What Is A Possible Effect Of Crossing Animals That Belong To Different Species?

Abstract

Social animals self-organise to create groups to increase protection against predators and productivity. I-to-1 interactions are the edifice blocks of these emergent social structures and may stand for to friendship, grooming, advice, amid other social relations. These structures should be robust to failures and provide efficient communication to compensate the costs of forming and maintaining the social contacts merely the specific purpose of each social interaction regulates the evolution of the respective social networks. We collate 611 animal social networks and bear witness that the number of social contacts East scales with grouping size N as a super-linear power-law $E=CN^\beta$ for various species of animals, including humans, other mammals and non-mammals. Nosotros identify that the ability-law exponent $\beta$ varies according to the social function of the interactions every bit $\beta = 1+a/4$, with $a \approx {one,ii,3,4}$. Past fitting a multi-layer model to our data, we observe that the cost to cross social groups likewise varies according to social function. Relatively depression costs are observed for physical contact, grooming and group membership which lead to small-scale groups with high and constant social clustering. Offline friendship has like patterns while online friendship shows weak social structures. The intermediate case of spatial proximity (with$\beta =i.five$ and clustering dependency on network size quantitatively similar to friendship) suggests that proximity interactions may exist as relevant for the spread of infectious diseases equally for social processes similar friendship.

Introduction

Social animals including humans alive in groups to optimise the multiplicative benefits of social interactions such every bit protection, coordination, cooperation, access to information, and fettle, while balancing the competition, disease run a risk, and stress costs of grouping living^ane,two,3. Social interactions are fundamentally dyadic yet sufficiently diverse to link multiple animals or humans in connected social structures^one,4,5. The purpose of social interactions is also diverse and spans a range of processes including communication, trust, grooming, dominance, or simply the loosely defined idea of friendship^i,4,6. Correlations betwixt social interactions, as for example dominance and physical contact, friendship ties maintained through communication, or the intertwined relation betwixt trust and spatial proximity, reveal the complication of social phenomena and suggest that mutual principles may underlie the formation of social ties.

A primal question concerns how the number of social connections depends on grouping size, and whether at that place are any emerging patterns in this relationship. The respond may reveal whether interaction patterns go more circuitous with size in order to maintain efficient social structures within the group. The price to establish and maintain social contacts in small groups is relatively low but increases in larger groups⁷. This increasing costs leads to peer selection, either by necessity or affinity, upward to a species-specific cognitive saturation point in the number of contacts one tin can manage⁸. Bold that all members of a social group are reachable via social ties, in the limiting scenarios, a grouping of size N individuals may take a fragile star-like structure with $E=N-ane$ social ties to minimise social interactions (lowest cost) or a fully continued clique with $E=N(N-1)/2$ ties (highest price).

Evolutionary arguments support that social groups specialise and optimise social interactions to salvage resource while keeping or increasing the group efficiency^9,x,11, as for example in response to predators (ecological weather)¹² or to fitness^thirteen,14. There is as well the argument that human social networks take an optimal size to optimise information transfer inside groups¹⁵. Inquiry on urban systems shows that human societies also organise in groups (e.one thousand. cities) to optimise resources like infra-construction and to increase intellectual, social and economical outputs^16,17. These observations lead us to hypothesise that across species and social contexts, the number of social contacts Due east scales with group size N as $E = C N^{\beta }$, where C and $\beta$ are positive constants.

Until recently, measuring social interactions was laborious. Past research relied on observations of animal and human behaviour or self-reporting of social contacts through questionnaires⁴. A natural limitation of these techniques is the size of the observed populations and potential recalling errors, equally for example the disability to accurately place or quantify each interaction^1,4. Electronic devices (e.g. mobile phones^xviii or proximity sensors^19,20) and online platforms now provide means for passive and accurate recording of spatio-temporal location, communication between animals and between humans, among other forms of beast or human interactions. State-of-the-art electronic data collection is scalable merely its ability to detect authentic social interactions may be questioned and should be treated cautiously^21,22.

Nosotros collate extensive data to testify empirically that the number of social contacts scales super-linearly (i.e. $\beta >1$) with group size and that social interactions can exist categorised in different exponents $\beta$ independently of the animal species. We provide bear witness that this scaling is necessary to maintain fundamental complex network structures irrespective of existing grouping sizes. We too fit our information to a social network model and show that a multi-layer structure and the toll of crossing social layers may explain the estimated scaling exponents.

Results

The data sets were collated using online databases of brute and human social networks previously analysed by other authors. All networks were reviewed for consistency and the information sets were standardised such that only unique pairs of social contacts were counted, i.due east. self-loops, weighting, timings of contacts, and directions were removed. Social interactions were identified and labelled in the original studies past domain experts via direct ascertainment (animal interactions), questionnaires (offline friendship), electronic devices (spatial proximity), and online platforms (online friendship) (meet SI). To minimise potential ambiguities, each network was constructed based on the specific definition of social interaction in the respective original study. Table ane shows the number of networks for each type of social interaction and animal grade, including captive and free-ranging animals. The network size varies across species and social interactions considering of experimental settings, characteristics and limitations of the report populations, e.g. the observation chapters of researchers, price of technical devices, free-range vs. confined animals, online platforms, or animals living in small groups (see SI).

Tabular array 1 Number of networks for each type of social interaction and brute course.

Full size table

Scaling of social interactions

The networks of social interactions were grouped in categories following the type of social interactions as reported in the original studies (Table i). Figure 1 shows the scaling between the number of social contacts Due east and size N (i.eastward. the number of interacting individuals) for each of the 6 original categories. We presume that the scaling of social relations is independent of species and examination our hypothesis $E=CN^\beta$ by fitting a power-law to the data using logarithmic transformed variables to evenly distribute the data points:

$$\begin{aligned} \log Eastward = \log C + \beta \log North. \end{aligned}$$

(i)

The fitting do gives super-linear ability-law exponents (i.east. $\beta >ane$) and potent linear correlations ($0.55<r<0.99$) for all categories of social interactions (Tabular array two). Assuming a small error $\epsilon$ in ${\hat{\beta }}$, the exponents follow the full general law ($\beta =1+a/4$) with $a \approx i$ for online friendship ($\epsilon =12\%$), $a \approx 2$ for spatial proximity ($\epsilon =iv\%$), $a \approx iii$ for grouping membership ($\epsilon =2.6\%$) and offline friendship ($\epsilon =5\%$), and $a \approx iv$ for concrete contacts ($\epsilon =1\%$) and grooming ($\epsilon =6\%$), despite differences in species and sample sizes (Fig. one).

Tabular array 2 Best fitting exponents for the vi types of social interactions.

Full size table

This super-linear scaling indicates increasing densification of social contacts, that is, larger social groups have on boilerplate more than social contacts per-capita than the smaller ones. It is not surprising that $\beta >1$ because the number of social connections must scale at least linearly with grouping size ($E \propto Due north$) to maintain the social network connected; this is known as the percolation threshold in random networks²³. If $E \approx N$, modest perturbations may fragment the network, breaking down the grouping structure. Furthermore, $\beta >1$ suggests that a super-linear number of contacts are necessary to create and maintain the circuitous social network structures for the groups to role cost-efficiently irrespective of size.

Social network structure

We study the network structures for each of the six types of social interactions (run into "Methods"). The clustering coefficient $\langle cc \rangle$ is a local measure of the level of sociality between mutual contacts of a focal individual (i.e. the fraction of social triangles). Its intensity indicates an evolutionary group reward as for example fitness benefits^24,25. Networks with higher clustering are relatively more robust since the deletion of a social connexion would not significantly touch on interaction and advice amid close contacts. In our social networks, $\langle cc \rangle$ is constant for varying network size for all types of social contacts (Fig. two). In random networks, the clustering coefficient decays with increasing network size every bit $\langle cc \rangle = \langle k \rangle /N$, where $\langle k \rangle$ is the average number of contacts (or edges) in the network²³. The inset of Fig. 2F shows the results for the randomised versions of the same online friendship networks (see SI for the other categories). In all categories, at that place is a higher clustering coefficient than expected on the basis of randomised social contacts (come across caption Fig. 2). Since the average degree is defined as $\langle k \rangle = 2E/N$, we have $\langle cc \rangle = \langle k \rangle /Due north= 2E/N^2$ and thus would need $E \propto N^2$ to take constant clustering in random networks. Evolutionary theory implies that more complex structures may emerge in such social systems to optimise resources, e.one thousand. to reap the fettle related benefits, and thus relatively less social contacts become necessary to accomplish the same level of clustering across group sizes^24,25. For case, for some classes of random heterogeneous networks, $\langle cc_{\text {h}} \rangle = A/N$, where the proportionality constant A depends on the heterogeneity of the distribution of contacts among individuals and is lower than $\langle k \rangle$ ²³.

The average length of the shortest-paths $\langle l \rangle$ measures the average distance between any pairs of individuals in the social network and quantifies the communication potential between parts of the network²⁶. Shorter average distances (i.eastward. $\langle l \rangle \ll Due north$, resulting in the minor-world upshot²⁷) signal that data flows quickly over the network, which is a fundamental feature of efficient group arrangement²⁸. For physical contacts, grooming, and group membership, $\langle 50 \rangle$ is constant and slightly higher than one (Fig. 2A–C). For spatial proximity and offline friendship, the values increase with size following quantitatively similar trends (Fig. 2nd,Eastward). The results for online friendship suggest a constant trend (Fig. 2F). In all cases, the average path-length is $\langle l \rangle < 6$, which is the small-scale-globe horizon observed empirically²³. For all 6 categories, the random versions of the same networks give constant relations albeit mostly with lower values (see SI). In theoretical random networks, the average altitude increases slowly with the network size as $\langle l \rangle \approx \log (N) / \log (\langle k \rangle )$ ²³. Still, the average path-length $\langle fifty \rangle$ is approximately constant beyond grouping sizes if $Due east \propto North^{\beta }$ for $\beta >1$, since in this instance $\langle l \rangle \approx \log (Northward) / \log (2N^{\beta -1}) \approx i/(\beta -1)$. Smaller $\beta$ thus leads to higher $\langle fifty \rangle$, as observed in the analysed networks. In some classes of heterogeneous random networks, $\langle fifty \rangle$ is also nearly constant with network size²³. The density of contacts explains the low $\langle l \rangle$ for concrete contact, grooming and group membership. The discrepancy of spatial proximity, offline and online friendship with the random case indicates that more circuitous network structures are existence formed in larger groups for these types of social interactions. In sparse networks, like those, a high level of local clustering increases the distance between random pairs of network nodes because of local spots of connectivity redundancy²³. Taken together, the abiding clustering across network sizes (Fig. 2) implies that the average distance volition necessarily increase (Fig. 3), unless followed by a sufficient increase in the number of connections (to maintain low average distances equally the group increases). The growth in offline friendship followed by a seemingly constant pattern for online friendship (which has larger sizes) suggests a potential saturation in $\langle l \rangle$ for human friendship in line with the small-world horizon observed in previous studies²³. Although communication remains efficient (because $\langle l \rangle \ll North$), the benefits of forming larger groups exercise non compensate the costs of optimising certain network structures, as is the case for other types of social interactions involving concrete contact.

Multi-layer model

Multi-layer models can be used to stand for the underlying generative mechanisms through which individuals combine skills and affinity to build upwards more complex social groups. From single individuals to the entire population, individuals may exist stratified in layers (or levels) corresponding to different groups²⁹. For example, living in households (layer 1) within neighbourhoods (layer two) that in turn are function of cities (layer 3), and so on, seems natural for humans. While people mostly interact with those in the same group (due east.grand. within the same household), interactions across groups are less frequent³⁰ (e.chiliad. betwixt different households in the aforementioned neighbourhood). Interactions across groups at the same layer are necessary to define higher-order groups, i.e. a grouping at the next higher layer, as for example a neighbourhood is a upshot of interactions between individuals from different households. Multi-layer models take been used to explicate spatial relations in vascular³¹ and infrastructure¹⁷ systems. We contend that such models are also of value for social groups, non necessarily spatially bound, since multi-layer arrangement has been observed across animate being species in which a relation between grouping sizes in unlike layers vary from well-nigh 2.5 in primates to well-nigh 3 for other mammals including humans^14,32. This means that individuals are organised as multiples of iii, for example, in groups of 5 (layer 1), 15 (layer 2), 45 (layer iii), and so on. The model detailed below does not aim to reproduce all structures of the 611 analysed networks just focuses on the scaling exponents $\beta$.

The self-similar multi-layer group structure is mathematically represented as a branching tree with a group at layer h separate into b sub-groups at layer $h-1$ (Fig. four). At the highest layer $h_{\text {max}}$, all individuals vest to a single social group, i.e. $N=b^{h_{\text {max}}}$, and at the everyman layer ($h_{\text {min}} = 0$), each group is formed past a single unique private. In this model, individuals i and j make a social contact (i,j) with probability $p_{\Delta h}(i,j)$ dependent on the altitude $\Delta h$ between the layers that separate them. Closer individuals (e.g. at distance $\Delta h=1$ because they are living in the same neighbourhood or belonging to the same social group) are more likely to interact than individuals living far autonomously (due east.one thousand. at altitude $\Delta h=2$ because they are living in different cities or belonging to different social groups), i.east. $p_{\Delta h}(i,j)$ decreases with $\Delta h$. The multi-layer tree-like structure only defines the altitude $\Delta h$ between the groups that is in turn used to course contacts in the social network (Fig. 4); the resulting social network only has tree-like construction for sparse networks, i.eastward. when $E \ll North^2$. The self-similarity betwixt layers implies that $p_{\Delta h}(i,j)/p_{\Delta h-ane}(i,j)=const$ ³³. A power-police force of the course $p_{\Delta h} \propto c^{-\Delta h}$, with $c>1$, satisfies this human relationship. The parameter c represents the toll to make social interactions across layers, that we assume is lower than the toll to create a new layer, i.due east. $c < b$, because multiple contacts are necessary to establish a new layer. For a given individual i, the expected number of social connections $\langle e \rangle _i$ is:

$$\brainstorm{aligned} \langle due east \rangle _i&= \sum _{j \ne i} p_{\Delta h}(i,j) = \sum _{\Delta h=1}^{h_{\text {max}}} (b-ane)b^{\Delta h-ane}c^{-\Delta h} = \frac{b-one}{c} \sum _{\Delta h=1}^{h_{\text {max}}} (b/c)^{\Delta h-1}. \terminate{aligned}$$

For $1 \le c < b$, the sum converges:

$$\begin{aligned} \langle eastward \rangle _i = \frac{b-1}{c} \times \frac{(b/c)^{h_{\text {max}}}-1}{(b/c)-one} \cease{aligned}$$

Since $N = b^{h_{\text {max}}}$ and $c^{h_{\text {max}}} = b^{h_{\text {max}} \log _b (c)}$, nosotros get:

$$\begin{aligned} \langle e \rangle _i&\propto N^{ane-\log _b (c)}. \end{aligned}$$

Therefore, the total number of social connections is:

$$\brainstorm{aligned} E \sim Northward \times \langle e \rangle _i \propto N \times Northward^{1-\log _b (c)} \sim Due north^{2-\log _b (c)}. \cease{aligned}$$

The multi-layer model implies that $\beta =2-\log _b (c)$. Assuming that $b=2.v$ ^xiv, the cost of connections is thus $c_{\text {A}} = 1$ for concrete contact ($\beta _{\text {A}} \approx 2$), $c_{\text {B}} = i$ for grooming ($\beta _{\text {B}} \approx 2$), $c_{\text {C}} = ane.26$ for group membership ($\beta _{\text {C}} \approx 1.75$), $c_{\text {D}} = 1.58$ for spatial proximity ($\beta _{\text {D}} \approx 1.5$), $c_{\text {E}} = 1.26$ for offline friendship ($\beta _{\text {Eastward}} \approx one.75$) and $c_{\text {F}} = 1.99$ for online friendship ($\beta _{\text {F}} \approx 1.25$). This cost is associated to crossing (virtual) barriers betwixt social groups that might crusade the creation of larger groups. The low cost ($c = 1$) for concrete contact and grooming means that $p_{\Delta h} = one$, i.e. the probability to form connections is independent of the social altitude $\Delta h$, collapsing the assumption of multi-layer structure. For such types of social interactions, the connections within the same social group are favoured; individuals do not groom in dissimilar social groups nor brand persistent concrete contacts, except physical contacts for disharmonize that would not be reflected in our data. This effect is related to the loftier clustering coefficient reported in previous sections and may explain the relatively minor size of such networks. The number of social contacts for such activities is limited within the same social group.

Online friendship, on the other hand, is costly ($c \approx 1.99$) in terms of crossing social boundaries to connect individuals from different social groups³⁴, e.yard. with different tastes, ideas, location, age, and so on. Socially closer individuals would be favoured hither as well since it is harder to be friends with dissimilar people than with those similar to each other³⁰. However, given that online connections are cheap to establish and maintain (i.due east. practice non need nurturing and resources), the multi-layer structure becomes relevant with a non-negligible number of socially distant connections being formed. Furthermore, online friendship typically mixes (real) friends, acquaintances, relatives, and co-workers, each belonging to different social groups, with some individuals acting as social brokers. For instance, online friendship is more easily established between those studying in the same schoolhouse than at different schools; however, inter-school friendship is facilitated past the online platform, though socially costly (lack of contiguous interactions, no mutual friends, building trust). For networks derived from mobile phone communication in urban populations, a scaling exponent $\beta =1.xv$ has been reported^35,36,37. Such mobile communication data sets mix professional person and personal relations which possibly also leads to higher costs in the sense of crossing social boundaries. In 1 report, a constant clustering coefficient has been also observed suggesting that similar underlying principles may explain the formation of such social or communication structures³⁷. The multi-layer structure becomes less relevant for offline friendship ($c \approx i.32$) that are typically more spatially constrained in our data. For case, students or prison house inmates volition report friendship with those effectually them. In schools, from where almost of our data come from, the social construction is seen at the class and schoolhouse layers simply. Given experimental limitations, it is often not possible to report friends exterior the study setting, which could reveal higher social layers, e.thousand. neighbourhood friends. It is possible that the exponent $\beta$ for friendship is thus betwixt what nosotros estimated for offline and online friendship if all layers of friends and non only those in the same study setting were reported.

Our analysis finds an intermediate exponent ($\beta = 1.five$) and cost ($c \approx 1.58$) for spatial proximity. Spatial proximity is a detail type of social interaction. Grooming, concrete contact and human friendship are well-defined interactions identified, respectively, by observing joint activities or by directly inquiring individuals. However, spatial proximity interactions are measured by sensors or direct ascertainment and capture a mixture of social situations. Spatial proximity might reflect analogousness, trust and friendship between individuals and animals sharing the same space³⁰, e.g. persistent spatial proximity betwixt pairs of cows³⁸, or behavioural or trait similarity, i.e. homophily, every bit for instance friends visiting a museum³⁹ or wellness-care workers in hospitals⁴⁰. On the other mitt, spatial proximity interactions might simply reflect spatial constrains forcing individuals and animals to be in close proximity during periods of fourth dimension, e.one thousand. a grouping of visitors of an art exhibition³⁹ or confined animals³⁸. Nevertheless, also in the later, affinity and trust are reflected in the proximity contacts. As discussed to a higher place, it is possible that friendship at the society layer likely follows patterns intermediate to those observed in the online ($\beta =ane.25$) and offline ($\beta =1.75$) categories. The existing literature associating friendship to time that individuals spent together³⁰ and the observation that spatial proximity contacts follow an intermediate exponent ($\beta =1.5$) propose a potential link between these social interactions. We cannot brand a stiff association betwixt the two types of social interactions due to lack of data of offline friendship in larger populations. Previous modelling exercises in urban populations suggest that $\beta _{B} = i.v$ tin be explained by mobility ($H=ii$, where H is the Hausdorff dimension of a path in space) over two dimensional ($D=2$) spaces based on the assumption that fully-mixed populations may fully explore a given area¹⁷. While this assumption may hold within, e.m. schools, museums or barns, it does not utilise on larger spatial areas since humans and animals are territorial and tend to spend most of time within certain locations⁴¹ or with certain individuals³⁰. On the other manus, the same model suggests that contacts per-capita scale as 0.25 (i.eastward. $\beta = 1.25$) under the same atmospheric condition (i.e. $H=D=ii$). This fits well to our findings for offline friendship, where people may about explore the whole social space and potentially interact with dissimilar individuals.

Conclusions

Our findings reveal key aspects of the system of animal social networks. Though primates and non-primates (including humans) are more represented than other animals in our information set, the universal scaling relations $E=CN^{\beta }$ between the number of social contacts East and size Northward suggest common organization principles across fauna species that tin can be explained by multi-layer models designed to maintain the operation of the social groups^14,32. Different scaling exponents following the general relation $\beta = one+a/four$, with $a \approx {1,two,iii,iv}$ allow us to distinguish types of social interactions and to infer network structures underlying those interactions. For all types of social interactions, the local clustering remains abiding for increasing network sizes albeit having unlike intensity in each instance. Physical contacts, training and group membership have similar constant median values that are higher than observed for spatial proximity, offline and online friendships. The average path-length is also constant and follow the pocket-sized-globe design (i.eastward. $\langle 50 \rangle \ll N$) for nearly cases with the exception of spatial proximity and offline friendship where a quantitatively like positive trend is observed with values below the small-world horizon of $\langle l \rangle \approx 6$ previously observed in social networks²³.

One may argue that humans differ from other animals past developing more efficient social network structures, with relatively less contacts for larger network sizes, and thus lowering the scaling exponents. In that location is a quantifiable relationship with brain and grouping sizes, along with the complexity of the interactions. Humans are able to process the cognitive demand of other forms of relationships such as friendship, rather than mating and authorization relations that often occur inside other animals and species⁴². The common scaling pattern observed beyond species and particularly for spatial proximity weakens the hypothesis that animals differ. Our results suggest that the type of social interaction, and to a lesser extent, the group size, are more relevant to determine the scaling exponents than the animal species. We reached this conclusion by combining information from different species. More statistical power could be achieved with a larger sample of network data for specific combinations of social interactions and species in social club to written report these relations separately. Given the multi-layer structure of social networks and experimental constraints, offline friendship data sets are express to relatively small social circles³⁰. If one could map college social layers, the scaling exponent could decrease, likely to the same value as observed for spatial proximity. If this is confirmed in time to come studies, nosotros volition be able to infer that spatial proximity is a proxy of friendship across animals species³⁰.

Physical contact, grooming and grouping membership are associated with more than robust and topologically efficient networks (since clustering is higher and path-lengths are shorter) than friendship and proximity interactions. This social cohesion is a result of homophily and coordination to maintain group functioning, which likely creates smaller groups in these categories relative to friendship and proximity categories because of the cost of nurturing contacts. The frequency and number of social interactions leading to stable social contacts are also important to regulate diffusion processes such equally communication²⁶, innovation^16,17, infectious diseases^19,43 and social phenomena^thirty,44. Our results suggest that physical contacts and grooming are more efficient than proximity to facilitate spread phenomena at the population (network) level. Online friendships are associated to looser social structures easier to fragment as the groups increase in size. The relatively high cost of nurturing too many online social contacts across social layers restrains the opportunities to generate college clustering or common friends, and create redundant structures, equally observed in the smaller networks related to activities necessary to continue the grouping functioning.

Although we focus on temporally stable social networks⁴⁵, the availability of temporal information and intensity of certain social interactions could likewise assist to sympathise the formation and dissolution of social contacts and how particular network structures are formed. Future research should add a quality measure to social interactions (east.g. via weights or temporal dynamics) to investigate the varying importance of creating and maintaining particular structures⁴⁶. Potent super-linear scaling implies prohibitive social costs to maintain larger groups for some types of social interactions. The questions on whether there is a maximum or optimal group size in which efficient groups can exist and fitness is maximised⁴⁷, or whether more complex network structures are necessary to sustain larger groups, remain open up.

Methods

Information

The data sets used in this study were nerveless using public network data repositories. A listing of repositories and a full list of the original references for the 611 data sets are bachelor in the SI. The 6 types of social interactions: concrete contact, training, group membership, spatial proximity, offline friendship and online friendship were identified and labelled in the original studies by domain experts via directly observation (animal interactions), questionnaires (offline friendship), electronic devices (spatial proximity), and online platforms (online friendship). All 611 networks were standardised for the assay, including the removal of self-loops, edge directions, and border weights.

Networks

A network G of size Northward is divers as a fix of N nodes i and a gear up of E edges (i,j) connecting nodes i and j. A node represents either a person or an animal. An edge represents a social connection of a specific type. In an undirected network, edges are reciprocal, i.e. $(i,j)=(j,i)$. In a network without self-loops, in that location is no edge (i,i).

The clustering coefficient of a node i is given by:

$$\begin{aligned} cc_i = 2e_i/(n_i (n_i-1)) , \end{aligned}$$

(two)

where $e_i$ is the number of edges (connections) between the $n_i$ nodes straight connected to node i. The average clustering coefficient of the network Yard is thus:

$$\begin{aligned} \langle cc \rangle = \frac{1}{N} \sum _{i=1}^N cc_i . \end{aligned}$$

(3)

The topological distance betwixt the nodes i and j is the length of the shortest-path $l_{ij}$ in number of edges. Information technology is calculated within the largest connected component of the network G. In the largest connected component, in that location is at to the lowest degree one path between whatsoever pairs of nodes i and j. The average shortest-path length is:

$$\brainstorm{aligned} \langle l \rangle = \frac{1}{N(N-1)} \sum _{i,j=1}^{N} l_{ij} . \end{aligned}$$

(4)

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Acknowledgements

The authors thank Luana de Freitas Nascimento for helpful discussions.

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Affiliations

Department of Economics, Ghent University, Ghent, Belgium

Luis E. C. Rocha & Koen Schoors
Department of Physics and Astronomy, Ghent University, Ghent, Belgium

Luis East. C. Rocha & Jan Ryckebusch
The Business School, Edinburgh Napier University, Edinburgh, UK

Matthew Smith
College School of Economics, National Research University, Moscow, Russia

Koen Schoors

Contributions

L.R. designed the enquiry, made the assay and wrote the draft; J.R., K.S., Yard.South. contributed with methods; All authors revised the manuscript.

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Correspondence to Luis Due east. C. Rocha.

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Rocha, L.E.C., Ryckebusch, J., Schoors, K. et al. The scaling of social interactions across animal species. Sci Rep 11, 12584 (2021). https://doi.org/10.1038/s41598-021-92025-1

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Received: 02 Feb 2021
Accepted: 01 June 2021
Published: xv June 2021
DOI : https://doi.org/x.1038/s41598-021-92025-1

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