Faculties
DE / EN  ## Evolution Models for Historical Networks

### Project description

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:

Modeling assumptions
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 $$\alpha \in [0, 1]$$:

Network formation
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

• Spatiotemporal Reconstruction of Ancient Road Networks Through Sequential Cost–Benefit Analysis (, , and )
PNAS Nexus, 2(2), .
@article{StahlbergSagnolDucke+2022,
author = {Maximilian J. Stahlberg and Guillaume Sagnol and Benjamin Ducke and Max Klimm},
title = {Spatiotemporal Reconstruction of Ancient Road Networks Through Sequential Cost--Benefit Analysis},
journal = {PNAS Nexus},
year = {2023},
volume = {2},
number = {2},
accepted = {20.12.2022},
doi = {10.1093/pnasnexus/pgac313},
}
The construction of ancient road networks spanned generations and exhibits temporal path dependence that is not fully captured by established network formation models that are used to support archaeological reasoning. We introduce an evolutionary model that captures explicitly the sequential nature of road network formation: A central feature is that connections are added successively and according to an optimal cost–benefit trade-off with respect to existing connections. In this model, the network topology emerges rapidly from early decisions, a trait that makes it possible to identify plausible road construction orders in practice. Based on this observation we develop a method to compress the search space of path-dependent optimization problems. We use this method to show that the model's assumptions on ancient decision making allow the reconstruction of partially known road networks from the Roman era in good detail and from sparse archaeological evidence. In particular, we identify missing links in the major road network of ancient Sardinia that are in good agreement with expert predictions.

#### Outreach

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