^{*}

Conceived and designed the experiments: BB AD. Performed the experiments: BB. Analyzed the data: BB. Contributed reagents/materials/analysis tools: AD. Wrote the paper: BB AD. Developed theory: BB.

The authors have declared that no competing interests exist.

An important function of many complex networks is to inhibit or promote the transmission of disease, resources, or information between individuals. However, little is known about how the temporal dynamics of individual-level interactions affect these networks and constrain their function. Ant colonies are a model comparative system for understanding general principles linking individual-level interactions to network-level functions because interactions among individuals enable integration of multiple sources of information to collectively make decisions, and allocate tasks and resources.

Here we show how the temporal and spatial dynamics of such individual interactions provide upper bounds to rates of colony-level information flow in the ant

Using thousands of time-stamped interactions between uniquely marked ants in four colonies of a range of sizes, we demonstrate that observed maximum rates of information flow are always slower than predicted, and are constrained by regulation of individual mobility and contact rate. By accounting for the ordering and timing of interactions, we can resolve important difficulties with network sampling frequency and duration, enabling a broader understanding of interaction network functioning across systems and scales.

An important function of many complex networks (e.g. HIV infections, power grids, mobile phone calls) is to inhibit or promote the transmission of disease, resources, or information between individuals

Ant colonies are a model comparative system for understanding general principles linking individual-level interactions to group-level functions. Local interactions between individuals via direct antennal contact are known to be functionally important

We developed two broadly applicable tools to understand network dynamics: first, a diffusion model to predict bounds to rates of information flow in groups of different size; and second, a ‘time ordered network’ framework for empirically tracing potential pathways of information flow through dynamic interaction networks. We used the ant

We measured information flow by constructing time-ordered networks that preserve the temporal sequence of interaction events, enabling the enumeration of all paths for potential information flow that are causally permitted (i.e. no back-propagation in time;

Because proximity is defined to be a prerequisite for interaction in this and many other networks

We hypothesized that ant colonies would achieve this bound based on a null expectation of no systematic differences between individuals. In this scenario, individual interactions would promote rapid spread of information throughout the colony, potentially enabling efficient task allocation. In several social insect species, individuals do vary in their interaction rate and task assignments, but the consequences for information flow remain unclear

We find that maximum information flow in all our ant networks is significantly slower than predicted by the model at long time scales (^{−6} for all colonies and filming sequences, all 95% CIs within 0.15–0.59). However, at short time scales, information flows faster than predicted. To also determine the minimum average time delay between information reaching an individual from any other individual through direct or indirect paths we used vector clock latencies

We used a diffusion model to predict a theoretical bound to the number of individuals reached over time by a message from a focal individual. The model assumes that individuals interact like in a kinetic gas. We also used empirical time-ordered networks to determine an empirical upper bound to rates of information flow, assuming perfect communication. On rescaled axes determined from Eq. 1, empirical data are predicted to reach theoretical values, falling on the universal 1∶1 line. We found that colony-level information flow is significantly slower than predicted by individual mobility in the diffusion model.

We tested whether this pattern was the result of the specialization of some individuals on interaction tasks. The model assumed that all individuals are unspecialized, which is equivalent to interaction rate being a Poisson-distributed random variable (Eq. S1 in ^{−6}, all slopes within 0.92–1.24). We also tested the hypothesis that only the queen ant might have a preferred or special role in each colony's interaction network. Across colonies in temporal and aggregated networks, we found that, relative to all other ants, the queen did not have a lower latency (mean quantile 35%±9 s.e.), higher out-degree (mean quantile 43%±12 s.e.), or higher betweenness centrality (mean quantile 54%±15 s.e.). These results support the central assumptions of the diffusion model and indicate that individual specialization cannot explain patterns of information flow.

We found that individual ants with high degree in time-aggregated networks from over one filming session did not have high degree in subsequent filmings. This indicates that individual ants do not have long-term specialization in interaction, and that the colony-level network structure emerges from the regulated interactions of many individuals.

These results for specialization have implications for the topology of the interaction network. Scale-free networks, which characterize many human systems, show consistent individual specialization, with a power-law distribution of high- and low-degree individuals. In contrast, the diffusion model predicts that the out-degree distribution ^{−6}, all slope C.I.s within 0.17–0.70). Moreover, the form of the degree distribution for fully aggregated networks in five of eight cases was consistent with the binomial-exponential model in Eq. 2 (maximum likelihood fits of Pareto and binomial distributions: all ΔAIC>4). This slowly converging degree distribution is inconsistent with the properties of scale-free networks, and indicates that sampling window fundamentally affects the properties of time-aggregated networks.

The distribution is binomial and the mean degree increases with time. Data are kernel-smoothed to ease interpretation.

Although the overall shape of each distribution is approximated by the binomial-exponential model, degree increases significantly more slowly than predicted by the diffusion model. This is consistent with the limited information flow observed in

We also tested if information flow was limited by long time delays between interactions. Many human networks are known to have heavy-tailed (power-law) delay distributions

Consistent with many human networks, we found a long-tailed waiting time distribution that was often best fit by a power-law, in contrast to the exponential distribution predicted by the diffusion model (Eq. 3). These long waiting times contribute to limitations to information flow.

Another possible mechanism for limiting information flow is reduced individual movement speed or spatial fidelity to certain regions. We found that individual spatial displacement as a function of time increased more slowly than predicted by the diffusion model (^{−6}, all slopes in 0.67–0.91). Thus, this study suggests that individual ants limit their mobility, resulting in long delays between interactions and limited information flow. Previous studies of individual spatial fidelity in related species are consistent with this finding and provide support for this mechanism

As in a diffusion process, the displacement of individuals from their initial position is predicted to increase with the square root of time (Eq. S4). Across colonies, displacements increase more slowly than expected but do scale with model predictions (test of slope = 1 for SMA regression of transformed data: all p<10^{−6}, all slopes in 0.67–0.91).

Our results call for a deeper general understanding of the adaptive significance of different network structures. Variable rates of information flow may control efficient group function

We have provided strong bounds to information flow in ant networks that are set by constrained mobility and regulation of interactions between unspecialized individuals. These results provide a unique perspective on the organization of ant colonies of a range of sizes and contrast strongly with the common ‘scale-free’ nature of many human systems, challenging notions of structural universality in self-organized networks. A dynamic approach using our framework and model will provide important insights into the link between individual behaviors and group function in other biological networks like food webs

We collected whole colonies of the ant

In order to distinguish individuals, we marked every worker and queen. Each ant was removed from its colony, anesthetized with CO_{2}, and given a unique set of marks with acrylic paints applied with a thin wire under a dissecting microscope. Paint of red, yellow, white, green, or blue color was applied in four locations: head, thorax, left gaster, and right gaster. After the paint dried, ants were immediately revived and returned to their colony. This paint-marking technique previously has been shown to have minimal long-term effect on ant behavior. Colonies were given two days to recover after all individuals were marked.

Colonies were filmed in high definition with a digital camcorder (Canon, HV20). Conditions were standardized to mid-afternoon and ambient lighting, with the camera located above the upper glass surface of the nest. Filming occurred for approximately 30 minutes and was repeated under identical conditions approximately three weeks later. Films were converted to MPEG4 videos and stored on a computer for later analysis.

We recorded all interactions between all individual ants in each video (

We constructed time-aggregated and time-ordered interaction networks using the ‘timeordered’ and ‘igraph’ packages in R (

Because we defined interactions as requiring physical contact between ants, network structure and dynamics can also be understood by studying spatial mobility patterns. We hypothesized that random movement - kinetic gas dynamics - would be sufficient to explain many patterns of interactions in ant colonies. Such models have been previously used for biological studies of animal mobility, but only recently for the purposes of understanding information flow and collective behavior. Here we have assumed that interaction and information flow occurs only during collisions between individuals. In this model, ants behave like particles in a two-dimensional gas: all individuals have identical sizes and interaction rules, such that they walk in straight lines until they touch another ant or a wall, after which they bounce elastically off the obstacle. This model is particularly useful because it depends only on five independently measurable parameters:

Following the derivation of Hutchinson & Waser, the total number of interactions _{G}

The distribution of waiting times between interactions for one individual,

The expected resultant displacement

The previously-discussed out-degree distribution

The model makes several other quantitative predictions. First, touches should be distributed completely spatially randomly within the region, because individual ants are found with equal probability in all locations. Second, the number of touches given by an ant should be equal to the number of touches received by that ant, because the mobility model does not distinguish between the initiator and target of an interaction. Third, for long sampling intervals, the diameter of the network (constructed of links between individuals that have touched at least once) should decrease to one, because all ants will eventually have at least one interaction with each other ant. Fourth, an individual's touch rate should be predictive of its touch rate in the future only for short sampling intervals - that is, over long time scales, no ant should maintain a role as a high-functioning or low-functioning communicator, because touch rate is a Poisson-distributed random variable, and measured touch rate is necessarily averaged over some finite time interval. Lastly, the model also suggests that the queen should have no special role with respect to information flow if she moves in the same way as all other ants.

Time-aggregated networks for all colonies and filming sessions.

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Time-ordered networks for all colonies and filming sessions.

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Out-degree distributions over time for all colonies and filming sessions. Mean degree increases over time (larger percentages of data aggregated) as predicted by the diffusion model.

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Spatial controls on interactions and information latency. Information latency, the minimum delay time for a message to propagate from one individual to another through direct or indirect paths, increases with distance from the center of the nest (test of slope = 0 for OLS regression: p<0.01 for three of four filmings). Ants located in the center of the nest are relatively better informed, indicating spatial structure to the interaction network not predicted by the diffusion model.

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Distribution of interactions in space and time. Interactions are shown as colored dots. More recent interactions are shown in yellow; less recent ones in red. The nest entrance is located at the top center of each plot. Interactions are clustered (KS test of complete spatial randomness for x- and y- covariates, all p<10^{−6}) and appear to propagate in traveling waves through the colony.

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No evidence for positive feedback for interactions. A proposed hypothesis for traveling waves of interactions in colonies is activation of individuals by interaction events (S. Boi,

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Summary of data collected. Asterisks (*) denote filmings in which spatial data were collected.

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Complete network data for all colonies.

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Tuan Cao, Milan Curry, Han Jing, Kayla Lauger, and Daniel Wolf assisted with observations. Emma Wollman drew