Behav Ecol Sociobiol (2010) 64:1449–1459 DOI 10.1007/s00265-010-0960-x

ORIGINAL PAPER

Simulation of information propagation in real-life primate networks: longevity, fecundity, fidelity Bernhard Voelkl & Ronald Noë

Received: 12 November 2009 / Revised: 15 March 2010 / Accepted: 14 April 2010 / Published online: 30 April 2010 # Springer-Verlag 2010

Abstract In many vertebrate species, we find temporally stable traditions of socially learned behaviors. The social structure of animal populations is highly diverse and it has been proposed that differences in the social organization influence the patterns of information propagation. Here, we provide results of a simulation study of information propagation on real-life social networks of 70 primate groups comprising 30 different species. We found that models that include the social structure of a group differ significantly from those that assume random associations of individuals. Information spreads slower in the structured groups than in the well-mixed groups. While we found only a minor effect on the path lengths of the transmission chains, robustness against information extinction was strongly influenced by the group structure. Interestingly, robustness against information loss was not correlated with propagation speed but could be predicted reasonably well by relative strength assortativity—a structural network metric. In those groups where highly pro-social individuals preferentially interact with other pro-social individuals, Communicated by J. Krause Electronic supplementary material The online version of this article (doi:10.1007/s00265-010-0960-x) contains supplementary material, which is available to authorized users. B. Voelkl (*) : R. Noë Ethologie des Primates, Département Ecologie, Physiologie & Ethologie, IPHC (UMR 7178), CNRS–Université de Strasbourg, 23 rue Becquerel, 67087 Strasbourg, France e-mail: [email protected] R. Noë Faculté Psychologie, Université de Strasbourg, 12 rue Goethe, 67000 Strasbourg, France

information was more likely to be lost. Our results show that incorporating group structure in any social propagation model significantly alters predictions for spreading patterns, speed, and robustness of information. Keywords Social learning . Information transmission . Propagation . Tradition . Epidemic modeling

Introduction While most inherited information in animals is assumed to be transmitted genetically from one generation to the next, behavioral traditions are the product of a distinctly nongenetic inheritance mechanism. The most famous examples for behavioral traditions in animals include birdsong dialects (Marler 1952), milk bottle opening in British tits (Hinde and Fisher 1951), tool processing in new Caledonian crows (Hunt and Gray 2003), potato washing in Japanese macaques (Kawai 1965), and termite fishing and nut cracking in chimpanzees (Goodall 1968; Sugiyama and Koman 1979). In all these cases, a novel behavior was presumably invented by one or a few individuals and afterwards adopted by other individuals either by imitation or by other social learning mechanisms like emulation learning or local and stimulus enhancement (Heyes and Galef 1996). Behavioral innovations can spread horizontally (within one generation) and vertically (from one generation to the next). In this paper, we focus on the initial horizontal diffusion process. To support the claim that certain behaviors were in fact acquired socially, behavioral ecologists took a closer look at the spreading pattern of putative behavioral traditions in free-ranging populations. It was argued that a distinction between social and asocial learning processes can be made

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on the basis of the diffusion curve (Cavalli-Sforza and Feldman 1981; Boyd and Richerson 1985; Laland et al. 1996). Social learning should result in accelerating diffusion curves, such as the logistic, the exponential, or the hyperbolic sine as the number of informed individuals increases over time, whereas asocial learning (e.g., individual learning by trial and error or instant learning) should be characterized by linear diffusion curves. However, Laland and van Bergen (2003) argued that if variation in the individuals’ learning time is incorporated in the model, even asocial learning will result in accelerating diffusion curves. Consequently, examinations of available datasets— mostly from primate groups—lead to conflicting results. While Lefebvre (1995) claimed to have found support for the social-transmission hypothesis, Reader (2004) reported that the curves were quite variable and often did not fit the predictions sufficiently. Conventional models treat the proportion of informed individuals as a continuous variable which allows representing the process by ordinary differential equations. This evokes two basic problems, however. Firstly, primate groups are usually small, often consisting of only a dozen individuals or even less. With such small groups, modeling the numbers of informed individuals as continuous variables might give only poor approximations. In such a case, a Markov process with discrete group size and time would be more appropriate. Secondly, and as we think more importantly, primate groups might not represent well-mixed populations with respect to social learning opportunities, which means that the likelihood that an individual learns from another one differs between dyads (Coussi-Korbel and Fragaszy 1995). To take unequal interaction rates into account, researchers have started to model animal groups as social networks. By relaxing the assumption of wellmixed populations, network approaches to information transmission produced more accurate predictions for the expected propagation patterns (Watts and Strogatz 1998; Kincaid 2000; Valente 2005; Voelkl and Noë 2008; Franz and Nunn 2009; Vital and Martins 2009). Summing up the pre-requisites of an entity to qualify as a good replicator Dawkins (1982) coined a slogan reminiscent of the one brought to fame during the French Revolution: “Longevity, Fecundity, Fidelity”. While these requirements were originally postulated for genetic inheritance, they apply equally well to non-genetic, cultural inheritance systems. Thus, a novel behavior has to meet three basic criteria in order to lead to a new social tradition: (1) it should evoke copying by social learning (fecundity), (2) the resulting behavior should closely resemble the model being copied (fidelity), and (3) the behavior should be shown over a considerable time span (longevity). While these three properties are basically determined by the species’ learning capacity and the nature of the behavior,

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it has been postulated that the social structure of a group might have a strong influence as well (Coussi-Korbel and Fragaszy 1995). In this study, we put this claim to the test after complementing the verbal arguments with quantitative predictions. For this purpose, we simulated information flow on social networks based on empirical data derived from observations of primate groups. We compare these results with the predictions for well-mixed groups with respect to propagation speed, path lengths of the transmission chains, and resilience against information extinction. We show that different assumptions lead to different results and discuss why this should change our expectations for the outcome of social transmission studies.

Method Data collection We studied information propagation on interaction networks of 70 primate groups. The networks were derived from dyadic interaction matrices that were partly taken from the literature and partly from unpublished material either collected by the authors or made available by colleagues. The dataset comprises 30 species of 17 different genera, including lemurs, New World and Old World monkeys, and apes. Thirty-four groups were kept in captivity, six in parks or larger outdoor enclosures (semifree-ranging), and 30 were observed in the wild. A list with details of all groups (species, group size, living conditions, and source) is provided online (Online Resources) and a detailed network analysis of these groups is given in Kasper and Voelkl (2009). The group sizes in our sample range from four to 35, with a median of nine individuals per group. The behaviors that we considered were sociopositive interactions (grooming, social play, and body contact), which are indicative of high levels of social tolerance. These behaviors have in common that they imply close proximity of the involved individuals, usually over a prolonged time period. For our purpose, we defined a group as the set of those individuals who are directly or indirectly connected with each other through socio-positive interactions. That means that single individuals or small subgroups of individuals that never showed socio-positive interaction with other group members were excluded from the sample. In two cases, this procedure led to the removal of a single individual, in two cases of two individuals, and in one case of an isolated subgroup of four individuals. For the calculation of independent contrasts, phylogenetic distances of the primate species were taken from Purvis (1995) and completed with data from Schneider et al. (2001), CortésOrtiz et al. (2003) and Steiper and Ruvolo (2003).

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Simulation The information transmission scenario is basically equivalent to an SI model (Hamer 1906 reviewed in Hethcote 2000) in epidemic modeling, with only two classes of individuals, susceptible and infected, whereby individuals can only move from the former to the latter class. For the case of well-mixed groups, this can be modeled as a Markov process (Online Resources). However, this approach is not feasible for the heterogeneous group structures found in real primate groups. To model information flow in the primate networks we used, therefore, the graph-based simulation approach first suggested in Voelkl and Noë (2008). Following this approach, we mapped the social networks as graphs where the vertices represent individuals and edges between the vertices represent the “connections” between the individuals (Wasserman and Faust 1994; Newman 2003). In our case, the edges represent spatiotemporal associations of a kind that enable the transmission of a specific information unit. We assume that dyadic sociopositive interactions qualify as such associations or are proportional to the rate at which such associations are formed (see “Discussion” section). The weight of an edge represents the likelihood that such an association is formed between the two individuals of a specific dyad. The graphs, which are based on the empirical data of real primate groups, represent our “structured” groups. We constructed three further types of graphs for each group. (1) A complete graph with edge weights of unity. This graph represents the case of a “well mixed” group, where all individuals are equally likely to interact with each other. (2) A “topological” version, with the same connections but without edge weights. (3) A “randomized” version, where edge weights were conserved but connections were randomly reshuffled (with the restriction that the resulting graph was still connected, Fig. 1). Using the latter two graph types allowed us to split up the heterogeneity present in the empirical data into two components: heterogeneity due to the pattern of the connections (preserved in the topological version) and heterogeneity in the strength of the connections (preserved in the randomized version). The simulations were always carried out on all four graph types with 1,000 repetitions per group and graph type. We simulated information transmission as a random process with the following rules: (1) We represent the knowledge status of the individuals by assigning weights to the vertices of a graph G. At the onset of the simulation, all individuals are naïve, hence their vertex weights w(v) =0. (2) Thereafter, we randomly choose one individual as inventor by assigning one bit of information to that

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(a)

(b)

(c)

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Fig. 1 The four graph types: a structured graph based on the empiric interaction data with edge weights indicating interaction frequencies, b topological graph with the same edge set as the structured graph but edge weights of unity, c randomized graph with randomly reshuffled edges but preserved edge weights, and d complete graph with edge weights of unity representing the well-mixed group

individual, w(vr) =1. Individuals with a vertex weight of 1 are henceforth called “informed individual”. We assume that this invention is made only once, as this is suggested to be often the case for inventions in animal groups (Laland and Reader 1999; Laland and Van Bergen 2003). Scenarios with multiple inventions might also be of interest but are beyond the scope of this study, as—for heuristic reasons—we want to keep the model as simple as possible. (3) In each round, an interaction will take place with probability p(int) =0.01 n, where n is the number of individuals in the group. (4) If indeed an interaction takes place, two individuals are randomly selected to interact with each other, with the likelihood that this specific dyad is selected being determined by the weight of the edge connecting the corresponding vertices of the graph. If both individuals are naïve, they will stay naïve. If one individual is naïve and the other individual is informed, then the naïve individual will learn the skill and thereafter both individuals are informed. If both individual are informed, nothing changes.
w ðv Þ ¼

0; 1;

if wðuÞ ¼ wðvÞ ¼ 0 if wðuÞ ¼ 1 _ wðvÞ ¼ 1

8fu; vg 2 V ðG; wÞ

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(5) The simulation is terminated as soon as all individuals have acquired the information. To study the longevity of traditions we ran another set of simulations with an additional rule (4b): After each round a randomly chosen individual will forget the information with a probability of p(forget) =0.1 p(int) and become a naïve individual again. The value for p(forget) was chosen because it is small enough that the information will not always die out immediately but large enough to make this happen at least from time to time. This scenario corresponds to the SIS model (Kermack and McKendrick 1927) in epidemic modeling. Furthermore, we had to adapt the stopping rule (5): instead of terminating the simulation as soon as all individuals are informed (a criterion that might not be reached when the forgetting rate is reasonably high), the simulation was terminated after 10,000 rounds.

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respectively, minus the average propagation time for the well-mixed group. Proportions of forgetting were arcsine transformed (Sokal and Rohlf 1997). We constructed a propagation tree with the inventor as the root for each simulation run. We calculated the average P 1 path length for the propagation tree as apl ¼ n1 d ðviÞ where n is the number of vertices and d(vi) is the distance of vertex vi from the root. The average path length is the average number of transmission processes the information needed to reach any of the other group members. To quantify the structuring of the primate networks into subgroups, we used the community modularity metric Q (Table 1) in combination with an agglomerative algorithm suggested by Clauset (2005). A discussion of the robustness of this measure for the presented dataset is given in Kasper and Voelkl (2009).

Statistical analysis

Results

The simulation and all statistical analysis were carried out with Mathematica 6.0 from Wolfram Research Inc. As our dataset stems from interaction matrices from 30 primate species that are phylogenetically related to each other to different extents, the data cannot be regarded as statistically independent. We therefore calculated phylogenetic independent contrasts using the method proposed by Felsenstein (1985). We assumed a phylogenetic distance of zero for groups belonging to the same species. For all main results, the units of measurement were therefore independent species contrasts (n=29). In addition, we performed several post hoc analyses of the influence of certain network characteristics on information flow and on the stability of acquired skills using individual networks (n=70) as data points. Whenever we made multiple means comparisons, we performed a correction for the critical alpha level of 0.05 using the Dunn-Šidák method (Sokal and Rohlf 1997). The corrected alpha level is given together with the test statistics and contrasts were reported significant only if they were below α′. Mean values are given together with the standard deviation. Propagation time was expressed in rounds of the simulation. After we ran 1,000 simulations for each condition, we calculated the arithmetic means for all N-1 transmissions to generate averaged propagation curves. We took the propagation half-time, which is the average time in rounds needed until half of the group acquired the information, to arrive at a single measure describing the propagation speed within a group. For even group sizes, we interpolated this value by averaging over the two adjacent values. To facilitate the comparison of the four different graph types, propagation time is expressed as the net propagation time, which is the average propagation time for the structured, topological, or random graph,

Propagation speed Information spread faster in the well-mixed groups than in the structured groups (paired t test, t=2.52, n=29, p= 0.018). The median propagation half-time was 1.39 times longer in the structured systems than in the corresponding well-mixed groups. A group-level comparison of the topological and the randomized scenario with the structured and the well-mixed groups is very informative in two respects. First, it shows that information spreads slower in both the topological (paired t test, t=6.40, p<0.0001, α’= 0.01) and the randomized (paired t test, t=8.94, p<0.0001, α’=0.01) groups than in the well-mixed groups. This indicates that both the unequal distribution of interaction links and the unequal distribution of interaction frequencies observed in the primate groups had a negative effect on the spread of information. At the same time, information spread significantly faster in the topological (paired t test, t=3.58, p=0.0006, α’=0.01) and the randomized (paired t test, t= 3.06, p=0.003, α’=0.01) groups than in the structured groups, indicating that the previous effects do not cancel out but add up, making the structured group the least effective in terms of information propagation (Fig. 2). For a well-mixed group, one would expect that the propagation time increases proportional to the log of the group size (Hethcote 2000). A regression analysis of the simulated data for the well-mixed scenario against group size confirms this by giving a fit of 99.9% for the log-linear model. For the structured groups, however, we find only a modest fit that does not reach significance after Dunn-Šidák correction for multiple tests (analysis of variance; ANOVA: p=0.03, df=69, F=4.85, R2 =0.05; α’=0.01). As this is a striking difference, we tried to

At the individual level, social interactions can be distributed evenly between all interaction partners or unevenly—with some partners receiving more attention than others. This inequality is quantified by the edge weight disparity Closeness centrality measures how close an individual is to others in the network in terms of its ability to quickly interact with others. It is assessed on the basis of geodesic distances and does not only depend on direct but also indirect edges The eigenvector centrality of an individual is based on the sum of the centralities of its neighbors which enables an individual to gain high centrality in two different ways: it can itself have strong relationships with many other individuals or be in contact with those individuals that are most central. The eigenvector centrality is the eigenvector of the edge weight matrix corresponding to the largest Eigenvalue The clustering coefficient characterizes the local group cohesiveness by evaluating the extent to which vertices adjacent to any vertex are also adjacent to each other. It can be interpreted as a measure of individual sociality Community modularity measures the degree of fragmentation of a group into subgroups by comparing interaction frequencies within and between subgroups Relative strength assortativity measures to what extent individuals interact with other individuals with a vertex strength similar to their own (i.e., if individuals which have many social interactions preferentially interact with other individuals who also have many social interactions) The concept of network flow can be visualized as a system of pipes where edges are pipes with diameters given by the edge weights. The maximum flow between any two vertices is the amount of liquid that could flow through any pipes from one vertex to the other Resilience is a concept closely related to network flow. It measures how much the network flow will be reduced by randomly (or systematically) removing one vertex at a time

Edge weight disparity

j¼1

aij wij sj

Evaluation based on the max-flow/min-cut theorem (Ford and Fulkerson 1956)

bws;i ¼ si

PN 1

ccwi ¼ si ðk1i 1Þ  P ðwij þwil Þ aij ail ajl  j;l P 2 Q ¼ ji pii  oi 2

0

Albert et al. 2000

Newman 2004

Kasper and Voelkl 2009

Newman and Girvan 2004

Barrat et al. 2004

Bonacich 1972

Barthelemy et al. 2005

P ci ¼ 1l Nj¼1 wij cj l ¼ const:

wij si

Reference

Wasserman and Faust 1994

j¼1

PN

 2

Cc ðvi Þ ¼ PNðN 1Þ d ðvi ;vj Þ j¼1

Y2 ðvi Þ ¼

Definition

The social network is represented as a graph G(E,V) with an adjacency matrix A with elements aij equalling one if there exists a connection (observed interaction) between individuals i and j and zero P otherwise. N ¼ jV j: Edge weights P representing the frequency of interactions between two individuals are given by the matrix W with elements wij. The strength of a vertex i is given by si ¼ Nj¼1 aij wij and its degree as ki ¼ Nj¼1 aij . Closeness centrality: d(i,j) is a distance function giving the number of edges in the geodesic linking individuals i and j. Community modularity: pii is the proportion of edges within subgroup Si and oi is the proportion of edges that start from subgroup Si. For the partitioning of a graph into subgroups several algorithms have been suggested (e.g., Clauset 2005; Newman 2006).

Resilience

Network flow

Strength assortativity

Community modularity

Clustering coefficient

Eigenvector centrality

Closeness centrality

Description

Measure

Table 1 Network measures

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T ime (rounds)

Fig. 2 Graph representations (a, c, e, g) and diffusion curves (b, d, f, h) for four selected primate interaction networks. Graph representations: vertices represent individuals and edges represent socio-positive interactions between individuals. The thickness of the edges is proportional to the interaction frequencies. Networks are based on socio-matrices of a captive Samiri sciureus (Vaitl 1977), c wild Papio anubis x hamadryas (Sugawara 1979), e wild Pan troglodytes (Sugiyama 1969), and g semi-free ranging Macaca fuscata (Hasegawa and Hiraiwa 1980). Diffusion curves give the average number of informed individuals in a given round for the structured graph (solid line), the complete graph (dotted line), the randomized graph (dashed line), and the topological graph (gray line)

identify descriptors of network structure that could explain the variance in propagation speed by performing regression analyses with a set of network measures (eigenvector centrality, closeness centrality, network flow, resilience, community modularity, weighted clustering coefficient, disparity, and relative strength assortativity). However, we could not find any single parameter that would account for more than 15% by itself, or any linear combination of

parameters that would account for more than 21% of the total variance. Thus, for the structured groups, neither group size nor any of the applied network metrics could qualify as a useful predictor for propagation time. For 13 groups, net propagation time curves (Fig. 3) were slightly U shaped in the sense that they showed a local minimum close to xN/2 (Fig. 3c, d). In nine cases, this local minimum was also the absolute minimum. Such a U-shaped curve can be expected for strongly centralized or star-like networks with one highly attractive individual in the middle to which all other individuals are connected (Voelkl and Noë 2008). Centralized networks are characterized by a highly skewed vertex strength distribution. Taking the ratio of the second to the third central moment as a measure for skewness, we found the skewness of those groups that produced U-shaped propagation time curves to be significantly higher than for the remaining groups (MannWhitney test, p=0.006, two-tailed, n1=13, n2=57, U= 551). Thus, as predicted, centralized networks with one or a few highly prominent individuals in the center did indeed show increased net propagation speed in the middle phase of the propagation process. In other words, these central individuals seem to function as accelerators for information propagation. Path length The path length is the number of transmission processes along a transmission chain from the inventor of a new trait to a specific target individual. Average path lengths, i.e., the median of the path lengths from the inventor to all other group members, were significantly longer in the structured scenario than in the well-mixed scenario of equal size (paired t test, n=29, t=2.61, p=0.014).

2

Net propagation time (100 rounds)

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Fig. 3 Net propagation curves for the four groups depicted in Fig. 2. Net propagation times were retrieved by subtracting average propagation times for the well-mixed scenario from average propagation times for the structured scenario

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Information loss In the second simulation experiment, we introduced a propensity to forget the acquired information after some time. As we model the case of a single invention, this means that if there is at some point of time only a single informed individual and this individual forgets the information, then the information is lost. To quantify the risk of information loss for a group, we calculated the proportion of simulation runs in which the information went extinct within 10,000 rounds. We found that in the structuredscenario groups were much more prone to loosing the information than in the well-mixed scenario (paired t test, t=4.05, n=29, p=0.0004).

average path length

5

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group size residual average path length

The mean average path length of all 70 groups was 2.85 (±0.56) for well-mixed groups, 2.95 (±0.61) for the topological versions of the systems, 2.90 (±0.61) for the randomized graphs, and 3.07 (±0.65) for the structured systems. Although the differences between the four scenarios were quite small, they were relatively consistent and group comparisons revealed significant differences between, structured groups and their topological equivalents (paired t test, t=9.91, p<0.0001, α’=0.01), structured and randomized groups (t=6.17, p<0.0001, α’=0.01) as well as between the topological variants and well-mixed groups (t=6.25, p<0.0001, α’=0.01) and randomized and well-mixed groups (t=10.94, p<0.0001, α’=0.01). Thus, both reduced connectivity and variation in interaction frequencies increased the average path length, but the effect was even stronger in the structured systems with both types of heterogeneity. The larger a group is, the more likely there are long transmission chains. More precisely, we would expect that the average path length increases linearly with the log of the group size. A log-linear regression of the average path length for the structured groups against the group size suggests that group size can in fact explain over 95% of the variation in the average path length (ANOVA: p<0.0001, df=69, F=1410.7, R2 =0.95, Fig. 4a). As this result is not surprising, we were more interested in finding an explanation for the remaining variance. Taking the residuals of this regression, we found that community modularity explained over 35% of the remaining variance (ANOVA: p<0.0001, df=69, F=38.92, R2 =0.35, Fig. 4b). Community modularity is a network measure that quantifies the structuring of a group into subgroups. That means that the average length of the transmission chain is influenced not only by group size but also by the group structuring, with groups containing clearly distinctive subgroups producing longer transmission chains. Note, however, that predicted differences are quite small and might be difficult to detect in noisy data.

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community modularity Fig. 4 a Average path length for the 70 primate networks plotted against group size. b To account for the influence of group size, the residuals of the linear regression of average path length against group size are plotted against community modularity

A group-level comparison shows higher levels of information loss in both the topological (paired t test, t= 5.07, p<0.0001, α’=0.01) and the randomized (t=7.84, p< 0.0001, α’=0.01) than in the well-mixed scenario. At the same time, information extinction was more likely in the structured groups than in the topological scenario (t=6.37, p<0.0001, α’=0.01) and the randomized scenario (t=3.58, p=0.0006, α’=0.01). Thus, the picture is basically the same as for propagation speed and average path length. However, linear regressions of the proportion of runs with information extinction with the parameters group size (ANOVA: p=0.006, df=69, F=7.96, R2 =0.09, α’=0.017), path length (ANOVA: p=0.019, df=69, F=5.77, R2 =0.06, α’=0.017) and propagation speed (ANOVA: p<0.0001, df=69, F= 26.96, R2 = 0.27, α’= 0.017) point to relatively weak relationships only. Before analyzing the data, we expected to find that the risk of information getting lost would be strongly correlated with the propagation half-time, i.e., the slower the spread of information, the higher the risk of irreversible loss. It came therefore as a surprise that the propagation half-time accounted for 27% of the variance in the proportion of

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extinctions only. To see whether there are other features of the structured graphs that can explain the variance in the extinction risk of a system, we plotted the extinction data against the eight different network descriptors listed above. Out of these network measures, only the relative strength assortativity showed a reasonably high correlation with forgetting. For the 70 groups, relative strength assortativity could explain 49% of the variation in the proportion of extinctions (ANOVA: p<0.0001, df=69, F=66.85, R2 =0.49, α’=0.006, Fig. 5). That means that in systems in which individuals with high interaction rates preferentially interact with other individuals with high interaction rates it is more likely that information is lost again.

Discussion In this study, we investigated the social propagation of information within animal groups addressing three basic questions: (a) the efficiency of propagation, measured by its propagation speed, (b) its consistency in appearance, estimated on the basis of the path length, and (c) its persistence over time, investigated by simulating forgetting. We will discuss all three questions in this order. (a) Efficiency The simulation experiment showed that information propagation was markedly slower in the structured groups than in well-mixed groups of the same size. After simulating information flow on the structured networks we tried to reveal the effect of different components of structure: (1) by using un-weighted topological versions of the networks (thus removing the variance in weight of the edges) and (2) by randomly reshuffling the edges while keeping their number and

weight unaffected (thus removing the topology specific to primate groups). In both scenarios, propagation speed slowed down, but not as strongly as in the original structured systems that contained both types of heterogeneity at the same time. Interestingly, propagation speed could not be predicted by group size. It seems that the effects of structural heterogeneity were so strong that they completely overshadowed any influence that group size could have had on propagation speed. This was at least the case for our sample that included groups from group size four to 35. It is of course possible that we would find at least a weak relationship if we include data from much larger groups. (b) Consistency The path length was also influenced by the group structure and, as with propagation speed, the effect was strongest for the original weighted structures. We were interested in the average path length, because we hypothesized that it might have important consequences for the fidelity of the transmitted information. If we assume that the transmission process from one individual to the other is a source for “transcription errors”, then the length of a transmission chain will influence the extent to which the behavior is modified. Therefore, the average path length of the information chain will provide us with an estimate for the expected variability of a socially learned behavior. Although we found statistically significant differences in the average path length between the well mixed and the structured groups, we want to emphasize that the effect size was very small. That means that for any two groups of the same size we would expect very similar levels of consistency. In this respect, all four scenarios resemble “small worlds” (Watts and Strogatz 1998) where the mean path length increases proportionally with the logarithm of group size.

relative strength assortativity

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Information could go extinct at the group level when we introduced the possibility that individuals sometimes ‘forget’ information. We get a measure for the expected longevity of an acquired piece of information by counting how frequently this happens. While the risk of extinction was found to be higher in the structured systems than in the well-mixed scenarios neither group size nor propagation speed turned out to be good predictors for a group’s resilience against information extinction. Searching for a structural network measure, we found that relative strength assortativity could explain 49% of the variance in the extinction data. However, as this relationship was found post hoc by regressing extinction risk against a set of network measures, this outcome has to be interpreted with care and requires further validation.

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General discussion The largest primate group in our sample comprised 35 individuals. While average group sizes of most primate species are much smaller, there are a few species (e.g., macaques, baboons, mandrills, or snub-nosed monkeys) with group sizes of more than 100 or 200 animals. We cannot say with any certainty how these large groups would fit into the picture, but as such groups are often hierarchical organized and subdivided into distinct matrilines, we would expect high levels of structuring and consequently strong effects on information propagation and stability. This should also be true for the perhaps most famous example of cultural propagation of a behavioral skill in primates: the sweet potato washing of Japanese macaques originally reported by Kawai (1965). Researchers that re-analyzed the data raised doubts that this behavior was in fact socially transmitted, because the observed propagation patterns did not fit predictions of simple SI-models (Galef 1992). However, if we consider that macaque bands are usually strongly structured into matrilines, which are themselves structured, it is not surprising that propagation patterns differ from predictions for well-mixed groups. It might not be possible to re-analyze these historical data as the information about the social relationships might be incomplete. However, future analyses of information propagation studies should not base their predictions on the assumption of equal mixing but should take effects of heterogeneous structuring into account. According to Hinde, the social structure of a group can be described “in terms of the properties of the constituent relationships and how those relationships are patterned” (Hinde 1983, p.6). Of course, this social structure is not entirely static and changes over time. As we did not incorporate dynamical changes in the group structure, our model is basically meant for social adoption processes that take place on a shorter time scale than structural changes— this can be anything from a few days to a few months. We took data of social interactions that require high levels of social tolerance (grooming, body contact, and social play) and we assumed that information will be transmitted between individuals with likelihoods proportional to the frequencies of these socio-positive interactions. These interactions have been shown to be good predictors for overall spatial proximity (e.g., Troisi et al. 1989; Arnold and Whiten 2003; Ferreira et al. 2006) and evidence that proximity can indeed foster social learning comes from both observational and experimental studies (Kawai 1965; Hausberger et al. 1995; Terkel 1996; Smith et al. 2002; Perry et al. 2003; van Schaik et al. 2003; Bonnie and de Waal 2006). Testing this hypothesis in starlings, Boogert and colleagues (2008) found no significant effect of social association on the propagation pattern of a novel foraging

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technique. However, the authors concede that their negative result might also be due to the low statistical power (only three groups of five individuals), the relative homogeneous group structure or the restricted space of the laboratory condition. Thus, while the “proximity conjecture” is a commonly used premise in models of information propagation (e.g., Coussi-Korbel and Fragaszy 1995; van Schaik et al. 1999; Croft et al. 2005; Nunn et al. 2009) and while there is certain supportive evidence for it, this evidence is not unambiguous and needs further empirical confirmation. Of course, social proximity might not be necessary for all types of social learning. For example, information about the presence of predators conveyed through loud alarm calls might be equally accessible for all group members (Hauser 1997). However, there is also social information that requires high levels of proximity or social tolerance between sender and receiver for its transmission. Imitation of details of a novel foraging technique might require observation at a close distance (Voelkl and Huber 2007), or information about ingested food might require direct body contact (Drapier et al. 2002; Kasper et al. 2008). For such cases, we argue that transmission likelihoods should accord with frequencies of socio-positive behavior due to the simple logic: “the more time two individuals spend together, the more likely that one individual will have the opportunity to learn something from the other”. With this simulation study, we could demonstrate that group structure can have significant influence on spreading patterns and stability of socially transmitted behavior. Not all observed effects were anticipated—some are even counterintuitive. This corroborates our notion that incorporating details about the social structure into models might be essential to generate realistic predictions. As social structure differs considerably among species, populations and groups, this inevitably comes at a cost of loss of generality, because we will have to revert to numeric simulations tailored to each specific case rather than using a more general analytic approach. We first developed a method that allowed us to make predictions how the social structure of a group should influence the propagation patter of socially transmitted information (Voelkl and Noë 2008). Thereafter, we investigated whether group structures of a well studied taxon— primates—are diverse enough to detect such differences and how the expected propagation patterns relate to other social network metrics of these groups (this study). The logical final step will be to test the predictions stemming from this model empirically. How this can be done experimentally has already been demonstrated by Boogert and colleagues (2008), though preferentially testing should take place under more natural settings and encompass larger groups.

1458 Acknowledgments We thank Filippo Aureli, Liesbeth Sterck, Claudia Kasper, Christine Schwab and three anonymous reviewers for helpful comments on the manuscript. Financial support: EU-FP6 NEST project GEBACO (28696) and Austrian Science Fund (FWF) project J-2933-B17.

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