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Reciprocity and behavioral heterogeneity govern the stability of social networks

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Abstract

The dynamics of social networks can determine the transmission of information, the spread of diseases, and the evolution of behavior. Despite this broad importance, a general framework for predicting social network stability has not been proposed. Here, we present longitudinal data on the social dynamics of a cooperative bird species, the wire-tailed manakin, to evaluate the potential causes of temporal network stability. We find that when partners interact less frequently, and when the breadth of social connectedness within the network increases, the social network is subsequently less stable. Social connectivity was also negatively associated with the temporal persistence of coalition partnerships on an annual timescale. This negative association between connectivity and stability was surprising, especially given that individual manakins who were more connected also had more stable partnerships. This apparent paradox arises from a within-individual behavioral trade-off between partnership quantity and quality. Crucially, this trade-off is easily masked by behavioral variation among individuals. Using a simulation, we show that these results are explained by a simple model that combines among-individual behavioral heterogeneity and reciprocity within the network. As social networks become more connected, individuals face a trade-off between partnership quantity and maintenance. This model also demonstrates how among-individual behavioral heterogeneity, a ubiquitous feature of natural societies, can improve social stability. Together, these findings provide unifying principles that are expected to govern diverse social systems. Significance Statement In animal societies, social partnerships form a dynamic network that can change over time. Why are some social network structures more stable than others? We addressed this question by studying a cooperative bird species in which social behavior is important for fitness, similar to humans. We found that stable social networks are characterized by more frequent interactions, but sparser connectivity throughout the network. Using a simulation, we show how both results can be explained by a simple model of reciprocity. These findings indicate that social stability is governed by a trade-off whereby individuals can either maintain a few high-quality partners, or increase partner number. This fundamental trade-off may govern the dynamics and stability of many societies, including in humans.
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Reciprocity and behavioral heterogeneity govern the stability of social
networks
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Roslyn Dakin1,2* and T. Brandt Ryder1,3§
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1Migratory Bird Center, Smithsonian Conservation Biology Institute, National Zoological Park,
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Washington, DC 20013, USA
2Department of Biology, Carleton University, Ottawa, ON, K1S 5B6, Canada
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3Bird Conservancy of the Rockies, Fort Collins, Colorado, 80525, USA
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Corresponding authors:
* roslyn.dakin@gmail.com
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§ ryder@si.edu
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Abstract
The structure of social network interactions can determine the transmission of information, the
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spread of diseases, and the stability of diverse systems1–6. Despite this broad importance, a
general framework for predicting social network stability has not been proposed7,8. Here, we use
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repeated measures of social structure in a cooperative bird species9,10, the wire-tailed manakin, to
evaluate the causes of temporal network stability. We find that social networks become
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destabilized when partners interact less frequently, and when social connectedness within the
network increases. This negative effect of connectivity also predicts the temporal persistence of
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social partnerships on an annual timescale. These results were surprising11, especially given that
individuals who were more connected in the manakin population had more stable social
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partnerships. This apparent paradox arises from a within-individual behavioral trade-off between
the maintenance of partner quantity and quality. Crucially, the trade-off is easily masked by
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substantial among-individual heterogeneity in sociality12,13. Using individual-based simulations,
we show that these results are explained by a simple model that combines among-individual
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behavioral heterogeneity and reciprocity within the network. This model also demonstrates how
among-individual heterogeneity, a ubiquitous feature of natural societies13,14, can stabilize a
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social network. However, simulations including only heterogeneity or reciprocity alone were
insufficient to generate the empirical patterns of stability. Together, these findings provide
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unifying principles that are expected to govern the stability of diverse social systems.
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2
Social network structure – or, the way individuals are linked by repeated social interactions – can
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influence the transmission of information, culture, resources, and diseases1–6. Recent work has
begun to demonstrate how changes to social network topology can have diverse costs2,1517 and
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benefits1820. Theory is well established on the formation of biological networks and the
resilience of ecological communities and gene co-expression systems to the loss of
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connections14,21,22. However, we lack a general framework to explain how behavioral processes
within a social network govern the stability of partnerships7,8. A mechanistic understanding of
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these processes is essential to explain the diversity of network structures, predict the downstream
fate of social interactions, and engineer societies.
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Here, we combine repeated measures of social structure and mechanistic models to
elucidate the drivers of social network stability in a cooperative system. Our empirical approach
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was based on autonomous biologging of a neotropical bird species, the wire-tailed manakin
Pipra filicauda. Cooperative partnerships are a key part of the manakin social system5,9,10,
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similar to humans23. Male wire-tailed manakins cooperate by forming display coalitions which
are the basis of dynamic social networks12,24. Cooperation occurs at spatial locations known as
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leks, each of which typically has between 4-14 display territories and about twice as many males
that visit the lek (each of whom limits his interactions to particular coalition partners25). The
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male-male coalitions that form at the lek territories are a prerequisite for an individual to attain
territory ownership, and ultimately, sire offspring9,10.
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By tracking a population of 180 male manakins and 36,885 social interactions over three
years, we took repeated measures of weighted social networks at 11 different lek sites (on
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average, repeated measures of the same lek were 21 days apart; IQR 17-24). To analyze the
temporal dynamics of network topology from time t1 to t2, we define network stability as the
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number of male-male partnerships (network edges) shared by both time points divided by the
number of partnerships at either time point (i.e., the intersection divided by the union; Fig. 1a).
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This metric captures the persistence of social partnerships at a collective scale, after accounting
for any changes in the representation of particular individuals (nodes)26. We found that the
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manakin social networks were more stable than expected by chance (Fig. 1a), similar to other
social animals15,27,28. However, stability was not constant, because each network fluctuated
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across a range of values (mean stability 0.43 ± SD 0.23; repeatability of stability 0, 95% CI 0–
0.22).
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To test how the social structure at t1 might predict subsequent network stability, we used
a mixed-effects modelling framework. We found that three properties of the social network
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structure could explain 28% of the variation in network stability in the best-fit statistical model:
network size (the number of individuals or nodes in the network), network weight (the average
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frequency of social interactions), and network density (the proportion of possible partnerships
that actually occurred, which is a measure of connectivity; Fig. 1b–d). Our analysis also
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controlled for the timing (year and mean Julian date) when each sample of a network was taken.
All else being equal, when more individuals within the network interacted (greater connectivity),
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and when dyads in the network interacted less frequently, the social structure destabilized over
the subsequent weeks (Fig. 2 and Extended Data Tables 1-2).
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The result that network connectivity decreases stability was unexpected, given that
connectedness is thought to foster social cohesion11. To provide a mechanistic explanation for
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these results, we built a simulation model based on the hypothesis that individual behaviors
would drive emergent properties of the network14. Our individual-based model assumed simple
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rules that describe a scenario of reciprocity23 with among-individual heterogeneity13,14: (i) social
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partnerships are formed through reciprocal partner choice, wherein both individuals must choose
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each other; (ii) individuals prefer social partners with whom they have interacted in the recent
past29; and (iii) individuals differ in the expression of behavioral phenotype (sometimes referred
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to as among-individual heterogeneity, or personality). This third assumption of heterogeneity has
been found to be ubiquitous in humans and animals13 and has been shown to influence collective
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performance30 and the evolution of cooperation31,32. We generated 3,000 initial networks de novo
across a broad range of network sizes, weights, and densities, and used these initial random
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networks to parameterize (ii) and (iii). We then allowed the individual nodes to interact
repeatedly. Finally, we computed the stability of each network, by comparing the initial structure
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to the one that resulted from the newly-simulated interactions.
Similar to the manakin data, we found that well-connected networks (high density) with
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low frequency of cooperation (low weight) were less stable (Fig. 2). Hence, the simple model of
reciprocity plus heterogeneity was sufficient to recreate patterns of temporal change observed in
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the empirical networks. Moreover, we found that null model simulations lacking all three
assumptions (i–iii), or that included only (i), (ii), or (iii) alone, were insufficient to generate the
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empirical patterns of stability. In null models that lacked reciprocity, denser networks were also
consistently more stable, making the negative effect in the empirical data particularly striking.
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Overall. our findings indicate that both behavioral processes, reciprocity and heterogeneity, are
necessary to recreate the weight and density effects. Finally, we found that the larger simulated
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networks with more individuals were also significantly less stable, independent of network
weight and density. This effect of network size was also consistent with the manakin data
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(although in the empirical analysis, it was not quite statistically significant; see Extended Data
Table 3).
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Why are some social partnerships able to persist through time15,28? To understand how
social structure might influence the fidelity of particular bonds over longer timescales, we
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analyzed the annual persistence of 669 manakin partnerships from one season to the next (Fig.
3a–b). The analysis accounted for the identities of the partners, the year, the lek network where
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the partnership occurred, and other factors including the spatial overlap of the individuals. Two
features predominantly explained the variation in partnership persistence: the interaction
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frequency (edge weight), and the local social density (edge connectivity, which quantifies the
number of alternative paths that can connect two partners in a social network). Specifically, a
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partnership was more likely to persist if the two individuals interacted more frequently, but had
lower connectivity in their social neighborhood. These results are consistent with the phenomena
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observed at the network level over shorter weekly timescales (Fig. 2). Moreover, we found that
the simulation of reciprocity and heterogeneity could also recreate our empirical results on
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partnership persistence (Fig. 3c–d).
These negative effects of social connectivity suggest that social stability is governed by a
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fundamental trade-off between the quantity and quality of social partnerships. Contrary to the
trade-off hypothesis, however, the manakins with more partners (i.e., those with higher average
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degree centrality) formed coalitions that were more likely to persist through time (Fig. 4a). This
apparent paradox is resolved by partitioning the variation among- and within-individuals (Fig.
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4b-c). Among individuals, the males who were more connected were better able to maintain their
partnerships (Fig. 4b). However, when a given male had more partners than his average, he was
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less able to maintain them (Fig. 4c). Thus, each individual may have a different threshold for the
number of stable coalition partnerships he is able to maintain. This explains why densely
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connected social networks are less stable (Fig. 2) even though well-connected individuals are
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better at maintaining partnerships (Fig. 4a-b). In wire-tailed manakins, the proximate causes of
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this among-individual heterogeneity are not yet well understood33, but hypotheses for future
study may include a male’s quality, social experience, and/or his compatibility with his partners.
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An important conclusion from our results is that among-individual heterogeneity can often mask
behavioral trade-offs13, including in humans, emphasizing the importance of longitudinal studies.
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How might the magnitude of behavioral heterogeneity influence the stability of
cooperative networks30? Our simulation model provided an opportunity to explore this question.
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To measure heterogeneity, we computed the coefficient of variation in degree centrality
(CVdegree) in each of the initial networks; higher values indicate greater behavioral heterogeneity
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in the system21. We found that CVdegree had a significant positive effect on subsequent network
stability (Fig. 5), demonstrating that individual variation in sociality can foster stable social
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networks. This is similar to the way some ecological systems are affected by heterogeneity (e.g.,
CV of connectedness (degree) and edge weights)21,22. In social systems, behavioral heterogeneity
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can also include suites of correlated traits such as dispersal, risk-taking, and cognitive ability, in
addition to variation in sociality7,13,34. Further study is needed to understand how this covariation
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influences social network stability and the evolution of complex social behavior.
In human societies, social media platforms are rapidly transforming our social networks
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by increasing the density of online relationships at a potential cost to face-to-face interactions6,35.
Here, we show how these changes can broadly destabilize emergent social structure. Our results
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explain why increased connectedness is detrimental for the stability and maintenance of
cooperative networks. When individuals become too connected, time and energy constraints can
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limit the ability to reciprocate social partnerships. An important question is how much
topological changes affect other dynamics, such as the spread of emotions, cultural evolution,
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and disease transmission. Although our study focused on one type of cooperative system, many
other social networks are formed as a result of competitive, aggressive, mating, and information-
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sharing interactions7. As a unifying framework, we propose that social stability in these other
contexts will also be determined by the simple behavioral processes that generate heterogeneity,
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partner preferences, and the symmetry of partner choice.
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METHODS
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Field methods
Observed social networks were based on a study of wire-tailed manakins, Pipra filicauda, at the
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Tiputini Biodiversity Station in Ecuador (0º 38’ S, 76º 08’ W, 200 m elevation). Male wire-tailed
manakins perform cooperative courtship displays at exploded leks, where males are in acoustic
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but not visual contact36. The population at Tiputini has been monitored since 2002 to study the
fitness benefits of cooperative behavior9,10. The present study spanned three field seasons
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(December-March) in 2015-16, 2016-17 and 2017-18, and used an automated proximity data-
logging system to record cooperative interactions among males, as described in previous
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protocols that provide detailed algorithms12,24. Briefly, manakins were captured using mist-nets
and each male was outfitted with unique color bands and a coded nano-tag transmitter (NTQB-2,
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Lotek Wireless; 0.35 g). To record the social network at a given lek, proximity data-loggers
(SRX-DL800, Lotek Wireless) were deployed in each territory to record all tag detections within
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the territory from 06:00 to 16:00 for ~6 consecutive days (± SD 1 day)12,33. Because the social
interactions were measured using this automated system, the networks were constructed blind to
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the sociality of particular individuals and/or leks. Territory ownership was assigned using direct
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observation of color-banded males at the display sites9, and was subsequently verified in the
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proximity data. Sample sizes were not predetermined because our aim was to track all
individuals within the studied leks33. All animal research was approved by the Smithsonian
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ACUC (protocols #12-23, 14-25, and 17-11) and the Ecuadorean Ministry of the Environment
(MAE-DNB-CM-2015-0008).
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Data processing
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All data processing and statistical analyses were performed in R 3.5.137. Male-male cooperative
interactions on the display territories were determined using spatiotemporal overlap of tag
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detections in the proximity data (see our previous study12 for detailed algorithms and ground-
truthing data). We used these dyadic interactions to build a weighted social network for each lek
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recording session, with nodes representing male identities, and the edges weighted by the
frequency of dyadic interactions summed over the recording session. In total, we characterized
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86 repeated measures of the social networks at 11 leks (mean 7.8 sessions per lek, ± SD 3.7)
from 29,760 sampling session hours and 36,885 unique social interactions among 180
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individuals. We used a clustering analyses in the igraph package38,39 to verify that our sampling
design was well-matched to the inherent social structure of the population (Extended Data Fig.
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1).
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Network stability
The stability of social network topology is determined by both the gain and loss of associations
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over time. We therefore defined a bidirectional metric of social network stability that compares
two repeated measurements of a social network structure, N, at times t1 and t2. The stability of N
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over the period
𝑡"↔ 𝑡$
is defined as the number of social partnerships (i.e., edges) shared by N1
and N2 (i.e., intersection Ç ), divided by the total number of unique edge connections in either N1
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or N2 (i.e., union È). Using E to represent network edges, stability is thus defined as:
𝑆𝑡𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =
(
𝐸"
𝐸$
)
(
𝐸"
𝐸$
)
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In most social networks, individuals (or nodes) can also be gained or lost over time, which alters
the set of possible interactions that could occur. To ensure that our measure of social network
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stability was based on edges that could have occurred at both time points, only individuals who
were present at both t1 and t2 were included in the calculation26. Therefore, our definition
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captures stability of the network topology after accounting for changes in the presence of
individual nodes. To further account for measurement error40, we also filtered the stability
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calculation to include only significant edges that met two criteria in our empirical data: (1) they
occurred more often than the average occurrence in 1,000 random permutations of the interaction
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data, and (2) they occurred at least six times during the recording session (i.e., on average, about
once per day). The second criterion ensured that rare interactions were not easily deemed
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significant. The value of six was chosen to correspond to the average length of the recording
sessions, but we also verified that other thresholds >2 did not influence our results.
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The average stability score for the manakin networks was 0.43 (± SD 0.23, n = 60
networks at 11 leks). Note that the sample size of 60 is smaller than the total number of
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recording sessions, because the stability dataset is limited to networks that were also sampled at
t2 within the same season. The observed networks were also more stable than expected by chance
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(paired t-test, t = 12.08, p < 0.0001), as determined by random network rewiring (100 edge
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permutations for each of the 60 measurements; grand mean null expected stability 0.07 ± SD
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0.05).
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Network-level analyses
To determine how network-level properties at t1 predict subsequent network stability, we fit
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mixed-effects regression models using the package lme441 (n = 60 networks at 11 leks). The
analysis included lek identity as a random effect, and to account for potential temporal trends, we
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also included field season (categorial) and mean Julian date of the social interactions
(continuous) as two fixed effects. We considered five other fixed-effects that characterize the
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network structure: (1) network weight, a measure of the average relationship frequency,
calculated as the mean of the log-transformed edge weights; (2) network density, a measure of
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connectivity calculated as the proportion of relationships that actually occurred relative to a
completely connected network; (3) network transitivity, a measure of the probability that a given
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individual/node’s social partners are also connected; and (4) network modularity, a measure of
how well the network can be subdivided into separate communities using the random-walk
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algorithm38,39. To account for the fact that network-level properties often scale with network
size14,42 (Fig. 1e–g), we also included (5) the log-transformed number of individuals/nodes in the
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social network. Because density, transitivity, and modularity were all similar measures of
network connectivity, and because the sample size was relatively small, we used a model
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selection procedure to compare candidate models that included either (2), (3) or (4), and/or (1),
or none of (1)-(4); full results are provided in Extended Data Tables 1-2. Finally, we evaluated
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whether our measures of network stability were influenced by two logistical factors: first,
sampling effort, and second, a testosterone manipulation experiment that was conducted for a
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separate study in 2016-17 and 2017-18 (n = 9 individuals out of 180 that were implanted with
testosterone33). To verify that these two logistical factors did not influence our results, we added
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additional fixed effects for the number of recording hours (median 75, mean 73 ± SD 10) and/or
the number of hormone-manipulated individuals in a given network (median 0, mean 0.10 ± SD
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0.41), neither of which had a significant effect on network stability (all p > 0.43). We also
verified that all conclusions of our network-level analysis were unchanged when accounting for
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either or both of these covariates. To determine the repeatability of network properties of the
leks, we calculated the proportion of total variation that was due to differences among the leks
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using mixed-effects models with lek as the random effect and field season and Julian date as
fixed effects41,43.
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Edge-level analysis
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The edge-level analysis examined the persistence of manakin social partnerships on an annual
timescale. This analysis considered 669 dyadic partnerships among 91 individuals wherein both
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individuals in the partnership were also present and tagged in the subsequent breeding season.
The binary response variable, partnership persistence, was defined as whether a partnership was
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both sustained and significant in the subsequent season (see Network stability for criteria for
significance). Because both individuals in a social partnership can contribute to its fate, and
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because they both had other partnerships in the dataset, we modelled persistence using a
multiple-membership structure in a binomial mixed-effects regression model, fit with the brms
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package44. This method can be used to account for multiple partner identities within a single
random effect structure12,4446. In our analysis, the two identities were weighted equally, because
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we assumed they could both determine partnership persistence. An additional random effect was
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included to account for the lek where each partnership occurred. The analysis also included three
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fixed effects to account for the initial field season (categorical), the territorial status of the pair
(categorical; either two territory-holders, a territory holder plus a floater, or two floaters9), and
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the initial spatial overlap of the pair, which can influence the probability of interaction28.
Because manakins use discrete display territories, we defined the spatial overlap of two males as
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the log-inverse of the chi-squared statistic comparing their distributions of territory detections
(pings) in the proximity data; larger values of this metric indicate greater spatial overlap. To test
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our main hypothesis that edge-level network properties would predict partnership persistence, we
also included the following fixed effects: (1) edge weight, or the log-transformed social
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interaction frequency; (2) edge betweenness, a measure of social centrality, defined as the log-
transformed number of shortest paths passing through that edge; and (3) edge connectivity, a
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measure of social density, defined as the minimum number of edges that must be removed to
eliminate all paths between the two individuals/nodes in a partnership39. We ran four
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independently seeded chains with default priors, storing 2,000 samples from each chain, and
verifying that the convergence statistics were all equal to one44 (Extended Data Table 4).
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Among-individual analysis
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To test whether partnership persistence could be attributed to behavioral differences among
individual manakins, we refit the analysis described above, but without accounting for (1)-(3)
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listed above. The random intercepts from this model provide an estimate of among-individual
variation in social stability12,43. We hypothesized that the following behavioral phenotypes12
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could affect this trait: (1) a male’s average daily effort, measured using his log-transformed
count of detections (pings) on the leks; (2) his average daily strength, using his log-transformed
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sum of interaction frequencies; (3) his average daily degree, using his log-transformed number of
social partnerships, and (4) and his average daily social importance, defined as the exclusivity of
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his partnerships (see our previous protocol12 for additional details). Because these four
phenotypes were also correlated12,33, we compared six candidate regression models, four of
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which included only one behavioral phenotype, one of which included all four phenotypes, and
one that included no behavioral phenotypes (n = 91 individuals; see Extended Data Table 5). All
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candidate models included a male’s status as either a territory-holder or floater.
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Quantity-quality trade-off analysis
We next sought to test the hypothesis that individuals in a network face a trade-off between the
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quantity (number of partners) and stability of those social partnerships. Because among-
individual variance can mask trade-offs that occur within-individuals47, testing this hypothesis
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requires a variance-partitioning approach. To achieve this, we defined repeated measures of
individual partnership maintenance as the proportion of a male’s coalition partners that were
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maintained from a given recording session to the next recording session (n = 565 repeated
measures of 152 individuals). Note that a male had to be present, tagged, detected, and not part
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of the hormone manipulation experiment in both the initial and subsequent recording sessions to
be included in this sample. We used within-group centering to partition the variation in the
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predictor variable, degree centrality, within- and among-individuals48. The first step was to
determine log-transformed degree for each male in each recording session; next, we took a single
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average value per male; and finally, we calculated relative degree in each recording session as a
male’s log-transformed degree minus his overall average. Thus, average and relative degree
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represent two orthogonal predictors that can be analyzed within the same regression model. The
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analysis was fit as a binomial mixed-effects model in lme441 with a random effect of individual
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identity, and it also included two categorical fixed effects to account for field season and
territorial status (Extended Data Table 6).
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Individual-based simulation models
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To provide a mechanistic explanation for how individual behavior scales up to influence social
network stability, we developed a simple individual-based simulation model. The model was
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based on the general principles of social reciprocity23 and among-individual behavioral
heterogeneity. There were three core assumptions: (i) individuals had to actively choose each
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other in order to form a partnership; (ii) each individual had a ranked set of preferences for social
partners, predicted by its previous social interaction frequencies, and (iii) individuals express
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consistent differences in their social behavior (referred to as behavioral phenotype). The second
rule (ii) is supported by strong evidence that social relationships are non-random and persist over
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long time-scales in human and nonhuman animals15,28. Together, (i) and (ii) also represent a form
of reciprocal altruism23, because prior interactions increase the probability that a partner will be
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re-chosen. Rule (iii) represents a phenomenon that is often referred to as among-individual
variation, heterogeneity, or personality; it has empirical support across vertebrates13, including in
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manakins12.
To experimentally test the effects of network size, weight, and density on network
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stability, we generated 3,000 initial networks with diverse properties that were within the range
of the observed data. Network size was first chosen from the range of 11-20 individuals or nodes
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(10 size bins). To manipulate network density along the same range observed in the manakin
data, we first generated completely connected networks, and then randomly removed edges until
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a target initial density was achieved (targets ranging from 0.2-0.8, for a total of 20 target density
bins). To generate a broad range of initial network weights, each edge weight was first sampled
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from the manakin data, and then multiplied by a weight constant ranging from 0.2-2.0 (15 weight
factor bins). The resulting edge weights were then rounded up, to a maximum of 500. We
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generated 3,000 networks with all possible combinations of these network properties (10 x 15 x
20 = 3,000).
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The simulation proceeded as follows. First, to satisfy rule (ii), we assigned a set of
preferences to each node based on that node’s partnerships in the initial starting network. The set
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of preferences included all other nodes, ranked by interaction frequency with the focal node in
the initial network. Hence, the probability of choice was correlated with initial interaction
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frequency. To satisfy rule (iii), each node was also allotted a specific number of interaction
attempts per time step (ranging from 1-4). This number was calculated by log-transforming the
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strength of the focal node in the initial network (also referred to as weighted degree) to obtain its
behavioral phenotype; higher values meant that a node could attempt more social interactions per
344
unit time. To satisfy rule (i), a partnership was only formed if both nodes chose each other within
a given time step. The simulation ran over five time steps and the final network was determined
346
by summing the new interactions that occurred (Extended Data Fig. 2). No filtering was applied
to calculate network stability in the simulation. For the null model, we followed the same
348
procedures above, except that each individual’s partner choice probabilities were assigned
randomly to the set of all other nodes, the number of attempted interactions per time step was
350
fixed across individuals, and reciprocal partner choice was not required for partnership formation
in the null model (i.e., assumptions ii, iii, and i were removed). We also tested models with either
352
(i), (ii), or (iii) alone. After running the simulations, we used linear models to statistically
.CC-BY-NC-ND 4.0 International licenseIt is made available under a
was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint (which. http://dx.doi.org/10.1101/694166doi: bioRxiv preprint first posted online Jul. 7, 2019;
9
analyze the variation in network stability and examine our three predictors of interest: network
354
size, weight, and density. To compare the results of this analysis with the statistical estimates
derived from the observed data, all predictors and response variables were standardized to have a
356
mean of 0 and SD of 1 (Extended Data Table 3). To test whether the simulation model of
reciprocity and heterogeneity could also explain our edge-level analysis, we used a binomial
358
mixed-effects regression of edge persistence in the simulation, with the identity of the initial
network as a random effect, and edge weight and edge connectivity as the predictors.
360
Data availability
362
The data and R code necessary to reproduce our results are available at:
https://figshare.com/s/470aeac186a9dab72860
364
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14
Supplementary Information
Extended Data Figures 1-2 and Tables 1-6
468
Acknowledgements
470
We thank Ben Vernasco, Camilo Alfonso, Brent Horton, Ignacio Moore, Brian Evans, David and
Consuelo Romo, Kelly Swing, Diego Mosquera, Gabriela Vinueza, the Tiputini Biodiversity
472
Station of the Universidad San Francisco de Quito, and Julie Morand-Ferron. Funding was
provided by National Science Foundation (NSF) IOS 1353085 and the Smithsonian Migratory
474
Bird Center.
476
Author Contributions
RD and TBR designed the study, analyzed the data, and wrote the paper. TBR collected the data.
478
RD wrote the simulations.
480
Data Deposition
The data and R code necessary to reproduce our results are available at:
482
https://figshare.com/s/470aeac186a9dab72860
484
Competing Interests
We declare no competing interests.
486
.CC-BY-NC-ND 4.0 International licenseIt is made available under a
was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint (which. http://dx.doi.org/10.1101/694166doi: bioRxiv preprint first posted online Jul. 7, 2019;
15
FIGURE LEGENDS
488
Fig. 1. The temporal stability of social networks. a, Variation in network structure is generated
490
by the temporal dynamics of social interactions. In a, two initial manakin networks are shown
(blue and green), with individuals depicted as nodes, and edge thickness weighted by the
492
interaction frequency on a log scale. When the same two networks were sampled a second time,
the edge structure of the blue network remained mostly stable, whereas the structure of the green
494
network had largely changed. Social networks in wire-tailed manakins were more stable than
expected by chance, as shown by the fact that nearly all of the observed stabilities in the grey
496
distribution exceed the 95% confidence interval of the null expectation (vertical black bar; the
null is based on an edge permutation). The observed networks also varied in properties such as:
498
b, the number of individuals (size), c, the proportion of possible relationships that occurred
(density), and d, the average frequency of interactions (weight). The illustrated networks in b–d
500
are colored according to the gradients along the x-axes. In the illustration for network weight,
edge thicknesses are also scaled to the average interaction frequency. e–f, Scaling of density and
502
weight with network size. g, Repeatability of network properties ±95% confidence intervals (n =
86 repeated measures, 60 for stability, of 11 lek networks).
504
Fig. 2. A model of reciprocity and behavioral heterogeneity predicts network stability. a–b,
506
The stability of a social network is positively associated with the average frequency of
interactions (weight), whereas stability is negatively associated with the proportion of possible
508
relationships that occurred (density). These effects were confirmed in an individual-based
simulation of reciprocity that combined three behavioral rules: (i) a requirement for reciprocal
510
partner choice, (ii) a preference for previous partners, and (iii) repeatable variation among
individuals in social behavior. The sample sizes are n = 60 repeated measures of 11 lek networks
512
in the observed data, and n = 3,000 simulated networks. Note that the standardized coefficients
derived from the simulation fall within the shaded 95% confidence intervals of the observed
514
data, unlike the null model. The 95% confidence intervals for all simulation estimates are not
visible because they are narrower than the data points. The right columns in a and b show partial
516
residual plots from the statistical analyses, after accounting for additional covariates (Extended
Data Tables 1-3). For clarity, shading is used to show the 95% central range for partial residuals
518
binned along the x-axis, instead of plotting all 3,000 data points for the simulations.
520
Fig. 3. Social structure predicts the long-term persistence of social partnerships. a–b, The
probability that a partnership persisted across years was greater when the two partners interacted
522
at a higher frequency (edge weight), but had fewer paths connecting them in the social network
(edge connectivity). Data points show how edge weight and connectivity (x-axes) determine the
524
predicted probability of partnership persistence (y-axis) in a multiple-membership analysis (n =
669 partnerships among 91 individuals). c–d, The influence of edge weight and connectivity is
526
also found in the individual-based model of reciprocity described in Fig. 2. Shaded areas in c–d
show the 95% central range for partial residuals binned along the x-axis.
528
Fig. 4. Behavioral heterogeneity and social stability. Social behavior varies both within and
530
among individuals. a, At the among-individual level, manakins who were consistently more
gregarious fostered greater long-term social stability. The plot in a shows how an individual’s
532
behavior (x-axis) predicts his effect on annual partnership persistence (y-axis, ±95% confidence
.CC-BY-NC-ND 4.0 International licenseIt is made available under a
was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint (which. http://dx.doi.org/10.1101/694166doi: bioRxiv preprint first posted online Jul. 7, 2019;
16
interval of the prediction line). Males who consistently interacted with more partners per day
534
(high average daily degree) promoted long-term coalition persistence (n = 90 individuals). b–c,
A trade-off is revealed when examining repeated measures within-individuals. The plots in b and
536
c show partial residuals from a variance partitioning analysis of within-season partnership (or
edge) maintenance (n = 565 repeated measures of 152 individuals). Despite the positive among-
538
individual effect shown in b, at times when a given male had more partners than his average in c,
he was less able to maintain stable partnerships. To visualize among- and within-individual
540
variation, a single average is plotted for each male in b (± SE if a male had >3 measurements),
whereas a separate linear fit is shown for individuals with >3 measurements in c. d, Social
542
networks with greater among-individual behavioral heterogeneity (CVdegree) were also more
temporally stable in an individual-based simulation. The y-axis shows partial residuals from an
544
analysis that also accounts for the effects of network size, weight, and density (n = 3,000).
Shading indicates the 95% central range for partial residuals binned along the x-axis.
546
.CC-BY-NC-ND 4.0 International licenseIt is made available under a
was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint (which. http://dx.doi.org/10.1101/694166doi: bioRxiv preprint first posted online Jul. 7, 2019;
Figure 1
a b
e
t
1
Stability
0.88
0.30
0.0
0.2
0.4
0.6
0.8
1.0
Repeatability
Size Density Weight Stability
Frequency
0
10
20
30
# Individuals
2 5 10 20 40
Proportion
0.0 0.2 0.4 0.6 0.8 1.0
Mean log(edge weight)
0123
f
0.2
0.4
0.6
0.8
1.0
# Individuals
Network density
5 10 20 40 0
1
2
Network weight
5 10 20 40
Network size Network density Network weight
c d
Time
g
Stability
0 1
Fig. 1. The temporal stability of social networks. a, Variation in network structure is generated
by the temporal dynamics of social interactions. In a, two initial manakin networks are shown
(blue and green), with individuals depicted as nodes, and edge thickness weighted by the interac-
tion frequency on a log scale. When the same two networks were sampled a second time, the
edge structure of the blue network remained mostly stable, whereas the structure of the green
network had largely changed. Social networks in wire-tailed manakins were more stable than
expected by chance, as shown by the fact that nearly all of the observed stabilities in the grey
distribution exceed the 95% confidence interval of the null expectation (vertical black bar; the
null is based on an edge permutation). The observed networks also varied in properties such as:
b, the number of individuals (size), c, the proportion of possible relationships that occurred
(density), and d, the average frequency of interactions (weight). The illustrated networks in b–d
are colored according to the gradients along the x-axes. In the illustration for network weight,
edge thicknesses are also scaled to the average interaction frequency. e–f, Scaling of density and
weight with network size. g, Repeatability of network properties ±95% confidence intervals (n =
86 repeated measures, 60 for stability, of 11 lek networks).
.CC-BY-NC-ND 4.0 International licenseIt is made available under a
was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint (which. http://dx.doi.org/10.1101/694166doi: bioRxiv preprint first posted online Jul. 7, 2019;
Null model
Network stability
0.5 1.5 2.5
0.0
0.5
1.0
Network weight
0.2 0.5 0.8
0.0
0.5
1.0
Network density
Network stability
Observed
networks
a b
Individual-based model
Network weight
effect (std. coef.)
Network density
effect (std. coef.)
+ +
Figure 2
i ii iii
0.5 1.5 2.5
0.0
0.5
1.0
0.2 0.5 0.8
0.0
0.5
1.0
0.5 1.5 2.5
0.0
0.5
1.0
0.2 0.5 0.8
0.0
0.5
1.0
0 0.4 0.8 -1 0 1
Fig. 2. A model of reciprocity and behavioral heterogeneity predicts network stability. a–b,
The stability of a social network is positively associated with the average frequency of interac-
tions (weight), whereas stability is negatively associated with the proportion of possible relation-
ships that occurred (density). These effects were confirmed in an individual-based simulation of
reciprocity that combined three behavioral rules: (i) a requirement for reciprocal partner choice,
(ii) a preference for previous partners, and (iii) repeatable variation among individuals in social
behavior. The sample sizes are n = 60 repeated measures of 11 lek networks in the observed data,
and n = 3,000 simulated networks. Note that the standardized coefficients derived from the
simulation fall within the shaded 95% confidence intervals of the observed data, unlike the null
model. The 95% confidence intervals for all simulation estimates are not visible because they are
narrower than the data points. The right columns in a and b show partial residual plots from the
statistical analyses, after accounting for additional covariates (Extended Data Tables 1-3). For
clarity, shading is used to show the 95% central range for partial residuals binned along the
x-axis, instead of plotting all 3,000 data points for the simulations.
.CC-BY-NC-ND 4.0 International licenseIt is made available under a
was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint (which. http://dx.doi.org/10.1101/694166doi: bioRxiv preprint first posted online Jul. 7, 2019;
Probability of annual partnership persistence
a b
c d
Edge weight (log) Edge connectivity (log)
Figure 3
0123456
0.0
0.2
0.4
0.6
0.8
1.0
1.0 2.0 3.0
0.0
0.2
0.4
0.6
0.8
1.0 Individual-based
model
0123456
0.0
0.2
0.4
0.6
0.8
1.0
1.0 2.0 3.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 3. Social structure predicts the long-term persistence of social partnerships. a–b, The
probability that a partnership persisted across years was greater when the two partners interacted
at a higher frequency (edge weight), but had fewer paths connecting them in the social network
(edge connectivity). Data points show how edge weight and connectivity (x-axes) determine the
predicted probability of partnership persistence (y-axis) in a multiple-membership analysis (n =
669 partnerships among 91 individuals). c–d, The influence of edge weight and connectivity is
also found in the individual-based model of reciprocity described in Fig. 2. Shaded areas in c–d
show the 95% central range for partial residuals binned along the x-axis.
.CC-BY-NC-ND 4.0 International licenseIt is made available under a
was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint (which. http://dx.doi.org/10.1101/694166doi: bioRxiv preprint first posted online Jul. 7, 2019;
0 0.4 0.8
0.3
0.5
0.7
Figure 4
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
Average session degree
(# partners/session, log)
Proportion of edges
maintained
-2 -1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
Relative session degree
0.5 1.0 1.5 2.0
-1.0
-0.5
0.0
0.5
1.0
Average daily degree
(# partners/day, log)
Individual’s effect on annual
partnership persistence
a Among-individual b Among-individual c Within-individual
CVdegree
Network stability
d Network-level
More heterogeneous
Fig. 4. Behavioral heterogeneity and social stability. Social behavior varies both within and
among individuals. a, At the among-individual level, manakins who were consistently more
gregarious fostered greater long-term social stability. The plot in a shows how an individual’s
behavior (x-axis) predicts his effect on annual partnership persistence (y-axis, ±95% confidence
interval of the prediction line). Males who consistently interacted with more partners per day
(high average daily degree) promoted long-term coalition persistence (n = 90 individuals). b–c, A
trade-off is revealed when examining repeated measures within-individuals. The plots in b and c
show partial residuals from a variance partitioning analysis of within-season partnership (or
edge) maintenance (n = 565 repeated measures of 152 individuals). Despite the positive among-
individual effect shown in b, at times when a given male had more partners than his average in c,
he was less able to maintain stable partnerships. To visualize among- and within-individual
variation, a single average is plotted for each male in b (± SE if a male had >3 measurements),
whereas a separate linear fit is shown for individuals with >3 measurements in c. d, Social
networks with greater among-individual behavioral heterogeneity (CVdegree) were also more
temporally stable in an individual-based simulation. The y-axis shows partial residuals from an
analysis that also accounts for the effects of network size, weight, and density (n = 3,000).
Shading indicates the 95% central range for partial residuals binned along the x-axis.
.CC-BY-NC-ND 4.0 International licenseIt is made available under a
was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint (which. http://dx.doi.org/10.1101/694166doi: bioRxiv preprint first posted online Jul. 7, 2019;
Extended Data Fig. 1
1
H3-1000
10
2
Chichico
Tower
3
P1050
4
GL
5Al Rio
6
HL
7
8
9
P75
Proportion of a male’s
interactions on
his primary lek
Frequency
0
50
100
150
0.0 0.5 1.0
ba c
2017-18
Individual
(colored by
primary lek) Cluster
n = 82
Al Rio
Chichico
GL
H3-1000
HL
MQ3600
P1050
P2-200
P2.5-800
P75
Tower
Leks
Cluster Primary lek
Extended Data Figure 1 | Social structure corresponds to lek structure. We performed a
clustering analysis to verify that the lek units captured the natural social structure of the manakin
population. The first step was to find the natural community structure of each annual social
network (i.e., defined as the aggregation of all recording sessions and all leks in a given field
season). To do this, we used the random walk clustering algorithm in the igraph package. The
three annual social networks were highly modular (modularity coefficients of 0.70, 0.73, and
0.83, respectively), demonstrating strong clustering of interactions into communities. a, Example
annual social network from 2017-18. Each node represents an individual male and is colored
according to his primary lek, defined as the site where he engaged in the most cooperative
interactions. The overlaid color polygons indicate cluster membership. The layout of the network
was determined using the force-directed LGL algorithm, with edges weighted by the interaction
frequencies. Next, we compared annual cluster membership to each male’s primary lek designa-
tion, defined as the lek where the male engaged in the greatest number of social interactions that
year. The probability that a male’s primary lek matched the majority of his cluster was 90%
(266/296 male-years). b, Example Sankey diagram to compare the clustering algorithm with
primary lek usage. The relatively low proportion of crossings demonstrates strong correspon-
dence. c, Histogram showing how males typically limited nearly all of their interactions to a
single lek. This spatial patterning of interactions creates structure (modularity) in the network,
resulting in the strong correspondence between lek structure and the clustering algorithm. Note
that the same color schemes are used throughout a–b. Finally, the assortativity coefficient
provides another way to compare lek and cluster structure in the social networks. The three
annual networks were highly assorted by primary lek (coefficients 0.51, 0.51, and 0.69) to a level
that was statistically indistinguishable from the assortment based on cluster membership
(coefficients 0.47, 0.52, and 0.69, respectively). Based on this strong correspondence between
lek structure and the structure revealed by the clustering algorithm, we conclude that our sam-
pling design was well-matched to the inherent social structure of the population.
.CC-BY-NC-ND 4.0 International licenseIt is made available under a
was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint (which. http://dx.doi.org/10.1101/694166doi: bioRxiv preprint first posted online Jul. 7, 2019;
Extended Data Fig. 2
Sparse Dense
Low High
Initial network weight
Initial network density
a b
c d
N1 initial N2
Σ
IBM
Stability = 0.53
+
+
+
+
N1 initial N2
Σ
IBM
Stability = 0.55
+
+
+
+
N1 initial N2
Σ
IBM
Stability = 0.70
+
+
+
+
N1 initial N2
Σ
IBM
Stability = 0.80
+
+
+
+
Extended Data Figure 2 | Procedure for the individual-based simulation models. a–d, Four
examples illustrate the general approach and the data generated. The initial social network (N1)
represents the starting point used to determine the probabilities of subsequent social interactions.
Each individual-based model (IBM) was run for five time-steps, akin to measuring interactions
across multiple days. During this time, new partnerships were formed, as shown. The subsequent
network (N2) represents the sum of new interactions that occurred during the IBM time-steps.
The stability score is calculated by comparing the topology of N2 with that of N1. Note that
within each example a–d, the layout of particular nodes is kept constant throughout. Edge thick-
nesses are scaled to the log-transformed interaction frequency in a consistent manner throughout
the figure. Note that the N2 networks have a lower network weight as compared to N1, because
the time interval for the simulation was relatively short (albeit sufficient to develop a wide range
of network stability values). The four examples a–d were chosen because they vary in the initial
network weight (increasing left to right) and density (increasing bottom to top). The network size
of eight was used in this figure for visual clarity in the illustration; however, in the actual simula-
tions, network sizes ranged from 11-20 individuals.
.CC-BY-NC-ND 4.0 International licenseIt is made available under a
was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint (which. http://dx.doi.org/10.1101/694166doi: bioRxiv preprint first posted online Jul. 7, 2019;
Preprint
Full-text available
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Connected: The surprising power of our social networks 378 and how they shape our lives
  • N A Christakis
  • J H Fowler
Christakis, N. A. & Fowler, J. H. Connected: The surprising power of our social networks 378 and how they shape our lives. (Little, Brown and Company, 2009).