Exploring Maps with Greedy Navigators

During the last decade of network research focusing on structural and dynamical properties of networks, the role of network users has been more or less underestimated from the bird’s-eye view of global perspective. In this era of global positioning system equipped smartphones, however, a user’s ability to access local geometric information and find efficient pathways on networks plays a crucial role, rather than the globally optimal pathways. We present a simple greedy spatial navigation strategy as a probe to explore spatial networks. These greedy navigators use directional information in every move they take, without being trapped in a dead end based on their memory about previous routes. We suggest that the centralities measures have to be modified to incorporate the navigators’ behavior, and present the intriguing effect of navigators’ greediness where removing some edges may actually enhance the routing efficiency, which is reminiscent of Braess’s paradox. In addition, using samples of road structures in large cities around the world, it is shown that the navigability measure we define reflects unique structural properties, which are not easy to predict from other topological characteristics. In this respect, we believe that our routing scheme significantly moves the routing problem on networks one step closer to reality, incorporating the inevitable incompleteness of navigators’ information.

💡 Research Summary

The paper addresses a gap in network science: most studies focus on global structural and dynamical properties while largely ignoring the behavior of individual users who navigate with only partial, local information. In the era of GPS‑enabled smartphones, people rely on directional cues and memory of previously visited routes rather than on globally optimal paths. To model this realistic navigation, the authors introduce a Greedy Spatial Navigator (GSN). In a spatial graph where each node i has coordinates r_i, the navigator at node i computes the angle θ_j between the vector toward the target t and each neighbor j. It moves to the unvisited neighbor with the smallest θ_j; if all neighbors have been visited, it backtracks to the previous node. This simple back‑tracking rule prevents dead ends that plague pure greedy strategies based solely on geometric proximity.

Two performance metrics are defined. Let d be the average shortest‑path length (the global optimum). Let d_g be the average number of edges traversed by GSN over all source–target pairs, and d_r the analogous average for a random depth‑first search (DFS) that chooses a random neighbor at each step. The “greedy navigability” ν = d / d_g and the “random navigability” ζ = d / d_r are normalized to the interval (0,1]; higher values indicate routes closer to optimal. Empirical evaluation on several real‑world spatial networks—downtown road excerpts of Boston and New York City, European railway networks, and a maze (Leeds Castle Maze)—shows ν consistently larger than ζ, confirming that real infrastructure is designed to be friendly to direction‑based navigation.

Beyond path length, the authors propose new centrality measures tailored to GSN. Navigator centrality n(x) counts how often node or edge x lies on GSN routes, analogous to betweenness but based on greedy paths. They find strong correlation between n and traditional betweenness for nodes, but a much weaker relationship for edges, indicating that edge importance depends heavily on the greedy rule.

A particularly novel contribution is the definition of edge “essentiality” e(l) = d_g