Emmanuel Fureix

Le Journal pour rire, en pleine révolution de 1848, se gausse des préfets-girouettes capables de s'adapter en un tour de main au nouvel ordre politique -ou, si, l'on préfère, d'assurer la continuité de l'État [doc. 1]. L'un d'entre eux, après avoir reçu une dépêche télégraphique annonçant la proclamation de la République, range soigneusement le buste du roi déchu Louis-Philippe dans une sorte d'armoire à bustes -archives visuelles des régimes successifs depuis la Révolution. Il en extrait un buste miraculeusement préservé de la Première République, celle de 1792 1 . La lettre de la caricature explicite la charge : « Un préfet intelligent. "Rentre là-dedans, tyran, et vive la République !" ». Le préfet préserve sa position sociale en affichant par l'image son adhésion supposée au nouveau régime, tout en ménageant d'autres issues possibles. Au-delà de la critique des girouettes et autres « républicains du lendemain », la satire graphique nous fait rentrer dans la boîte noire d'un siècle d'assemblage et de composition, de juxtaposition et de réemploi, « siècle de plâtre » aussi visuel que politique 2 . L'armoire à bustes conserve des souverainetés passées, mais aussi des souverainetés alternatives -on distingue ainsi une effigie d'Henri V, disponible en cas de restauration bourbonienne.

Cette caricature nous invite à prolonger l'enquête autour de l'objet éminemment politique qu'est le buste de souveraineté à l'âge de la reproductibilité technique et de la fragilité des régimes. Les bustes officiels, qui se diffusent au XIX e siècle à une échelle jusque-là inédite, incarnent une souveraineté personnifiée, mais aussi, du fait de leur taille et de leur dissémination dans l'espace, une souveraineté accessible et manipulable. Au fond, ces bustes permettent de saisir, dans leur cycle de vie et éventuellement de mort, le rapport concret d'une société postrévolutionnaire à la souveraineté politique en tant que source visuelle d'où procède l'autorité. Dans un moment de très forte politisation, mais aussi de très forte incarnation du pouvoir et de sa légitimation, les bustes de monarques -de Napoléon I er à Napoléon III -cristallisent les passions et les gestes d'adhésion, de dissidence ou de profanation.

Or l'intérêt historiographique pour les images de souverains et les allégories politiques s'est longtemps concentré sur l'iconologie, les codes visuels et les imaginaires sociaux et politiques qu'ils charrient 3 . On ne s'est pas suffisamment interrogés sur les pratiques qui s'emparent de ces objets à effigie, de la monstration à la destruction, ni sur les regards portés sur eux, ni sur leur rapport à l'espace, qu'il soit public, semi-public ou privé. C'est ainsi la vie sociale et spatiale de ces bustes de souveraineté qui mérite d'être exhumée, à la faveur d'un « tournant matériel » qui permet de restituer aux objets politiques l'agentivité qui était la leur à l'âge des révolutions 4 . L'article portera ici sur l'espace français sur un long cycle politique jalonné de six régimes successifs, de la chute du Premier Empire à l'instauration de la Troisième République, cycle qui marque l'apogée des luttes autour des objets de souveraineté 5 .

Inscrire la souveraineté dans l'espace

Au XIX e siècle, les bustes officiels des monarques constituent, après le drapeau (tricolore ou blanc), la principale marque de souveraineté politique visible dans les édifices publics. Ils ornent ainsi les hôtels de préfectures et de sous-préfectures, les mairies, les tribunaux, les casernes et corps de garde, mais aussi, parfois, des écoles, musées, bibliothèques, halles et marchés, fontaines publiques, voire certaines églises 6 en l'absence de mairie. Aucune disposition légale n'encadre un tel usage, qui n'est pas systématique, en particulier dans les communes rurales faiblement équipées en mobilier administratif. Mais de fait, on assiste au fil du siècle à une multiplication et à une dissémination de ces bustes, liées à une véritable politique des images, consciente et instrumentale, visant à accroître la visibilité de la souveraineté. Cette politique atteint son acmé sous le Second Empire, destinée à graver dans l'esprit des citoyens -y compris des enfants -la personnification de l'État, mais aussi à légitimer le pouvoir par le faste -en particulier dans les hôtels de préfecture 7 .

La transformation, au regard des pratiques d'Ancien Régime, est substantielle. L'effigie du souverain régnant n'est plus visible sous la forme de la monumentale statue de place publique [START_REF]exemple la statue d'Henri IV sur le Pont-neuf à Paris), mais ne construit pas de statue de place publique à l'effigie de Louis XVIII ou de Charles X[END_REF] , ou « place royale », mais sous celle de bustes disséminés dans une multitude de bâtiments et monuments civils [START_REF]Les bustes de souverains étaient alors cantonnés à de rares édifices[END_REF] . Moins imposante, l'image du souverain est bien plus présente dans les lieux de pouvoir et de micro-pouvoir, conduisant à un quadrillage progressif de l'espace public. L'effigie est surtout visible dans le cadre d'une relation voire d'une intégration à l'État : l'exercice de la justice (au tribunal), le prélèvement de l'impôt, l'instruction (dans la salle de classe), le vote (à la mairie), le tirage au sort des conscrits et le conseil de révision. Il ne s'agit plus d'assurer l'union charnelle des sujets et de la Couronne dans l'espace urbain, ni de contempler « la froide impassibilité du roimachine » : la mécanique théologico-politique du XVII e siècle, décortiquée par Louis Marin, n'est plus vraiment d'actualité [START_REF] Marin Louis | Le portrait du roi[END_REF] . La dissémination iconique du souverain vise désormais à occuper le « lieu vide » du pouvoir post-révolutionnaire [START_REF] Lefort | Essais sur le politique, 19 e -20 e siècles[END_REF] , à conjurer les souverainetés alternatives [START_REF]un contremaître de filature au chômage saisit un buste de Louis-Philippe dans l'hôtel de préfecture, puis le promène à travers la ville. L'effigie royale est insultée par la foule, couverte de crachats, puis jetée dans la Deûle 32[END_REF] , et à incarner l'extension progressive du domaine de l'État administratif dans la vie sociale.

Cette dissémination est rendue possible par l'industrialisation de la production statuaire et la réduction du coût des bustes au cours du XIX e siècle. À la duplication classique des bustes par moulage en plâtre d'originaux en marbre, s'ajoute le procédé de réduction mécanique des statues, dit procédé Collas, breveté en 1837. Sous le Second Empire, la maison Barbedienne propose des bustes en plâtre à 25 francs, quand ils coûtaient 150 francs sous la Restauration. Ils peuvent désormais être choisis à distance, sur catalogue -tout comme les éléments de décor éphémère des fêtes publiques. Les bustes de souverains entrent ainsi graduellement dans la culture industrielle, sérielle et marchande. Ils peuvent aussi, plus classiquement, être commandés à une échelle locale auprès d'artistes du cru, ou à une échelle nationale par le bureau des Beaux-Arts, puis déposés et exposés dans les bâtiments civils concernés. La visite officielle du souverain peut enfin s'accompagner de l'offre de bustes, par la préfecture, aux petites communes traversées qui en étaient jusque-là dépourvues. Si quelques modèles dominent quantitativement -ainsi, sous le Second Empire, les bustes de Napoléon III par Émilien de Nieuwerkerke et par Jean-Auguste Barre -il n'existe pas de buste officiel unique, au sens où nous l'entendons aujourd'hui pour les photographies de présidents de la République.

La dissémination des bustes de souverains déborde l'espace des bâtiments civils pour gagner aussi des espaces privés ou semi-privés : façades de demeures particulières, intérieurs aristocratiques ou bourgeois, salles de banquets, cabarets, tavernes, goguettes. Le buste peut alors être lu comme signe d'allégeance au régime, plus ou moins formelle et sincère selon les cas. Dans nombre de portraits nobiliaires sous la Restauration, trône un buste à proximité de la personne représentée, incarnant sa fidélité au pouvoir mais aussi une « nouvelle légitimité de service » de la part de l'ancienne noblesse comme de la noblesse d'Empire 13 . De la même manière, un royaliste bordelais, Partarrieu, n'hésite pas à exhiber sur la façade de son immeuble, à l'occasion de la visite du duc et de la duchesse d'Angoulême (en mars 1815), pas moins de trois bustes à l'effigie de Louis XVIII, du duc d'Angoulême et d'Henri IV, accompagnés d'inscriptions latines 14 . L'adhésion par les bustes semble ici s'inscrire dans une logique de don et de contre-don : le dénommé Partarrieu accompagne une demande de poste, celui de lieutenant général de police, d'une description soigneuse de ce décor sculpté, comme si ce dernier imposait une reconnaissance -qu'il n'obtient d'ailleurs pas. Dans d'autres cas, le buste de souverain vient simplement donner le change, signe de convenance et non d'adhésion -ainsi lors de la campagne de banquets protestataires au printemps 1830, où figure l'effigie du très contesté Charles X, « corollaire presque obligé du toast au roi 15 ».

La dissémination des bustes de souverains dans l'espace se traduit aussi par le commerce de statuettes de petite taille, miniaturisées, à faible coût, en ville comme à la campagne, par des figuristes en plâtre, souvent italiens 16 . Mais en entrant en concurrence avec d'autres effigies plus ordinaires, celles d'hommes célèbres ou celles, séditieuses, de souverains alternatifs, les bustes de monarques perdent alors autant en sacralité qu'ils gagnent en visibilité. La technique de vente de la plupart des marchands et colporteurs consiste d'ailleurs à mêler dans leur bric-à-brac d'objets des signes contradictoires pour mieux les dépolitiser : sous la Restauration des statuettes de saints, de membres de la famille royale et de Napoléon 17 .

Activer l'aura des effigies

Le buste du souverain est inscrit dans une vie sociale et rituelle très intense, qui active l'aura de l'effigie dans l'espace civique : processions, expositions, inaugurations solennelles. L'analogie est forte avec les statues de grands hommes à l'âge de la « statuophilie » républicaine 18 : les effigies sculptées sont placées au coeur de fêtes civiques, de gestes, de regards qui seuls permettent d'en saisir les appropriations sociales. Une fois ritualisés, exposés et acclamés, les bustes participent d'une modalité oubliée de la représentation politique, l'acclamatio 19 iconique. Les processions de bustes, singulièrement dans des contextes de transition politique, contribuent au marquage de la souveraineté dans l'espace urbain, marquage informel autant qu'officiel. Ainsi à Marseille, au tout début des deux Restaurations, en 1814 comme en 1815, des femmes des classes populaires, poissonnières, vendeuses de légumes, ouvrières, organisent des « promenades royales », soit des processions de bustes du roi dans la ville, avec force embrassades, jets de fleurs, bénédictions. Le 11 mai 1814, à l'occasion d'une de ces processions de bustes, les poissonnières « ont fait leurs farces, elles ont harangué le roi à leur manière, elles l'ont couronné d'une belle couronne d'argent 20 ». Un an plus tard, pas moins d'une dizaine de « promenades royales » ont eu lieu en un mois (entre juillet et août 1815) dans la ville de Marseille, prolongées par des bals publics. Le 27 juillet, les regrattières déploient une effigie du roi « grandeur demi-nature », recouverte de vêtements, avec cordon bleu, portée sur un brancard et déposée « sur une espèce de trône ». L'image du roi, d'une ressemblance jugée troublante, croise celle de la Vierge, processionnée au même moment. Elle semble comme douée de vie :

Tous les tambours ont battu aux champs devant elle [l'effigie du roi], tous les drapeaux l'ont saluée, tous les militaires lui ont présenté les armes comme si le roi avait été réellement présent… Enfin, lorsque la Sainte Vierge est arrivée, le roi [sic] s'est mis en marche pour aller vers elle (porté par les revendeuses) et portant un cierge qu'il a offert lui-même à Notre-Dame de la Garde 21 .

L'effigie du roi, en l'absence de son corps physique, est activée politiquement par des gestes d'adhésion quasi religieuse.

Les « inaugurations » de bustes de souverains se multiplient au début de la Restauration puis sous la monarchie de Juillet, ainsi qu'à la fin de la Deuxième République et au début du Second Empire. Ces campagnes d'inauguration fonctionnent comme des rituels d'acclamation iconique du nouveau souverain. La plus intense suit immédiatement le plébiscite consécutif au coup d'État de décembre 1851 : les communes s'empressent alors de commander des bustes du prince-président, en guise de confirmation de l'onction plébiscitaire. Selon la logique des « actes d'image par substitution 22 », les discours d'inauguration reposent rhétoriquement sur la présence réelle de l'image, et interpellent le prince comme s'il était présent. À Condé-sur-Vesgre (Seine-et-Oise), le 18 janvier 1852, six mille citoyens des environs affluent pour inaugurer en grande pompe le buste Louis-Napoléon, offert à la commune par un compagnon de Napoléon I er à Sainte-Hélène, Achille Archambault. Ce dernier porte alors un toast au buste du prince-président, après qu'il ait été processionné, bénit à l'église, touché à la mairie, ovationné dans la salle de banquet : « C'est le buste pour ainsi dire vivant de son neveu lui-même, de son héritier, que sept millions et demi de voix populaires viennent de rappeler aux Tuileries 23 . » Lors de la fête du monarque -de la Saint-Louis (le 25 août) à la Saint-Napoléon (le 15 août) -le buste du souverain occupe aussi une place centrale, livré aux regards des sujets-citoyens. Sous le Second Empire, la fête nationale du 15 août, qui associe la fête de l'Assomption et la célébration de l'empereur, voit ainsi le buste de Napoléon III exposé sur des autels-reposoirs. L'exposition du buste du souverain doit contribuer à l'effervescence civique, et à l'identification de la majesté de l'État avec la personne de l'empereur 24 . Le photographe Charles Nègre a immortalisé à des fins de propagande cette fiction de dialogue cérémoniel par l'image, le 15 août 1859 à l'asile d'ouvriers de Vincennes [doc. 2]. Une foule composée d'ouvriers en convalescence et de membres du personnel se découvre la tête en guise de salut à l'empereur, présent en effigie. Un même dispositif visuel peut être observé à l'occasion d'un banquet de la Saint-Napoléon à la Garde de Paris 25 : le buste occupe le centre de l'espace, salué par un toast des officiers présents26 .

Manipuler la souveraineté : les bustes au coeur de la tourmente

L'agentivité des objets politiques, c'est-à-dire leur capacité performative dans certaines configurations historiques, est réversible. Si les bustes de monarques contribuent à marquer leur souveraineté et asseoir leur légitimité, ils sont aussi l'objet de rites de destruction ou de profanation qui en sapent les fondements. L'aura politique activée par les rituels d'acclamatio devient disponible, en contexte révolutionnaire, pour l'iconoclasme. Dans des conjonctures fluides -révolutions mais aussi restaurations -où la souveraineté apparaît disponible, les bustes de souverains sont, de fait, placés au coeur de la tourmente. Briser le buste du roi ou de l'empereur revient alors à affirmer que le trône est vide, et que la souveraineté appartient au peuple briseur d'images.

À Paris, au palais des Tuileries, dans le centre charismatique du pouvoir, les bustes de souverains sont particulièrement visés lors de « saccages rituels27 » qui scandent les révolutions de 1830, 1848 et 1870. Loin de faire table rase des images du passé, les insurgés trient, épurent soigneusement les signes jugés insupportables -et parmi eux figurent les bustes du monarque déchu. En 1830, les bustes du détesté Charles X sont détruits sans ménagement quand ceux de Louis XVIII, auteur de la Charte, sont couverts d'un voile noir, ou de l'inscription « respect aux propriétés nationales28 », dans un geste situé à mi-chemin entre destruction et conservation, proche de l'iconoclash (Bruno Latour)29 . En 1848, les bustes de Louis-Philippe sont impitoyablement abattus, quand les effigies du duc d'Orléans, son fils aîné décédé en 1842, sont respectées.

À l'échelle locale, dans les hôtels de préfectures ou de sous-préfectures, dans les « maisons communes » ou hôtels de villes, la destruction du buste du monarque fait pour ainsi dire partie du rituel de transfert de souveraineté, là où les rapports de force le permettent. Dans des moments d'incertitude, surtout avant 1848 où les informations relatives aux événements parisiens parviennent avec lenteur et parcimonie, l'attaque d'un des principaux signes de souveraineté vient accréditer la possibilité d'une révolution ou d'une restauration. L'iconoclaste, alors, « casse un certain présent -fût-ce à coups de marteau […] -et lève le bras vers le futur qui s'ouvre 30 ».

Il en est ainsi à Graveson, petite commune des Bouches-du-Rhône proche Saint-Rémy-de-Provence et de Maillane, en 1815. Le 6 avril, à trois heures du matin, une bande armée de 150 « bonapartistes » fait irruption dans ce gros bourg, au moment où le pouvoir parisien a déjà basculé du côté de Napoléon (depuis le 20 mars), mais pas encore le pouvoir local. Dans ce village, deux « factions » s'opposent frontalement depuis la révolution, autour de deux familles de notaires : les Raoulx, liés au ci-devant marquis de Graveson, et les Mercurin, identifiés aux « jacobins ». Dans cette « Vendée provençale » située entre Rhône, Durance et Alpilles, la municipalité est encore dominée par les « blancs ». Mais des rumeurs insistantes circulent, depuis quelques jours, d'une entrée de Napoléon à Paris et de l'exil de Louis XVIII. La souveraineté est donc flottante, incertaine. La scène inaugurale d'une révolution locale se joue autour du buste de Louis XVIII. Les insurgés, représentant le clan des Mercurin, viennent chercher des armes, et prendre par la force le pouvoir municipal. Ils se dirigent vers la mairie, détruisent les proclamations officielles, les registres d'état civil, et surtout le buste du roi, qu'ils jettent par la fenêtre en s'exclamant : « le bougre a ben petà ». Ils justifient leur geste par la vacance de la souveraineté ; l'un d'eux s'écrie : « Aujourd'hui nous n'avons pas besoin de maire, c'est moi qui le suis ; il n'y a plus de gouvernement ni roi pour l'épreuve : « Si tu es Dieu, lève-toi pour faire miracle ! », s'écriait ainsi un huguenot en 1562 au moment de jeter un crucifix au feu [START_REF] Olivier | Les affrontements religieux en Europe du début du XVIe au milieu du XVII e siècle[END_REF] . Rien de tel au XIX e siècle, y compris lorsque sont attaqués les bustes de rois très-chrétiens comme Charles X ou Louis XVIII. Plus exactement, c'est désormais la puissance proprement politique des images qui est défiée -sinon mise à l'épreuve -par les briseurs de bustes. La souveraineté présente dans l'effigie est vidée de sa substance, réduite à de la pure matière : en 1870, les paysans iconoclastes de l'Ariège que nous venons d'évoquer désignaient ironiquement les bustes de l'empereur comme des « plâtres ». À Courmonterral (Hérault) en décembre 1830, lorsque plusieurs centaines de « blancs » nostalgiques de Charles X s'attaquent au buste de Louis-Philippe à peine inauguré, ils s'écrient « Vive le roi ! À bas Philippe », dépossédant l'effigie de sa souveraineté au profit du roi en exil 41 .

Les bustes sont néanmoins insultés ou apostrophés comme si l'image était dotée de vie. Le 14 avril 1814 à Marseille, alors que la nouvelle de la restauration des Bourbons sur le trône ne circule qu'à l'état de rumeur, un boucher asperge le buste de Napoléon de sang animal puis l'interpelle ainsi : « Tiens, tu aimais tant le sang, bois celui-là 42 ». Le 24 février 1848 aux Tuileries, alors que la République n'est pas encore proclamée, un insurgé place un bandeau sur les yeux du buste de Louis-Philippe et l'apostrophe en ces termes : « C'est toi qui es aveugle 43 », allusion au dernier discours du trône et aux « passions ennemies et aveugles » dénoncées par le roi. On devine à ces mots le degré d'information et de politisation de cet insurgé iconoclaste.

De manière générale, la taille des bustes rend les interactions, interpellations, jeux de rôles plus aisés qu'avec une statue. Dans un cabaret de Tanus (Tarn) en 1817, un vétéran napoléonien achète deux bustes-figurines du duc et de la duchesse d'Angoulême, les couche sur une table, feint de les faire boire, puis insulte l'effigie du duc : « Bois, bois, bougre de c…, je veux te guillotiner », avant de lui couper le cou au couteau. Puis vient le tour du buste de la duchesse, qu'il glisse dans sa poche avec ces mots : « Entre là, putain, pour t'en faire autant quand il me plaira 44 ». La profanation des effigies politiques traverse toute une gamme qui va de la procession carnavalesque à l'exécution capitale, en passant par l'insulte, sexualisée ou non. Maurice Agulhon appelle « expressionnisme politique » cette manière « de parler [aux images et aux signes] ou de les faire parler, d'abattre les mauvais et de mettre en valeur les bons ». Il le caractérise comme « populaire » et estime qu'il tend à décliner à la fin du XIX e siècle [START_REF] Agulhon | Marianne au combat…[END_REF] . Le terrain empirique de l'iconoclasme confirme plutôt ces deux hypothèses : d'une part, les rares briseurs de bustes dûment identifiés appartiennent à des catégories populaires, et d'autre part, le buste de Marianne, à la fin du XIX e et au début du XX e siècle, semble moins faire l'objet d'« actes d'image » que ses prédécesseurs monarchiques.

On aurait tort, quoi qu'il en soit, de ne voir dans ces face-à-face avec les bustes qu'une mise à l'épreuve de la souveraineté. Les attaques iconoclastes s'inscrivent aussi dans des interactions beaucoup plus banales, ordinaires, où le jeu, la dérision, les bravades et les règlements de comptes peuvent compter autant que la politique. Il en est ainsi dans les tavernes, maisons de tolérance ou encore dans les corps de garde, où le buste royal ou impérial fait quelquefois les frais de soirées bien arrosées et de défis virils. Dans un corps de garde du port au vin de Nantes, en 1834, l'effigie de Louis-Philippe est ainsi barbouillée de noir, ornée de moustaches en étoupe, puis brisée et jetée dans la Loire par des gardes nationaux [START_REF][END_REF] . L'affaire suscite un scandale public, mais l'enquête judiciaire conclut étonnamment à l'impossibilité de prouver une intention séditieuse dans des gestes qui auraient été accidentels, et toutes les poursuites sont abandonnées. Le scandale de la publicité

  Revue française de science politique, 2015/3, p. 381-403.20 PELLIZZONE Julie, Souvenirs, 1787-1815, tome I, Aix-en-Provence, Indigo, 1995, p. 383. 21 PELLIZZONE Julie, Journal d'une Marseillaise, 1815-1824, tome II, Aix-en-Provence, Indigo, 2000, p. 93. 22 Sur cette notion d'« acte d'image par substitution », cf. BREDEKAMP Horst, Théorie de l'acte d'image, Paris, La Découverte, 2015, p. 159 et suiv. 23 Inauguration du buste du prince Louis-Napoléon, à Condé-sur-Vègre (Seine-et-Oise), le 18 janvier 1852, Paris, imp. de Schneider, 1852, p. 7. 24 HAZAREESINGH Sudhir, La Saint-Napoléon. Quand le 14 Juillet se fêtait le 15 août, Paris, Tallandier, 2007, p. 143 et suiv. 25 Cf. CARDONI Fabien, « La garde de Paris sous l'ère des transitions politiques », Annales de Bretagne et des Pays de l'Ouest, n° 114-2, 2007, p. 135-145.

  13 DE WARESQUIEL Emmanuel, « Portrait du roi et de ses élites sous la Restauration et la Monarchie de Juillet… une contribution à l'étude des représentations du pouvoir », Versalia. Revue de la Société des Amis de Versailles, 2006, n° 9, p. 178-194. 14 « Optato et optimo principi » pour Louis XVIII (« à notre prince désiré et excellent ») ; « A l'auguste ami des Bordelais » pour le duc d'Angoulême, « il fut de ses sujets le vainqueur et le père » pour Henri IV. Ce dernier souverain fait partie du panthéon glorieux de la Restauration, figure de réconciliation après une guerre civile. 15 ROBERT Vincent, Le temps des banquets. Politique et symbolique d'une génération (1818-1848), Paris, Publications de la Sorbonne, 2010, p. 130. 16 CARMINATI Pauline, « Du colportage au musée du Louvre. Les mouleurs italiens en France au tournant du XIX e siècle », Diasporas. Circulations, Migrations, Histoire, 2018, n° 32, p. 113-124 ; MORACHIOLI Sandro, « Not Just Print Culture : Small-Size Sculptures and Street Vendors in the Visual Culture of the Risorgimento », in Enrico FRANCIA et Carlotta SORBA (eds), Political Object since the Age or Revolutions. Material Culture, National Identities, Political Practices, Rome, Viella, 2021, p. 117-134. 17 FUREIX Emmanuel, "Resisting with Objects? Seditious Political Objects and their Agency in Restoration France (1814-1830)", in Christopher FLETCHER (ed.), Everyday Political Objects. From the Middle Ages to the Contemporary World, Londres, Routledge, 2021, p. 135-150. Voir aussi HAZAREESINGH Sudhir, La légende de Napoléon, Paris, Tallandier, 2005.

18

LALOUETTE Jacqueline, Un peuple de statues. La célébration sculptée des grands hommes (1804-2018), Paris, Mare et Martin, 2018.

19

Cf. IHL Olivier, « Une autre représentation. Sur les pratiques d'acclamatio dans la France de la Seconde à la Troisième République »,

La plupart de ces bustes ont disparu après la Révolution, notamment dans les premières années de la Restauration.

« L'assemblage est partout dans ce siècle de plâtre où triomphent la juxtaposition, la mosaïque, l'objet cloisonné et le plaisir que procure le foisonnement des références », écrit Alain Corbin dans « Le XIX e siècle ou la nécessité de l'assemblage », in Alain CORBIN et al., dir., L'invention du XIX e siècle, Paris, Klincksieck-Presses de la Sorbonne nouvelle, 1999, p. 153-159. Sur la figure de la girouette, cf. SERNA Pierre, La République des girouettes. 1789-1815, et au-delà : une anomalie politique, la France de l'extrême centre, Seyssel, Champ Vallon, 2005.

Approche représentée notamment par AGULHON Maurice, Marianne au combat. L'imagerie et la symbolique républicaines de 1789 à 1880, Paris, Flammarion, 1979.

Cf. l'article séminal de SORBA Carlotta, « Faire de l'histoire du politique avec les objets. Trajectoires et enjeux actuels », Revue d'histoire culturelle, XVIII e-XXI e siècles, 2022/4.

Nous nous permettons de renvoyer sur ce point à notre ouvrage, L'oeil blessé. Politiques de l'iconoclasme après la Révolution française,

Ceyzérieu, Champ Vallon 2019. La Commune de 1871 porte peu atteinte aux bustes de Napoléon III, déjà largement épurés par la révolution de septembre 1870. 6 Ainsi à Musculdy (Basses-Pyrénées) sous la Restauration, ou encore à Horgues (Hautes-Pyrénées) et à Marquefave (Haute-Garonne) sous la monarchie de Juillet.7 Cf. BOUJU Péric, « Architecture et lieux de pouvoir en Bretagne : XVIII e -XX e siècle », thèse de doctorat d'histoire sous la direction de Jean-Yves Andrieux, Université de Rennes 2, 2011, et MAUDUIT Xavier, « Le ministère du faste : la Maison de l'Empereur

La Saint-Napoléon à la caserne. Cantate militaire chantée au Banquet de la Garde de Paris à l'occasion de la fête de l'Empereur, estampe, Paris, imp. Trinocq, 1853.

Concept emprunté à GINZBURG Carlo, « Saccheggi rituali. Premesse a una ricerca in Corso », Quaderni storici, n° 65, 1987-2, p. 615-636. Cf. FUREIX Emmanuel, L'oeil blessé…, op. cit., p. 210-229.

 28 ROZET Louis, Chroniques de juillet 1830, Paris, Barrois et Duprat, 1832, volume II, p.

197.29 LATOUR Bruno, « Iconoclash. Au-delà de la guerre des images », Traduit de l'anglais par Aude Tincelin, paru en anglais sous le titre « What is Iconoclash? Or Is there a world beyond the image wars? » in Bruno LATOUR and Peter WEIBEL, Iconoclash, Beyond the Image-Wars in Science, Religion and Art, Cambridge, MIT Press, 2002.


domain: scco sde

text:
Péron Guillaume
email: guillaume.peron@univ-lyon1.fr

The frequency and position of stable associations offset their transitivity in a diversity of vertebrate social networks

Keywords: clustering coefficient, graph theory, intransitivity, socio-spatial structure, peer of a peer, epidemiology, strength of weak ties

When the estimated strength of social associations corresponds to the proportion of time spent together, strong links, those that take up most of the recorded time of individuals, are compulsorily transitive and tend to occur in clusters.

However, I describe three ways in which the frequency and position of strong associations apparently offset the expected transitivity of strong links in published association networks from 26 species of vertebrates. Instead of occurring in groups of three, strong links were mostly isolated. When they did occur in clusters, the clusters were small. The phenomena increased in intensity as the overall number of links of all strengths and the overall network transitivity increased. Since stable transitive motifs are beneficial to cooperation, these results can help explain why cooperative behaviors are not more frequent than they are in group-living vertebrates. Inversely, stable transitive motifs may be rare and small because the benefits of cooperation do not overcome the costs associated with these motifs. The summary statistics developed for this study captured information not conveyed by other network-level metrics; thus they may help quantify the socio-spatial structure of populations and potentially tease apart the environmental, species-specific, and individual drivers.

Introduction

 [START_REF] Granovetter | The Strength of Weak Ties[END_REF]observed that strong friendships between humans tend to be transitive. In other words, when individual A has two best friends B and C, then B and C are also usually best friends. The proximate reasons for this transitivity include the "theory of cognitive balance" stating that good friends want their feelings towards third parties to be congruent [START_REF] Heider | The Psychology of Interpersonal Relations[END_REF], and "network homophily", meaning that strong friendships tend to emerge from (and to further promote) similarity in cultural tastes etc. [START_REF] Mcpherson | Birds of a Feather: Homophily in Social Networks[END_REF]. As a result, the networks feature clusters or cliques of individuals that are tightly-knit together. There are still weak ties between these cliques, which do not obey the same transitivity rules. These weak ties then perform a major role as the connections between, and the gateways into, the cliques: this is the "strength of weak ties paradigm" [START_REF] Granovetter | The Strength of Weak Ties[END_REF].

In animal studies, strong ties are also expected to be more transitive than weak ties, but firstly because of the way social ties are measured as the proportion of time that dyads spend together [START_REF] Holekamp | Society, demography and genetic structure in the spotted hyena[END_REF][START_REF] Rubenstein | Similar but Different: Dynamic Social Network Analysis Highlights Fundamental Differences between the Fission-Fusion Societies of Two Equid Species, the Onager and Grevy's Zebra[END_REF]. Indeed, if individual A spends most of its time with B and with C, then B and C also spend most of their time together. In other words, strong associations cannot be intransitive and should thus be more transitive on average than weak associations. The resulting stable clusters of closely-knit individuals promote the evolution of cooperative behaviors and represent a fitness advantage when cooperation is effective [START_REF] Grinnell | Cooperation in male lions: kinship, reciprocity or mutualism?[END_REF][START_REF] Nowak | Five rules for the evolution of cooperation[END_REF][START_REF] Silk | The benefits of social capital: close social bonds among female baboons enhance offspring survival[END_REF][START_REF] Teunissen | Context-dependent social benefits drive cooperative predator defense in a bird[END_REF]. However, there are also costs to life in a tight group. Costs include the rapid spread of pathogens [START_REF] Morrison | Rapid transmission of respiratory infections within but not between mountain gorilla groups[END_REF], reproductive conflicts and other types of conflicts within the cliques [START_REF] Datta | The acquisition of dominance among free-ranging rhesus monkey siblings[END_REF][START_REF] Holekamp | Society, demography and genetic structure in the spotted hyena[END_REF], and impaired access to information during the periods when the weak links are not active [START_REF] Artime | Dynamics on networks: Competition of temporal and topological correlations[END_REF]. If these costs exceeded the benefits, the transitivity of strong links would operate as a constraint rather than an advantage.

The objective of the present paper is to apply several network statistics to decipher whether the transitivity of strong links is offset by the way strong links are distributed in animal association networks. For this purpose, I assembled a set of published animal association networks from 26 species (see methods). First, I verified that strong links were indeed more transitive than weak links. Second, I quantified three aspects of the distribution of link strength in the networks.

(i)

The network-level Gini coefficient of inequalities [START_REF] Gini | On the Measure of Concentration with Special Reference to Income and Statistics[END_REF] (method section 1.5). This metric is an indicator of the overall rarity of strong links, i.e., the occurrence of a few very strong links amidst mostly weak links. For example, a group where mother-offspring bonds are much stronger than other types of associations would exhibit a high Gini coefficient if the offspring were few. The alternatives are that strong links are not very different from weak links, or that most of the links are strong.

(ii)

The triadic dissimilarity between the three links in each triangle (a new metric, see method section 1.6). This metric indicates whether strong links more often occur in groups of three, or in an isolated fashion. For example, if breeding pairs travel together and they both avoid other pairs and evict single individuals from foraging spots [START_REF] Black | Agonistic behaviour in barnacle goose flocks: assessment, investment and reproductive success[END_REF], strong links (breeding pairs) would be mostly isolated, leading to an excess of weak triplets closed by a single strong link [START_REF] Péron | Transitivity scores that account for triadic edge weight similarity in undirected graphs[END_REF].

(iii)

The fragmentation of the subnetworks of strong links (method section 1.7). This metric indicates whether, when strong links are not isolated, the clusters of strong links are relatively small or large.

For example, if there is a core group of closely tied individuals amidst a cloud of individuals that spend most of their time alone, the subnetwork of strong links should appear less fragmented than the network as a whole. By contrast, if the network is made of family units in which recent offspring are more tightly linked to their mother than older offspring, then the subnetwork of strong links, made exclusively of the links between recent offspring and their mothers, would appear more fragmented than the network as a whole.

If strong links were rare (high Gini coefficient), isolated (high triadic dissimilarity), and if any cluster of strong links was small (high fragmentation of the subnetwork of strong links), then I concluded that the expected transitivity of strong links was in effect counter-balanced by the distribution of strong links in the network. In addition, I tested whether these patterns occurred more often or more intensely in networks that have many links and many transitive motifs compared to networks that have few links and few transitive motifs. If that was the case, this would suggest a functional response, i.e., a change in the probability to create specific patterns in the social network with a change in the availability of social partners. For example, the number of social partners could influence the perceived benefits obtained from stable partners, reinforce or weaken the effect of existing relationships on the probability to create new relationships, or the attraction exerted on bonded pairs by other bonded pairs. In practice, I first verified the expected transitivity of strong links. Next, I developed and applied the aforementioned three network-level statistics and correlated them to the edge density.

Material and methods

Definitions

Association networks (sometimes termed contact networks) refer to undirected social networks where connections occur through proximity between individuals. This definition excludes directed networks, such as dominance relationships, grooming, etc. The links (or ties, or edges) can be weighted by the relative dyadic association frequency [START_REF] Holekamp | Society, demography and genetic structure in the spotted hyena[END_REF][START_REF] Rubenstein | Similar but Different: Dynamic Social Network Analysis Highlights Fundamental Differences between the Fission-Fusion Societies of Two Equid Species, the Onager and Grevy's Zebra[END_REF], hereafter termed the link strength.

Following previous authors [START_REF] Sah | A multi-species repository of social networks[END_REF], I considered three types of associations in this study: (i) physical contact or staying any amount of time within touching distance of associates, (ii) close proximity, a category in which I pooled nearest neighbor data and data documenting the time spent within a given radius of associates (with cutoff distance and duration defined by the original authors of the source studies), and (iii) shared group membership, which mostly applies in a fission-fusion context.

Triplets correspond to situations where one individual A is associated with two different individuals B and C. In a transitive association network, most triplets are closed by a link between B and C, thereby forming a triangle [START_REF] Granovetter | The Strength of Weak Ties[END_REF][START_REF] Newman | Random graph models of social networks[END_REF]. The unweighted transitivity score of an association network, hereafter denoted 𝒞 0 , corresponds to the proportion of closed triangles among all the triplets [START_REF] Newman | Random graph models of social networks[END_REF]. Several adjacent triangles make a transitive motif or a transitive cluster or a clique. Examples of completely intransitive networks include grid-like and tree-like networks [START_REF] Newman | The mathematics of networks[END_REF]. Some researchers use the clustering coefficient instead of the transitivity score which differ from the transitivity score because the clustering coefficient averages an individual transitivity score whereas the transitivity score averages a triplet score [START_REF] Barrat | The architecture of complex weighted networks[END_REF]. Importantly, the interpretation of the transitivity score depends on whether the links are undirected, as is the case in this study, or directed. In dominance networks, the links are directed, and the transitivity score measures the linearity of the social hierarchy [START_REF] Mcdonald | Comparative transitive and temporal orderliness in dominance networks[END_REF]. In association networks which are the topic of the present study, the links are not directed, and the transitivity score measures the frequency and size of transitive motifs.

The triplets can be also weighed according to the strength of the links inside of them, in order to generate a weighted transitivity score [START_REF] Opsahl | Clustering in weighted networks[END_REF], hereafter denoted 𝒞 1 . If 𝒞 1 is larger than 𝒞 0 , the probability that a triplet is closed increases with the strength of the links inside of it, and inversely if 𝒞 1 is smaller than 𝒞 0 then weak triplets are on average more likely to be closed than strong triplets [START_REF] Opsahl | Clustering in weighted networks[END_REF].

Another approach to the quantification of transitive clusters is to consider them as modules of individuals that interact more among themselves than with the rest of the network. Strong links would then correspond to within-module links and weak links would correspond to cross-module links. The network modularity quantifies how distinct these modules are, i.e., how rare and weak are the links between modules [START_REF] Newman | Random graph models of social networks[END_REF][START_REF] Pons | Computing communities in large networks using random walks[END_REF]. In practice, I delineated the modules using the short random walk communityfinding algorithm (routine cluster_walktrap from igraph; [START_REF] Pons | Computing communities in large networks using random walks[END_REF]. Except when explicitly stated otherwise, I took into account the weight of the links when delineating the modules (argument weights = E(graph)$weight). I then computed the modularity score following the usual formula [START_REF] Newman | The mathematics of networks[END_REF].

Lastly the edge density, denoted 𝒟 0 , corresponds to the overall number of links in the network divided by the maximum possible number of links if all individuals were connected to each other. The edge density measures the overall probability that a link exists between any two individuals.

Literature search for network association data

This study uses published data only. Original data were collected in accordance with relevant institutional and national guidelines, as explained in the source articles listed in Table 1.

I focused on vertebrates that always or mostly forage in groups and/or always or mostly roost in groups, but without any criteria regarding the occurrence of cooperative behaviors within those groups. The data needed to be collected from free-ranging groups with naturally occurring kinship structure, and to document one of the aforementioned association types (see section 1.1). I first searched open data repositories: dryad.com and https://bansallab.github.io/asnr/about.html ("A social network repository" or ASNR; [START_REF] Sah | A multi-species repository of social networks[END_REF] on Jan 5, 2022. Next, I used the search engine googlescholar.com with the keywords "animal" and "social network". I searched the citation network of the first 60 hits upward and downward. The cutoff number 60 was chosen as a tradeoff between the risk of missing a poorly-cited study and the risk of oversight due to the sheer task at hand. If the title or abstract indicated that data corresponding to the above criterion existed, I elicited data sharing over email. This procedure yielded data from 26 species and three taxonomic classes.

These original studies varied in the way they quantified dyadic association rates. The most frequent method was to use the co-occurrence frequency relative to each associate's own frequency in the dataset. Several studies however reported indexes derived from activity time budget analyses, and a few reported a discretized index of association strength. To standardize the link strengths across datasets, I rescaled the link strengths between 0 and 1 using a logit-transformation so that the median point between the weakest and strongest links of each network was attributed strength 0.5.

The seasonal timing of data collection was decided by the original authors. I did not select data according to that criterion.

Verifying the natural transitivity of strong links

I used two tests of the natural transitivity of strong links. First, I compared the unweighted transitivity score 𝒞 0 and the weighted transitivity score 𝒞 1 [START_REF] Opsahl | Clustering in weighted networks[END_REF]. If strong links are more transitive than weak links, then triplets made of strong links are more likely to be closed than triplets made of weak links, and thus I expect 𝒞 1 > 𝒞 0 .

Second, I manipulated the networks by removing an increasing proportion of the weakest links. I com-

puted the quantity 𝑏(𝑄) = 𝒞 0 (𝑄)-𝒟 0 (𝑄) 1-𝒟 0 (𝑄)
where Q is the proportion of remaining links and 𝒟 0 is the edge density.

If 𝒟 0 (𝑄) = 1, then also 𝒞 0 (𝑄) = 1 and 𝑏(𝑄) = 1. Note that this is the only section where 𝒞 0 and 𝒟 0 depend on Q. In the rest of the paper, I report the values corresponding to Q = 100%, i.e., unmanipulated networks, meaning that 𝒟 0 = 𝒟 0 (100%) and 𝒞 0 = 𝒞 0 (100%).

The quantity 𝑏(𝑄) takes value 0 under the null hypothesis that the manipulated network is not more transitive than expected from the overall probability of a link occurring anywhere in the manipulated network.

Otherwise, 𝑏(𝑄) varies between -𝒟 0 (𝑄)

1-𝒟 0 (𝑄) for a completely intransitive network and +1 as the upper boundary value for completely transitive networks. I expected b to increase as Q decreased, i.e., as I removed links. I regressed the logit-transformed b against logit-transformed Q in interaction with the initial edge density, using a nonparametric spline model (function gam in R-package mgcv).

Cross-species regressions

In sections 1.5 to 1.7 below, I perform regressions between population-specific metrics. A potential source of spurious inference in such regressions is phylogenetic inertia, i.e., when the dependent variable is conserved along the phylogeny, and thereby pairs of closely related taxa essentially duplicate one another. If not corrected for, this mechanism can lead to over-estimate the effect sizes or the statistical significance of the results. Phylogenetic generalized models, which force the residuals of related species to be more similar than those of non related species, have become the default method to address the issue [START_REF] Paradis | APE: Analyses of Phylogenetics and Evolution in R language[END_REF]. However, this statistical method requests adequate sample sizes to perform, and is only necessary if there is indeed a problematic phylogenetic structure in the data. The present study has 26 species for 23 genera, and only 2.4 species per taxonomic order on average (median: 1). In that situation, deep phylogenetic divergences expectedly take precedence. The potential bias can be either major if e.g., all mammals or all primates exhibited the same values, or undetectable if e.g., the variance between the 8 primates was larger than the variance across taxonomic orders.

Given these constraints and predictions, and given that preliminary analyses indicated that the second scenario was most likely (Table 1), I replaced the correlated error structure by a simple random effect of the taxonomic order. Results were qualitatively similar if removing that effect altogether.

The rarity of strong links

I computed the network-level Gini coefficient of inequalities between link strengths as

𝐺 = 1 2𝑛 2 𝑥̅ ∑ ∑ |𝑥 𝑗 -𝑥 𝑖 | 𝑛 𝑗=1 𝑛 𝑖=1
where the 𝑥 𝑖 are the link strengths and n is the total number of links. The null hypothesis that strong links are not rare was represented by the uniform distribution of link strength, which corresponds to G = 0.33. If a few strong links were much stronger than the bulk of the links, G would tend towards 1. If all links were of equal strength, G = 0. I tested the prediction that strong links should be rarer in dense and transitive networks using a linear model with the logit-transformed Gini index as dependent variable and the logit-transformed edge density of the networks as predictor.

Dissimilarity between the three links in each triadic relationship

The objective here is to compare the three link strengths in each triad. Hereafter, a triad is any three individuals with at least one non-zero link between two of the three individuals. Triads therefore include closed triangles, open triplets, but also "false triplets" corresponding to a dyad plus a disconnected individual.

The coefficient of variation is inadequate for this purpose because it captures the average pairwise difference, and not the variability between the pairwise differences. Many different combinations of three link strengths can yield the same coefficient of variation. Therefore I derived new triad-level dissimilarity scores.

For each triad, I denoted a, b, and c the three link strengths in increasing order. They varied between 0 (no link) and 1 (maximum recorded strength). I measured the dissimilarity between them with 𝜔 the rescaled sum of proportional pairwise differences, and g the triad-level Gini coefficient [START_REF] Péron | Transitivity scores that account for triadic edge weight similarity in undirected graphs[END_REF]:

Eq. 1a 𝜔 = 𝑎 -𝑐 𝑎 + 𝑏 -𝑐 𝑎 -1 = 𝑏 -2𝑐 𝑎 Eq. 1b 𝑔 = 1 2 (𝑎 -𝑏) + (𝑏 -𝑐) + (𝑎 -𝑐) 𝑎 + 𝑏 + 𝑐 = 𝑎 -𝑐 𝑎 + 𝑏 + 𝑐
These formulae reduces the 3D space {𝑎, 𝑏, 𝑐} into a 2D space {𝜔, 𝐺}. For ease of interpretation, I suggest to divide the {𝜔, 𝑔} space into four quadrants corresponding to different triad configurations (explained in Figure 3).

Next, I computed graph-level triadic dissimilarity scores based on ω and g. They are simply the average of ω and g over all the triads in the network, yielding the new network-level statistics 𝒞 𝜔 and 𝒞 𝑔 .

Eq. 2a

𝒞 𝜔 = 1 𝒯 ̃∑ 𝜔 𝑡 𝒯 t=1
Eq. 2b

𝒞 𝑔 = 1 𝒯 ̃∑ 𝑔 𝑡 𝒯 t=1
The index t refers to the triad number (𝒯 ̃ triads in total). Eqs. 2a-2b are superficially similar to the weighed transitivity score [START_REF] Opsahl | Clustering in weighted networks[END_REF]. However I replaced the triplet strength by the triplet dissimilarity scores. The code to compute 𝒞 𝜔 and 𝒞 𝑔 from the adjacency matrix is provided (Appendix S1).

I then devised some permutation procedures to assess whether the observed dissimilarities exceeded the expectation at random (code in Appendix S1). If test A was not significant, I concluded that the dissimilarities were explained by variation in individual frequencies in the dataset, i.e., the dissimilarities were not large. If test B was not significant, I concluded that the dissimilarities were large but mostly explained by missing links, rather than by the variation in the strength of existing links. If both tests A and B were significant, I concluded that there was either an excess of weak triplets closed by one strong link or an excess of balanced triangles made of three equally strong links, which an examination of the ω and g scores could decipher.

I also devised a test that focused on the lack of "forbidden triads" sensu Granovetter, i.e., strong triplets closed by a weak link. For this test C, I simply shuffled the link strengths while conserving other network aspects [START_REF] Opsahl | Prominence and Control: The Weighted Rich-Club Effect[END_REF]. I recomputed the ω and g scores after the shuffle. I used a chi-squared test with four degrees of freedom to determine whether the observed quadrant distribution differed from the expectation from shuffled networks. I expected a significant lack of triads in quadrant 3 ("forbidden triads"). This could also complete tests A and B by pointing out whether quadrant 1 (balanced triangles made of three equally strong links) or quadrant 2 (weak triplets closed by one strong link) were in excess.

I compared the new transitivity scores across studies using linear mixed models with the logit-transformed 𝒞 𝜔 and 𝒞 𝐺 as dependent variables and the logit-transformed edge density as predictor. Because the occurrence of modules of individuals that interact more amongst themselves than with the rest of the network could be the cause of triad-level dissimilarities, I also considered the additive effect of the network modularity.

The fragmentation of the subnetwork of strong ties

After having delineated the aforementioned modules (see section 1.1), I computed the fragmentation score as the overall network size divided by the average size of the modules that contained more than one individual.

If the score is high, the network is fragmented into a large number of small modules, and inversely. I used notation ℱ 𝑤 and ℱ 0 for the weighted and unweighted versions of the fragmentation score, respectively. ℱ 𝑤 is computed when the community-finding algorithm accounts for link strength and ℱ 0 is computed when it does not. I then computed the fragmentation ratio

ℱ 𝑤 -ℱ 0 ℱ 0
. The ratio quantifies how much more fragmented the subnetwork of strong links is, compared to the network as a whole (Appendix S2 for a simulation). A large ratio indicates that strong links occur in many small modules, e.g., there is more than one module of strong links in each module of weak links (a module can contain a single link and two individuals). A small ratio indicates that there is on average about one module of strong links embedded in each module of weak links. A negative ratio indicates that some of the modules do not feature any strong link. In other words, the fragmentation ratio

ℱ 𝑤 -ℱ 0 ℱ 0
increases when the subnetwork made of only the stronger ties is more fragmented than the network as a whole.

I predicted that this ratio should be positive and increase with the edge density. I tested these predictions using a linear model with the fragmentation ratio as the dependent variable and the logit-transformed edge density as predictor.

Results

Verifying the natural transitivity of strong links

The weighted transitivity score 𝒞 1 was on average larger than the unweighted score 𝒞 0 but only by 2% (± SD: 6%) (Table 1). This suggests that strong triplets were too rare to influence the computation of 𝒞 1 (cf. next section).

In the second test, as I removed an increasing proportion of the weakest links, there was moderate evidence that the transitivity increased more than the edge density (Fig. 1a; McFadden's r² = 0.06). Statistically speaking the effect was however significant (likelihood ratio test against the intercept-only model: df = 8.3, deviance = 6.3, F-test P < 0.001). The evidence mostly came from networks of intermediate initial edge density (Fig. 1a: grey polygon; likelihood ratio test against the model without the interaction: df = 7.7, deviance = 5.8, P < 0.001,

McFadden r² for the interaction = 0.04). The predicted increase in b from about 0.45 to 0.6 (grey curve) would translate in approximately a +0.08 gain in transitivity. Thus, the increase in transitivity with Q was moderate on average across studies. However, the increase was clear-cut in at least some of the studies (e.g., Fig. 1b).

The rarity of strong links

56% of the studies reported distributions of link strength that were more inegalitarian than a uniform distribution (Fig. 2). However, most of the dense networks exhibited inegalitarian properties, as the Gini coefficient increased with the edge density (Wald's Z = 4.8, McFadden's r² = 0.3, ANOVA: P = 0.002; Fig. 2). In other words, strong links were rarer in dense than sparse networks.

Fig. 2: The Gini coefficient (y-axis) increases with the edge density 𝒟 0 of the network (x-axis). The color scale corresponds to the transitivity score 𝒞 0 . The Gini coefficient is a measure of dispersion based on the pairwise differences in link strength among all the connected dyads in the network. The highest Gini coefficient in the present study (acorn woodpecker Melanerpes formicivorus; data: [START_REF] Shizuka | Constructing social networks from automated telemetry data: A worked example using within-and across-group associations in cooperatively breeding birds[END_REF] means that 95% of the recorded associations occurred between 5% of the connected dyads. The dashed line corresponds to the null hypothesis that the distribution of link strength is uniform.

Dissimilarity between the three links in each triadic relationship

As the edge density increased, the networks dissimilarity scores went from quadrant 4 (mostly false triplets) to quadrant 1 (mostly balanced triangles) through quadrant 2 (mostly weak triplets closed by one strong link) while avoiding quadrant 3 (mostly strong triplets closed by one weak link; "forbidden triad") (Fig. 3).

Triplets in quadrants 1 and 2 were more frequent and quadrant 3 was rarer than expected from shuffled networks (test C) (Fig. 3). There was strong evidence that the excess of quadrants 1 and 2 was due to the occurrence of distinct modules, because both the 𝒞 𝜔 and 𝒞 𝑔 score correlated with the network modularity, even after accounting for the edge density (respectively, Z = -0.5, r²=0.01, P = 0.001 and Z = 6.2, r²=0.43, P <0.001; Table 1).

The 𝒞 𝜔 and 𝒞 𝑔 scores nevertheless captured information neither conveyed by the modularity score nor by the usual transitivity score (Appendix S3). In most studies, including all the studies that involved more than 35 individuals, both permutation tests A and B were positive (Table 1). This confirmed that the excess of quadrants 1 and 2 was not due to links missing at random or to variation in individual frequency of occurrence. Yet, in a few studies, the dissimilarities were as expected under one of the null models. In networks that were both small and dense, I found no evidence that the triads were any different from those of a full network (Table 1: Poecila reticulata, Macaca assamensis, Macaca fuscata: PA>0.05). In the proximity logs of barn swallows Hirundo rustica, of one of the subpopulations of elk

Cervus canadensis, and in the Nilgiri langur Trachypithecus johnii, the triadic dissimilarities could apparently be created by links missing at random (PB>0.05).

The fragmentation of the subnetwork of strong ties

Accounting for link strength increased the fragmentation score on average by 35% (±SD 50%) (Fig. 4 and Nearly false triplets 4 P = 0.003; Fig. 4). These results mean that most modules featured at least one strong link; and that within each module, strong links tended to be isolated or to form several small clusters rather than to occur as a single cluster of strong links per module. The phenomenon increased in intensity as the network edge density increased.

Fig. 4: The fragmentation ratio (y-axis) increases with the edge density 𝒟 0 of the network (x-axis). The color scale corresponds to the transitivity score 𝒞 0 . The fragmentation ratio is the proportional difference between the weighted and unweighted fragmentation scores. The highest fragmentation ratio in the present study (a low density population of elk Cervus canadensis; data: Webber & Vander Wal, 2020) means that the weighted network was 3 times more fragmented than the unweighted network.

Discussion

By reanalyzing 26 published datasets of animal association networks, I first retrieved a major result of the field, namely that strong links are more transitive than weak links (Fig. 1). However, the effect was maybe not as strong as expected. My explanation is that, in many of the studied networks, even the strongest associations were not strong enough to mechanistically force transitivity. In addition, a few of the networks that I included feature minimal variation in link strength. Nevertheless, in the studies that were not affected by any of these two issues, the increase in transitivity with link strength was clear-cut (e.g., Fig. 1b). Next, I observed three ways in which the frequency and position of strong links offset their natural transitivity. First, strong links became rarer as the edge density and overall transitivity increased (Fig. 2). Second, the three links in each triangle were

Edge density

Fragmentation ratio 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Group membership Spatial proximity Contact

mostly dissimilar, more dissimilar than expected, and increasingly dissimilar as the edge density and overall transitivity increased (Fig. 3). Third, the subnetworks of strong links were increasingly fragmented as the edge density and the overall transitivity increased (Fig. 4). These results do not challenge the strength of weak ties paradigm [START_REF] Granovetter | The Strength of Weak Ties[END_REF]. Indeed the paradigm described some of the networks very well (Fig. 1b). However, in many of the included studies, most ties were weak, making the strength of weak ties a somewhat tautological property.

These results may help explain why cooperative behaviors are not more widespread than they are among group-living vertebrates. First, cooperation requires stable relationships [START_REF] Nowak | Five rules for the evolution of cooperation[END_REF][START_REF] Teunissen | Context-dependent social benefits drive cooperative predator defense in a bird[END_REF].

I found these to be increasingly rare as the overall gregariousness, the number of association partners, increased (Fig. 2). This suggests that vertebrate social groups can either be dense or stable. Upper limits on the number of simultaneous partners, or a negative effect of existing relationships on the probability to create new ones, might be involved [START_REF] Dávid-Barrett | Processing power limits social group size: computational evidence for the cognitive costs of sociality[END_REF][START_REF] Stadtfeld | The Emergence and Stability of Groups in Social Networks[END_REF]. Second, I found that stable transitive motifs were fewer and smaller than what they could have been. Compared to intransitive networks, stable transitive motifs offer more pathways for the benefits of collaboration to be collected, e.g., via indirect reciprocity [START_REF] Block | Reciprocity, transitivity, and the mysterious three-cycle[END_REF] and contributions towards common goods [START_REF] Mielke | Snake alarm calls as a public good in sooty mangabeys[END_REF]. Stable transitive motifs might also facilitate the policing and coercion of selfish associates for the same reasons. The scarcity and small size of stable transitive motifs would not help cooperative behaviors to emerge. Overall, my observations help explain why many group-living vertebrates do not exhibit advanced cooperative behaviors, or why cooperative behaviors are rarely expressed. Note however the alternative interpretation that stable transitive motifs may be rare and small because cooperation does not bring enough benefits. In addition, as mentioned earlier, a functional response is also possible, in which individuals navigate their social environment in a way that is analogous to the way resource selection occurs in the physical environment [START_REF] Holling | The Components of Predation as Revealed by a Study of Small-Mammal Predation of the European Pine Sawfly[END_REF]. More precisely, the marginal benefit of an additional stable relationship might decrease with the number of pre-existing stable relationships, leading to a concave relationship between the number of potential partners (as proxied by the edge density) and the selection of specific partners. Formal tests would require dynamic data rather than aggregated data, so that a dynamic model of link creation, stability, and activation can be fitted, while taking individual attributes such as kin relationships into account (e.g, [START_REF] Snijders | Introduction to stochastic actor-based models for network dynamics[END_REF].

Importantly, the patterns that I report do not necessarily emerge from individual decisions alone. Environmental and demographic variation clearly contribute to network structure. For example, the fragmentation ratio changed from -0.3 to +2.1 between a dense subpopulation and a sparse subpopulation of elk Cervus canadensis (data: [START_REF] Webber | Heterogeneity in social network connections is density-dependent: implications for disease dynamics in a gregarious ungulate[END_REF]. On the other hand, the observed patterns can be created by relatively simple social behaviors. Kin-biased associations in particular, especially between mother and offspring, and associations between breeding male and female, can cause an excess of weak triplets closed by one strong link [START_REF] Péron | Transitivity scores that account for triadic edge weight similarity in undirected graphs[END_REF]. This point is important because different species have different hard-wired socio-spatial structures, e.g., in most mammals, offspring form strong bonds with their mothers that may last past weaning age, whereas in birds the dominant bond may be between the male and female of a pair. In other words, the patterns that I report stem from a wide variety of mechanisms. Some are environmental constraints, some are speciesspecific evolved social strategies, and others represent individual reactions norms.

In terms of caveats, a major one is that the datasets are not representative of all vertebrate social systems.

First, the literature is biased towards small group sizes, which are easier to monitor. Besides that, social network methods are not compulsorily used on the most social species. For example, the association network of lionesses Panthera leo can appear trivial because association is almost obligate within a pride and almost forbidden across

prides. Yet, it is in this context that the strength of weak ties paradigm applies the most [START_REF] Craft | Disease transmission in territorial populations: the small-world network of Serengeti lions[END_REF], that the transitivity expectedly increases the most with link strength, and that cooperation is expectedly the most beneficial.

Another caveat is that the criteria to determine what constituted an association undoubtedly influenced the recorded network structures [START_REF] Gazda | The importance of delineating networks by activity type in bottlenose dolphins ( Tursiops truncatus ) in Cedar Key, Florida[END_REF]. For example, the network drawn from the ritualized embraces of brown spider-monkeys Ateles hybridus [START_REF] Rimbach | Brown spider monkeys ( Ateles hybridus ): a model for differentiating the role of social networks and physical contact on parasite transmission dynamics[END_REF] was one of the few that scored almost into quadrant 3 (the "forbidden triad"; Fig. 3). This specific type of association lasts less than a minute [START_REF] Rimbach | Brown spider monkeys ( Ateles hybridus ): a model for differentiating the role of social networks and physical contact on parasite transmission dynamics[END_REF]. This would expectedly relax the physical constraint that strong links are compulsorily transitive.

This suggests that the triadic dissimilarity scores 𝒞 𝜔 and 𝒞 𝑔 can perform as an indicator of the intransitivity of strong links. Ritualized embraces may obey a principle of preferential attachment to keystone individuals [START_REF] Range | Can simple rules account for the pattern of triadic interactions in juvenile and adult female sooty mangabeys?[END_REF][START_REF] Schino | Grooming, competition and social rank among female primates: A meta-analysis[END_REF], which would create strong triplets featuring a keystone in the middle, closed by weak or no links between the subordinates. The rudimentary permutation tests A and B are however not suited to formally test such hypotheses. Here also, formal tests would require dynamic data rather than aggregated data. Nevertheless, the statistics developed for this study captured information not summarized in other network-level metrics (Appendix S3). They can therefore help quantify the socio-spatial structure of different species and populations, the extent to which this structure favors the evolution of cooperation, and the social network response to changes that influence the costs and benefits of sociality. Tables Table 1: Overall results of the weighted network analysis for the 26 species. Given are the taxonomic order, the type of association (ty = GM: shared subgroup membership, PR: spatial proximity, CO: physical contact), the number of study individuals N, the edge density 𝒟 0 , the modularity ℳ, the variance in rescaled edge weights Var, the Gini coefficient of inequality in link strength G, the fragmentation ratio FR quantifying the fragmentation of the subnetwork of strong links relative to the network as a whole, the unweighted transitivity coefficient 𝒞 0 , the transitivity coefficient weighted for triplet strength 𝒞 1 , and the triadic dissimilarity scores 𝒞 𝜔 and 𝒞 𝑔 . "q." gives the quadrants that

  Test A: I generated full random networks by drawing 10 groups with individual probability of occurrence proportional to their frequency in the original data and computing the link strength as the co-occurrence frequency in the 10 groups. Thus, the expected association strength between two individuals A and B was the frequency of A times the frequency of B. Test B: To represent the edge density of the original network, I removed some links at random. To do that, I drew an Erdős-Rényi graph with the same edge density as the original network. I then assigned link strengths line by line in the adjacency matrix, by drawing them from the observed link strengths departing from the focal individual. I then rescaled the link strengths so that the sum of the link strengths in each line was proportional to the observation frequency of the focal individual in the original dataset. This process generated a random matrix with an excess of zeros but the same magnitude of across-and within-row variation in link strength as the original data. Then I drew 100 groups by first drawing an individual and then its associates based on that matrix. From these 100 groups, I computed dyadic co-occurrence frequencies. These frequencies represented the final link strength of the simulation. Compared to the initial Erdős-Rényi graph, some links were created and some disappeared during the group drawing process. In both tests, I computed the Mahalanobis distance between the simulated and observed 𝒞 𝜔 and 𝒞 𝑔 scores. The Mahalanobis distance M measures the distance between a point P (here, the two observed dissimilarity scores 𝒞 𝜔 and 𝒞 𝑔 ) and a distribution characterized by a mean P0 and a variance-covariance matrix S, according to the formula 𝑀 = √(𝑃 -𝑃 0 ) 𝑇 • 𝑆 -1 • (𝑃 -𝑃 0 ). I computed P0 and S from the cloud of simulated 𝒞 𝜔 and 𝒞 𝑔 values. I then assessed the Mahalanobis distance against the chi-squared distribution with two degrees of freedom.

Fig. 1 :

 1 Fig. 1: (a) Model predictions illustrating that, when an increasing proportion of the weakest links is removed (x-axis), the network transitivity 𝒞 0 is increasingly larger than expected from the edge density 𝒟 0 (y-axis). (b) Raw data for a single population. The data document group membership in a killer whale (Orcinus orca) population (Weiss et al. 2020). Thick black lines indicate the 10% strongest links.

Fig. 3 :

 3 Fig. 3: (a) Graph-level triadic dissimilarity scores 𝒞 𝑔 and 𝒞 𝜔 in 26 species (see methods for the definition). The color scale corresponds to the edge density 𝒟 0 . The grey outline delineates the domain of possible (𝜔, 𝑔) values. The pictograms represent the six extreme cases of triads. Triplets correspond to triads where one individual A is associated with two different individuals B and C. If the triplet is closed by a link between B and C, it is called a triangle. If the three links in the triangle are equally strong, the triad falls in the lowerleft corner of the plot. Different triangular configurations correspond to different other sections of the plot, a delineated by the four quadrants 1 to 4. A false triplet is a triad made of a dyad plus a disconnected singleton, corresponding to the rightmost part of the plot. (b) Triad-level dissimilarity scores summarized over two categories of edge density and over the four quadrants. The black outline indicates the expected distribution from test C, demonstrating the lack of "forbidden triads" (quadrant 3) and the excess of triangles where the closing link is strong (quadrant 1 and quadrant 2).

Fig. 1b :

 1b Fig.1b: average fragmentation ratio 0.35). The fragmentation ratio increased with edge density (Z = 2.3, r² = 0.09,

  were more frequent than expected in test C. "Ref." is the original study and "ASNR" is the reference number in "A social network repository"[START_REF] Sah | A multi-species repository of social networks[END_REF]; https://bansallab.github.io/asnr/about.html).

	Trachypithecu s johnii	Nilgiri langur Primates	PR 10	0.29	0.2	0.1404 0.17	0.33	0	0	0.11	0.94	0.001	0.35	0.70	Nunn, 2012) -(Griffin	&	375
	Papio cynocephalus	Yellow baboon Primates	PR 16 (10; 25)	0.28 (0.10; 0.60)	0.28 (0.05; 0.46)	0.0297 0.1205) (0.0086;	0.08 0.15) (0.03-	0.21 (0-1)	0.27 (0; 0.52)	0.27 (0; 0.51)	0.18 0.32) (0.09;	0.85 0.95) (0.7;	0 -0.02 0.08-0.34	0.56-0.99	-(Franz et al., 2015)	261-285
	Macaca assamensis	Assam macaque	Primates	PR 13	0.95	-0.03	0.0246 0.6	1	0.95	0.95	-0.29	0.27	0.06	<0.001 0.47	Gonzalez et -(Puga-	765
																	al., 2018)
	Species fuscata Macaca	macaque Japanese	Tax. Order Primates	ty N PR 11	𝓓 𝟎 0.98	𝓜 -0.09	Var 0.0537 0.38 G	FR 0	𝓒 𝟎 0.98	𝓒 𝟏 0.99	𝓒 𝝎 -0.09	𝓒 𝒈 0.38	Test A Test B Test C q. Ref. Gonzalez et 0.07 0.001 0.89 -(Puga-	ASNR 760
	Poecilia reticulata Macaca tonkeana Melanerpes formicivorus Cacatua galerita Otospermophi lus beecheyi Syncerus caffer Acanthiza reguloides Zonotrichia atricapilla Cervus canadensis Orcinus orca Killer whale Trinidad guppy Tonkean macaque Acorn woodpecker Sulphur-crested cockatoo California ground-squirrel African buffalo Cetartiodactyl Cyprinodontif ormes Primates Piciformes Psittaciformes GM 77 PR 8.7 (8; 10) CO 25 GM 58 (69; 88) Rodentia PR 60.5 (60; 61) a GM 64 Buff-rumped thornbill Passeriformes GM 62 Golden-crowned sparrow Passeriformes GM 37 (27; 46) Elk Cetartiodactyl a PR 26 (23; 29) Cetartiodactyl a PR 72 Hirundo rustica Rustic swallow Passeriformes CO 17 (17; 17) Sousa sahulensis Australian humpback-Cetartiodactyl a GM 50 dolphin Macropus giganteus Eastern grey-kangaroo Marsupialia PR 17 Loxodonta africana African elephant Proboscidea GM 171 (134-122) Tursiops truncatus Bottlenose dolphin Cetartiodactyl a PR 189 (188; 190) Equus Onager Perissodactyla GM 350 hemionus Procavia capensis Rock hyrax Hyracoidea PR 32 (20; 44) Equus grevyi Grevy's zebra Perissodactyla GM 81 Ateles hybridus Brown spider-monkey Primates CO 17 Crocuta crocuta Spotted hyena Carnivora GM 35 (35; 36) Brachyteles arachnoides Southern muriqui Primates PR 22 Zalophus Galápagos Carnivora GM 405 wollebaeki sea-lion	0.97 (0.91; 1) 0.60 0.42 0.20 (0.13; 0.37) 0.17 (0.14; 0.19) 0.70 0.61 0.40 (0.34; 0.47) 0.26 (0.18; 0.37) 0.70 0.64 (0.39; 0.90) 0.74 0.67 0.03 (0.02; 0.04) 0.06 (0.06; 0.06) 0.15 0.31 (0.16; 0.52) 0.24 0.53 0.89 (0.86; 0.93) 0.72 0.09	-0.07 (-0.10; -0.05) 0.06 0.07 0.27 (0.12; 0.39) 0.36 (0.34; 0.39) 0.12 0.05 0.23 (0.22; 0.24) 0.12 (0.11; 0.12) 0.05 0.06 (-0.03; 0.14) 0.09 0.03 0.48 (0.31; 0.61) 0.44 (0.4; 0.48) 0.42 0.35 (0.10; 0.50) 0.58 0.01 0.02 (0; 0.03) 0.01 0.51	0.1003 (0.0382; 0.2035) 0.0074 0.48 0.35 (0.31-0.43) 5*10 -4 0.9 0.0035 (5*10 -4 ; 0.47 (0.38-0.53) 0.45 0.0059 (0.42-(0.0034; 0.48) 0.0083) 0.0684 0.53 0.0067) 0.0025 0.4 0.0125 (0.006; 0.5 (0.46-0.53) 0.77 0.0084 (0.73-(0.0046; 0.8) 0.0143) 0.0048 0.64 0.0189) 0.0087 (0.0063; 0.011) 0.37 (0.31-0.43) 0.0295 0.58 0.0089 0.57 0.0018 (9*10 -4 ; 0.0028) 0.2 (0.12-0.25) 0.11 0.0022 (0.1-(0.0018; 0.0026) 0.13) 0.118 0.37 0.0441 (0.0052; 0.1679) 0.35 (0.29-0.48) 0.1718 0.3 0.0325 0.63 0.4 (0.4-0.0058 0.41) (0.0046; 0.0072) 0.0097 0.49 6*10 -4 0.41	0.67 (0-1) 0.5 -0.21 0.19 (-0.09-0.41) 0.29 (0.15-0.43) 0.5 0.67 0.42 (0.33-0.5) 0.93 (-0.27-2.08) 1.5 0.25 (0-0.5) 0.5 1.27 -0.01 (-0.08-0.05) 0.06 (-0.08-0.2) 0.2 0.16 (-0.33-0.57) 0.2 0.88 0.32 (-0.03-1) 2 0.14	0.97 (0.92; 1) 0.63 0.78 0.60 (0.54; 0.64) 0.46 (0.44; 0.48) 0.84 0.76 0.64 (0.58; 0.69) 0.61 (0.56; 0.67) 0.79 0.70 (0.47; 0.92) 0.85 0.84 0.33 (0.32; 0.36) 0.52 (0.52; 0.53) 0.65 0.77 (0.5; 0.91) 0.90 0.72 0.92 (0.89; 0.96) 0.85 0.40	0.98 (0.94; 1) 0.63 0.78 0.61 (0.55; 0.65) 0.48 (0.45; 0.50) 0.90 0.77 0.67 (0.61; 0.74) 0.63 (0.6; 0.68) 0.80 0.70 (0.48; 0.93) 0.89 0.86 0.33 (0.32; 0.36) 0.53 (0.52; 0.54) 0.82 0.82 (0.54; 0.99) 0.78 0.74 0.93 (0.9; 0.97) 0.85 0.41	-0.02 (-0.13; 0.16) 0.2 -0.03 0.08 (0.04; 0.12) 0.09 (0.09; 0.09) -0.05 0.04 0.12 (0.1; 0.13) 0.09 (0.05; 0.12) -0.04 -0.06 (-0.36; 0.23) -0.02 -0.10 0.04 (0.02; 0.05) 0.03 (0.03; 0.04) 0.04 0.02 (-0.07; 0.12) 0.03 0.11 -0.32 (-0.43; -0.22) -0.10 0.06	0.48 (0.36; 0.59 0.68) 0.68 0.86 (0.74; 0.91) 0.90 (0.89; 0.91) 0.68 0.54 0.76 (0.75; 0.78) 0.82 (0.74; 0.89) 0.46 0.49 (0.24; 0.95 0.74) 0.49 0.97 (0.96; 0.96 (0.95; 0.96) 0.94 0.98) 0.86 (0.7; 0.93) 0.91 0.67 0.27 (0.23; 0.30) 0.45 0.94	0.06-0.24 <0.001 <0.001 0.42 0.004-0.03 0.38-0.45 <0.001 <0.001 <0.001 Q1 (Shizuka et al., al., 2018) -(Krause et al., 2017) -(Griffin & Nunn, 2012) 2022) <0.001 <0.001 <0.001 Q1 (Aplin et al., <0.001 0.004 <0.001 Q1-(Smith et al., Q2 2018) 2021) <0.001 <0.001 <0.001 Q2 (Cross et al., 2004) <0.001 <0.001 <0.001 Q1-Q2 (Farine & Milburn, 2013) <0.001 <0.001 <0.001 Q1-Q2 (Shizuka et al., 2014) 0 -0.02 0.03-<0.001 Q3 (Webber & 0.38 Vander Wal, 2020) <0.001 <0.001 <0.001 Q2 (Weiss et al., 2020) 0.002-0.005 0.06-0.45 0-0.3 Q1 (Levin et al., 2016) <0.001 <0.001 <0.001 Q2 (Hunt et al., 2019) <0.001 <0.001 <0.001 Q1-Q2 (Grant, 1973) 360 369 744,746 451 441-442 319-731 320 <0.001 Q1-Q3 <0.001 <0.001 <0.001 Q1 (Gazda et al., 336,339 2015) (Murphy et al., * * * * (Rubenstein et 2020) al., 2015) 0 -0.08 <0.001 <0.001 Q2 (Barocas et al., <0.001 <0.001 <0.001 Q1-(Rubenstein et Q2 al., 2015) 2011) <0.001 <0.001 0.002 Q1-Q2 (Rimbach et <0.001 <0.001 <0.001 Q1-(Holekamp et 357-Q2 al., 2012) 359 al., 2015) <0.001 <0.001 <0.001 Q1-Q2 (Griffin * * * * (Wolf et al., & Nunn, 2012) 2007)

Funding: This research received no external funding.

Ethical statement: This study uses published data only. Original data were collected in accordance with relevant institutional and national guidelines, as explained in the source articles listed in Table 1.

Data Availability Statement: This manuscript does not use new data. All data sources are listed in Table 1.

Acknowledgments: I thank all the people who contributed to the collection and curation of the association network data.

Author Contributions: NA

Conflicts of Interest:

The authors declare no conflict of interest