Uncovering complementary information sharing in spider monkey collective foraging using higher-order spatial networks

A driver of the structure and dynamics of networked systems is their adaptive capacity to collectively process information in ways that none of the components would be able to alone¹. A prevalent feature of this information processing is its distributed nature: different components complement one another in the information they contain, which they aggregate via the network of interactions. This complementarity in information aggregation is a hallmark of collectively intelligent systems^2,3.

In social animals, the sharing of information about resource availability between group members has long been considered one of the benefits of living in groups^4,5,6. Information transfer between individuals allows the group as a whole to gain better knowledge of the location of available food sources than any individual would gain on its own, especially in heterogenous, dynamic environments^4,7. This information transfer need not be through the use of active signals, and may simply require that naïve individuals follow knowledgeable ones to find food sources that the latter know uniquely (e.g. ^8,9,10). In species where group members disperse over wide foraging areas, knowledge could be distributed among all of them, such that each may act as naïve or knowledgeable with respect to different food sources^8,11. Sharing foraging information thus may allow the benefits of being in a group to outweigh the costs of competition caused by foraging with others⁴.

Fission-fusion dynamics are a property of many animal groups^12,13 that occupy a common area but are often divided into subgroups that frequently come together and split from each other. In species with a high degree of these dynamics, the entire group is never found together, and subgroups of varying sizes spread over large areas, travelling independently of other subgroups¹⁴ but also coming together often¹⁵. In terms of group-decision making, fission-fusion dynamics can be seen as the result of a lack of consensus between individuals who may have different priorities on their activities¹⁶. However, in heterogenous, dynamic environments, this lack of consensus may have advantages in terms of distributed foraging and information gathering: individuals may forage in different regions of the group home range, while in subgroups with different combinations of individuals¹⁷. Thus, a subgroup will often contain individuals who have been foraging in (and perhaps have knowledge about) different regions of their home range.

Spider monkeys (Ateles spp.) are frugivorous primates living in tropical forests, where fruit can be found in ephemeral patches that vary in time and space depending on the tree species¹⁸. They have a high degree of fission-fusion dynamics¹⁹ in terms of the variation in subgroup size¹⁵ and composition¹⁷, and they occupy partially overlapping home ranges²⁰. In previous work¹¹, we showed that spider monkeys share information about newly discovered fruiting trees: focusing our observations on relevant trees during their whole fruiting period, we showed that individuals that were naïve about the presence of fruit in the focal tree tended to arrive with other group members that already knew about it. Using a random arrival null model, we also showed that the group as a whole finds out about the fruiting tree in fewer visits than if each individual had to find it on its own¹¹.

Partly inspired by those results, Falcón-Cortés et al.²¹ used an agent-based model to explore the effects of information sharing and memory of visited patches on the collective estimation of food patch quality. When individual agents remember previously visited patches, move toward them depending on their quality, and copy others’ visiting patterns if not finding food on their own, the group as a whole is able to compute the relative rank of food patches in the environment, concentrating its activity in proportion to the size of patches and focusing foraging efforts around the largest patches²¹. These studies show that sharing complementary information about heterogenous environments can effectively lead to an improvement in the group’s overall knowledge, and thus could be considered a form of collective information processing, sensu Moussaid et al.²².

Here we assume that information sharing takes place between individuals, in the context of a fluid grouping pattern where knowledgeable and naïve individuals coincide temporarily in the same subgroups and locations, being able to share information with others about areas they uniquely know. We also assume that an individual’s core range represents, for a given season, the area where it knows the location of available fruiting trees relatively well. This assumption is supported by the species’ wide daily travel distance (between 500 and 4500 m with means between 1750 and 3311 m in different study sites²³) when an individual visits around 10 different fruiting trees²⁴. Individual core ranges vary across seasons as would be predicted based on fruit availability, with dry seasons showing larger and less overlapping core ranges than wet seasons²⁰.

The way in which individual core ranges overlap shows that, normally, at least two individuals (but not the whole group) coincide in most portions of a group’s range²⁰, such that range overlaps can be thought of as higher-order interactions between two or more individuals. Therefore, typical network approaches to describe the full space-sharing structure of the group are inadequate due to their inability to capture multibody (non-dyadic) interactions between individuals^25,26. Simplicial sets are increasingly applied to model such scenarios where non-dyadic interactions are key features of network structure^27,28,29,30. Simplicial sets are generalisations of ordinary networks which allow for arbitrarily many nodes or individuals to be connected by a single higher-order edge called a simplex (Fig. 1). For spatial problems, a particularly relevant class of simplicial structure is the simplicial complex²⁵ which imposes the additional assumption of downward closure; meaning that the simplicial set must contain all possible nested lower-order simplices of any simplex^26,31,32. For example, a simplicial complex that includes a triadic connection between three individuals must also contain the three dyadic connections between these individuals. This assumption of downward closure provides a suite of additional computational tools for studying higher-order structure³².

Simplicial complex structure is often described through the computation of Betti numbers. There is a Betti number for each dimension, denoted β_i, which gives the number of holes of a particular simplicial complex in the i-th dimension. For instance, β₀ is the number of connected components, β₁ the number of loops and β₂ the number of voids³². For higher dimensions, we cannot visualise the holes, so the Betti numbers are especially useful to describe higher dimensional topological structure. In the context of a higher-order network structure of spatial overlaps, the topological holes described by the Betti numbers might result from individuals, or subsets of them, using areas uniquely with respect to others while maintaining some degree of redundancy in the common intersection. Thus, Betti numbers may characterise complementarity in the information possessed by different subsets (represented here by simplices). In addition, simplicial centrality measures³¹ allow the identification of simplices that are well-connected to others in the simplicial complex.

We hypothesise that, by pooling complementary information about the location of feeding trees, the group as a whole acquires a more complete knowledge of a complex, dynamic environment during a given season than any individual on its own. In particular, given our knowledge of spider monkey socio-spatial ecology, we expect to find evidence of complementarity in the structure of the simplicial complex representing the overlaps of the individual core ranges. Our strategy, outlined in Fig. 1, involves first calculating individual core ranges and their relative intersections. We assume that complementary foraging information can be captured through the intersection/union ratio for any given set of individual core areas, representing a balance between redundant and uniquely known information. The intersection is where the set can coincide to share information about those areas known uniquely by subsets of individuals in the rest of their union.

We use varying thresholds of this intersection/union ratio to connect sets of individuals with decreasingly redundant overlaps into simplicial complexes. Then, we explore the structure of these complexes, searching for holes in various dimensions (given by the Betti numbers) and their persistence for different thresholds of redundancy. We interpret these holes and their persistence as evidence of complementarity in the foraging information shared by the whole network. We also compare the simplicial complexes between the dry and wet seasons, as the uncertainty about the foraging environment may be greater in the former. Thus, we expected to find evidence for more complementarity in the information being shared during the dry seasons. Finally, we analyse the centrality patterns in the structure of the simplicial complex, to identify those sets of individuals that are key in terms of sharing foraging information with others by participating in more simplices.

Individual core ranges overlap only partially

A visual inspection of the overlaid individual core ranges during a given season shows that different subsets of individuals predominantly use different parts of a group’s home range (Fig. 2). The area comprised by the union of the individual core ranges was between 20 and 70 ha throughout the study, with the exception of the dry season of 2015, when one individual occupied a core range that was much larger than anyone else’s (bringing the union to 145.2 ha). There was no statistically significant increase in the area of the union with the number of individuals overlaid in each season (ρ₁₀ = 0.06, p = 0.85) nor a significant difference in the union area between seasons (ANOVA F_1,10 = 0.875, p = 0.37).

Figure 3 shows the density distribution of individual core ranges for the different years and seasons. Individual ranges differed significantly between years and seasons (ANOVA interaction between years and seasons, F_1,135 = 1.94, p = 0.09; year: p < 0.001; season: p < 0.001). During a given year, individual spider monkeys occupied larger core ranges during the dry season than during the wet season. This is particularly evident in 2014, 2015 and 2016.

The extent of overlap varies predictably with the number of individual core ranges

We developed a theoretical prediction of the extent to which the core ranges of a set of individuals of size n should intersect in order to strike a balance between the opportunities to coincide in the common overlap and the amount of unique knowledge each member of the set should hold outside of that area. In order to develop this prediction (detailed in the Supplementary Information), we first assume that a particular spatial overlap structure in a given subset of individual core ranges has a degree of complementarity depending on the amount of information that can be transferred between individuals. This information transfer, in turn, is proportional to the area known uniquely by some individuals but not others, and to the opportunities they have to coincide in a common area in order to share their uniquely known information. The higher the transfer of information, the more complementary their relative use of areas is. We also assume that individuals are distributed uniformly within their own areas, and independently from each other. Then we constructed an objective function describing this transfer of information and optimised it under the additional assumption of an homogeneous foraging ability in the population, i.e. individual core ranges of identical sizes (see Supplementary Information for more details). We found that the optimal intersection/union ratio for a set of n-many individuals is w^* = 1/(n + 1).

The above relationship predicts that larger sets of individual areas should have a lower intersection/union ratio, according to the convex decreasing curve shown in Fig. 4. The empirical values of this ratio similarly show a convex decreasing trend for all overlapping sets, in many cases qualitatively fitting the theoretical prediction (Fig. 4). Overall, smaller sets of up to five individuals vary more widely in the value of w than larger sets. The qualitative fit between the optimal prediction and the observations is particularly good for some seasons (dry seasons in 2014 and 2017 and wet seasons in 2013, 2016 and 2017), but less so in others (dry season in 2016, wet season in 2015). At fixed n, there also appears to be a multimodal distribution of the observed values of w, with the lowest values resembling the theoretical prediction and others being higher than predicted, suggesting other processes that are not captured by our theoretical model. In sum, the prediction of an optimal value of w*, representing a complementary balance between redundant and unique information, is overall consistent with the observed patterns, but overlaps tend to be larger (i.e. more redundant) than predicted by our model alone.

The observed values of w did not vary significantly between seasons but they varied significantly between years (ANOVA interaction between years and seasons, F_1,3084 = 9.56, p < 0.001; year: p < 0.001; season: p = 0.24).

The structure of simplicial complexes shows holes at various dimensions

The previous results on the relevance of w^* led us to use it as a reference for constructing simplicial complexes depending on how far a given set of individual core ranges was to the predicted w^*. Each simplicial complex has an α value calculated from their value of w, taking into account how the theoretical optimal prediction w^* varies with number of individuals, n. At low values of α, simplicial complexes are constructed only containing sets of individual areas that overlap considerably (i.e. have highly redundant spatial interactions), while high values of α imply the additional inclusion of sets of individuals that overlap less (i.e. have relatively more unique spatial interactions).

When we use α as a filtration parameter, including in the simplicial complex only those simplices that are below the α threshold, we find evidence of structural holes at various dimensions (Fig. 5). Multiple connected components, representing holes in the lowest dimension, persist for values higher than α = 2, especially in the dry seasons. In some circumstances, only when including the overlaps with the lowest level of redundancy do the simplicial complexes become fully connected. The dry season of 2017, in particular, shows 3 connected components for all values of α. Loops (β₁) and voids (β₂) are common when including interactions beyond α = 3 and often persist for several units of α. Holes of dimension 3 (β₃) only appear in the dry seasons of 2015 and 2016, while two long-persisting holes of dimension 4 (β₄) appear in the wet season of 2012. In the dry season of 2015, holes of four different dimensions (β₀ to β₃) coexist at some α values, representing a particularly rich spatial structure across the filtration.

If we assume that a hole of any dimension in the simplicial complex is the result of unique areas being used by some individuals in the group but not by others, then we can interpret holes and their persistence over different values of the filtration parameter as evidence of complementarity in the foraging information shared by these subsets. The existence of these holes implies that pockets of unique knowledge may exist among some sets of individuals even as we include spatial overlaps with less redundant information (higher α). Overall, these results provide compelling evidence that there are many areas occupied uniquely by subsets of individuals of various sizes, and, if we interpret these patterns of overlaps as indications of partial information sharing, there is potentially a large amount of complementary information being shared.

A qualitative exploration of the persistence barcodes in Fig. 5 suggests that the dry season contained more topological holes, which were also more persistent, providing stronger evidence for knowledge complementarity in the dry seasons. To quantify the patterns in each persistence barcode, we developed a filtration complementarity index, which measures the interval over α for which different holes last over the filtration, and gives more weight to those of higher dimension. The values of this index show no clear evidence for simplicial complexes from the dry seasons to have higher complementarity than for those of the wet seasons (ANOVA F₁ = 2.2, p = 0.17; Fig. SI4), although our power to detect an effect is weak given our small sample size of 6 years. The pattern is similar for the correlation with the variation in fruit abundance across fortnights for each season. While seasons with higher variation in fruit abundance have slightly larger values of complementarity on average (Fig. SI1), we did not find statistical support for this pattern (ρ = 0.42 and p = 0.16).

Simplicial centrality increases with size of the simplex

To examine the structure of the simplicial complexes, we utilise the maximal simplicial degree centrality, a generalisation of degree centrality in dyadic networks³¹. Across the majority of seasons, we found a clear sigmoidal relationship between the centrality of a simplicial complex and its size (Fig. 6). Maximal degree centrality increases gradually with size of the simplices and then more steeply, levelling off or decreasing after the simplex is of size 5. This generally implies that simplices containing more individuals are involved in a larger and more variable number of spatial interactions with other simplices, while above a certain size there is a plateau in this effect. We found no clear trends in the relationship between the simplicial degree centrality of a simplex and the average age, the proportion of males or the proportion of immigrant individuals in the individuals comprising it (see Supplementary Information, Figs. SI2–4). Note that the increasing trend of centrality with simplex size is consistent when varying the parameter ({n}_{max }) (see Supplementary Information, Figs. SI5–9), implying that it is indeed larger simplices which are more central and not those of some intermediate size which we could not observe due to our choice of ({n}_{max }=6).

We have studied the higher-order spatial interactions between individual spider monkeys, who live in a highly variable foraging environment and have a high degree of fission-fusion dynamics. The majority of a group’s range is covered by different combinations of, quite often more than two, individual core ranges. Simplicial complexes allowed us effectively represent these higher-order interactions, providing a methodology to quantify the complementarity of different subsets of individuals of varying size. These simplicial complexes contained holes that persisted for various values of the redundancy of interactions. These pockets of unique knowledge among some sets of individuals remained even when including less redundant interactions than predicted by w^*. In other words, the spatial interaction networks did not become fully connected until we included very weakly connected sets of individuals, which are likely sharing a large amount of unique knowledge even if coinciding in small intersections. Overall, our results provide strong support for our prediction of complementary patterns of information sharing.

In agreement with our prediction of the optimal spatial arrangement for maximum information transfer, spider monkeys seem to be balancing the intersection/union ratio of their core areas, or the extent of redundant and unique knowledge that an interaction would provide to a given set. Optimality approaches to collective behaviour are scarce, due to the difficulty in identifying the currency and level of optimisation³³. Here, we have informed our representation of higher-order interactions as simplicial complexes with a theoretical optimal prediction of a convex decreasing function of the size of the subset. This optimality argument, developed in detail in the Supplementary Information, assumes that the intersections between individuals comprising each simplicial complex provide an opportunity to share unique information that each one (or subsets of them) possess. In other words, our prediction is simply based upon the balance between redundancy and uniqueness in the use of space by any given set of individuals. This balance is not what one would propose for a central place forager (e.g. hooded crows, Corvus corone cornix⁸) which would only share a small portion of redundant areas for sets of any size, as the opportunities for information sharing would only take place in a small, regularly used central area within their range (e.g. their roost). Our optimality argument also rests on the assumption that each group member benefits from information pooling in a way that surpasses the costs of feeding competition⁴ and the alternative benefit of gathering information individually⁶.

While other, alternative models could possibly explain the observed patterns, ours simply assumes a balance between the opportunities to coincide in overlapping areas and the maximum amount of knowledge that can be shared about areas outside of those overlaps, applicable to any number of individual areas. For example, the most redundant interactions could be due to individuals cohesively exploring common areas for other, social reasons. Conversely, values below the predicted line are rare: sets of individual ranges do not seem less redundant than would be predicted based on our optimality argument. This is consistent with previous findings on the same species³⁴ indicating that repulsion between individuals is less relevant than attraction as a mechanism behind fission-fusion dynamics.

Particularly for smaller subsets, between 2 and 6 individuals, the value of the intersection/union ratio falls steeply with n, both for the optimal w^* and the observed values of w. This suggests that the balance between redundant and unique knowledge at this range of subset sizes is more sensitive to the addition of new areas, contrary to the situation with subsets of 10 individuals or larger. It is interesting that we found a similar pattern in the simplicial centrality analysis: this metric increases sigmoidally as simplices increase in size, with a maximum rate of increase in simplices of intermediate size (4–6 individuals) and then levelling off at larger simplex sizes of 7-9, depending on the season and the maximum simplex size considered (Figs. SI5–9). From the overall patterns of simplicial centrality, we can speculate that simplices representing subsets of intermediate size could be key in maintaining updated levels of foraging information, as their centrality is more sensitive to simplex size and also have their intersection/union ratio poised at the most sensitive range with respect to the addition of new individuals (and are therefore the most dynamic ones in terms of the redundant and unique knowledge they possess).

Sueur et al.¹⁶ predicted that broker individuals, who connect different clusters of the association network, are particularly important in fluid groups with high degrees of fission-fusion dynamics. Our simplicial centrality results show that subsets of increasingly many individuals participate in increasingly many spatial interactions with other simplices. The fact that we did not find a clear relationship between simplicial centrality and the age and sex of the adults, or their recent immigration status, suggests that rather than being an attribute of individuals (as in killer whales, for example³⁵), the asymmetry in foraging information is predominantly the result of spider monkeys’ high mobility and fluid grouping patterns. This had already been suggested by previous work on the factors that determine an individual’s decision towards fission or fusion¹⁹ and is compatible with the idea that spider monkeys are collectively pooling their foraging information in a dynamic fashion¹¹. Of course, many other factors related to individual age and sex classes shape the structure of dyadic association networks³⁶, but here we are uncovering those aspects of fission-fusion dynamics that relate to foraging knowledge as a dynamic, distributed process which varies depending on all adult members’ knowledge about a dynamic environment. Future work analysing associations in subgroups as higher-order interactions will be useful to evaluate the relevance of foraging knowledge as we have uncovered it here as an additional influence on the structure of association networks.

We did not find stronger evidence for complementarity in the dry compared to the wet seasons. While it appeared as though the simplicial complexes had more holes of various dimensions during the dry seasons, we were unable to detect statistically significant differences using a composite index of filtration complementarity. During the dry seasons, spider monkeys tend to rely on less abundant tree species and thus exploit a more heterogenous environment. During the wet season, the hyperabundance of Brosimum alicastrum²⁰ would make it less valuable to share information about scarce sources of food. However, we did not find significant differences in the temporal variation of the biweekly fruit abundance between the seasons. While our sampled number of seasons is small, it is possible that rather than temporal variation, it is the spatial variation that is higher in the dry than in the wet seasons. During the dry seasons, fruit is more scarce and dispersed, subgroups tend to be smaller and home ranges larger²⁰. It would be interesting to test the information content of different regions of the home range, using more accurate tree distribution and abundance data than was available for this study.

More generally, finding resources in a heterogenous, dynamic environment can be seen as a problem of pattern recognition, which can be solved effectively by systems with some degree of distributed information processing. This capacity can be found in various natural systems, recently described as ‘liquid brains’³⁷, ranging from social insect colonies to the immune system. These systems process information via interactions between mobile agents that interact only temporarily when they happen to be within reach. As an example, Vining et al.³⁸ modelled a set of simple agents that could only communicate with others within a short radius but were able to move about, thus increasing the number of agents with whom they could exchange information over time. This system was able to solve a global problem of density classification (converging to the binary state that was more common in the beginning of the run amongst all agents). Crucially, agents were able to solve the most difficult versions of the problem (where the difference between the majority and minority binary state was very small) only when moving at intermediate speeds. This allowed them to balance the frequency of interactions with others in their current neighbourhood and those in other locations. In other words, an optimal degree of mixing between agents led the group to solve more difficult problems effectively³⁸. This optimal degree of mixing is analogous to the optimisation of w we used here: an optimal area of intersection between individual use areas allows monkeys to coincide to share information while exploring enough area on their own, which can then be shared. Unique information, gathered by individual members of the group or subsets of them, becomes complementary when monkeys coincide in redundant areas and can share what they uniquely know. The redundancy/uniqueness balance is also more generally related to the balance between exploitation, where more redundant information is gathered about the environment, and exploration, where more unique information is gathered^39,40.

Collective information processing can result from individual members sharing complementary information, as we have found here, or it may also include synergistic information⁴¹, whereby two or more individual estimations of a target variable not only are complementary in their estimation but combine their information in such a way as to produce new knowledge². An example would be if one subset of individuals would contribute the location of a food source and another subset the timing of the fruiting of that source. The latter could be done by individuals who happened to visit the same trees repeatedly, gathering unique information about their fruiting status (as has been observed in other high fission-fusion species like the chimpanzee (Pan troglodytes⁴²)). The resulting, combined knowledge by both subsets of individuals would be synergistic in the sense of allowing all of them to exploit the food source according to its location and timing. Further empirical and modelling work could explore these ideas and search for synergy in collective information processing about the foraging environment in these and other species.

Using a theoretical prediction of the optimal balance between the use of redundant and unique areas, as well as simplicial complexes to represent higher-order interactions, we have been able to show complementarity in information sharing, a hallmark of collective information processing^2,3. This work exemplifies the usefulness of higher-order methods, as we have been able to uncover details in the spatial structure which could not be found in a dyadic representation of the same structure. While these methods come with greater complexity over the dyadic approach, they are extremely well suited to tackling problems surrounding collective behaviour. Spider monkeys with different knowledge of their foraging area contribute with different pieces of information to group-level knowledge, in such a way that the group as a whole can access more foraging trees in a dynamic landscape than any individual could on its own. This complementary information sharing in higher-order spatial networks would be a compelling example of collective intelligence in natural conditions⁴³.

Data collection

Ranging and subgroup composition data were collected by experienced observers between January 2012 and December 2017 in a habituated group of Geoffroy’s spider monkeys (Ateles geoffroyi) living around the Punta Laguna lake in the Otoch Ma’ax yetel Kooh protected area, in the Yucatan peninsula, Mexico. This group was the subject of a long-term, continuous study from 1997 to 2020⁴⁴. We defined dry seasons from November to April and wet seasons from May to October²⁰. The spider monkeys’ foraging environment differs importantly between the two seasons, due mainly to a very abundant species (Brosimum alicastrum) fruiting during the wet season²⁰. Therefore, fruit during the wet season is more abundant and homogeneously distributed, while during the dry season spider monkeys use more species that are less abundant overall and thus face a more heterogenous and uncertain environment²⁰.

We tracked changes in fruit abundance by recording, every 2 weeks, the number of trees with fruit for each species in a phenological trail. This trail contained a sample of 10 individual trees from 16 species, including 12 which comprise >50% of the monkey’s diet^45,46. We estimated the density of each of these species using data collected in eight 0.25 ha plots (total of 2 ha) and fourty-eight 100 m × 2m line transects (total 0.96 ha) within the home range of the spider monkeys (see ref. ⁴⁷ for more details about the vegetation sampling).

During 4–8 h-long daily subgroup follows, we performed subgroup scans every 20 min, noting the subgroup composition and its position using a hand-held global positioning system located roughly in the centre of the subgroup (subgroups can be dispersed over a maximum diameter of 50 m). Individual core range estimation was based on all the subgroup locations where a given individual was present. We restricted the analysis to adults (individuals >5 years by January 1st of each year analysed) given that juveniles and infants do not travel independently of their mother and thus would not make any independent contribution to the information pool of a given season. We also confined the analyses to adult individuals with a minimum of 100 scan sample locations per season. Table SI1 contains the number of samples and observation days for each study individual that fulfilled these criteria in each season and year.

Data analysis

We used individual locations for the estimation of individual core ranges, defined as the 60% utilisation distribution generated with the adaptive Local Convex Hull (a-LoCoH) method ⁴⁸, using the T-LoCoH package⁴⁹ for the R programming environment⁵⁰ with a constant value of 15000 for the a parameter for hull construction across seasons. The use of the 60% utilisation distribution as a meaningful area is validated by previous work where we showed that it lied at the inflection point in the curve of isopleth values versus proportion of home range covered⁵¹. Core ranges defined in this manner represent the area where individuals concentrated their regular movements, excluding more sporadically used areas. For more details on the calculation of individual core ranges, see Smith-Aguilar et al.²⁰.

We calculated the intersections and unions between individual core ranges for each season using the Spatstat package⁵² for the R programming environment⁵⁰. For a set C = {C₁, …, C_n} of n individual core ranges, we used the ratio denoted as w ∈ [0, 1] of the areas of their intersection and their union:

We used the fruit abundance data from the phenological trail, together with the diameter at breast height of the trees monitored and the density for each species in the study site, to estimate fruit abundance and its variability in each season. We used the following formulation, based on²⁰:

where IFA_f ≥ 0 is the index of fruit abundance for the monitoring period f based on the 16 species monitored, Tf_i is the proportion of trees with fruit of species i, DBH_i is the sum of the diameter at breast height over all the sampled trees with fruit of species i during period f and Den_i is the tree density for species i in the study site. We considered the IFA_f values for all the monitoring periods in a season to obtain the mean and the coefficient of variation of the IFA_f per season.

Higher-order network analysis

For each dataset (i.e. each season), we represented the entire space-sharing network of the study group as a simplicial complex K. The nodes of the network were all the adult individuals in the group during the particular season of the dataset (Table SI1). Since all core ranges coincided in some small common area, we defined a maximal simplex size, ({n}_{max }). This maximal size was needed because including interactions among more than ({n}_{max }) individuals yielded a trivial simplicial complex structure (containing every possible simplex), and these larger interactions were less likely to be related to foraging activity, but rather to common sleeping areas in the centre of the group’s home range. Then there exists a simplex connecting a subset of individuals if their corresponding intersection/union ratio, w, is non-zero, and if the size of the subset is no more than ({n}_{max }). Upon observation, the value of ({n}_{max }=6) was the largest which consistently resulted in non-trivial structures across all seasons. We computed the Betti numbers of the simplicial complexes for all seasons.

We investigated how the final simplicial complex, K, was formed when adding simplices in decreasing order of redundancy. Between subsets of the same size, we assumed that a higher relative spatial overlap, quantified by w, corresponded to a more redundant interaction. However, what constituted a redundant spatial overlap for information transfer depends upon the number of individuals in the subset, and therefore we required a scaling of the w values with their corresponding subset size in order to make fair comparisons. To this end, we computed the w value from the spatial overlap of n individuals which optimised the information transfer when these individuals had equal domain knowledge and utilised their core ranges uniformly, balancing the size of the unique areas they could share information about with the opportunity to share them when coinciding with others. When this balance of redundant and unique information leads to an optimal information transfer, we define it as complementary. As detailed in the Supplementary Information, for n individuals this optimal overlap was computed as:

This expression describes the amount of overlap required for a group of n individuals to maximise their transfer of information under the given assumptions of uniformity and independence of movements. We then defined a parameter α to describe how far from w* a given overlap was. For the i-th non-zero spatial overlap in the data, consisting of n(i) many individuals and with relative overlap value w_i, the corresponding α value was given as the solution to:

where f(α) is some arbitrary positive and decreasing function. We picked f(α) = 5 − α as a linear function such that the corresponding α value for each set was non-zero, and each simplex has α > 0, thus ensuring we consistently begin our filtration with a trivial structure consisting only of 0-simplices. We varied α over the interval [0, 5). For α ≤ 5 each possible observation (including those with zero overlap, due to the inclusion of α = 5) is represented in the simplicial structure. Variation in the choice of the function f would influence the axis of the persistence diagrams, but the same topological features would be observed.

Then, we defined a sequence of simplicial complexes K₀ ⊆ K₁ ⊆ ⋯ ⊆ K_N = K. Here, K₀ is the set of individuals in the study group (without any simplices connecting them) and each K_j is a simplicial complex with those simplices with α below the corresponding j-th lowest α value, i.e. corresponding to the intersection/union ratio w above the corresponding w threshold (possibly adding additional lower-order simplices to satisfy downward closure). This last procedure is justified on the basis of the clear tendency of overlaps to occur toward the centre of the group’s range (see Fig. 2), which makes it unlikely that the areas of n individuals will overlap significantly but the overlaps between subsets of n will not. More generally, in higher-order spatial studies, it is standard to assume that if a higher-order interaction is significant, then all lower-order sub-interactions should be considered significant²⁵. This sequence is called a filtered simplicial complex, and α is called the filtration parameter³². Simplicial complexes which are lower in the sequence will consist of more redundant interactions in terms of their information sharing. We computed the Betti numbers for each simplicial complex in the sequence (the persistent homology) for all available years and seasons in order to examine how the higher-order structure varied with the redundancy of interactions. Analysis was performed using the Gudhi package for Topological Data Analysis, implemented in Python⁵³.

We quantified how much evidence of complementarity in information sharing there was in a filtered simplicial complex through the filtration complementarity index. Structures with greater evidence of complementary sharing would be those with a larger number of higher order features (i.e. larger Betti numbers of high order), which persisted for longer throughout the filtration. Simplicial complexes with holes of higher dimensions may be described as more complex and present more evidence of complementary information sharing among a wider variety of simplices. We therefore defined the filtration complementarity index for season i as:

Here, ({B}_{i}^{(d)}(alpha )) is the d-th Betti number at the filtration value of α in the i-th season, n_i is the number of individuals in the data in this season, and 〈⋅〉_{[0, 5)} is the average operator over the filtration interval [0, 5). We scaled the number of connected components by the number of individuals as to not give seasons with larger sample sizes a higher score automatically (as the number of connected components is highly dependent upon n_i). We did not scale the higher-order features as we did not expect them to be (directly) impacted by sample size differences observed in our data. Higher-order features were given greater importance through the scaling by d + 1. We also tested other definitions of the filtration complementary index, including non-weighted or which did not consider the connected components. All of them yielded similar results.

We additionally studied which subsets were the most central in the simplicial complex across years and seasons. For this, we implemented the maximal simplicial degree centrality measure as defined in³¹. This is a generalisation of the standard degree centrality used in dyadic networks to simplicial complexes, which identifies those subsets that appear in the largest number of simplices. This metric is designed so that the downward closure assumption does not produce biases towards higher or lower order simplices, which are not automatically classed as more or less central just for being part of a nested structure. We computed the centrality value for each simplex in the simplicial complex for each year and season at a filtration value of α = 4 (meaning we do not consider subsets with more unique interactions). Other values of α yielded similar results. We analysed how simplicial centrality varied with simplex characteristics (size, average age, proportion of males and proportion of immigrants). When analysing how simplicial centrality varies with simplex size, we considered variation in the parameter ({n}_{max }in {4,5,6,7,8,9}) to ensure our results were qualitatively robust to variations in the distribution of simplex sizes.