Our knowledge of prehistoric and ancient road networks is fragmentary: Contemporary maps are rarely available and far from modern standards of precision. Some roads may be textually referenced and some are found to be still in use today while the remains of others can only be spotted from the air or sparsely sampled via ground-penetrating radar or excavation. Modern maps of ancient networks thus emerge as a conglomeration of data from many sources.

To predict missing links in incomplete networks and to better understand the principles that drive the evolution of ancient road networks, we develop and study domain-specific mathematical models.

A sequential model of network formation

We propose a novel model based on three simplifying assumptions concerning the growth of ancient road networks:

1. Roads are added in sequence as their construction is fast compared to the lifespan of a network.
2. Traveling through the wilderness is prohibitively difficult, so the road network will eventually contain adequate connections between all pairs of significant settlements.
3. The course of connections optimizes a trade-off between road construction and travel costs.

Our model assumes a maximal network \(G = (V, E)\) of possible road segments equipped with a terrain cost \(c \colon E \to \mathbb{R}_{> 0}\), and a set of connections \(K \subseteq \mathcal{P}_2(V)\) to establish, normally given as all pairs \(K = \mathcal{P}_2(S)\) over a set of settlements \(S \subseteq V\). If these connections are brought into an order \(\pi \in S(K)\), then a subgraph of \(G\) emerges from the following procedure, parameterized by \(\alpha \in [0, 1]\):

1. Initialize the set of roads \(R := \emptyset\).
2. For every connection \(\{u, v\} = \pi_i\) in order of \(\pi\):
   1. Find a least-cost \(u\)-\(v\)-path \(P \subseteq E\) in \(G\) according to edge costs \(c\).
   2. For every edge \(e \in P\) with \(e \notin R\):
      1. Reduce the edge's cost to \(c(e) := \alpha c(e)\).
      2. Update \(R := R \cup \{e\}\).
3. Return the road network \(G[R]\).

References

Outreach

Challenge 20 of the MATH+ Advent Calendar 2021 is a winter-themed riddle about spatiotemporal reasoning, aimed at school students.

Data and source code

Available at GitLab.