andreoni | Predicting social tipping and norm change in controlled experiments[END_REF] . These deterministic models assume that agents can be in two states (inert or activated), and that a particular agent becomes activated if a fraction of its neighbors (in the network sense) larger than a given threshold are themselves activated [START_REF] Watts | A simple model of global cascades on random networks[END_REF][START_REF] Granovetter | Threshold models of collective behavior[END_REF] . These threshold models fit perfectly Centola & Macy's original definition of a complex contagion, whereby a transition from simple to complex contagion takes place when increasing the threshold beyond a value corresponding to having more than one activated neighbor [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF][START_REF] Centola | Cascade dynamics of complex propagation[END_REF][START_REF] Watts | A simple model of global cascades on random networks[END_REF] . However, the key concept of activation may not be as straightforward when considering models lacking a threshold, which can become an issue when trying to use the existing definition of a complex contagion.

Here, we show theoretically and experimentally that such a transition from simple to complex contagion, as originally identified in threshold-based models, can also be exhibited by another general class of collective decision-making processes; specifically, a class of models based on consensus and devoid of any thresholds or nonlinearities. Using a new way of characterizing complex contagions, we uncover their existence in consensus-based dynamics. Specifically, we shed a new light on some fundamental mechanisms underpinning networked systems, which may support the study of a vast range of collective behaviors in both the animal and social worlds.

Unsurprisingly, the network topology plays a pivotal role in this study 2, 3, 10-13, 15, 16 . It is known that it strongly affects both spreading types in threshold models, albeit in fundamentally different ways. While simple contagions are enhanced by short network distances [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] , complex ones are amplified by high levels of clustering [START_REF] Centola | How Behavior Spreads: The Science of Complex Contagions[END_REF][START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF][START_REF] Olfati-Saber | Ultrafast consensus in small-world networks[END_REF][START_REF] Centola | Cascade dynamics of complex propagation[END_REF] . Within the framework of threshold models, the behavior or state being transmitted is of a binary nature: 'active' or 'inactive'. This simplification clearly facilitates the tracking of behavioral cascades from a source (or multiple sources) to the entire system (or parts of the system). This feature serves well the purpose of studying collective decision-making processes involving two options, such as voting, adoption of innovations, binary opinion dynamics [START_REF] Centola | How Behavior Spreads: The Science of Complex Contagions[END_REF][START_REF] Centola | The spread of behavior in an online social network experiment[END_REF] . However, numerous collective decisions are more complex and involve a continuum of options rather than just a binary set [START_REF] Friedkin | Network science on belief system dynamics under logic constraints[END_REF][START_REF] Tian | Social power evolution in influence networks with stubborn individuals[END_REF][START_REF] Proskurnikov | A tutorial on modeling and analysis of dynamic social networks. part i[END_REF] . A full understanding of the influence of network metrics on consensus-based decision-making involving behavioral propagation and/or external perturbations to the consensus is lacking. Such knowledge would help gain insight into the disturbed collective dynamics of social and animal groups, e.g., when responding to a predator's attack or to misleading information on social networks.

Biologists have indeed recently acknowledged the profound similarities between human and animal social behaviors. For instance, Sosna et al. recently reported a study on the "fear response" of a school of fish collectively making fast decisions under risky conditions [START_REF] Sosna | Individual and collective encoding of risk in animal groups[END_REF] . They found that the properties of the network (their "social connectivity") are the primary factors responsible for the high collective responsiveness of the school in terms of number of behavioral cascades and their sizes. Furthermore, Firth [START_REF] Firth | Considering complexity Animal social networks and behavioural contagions[END_REF] has made the case that complex contagions might be key to explaining some specific collective animal dynamics, especially those with socially transmittable behaviors. In the case of direct behavioral transmission in mobile animal groups-schooling fish, flocking birds, swarming insects [START_REF] Handegard | The dynamics of coordinated group hunting and collective information transfer among schooling prey[END_REF][START_REF] Couzin | Self-organization and collective behavior in vertebrates[END_REF][START_REF] Krause | Living in Groups[END_REF][START_REF] Kastberger | Social waves in giant honeybees repel hornets[END_REF] -a full understanding of the nature of the propagation is still lacking [START_REF] Rosenthal | Revealing the hidden networks of interaction in mobile animal groups allows prediction of complex behavioral contagion[END_REF][START_REF] Sosna | Individual and collective encoding of risk in animal groups[END_REF] . The markedly fast spread of behaviors within animal groups-such as waves of response, evasive maneuvers in schools of fish [START_REF] Rosenthal | Revealing the hidden networks of interaction in mobile animal groups allows prediction of complex behavioral contagion[END_REF][START_REF] Sosna | Individual and collective encoding of risk in animal groups[END_REF][START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF][START_REF] Radakov | Schooling in the Ecology of Fish[END_REF] , and collective turns in flocks of starlings [START_REF] Attanasi | Information transfer and behavioural inertia in starling flocks[END_REF] -has been a source of inquiry for a long time [START_REF] Radakov | Schooling in the Ecology of Fish[END_REF] . Recent largescale empirical evidence with fish and birds have revealed the intricate patterns of interaction among individuals [START_REF] Rosenthal | Revealing the hidden networks of interaction in mobile animal groups allows prediction of complex behavioral contagion[END_REF][START_REF] Sosna | Individual and collective encoding of risk in animal groups[END_REF][START_REF] Attanasi | Information transfer and behavioural inertia in starling flocks[END_REF][START_REF] Cavagna | The physics of flocking: Correlation as a compass from experiments to theory[END_REF][START_REF] Attanasi | Collective behaviour without collective order in wild swarms of midges[END_REF][START_REF] Cavagna | Flocking and turning: a new model for self-organized collective motion[END_REF] , which underpin the behavioral cascades throughout the group. A key element to these

2/21

inter-agent interactions is alignment-metric or topological-that introduces a consensus component to the collective decision-making process. Unlike binary threshold-based models, such orientation-consensusbased ones do not lend themselves well to the tracking of behavioral cascades given the non-binary nature of the state-variable. Moreover, as with all ethological results, even if it is possible to modify the interaction among agents in some ways [START_REF] Sosna | Individual and collective encoding of risk in animal groups[END_REF] , it is virtually impossible to fully control all aspects of it.

However, biologists have started using robotic agents in place of animals to be able to measure and quantify some features of interest [START_REF] Li | Vortex phase matching as a strategy for schooling in robots and in fish[END_REF][START_REF] Li | Using a robotic platform to study the influence of relative tailbeat phase on the energetic costs of side-by-side swimming in fish[END_REF] . Hence, by following a similar approach with a multi-robot system, one could compare the effectiveness of the social transmission of information when changing the local interaction rule, i.e. when changing the topology of the interaction network.

Here, we specifically consider a collective decision-making process reminiscent of a group escape response. Using the leader-follower consensus (LFC) dynamics-a particular instance of the general control-theoretic framework of the Taylor model [START_REF] Proskurnikov | A tutorial on modeling and analysis of dynamic social networks. part i[END_REF][START_REF] Pirani | Coherence and convergence rate in networked dynamical systems[END_REF][START_REF] Baumann | A laplacian approach to stubborn agents and their role in opinion formation on influence networks[END_REF] -we are able to study the behavioral contagion within a group of networked agents driven by a single leader acting as a stimulus of tunable frequency [START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF][START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF] .

With this linear-time invariant (LTI) dynamics, by varying the network topology-specifically the clustering coefficient, average shortest path and Kirchhoff index-we observe that slow-paced (resp. fast-paced) stimuli propagate in ways reminiscent of a simple (resp. complex) contagion. Furthermore, we uncover a transition from simple to complex contagion when varying the pace of the stimulus-i.e, its frequency. This transition is made apparent by measuring the effectiveness of the behavioral propagation-quantified here by means of the concept of collective frequency response [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF] -when varying the topological features of the interaction networks (e.g., clustering coefficient, average shortest path, etc.). In addition, using a robotic experimental test-bed comprising 10 networked agents performing an angular heading consensus similar to those observed in collective turns of flocking starlings [START_REF] Cavagna | Flocking and turning: a new model for self-organized collective motion[END_REF] , we unambiguously confirm that the behavioral propagation has the features of a complex contagion when driven by fast-paced stimuli.

These results have far-reaching implications for several reasons. First, they extend the concept of transition from simple to complex contagion-heretofore limited to binary threshold-based models [START_REF] Centola | Cascade dynamics of complex propagation[END_REF][START_REF] Watts | A simple model of global cascades on random networks[END_REF] to the continuous class of consensus-based models. It is worth highlighting that the original linear threshold model (LTM) with binary options has been extended to a continuous threshold model (CTM) of cascade dynamics, which involves nonlinearities and a threshold [START_REF] Zhong | A continuous threshold model of cascade dynamics[END_REF] . However, no evidence of a transition from simple to complex contagion has been reported for the CTM. Second, these results reveal that the nature of the contagion-simple or complex-is directly related to the type of behavior spreading, and more specifically to the pace of its intrinsic dynamics-e.g., slow external perturbations vs. collective startle response. Lastly, the insights gained from this study could offer new directions for biologists and social scientists to explore and experiment with animal and human groups respectively. They could also be harnessed to improve the design and robustness of engineered networked systems (e.g. Internet of Things, sensor networks, swarm robotics).

Results

Polarization speed in threshold models

Complex contagions have been originally uncovered and studied using the archetypal LTM [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF][START_REF] Centola | Cascade dynamics of complex propagation[END_REF][START_REF] Watts | A simple model of global cascades on random networks[END_REF][START_REF] Granovetter | Threshold models of collective behavior[END_REF] , where nodes become active when the fraction of their active neighbors crosses a certain threshold θ . Specifically, the LTM dictates that for any agent i, the binary state-variable s i (t) is updated according to

s i (t + 1) = 1 if s j (t) j∼i > θ , 0 otherwise, (1)

3/21

where • j∼i is the average over all neighbors of i (Methods). Significant attention has been dedicated to understanding the interplay between θ and network topology for global cascades to occur (i.e. yielding an activation of 99% of nodes) [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF] . It has been repeatedly reported that complex contagions spread "faster and further" [START_REF] Centola | Cascade dynamics of complex propagation[END_REF][START_REF] Centola | The spread of behavior in an online social network experiment[END_REF] on highly clustered networks, as compared to simple contagions [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF][START_REF] Centola | Cascade dynamics of complex propagation[END_REF][START_REF] O'sullivan | Mathematical modeling of complex contagion on clustered networks[END_REF] . It is worth stressing that the investigation of this important statement remains limited to the long-term dynamics of global cascades [START_REF] Peng | Network modularity controls the speed of information diffusion[END_REF] . On the other hand, the early dynamics of contagion affecting a smaller fraction of nodes-say cascades of 30% activated nodes, which is still macroscopic-has been relatively overlooked.

There has been no attempt to relate the actual speed of contagion with the transition from simple to complex contagion for incomplete-i.e., non-global-cascades. This speed of contagion, also known as diffusion speed, is the number of infected nodes per unit time and is increasingly recognized as an important indicator of the contagion dynamics [START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF][START_REF] Peng | Network modularity controls the speed of information diffusion[END_REF][START_REF] Barthelemy | Dynamical patterns of epidemic outbreaks in complex heterogeneous networks[END_REF][START_REF] Karsai | Small but slow world: How network topology and burstiness slow down spreading[END_REF][START_REF] Delvenne | Diffusion on networked systems is a question of time or structure[END_REF] .

A useful metric to analyze this is given by the so-called polarization speed v = (P(t) -P(0))/t at instant t, where P(t) measures the polarization of the system (Methods). This quantity gives an indication of the speed at which a random activator node and its neighbors can activate a given fraction of nodes (here 30%; Methods and Supplementary Fig. S1 for other fractions) [START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF][START_REF] Watts | A simple model of global cascades on random networks[END_REF][START_REF] Karsai | Complex contagion process in spreading of online innovation[END_REF][START_REF] Granovetter | Threshold models of collective behavior[END_REF] . Here, we consider this polarization speed, which is identical to the average speed of diffusion considered in Ref. [START_REF] Peng | Network modularity controls the speed of information diffusion[END_REF] and is also somehow related to the concepts of spreading speed, half prevalence time, or time until half the network is infected [START_REF] Karsai | Small but slow world: How network topology and burstiness slow down spreading[END_REF][START_REF] Starnini | Immunization strategies for epidemic processes in time-varying contact networks[END_REF][START_REF] Lawyer | Understanding the influence of all nodes in a network[END_REF] , and we study how it relates to some network descriptors used for the study of the spread of misinformation in temporal network epidemiology [START_REF] Holme | Impact of misinformation in temporal network epidemiology[END_REF] .

Following in the footsteps of Centola & Macy, we use their version of the LTM [START_REF] Centola | Cascade dynamics of complex propagation[END_REF] to study the transition from simple to complex contagion. Simple contagions are observed in LTMs with low thresholds θ , which lead to an increased spreading rate on networks with smaller values of the average shortest path [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF][START_REF] O'sullivan | Mathematical modeling of complex contagion on clustered networks[END_REF] .

As θ increases, the contagion becomes complex and for the spreading process to endure, the network must possess a sufficiently high value of the clustering coefficient C [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF][START_REF] Watts | A simple model of global cascades on random networks[END_REF] . Here, we consider the classical small-world Watts-Strogatz (WS) networks [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] , which are constructed on the basis of a single free parameter, namely the rewiring probability p. By varying p, we can effectively tune C, , or the Kirchhoff index R g of the network (Methods). The Kirchhoff index R g = N ∑ N i=2 λ -1 i , is a distance metric based on the eigenvalues 0 = λ 1 < λ 2 < • • • < λ N of the Laplacian matrix [START_REF] Ellens | Effective graph resistance[END_REF] (Methods).

The variations of the polarization speed v with θ are reported in Fig. 1. At low threshold, the polarization speed increases when decreases (and C decreases as well), which is characteristic of simple contagions (Fig. 1(a)). At higher θ values, this trend is inverted and we retrieve the well-known complex contagion phenomenology in which v increases with C (and also with ). These trends can also be appreciated by observing the particular network topology corresponding to the purple curve (high C, and R g ) in Fig. 1(a). It goes from producing the worst performing simple contagion (out of the 4 topologies considered here) at low θ , to generating the best performing complex contagion at higher threshold values.

As expected, the transition region corresponds to intermediate values of the threshold such that the ordering of the different networks does not reveal an unambiguous simple or complex contagion. It is worth noting that highly clustered networks can sustain a complex contagion at higher thresholds compared to the more rewired networks, which lack ample clustering to sustain spreading [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF] .

To analyze this transition, we calculate the Spearman's correlation coefficient r s between the polarization speed v and each network metric χ ∈ {C, , R g }, for each threshold value θ (Methods and Supplementary Fig. S2). There is a marked transition from r s ≈ -1 for θ < 0.1, to r s ≈ +1 when θ > 0.28 (Fig. 1b). These thresholds mark the cutoff of purely simple and complex contagion-used to draw the shaded regions in Fig. 1a-with the transition region in between experiencing a more complex interplay of the network parameters and the resulting polarization speed. Note that the correlation between the three network parameters and the polarization speed are extremely similar, owing to the well-known fact that the

4/21

WS network model has only one free parameter (the rewiring probability p; see Ref. [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] ). These results are in good agreement with other observations of this transition [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF][START_REF] Centola | Cascade dynamics of complex propagation[END_REF][START_REF] Watts | A simple model of global cascades on random networks[END_REF][START_REF] O'sullivan | Mathematical modeling of complex contagion on clustered networks[END_REF] , thereby suggesting that v, characteristic of the early contagion dynamics, is both an adequate and effective indicator of the nature-simple or complex-of the behavioral propagation.

0.0 0.1 0.2 0.3 0.4 0.5
Threshold (θ )

10 -3 10 -2 10 -1 Polarization Speed (v) (a) Simple Complex C | | R g 0.34|4.2 |0.7×10 3 0.51|4.8 |0.8×10 3 0.66|6.9 |1.2×10 3 0.70|313|41×10 3 0.0 0.1 0.2 0.3 0.4 0.5 Threshold (θ ) -1.0 -0.5 0.0 0.5 1.0 Correlation r s (v, χ) (b) Simple Complex χ C R g Figure 1.
Linear threshold model on WS networks of N = 10, 000 nodes, with fixed average degree k = 16 and uniform threshold θ . Initially a single randomly selected seed node and its neighbors are activated. (a) Average polarization speed v when the cascade size is 30% active nodes on the network, with lines stopping prematurely if the fraction is never reached, shown for a representative set of network metrics (Supplementary Fig. S1 for other cascade sizes). The WS rewiring probability p is used to generate network samples having specific values of χ ∈ {C, , R g }. (b) Spearman's correlation coefficients r s between the polarization speed v and each network property χ ∈ {C, , R g }.

Transition from simple to complex contagion in consensus models

In the previous section, we showed that with the classical LTM, the transition from simple to complex contagion can be analyzed and understood from a new angle-the speed of contagion-using v as an indicator of such speed. The question now is whether such a transition can be observed with other collective decision-making processes that do not involve binary state-variables with cascades of changes, nor nonlinear mechanisms /thresholds. Specifically, we consider the canonical linear time-invariant Taylor model [START_REF] Proskurnikov | A tutorial on modeling and analysis of dynamic social networks. part i[END_REF][START_REF] Pirani | Coherence and convergence rate in networked dynamical systems[END_REF][START_REF] Baumann | A laplacian approach to stubborn agents and their role in opinion formation on influence networks[END_REF] , which has been widely used to characterize a vast breadth of collective behaviors [START_REF] Swain | Coordinated speed oscillations in schooling killifish enrich social communication[END_REF][START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF][START_REF] Cavagna | The physics of flocking: Correlation as a compass from experiments to theory[END_REF][START_REF] Attanasi | Collective behaviour without collective order in wild swarms of midges[END_REF][START_REF] Young | Starling flock networks manage uncertainty in consensus at low cost[END_REF][START_REF] Vicsek | Collective motion[END_REF] and decision making [START_REF] Proskurnikov | A tutorial on modeling and analysis of dynamic social networks. part i[END_REF][START_REF] Amelkin | Polar opinion dynamics in social networks[END_REF] . In the Taylor model, the agents (nodes) seek to reach a consensus by performing some average of their own state along with those of their neighbors in the network sense. However, flocking birds and schooling fish in the wild seldom reach a complete consensus given their incessant collective maneuvering: the convergence to a stationary state clearly does not apply to their dynamics. This is even more true when these animal groups are dealing with predator attacks or other external perturbations. Indeed, accumulating empirical evidence shows that swift behavioral propagation is the true hallmark of collective behavior, rather than high consensus or polarization [START_REF] Cavagna | The physics of flocking: Correlation as a compass from experiments to theory[END_REF] . It can therefore be said that although consensus is at the root of their collective actions, these systems effectively tend to operate away from consensus [START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF][START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF] .

Considering the particular Taylor model corresponding to the LFC dynamics, one can drive the system away from consensus by imposing a given dynamics to the "leader" (also known as "stubborn", "zealot",

5/21

"informed" agent in some contexts [START_REF] Proskurnikov | A tutorial on modeling and analysis of dynamic social networks. part i[END_REF] ). The leader's behavior then propagates to the neighboring agents, and further to the entire system, thereby determining the emergent collective response. From the controltheoretic perspective, this leader introduces a time-varying input signal into the system. However, this behavioral propagation intricately depends on the network topology, as well as on the leader-follower consensus dynamics considered. Here, we consider N agents with state-variable x i (t) seeking to follow the arbitrary trajectory x 0 (t) = u(t) = sin ωt of the leader agent i = 0, by means of the following linear distributed consensus:

dx i dt = N ∑ j=1 w i j x j (t) + w i0 u(t), (2)
where w i j is a weight related to the interaction between agents i and j (Methods). The collective frequency response of the system, H 2 (ω) (Eq. ( 7) in Methods), can be interpreted as the number of agents that are able to respond or follow the leader's behavior, as a function of its frequency ω [START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF][START_REF] Mastroeni | Agent-based models for opinion formation: a bibliographic survey[END_REF] .

The LFC dynamics at low frequency (ω → 0) has been comprehensively studied. For instance, it is well known that the collective response increases as decreases [START_REF] Olfati-Saber | Ultrafast consensus in small-world networks[END_REF][START_REF] Baumann | A laplacian approach to stubborn agents and their role in opinion formation on influence networks[END_REF][START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF] . This phenomenology is analogous to that of a simple contagion (cf. the increase of the polarization speed v as decreases for the LTM at low threshold θ , Fig. 1(a)). Given the transition from a simple to a complex contagion when increasing θ in the LTM, one is naturally led to consider the possible existence of a transition in the LFC when increasing the frequency ω of the leader's dynamics.

To investigate if the LFC indeed exhibits such a transition, we follow the same approach as for the LTM.

We analyze the collective response on networked systems having 240 nodes with a fixed average degree of k = 16. Using the same family of small-world WS networks [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] (Methods), we are able to compute analytically H 2 (ω) (Fig. 2(a)) for the same values of the clustering C as the ones previously used for the LTM (Fig. 1). It is worth adding that similar results are obtained with a family of scale-free networks (Supplementary Fig. S4). Unsurprisingly, at low frequency (ω 10 -2 ) we observe a phenomenology consistent with a simple contagion, namely H 2 increases as decreases. Upon increasing ω, this trend is reversed and H 2 grows with C in ways that are reminiscent of a complex contagion. However, to ascertain that this phenomenology is indeed a transition from a simple to a complex contagion, one has to verify that the simple contagion at low frequency is driven by or R g , while the complex one at high frequency is controlled by C. To this aim, we calculate the Spearman's correlation coefficient r s between H 2 and χ ∈ {C, , R g } (Fig. 2(b)). For these three network metrics, r s exhibits a clear sigmoidal trend from -1 at low frequency to +1 at high frequency. This trend echoes the one observed with the LTM when varying θ (Fig. 1(b)), with a transition region in the middle. Let us note that the important element here is the presence of a transition regardless of the actual values of the upper (resp. lower) bound of the simple (resp. complex) contagion region. Although beyond the scope of this study, a thorough analysis of the profound nature of this transition-e.g., cross-over, phase transition-might help in systematically defining the extent of this transition region.

However, we are still unable to conclude that C is fully responsible for the observed trend at high frequency with the LFC, although this fact is well known for the LTM at high θ . To reach this conclusion, we have to address a well-known structural constraint with the WS networks, namely the fact that they are constructed by means of a single parameter-the rewiring probability p [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF][START_REF] Watts | Networks, dynamics, and the small-world phenomenon[END_REF][START_REF] Watts | Networks, dynamics, and the small-world phenomenon[END_REF] . This is clearly visible in the insert of Fig. 2(b), where C monotonically increases with R g and (see Supplementary Fig. S9). To overcome this issue, we include additional WS networks-with different values of the average degree-and select a subsample of these networks having uncorrelated network metrics (Methods and insert of Fig. 2(c)).

Given this extended network sampling, we need to account for the effects of degree variations [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF][START_REF] Ellens | Effective graph resistance[END_REF][START_REF] Kooij | Tuning the clustering coefficient of generalized circulant networks[END_REF] ,

6/21 10 -3 10 -2 10 -1 10 0 -1 0 1 Correlation r s ( H2 (ω), χ) (b) Simple Complex χ C ¯ Rg 10 -3 10 -2 10 -1 10 0
Frequency (ω)

7/21

and as such we impose a normalization procedure (overbar notation) for all quantities of interest: H2 , C, ¯ and Rg (Methods). Using this methodology, we obtain the following key result: at low frequency, H2 is highly (negatively) correlated with Rg and practically uncorrelated with C, while the opposite is true at high frequency. This unambiguously confirms our hypothesis that the consensus-based behavioral propagation at low (resp. high) frequency is of the simple (resp. complex) contagion type.

It is worth emphasizing that the only commonality between the LTM and the LFC is that both constitute a collective decision-making protocol exhibiting a transition from simple to complex contagion. Although there are obvious similarities between the phenomenology of this transition in both cases-as illustrated by the behaviour respectively of the polarization speed (Fig. 1) and of the collective response (Fig. 2), we do not seek here to establish any formal equivalence between their respective control parameters (θ for the LTM and ω for the LFC).

Complex contagion with networked robots governed by nonlinear heading consensus

The theoretical result described above, concerning the canonical LFC, is compelling in several ways. First, it reveals the existence of a transition from simple to complex contagion in the absence of threshold-based mechanisms. This transition in the behaviour of consensus-based decision-making processes occurs when varying the inherent pace of the behavior that is propagated through the system. Second, it has significant ramifications for the understanding of some collective behaviors subjected to external fast-pace perturbations, e.g., following a predator's attack leading to collective evasive maneuvers by schools of fish or flocks of birds. As recently stressed by Firth [START_REF] Firth | Considering complexity Animal social networks and behavioural contagions[END_REF] , the profound nature of these socially transmittable

behaviors has yet to be fully understood. Firth makes clear that the concept of complex contagion has been relatively overlooked by biologists, although it might help explain a vast breadth of collective animal behaviors [START_REF] Firth | Considering complexity Animal social networks and behavioural contagions[END_REF] . In addition, there has been emerging evidence of complex contagion in some schooling behaviors of golden shiners exhibiting highly clustered interaction networks 2 . As already mentioned, dealing with wild animal groups is not only challenging from the practical standpoint, but it also restricts the ability to analyze the influence of the deeply-ingrained nature of the interaction among agents.

The use of robotic systems-in lieu of biological ones-has been considered to overcome some of these challenges and to offer a new toolkit to deepen our understanding of collective animal behaviors [START_REF] Li | Vortex phase matching as a strategy for schooling in robots and in fish[END_REF][START_REF] Li | Using a robotic platform to study the influence of relative tailbeat phase on the energetic costs of side-by-side swimming in fish[END_REF] .

Although simulations offer unique ways to systematically analyze algorithms of collective dynamics, they inevitably reduce the fidelity of the model to achieve computational tractability. Indeed, the simulationreality gap in robotics is known to be exacerbated with multi-robot systems [START_REF] Dorigo | Swarm robotics: Past, present, and future[END_REF] . For instance, simulations can fail to adequately capture: (1) the complex physical interactions among agents, (2) the inherent variability among units forming the group, and (3) the fine details of real-world settings in which the agents are embedded. As a matter of fact, achieving high-fidelity simulations often requires the input or feedback from physical experiments.

Data gathered in robotico, with a highly controllable and controlled environment, enable a rapid investigation of a number of hypotheses about collective behaviors. In turn, the outcome of such investigations can serve biologists to identify new directions to explore, test and validate with empirical data. Following this strategy, we use a networked robotic system to assess various socially transmittable behaviors when changing the topology of the interaction network in the presence of a collective decision-making with a nonlinear component. It is worth stressing that the results obtained with the LFC (Fig. 2) are for a linear system dynamics, yet they reveal a surprisingly complex phenomenology. Nonetheless, some collective decision-making processes among moving animals are based on a consensus associated with the direction of travel, and are inherently nonlinear. Such nonlinear interactions among conspecifics have recently been shown to be responsible for sudden directional switches in groups of pigeons [START_REF] Chen | Coordinating directional switches in pigeon flocks: the role of nonlinear interactions[END_REF] .

Therefore, we carried out a series of experiments on nonlinear leader-follower heading consensus with 8/21 a collective of ground robots where each one aligns its direction of travel with that of their neighbors in the network sense (Methods). As a consequence, changing the structure of the interaction network-and in particular its properties such as χ ∈ {C, , R g }-modifies the nature of the neighborhoods involved in the nonlinear heading consensus. This type of collective decision-making-closely related to Vicsek's canonical model of collective motion [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF] -bears some resemblance with the Taylor model analyzed previously. However, a physical embodiment of such a complex system involves significant deviations from the ideal scenario corresponding to the theoretical calculations obtained for the LFC. For instance, while the dynamics of the robots are ultimately governed by physical processes that are continuous in time, the units sense each other's state using asynchronous, discrete communications with stochastic delays and communications dropouts [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF] .

The networked robotic system comprises 10 robots (1 leader and 9 followers, Methods) equipped with a "swarm-enabling" unit-providing on-board data-processing, computing and distributed communication capabilities-that allows the system to perform a collection of decentralized cooperative control strategies [START_REF] Chamanbaz | Swarm-enabling technology for multi-robot systems[END_REF] . The leader continuously rotates at a fixed frequency ω in the range 0.03 Hz < ω ≤ 0.3 Hz. Each robot, including the leader, periodically transmits its heading information to its direct neighbors as per the specific network topology considered. Three distinct topologies ("Random", "Ring" and "Caveman" as shown at the bottom of Fig. 3) are selected on the basis that they have the same average degree k = 4, yet notably different clustering coefficients C (Methods). Collective response (H/N )

(a) (a) (a) k = 4 Random C ≈ 0.42 ≈ 1.49 R g ≈ 23.3 Ring C ≈ 0.5 ≈ 1.67 R g ≈ 25.0 Caveman C ≈ 0.73 ≈ 1.82 R g ≈ 34.2 0.03 0.1 0.3 0.6
Leader frequency (ω)

10 -3 10 -2 10 -1
Collective response (H/N )

(b) (b) (b)
Figure 3. Experimental analysis with a leader-follower networked robotic system networked by means of three topologies: "Random", "Ring" and "Caveman" with k = 4 and 9 + 1 agents (Methods). The graphs represent the network topologies used in the series of experiments, with the selected leader node depicted in yellow: (a) Experimental results of angular consensus dynamics (mean value and associated standard deviation; Methods), (b) simulation results of the nonlinear angular consensus dynamics.

It is worth highlighting that this experimental setup and methodology are identical to the one reported 9/21 in Ref. [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF] , except for the fact that different network topologies are considered here. According to the analysis of the LFC, the collective response of this networked robotic system is expected to go down when increasing the frequency ω of the leader agent. This appears clearly in Fig. 3(a) for all three networks considered. The decrease in the collective response is essentially the same at low frequency (ω ≤ 0.06 Hz) for all cases. Above that frequency however, we observe a strong difference between all 3 topologies in the range 0.06 Hz ≤ ω ≤ 0.2 Hz. This intermediate range of frequencies shows a rich collective behavior since the dynamics of the leader is rather "fast" thereby preventing any form of heading consensus to be achieved. This places us in a regime similar to that of animal groups dealing with fast-pace perturbations.

Beyond 0.2 Hz, the highest collective response, achieved with the "Caveman" network, experiences a sharper decline, which we suspect would go down to the same low level as for the "Random" and "Ring" networks at higher frequencies. Unfortunately, increasing the frequency above 0.3 Hz is not possible in practice due to a limit in the achievable rotation frequency of the leader.

To compensate for this experimental limitation, we perform simulations of the leader-follower nonlinear heading consensus dynamics with 9 + 1 agents by integrating the system of Eqs. ( 15) (Fig. 3(b)).

Unsurprisingly, these simulations show higher levels of collective response at low frequency, compared to the experimental ones, as they correspond to an idealized communication between agents. In addition, the frequency at which the differences between the different networks become apparent is higher in the simulation case with respect to the experimental one. Finally, we can reach higher frequency in the simulations than in the experiments, and the results do show the expected merging of the three collective responses at the highest frequency values (that were not reached in Fig. 3(a)).

Following the analytical results with the LFC (Fig. 4), we are able to further analyze the robotic experiments. The complex contagion induced by the leader unit to the 9 followers is clearly visible in the intermediate frequency range 0.06 Hz ≤ ω ≤ 0.2 Hz. The results are indeed in excellent agreement with the complex contagion phenomenology studied previously in much larger networks ("complex" hatched region of Fig. 4(a)). When the robotic units are interconnected by means of the most clustered network ("Caveman"), the behavioral spread is the most effective and the collective response the highest. On the other hand, when decreasing the clustering between robotic units-from "Caveman" to "Ring" and ultimately to "Random"-the effectiveness of the spread of the leader's behavior is reduced, and so is the ensuing collective response at the group level.

We note on the other hand that, given that the experiment is limited to 10 robotic units with an average degree k = 4, the diameters and Kirchhoff indices for all 3 networks are small. Therefore, the simple contagion process at low frequency (ω ≤ 0.06 Hz) is simply too rapid and no meaningful distinction between the considered topologies can be observed experimentally (Fig. 3(a)).

Discussion

In summary, we have shown theoretically and experimentally that complex contagions are more general and prevalent than originally thought, and that transitions from simple to complex contagion are not limited to threshold-based models. Instead, our results suggest that such a transition might be a general feature in some classes of collective decision-making processes.

Network science has been instrumental in uncovering the existence of complex contagions by Centola & Macy [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF][START_REF] Centola | Cascade dynamics of complex propagation[END_REF] in the social sciences. Since these seminal works, complex contagions have been observed in other spreading dynamics over complex networks, including epidemic-like stochastic models [START_REF] Karsai | Complex contagion process in spreading of online innovation[END_REF][START_REF] Dodds | Universal behavior in a generalized model of contagion[END_REF][START_REF] Böttcher | Failure and recovery in dynamical networks[END_REF][START_REF] Böttcher | Critical behaviors in contagion dynamics[END_REF] .

However, and as already mentioned, the study of the transition from simple to complex contagion has still been limited to the original LTM used by Centola & Macy, as well as social network experiments with binary options in decision-making. Hence, our discovery of this very transition within a fully linear 10/21

decision-making protocol devoid of any thresholds greatly expands the relevance of this phenomenon to a vast breadth of collective decision-making processes beyond the social sciences, and including collective animal behavior and collective robotics.

At this point, it is critical to realize that ascertaining the exact nature of a contagion-simple or complex-remains challenging. Because of that, some scientists still cautiously use the term "complex contagion" or even refer to it obliquely: e.g., "complex contagion flavor" [START_REF] Iacopini | Simplicial models of social contagion[END_REF] . Even with the full details of:

(1) a given collective dynamics-e.g, linear threshold dynamics for a given value of θ , or a leader-follower consensus dynamics for a given value of ω-as well as (2) the interaction network topology among agents, one is lacking a definite way of characterizing the behavioral contagion. This work offers a key conceptual advance in overcoming this challenge. Indeed, the network-based classification proposes to vary the topology with the goal of either increasing or reducing both the clustering coefficient C and the Kirchhoff index R g . In the event that the effectiveness of the behavioral propagation is found to be positively correlated with C and uncorrelated with R g , then the contagion is complex. Conversely, if a behavioral propagation is negatively correlated with R g and uncorrelated with C, then it can be classified as a simple contagion (Fig. 2(c)). It is therefore worth stressing that this complex networks characterization of simple/complex contagions is agnostic to the actual details of the collective decision-making process.

In all cases investigated here, the simple or complex nature of a behavioral propagation is determined by the value of a control parameter of the dynamical process taking place at the node level: e.g., the threshold θ for the LTM and the inherent pace of the behavior ω for the LFC. The proposed characterization allows us to identify the intervals of the control parameter associated with simple or complex contagions, with a transition region in between (Figs. 1 &2).

Despite the apparent relative simplicity of this complex networks characterization of contagions, one should not underestimate the associated practical challenges for scientists studying a specific collective decision-making dynamics among humans, within animal groups, or with networked robots. Varying the network topology might not always be achievable as previously stressed in the case of flocks of birds or schools of fish. This approach was central to Centola's study of humans involved in an online social experiment, in which the topology of the social network was controlled and manipulated [START_REF] Centola | The spread of behavior in an online social network experiment[END_REF] . This groundbreaking experiment has been made possible thanks to the emergence of online social networks and the related technology. Clearly, such an approach could not be considered and implemented with animal groups. However, as our networked robotic experiment shows, the use of artificial agents mimicking animal behaviors offers a new toolkit to analyze and gain insight into collective animal behavior. With this approach, one can carry out the proposed complex networks characterization and ascertain the naturesimple or complex-of the behavioral propagation. Last but not least, one can also modify the triggering behavior-in our experiment the stimulus associated with the frequency of the leader-to possibly uncover a transition from simple to complex contagion. Such an approach opens new doors toward understanding the mechanisms underpinning such collective behaviors.

Given how ubiquitous collective decision-making is in human societies, animal groups and networked multi-agent systems, these new results will have profound ramifications for our understanding of numerous phenomena in these fields. The importance of these results goes beyond the class of consensus-based protocols considered here, although studying a second-order leader-follower consensus 66 with an underdamped dynamics seems like a natural extension to this work given its acknowledged relevance to collective information transfer in flocks of birds [START_REF] Attanasi | Information transfer and behavioural inertia in starling flocks[END_REF][START_REF] Cavagna | The physics of flocking: Correlation as a compass from experiments to theory[END_REF] . The novelty of these results effectively has implications for a vast breadth of collective decision-making protocols involving a continuum of options to choose from.

Firth [START_REF] Firth | Considering complexity Animal social networks and behavioural contagions[END_REF] recently speculated the importance of complex contagions in animal social networks and behavioral contagions. Interestingly, Rosenthal et al. [START_REF] Rosenthal | Revealing the hidden networks of interaction in mobile animal groups allows prediction of complex behavioral contagion[END_REF] had the intuition that complex contagions occur in schools of golden shiners owing to the high levels of clustering exhibited by their network of interaction. This led They also speculate that the fish might actively control their interactions to achieve a higher collective responsiveness [START_REF] Sosna | Individual and collective encoding of risk in animal groups[END_REF] . Our results offer a theoretical backing to this idea, and it might provide biologists with new directions to explore and experiment.

Besides shedding a new light on our understanding of collective behaviors, these results have also clear implications for the design of man-made networked systems: e.g., Internet of Things, multi-robot systems, dynamic sensor networks. Finally, we hope the present analysis of these rich phenomena of transition from simple to complex contagion will be extended to more complex networks, including heterogeneous, temporal and/or multi-layer networks.

Methods

Linear threshold model

The LTM is a binary-option model of collective decision-making widely used in the study of complex contagion [START_REF] Centola | Complex contagions and the weakness of long ties[END_REF][START_REF] Centola | Cascade dynamics of complex propagation[END_REF] . As described in Refs. [START_REF] Centola | Cascade dynamics of complex propagation[END_REF][START_REF] Watts | A simple model of global cascades on random networks[END_REF][START_REF] Granovetter | Threshold models of collective behavior[END_REF] , it consists of a connected network of N nodes, with each node i characterized by a state-variable with two possible discrete values s i (t) ∈ {0, 1}-inactive or active-and a fixed threshold value θ i ∈ [0, 1]. At t = 0, all nodes start in the inactive state except for a random seed node l and its neighbors, denoted by { j | j ∼ l}, whose state is set to active. Whenever the fraction of an agent's neighbors that are in the active state is larger than or equal to the agent's threshold θ i , that node will switch to the active state, i.e. s i (t + 1) = 1 when s j∼i (t) ≥ θ i . The LTM dynamics on a given network can be characterized by the average polarization or activity, defined as P(t) = s i (t) {i=1,...,N} , and the associated polarization speed [START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF][START_REF] Peng | Network modularity controls the speed of information diffusion[END_REF] v(t) = P(t) -P(0) t ,

which effectively is the average propagation speed until instant t.

Note that when P(t) = 1, all nodes are in the active state and a global cascade has taken place.

Particular emphasis has been put on characterizing which network topologies lead to global cascades for certain values of θ i 13, 16, 17 , and the realisation that complex contagions spread faster on networks with high levels of clustering [START_REF] Centola | Cascade dynamics of complex propagation[END_REF][START_REF] Centola | The spread of behavior in an online social network experiment[END_REF][START_REF] O'sullivan | Mathematical modeling of complex contagion on clustered networks[END_REF] . However, numerous empirical observations attest that behavioral cascades are rarely global: evasive maneuvers in schooling golden shiners involve 10 ∼ 30% of the individuals 2 and the adoption of new health behavior in an online social network experiment only reached 30 ∼ 60% of participants [START_REF] Centola | The spread of behavior in an online social network experiment[END_REF] . Since complex contagions exhibit faster spreading in their early stage of propagation [START_REF] Centola | The spread of behavior in an online social network experiment[END_REF][START_REF] O'sullivan | Mathematical modeling of complex contagion on clustered networks[END_REF] that in their late near-global stage, we focus here on the early dynamics of the cascade and therefore consider the polarization speed v(t f ) for which 30% of the nodes are activated, i.e. for the smallest time t f where P(t f ) > 0.3 (Supplementary Fig. S1 for other cascade sizes).

To study the transition from simple to complex contagion with the LTM, we consider fairly large networks of N = 10 4 nodes, with identical thresholds θ i = θ ∀ i, in order to ensure that the active state propagates for long enough at low values of θ . We let the active state propagate from a randomly selected seed node and its neighbors until the system reaches equilibrium where P(t end + 1) = P(t end ). In our analysis we only consider networks where P(t end ) is at least as large as the desired cascade size, and we omit from our analysis the simulations where the active state failed to propagate.

12/21

Leader-follower consensus model

The leader-follower consensus model (LFC) [START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF][START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF][START_REF] Pirani | Coherence and convergence rate in networked dynamical systems[END_REF] -a particular version of the Taylor model [START_REF] Proskurnikov | A tutorial on modeling and analysis of dynamic social networks. part i[END_REF] -is commonly used by the control community to study opinion dynamics and formation control. Here, N identical agents perform a distributed consensus protocol on their state-variable x i (t), and one particular agent i = 0-the leader-follows an arbitrary trajectory x 0 (t) = u(t) instead. The dynamics of this linear system is governed by the following set of first-order ODEs:

dx i dt = N ∑ j=1 w i j x j (t) + w i0 u(t), (4)
where w i j = ω 0 (a i j /k iδ i j ) is the inter-agent consensus protocol weight for the interaction between agents i and j. The natural response frequency ω 0 is assumed to be the same for all agents. The degree of agent i is given by k i = ∑ N j=0 a i j , and the adjacency matrix entry is a i j = 1 if agent i is connected to j and 0 otherwise-classically δ i j is the Kronecker delta. The solution to Eq. ( 4) in the frequency domain can be expressed in matrix form as 12

X(ω) = ( jωI -W F ) -1 W L u(ω), (5)
with I the appropriately sized identity matrix, W F = (w i j ) N×N = ω 0 L G where L G is the grounded Laplacian matrix [START_REF] Pirani | Coherence and convergence rate in networked dynamical systems[END_REF] and W L = (w i0 ) {i=1,...,N} is the consensus protocol between the N followers and the leader i = 0. The frequency response of the multi-agent system measures the ability of the agents to follow the leader's trajectory, u(t), and can be expressed in the frequency domain as the transfer function along the jω axis in the s-plane 12, 67

H(ω) = δ X δ u (ω) = ( jωI -W F ) -1 W L , (6)
with the entries of the vector H = (h i ) {i=1,••• ,N} corresponding to the individual agent's frequency response. It is clear from Eq. ( 6) that the response function has a nontrivial dependency on the topology of the agents' connectivity through the adjacency matrix, and therefore the entries of W F and W L . We define the collective frequency response of the system as 12

H 2 (ω) = H † (ω)H(ω) = σ 2 (ω), (7)
with † being the adjoint matrix operator and σ being the singular value of the system with a single leader-classically used in the analysis of multiple-input and multiple-output (MIMO) systems. When all agents in the system perfectly follow the leader's behavior the response reaches its maximum value, H 2 = N. Thus the quantity H 2 /N can be interpreted as the fraction of agents in the system following the leader. It is worth stressing that since H 2 is not constant under leader selection, the results we present in this paper are obtained by averaging over all possible leader placements in the network.

To study the transition from simple to complex contagion in the LFC, we consider networks of N + 1 = 240 agents. This is significantly smaller than for the LTM, but the LFC shows a much higher sensitivity to variations in the network metrics, which yields a range of H 2 large enough to perform our analysis, even with relatively small network sizes.

Networks and metrics

To study the influence of the network topology on the behavioral propagation in the LTM and the LFC, we use the small-world Watts-Strogatz (WS) network model [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] . These networks enable us to vary key network metrics by changing a single parameter, namely the rewiring probability 5 p. To quantify the generated network topologies, we use the following metrics: the clustering coefficient C, the average shortest path and the Kirchhoff Index R g defined below. We use an average degree k = 16 for Fig. 1 and Fig. 2(a-b)

to ensure high levels of C before rewiring [START_REF] Kooij | Tuning the clustering coefficient of generalized circulant networks[END_REF] and a large range in and R g .

Clustering coefficient

To measure the level of community structure, and thus the potential for reinforcement of the behavior that is propagating, we use the average of the local clustering coefficient as defined in Ref. [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] , and we simply refer to it as "clustering coefficient" or C throughout the paper.

Average shortest path

The shortest path between a given pair of nodes is the minimum number of hops needed to connect the two. The average shortest path is the average between all pairs of nodes. It is well known that for WS networks, decreasing the distance between nodes both increases the effectiveness of simple contagion [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF][START_REF] Centola | Cascade dynamics of complex propagation[END_REF] and the time it takes to reach consensus [START_REF] Olfati-Saber | Ultrafast consensus in small-world networks[END_REF] .

Kirchhoff index

The Kirchhoff index R g -also known as the resistance distance-is a distance metric obtained by replacing every connection with a 1 Ω resistor and averaging the resistance between all node pairs [START_REF] Ellens | Effective graph resistance[END_REF] , thus considering all paths on thep network. Further, R g is directly related to the eigenvalues 0 = λ 1 < λ 2 < • • • < λ N of the Laplacian matrix [START_REF] Ellens | Effective graph resistance[END_REF] and can be expressed as

R g = N N ∑ i=2 1 λ i . (8)
The cohesion of the follower states-under a white noise disturbance-can be measured by the H ∞ -norm of the dynamics of Eq. ( 4)-which is H 2 (ω = 0)-and it has been proven that maximizing the cohesion is done by minimizing the Kirchhoff index R g 42, 43 .

Correlations

We consider the Spearman's rank correlation r s to investigate the impact of network metrics on a given collective behavior. We use the rank correlation due to the expected nonlinear relationship between the performance of the collective behavior and network metrics for both the LTM and LFC [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF][START_REF] Olfati-Saber | Ultrafast consensus in small-world networks[END_REF] (Supplementary Figs. S2, S5 & S6 for point distributions). The Spearman's correlation is defined by

r s = cov(rg X , rg Y ) std(rg X )std(rg Y ) , (9)
with rg X being the rank of the metric, cov(X) and std(X) the covariance and standard deviation respectively.

Given the fact that the WS networks are generated by varying one single free parameter (the rewiring probability p), the associated network metrics are inevitably correlated. As a consequence, we are unable to distinguish the individual effects of each network metric for a given average degree k (insert of Fig. 2(b)). To overcome this, we generate networks of different average degrees k ∈ {4, 6, . . . , 32}, and consider a sampled set of these networks for which C and R g vary independently of each other (insert of Fig. 2(c)). However, as shown in Refs. [START_REF] Mateo | Effect of correlations in swarms on collective response[END_REF][START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF] , the collective frequency response H 2 is notably influenced by the degree distribution, and so are the network metrics considered χ ∈ {C, , R g } [START_REF] Olfati-Saber | Ultrafast consensus in small-world networks[END_REF][START_REF] Ellens | Effective graph resistance[END_REF][START_REF] Kooij | Tuning the clustering coefficient of generalized circulant networks[END_REF] . To account for that degree dependency, we normalize all quantities involved, with the normalized quantity denoted by an 14/21 overbar. First, the normalized collective frequency response is given by H2 (ω, k, p) = H 2 (ω, k, p) H 2 (ω, k, p max ) , (10) in which p max (ω, k) is the rewiring probability that yields the largest H 2 (ω, k, p). Note that H2 (ω) ≤ 1. Second, the normalized clustering coefficient is defined as

C(k, p) = C(k, p) C(k, p = 0) , (11)
such that for any k the most clustered networks, obtained for p = 0, have C = 1 (Supplementary Fig. S5) Third, since the distance metrics also change with k [START_REF] Olfati-Saber | Ultrafast consensus in small-world networks[END_REF][START_REF] Ellens | Effective graph resistance[END_REF] , we introduce the following normalized distance metrics

¯ (k, p) = (k, p) (k, p = 1) , (12)
Rg (k, p) = R g (k, p) R g (k, p = 1) ,

which are based on the smallest distances, obtained with p = 1.

From all the networks generated with different p and k , we construct a sample that almost fully decorrelates the clustering coefficient and the Kirchhoff index: i.e., such that r s ( C, Rg ) ≈ 0 (insert of Fig. 2(c), and Supplementary Fig. S3 for other subsamples of networks).

Nonlinear heading consensus experiments

To empirically measure the collective response-with differently paced leaders-we use the same experimental multi-robot setup as Ref. [START_REF] Mateo | Optimal network topology for responsive collective behavior[END_REF] with N = 10 robotic units (called 'eBots') communicating by means of their so-called "swarm enabling unit" [START_REF] Chamanbaz | Swarm-enabling technology for multi-robot systems[END_REF] . These ground differential-drive robots are collectively moving about a two-dimensional domain (Supplementary Fig. S10). The motion of the units is the superposition of a translational motion and a rotational one. In this setup, the leader agent undergoes a constant rotational motion such that its heading α L (t) is governed by dα L (t) dt = ω, with ω denoting the frequency of the leader.

Its 'leader' status comes from the fact that its behavior does not depend on the follower agents. Note that the agents have no way of distinguishing the leader from any other agent in the system. The robot's heading is the only state-variable driven by the robot's controller regardless of the speed. Specifically, the N F = 9 follower eBots seek to align their heading α i (t) with that of their neighbors α j∼i (t). The nonlinear heading consensus algorithm determines a target heading α for each unit according to

α i = α j∼i = arctan ∑ j∼i sin α j ∑ j∼i cos α j , (14)
where j ∼ i denotes the set of topological neighbors of i in the network sense, and • is an angular average. The nonlinear heading consensus (14) is identical to the one considered in Vicsek's model [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF] except for the fact that we use a static topological neighborhood while a metric one was originally considered in Ref. [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF] . Each follower unit updates its target heading α i asynchronously every ∆T = 0.1 s. The nonlinear heading consensus algorithm used in the experiments is a discrete-time equivalent of

dα i dt = ω 0 α j∼i -α i , (15)
15/21

with ω 0 ∆T 1 and for times t ∆T . Here, ω 0 is the natural frequency of angular rotation of the eBots.

The information of each neighbor's state is also updated with the same sampling rate, but not necessarily concurrently with each other or with the update of Eq. ( 14).

The robots are interconnected according to three fixed network topologies with k = 4 due to size limitations of the network N = 10, namely: the connected caveman network [START_REF] Watts | Networks, dynamics, and the small-world phenomenon[END_REF] (for maximal clustering coefficient), a k-regular random network (for low values of C), and the 1D ring lattice (WS networks with p = 0 offering an intermediate level of C). These three fixed network topologies provide a wide range of values for the clustering coefficient as shown in Fig. 3.

Each run of the experiment starts at t = 0 with all eBots aligned with the leader: i.e., α i (0) = α L (0) = 0 ∀ i. Each run lasts for a duration of T = 10 minutes to make sure the leader performs a meaningful number of rotations for all frequencies considered. The capacity of the N F followers to maintain their heading aligned with that of the leader is measured by the empirical collective response 12

H = N F ∑ i=1 1 T T 0 cos(α i (t) -α L (t))dt, (16)
which is averaged over three repeated runs, for each angular speed of the leader considered.

1 CFigure 2 .

 12 Figure 2. Leader-follower consensus model on WS networks of N = 240 nodes. The collective response is averaged over all nodes as leader. (a) Collective response H 2 , for a subset of the networks with average degree k = 16; (b-c) Spearman's correlation coefficient r s between the collective response H2 and normalized network metrics χ ∈ { C, ¯ , Rg }. (b) Networks with average degree k = 16. (c) Networks with degrees k ∈ {4, 6, . . . , 32}, limited to a range of uncorrelated network parameters. Note that the inserts show the distribution of the normalized clustering coefficients C and Kirchhoff indexes Rg used to compute r s (Supplementary Figs.S5-S6). The purple points in the inserts are for networks with k = 16, while those in green in panel (c) are for other values of k . The red box in the insert of panel (c) highlights the dots corresponding to the subsample of networks used to generate the Spearman's correlation coefficient in panel (c), with r s ( C, Rg ) ≈ 0.03 (Supplementary Fig.S3for other possible subsampling of networks).

11 / 21 them

 1121 to consider a (fractional) threshold-based model-generating complex contagions-to characterize behavioral cascades in this particular schooling setting 2 . While Rosenthal et al. limited their work to threshold-based models to analyze this phenomenon, our results would allow their study to be expanded upon, as the decision-making process in golden shiners is clearly based on a continuum of options. From the same research group and with the same animal groups, Sosna et al. provided additional empirical evidence about the importance of the topology of the interaction network on collective responsiveness 3 .

Acknowledgements (not compulsory)

This work is partially supported by the National Research Foundation (NRF) under its National Cybersecurity R&D Programme (Award No. NRF2014NCR-NCR001-040). The authors also acknowledge the support from the SUTD-MIT International Design Center under the Grant #IDG31900101. A.B. is partially supported by the Agence Nationale de la Recherche (ANR) project DATAREDUX (ANR-19-CE46-0008).

Data availability

All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Information. Additional data related to this paper may be requested from the authors.

Code availability

All codes developed for this study are available at https://github.com/Horsevad/Simple_to_ complex_contagion_in_collective_decision-making 20/21

Author contributions statement

Competing interests

The authors declare no competing interests

Supplementary Information

Supplementary Information accompanies this paper.