A distributed algorithm to maintain and repair the trail networks of arboreal ants

Abstract

We study how the arboreal turtle ant (Cephalotes goniodontus) solves a fundamental computing problem: maintaining a trail network and finding alternative paths to route around broken links in the network. Turtle ants form a routing backbone of foraging trails linking several nests and temporary food sources. This species travels only in the trees, so their foraging trails are constrained to lie on a natural graph formed by overlapping branches and vines in the tangled canopy. Links between branches, however, can be ephemeral, easily destroyed by wind, rain, or animal movements. Here we report a biologically feasible distributed algorithm, parameterized using field data, that can plausibly describe how turtle ants maintain the routing backbone and find alternative paths to circumvent broken links in the backbone. We validate the ability of this probabilistic algorithm to circumvent simulated breaks in synthetic and real-world networks, and we derive an analytic explanation for why certain features are crucial to improve the algorithm's success. Our proposed algorithm uses fewer computational resources than common distributed graph search algorithms, and thus may be useful in other domains, such as for swarm computing or for coordinating molecular robots.

Figure 1

Turtle ant habitat and trail network. (A) The photograph shows the highly tangled forest canopy in which turtle ants forage. (B) Experiments were performed in which an edge in the path was cut, to observe how the ants respond and repair the break. (C) Modeling the trail network as a graph, with junctions as nodes and connecting branches and twigs as edges. The diagram on the right from shows a detailed depiction of a large portion of the trail network. Each day’s path is shown in a different color (see legend), and additional repair paths are shown in a distinct color. Solid lines connect two nodes that are on the same plant (e.g. node 36 and node A are on the same plant). Dashed lines connect two nodes that are on a different plant (e.g. nodes B and C are on different plants).

Figure 2

Maximum likelihood computation. (A,B) Example node junction and edge choices for turtle ants. All ants arrive at node 1 from a different node that is not shown. In the example, we assume pheromone has been deposited at previous time-points, and we now compute the likelihood of the next ant choice. Under the RankEdge algorithm, the likelihood of choosing edges 1 → 3 or 1 → 2 is (1 − q_explore)(1/2); the likelihood of edge 1 → 4 is q_explore(1 − q_explore); and the likelihood of 1 → 5 is $q_{\mathrm{explore}}^2$. Under the Weighted algorithm, the likelihood of choosing edge 1 → 5 is q_explore; the likelihood of edge 1 → 4 is (1 − q_explore)(1/(1 + 2 + 2)); and the likelihood of edges 1 → 2 or 1 → 3 is (1 − q_explore)(2/(1 + 2 + 2). Under the Unweighted algorithm, the edge weights are disregarded, and the likelihood of taking any one of the four edges is (1/4). (C,D) For each combination of q_explore (x-axis) and q_decay (y-axis) values, we determined the pair’s likelihood of producing the choices observed in turtle ants. Each heatmap shows the likelihood for each algorithm with a zoom-in below around the highest likelihood region. The optimal parameter values for each algorithm, depicted in white, are shown in Table 1.

Figure 3

The maximum likelihood parameters closely match the best simulation parameters: (A) The color of each square in the heatmap corresponds to the robustness (Methods) of the success rates for the RankEdge algorithm for each combination of q_explore (x-axis) and q_decay (y-axis) values. Results are aggregated over the six simulated and real-world networks presented in Figs 4 and 5. (B) The maximum likelihood parameter estimates for RankEdge from observations of turtle ants. The black rectangle in both panels shows that the parameter values that best explain the turtle ants’ behavior also perform best for solving the network repair problem.

Figure 4

Success rates for each network. (A–E) For each network we show the initial graph (left), an example of the final graph after running the RankEdge algorithm using the maximum likelihood parameters (middle), and the algorithm’s success rate for each parameter combination (right). In each panel, black dots indicate nodes in the network, and solid lines indicate edges that may be traversed. If two adjacent nodes are not connected by an edge, there is a space between them. In the initial graphs, the ‘X’ marks the edge that is broken. The x-axis of the heatmap (right column) shows q_explore, and the y-axis shows q_decay under the range close to the MLE parameters. Darker shades of red indicate success rates closer to 1, and thus are better.

Figure 5

Repairing road closures in the Europe road graph. Analysis of how well the turtle ant algorithm translates to repair simulated breaks in a real-world transport network. (A) An example of a path in the European E-road network connecting Munich to Berlin, Germany. The roads and junctions form a graph. On the left, the black ‘X’ shows a road that has been broken or closed along the path. On the right, we show an alternative path that avoids the broken road. (B) The success rate of the turtle ant algorithm (RankEdge) applied to this network. Map data: Google, DigitalGlobe.

Figure 6

Poor path entropy for Weighted. The initial (left) and final (right) networks for the (A) Full grid and (B) Spanning grid. In both cases, the MLE parameter values (q_explore = 0.05, q_decay = 0.01) for Weighted did not find a low path entropy solution.

Figure 7

Analysis in the absence of a break. (A) Initial Spanning grid, with no break. (B) The final network produced using Weighted, which does not find a low entropy solution. (C) The final graph using RankEdge, which finds a low path entropy solution.

Figure 8

Turtle ants prune paths. The diagram from shows the results of an experiment in which an edge was cut. Left: The initial trail is shown in grey. The edge connecting nodes 5 and 6 was cut. After 75 minutes, the turtle ants explored several new paths (red). Center: Five hours after the cut, some of the red paths were pruned (transparent grey). Ants traveling down from node 7 took one trail, consisting of nodes 6, 15, 16, 17, 18, 4, and 3, because they could not use 12 in this direction. Ants traveling in the other direction took another trail, consisting of nodes 3, 4, 5, 12, 13, and 14, or the trail consisting of 11, 13, and 14. Right: The next day, there was additional pruning. Because node 12 could now be used in both directions, ants traveled both ways on the indicated trail.