Abstract
The social network structure of animal populations has major implications for survival, reproductive success, sexual selection, and pathogen transmission of individuals. But as of yet, no general theory of social network structure exists that can explain the diversity of social networks observed in nature, and serve as a null model for detecting species and population-specific factors. Here we propose a simple and generally applicable model of social network structure. We consider the emergence of network structure as a result of social inheritance, in which newborns are likely to bond with maternal contacts, and via forming bonds randomly. We compare model output to data from several species, showing that it can generate networks with properties such as those observed in real social systems. Our model demonstrates that important observed properties of social networks, including heritability of network position or assortative associations, can be understood as consequences of social inheritance.
The transition to sociality is one of the major shifts in evolution, and social structure is an important and ever-present selective factor, affecting both reproductive success1 and survival2,3. Sociality affects individual health, ecological dynamics, and evolutionary fitness via multiple mechanisms in humans and other animals, such as pathogen transmission e.g.4,5 and promoting or hindering of particular social behaviors6,7,8. The social structure of a population summarizes the social bonds of its members9. Hence, understanding the processes generating variation in social structure across populations and species is crucial to uncovering the impacts of sociality.
Recent years have seen a surge in the study of the causes and consequences of social structure in human and animal societies, based on theoretical and computational advances in social network analysis 10,11,12,13,14,15,16. The new interdisciplinary network science provides many tools to construct, visualize, quantify and compare social structures, facilitating advanced understanding of social phenomena. Researchers studying a variety of species, from insects to humans, have used these tools to gain insights into the factors determining social structure 17,18,19,13. Using social network analysis provided evidence for the effects of social structure on a range of phenomena, such as sexual selection20 and cultural transmission21,22.
At the same time, most applications of social network analysis to non-human animals have been at a descriptive level, using various computational methods to quantify features of social structure and individuals’ position in it. These methods, combined with increasingly detailed data “reality mining”23 about social interactions in nature, provided valuable insights about the complex effects of social interaction on individual behaviors and fitness outcomes. Yet, we still lack a comprehensive theory that can explain the generation and diversity of social structures observed within and between species. There have been only a few efforts to model animal social network structure. Notably, Seyfarth24 used a generative model of grooming networks based on individual preferences for giving and receiving grooming, and showed that a few simple rules can account for complex social structure. This model and related approaches e.g.,25 have been very influential in the study of social structure and continue to drive empirical research. At the same time, they mostly focused on primates and were geared towards specific questions such as the effects of relatedness, social ranks, or ecological factors in determining social structure.
Independently, a large body of theoretical work in network science aims to explain the general properties of human social networks through simple models of how networks form. Yet these models tend to focus either on networks with a fixed set of agents26, or on boundlessly growing networks27, with few exceptions28,29. These network formation models therefore have limited applicability to animal (and many human) social groups where individuals both join (through birth of immigration) and leave (through death or emigration) the network. Furthermore, most work in network science concentrates on the distribution of number of connections individuals have (the degree distribution). Models that fit the degree distribution of real-world networks tend to be a poor fit to other important properties, notably the tendency of social connections to be clustered30,27, i.e., two individuals to be connected with each other if they are both connected to a third individual. Real-world human and animal networks exhibit significantly more clustering than random or preferential attachment models predict27,13.
Simple generative models of complex systems have been highly useful in other fields, such as metabolic networks31 and food webs32, but there has been little effort to build such models applicable to animal social networks. In this paper, we provide a widely applicable network formation model based on simple demographic and social processes. Our model assumes a simple neutral demography and focuses on a central social process that is in operation in many social species: the “inheritance" of social connections from parents. This central component of our model is based on the observation that in many species with stable social groups, individuals interact with the social circle of their parents. This is essentially the case in all mammals, where newborns stay close to their mothers until weaning, but also found in many other taxa, such as birds33, fish34, and arthropods35. After positively interacting with the parents’ social contacts, young individuals are likely to form social bonds with these conspecifics, as was found in African elephants, Loxodonta africana 36.
We demonstrate that this simple social inheritance process can result in networks that match both the degree and local clustering distributions of real-world animal social networks, as well as their modularity (which measures the strength of division of a network into modules, or subgroups). We also show that social heritability of connections can result in the appearance of genetic heritability of individual social network traits, as well as assortativity in the absence of explicit preference for homophily. Our approach highlights commonalities among groups, populations, and species, and uncovers a general process that underlies variation in social structure.
Results
Our departure point is the model by Jackson and Rogers27, in which “role models” in a network introduce their new contact to their other contacts. This model can reproduce many attributes of large-scale human social networks. Similar models reconstruct the structure of other systems, such as protein interaction networks37, and the World Wide Web38. However, Jackson and Rogers’ model (like most other models in this family) is based on a constantly growing network with no death or emigration of agents and their results hold asymptotically for very large networks. Since we are interested in small-scale animal networks that do not grow unboundedly, we model a population where existing individuals die and get replaced at an equal rate with newborn individuals28 (see SI 8 for results for slowly growing and shrinking networks). We model binary and undirected networks, so implicitly assume social bonds are neutral or cooperative, but our model can be extended to weighted networks that describe the strength of each social bond, and directed ones, such as agonistic networks.
Consider a social group of size N. Suppose that each time step, an individual is born to a random mother, and one individual is selected to die at random. With probability pb, the newborn will meet and connect to its mother (generally, pb will be close to one, but can be low or zero in species such as some insects, where individuals might not meet their mothers). A crucial component of our model is the general assumption that the likelihood of a newborn A connecting with another individual B depends on the relationship between A’s mother and B: the probability A will connect to B is given by pn if A’s mother is connected to B, and pr if not (n and r stand for neighbor and random node, respectively; Figure 1). Hence, pn is the probability an offspring “inherits” a given connection of its parent. Here, we focus on the case A always connects to its mother (pb = 1), but the model can be extended to include a lower probability to connect to the contacts of A’s mother if A does not connect to its mother, when pb < 1. If pn > pr, the population exhibits a tendency for clustering, a well-established and general phenomenon in social networks39,13. In the Supporting Information section we present an extension of this basic model to account for two sexes, where only females reproduce. We show that if newborns are likely to copy only their mothers, the resulting social network is similar.
We simulated social network dynamics to test how social inheritance and stochastic social bonding affect network structure, heritability, and assortativity (see Methods for simulation details). We also provide analytical expressions for the degree distribution, and approximations for mean degree and mean local clustering coefficient in the Methods section and in the Supplementary Information (SI; section SI 1). For all of our numerical results, we assume pb = 1. As expected, the network density (the number of edges out of all possible edges) depends on pn and pr. The mean clustering coefficient, a measure of the extent to which nodes tend to cluster together, also depends on these parameters, but not monotonically; high levels of clustering were observed in simulations with low or high pr, but not at intermediate levels (Fig. 2). We also tested how changes in network size, caused by increased or decreased probabilities of death during the simulations, affected its properties. These tests did not provide a general conclusion, but suggested that the network structure might be moderately influenced by whether the network is growing or not (see SI 8).
We then compared the output of our model with observed animal social networks of four different species, namely spotted hyena (Crocuta crocuta13), rock hyrax (Procavia capensis40), bottlenose dolphin (Tursiops spp.41), and sleepy lizard (Tiliqua rugosa42). We used two independent ways to estimate model parameters using data from each of the four species: a computational dimensionality reduction approach (partial least squares regression, PLS) and analytical approximations for the mean degree and local clustering coefficients (see Methods). When we run our model using pn and pr estimated from the data using either method, we recapture the distributions of degree and local clustering coefficient, as well as the network modularity. Figure 3 illustrates that our model of social inheritance can produce networks with realistic social structure (see SI 4 for fitting the two-sex model to observed networks). Our model’s good match of local clustering distributions distinguishes it from other network growth models, based on assortative or generalized social preferences, as well as the preferential attachment models that are popular in network science27 (see also SI 6 and SI 7). Furthermore, our model generated networks with realistic modularity values (see SI, figure S5). The values we found suggest that social inheritance is stronger in hyena and hyrax than in dolphins and sleepy lizards (Table 1).
Next, we tested if social inheritance can result in heritability of indirect network traits in social networks. Direct network traits (individual network traits that depend only on direct association with others, i.e. on the immediate social environment), such as degree, will by definition be heritable when pn is high and pr low. To see if this also holds for indirect network traits (traits that may depend also on associations between other individuals), we measured the correlation between parent and offspring betweenness centrality (which quantifies the number of times a node acts as a bridge along the shortest path between two other nodes; see Methods) for a set of social inheritance (pn) values. As Fig. 4 shows, high probabilities of social inheritance results in a pattern of heritability. In other words, when individuals are likely to copy their parents in forming social associations, the resulting network will exhibit heritability of centrality traits, although the only heritability programmed into the model is that of social inheritance and stochastic bonding. Similar patterns obtain for local clustering coefficient and eigenvalue centrality (Figures S7 and S8).
Finally, we tested the effect of social inheritance on assortativity, i.e. the preference of individuals to bond with others with similar traits. We simulated networks where each individual had one trait with an arbitrary value between 0 and 1. Newborns inherited their mother’s trait with probability 1-μ, where μ is the rate of large mutations. If a large mutation happened, the newborn had a random uniformly distributed trait value; otherwise, its trait was randomly picked from a Gaussian distribution around the mother’s trait, with variance σ2. Individuals followed the same rules of the basic model when forming social bonds. Hence, individuals did not explicitly prefer to bond with others with the same trait value. Nevertheless, the assortativity coefficient was significantly higher than in random networks, in which the trait values were re-assigned randomly (Figure 5).
As an alternative model generating assortativity, we considered an explicit as-sortativity model, in which newborns explicitly prefer bonding with those with similar traits. Although this model unsurprisingly generated networks with high assortativity (mean assortativity coefficient±SEM: 0.53±0.006 compared to -0.01±0.002 in networks with randomly shuffled trait values), it failed to recover the high clustering and modularity observed in networks generated by social inheritance and in the data (Supporting Information, figures S9 and S10). This result further suggests that assortativity might be a byproduct of social inheritance rather than a driving force of social network structure. A more generalized preferential attachment model, described in section SI 7, shows the converse is not true, i.e., that network patterns generated by social inheritance do not arise as a byproduct of genetically inherited traits and association preferences (see Discussion for more).
Discussion
Our model provides a step towards a general theory of social structure in animals that is grounded in social and demographic processes. Our approach is to use dynamic generative models based on simple processes to predict network-scale patterns that those processes are expected to produce, and compare them to observed networks. Such an approach has been widely and productively used in network theory and social sciences43,44,27, as well as other subfields of ecology31,32 but not in animal social networks. Our work addresses this gap. Our main result is that the combination of neutral demography and social inheritance can replicate important properties of animal social networks in the wild.
In particular, we show that our model can capture essential features of social networks of four different species in the wild, including not just the degree distribution and modularity, but also the clustering coefficient distribution, in contrast to most studies of social network formation. Clustering is an important feature of social networks, distinguishing them from other types of networks, such as transportation networks and the internet30. Theory predicts that clustered networks are more conducive to cooperation45, and empirical studies document a tendency to close triads40,13, suggesting that it might be a generally adaptive feature of social structure. Nevertheless, many previous models of sociality and network formation fail to account for the high clustering observed. For example, whereas preferential attachment can reconstruct the degree distribution of social networks, it fails to capture their high degree of clustering27. The social inheritance process is crucial to the formation of cohesive clusters in social networks because it biases newly formed connections to those that close triads of relationships.
Social inheritance requires a behavioral mechanism that facilitates introduction of newborns to their mother’s social partners. As in many species young individuals tend to follow their mothers, it is easy to think about such a passive mechanism: young individuals are introduced to other individuals by spending time with their mother’s partners. This process is consistent with the long-held view that mother-offspring units are fundamental to social structure46. Direct evidence for social inheritance comes from Goldenberg et al.36, who documented the tendency of female African elephants to “inherit” the social bonds of their mothers, driving network resilience. Moreover, in many species group members show active interest in newborns47, promoting the initiation of a social bond between newborns and their mother’s partners. Further work can test if initial interest in newborns later translates to stronger social bonds with individuals reaching adulthood. We note that social inheritance does not necessarily require an active process of “introductions” but can also happen passively, for example as a result of spatial fidelity among group members. Our model is agnostic with regard to the mechanism of social inheritance. That being said, the fitted model parameters for the four networks vary in ways that are suggestive for socio-ecological factors: for hyenas and hyraxes, we find high pn values, which may reflect the strong philopatry in these societies. In contrast, the relatively low fitted value of pn in dolphins may reflect their multi-level society featuring mother-son avoidance48.
We make a number of simplifying assumptions, such as no individual heterogeneity, or age - or stage-structure in our demography. Models of this type have a long and distinguished history in ecology and evolution49, and in the same spirit, we do not believe that nature is actually as simple as we model it. Nonetheless, the fact that this very simple model (but not other simple models, e.g., see SI 6, SI 7) can reproduce important aspects of real networks suggests that the social inheritance of connections is likely to be important in structuring social networks. Even though the details will no doubt vary across species and contexts, this simple, quantifiable process can explain observed variation in social networks. For example, our model does not treat sex-specific dispersal, a mechanism that results in different social environments for the two sexes. Nevertheless, there is evidence for social bonding with familiar individuals after dispersal50. This suggests that even after dispersal, individuals may “inherit” the social bonds of certain conspecifics serving as role models. Another use of simple models such as ours is to serve as a base model to test the effect of additional factors. For instance, after fitting the model to an observed social network, one could test whether personality can explain the variance not explained by social inheritance and stochasticity. This can be attained by adding personality to the agent-based model as a factor that influences individual bonding decisions.
Our model has implications for how the inheritance of positions in social networks, which has important implications for social dynamics, is to be interpreted. For example, Fowler et al. 51 found that in humans, network traits such as degree and transitivity were heritable. In rhesus macaques, Brent et al.52 found that indirect network traits such as betweenness are more heritable than direct ones. In contrast, a study of yellow-bellied marmots, Marmota flaviventris, presented evidence for heritability of social network measures based on direct interactions53, but not indirect interactions. Taken together, these studies show that network position can be heritable, but have not been able to elucidate the mechanism of inheritance. It is not unlikely that some social network traits are genetically inherited; for example, individuals might genetically inherit social preferences from their parents that lead them to connect to the same individuals. In SI 7 we show that such a mechanism is unlikely to account for the observed levels of clustering. Therefore, our work suggests that at least some of the heritability of network traits might be social (as opposed to genetic), from individuals copying their parents. This prediction is borne out by recent studies in elephants36. Importantly, while these previous studies attempt to control for effects of the social environment at the group or lineage level using quantitative genetics methods e.g.54, they were not designed to distinguish social inheritance at the individual level from genetic inheritance. Studying the dynamics of social bond acquisition can be a way to separate genetic and social inheritance.
Another robust finding in network science and animal behavior is that individuals tend to connect to others with traits similar to themselves (e.g.,55,56,57). This assortativity is crucial for social evolutionary theory, as the costs and benefits of social interactions depend on partner phenotypes. Recent work has found that as-sortative mating can arise without assortative preferences, as a result of dynamic processes in a closed system58. Our results show that social inheritance can lead to high assortativity in the absence of explicitly assortative preferences for social bonding. Indeed, an alternative model based on explicit assortativity failed to reconstruct topological features of observed networks. Empirically, our results call for a careful assessment of networks with apparent phenotypic assortment, and controlling for social inheritance. This will be difficult to do with only static network data, but will be feasible for species with long-term data on the network dynamics.
Our work points to several interesting avenues to be explored in future research. First, we used binary networks to describe the strength of social bonds that are inherently on a continuous scale11,59. Weighted networks that can describe the delicate differences in the strength of social bonds between individuals would be more relevant in some cases. Future generative models can consider varying bond strength by coupling a weighted network model with a model of behavioral dynamics of social bond formation for pairs of individuals. Second, even though our model is extremely simplistic, most of its mathematical properties (including probability distributions over network measures such as the degree distribution) are analytically intractable, which makes model-fitting a challenge. Methods such as approximate bayesian computation60, coupled with dimensionality reduction techniques61 can be used to develop algorithms for estimating parameters of the model and also incorporate more information about individual variation and environmental effects (See SI 2 for more). Additionally, long-term datasets on social network dynamics can allow estimation of the social inheritance and random bonding parameters pn and pr directly. Lastly, our model does not consider changes in social bonds after these were established. Although this assumption is supported by empirical findings concerning bond stability in some species13,12, future models in which this assumption is relaxed should be developed. We also assume a single type of bond between individuals, whereas in nature, different social networks exist for different kinds of interactions (e.g., affiliative, agonistic, etc.). Such “multilayer networks”62 represent an important future direction.
In conclusion, the theory we present here is based on the idea that social networks should be regarded and analyzed as the result of a dynamic process63 that depend on environmental, individual, and structural effects13. Our work represents a first step in developing a theory for the structure of social networks and highlights the potential of generative models of social and demographic processes in reaching this goal.
Methods
Expected mean degree and clustering coefficient
In this section and the next, we characterize some important aspects of our model analytically. First, we can write a simple approximation of the expected mean degree, d̅, of a network changing according to our model at stationarity. To do that, we note that at stationarity, killing an individual at random is expected to remove d̅ connections from the network. After this individual is removed, the average degree of the network becomes: . The expected degree of the connections made by the newborn is then: pb + d̅pn + (N – 2 – d̅′)pr. At stationarity, the links destroyed and added need to be the same on average, so we can write: and solve for d̅ to obtain: This approximation gives an excellent fit to simulated networks across all ranges of mean degree (Figure 2).
We can also approximate the expected mean clustering coefficient of a network at stationarity using a similar stationarity argument (see SI 1 for the derivation). Our simulations (Figure 2) show this approximation is very good except for the combination of very low pr and low to moderate pn, where it significantly overpredicts clustering.
Using the approximations for the mean degree and clustering coefficients, assuming pb = 1, and taking N to be the observed network size, we can estimate the pn and pr values for an observed network. In simulated networks, this approach generally yields accurate predictions except for the combinations of high pn and high pr (where it underestimates pr) and low pn and very low pr (where it overestimates pr). Our estimates of pn and pr for the four empirical networks from the analytical approximation are given in Table 1.
Expected degree distribution
Finally, we characterize the expected degree distribution in our networks using a mean-field model. We denote the degree distribution by ϕd for 0 ≤ d ≤ N – 1. In other words, ϕd is the probability that a randomly selected individual in the population has degree d.
Consider a focal individual that has degree d at time period t. In period t+1, the probability that this individual increases its degree by one, , is: The first fraction in (2) is the probability that the individual selected to die is not connected to the focal individual, while the second fraction is the expected probability that the newborn individual born to one of the remaining N – 1 individuals becomes connected to the focal individual.
The probability of a focal individual’s degree d (> 0) going down by one, is likewise given by which is simply the probability that the individual selected to die is connected to the focal individual, multiplied by the probability that the newborn individual does not connect to the focal individual.
Next, we need the probability that a newborn is born with d connections, denoted by bd. To compute this probability, we assume pb = 1 (the extension to pb < 1 is trivial), so that the newborn always connects to its parent, then bd(ϕ) is given by (for d ≥ 1; b0 = 0 in that case): where the inner sum is the probability that an offspring of a parent of degree l is born with degree d, and the outer sum takes the expectation over , the expected degree distribution after the death of a random individual, which for 0 ≤ l ≤ N – 1 is given by: reflecting the facts that the death of a random individual does not change the expected frequency of individuals that had degree d before the death, but with each death, an individual with degree d has a probability d/N of becoming degree d – 1.
Putting everything together, we can write the rate equation for the mean-field dynamics of the degree distribution28: where the first termis the rate of change in the frequency of degree d caused by the replacement of individuals of degree d by death and birth, and the rest of the terms give rates of degree changes due to losing and gaining connections.
Setting equation (6) equal to zero for all d and solving the resulting N equations, we can obtain the stationary degree distribution. We were unable to obtain closed-form solutions to the stationary distribution, but numerical solutions display excellent agreement with simulation results (see Figure 3). It is worth noting that although the and terms are similar to models of preferential attachment with constant network size e.g.28, these models assume that each new addition to the network has exactly the same degree, whereas in our model, the number of links of a newborn is distributed according to equation (4). Furthermore, the degree distribution does not capture the clustering behavior of preferential attachment models, which generate much less clustering than our model for a similar mean degree (results not shown), consistent with results in growing networks27.
Simulation process
We initialized networks as random graphs, and ran them long enough to converge to steady state, which we evaluated by the mean degree distribution of ensembles matching the expected degree distribution, mean degree and clustering values derived analytically. The time to convergence to steady state depends on the network size, pn, and pr: we found as a rule of thumb that 10 times the network size (i.e., 10 complete population turnovers on average) is enough for networks to come to stationarity, hence our choosing of 2000 steps for network size of 100. The only exception is with pn close to 1 (and to a lesser extent, pr very close to zero), where we find that convergence can take significantly longer.
Fitting models to observed networks
To obtain estimates of parameter values pn and pr from observed networks, we used two methods: (i) a computational approach using dimensionality reduction on the degree and local clustering distributions of simulated networks, and (ii) an analytical approach using approximations of the mean degree and local clustering coefficients. In this subsection, we describe the dimensionality reduction approach. For each empirically observed network, we ran the model with 10000 random values of pn and pr between 0 and 1, and the network size was set to match the observed network. We then used partial least squares regression, using the R package pls (version 2.4-3), to obtain a regression of the network degree and clustering coefficient distributions on pn and pr. Based on the regression formula, we predicted the values of pn and pr. The values predicted by the regression were sufficient to simulate networks that were close in their degree and clustering coefficient distributions to the observed networks. The values given in Table 1 are the result of the PLS fit. They are meant to demonstrate the ability of the model to generate realistic looking networks. In the SI, we provide a verification of the method’s ability to obtain the values of pn and pr.
Data
We compared the output of our model with observed animal social networks of four different species. For this analysis we used data from published studies of spotted hyena (Crocuta crocuta13), rock hyrax (Procavia capensis40), bottlenose dolphin (Tursiops spp.41), and sleepy lizard (Tiliqua rugosa42).
The hyena social network was obtained from one of the binary networks analyzed by13, where details on social network construction can be found. Briefly, the network is derived from association indexes based on social proximity in a spotted hyena clan in Maasai Mara Natural Reserve, Kenya, over one full year (1997). The binary network was created using a threshold retaining only the upper quar-tile of the association index values. Similarly, the hyrax network was described by40, and is based on affiliative interactions in a rock hyrax population in the Ein Gedi Nature Reserve, Israel, during a five-months field season (2009). The same upper quartile threshold on the association indices was used to generate a binary network. The dolphin network was published in41, and is based on spatial proximity of bottlenose dolphins observed over 12 months in Doubtful Sound, Fiordland, New Zealand. “Preferred companionships” in the dolphin network represent associations that were more likely than by chance, after comparing the observed association index to that in 20000 permutations. The lizard social network was published by42, and is also based on spatial proximity, measured using GPS collars. To get a binary network, we filtered this network to retain only social bonds with association index above the 75% quartile.
Network measures
To study the networks produced by our model and compare them to observed networks, we used a number of commonly used network measures. Network density is defined as where T is the number of ties (edges) and N the number of nodes. The global clustering coefficient is based on triplets of nodes. A triplet includes three nodes that are connected by either two (open triplet) or three (closed triplet) undirected ties. Measuring the clustering in the whole network, the global clustering coefficient is defined as The local clustering coefficient measures the clustering of each node: The betweenness centrality of a node v is given by where σst is the total number of shortest paths from node s to node t and σst(v) is the number of those paths that pass through v.
We detected network modules (also known as communities or groups) using the walktrap community detection method64. We used the maximal network modularity across all partitions for a given network. The modularity measures the strength of a division of the network into modules. The modularity of a given partition to c modules in an undirected network is where eii is the fraction of edges connecting nodes inside module i, and is the fraction of edges with at least one edge in module i.
Finally, we used the assortativity coefficient to measure how likely are individuals to be connected to those with a similar trait value65. For an undirected network, this coefficient is given by: where exy is the fraction of all edges in the network that connect nodes with traits x and y, ax is defined as , and is the variance of the distribution ax.
Acknowledgments
We are thankful to Kay E. Holekamp and Eli Geffen for sharing data, and to Robert Seyfarth, Elliot Aguilar, Çağlar Akçay, Jeremy Van Cleve, Slimane Dridi, and Tim Linksvayer for valuable comments. Comments from three anonymous reviewers helped improve the paper and are gratefully acknowledged. This study was supported by the University of Pennsylvania and NSF Grant EF-1137894.