Maxime Derex
email: maxime.derex@iast.fr

Alex Mesoudi

Cumulative cultural evolution within evolving population structures

Keywords: cultural evolution, cumulative culture, demography, group problem-solving, population size, population structure, social learning

Our species possesses the peculiar ability to accumulate cultural innovations over multiple generations, a phenomenon termed cumulative cultural evolution (CCE). Recent years have seen a proliferation of empirical and theoretical work exploring the interplay between demography and CCE. This has generated intense discussion about whether demographic models can help explain historical patterns of cultural changes. Here, we synthesise empirical and theoretical studies from multiple fields to highlight how both population size and structure shape the pool of cultural information that individuals can build upon to innovate, present the potential pathways through which humans' unique social structure might promote CCE, and discuss whether humans' social networks might partly result from selection pressures linked to our extensive reliance on culturally accumulated knowledge.

Problem-solving in populations over multiple generations

. This capacity can be viewed as a form of problem-solving by which humans have successfully solved complex ecological challenges.

.

.

. While this research has advanced our understanding of the link between demography and CCE and opened up promising new

avenues, it has also revealed a need to better articulate empirical research and theoretical models. Here we present the theory, discuss misconceptions, outline future challenges, and highlight new directions in research on demography and CCE.

Strength in numbers

Demography has long been considered a potential explanation for cultural changes documented in the archaeological record [START_REF] Jones | Emerging Picture of Pleistocene Australians[END_REF][START_REF] Cavalli-Sforza | Models for cultural inheritance I. Group mean and within group variation[END_REF][START_REF] Neiman | Stylistic Variation in Evolutionary Perspective: Inferences from Decorative Diversity and Interassemblage Distance in Illinois Woodland Ceramic Assemblages[END_REF], but it is with the theoretical work of Shennan [START_REF] Shennan | Demography and Cultural Innovation: a Model and its Implications for the Emergence of Modern Human Culture[END_REF] and Henrich [START_REF] Henrich | Demography and cultural evolution: How adaptive cultural processes can produce maladaptive losses -The Tasmanian case[END_REF] that the idea gained prominence among evolutionary human scientists. The main idea behind demographic models of cultural evolution is that, given that CCE only operates when at least some information is transmitted socially between generations [START_REF] Tomasello | Cultural Learning[END_REF][START_REF] Tennie | Ratcheting up the ratchet: on the evolution of cumulative culture[END_REF][START_REF] Lewis | Transmission fidelity is the key to the buildup of cumulative culture[END_REF], the effective population size (which depends on both population size and interconnectedness) can buffer the risk of losing cultural information (see Box 1). In Henrich's seminal model [START_REF] Henrich | Demography and cultural evolution: How adaptive cultural processes can produce maladaptive losses -The Tasmanian case[END_REF], for instance, individuals belong to a population of constant size and possess a psychological propensity to learn from successful individuals. This propensity creates a selective force that promotes the transmission of beneficial cultural traits and outweighs the degrading effects of learning errors when populations are large enough (Figure 1). These results suggest that decreases in effective population size (due to phenomena such as plagues, war or volcanic eruptions) might result in losses in individuals' level of skills (often proxied in the archaeological literature as the number of tools, or toolkit complexity) by constraining CCE. Several regional losses of cultural traits documented in the archaeological record, such as prehistoric Tasmania, have consequently been attributed to decreases in population size and connectedness [START_REF] Henrich | Demography and cultural evolution: How adaptive cultural processes can produce maladaptive losses -The Tasmanian case[END_REF][START_REF] Riede | The loss and re-introduction of bow-and-arrow technology: a case study from the Northern European Late Paleolithic[END_REF]. Conversely, the emergence of more complex cultural traits have been hypothesized to result from increases in population sizes and/or densities [START_REF] Powell | Late Pleistocene Demography and the Appearance of Modern Human Behavior[END_REF][START_REF] Marquet | Emergence of social complexity among coastal hunter-gatherers in the Atacama Desert of northern Chile[END_REF].

Experimental tests of the relationship between population size and CCE

One approach that has been used to evaluate the plausibility of demographic models of CCE involves lab experiments. Typically, participants who are part of groups of different sizes are tasked to improve a piece of technology. To date, 5 experiments from 4 different research groups provide support for a positive effect of group size on cultural complexity [START_REF] Derex | Experimental evidence for the influence of group size on cultural complexity[END_REF][START_REF] Wisdom | Social Learning Strategies in Networked Groups[END_REF][START_REF] Muthukrishna | Sociality influences cultural complexity[END_REF][START_REF] Kempe | An experimental demonstration of the effect of group size on cultural accumulation[END_REF][START_REF] Derex | The foundations of the human cultural niche[END_REF] (but see [START_REF] Caldwell | Human cumulative culture in the laboratory: Effects of (micro) population size[END_REF][START_REF] Fay | Increasing population size can inhibit cumulative cultural evolution[END_REF]). One study, for instance, exposed naïve participants in groups of 2, 4, 8 and 16 to demonstrations showing how to produce virtual arrowheads and fishing nets, and tracked the efficiency of those tools across time [START_REF] Derex | Experimental evidence for the influence of group size on cultural complexity[END_REF]. The larger the group, the less likely tools were to deteriorate, the more likely they were to improve, and the more likely a diversity of tool types were to be maintained. Using chains of participants and alternative tasks involving image-editing and knot-tying techniques, another study similarly showed that the deterioration of a technique is less likely (and its improvement more likely) in larger groups [START_REF] Muthukrishna | Sociality influences cultural complexity[END_REF]. Additionally, these experiments show that individuals use cues such as success to choose from whom they learn, lending plausibility to the assumption of Henrich's model that individuals selectively learn from successful demonstrators.

Importantly, some of these experiments relied on designs that only loosely reflect Henrich's initial assumptions (Box 2). Most, for instance, provide individuals with the opportunity to simultaneously learn and combine information from multiple demonstrators (a several-among-many design) [START_REF] Wisdom | Social Learning Strategies in Networked Groups[END_REF][START_REF] Muthukrishna | Sociality influences cultural complexity[END_REF][START_REF] Kempe | An experimental demonstration of the effect of group size on cultural accumulation[END_REF][START_REF] Derex | The foundations of the human cultural niche[END_REF] while Henrich's model assumes that individuals always select a single source of information from a larger pool of demonstrators. Some experiments that have relied on the former design, however, allowed participants to allocate their learning time strategically, which means that individuals' learning strategies might still, in practice, be consistent with Henrich's assumptions [START_REF] Muthukrishna | Sociality influences cultural complexity[END_REF]. Yet mechanisms that are not part of Henrich's model, such as combining information from multiple demonstrators to generate new solutions, certainly did play a role in these experiments [START_REF] Muthukrishna | Sociality influences cultural complexity[END_REF][START_REF] Kempe | An experimental demonstration of the effect of group size on cultural accumulation[END_REF][START_REF] Derex | The foundations of the human cultural niche[END_REF]. Due to this disconnect between experimental tests and theoretical models, it is not always clear whether experimental studies showing positive effects of demography offer genuine support for specific theoretical claims, nor whether purported failures to detect any effect of demography are valid challenges to theoretical models (see Box 2 for further discussion).

Real-world tests of the relationship between population size and CCE

A complementary and more direct approach to test the relationship between population size and CCE is to look for a correlation between toolkit size and population size using real-world ethnographic and archaeological data. Results with this approach have been mixed. Some studies support the hypothesis [START_REF] Powell | Late Pleistocene Demography and the Appearance of Modern Human Behavior[END_REF][START_REF] Marquet | Emergence of social complexity among coastal hunter-gatherers in the Atacama Desert of northern Chile[END_REF][START_REF] Kline | Population size predicts technological complexity in Oceania[END_REF][START_REF] Collard | Population Size and Cultural Evolution in Nonindustrial Food-Producing Societies[END_REF], but others do not [START_REF] Collard | Causes of Toolkit Variation Among Hunter-Gatherers: A Test of Four Competing Hypotheses[END_REF][START_REF] Collard | Risk, mobility or population size? Drivers of technological richness among contact-period western North American hunterâ€"gatherers[END_REF][START_REF] Collard | Population Size as an Explanation for Patterns in the Paleolithic Archaeological Record: More Caution Is Needed[END_REF][START_REF] Buchanan | Drivers of technological richness in prehistoric Texas: an archaeological test of the population size and environmental risk hypotheses[END_REF] (although [START_REF] Henrich | Understanding cumulative cultural evolution[END_REF] point out that some of these studies rely on the same datasets, and should not count as independent tests).

The difficulty with testing demographic models using real-world data is that human populations are typically embedded within extended networks of cultural exchange, making it difficult to gather meaningful estimates of population size. This constitutes a major obstacle for anthropologists and archaeologists because theoretical models explicitly link cultural complexity to the size of the population that shares information (i.e. the effective cultural population size) [START_REF] Henrich | Demography and cultural evolution: How adaptive cultural processes can produce maladaptive losses -The Tasmanian case[END_REF]. This implies that tests of demographic hypotheses should control for contact rates between inter-connected populations, which is typically challenging (but see [START_REF] Kline | Population size predicts technological complexity in Oceania[END_REF]). Proponents of demographic hypotheses have therefore argued that studies which reported null results are invalid because they do not take contact rates into account and typically treat culturally connected groups as independent, culturally isolated populations [START_REF] Henrich | Understanding cumulative cultural evolution[END_REF] (see Box 3 for other mismatches between models and empirical tests).

Other studies have tested demographic effects where they may not be predicted to occur. One study, for instance, found no evidence that larger populations support more complex folk tales, with complexity operationalised as number of tale types, number of narrative motifs within tales, and number of component details within tales [START_REF] Acerbi | Cultural complexity and demography: The case of folktales[END_REF]. Yet folk tales are very different to the technology that is the focus of most demographic models.

Tools that are more efficient and have higher payoffs are typically associated with an increasing number of component elements [START_REF] Arthur | The nature of technology: What it is and how it evolves[END_REF], which means that they tend to be more complex. However, if complexity is not associated with higher payoffs, then theoretical models do not predict that population size should necessarily affect it. The function of folktales, for instance, is to convey meaning. If similar meaning can be conveyed by simpler folktales, we should not necessarily expect to observe the most complex folktales in larger populations. The same line of reasoning applies to the evolution of language, which functionally adapts to the needs of efficient communication [START_REF] Nettle | Segmental inventory size, word length, and communicative efficiency[END_REF]. Studies that have investigated the relationship between speaker population sizes and phoneme inventory sizes [START_REF] Hay | Phoneme Inventory Size and Population Size[END_REF][START_REF] Moran | Revisiting population size vs. phoneme inventory size[END_REF][START_REF] Wichmann | Phonological diversity, word length, and population sizes across languages: The ASJP evidence[END_REF] or rates of language change [START_REF] Wichmann | Population Size and Rates of Language Change[END_REF][START_REF] Bromham | Rate of language evolution is affected by population size[END_REF][START_REF] Greenhill | Population Size and the Rate of Language Evolution: A Test Across Indo-European, Austronesian, and Bantu Languages[END_REF] have yielded mixed results. However, because language also evolves to become more learnable [START_REF] Kirby | Cumulative cultural evolution in the laboratory: An experimental approach to the origins of structure in human language[END_REF], we should not necessarily expect larger populations to produce more new words nor have larger phoneme inventory size.

Furthermore, folk tales and other forms of expressive culture may serve as markers of group membership and some models have suggested that smaller groups will have more exaggerated markers [START_REF] Mcelreath | Shared norms and the evolution of ethnic markers[END_REF]. This suggests that a clearer picture about the relationship between demography and the evolution of expressive cultural traits might emerge by moving away from arbitrarily chosen measures of complexity and by taking into account that functional and symbolic cultural traits exhibit different evolutionary dynamics [START_REF] Rogers | Natural selection and cultural rates of change[END_REF].

It is also worth stressing that, contrary to recent claims [START_REF] Vaesen | Population size does not explain past changes in cultural complexity[END_REF], no theoretical work ever predicted that population size should solely determine the number of tools (or any other measure of cultural complexity) found in human populations. Many factors are expected to affect toolkit complexity in natural populations, including mobility, subsistence practices and ecological factors. The risk hypothesis, for instance, holds that populations living in harsh environments create more numerous and specialised tools to mitigate the risk of resource failure due to stochastic variation [36-39, 54, 55]. Importantly, the risk hypothesis and the population size hypothesis differ in what they aim to explain [START_REF] Grove | Hunter-gatherers adjust mobility to maintain contact under climatic variation[END_REF]. The risk hypothesis explains what determines the size and complexity of toolkits (i.e. what creates the need for cultural complexity). The population size hypothesis is about the constraints imposed on CCE.

Claims that the absence of correlation between population size and toolkit complexity disprove demographic models are based on misconceptions about those models (see Box 3).

Inconclusive studies about the relationship between population size and CCE have had the merit of stimulating new work and led to important refinements to early theoretical work. Models with different assumptions have shown that the effects of effective population size hold when more conservative or alternative assumptions are considered (e.g. restricting potential demonstrators to a limited number of acquaintances [START_REF] Kobayashi | Innovativeness, population size and cumulative cultural evolution[END_REF]; conformist transmission [START_REF] Aoki | On the absence of a correlation between population size and 'toolkit size' in ethnographic hunter-gatherers[END_REF][START_REF] Lehmann | On the number of independent cultural traits carried by individuals and populations[END_REF] but see [START_REF] Vaesen | Cumulative Cultural Evolution and Demography[END_REF]; adding costs to acquiring knowledge [START_REF] Mesoudi | Variable Cultural Acquisition Costs Constrain Cumulative Cultural Evolution[END_REF]; and alternative pathways to innovation [START_REF] Kolodny | Evolution in leaps: The punctuated accumulation and loss of cultural innovations[END_REF]). However, recent studies also suggest that the relationship between effective population size and CCE can be mediated by numerous factors ( [START_REF] Aoki | On the absence of a correlation between population size and 'toolkit size' in ethnographic hunter-gatherers[END_REF][START_REF] Kolodny | Evolution in leaps: The punctuated accumulation and loss of cultural innovations[END_REF][START_REF] Fogarty | The niche construction of cultural complexity: interactions between innovations, population size and the environment[END_REF][START_REF] Fogarty | The Driving Forces of Cultural Complexity[END_REF][START_REF] Aoki | Cultural bistability and connectedness in a subdivided population[END_REF][START_REF] Bentley | The selectivity of social learning and the tempo of cultural evolution[END_REF]), and that there are numerous challenges in detecting demographic effects on CCE in real-world data (see Box 3). Despite these challenges, there is little doubt that changing the effective size of a population will alter the cultural information available to subsequent generations of learners, which will most likely constrain what can be achieved by individuals. In this context, promising new work has started to investigate more broadly how constraints on information flow within populations can further promote or hinder the gradual accumulation of cultural innovations.

Beyond numbers: CCE in social networks

Human populations do not consist of a collection of isolated groups of varying sizes. Multiple groups are typically connected by migratory and trade activities, which results in wide, heterogenous social networks. The role of connectedness on CCE was already acknowledged in early theoretical models [START_REF] Henrich | Demography and cultural evolution: How adaptive cultural processes can produce maladaptive losses -The Tasmanian case[END_REF][START_REF] Powell | Late Pleistocene Demography and the Appearance of Modern Human Behavior[END_REF]. A simulation model that explicitly implemented migratory activity among subdivided populations, for instance, showed that increasing the migration rate has a similar effect to increasing the size of an isolated population [START_REF] Powell | Late Pleistocene Demography and the Appearance of Modern Human Behavior[END_REF]. This is because increases in both population size and migratory activity increase the effective number of individuals available as demonstrators, and so reduce the risk of losing cultural information.

More recent work, however, has started to investigate in greater detail how the structure of the population impacts the accumulation of cultural information. Unlike early models, recent studies decouple the maintenance of existing traits and the production of new traits, more explicitly modelling the pathways that give rise to innovation [START_REF] Kolodny | Evolution in leaps: The punctuated accumulation and loss of cultural innovations[END_REF][START_REF] Derex | Partial connectivity increases cultural accumulation within groups[END_REF][START_REF] Creanza | Greater than the sum of its parts? Modelling population contact and interaction of cultural repertoires[END_REF][START_REF] Derex | Divide and conquer: intermediate levels of population fragmentation maximize cultural accumulation[END_REF].

Recent models, for instance, assume that existing traits can not only be refined but also combined with other existing cultural traits. When recombination between existing traits is incorporated as a pathway towards innovation, increases in population size and connectedness can have different effects on CCE [START_REF] Creanza | Greater than the sum of its parts? Modelling population contact and interaction of cultural repertoires[END_REF][START_REF] Derex | Divide and conquer: intermediate levels of population fragmentation maximize cultural accumulation[END_REF]. This is because, while increases in population size systematically benefit CCE by reducing the risk of cultural loss, increases in connectedness can reduce opportunities for innovation by homogenising cultural behaviours. This effect is illustrated by a recent lab experiment in which individuals could innovate by producing incremental changes within path-dependent technological trajectories (refinement) and by combining traits that have evolved along different trajectories (recombination) [START_REF] Derex | Partial connectivity increases cultural accumulation within groups[END_REF]. Results show that high levels of connectedness make individuals more likely to converge on similar solutions, which results in lower levels of cultural diversity and slower rates of innovation compared with less connected groups.

These results suggest that understanding the effect of demography on CCE requires us to consider not only how changes in connectedness affect the number of individuals available as demonstrators, but also how it shapes the cultural diversity to which individuals are exposed. When these two effects are considered simultaneously, models show that optimal rates of accumulation are reached for intermediate levels of connectedness [START_REF] Creanza | Greater than the sum of its parts? Modelling population contact and interaction of cultural repertoires[END_REF][START_REF] Derex | Divide and conquer: intermediate levels of population fragmentation maximize cultural accumulation[END_REF]. This is because low levels of connectedness increase the risk of cultural loss by decreasing access to demonstrators, while high levels of connectedness reduce opportunities to innovate by homogenising cultural behaviours. At intermediate levels of connectedness, groups can accumulate cultural information while remaining culturally distinct, which keeps fueling innovation.

These results have implications for CCE both at the macroscale and the microscale. At the macroscale, human population have been historically fragmented due to geographic barriers, conflicts and other factors, resulting in long-standing culturally differentiated subpopulations. In this context, increased levels of between-group connectedness are unlikely to homogenise cultural behaviours. Nevertheless, recent models suggest that, because of new opportunities for recombination, contacts between culturally differentiated groups should result in rapid cultural changes whose magnitude far exceed what is predicted by models that incorporate cultural loss alone [START_REF] Creanza | Greater than the sum of its parts? Modelling population contact and interaction of cultural repertoires[END_REF]. This also suggests that population structures that allow for contacts between culturally differentiated groups might act as endogenous drivers of cultural change [START_REF] Derex | Partial connectivity increases cultural accumulation within groups[END_REF][START_REF] Creanza | Greater than the sum of its parts? Modelling population contact and interaction of cultural repertoires[END_REF], even though it should not be assumed that populations will develop and maintain more complex cultural repertoires without appropriate incentives to do so (Box 3).

Patterns of connectedness might also affect CCE at the microscale by influencing individuals' exploration of the design space. Network and organization scientists, for instance, have jointly shown that behaviours are more likely to become homogeneous in well-connected than in partially-connected groups when learners preferentially acquire information from the same demonstrator [START_REF] Lazer | The Network Structure of Exploration and Exploitation[END_REF][START_REF] Fang | Balancing Exploration and Exploitation Through Structural Design: The Isolation of Subgroups and Organizational Learning[END_REF][START_REF] Mason | Propagation of innovations in networked groups[END_REF] (but see [START_REF] Mason | Collaborative learning in networks[END_REF][START_REF] Barkoczi | Social learning strategies modify the effect of network structure on group performance[END_REF]). Sociologists have similarly argued that behaviors tend to be more homogeneous within groups than between groups and that individuals with ties to otherwise unconnected groups have greater opportunities to develop new ideas because they are exposed to a broader diversity of information [START_REF] Burt | Structural Holes and Good Ideas[END_REF].

These studies illustrate how patterns of connectedness impact the quantity and diversity of information that individuals are exposed to and can draw on to make inferences, which in turn can impact populations' abilities to develop and maintain cultural traits. The benefits of sparsely interconnected networks on CCE in natural populations, however, remain to be properly evaluated. Complex cultural traits are typically hard to learn and several experiments have stressed the importance of multiple demonstrations and multiple learning attempts in the acquisition of complex skills [START_REF] Derex | Experimental evidence for the influence of group size on cultural complexity[END_REF][START_REF] Flynn | Studying children's social learning experimentally "in the wild[END_REF]. This suggests that occasional contacts between different individuals/groups might not allow complex skills to spread properly.

Additionally, network scientists have stressed the importance of the number of sources of exposures for the adoption of unproven new solutions [START_REF] Centola | Complex Contagions and the Weakness of Long Ties[END_REF]. Experiments typically provide participants with accurate information about alternative solutions, which allows them to confidently adopt the most rewarding ones. In noisy environments, however, interactions with multiple carriers might be critical for individuals to adopt alternative solutions [START_REF] Centola | Complex Contagions and the Weakness of Long Ties[END_REF] (see also [START_REF] Carja | The evolutionary advantage of cultural memory on heterogeneous contact networks[END_REF] for an example of how the mean number of connections within a network affects the spread of cultural traits). Future research should test whether the optimal level of connectedness differs depending upon the characteristics of the cultural traits one is looking at. Dense networks, for instance, might be critical for the cultural evolution of hard to learn traits (for which transmission is the key bottleneck), while the cultural evolution of easy to learn traits whose efficiency can be readily assessed might be faster in sparsely connected networks.

Characterizing human social networks in the wild

The effects of population interconnectedness on CCE suggests that cultural changes might be better understood by paying greater attention to the structure and evolution of human social networks. Mapping past, or even recent, social networks, however, is challenging.

Archaeologists and geneticists are still struggling to infer past population sizes [START_REF] French | Demography and the Palaeolithic Archaeological Record[END_REF][START_REF] Malinsky-Buller | One size does not fit all: Group size and the late middle Pleistocene prehistoric archive[END_REF][START_REF] Mazet | On the importance of being structured: instantaneous coalescence rates and human evolution-lessons for ancestral population size inference?[END_REF], let alone population structures [START_REF] Kovacevic | Simulating Geographical Variation in Material Culture: Were Early Modern Humans in Europe Ethnically Structured? In Learning Strategies and Cultural Evolution during the Palaeolithic[END_REF]. In recent years, approaches relying on social network analyses have seen a rise among archaeologists, but many challenges have still to be solved before being able to distinguish spatio-temporal patterns in social interactions from noise in archaeological data [START_REF] Mills | Social Network Analysis in Archaeology[END_REF][START_REF] Peeples | Finding a Place for Networks in Archaeology[END_REF][START_REF] Romano | A multilevel analytical framework for studying cultural evolution in prehistoric hunter-gatherer societies[END_REF].

Comparative and ethnographic studies, however, are already providing valuable information about human population structure. Comparisons between human huntergatherer societies and non-human primate societies, for instance, have shed light on what has been called the deep social structure of human societies [START_REF] Chapais | Monogamy, strongly bonded groups, and the evolution of human social structure[END_REF]. Contrary to most non-human primate societies, which are composed of independent, single-group structures, human societies are federations of multifamily groups [START_REF] Chapais | Monogamy, strongly bonded groups, and the evolution of human social structure[END_REF][START_REF] Hill | Co-Residence Patterns in Hunter-Gatherer Societies Show Unique Human Social Structure[END_REF]. This unique multigroup structure results in extensive networks of unrelated individuals that might be conducive to CCE [START_REF] Hill | Hunter-Gatherer Inter-Band Interaction Rates: Implications for Cumulative Culture[END_REF]. Data on interactions between same-sex adults from two hunter-gatherer populations, for instance, reveal that individuals typically interact with more than 300 samesex adults in a lifetime (although including opposite-sex adults and children results in estimates as high as 1000). In comparison, male chimpanzees are estimated to interact with only about 20 other males in a lifetime [START_REF] Hill | Hunter-Gatherer Inter-Band Interaction Rates: Implications for Cumulative Culture[END_REF] (see also [START_REF] Bird | Variability in the organization and size of hunter-gatherer groups: Foragers do not live in small-scale societies[END_REF] for a discussion on the large-scale social networks of hunter-gatherer groups).

Other studies among hunter-gatherer populations have started to more finely characterize hunter-gatherer networks. One study, for instance, used trackers to map incamp networks in two hunter-gatherer populations and showed that individuals invest early in their childhood in a few close friends who bridge densely connected families [START_REF] Migliano | Characterization of hunter-gatherer networks and implications for cumulative culture[END_REF]. These strong friendships increase the global efficiency of hunter-gatherer in-camp networks, which might facilitate the flow of social information (Figure 2). More recently, characterization of hunter-gatherer networks has been extended to between-camp interactions and has been used to simulate the accumulation of cultural innovations over real networks [START_REF] Migliano | Hunter-gatherer multilevel sociality accelerates cumulative cultural evolution[END_REF]. Results confirm that hunter-gatherers' social structures are made of multiple levels of clustering, and simulations suggest that this sparsely interconnected hierarchical network structure might accelerate CCE by allowing the coexistence of multiple cultural lineages and promoting the emergence of innovations (but see Box 4).

The few studies that have investigated networks in hunter-gatherers, however, have been limited to interview data and proximity measures [START_REF] Hill | Hunter-Gatherer Inter-Band Interaction Rates: Implications for Cumulative Culture[END_REF][START_REF] Migliano | Characterization of hunter-gatherer networks and implications for cumulative culture[END_REF][START_REF] Migliano | Hunter-gatherer multilevel sociality accelerates cumulative cultural evolution[END_REF]. Actual measurements of cultural transmission remain scarce, and the extent to which proximity networks accurately reflect transmission networks is currently unknown. Investigation of the co-occurrence of plant uses in dyads in one hunter-gatherer population, for instance, showed that not all knowledge is equally shared [START_REF] Salali | Knowledge-Sharing Networks in Hunter-Gatherers and the Evolution of Cumulative Culture[END_REF]. More specifically, results show that medicinal plants were mostly shared between spouses and kin, while plants that serve other functions were shared much more widely. This suggests that knowledge-sharing networks are content-specific and supports the idea that hunter-gatherer multi-level social structure enables culturally differentiated units to remain stable despite occasional co-residence [START_REF] Migliano | Hunter-gatherer multilevel sociality accelerates cumulative cultural evolution[END_REF]. This work also suggests that both structural barriers (i.e. lack of contact between individuals) and behavioral barriers (i.e. unwillingness to share cultural knowledge) have to be taken into account to properly evaluate the effects of population structure on CCE. Indeed, structural and behavioral barriers combine to result in an effective population structure that ultimately determines opportunities for cultural transmission. Contact between different ethnolinguistic groups, for instance, can potentially bring different cultural traits together due to significant between-group cultural distance. However, language barriers, endogamy, rivalry and other behavioural barriers such as in-group conformity might limit opportunities for cultural exchange between those groups [START_REF] Durham | Applications of Evolutionary Culture Theory[END_REF][START_REF] Tehrani | Do Transmission Isolating Mechanisms (TRIMS) Influence Cultural Evolution?[END_REF].

These results suggest that our understanding of the relationship between demography and CCE would benefit from a better understanding of how and why individuals form social ties both within-and between-groups and the extent to which different types of ties (such as kin-based, affine-based and friendship-based) are conducive to cultural transmission. This will permit more realistic implementation of cultural transmission into theoretical models. Indeed, while the combination of vertical cultural transmission (i.e. learning from parents) and success-biased learning is empirically supported and provides a useful first approximation of the dynamics of social learning in groups [START_REF] Henrich | Understanding cumulative cultural evolution[END_REF], multiple factors are likely to affect opportunities for social learning. Anthropological studies, for instance, have shown that social ties are more likely to form between people who share similar traits (i.e. homophily [START_REF] Apicella | Social networks and cooperation in hunter-gatherers[END_REF][START_REF] Centola | Homophily, cultural drift, and the co-evolution of cultural groups[END_REF]). Furthermore, understanding how individuals form social ties is an important avenue for future research because the way individuals form ties ultimately feeds back into the evolution of social networks (homophily, for instance, is known to introduce local structure into networks [START_REF] Centola | Homophily, cultural drift, and the co-evolution of cultural groups[END_REF][START_REF] Kandler | Networks, homophily, and the spread of innovations[END_REF]).

How did human social networks get their shape?

Even if questions remain regarding the effects of specific network properties on CCE, it seems clear that humans live within unusually large and uniquely structured social networks.

This raises questions about how and why humans have come to form large networks of unrelated or weakly related individuals.

Recently, it has been argued that, because individuals from culturally differentiated groups might have greatly benefited from increased between-group interactions, selection might have acted at the individual level to affect individuals' propensity to interact with outgroup members [START_REF] Greenbaum | Was inter-population connectivity of Neanderthals and modern humans the driver of the Upper Paleolithic transition rather than its product?[END_REF]. This might have involved changes in conscious behavioural choices (e.g. adjustments to out-group contacts due to perceived immediate benefits) and/or unconscious influence on behaviour (e.g. decreased fear of foreigners or tendency to disperse) [START_REF] Greenbaum | Was inter-population connectivity of Neanderthals and modern humans the driver of the Upper Paleolithic transition rather than its product?[END_REF].

Congruently, a recent simulation model that investigated whether network structure itself can evolve as a result of ecological pressures related to skill acquisition confirmed that selection can impact individuals' propensity to form random ties (such as non kin ties) [START_REF] Smolla | Cultural selection shapes network structure[END_REF]. Yet, it is not clear whether the acquisition of social information creates sufficiently strong incentives for individuals to overcome rivalry and other behavioural barriers that tend to reduce opportunities for cultural transmission between unrelated individuals. Moreover, increasing contacts is only one part of the problem, as many cultural traits are unlikely to be properly acquired without a demonstrator's willingness to share information [START_REF] Henrich | The evolution of prestige -Freely conferred deference as a mechanism for enhancing the benefits of cultural transmission[END_REF][START_REF] Mesoudi | An experimental simulation of the "copy-successfulindividuals" cultural learning strategy: adaptive landscapes, producer-scrounger dynamics, and informational access costs[END_REF][100].

Another possible way by which selection might have promoted the emergence of networks that are conducive to CCE is by acting on variation that exists at the group level [START_REF] Greenbaum | Was inter-population connectivity of Neanderthals and modern humans the driver of the Upper Paleolithic transition rather than its product?[END_REF]101]. Indeed, anthropologists have long stressed the role of cultural institutions in promoting both information sharing and interactions between non-kin [START_REF] Hill | Hunter-Gatherer Inter-Band Interaction Rates: Implications for Cumulative Culture[END_REF][101][102][103]]. Among the Ache and Hadza, for instance, ritual relationships, mediated by activities such as club fight rituals, have been shown to promote inter-band interaction. Quantitative analyses have revealed that ritual relationship is a more important predictor than kinship for different types of interactions, including opportunities for cultural transmission (such as observing tool making skills) [START_REF] Hill | Hunter-Gatherer Inter-Band Interaction Rates: Implications for Cumulative Culture[END_REF]. Furthermore, anthropologists have stressed that certain groups have cultural beliefs that connect envy and harm, which make successful individuals more likely to hide information from other group members, thus inhibiting CCE compared to other groups [101]. This suggests that groups that possess cultural institutions that promote information sharing and/or mobility might have attained higher cultural complexity and outcompeted groups with cultures less conducive to CCE [START_REF] Greenbaum | Was inter-population connectivity of Neanderthals and modern humans the driver of the Upper Paleolithic transition rather than its product?[END_REF]101]. It is also worth noting that the maintenance of large networks of unrelated or weakly related individuals might have been further supported by the emergence of cultural innovations such as kin naming systems and stylistic markers of group identity that typically promote cooperative interactions between unrelated individuals The question of whether humans' social structure might in part result from selection pressures linked to our extensive reliance on culturally accumulated knowledge will have to be carefully evaluated. Indeed, chimpanzees also live among nonrelatives [105] and humans' propensity to form ties with non-kin might be due to reasons unrelated to CCE and that just happened to be conducive to the accumulation of cultural innovations. Archeologists, for instance, noted that an incest avoidance rule would give rise to the same kind of sparsely connected networks that might benefit CCE [START_REF] Grove | Hunter-gatherers adjust mobility to maintain contact under climatic variation[END_REF]. Alternative determinants of outgroup contacts include resource distribution [START_REF] Grove | Hunter-gatherers adjust mobility to maintain contact under climatic variation[END_REF], reciprocal cooperative exchange [106] and coalition formation [107], among others. Specific predictions should be formulated and properly tested to disentangle the respective effects of these various mechanisms on network structure. The hypothesis that CCE directly shapes network structure by acting on conscious behavioural choices, for instance, would predict that individuals should flexibly reinforce or weaken their investment in non-kin ties depending on the usefulness of the information they provide.

Concluding remarks and future directions

The proliferation of work exploring the interplay of demography and CCE has recently led to many misconceptions due to loose interpretations of early theoretical models (Box 2 and 3).

Empirical tests that operationalize models in ways that are consistent with theoretical assumptions provide support for the hypothesis that effective population size constrains CCE. However, testing these models using real-world data remains difficult because multiple factors combine with demography to determine CCE and human populations are typically embedded within extended networks of cultural exchange. While these extended networks of contacts make it difficult to gather meaningful estimates of population size, recent research suggests that they might also affect CCE in ways that are not yet fully appreciated.

Understanding how population structure affects CCE will require us to understand precisely how structural and behavioral barriers constrain information flow in natural populations (Box 4).

The effects of connectedness on the accumulation of cultural information raise many questions about the relationship between humans' unique social structure and CCE (see Outstanding Questions). Through the study of the nature and the emergence of non-kin ties, both within groups and between groups, as well as knowledge-sharing networks in natural populations, it will be possible to illuminate how humans have managed to accumulate cultural information in such an unprecedented way and determine whether our unique social structure results in part from selection pressures linked to our extensive reliance on culturally accumulated knowledge.

Box 1: Demographic models of cultural change

Cultural drift. Some of the earliest cultural evolution models adapted early 20 th century models of genetic drift to the cultural case [START_REF] Cavalli-Sforza | Cultural transmission and evolution: A quantitative approach[END_REF][START_REF] Neiman | Stylistic Variation in Evolutionary Perspective: Inferences from Decorative Diversity and Interassemblage Distance in Illinois Woodland Ceramic Assemblages[END_REF][START_REF] Shennan | Demography and Cultural Innovation: a Model and its Implications for the Emergence of Modern Human Culture[END_REF]108]. Drift, whether genetic or cultural, is essentially sampling error. Drift models typically assume 'unbiased transmission' or 'random copying': each of N individuals within a finite and fixed-sized population possesses one of a set of discrete cultural traits. Each generation or timestep, individuals select another individual at random and acquire their cultural trait. This process results in the inevitable loss of trait variation. The speed with which traits are lost is dependent on N: smaller populations lose variation quicker. This is a highly simplistic model, but provides a useful base for exploring the effects of processes such as innovation and complex population structures such as island chains or bottlenecks on CCE, and has been used to explain archaeological assemblage diversity [START_REF] Neiman | Stylistic Variation in Evolutionary Perspective: Inferences from Decorative Diversity and Interassemblage Distance in Illinois Woodland Ceramic Assemblages[END_REF]108].

The 'Tasmanian' model. Perhaps the most influential demographic model of cultural evolution was formulated by Henrich [START_REF] Henrich | Demography and cultural evolution: How adaptive cultural processes can produce maladaptive losses -The Tasmanian case[END_REF]. This model was inspired by the empirical case of prehistoric Tasmania, which apparently lost complex technological traits (e.g. bone tools, warm clothing) around 10-12kya when Tasmania was cut off from the Australian mainland, thus decreasing the effective population size [START_REF] Jones | Emerging Picture of Pleistocene Australians[END_REF]. The model incorporates more psychologically plausible processes than simple drift models. Each of N individuals possesses a value of culturally transmitted 'skill' (e.g. basket-making), represented by a continuous variable z. Each timestep, each individual attempts to learn the skill value zh of the highestskilled member of the previous timestep, h (i.e. success biased transmission). Learning is imperfect, and affected by two kinds of processes. Learning error, determined by ɑ, always results in worse skill than zh. Another parameter, β, determines the extent of inferences, experiments, luck and other factors that on average make skill levels worse, but sometimes better, than zh. Combining these, Henrich assumed that the skill of a naive individual is drawn from a Gumbel distribution (Figure 1). N interacts with the latter β term: the more individuals there are, the more likely one of those individuals is to exceed zh, representing an increase in cumulative cultural knowledge/skill. If N is too small, then all learners will acquire values around the mode of the distribution, which is less than zh, resulting in a decrease in cultural complexity. Subsequent empirical work has shown that this Gumbel distribution is a reasonable approximation of social learning dynamics [109] (but see [110] for a critique of this model).

Population structure and trait recombination. Subsequent models have extended the

Tasmanian model to investigate in greater detail how the structure of the population impacts both the maintenance and the production of cultural traits. Stochastic simulations of the Tasmanian model with multiple sub-populations show that increasing the migration rate has a similar effect to increasing the size of an isolated population on CCE, because both increase variation within sub-populations and so reduce the risk of losing cultural information [START_REF] Powell | Late Pleistocene Demography and the Appearance of Modern Human Behavior[END_REF].

Recent studies have more explicitly modelled the pathways that give rise to innovation and revealed that the effect of migration can even be more pronounced when cultural traits can combine to form innovations that are "greater than the sum of their parts" [START_REF] Creanza | Greater than the sum of its parts? Modelling population contact and interaction of cultural repertoires[END_REF]. However, too frequent contact might not be beneficial to CCE because it prevents populations from remaining culturally distinct, and reduces opportunities to innovate [START_REF] Creanza | Greater than the sum of its parts? Modelling population contact and interaction of cultural repertoires[END_REF][START_REF] Derex | Divide and conquer: intermediate levels of population fragmentation maximize cultural accumulation[END_REF]. zi relative to zh. On the left, a small population (N=20) results in a population-level decline in skill, as no learner matches or exceeds zh. On the right, a large population (N=100) features some learners who exceed zh, resulting in an improvement in the next generation.

Box 2: Linking models and data in the lab

Experimental approaches are useful for investigating the relationship between demography and CCE because essential elements of theoretical models can be implemented under tightly controlled conditions, and tested against actual human behaviour (rather than modellers' assumptions about human behaviour) [111,112].

As noted in the main text, the majority of experimental studies have found support for the general predictions of demographic models [START_REF] Derex | Experimental evidence for the influence of group size on cultural complexity[END_REF][START_REF] Wisdom | Social Learning Strategies in Networked Groups[END_REF][START_REF] Muthukrishna | Sociality influences cultural complexity[END_REF][START_REF] Kempe | An experimental demonstration of the effect of group size on cultural accumulation[END_REF][START_REF] Derex | The foundations of the human cultural niche[END_REF]. This is all the more surprising given that these studies are remarkably diverse in experimental tasks, group sizes and interindividual interactions. Yet, it is worth highlighting that most experimental designs significantly deviate from the models they claim to test. In the main text we discuss one example, where experiments offer social learners the opportunity to combine information from multiple cultural demonstrators [START_REF] Muthukrishna | Sociality influences cultural complexity[END_REF][START_REF] Kempe | An experimental demonstration of the effect of group size on cultural accumulation[END_REF][START_REF] Derex | The foundations of the human cultural niche[END_REF], rather than learn from a single successful demonstrator as in the most-cited demographic models (see Box 1). The role of recombination across existing cultural traits has been stressed by scholars from multiple fields [113][114][115], and increased opportunities for recombination certainly is one pathway by which effective population size might affect CCE [101]. Yet, most experiments are presented as tests of models that do not feature recombination between existing traits and in which effective population size mostly affects CCE by buffering the risk of losing cultural information (see Box 1). Still other experiments have relied on tasks in which cultural loss is unlikely to occur [START_REF] Derex | The foundations of the human cultural niche[END_REF]. Thus, even though these experiments support the population size hypothesis, it is not always clear whether they provide appropriate tests of the theoretical models which they cite.

Maybe more problematic are experiments where results showing no relationship between demography and CCE are used to question the validity of theoretical models despite featuring different assumptions to those models. A recent experiment, for instance, had chains of participants make and throw paper airplanes, with each participant able to learn from 1, 2 or 4 previous participants [START_REF] Fay | Increasing population size can inhibit cumulative cultural evolution[END_REF]. Apparently contrary to the demographic hypothesis, flight distance only increased in the 1-demonstrator condition, not the 2-and 4demonstrator conditions. Yet this experimental design prevented participants from learning from the demonstrator of their choice. Instead participants were forced to attend to multiple, randomly ordered demonstrators for 1.5 minutes each. Yet, Henrich's model explicitly holds that it is the combination of the amount of beneficial cultural information (which increases in larger groups) and the selective choices of cultural learners that promotes CCE. Fay et al.'s results are consistent with the former in showing that larger groups produce greater variation in distance flight and give participants access to more efficient planes. But the constraints imposed on social learning strategies inhibited CCE in large groups by making learning more difficult in those groups.

Discrepancies between experiments and models are not inherently a problem: the assumptions of models can always be challenged and mechanisms other than those considered in theoretical models are worth investigating. Yet, the experimental literature would benefit from being more explicit about the theoretical basis underpinning the specifics of experimental designs and how they relate to theoretical models.

Box 3: Linking models and data in the wild

Several studies have investigated whether there exists a correlation between toolkit size or composition and population size in natural populations [START_REF] Powell | Late Pleistocene Demography and the Appearance of Modern Human Behavior[END_REF][START_REF] Marquet | Emergence of social complexity among coastal hunter-gatherers in the Atacama Desert of northern Chile[END_REF][START_REF] Kline | Population size predicts technological complexity in Oceania[END_REF][START_REF] Collard | Population Size and Cultural Evolution in Nonindustrial Food-Producing Societies[END_REF][START_REF] Collard | Causes of Toolkit Variation Among Hunter-Gatherers: A Test of Four Competing Hypotheses[END_REF][START_REF] Collard | Risk, mobility or population size? Drivers of technological richness among contact-period western North American hunterâ€"gatherers[END_REF][START_REF] Collard | Population Size as an Explanation for Patterns in the Paleolithic Archaeological Record: More Caution Is Needed[END_REF][START_REF] Buchanan | Drivers of technological richness in prehistoric Texas: an archaeological test of the population size and environmental risk hypotheses[END_REF], but there remain serious challenges in testing demographic effects on CCE in real world data.

One difficulty concerns limitations in what can be measured [START_REF] Aoki | On the absence of a correlation between population size and 'toolkit size' in ethnographic hunter-gatherers[END_REF]. Henrich's model (see Box 1) describes the level of skill of an individual within a population, a variable that in an archaeological context can be interpreted as the number of tools or tool components attributable to an individual. Yet, archaeological studies typically only have access to population-level rather than individual-level data. This makes purported tests that use population-level assemblage measures largely irrelevant to Henrich's predictions [START_REF] Aoki | On the absence of a correlation between population size and 'toolkit size' in ethnographic hunter-gatherers[END_REF]. Even though a recent model incorporating the appropriate population-level variable does predict a positive relationship between population size and toolkit size [START_REF] Aoki | On the absence of a correlation between population size and 'toolkit size' in ethnographic hunter-gatherers[END_REF], these discrepancies illustrate the need to use appropriate measures when attempting to test a model and/or to adapt models so they can properly be tested using empirical data.

A second difficulty is that demography has multiple aspects that can be difficult to fully take into account in ethnographic and archaeological studies. In the main text we discuss one example of this, where empirical data regarding census population sizes are used to test (and purportedly fail to support) the Tasmanian model without taking contact rates into account. Furthermore, recent models suggest that historical variations in population size and connectedness are as important as immediate demographic contexts in determining cultural complexity in a population [START_REF] Aoki | On the absence of a correlation between population size and 'toolkit size' in ethnographic hunter-gatherers[END_REF][START_REF] Fogarty | The Driving Forces of Cultural Complexity[END_REF][START_REF] Creanza | Greater than the sum of its parts? Modelling population contact and interaction of cultural repertoires[END_REF]. Some models, for instance, show that the number of traits in a population should depend not only on the current population size but also on the history of population growth and decline [START_REF] Aoki | On the absence of a correlation between population size and 'toolkit size' in ethnographic hunter-gatherers[END_REF][START_REF] Fogarty | The Driving Forces of Cultural Complexity[END_REF]. This can blur the relationship between population size and CCE because growing populations can have fewer cultural traits than smaller, declining populations. Similarly, two populations of the same size might be associated with toolkits of different sizes due to different demographic trajectories. Models also suggest that changes in interconnectedness can result in different outcomes including transient increases in cultural complexity [START_REF] Creanza | Greater than the sum of its parts? Modelling population contact and interaction of cultural repertoires[END_REF]. The effects of population histories represent a challenge for archaeologists whose data represent a record of aggregated events spanning long periods of time during which both population size and interconnectedness might have varied. Further models are needed to determine what testable signatures these dynamics might have left in the past for archaeologists and historians to detect.

Finally, demographic factors determine an upper boundary to the level of cultural complexity that can be reached by a population, but do not entirely determine the actual level reached by a population. Assuming that increased cultural complexity is beneficial, increases in population size should result in increases in cultural complexity but only because this relaxes constraints on CCE. A full understanding of CCE in natural populations requires both drivers of CCE and constraints to be taken into account. To that end, more research is needed to identify the factors that combine with demography to determine CCE in natural populations, such as environmental harshness [START_REF] Torrence | Re-tooling: towards a behavioral theory of stone tools[END_REF] and instability [116] or accumulated cultural traits themselves [START_REF] Mesoudi | Variable Cultural Acquisition Costs Constrain Cumulative Cultural Evolution[END_REF]117,118].

Box 4: Is human multilevel social structure beneficial to CCE?

Recent theoretical and experimental studies have challenged the assumption that anything that maximizes the flow of cultural information should positively impact innovation rates (Figure 2A-B). These results have led scholars to wonder whether CCE in human populations has benefited from our unique multilevel social structure via the partial constraints it imposes on information flow [START_REF] Derex | Partial connectivity increases cultural accumulation within groups[END_REF]. A recent simulation study provided support for this by showing that real hunter-gatherers' social networks allow the coexistence of multiple cultural lineages, thus promoting the emergence of innovations [START_REF] Migliano | Hunter-gatherer multilevel sociality accelerates cumulative cultural evolution[END_REF]. However, while characterizing actual networks is useful for understanding how cultural information is expected to spread, many (still largely unknown) parameters need to be taken into account before establishing whether, and if so why, human multilevel social structure promotes CCE. Previous work has shown, for instance, that the effect of network structure on CCE is mediated by factors such as individuals' probabilities of innovating (because even strong constraints on information flow prevent cultural diversification if innovation rates are low [START_REF] Derex | Divide and conquer: intermediate levels of population fragmentation maximize cultural accumulation[END_REF]) and the extent to which innovation depends on cultural diversity (because constraints on information flow both slow down and limit CCE when innovation does not depend on recombination [START_REF] Derex | Divide and conquer: intermediate levels of population fragmentation maximize cultural accumulation[END_REF]). In the aforementioned simulation study [START_REF] Migliano | Hunter-gatherer multilevel sociality accelerates cumulative cultural evolution[END_REF], both individuals' opportunity to innovate, and possibilities for recombination, were determined by the properties of an artificial cultural fitness that was designed to permit innovation through incremental improvement and recombination [START_REF] Derex | Partial connectivity increases cultural accumulation within groups[END_REF], but whose relevance to rates of CCE in natural populations is uncertain.

Maybe more importantly, the effect of network structure on cultural loss was not considered in those simulations [START_REF] Migliano | Hunter-gatherer multilevel sociality accelerates cumulative cultural evolution[END_REF]. When cultural loss is not taken into account, constraints on information flow necessarily benefit CCE by promoting cultural diversification.

In more realistic situations, constraints on information flow expose populations to higher rates of cultural loss, which can prevent cultural diversification [119]. Moreover, even if they have diverse cultural repertoires, sparsely connected populations can be unlikely to reach high levels of cultural complexity because of their inability to maintain complex cultural traits [START_REF] Derex | Divide and conquer: intermediate levels of population fragmentation maximize cultural accumulation[END_REF]. Thus, given our current limited knowledge about rates of loss and innovation, and opportunities for recombination, in real-world populations, it is not clear whether the network structure documented in [START_REF] Migliano | Hunter-gatherer multilevel sociality accelerates cumulative cultural evolution[END_REF] positively affects CCE or whether cultural complexity in hunter-gatherer populations would benefit from more connectedness by being less susceptible to cultural loss. Answering this question will require an evaluation of how sparse networks made of strong ties (e.g. kin and friendship ties) balance cultural loss and cultural diversity (Figure 2C). results show that moderately connected populations are slower at accumulating innovations but eventually reach higher levels of cultural complexity than highly connected populations when innovation depends on cultural diversity. Adapted from [START_REF] Derex | Partial connectivity increases cultural accumulation within groups[END_REF]. (B) Simulation models show that optimal rates of accumulation are reached for intermediate levels of connectedness when populations are exposed to cultural loss. Relative rates of accumulation between variously connected populations depend on parameters such as rates of innovation and cultural loss, and the extent to which innovation depends on cultural diversity (not shown). Adapted from [START_REF] Derex | Divide and conquer: intermediate levels of population fragmentation maximize cultural accumulation[END_REF]. (C) Patterns of connectedness affect both cultural loss and diversity. (i) In fully connected networks made of permanent links (solid lines), the average number of steps required to connect any two individuals (i.e. path length) is minimal and the efficiency with which information spreads is maximal. This reduces the risks of cultural but decreases cultural diversity. (ii) Removing ties increases the average path length between individuals and results in less efficient networks (e.g. from i to ii). (iii) Networks composed of individuals tied to the same number of neighbors can also vary in efficiency due to differences in average clustering coefficients (a measure that reflects the "cliquishness" of a network [120]). Increasing the average clustering coefficient results in less efficient networks (e.g. from ii to iii). (iv) Intermittent links between different parts of a network (dotted lines) further constrain information flow and result in substructures that are more likely to culturally diverge by isolation (illustrated by different colors) but also more likely to suffer from cultural loss.

[

  103]. Kin naming systems, for instance, allow familial relationships to extend to affine, distant kin and even non-kin [103] and might permit individuals to maintain privileged relationships with large numbers of individuals without requiring much cognitive effort nor physical cohabitation [104].

Figure 1 :

 1 Figure 1: Gumbel distribution from Henrich's Tasmanian model

Figure 2 :

 2 Figure 2: Trading cultural loss and diversity in structured populations. (A) Experimental

Acknowledgments

We thank four reviewers and Rob Boyd for valuable comments on the manuscript. MD thanks Alex Alvergne, Slimane Dridi and Vivek Venkataraman for helpful discussions. This work was supported by the French Agence Nationale de la Recherche (under the Investissement d'Avenir programme ANR 17-EURE-0010) and the European Union's Horizon 2020 research and innovation programme (under Marie Sklodowska-Curie grant agreement number 748310).

Glossary

Behavioural barriers: blocks on information flow due to behavioural tendencies such as an unwillingness to teach hard-to-learn skills, despite contact


Alain Dargelos

Claude Pouchan
email: claude.pouchan@univ-pau.fr

of the IR spectrum of cyanoacetylene HC3N beyond the harmonic approximation

Keywords:

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

Introduction

Cyanopolyynes, linear molecules of the raw formula H-(CC)n-CN, are among the most common species found in astronomical environments. The first members of this series have been detected and identified in the interstellar medium (ISM) since the 1970s [START_REF] Turner | Detection of interstellar cyanoacetylene[END_REF] and have been the subject of Voyager [START_REF] Kunde | C4H2, HC3N and C2N2 in Titan's atmosphere[END_REF] and Cassini [START_REF] Teanby | Latitudinal variations of HCN?HC3N and C2N2 in Titan's stratosphere derived from Cassini CIRS data[END_REF] missions in Titan's atmosphere.

The simplest of these cyanopolyynes, propynenitrile, more commonly known as cyanoacetylene, detected in 1971 by Turner [START_REF] Turner | Detection of interstellar cyanoacetylene[END_REF] was quickly the subject of great interest and its presence noticed in various astrophysical objects (Milky way, starless cores, solar-type protostar, comets, protoplanetary disks, external galaxies...), as recently reported in the article by Bizzochi et al. [START_REF] Bizzochi | Rotational and High-resolution Infrared Spectrum of HC3N: Global Rovibrational Analysis and Improved Line Catalog for Astrophysical Observations[END_REF], has stimulated a large number of laboratory experiments. Among these experiments, we can note the numerous investigations to better understand the rotational [START_REF] Chen | New Far Infrared Measurements of the rotational spectrum of cyanoacetylene HC3N[END_REF][START_REF] Thorwirth | The Millimeter and Submillimeter wave spectrum of HC3N in the ground and vibrationally excited states[END_REF][START_REF] Mbosei | Millimeter Wave spectra of HC3N[END_REF], vibrational [START_REF] Uyemura | Infrared intensities of Bending Fundamentals in Gaseous HCCCN and DCCCN[END_REF][START_REF] Khlifi | Absolute Intensity of the IR Bands of Propynenitrile[END_REF][START_REF] Khlifi | Integrated band intensity versus temperature for v1,v2, v5, and v6 bands of cyanoacetylene[END_REF][START_REF] Arie | Analysis of cyanoacetylene spectra in the Mid-Infrared[END_REF][START_REF] Jolly | New infrared integrated bands intensities for HC3N and extensive line list for the v5 aand v6 bending modes[END_REF] and ro-vibrational [START_REF] Bizzochi | Rotational and High-resolution Infrared Spectrum of HC3N: Global Rovibrational Analysis and Improved Line Catalog for Astrophysical Observations[END_REF] spectra and more precisely to understand the positions and intensity of the IR bands with the work of Uyemura et al. [START_REF] Uyemura | Infrared intensities of Bending Fundamentals in Gaseous HCCCN and DCCCN[END_REF], Khlifi et al. [START_REF] Khlifi | Absolute Intensity of the IR Bands of Propynenitrile[END_REF][START_REF] Khlifi | Integrated band intensity versus temperature for v1,v2, v5, and v6 bands of cyanoacetylene[END_REF], Jolly et al. [START_REF] Arie | Analysis of cyanoacetylene spectra in the Mid-Infrared[END_REF][START_REF] Jolly | New infrared integrated bands intensities for HC3N and extensive line list for the v5 aand v6 bending modes[END_REF] and more recently the work of Bizzochi et al. [START_REF] Bizzochi | Rotational and High-resolution Infrared Spectrum of HC3N: Global Rovibrational Analysis and Improved Line Catalog for Astrophysical Observations[END_REF] exploring between 450 and 1350 cm -1 the ro-vibrational spectrum by an exhaustive study using a high resolution IR spectra recording. Despite the importance of these experimental works, few theoretical calculations on the vibrational spectrum of this molecule have been undertaken, often limited to a simple harmonic approach [START_REF] Borget | Cyanoacetylene Adsorption on Amorphous and Crystalline Water Ice Films: Investigation through Matrix Isolation and Quantum Study[END_REF] or not taking into account at the anharmonic level the totality of the potential by limiting the study only to the four stretching modes treated without coupling with the bending vibrations [START_REF] Botschwina | Vibrational frequencies from anharmonic ab-initio/empirical potential energy functions. III-Stretching vibrations of hydrogen cyanide and acetylenes[END_REF][START_REF] Botschwina | Ab initio calculations of stretching vibrational transitions for the linear molecules HCN, HNC, HCCF and HC3N up to high overtones[END_REF]. Very recently C. Pouchan [START_REF] Pouchan | Quantum molecular dynamic approach to modeling the vibrational spectra of propynenitrile and its 1:1 complexes with water[END_REF] published the theoretical modelling of the HC3N vibrational spectrum by comparing a DFT molecular dynamics (DFT/MD) approach with a static anharmonic approach under the assumption of a vibrational second order perturbative treatment (VPT2). This study, limited to fundamental frequencies only, showed that the DFT harmonic calculations from a B3LYP functional [START_REF] Becke | Density-functional thermochemistry. III. The role of exact exchange[END_REF] associated with a 6-31+G (d,p) base did not reflect, unlike many other systems previously studied [START_REF] Carbonniere | Vibrational analysis of glycine radical: a comparative ab initio static and dynamic study[END_REF][START_REF] Carbonniere | Vibrational spectra for P(4)O(6) and P(4)O(10) systems: Theoretical study from DFT quartic potential and mixed perturbation-variation method[END_REF][START_REF] Barone | Accurate vibrational spectra and magnetic properties of organic free radicals: The case of H2CN[END_REF][START_REF] Dargelos | Ab-initio calculations of the IR spectra of dicyanodiacetylene (C6N2) beyond the harmonic approximation[END_REF][START_REF] Dargelos | Theoretical Investigation of the Infrared Spectrum of 5-Bromo-2,4-pentadiynenitrile from a CCSD(T)/B3LYP Anharmonic Potential[END_REF][START_REF] Dargelos | Ab Initio Modeling of the IR Spectra of Dicyanoacetylene in the Region 100-4800 cm -1[END_REF][START_REF] Benidar | Gas-Phase Infrared Spectra of Three Compounds of Astrochemical Interest: Vinyl, Allenyl and Propargyl Isocyanides[END_REF][START_REF] Marchal | Vibrational spectra of vinylarsine: A joint IR experimental and anharmonic theoretical study[END_REF], the harmonic results obtained at the CCSD(T) [START_REF] Ragavarachri | A fifth-order perturbation comparison of electron correlation theories[END_REF]/aug-cc-pVTZ [START_REF] Kendall | Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions[END_REF]level, and that the potential function limited to the order 4 did not allow to give accurate VPT2 anharmonic wavenumbers to reproduce the first harmonic bands of the lowest frequencies and their combinations.

It seemed interesting under these conditions to revisit the IR spectrum modelling by considering a new CCSD(T) approach in combination with a basis set of triple or quadruple zeta quality with adaptive accurate multidimensional potential energy surfaces followed by a vibrational analysis using a vibrational configuration interaction method (VCI). The bottleneck of this procedure remains the accurate calculation of extended areas of the potential energy surface by high-level ab-initio method that is a very time consuming task.

The major objectives of the theoretical work presented here to describe and predict the vibrational spectrum of HC3N are threefold and consist of:

-on the one hand to give a computational contribution to the experimental works carried out for the fundamental modes on the positioning and intensity of IR bands.

-on the other hand to analyse and assign by quantum calculation the combinations and hot bands recently identified by Bizzochi et al.[4] between 450 and 1350 cm -1 -finally, to predict for regions not fully studied experimentally the active bands expected to help in their future identification.

The paper is structured as follows: Sec. 2 summarizes the computational methods applied and the details of our calculations, the outline of the theory being given in more details in related papers [START_REF] Rauhut | Efficient calculation of potential energy surfaces for the generation of vibrational wave functions[END_REF][START_REF] Hrenar | Accurate calculation of anharmonic vibrational frequencies of medium sized molecules using local coupled cluster methods[END_REF][START_REF] Rauhut | A combined variational and perturbational study on the vibrational spectrum of P2F4[END_REF].

Sec. 3 provides the results obtained to describe, assign, and predict in spectral region not fully explored, the IR spectrum of propynenitrile HC3N in vacuo. All the results will be discussed and compared, when possible, with experimental data.

2-Methods and computational details. 2-1 Methods for calculating PES and VCI calculations

The calculation of extended areas of potential energy surface (PES) for a polyatomic linear system by coupled-cluster methods is a great challenge in quantum chemistry and needs very time-consuming resources. Different techniques and algorithms developed in our lab and reported in literature [START_REF] Rauhut | A combined variational and perturbational study on the vibrational spectrum of P2F4[END_REF][START_REF] Richter | An adaptive potential energy surface generation method using curvilinear valence coordinates[END_REF][START_REF] Richter | Toward linear scaling: Locality of potential energy surface coupling in valence coordinates[END_REF] offer an accurate representation of PES. They remain however a little limited to deal with the case of linear molecules and so we preferred to use the formalism developed by Rauhut et al. [START_REF] Rauhut | Efficient calculation of potential energy surfaces for the generation of vibrational wave functions[END_REF][START_REF] Hrenar | Accurate calculation of anharmonic vibrational frequencies of medium sized molecules using local coupled cluster methods[END_REF][START_REF] Rauhut | A combined variational and perturbational study on the vibrational spectrum of P2F4[END_REF] reporting an algorithm for an efficient and fully automatic generation of PES around an equilibrium structure. This method uses an incremental scheme of difference potentials (Vi, Vij, Vijk…called respectively 1D, 2D , 3D…terms) spanned by the normal coordinates qi of the system as originally proposed and described by Carter et al. [START_REF] Carter | Vibrational Self-Consistent Field Method for many-mode systems: A new approach and application to the vibration of CO adsorbeb on Cu (100)[END_REF][START_REF] Carter | Ab-initio calculation of force fields for H2CN and ClHCN and vibrational energies of H2CN[END_REF]. In our case the many body expansion of the PES is truncated after the 3D terms.

V (q1,….,qn) = V0 + ∑i Vi(qi) + ∑i<j Vij(qi,qj) + ∑i<j<k Vijk (qi,qj,qk) + …….

(

) 1
with

Vi (qi) = Vi 0 (qi) -V0 (2)
Vij (qi,qj) = Vi j 0 (qi,qj) -∑r {i,j} Vr(qr) -V0

Vijk (qi,qj,qk) = Vijk 0 (qi,qj,qk) -∑r>s {i,j,k} Vrs(qr,qs) -∑r {i,j,k} Vr(qr) -V0

V0 represents the total energy at the equilibrium geometry and the quantities with the superscript 0 denote the total energy obtained from the electronic calculations at distorted nuclear configurations.

The calculation of the PES, that can be computed by using different methods for the different increments, needs 9288 grid points resulting in 48 single-point calculations for the 1D terms, 956 energy calculations and 8284 energy determinations respectively for the 2D and the 3D terms contributions in the case of HC3N treated as belonging to the point group of C2V symmetry. Taking into account the small size of the system all determinations are made at the highest computational level i.e. CCSD(T).

VCI calculations account for vibrational correlation effects and the use of PES as generated from the SURF program and a basis of VSCF modals in which all modals were generated from 16 distributed Gaussians. VCI calculations include single (S), double (D), triple (T) and quadruple (Q) excitations. All calculations are carried out in C2v symmetry. For the irreducible representation A1 27019 configurations (32 S, 504 D, 4820 T and 21662 Q) have been selected while 22387 configurations( 9S, 270 D, 3464 T, 18644 Q) are required to describe each irreducible representation B1 and B2.

All calculations presented here have been performed with the MOLPRO suite of ab-initio programs [START_REF] Molpro | Coupled cluster method with perturbative inclusion of explicity correlated terms: a preliminary investigation[END_REF] 2-2 Computational details.

The CCSD(T) coupled-cluster electronic method is a standard tool to provide reliable values of molecular energies and properties as in our case Infra-red (IR) spectra .Associated at large number of orbital basis functions it gives chemically useful accuracy. To improve this method, explicitlycorrelated R12 approach that use additional two-particules basis functions depending explicitly on the inter-electronic distance has been proposed increasing the accuracy of reliable properties predictions [36][START_REF] Valeev | Simple coupled cluster singles and doubles method with perturbative inclusion of triples and explicity correlated geminals: The R12 CCSD(T) model[END_REF]. This method including correlated germinal is known as the CCSD(T) R12 model [START_REF] Valeev | Simple coupled cluster singles and doubles method with perturbative inclusion of triples and explicity correlated geminals: The R12 CCSD(T) model[END_REF] or the CCSD(T)-F12x model [START_REF] Knizia | Simplified CCSD(T)-F12x methods: Theory and benchmarks[END_REF] in which the linear r12 correlation factor is replaced by a short range nonlinear function F12(r12) that leads to much better results and numerical stability. . We will adopt this latter model in accordance with the MOLPRO software used in our calculations. In the case of HC3N all calculations are carried out at the CCSD(T)-F12 level associated with an aug-cc-pVTZ basis set to compute structural parameters and vibrational frequencies. Details of the VCI calculations to obtain anharmonic wavenumbers are given above while the intensities are calculated from dipole surfaces at MP2/aug-cc-pVTZ quality level.

3-Results and Discussions

We report in figure 1 the atomic numbering adopted to describe the propynenitrile molecule. The structural parameters as well as the main assignments of the modes reported in tables 1 and 2 refer to this figure.

3-1 Structural parameters and harmonic wavenumbers

We report in table 1 the geometries obtained at different levels of CCSD(T) theory. As expected we can observe for all distances the remarkable agreement between the values obtained at levels CCSD(T)-F12/ aug-cc-pVTZ and CCSD(T)/aug-cc-pVQZ. In all cases the distances calculated at these two levels are 0.002 to 0.004 angstroms shorter than those obtained at the CCSD(T)/aug-cc-pVTZ level often used to correctly deal with the geometrical structure of larger molecular systems.

Table 2 reports the harmonic vibrational wavenumbers obtained at the same different levels. The main assignments are also described for the description of the modes. Although not directly comparable, the experimental values are also recorded in the same table because of their expected low anharmonicity.

As for the structural parameters, the same behaviour can be observed for the calculation of the harmonic frequencies, with the CCSD(T)-F12/aug-cc-pVTZ values are closer to those of CCSD(T)/augcc-pVQZ. All the frequencies calculated at the highest calculation levels, with the exception of v4, being on average about ten cm -1 higher than those calculated at the CCSD(T)/aug-cc-pVTZ level. This is particularly important for the lowest set of modes where this difference is greater than the expected anharmonicity values.

3-2 Anharmonic wavenumbers and IR intensities for the fundamental modes

We report in Table 3 the anharmonic wavenumbers and the IR intensities obtained for the fundamentals. For comparison the results obtained, with the same potential, at VCI and VPT2 levels are compared too with the experimental values. Looking at the table 3, even though the VPT2 values are of very good quality, it appears that the wave numbers calculated at the VCI level remarkably reflect all the experimental values. The mean deviation between the VCI values and the experiment is less than 2 cm -1 (1.6 cm -1 ) with a maximum deviation of 3 cm -1 for the frequency v2. Concerning the IR intensities it is important to note that the most intense bands experimentally are also the most intense in the calculation, that their values are very close to the measured quantities except for the v2 mode for which the VCI and the VPT2 determinations are overestimated with respect to the experiment al value.

Considering the quality of the results obtained at the VCI level on the fundamental modes, it seemed interesting to us to propose a complete theoretical analysis of the IR spectrum of HC3N between 200 and 4600 cm -1 . This analysis will be carried out both on the calculation of frequencies and intensities and will be as far as possible compared to the experimentally observed values.

3-3 Analysis of the IR spectrum between 200 to 4600 cm -1

We have reported in Table 4 the complete IR spectrum of HC3N from the values of IR wavenumbers and intensities calculated at the VCI level. The spectrum thus calculated allows the assignment of all the bands (fundamentals, overtones, combinations, hot bands) expected in the region from 200 to 4600 cm -1 . This calculation is therefore an essential tool for the identification of all the bands observed experimentally. It therefore complements the IR analysis we will do by spectral region since these have either been carefully experimentally studied or partially or not studied at all. In order to propose an assignment to the observed bands, these are shown in Table 4 in correspondence with the theoretical values on the basis of the calculated wavenumbers and intensities.

The analysis of our calculations will be carried out in four spectral regions:

-the first one between 200 and 1350 cm -1 which has been well studied experimentally, -the second between 1400 and 2800 cm -1 ,less well studied experimentally than the first one, and which presents combination bands of not negligible intensity, -the third between 2900 and 4000 cm -1 where, in addition to the intense band characteristic of the CH valence mode, other weak absorption bands appear.

-the fourth one finally beyond 4000 cm -1 never studied experimentally.

3-3-1 Region 200-1350 cm -1

With the exception of the lowest low-intensity band v7 (222 cm -1 ), this region has been particularly well studied experimentally, in particular the v6 and v5 modes with high and medium intensities respectively centred around 499 and 663 cm -1 . Our calculations remarkably reproduce these three bending modes v7, v6 and v5 at 222, 496 and 662 cm -1 with intensities of 0.02, 12.2 and 95.1 km/mol respectively in good agreement with the experimental values of Jolly et al [START_REF] Jolly | New infrared integrated bands intensities for HC3N and extensive line list for the v5 aand v6 bending modes[END_REF].In this region recently studied by Bizzochi et al. [START_REF] Bizzochi | Rotational and High-resolution Infrared Spectrum of HC3N: Global Rovibrational Analysis and Improved Line Catalog for Astrophysical Observations[END_REF] our calculations show the existence of 2v7 and 3v7 overtones at 451 and 676 cm -1 respectively with mean intensities and that correspond to the vibration-rotation bands observed around 443 and 633 cm -1 , this latter very close to the fundamental band v5 and which therefore borrows from it a part of the IR intensity. Around these bands our calculations also predict the existence of the hot bands v7--->v6+v7 and v6--->2v6 located at 495.4 and 497.9 cm -1 and v5-->2v5 and v7-->>v5+v7 calculated at 649.2 and 689.7 cm -1 respectively. These bands could correspond to the bands of the spectrum simulated by Jolly et al. respectively at (497.9 and 514.4 cm -1 ) and (649.8 and 669.8 cm -1 ). The v6+v7 combination band calculated at 721 cm -1 with an intensity of 8.5 km/mol. corresponds to that observed at 720 cm -1 with a lower intensity of the order of 3 km/mol. Below 1000 cm -1 , in addition to the fundamental band v4 associated with the v(C-C) valence mode, very well described by our calculations (860 cm -1 ; 0.16 km/mol vs. 862 cm -1 ; 0.17 km/mol experimentally), we also propose the band of mean intensity attributed to the 3v7 overtone at 676 cm -1 and the combination bands v5+v7 and v6+2v7 of near-zero intensity at 901 and 959 cm -1 respectively. These three bands could correspond to the three signals observed by Bizzochi et al [START_REF] Bizzochi | Rotational and High-resolution Infrared Spectrum of HC3N: Global Rovibrational Analysis and Improved Line Catalog for Astrophysical Observations[END_REF] at 663, 885 and 946 cm -1 respectively. The overtone 2v6 calculated at 994 cm -1 with a very low intensity of 0.02 km/mol could explain the two bands observed respectively at 1013 and 998 cm -1 by Jolly et al. [START_REF] Jolly | New infrared integrated bands intensities for HC3N and extensive line list for the v5 aand v6 bending modes[END_REF] and Bizzochi et al. [START_REF] Bizzochi | Rotational and High-resolution Infrared Spectrum of HC3N: Global Rovibrational Analysis and Improved Line Catalog for Astrophysical Observations[END_REF].Above 1000 cm -1 , still in this same spectral area, our calculations confirm the presence of the combination band v5+v6 and the overtone 2v5 respectively calculated at 1176 and 1311 cm -1 corresponding to the bands observed at 1163 and 1314 cm -1 but with much higher intensities. As for the band observed by Jolly et al. at 1090 cm -1 , it could correspond to one of the two bands calculated at either 1084 or 1095 cm -1 and respectively assigned to the v5+ 2v7 or v4+v7 combination bands, the former being much more intense than the latter. To finish with this first spectral region, our calculations indicate the presence at 1232 cm -1 of a low-intensity v7+2v6 combination band, a band not observed by the experimentalists so far because of its very low intensity.

3-3-2 Region 1400-2800 cm -1

In the first half of this spectral region, up to 2000 cm -1 , a single band at 1970 cm -1 of very low intensity was observed. Our calculations place it exactly at 1970 cm -1 with an intensity of 0.04 km/mole and assign it to the 3v5 overtone. Very close, but not active, our calculations predict the presence of the 2v4+v7 combination band at 1962 cm -1 . Finally, two other bands not observed so far, are predicted by our calculations at 1543 and 1717 cm -1 and are respectively assigned to the v4+v5 combination mode and the 2v4 overtone. If the first band of very low intensity could not be observed, it would seem desirable that the second band with a calculated intensity of 0.29 km/mole could be sought around 1720 cm -1 .

Beyond 2000 cm -1 and up to 2800 cm -1 our calculations predict the existence of ten bands which should be observable whereas only five are observed among that the two fundamental bands v3 and v2 corresponding respectively to the valence modes of the CC and CN triple bonds. Even if the intensities are not perfectly reproduced our calculations locate for v3 and v2 the bands at 2079 and 2277 cm -1 in excellent agreement with the experimental observations at 2079 and 2274 cm -1 .

Excellent agreement can also be noted between the bands observed at 2494, 2575 and 2766 cm -1 with respective intensities of 0.34, 0.17 and 0.31 km/mole and the bands calculated at 2505 cm -1 (0.14 km/mole) and the two doublets (2588 and 2569 cm -1 ) and (2781 and 2753 cm -1 ) with respective overall intensities of 0.37 and 0.48 km/mole. Thanks to the calculation, these bands can respectively be allocated to the combination v2+v7 for the first one, and to (3v4; v3+v6) and (v3+v5; v2+v6) for the two doublets. Finally, not corresponding to any experimentally observed signal, our calculations predict in this region three additional bands at 2211 cm -1 (0.44 km/mole), 2307 cm -1 (0.06 km/mole) and 2534 cm -1 (0.06 km/mole) assigned to the combinations v4+2v5, v3+v7 and v3+2v7 respectively.

3-3-3 Region 2900-4000 cm -1

Except for the fundamental band v1 associated with the very intense CH valence mode (61.4 km/mole) observed at 3327 cm -1 , two other bands of low intensity are reported by Jolly et al. [START_REF] Jolly | New infrared integrated bands intensities for HC3N and extensive line list for the v5 aand v6 bending modes[END_REF] in this region at 3133 and 3549 cm -1 . Our calculations locate the fundamental remarkably well at 3329 cm -1 with a high intensity of 84.1 km/mole and predict for the other two signals the frequencies of 3151 and 3562 cm -1 respectively assigned to the combinations v2+v4 and v1+v7 in accordance to the experimental assignments. Five other bands of low and very low intensity proposed by our calculations were not observed. Two of them assigned to the simple combinations v3+v4 and v1+v5 are respectively calculated at 2952 and 3988 cm -1 with intensities of 0.07 and 1.60 km/mole respectively. The other three bands are associated with the complex combinations v2+v6+v7, v2+v5+v7 and v3+v5+v6 calculated at 3034, 3183 and 3291 cm -1 respectively with very low intensity signals (0.32, 0.21 and 0.12 km/mole).

3-3-4 Region 4000-4800 cm -1

This last region studied theoretically has not been explored experimentally. One can expect to find signals involving the first overtones of the modes associated with the CC and CN triple bonds (v3; v2) as well as combination bands involving the very intense modes due to the CH valence vibrations (v1) and CCH bending deformation (v5). The two overtones 2v3 and 2v2 are respectively calculated at 4148 and 4535 cm -1 with intensities of 0.03 and 0.16 km/mole. The most intense combination bands involving the v1 and v5 mode are the complex bands v1+v5+v7 on the one hand and 2v5+v1 on the other hand calculated at 4235 and 4636 cm -1 with non-negligible respective intensities of 2.02 and 3.80 km/mole. Finally, our calculations predict that two other bands could be observable in this spectral region. These are the complex combination v1+v6+v7 calculated at 4048 cm -1 and the combination band involving the modes associated with the CC and CN triple bonds, v2+v3, calculated at 4354 cm -1 , the first signal being however much more intense than the second (0.78 km/mole vs 0.01 km/mole).

4-Conclusions

In this study we have carried out, for the first time, accurate theoretical computations concerning the vibrational spectrum of cyanoacetylene HC3N in the mid-Infrared region. On the basis of CCSD(T)-F12/ aug-cc-pVTZ potential and by means of a pure variational method we propose a complete ab-initio modelling of the IR spectrum of HC3N which allows to assign fundamental, combinations, overtones and hot bands between 200 to 4800 cm -1 . Our original results give a new interpretation of the assignments of all the observed bands. We hope that this accurate theoretical work will attract the interest of experimentalists to analyse more precisely the spectrum of HC3N 2 ν7 ν7 ν7 --> ν6+ ν7 ν6 --> 2 ν6 ν5 --->2 ν5 ν7 --->ν5 + ν7 Table 4: Observed and anharmonic (ν(cm -1 ) wavenumbers calculated at CCSD(T)-F12/ aug-cc-pVTZ level from a VCI approach for cyanoacetylene in vacuo. Intensities (km/mol) are calculated within the anharmonic mechanical and electrical scheme. The main assignment is given from the variational method. Observed values are from Jolly et al. [START_REF] Jolly | New infrared integrated bands intensities for HC3N and extensive line list for the v5 aand v6 bending modes[END_REF], values in italic are from Bizzocchi et al. [START_REF] Bizzochi | Rotational and High-resolution Infrared Spectrum of HC3N: Global Rovibrational Analysis and Improved Line Catalog for Astrophysical Observations[END_REF]

Fig 1 :

 1 Fig 1: Propynenitrile atomic numbering

Table 1 :

 1 beyond 1350 cm -1 . Computed geometrical parameters (in Angstroms) for propynenitrile (or cyanoacetylene) in vacuo at different CCSD(T) levels of theory.

		Equilibrium	CCSD(T)/	CCSD(T)-F12 /	CCSD(T)/
		structure
			aug-cc-	aug-cc-pVTZ		aug-cc-
			pVTZ			pVQZ
		C1-C4	1.383	1.381		1.381
		C1≡C2	1.212	1.208		1.209
		C4≡N5	1.167	1.163		1.163
		C2-H 3	1.065	1.063		1.063
	Mode	Main	CCSD(T)/	CCSD(T)-		CCSD(T)/
		Assignment	aug-cc-	F12 /	aug-cc-	Experimental
			pVTZ	Aug-cc-	pVQZ
				pVTZ
			ω	ω		ω	ν

Table 2 :

 2 Computed harmonic wavenumbers (in cm -1 ) for propynenitrile (or cyanoacetylene) in vacuo at different CCSD(T) levels of theory. Comparison with experimental values

	Experimental values	ν(VCI)	ν(VPT2)	Assignment
	222.4 (0.18)	222 (0.2)	224 (0.4)	ν7
	498.8 (9.6)	496 (12.2)	496 (12.8)	ν6
	663.2 (66.1)	662 (95.0)	672 (88.6)	ν5
	862 (0.17)	860 (0.2)	860 (0.1)	ν4
	2079 (2.0)	2079 (0.1)	2077(0.2)	ν3
	2274 (10.0)	2277 (51.1)	2276 (42.2)	ν2
	3327 (61.4)	3329 (84.0)	3309 (51.2)	ν1

a Double degenerated modes b Ref

[START_REF] Bizzochi | Rotational and High-resolution Infrared Spectrum of HC3N: Global Rovibrational Analysis and Improved Line Catalog for Astrophysical Observations[END_REF]

; c Ref

[START_REF] Jolly | New infrared integrated bands intensities for HC3N and extensive line list for the v5 aand v6 bending modes[END_REF]

; d Ref

[START_REF] Borget | Cyanoacetylene Adsorption on Amorphous and Crystalline Water Ice Films: Investigation through Matrix Isolation and Quantum Study[END_REF]

Table 3 :

 3 Anharmonic (ν(cm -1 ) wavenumbers and IR intensities (in parenthesis) expressed in km/mole calculated for cyanoacetylene at CCSD(T)-F12/ aug-cc-pVTZ level from VPT2 and VCI methods within an analytical representation. Intensities are calculated within the anharmonic mechanical and electrical scheme. Comparison is made with the experimental work of Jolly and al.[START_REF] Jolly | New infrared integrated bands intensities for HC3N and extensive line list for the v5 aand v6 bending modes[END_REF]

	Observed	ν(cm -1 )	I(km/mol)	Main assignement
	frequencies
		4636	3.80	2 ν5 + ν1
		4535	0.16	2 ν2
		4354	0.01	ν2 + ν3
		4235	2.02	ν1 + ν5 + ν7
		4148	0.03	2 ν3
		4048	0.78	ν1 + ν6 + ν7
		3988	1.60	ν1 + ν5
	3549 (0.36)	3562	0.08	ν1 + ν7
	3327 (61.4)	3329	84.1	ν1
		3291	0.12	ν3 + ν5 + ν6
		3183	0.21	ν2 + ν5 + ν7
	3133 (0.64)	3151	1.55	ν2 + ν4