The scaling of social interactions across animal species
SScaling of social interactions across animal species
Luis E C Rocha a,b,* , Jan Ryckebusch b , Koen Schoors a , and Matthew Smith c a Department of Economics, Ghent University, Ghent, Belgium b Department of Physics and Astronomy, Ghent University, Ghent, Belgium c Department of Business, Edinburgh Napier University, Edinburgh, UK * [email protected] ABSTRACT
Social animals self-organise to create groups to maximise protection against predators, productivity and fitness. One-to-one interactions are the building blocks of these emergent social structures and may correspond to friendship, grooming,communication, among other social relations. These structures should be robust to failures and provide efficient communicationto compensate the costs of forming and maintaining the social contacts but the specific purpose of each social interactionregulates the evolution of the respective social networks. We collate 637 animal social networks and show that the numberof social interactions scales with group size as a super-linear power-law for various species of animals, including humans,other mammals and non-mammals. We identified that the super-linear exponents fall in 3 broad classes depending on thesocial function, i.e. cooperation, spatial proximity, and friendship. By fitting a hierarchical model to our data, we observedthat social hierarchy must be absent to create efficient network structures for cooperation. This is not the case for friendshipties where network performance is not driven by group survival. Social interactions measured by spatial proximity fall in anintermediary category, not as efficient as cooperation, and is likely more relevant for the spread of infectious diseases than forsocial phenomena.
Introduction
Social animals including humans live in groups to optimise the multiplicative benefits of social interactions such as protection,cooperation, access to information, and fitness, while balancing the competition, disease risk, and stress costs of group living .Social interactions are fundamentally dyadic yet sufficiently diverse to link multiple animals or humans in connected socialstructures . The purpose of social interactions is also diverse and spans a range of processes including communication,dominance, trust, grooming, or simply the loosely defined idea of friendship . Correlations between social interactions,as for example dominance involving physical contact, friendship ties maintained through communication, or the intertwinedrelation between trust and spatial proximity, reveal the complexity of social phenomena and suggest that common principlesmay underlie the formation of social ties.One fundamental question is whether the number of social interactions scales with group size. The answer may revealwhether interaction patterns become more complex with size in order to maintain efficient social structures within the group. a r X i v : . [ phy s i c s . s o c - ph ] F e b he cost to establish and maintain social contacts in small groups is relatively low but increases in larger groups. This increasingcosts leads to peer selection, either by necessity or affinity, up to a species-specific cognitive saturation point in the number ofcontacts one can manage . Assuming that all members of a social group are reachable via social ties, in the limiting scenarios,a group of size N individuals may have a fragile star-like structure with E = N − E = N ( N − ) / , asfor example in response to predators (ecological conditions) or to fitness (phylogenetics) . Research on urban systemsshows that human societies also organise in larger groups (e.g. cities) to optimise resources like infra-structure and to increaseintellectual, social and economic outputs . These observations lead us to hypothesise that across species and social contexts,the number of social interactions scales with group size as E = CN β , where C and β are positive constants.Until recently, measuring social interactions was laborious. Past research relied on observations of animal and humanbehaviour or self-reporting of social contacts through questionnaires . A natural limitation of these techniques is the size ofthe observed populations and potential recalling errors, as for example the inability to accurately identify or quantify eachinteraction . Electronic devices (e.g. mobile phones or proximity sensors ) and online media now provide means forpassive and accurate recording of spatio-temporal location, communication between animals and between humans, among otherforms of animal or human interactions. State-of-the-art electronic data collection is scalable but its ability to detect authenticsocial interactions may be questioned and should be treated cautiously .We collate extensive data to show empirically that the number of social interactions scale super-linearly (i.e. β >
1) withgroup size and that social interactions can be categorised in 3 general classes with different exponents β which are independentof the animal species. We provide evidence that this scaling is necessary to maintain fundamental complex network structuresirrespective of existing group sizes. We also fit our data to a social network model and show that an hierarchical structure andthe cost of crossing social hierarchies may explain the estimated scaling exponents. Results
The data sets were collated using online databases of animal and human social networks previously analysed by other authors.All networks were reviewed for consistency and the data sets were standardised such that only unique pairs of social contactswere counted, i.e. self-loops, weighting and directions were removed. Social interactions were identified and labelled by expertobservation, questionnaires, or electronic devices (see SM). Table 1 shows the number of data sets for each type of socialinteraction and animal class, including captive and free-ranging animals.
Universal classes of social interactions
The original social interactions are further grouped in 3 categories: (A) cooperative activities including group membership,physical contact, grooming and foraging; (B) spatial proximity that includes dominance relations and a variety of socialinteractions, e.g. friends hanging out or animals grooming together; and (C)
Human friendship including online and offline able 1.
Number of data sets for each type of social interaction and animal class. In a total of 637 data sets, there are 186cases of human and 451 cases of non-human social interactions, including 290 captive and 161 free-ranging animals.Mam. non-primates Mam. Non-hum. primates Humans Actinopterygii Aves Insecta Reptilia Totalgroup membership 4 0 0 7 5 0 0 16physical contact 0 4 0 0 2 244 0 250grooming 0 23 0 0 0 0 0 23foraging 0 0 0 0 2 0 0 2spatial proximity 63 58 88 9 0 12 1 231dominance 10 3 3 0 2 2 0 20offline friendship 0 0 71 0 0 0 0 71online friendship 0 0 24 0 0 0 0 24Total 77 88 186 16 11 258 1 637self-defined friendship ties. N E N N A B C fit humansavesinsectamammaliareptiliaactinopterygii
Figure 1.
Scaling relations across species and social interactions. Empirical data (discs) and regression line (dark grey dashedlines) for three categories of social interactions: (A) cooperation ( ˆ β A = .
08, 95%CI [ . , . ] , r = . p < . n = spatial proximity ( ˆ β B = .
54, 95%CI [ . , . ] , r = . p < . n = Human friendship ( ˆ β C = . [ . , . ] , r = . p < . n = β B = .
52, 95%CI [ . , . ] , r = . p < . n = E and size N (i.e. the number of interacting individuals)for each of the 3 categories. We test our original hypothesis by fitting a power-law to the data using logarithmic transformedvariables to evenly distribute the data points:log ( E ) = log ( C ) + β log ( N ) (1)The fitting exercise gives power-law exponents β > r for all 3 categories of social interactions:(A) cooperative activities ( ˆ β A = . ∼ β B = . ∼ / β C = .
15) (Fig. 1).The approximations ˆ β A ∼ β B ∼ . . uper-linear scaling indicates increasing densification of social interactions, that is, larger social groups have on averagerelatively more social interactions than the smaller ones. It is not surprising that β > E ∝ N ) to maintain the social network connected; this is known as the percolation threshold inrandom networks . If E ∼ N , small perturbations may fragment the network, breaking down the group structure. Furthermore, β > Social network structure
The clustering coefficient (cid:104) cc (cid:105) is a local measure of the level of sociality between common contacts of a focal individual (i.e.the fraction of triangles). Its intensity indicates an evolutionary group advantage as for example fitness benefits . Networkswith higher clustering are relatively more robust since the deletion of a social tie would not significantly affect interaction andcommunication among close contacts. In random networks, the clustering coefficient decays with increasing network size as (cid:104) cc (cid:105) = (cid:104) k (cid:105) / N , where (cid:104) k (cid:105) is the average number of contacts (or edges) in the network . In our social networks, (cid:104) cc (cid:105) is typicallyconstant for varying network size (Fig. 2A-C), with the exception of the friendship category where offline friendships (withsmaller N ) have higher clustering than those online (Fig. 2C). Since the average degree is defined as (cid:104) k (cid:105) = E / N , we have (cid:104) cc (cid:105) = (cid:104) k (cid:105) / N = E / N and thus need E ∝ N to have constant clustering. Evolution implies that more complex structuresmay emerge in such social systems to optimise resources, e.g. to reap the fitness related benefits, and thus relatively lesssocial contacts become necessary to reach the same level of clustering across group sizes . For some classes of randomheterogeneous networks, (cid:104) cc h (cid:105) = A / N , where the proportionality constant A depends on the heterogeneity of the distribution ofcontacts among individuals and is lower than the average (cid:104) k (cid:105) . In these networks, relatively less contacts are sufficient to keepclustering constant.The average length of the shortest-paths (cid:104) l (cid:105) measures the average distance between any pairs of individuals in the socialnetwork and quantifies the communication potential between parts of the network . Shorter average distances (small-worldeffect, i.e. (cid:104) l (cid:105) (cid:28) N ) indicate that information flows quickly over the network, which is a fundamental characteristic of efficientgroup organisation. In random networks, the average distance depends on the network size as (cid:104) l (cid:105) ∼ log ( N ) / log ( (cid:104) k (cid:105) ) . Incooperative activities, (cid:104) l (cid:105) is approximately constant across group sizes (Fig. 2D) which is justified again if E ∝ N since (cid:104) l (cid:105) ∼ log ( N ) / log ( E / N ) ∼ const . In some classes of heterogeneous random networks, (cid:104) l (cid:105) is nearly constant with networksize . For the other two categories, the average distance increases only slightly since (cid:104) l (cid:105) ∼ log ( N ) / log ( N β − ) for β <
2. Inthese cases, although communication remains efficient ( (cid:104) l (cid:105) (cid:28) N ), the benefits of forming larger groups do not compensate thecosts to optimise this particular network structure, as is the case for cooperation. Hierarchical model
Evolutionary processes generate diversity and adaptation in animal populations . Social hierarchy emerges in social populationsas a form of status, often associated to dominance and beneficial to sexual selection and social rank, or prestige in humans .
24 96 N 〈 l 〉 N N N 〈 cc 〉 N N A B CD E F
Cooperation Spatial Proximity Friendship
Figure 2.
Social network structures. Distribution of the average clustering coefficient (cid:104) cc (cid:105) among close contacts for (A) cooperation (Median M = .
79, dashed horizontal line); (B) spatial proximity ( M = . Human friendship ( M = . (cid:104) l (cid:105) for (D) cooperation ( M = . spatial proximity ( M = . Human friendship ( M = . . Hierarchical models can be used to represent the underlyingmechanisms through which individuals combine skills and affinity to build up social groups. From single individuals to the entirepopulation, individuals may be stratified in levels or hierarchies . For example, living in households within neighbourhoodsthat in turn are part of cities, and so on, seems natural for humans. While people mostly interact with those in the samesub-group (e.g. within the same household), less frequent interactions across sub-groups (e.g. between different householdsin the same neighbourhood) often take place. Such hierarchies have been observed across animal species in which a relationbetween group sizes in different levels vary from nearly 2.5 in primates to about 3 for other mammals including humans. Thismeans that individuals are organised, for example, in groups of 5 (level 1), 15 (level 2), 45 (level 3), and so on .The self-similar hierarchical group structure is mathematically represented as a branching tree with a group at hierarchy h split into b sub-groups at hierarchy h − h , a single social group contains all individuals, i.e. = b h (thus h max = log b N ), and at the lowest level ( h min = i and j make a social contact ( i , j ) with probability p h dependent on the hierarchical distance h between them.Closer individuals (e.g. living in the same neighbourhood) are more likely to interact than individuals living far apart (e.g. indifferent cities), i.e. p h ( i , j ) decreases with h . The self-similarity between hierarchies implies that p h ( i , j ) / p h − ( i , j ) = const .A power-law of the form p h ∝ c − h , with c >
1, satisfies this relationship. The parameter c represents the cost to make socialinteractions across hierarchies, that we assume is lower than the cost to create a new hierarchical level, i.e. c ≤ b . For a givenindividual i , the expected number of social contacts (cid:104) e (cid:105) is: (cid:104) e (cid:105) i = ∑ j (cid:54) = i p h ( i , j ) , (cid:104) e (cid:105) i = log b N ∑ h = ( b − ) b h − c − h (cid:104) e (cid:105) i = b − c log b N ∑ h = ( b / c ) h − . For 1 ≤ c < b , the sum converges and thus: (cid:104) e (cid:105) i ∝ N − log b ( c ) . hh-1h-2 Social NetworkUnderlying Hierarchy c=1c=2A B C D AA BBCC DDb=2 Figure 3.
Hierarchical model. The underlying hierarchical structure (left) defines the probability p h ( i , j ) ∝ c − h of formingcontacts between individuals i and j in the social network (right). If c =
1, everyone interacts with everyone else whereas forhigher c , interactions between hierarchically closer individuals (lower h ) are more common.Therefore, the total number of social contacts is: E = N × (cid:104) e (cid:105) i ∝ N × N − log b ( c ) E = N − log b ( c ) . he hierarchical model implies that β = − log b ( c ) . Assuming that b = . , the cost is thus c A = β A = c B = .
58 for spatial proximity ( β B = . c C = . β C = . c A = p h =
1, i.e. the probability to form social contacts is independent of the hierarchical level.From an evolutionary perspective, low costs to form contacts promote cooperation and allow the optimisation of social networkstructures necessary to strengthen the group. Friendship is part of social life and also responsible for creating a sense ofbelonging to a group. For humans, friendship is often measured by self-response questionnaires, is asymmetric betweencontacts, and typically ambiguous. The relatively high cost ( c c ∼ b = .
5) indicates that social hierarchies play a stronger rolein forming friendship ties than in cooperation. This is a result of friendship being driven by similarity on ideas and tasteswhereas cooperation is driven by survival . Similar patterns to friendship are observed in human communication measured viamobile phone calls in urban populations .The analysis gives an intermediate exponent ( β B = .
5) and cost ( c B = .
58) for social interactions defined by spatialproximity. This raises a fundamental question on the real meaning of these spatial interactions. Cooperation and humanfriendship are well-defined interactions identified, respectively, by observing joint activities or by directly inquiring individuals.The spatial interactions, on the other hand, are measured by sensors or by direct observation and capture a mixture of socialevents. They span from brief contacts, sometimes limited by space, to collaboration and dominance, as for example interactionsbetween university students or school pupils, animals in barns, domestic dogs wandering in the neighbourhood, or visitors of anart exhibition. Previous modelling exercises in urban populations suggest that scaling of some forms of social interactionsmay be a result of spatial constraints . While those models may explain communication and friendship patterns ( β ∼ . β ∼ . = / β ∼ . Proxies for human cooperation aretypically indirect, such as firms outputs (e.g. sales, value) or innovation, and their scaling with population size show lowerexponents 0 . < β < . . However, the common pattern across species in the spatial category and the fact that othermammals are represented in the cooperative category weaken this hypothesis. Conclusions
Our findings reveal key aspects of the organisation of animal social networks. Though primates and non-primates (includinghumans) are more represented than other animals, the scaling relations indicate common organisation principles across animalspecies that can be explained in terms of hierarchical models designed to maintain the functioning of the social groups. Differentscaling exponents allow us to distinguish between cooperation and friendship, and to infer that fundamental network structures ecessary to group existence are not as significantly present in friendship as they were in cooperation. Cooperation forms robustand efficient networks whereas friendship forms looser social structures that are easier to fragment as the groups increase insize. Efficient cooperation is only possible in the absence of social hierarchy. We also singled out spatial proximity interactionsand observed that they capture mixed social interactions with a scaling exponent ( β ∼ /
2) higher than previously derived viaindirect measures . These proximity or spatial interactions are likely more relevant for the spread of infectious diseases than todescribe social phenomena . Although such interactions increase disproportionately with group size, our results suggest thatthey are not as efficient as cooperative networks. Future research should add a quality measure to social interactions (e.g. viaweights or temporal dynamics) to investigate the varying importance of creating and maintaining particular structures . Strongsuper-linear scaling implies prohibitive social costs to maintain larger groups. The questions on whether there is a maximum oroptimal group size in which efficient groups can exist and fitness is maximised or whether more complex network structuresare necessary in larger groups remain open. Methods
Data
The data sets used in this study were collected using public network data repositories. A list of repositories and a full list of theoriginal references for the 637 data sets are available in the SI. All networks were standardised for the analysis, including theremoval of self-loops, edge directions, and edge weights.
Networks
A network G of size N is defined as a set of N nodes i and a set of E edges ( i , j ) connecting nodes i and j . A node representseither a person or an animal. An edge represents a certain social tie. In an undirected network, edges are reciprocal, i.e. ( i , j ) = ( j , i ) . In a network without self-loops, there is no edge ( i , i ) .The clustering coefficient of a node i is given by: cc i = e i / ( n i ( n i − )) (2)where e i is the number of edges between the n i nodes directly connected to node i . The average clustering coefficient of thenetwork G is thus: (cid:104) cc (cid:105) = N N ∑ i = cc i (3)The distance between the nodes i and j is the length of the shortest-path l i j in number of edges. It is calculated within thelargest connected component of the network G . In the largest connected component, there is at least one path between any pairs f nodes i and j in the component. The average shortest-path length is: (cid:104) l (cid:105) = N ( N − ) N ∑ i , j = l i j (4)Topical subheadings are allowed. Authors must ensure that their Methods section includes adequate experimental andcharacterization data necessary for others in the field to reproduce their work. References Krause, J. & Ruxton, G.
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Acknowledgements
The authors thank Luana de Freitas Nascimento for helpful discussions.
Author contributions statement
L.R. designed the research, made the analysis and wrote the draft; J.R., K.S., M.S. contributed with methods; All authorsrevised the manuscript.
Additional information