A visually realistic SIRSN

Definitions and mathematical statement

A scale-invariant random spatial network is a random family of spatial routes $\mathcal R(z,z')$ between planar points, subject to the source's consistency and finiteness axioms. Its central symmetry requirement is joint invariance under Euclidean similarities: for every $h(x)=cRx+a$, $c>0$, the finite collections \[ \left(h\mathcal R(z_i,z'_i)\right)_i \quad\text{and}\quad \left(\mathcal R(hz_i,hz'_i)\right)_i \] have the same law. This is a distributional statement for routes, not a demand that each realized network be fixed by every scaling.

Core finiteness requirements concern finite route length, finite expected length between unit-separated endpoints, and finite expected length intensity in a network sampled at Poisson points. For example, with an independent unit-intensity Poisson set $\Pi$, the sampled network is \[ \mathcal S(\Pi)=\bigcup_{z,z'\in\Pi}\mathcal R(z,z'). \] The exact route-consistency, measurability, and long-route intensity axioms defining SIRSN in the linked paper are not supplied; the displayed core is not substituted for a complete definition.

Open Problem 28 asks to construct a SIRSN that is visually realistic, meaning it does not look very different from a real-world road network. No reference data, statistical distance, tolerance, or formal visual predicate is specified. Hence the task is a model-design programme with qualitative assessment, not a precise extra axiom that can be inferred from scale invariance. Constructing a technically valid SIRSN alone does not certify the requested realism.

The supplied research summary reports Poisson-line constructions and later metric-structure progress, while explicitly not identifying a construction documented to meet Aldous's visual goal. Those technical construction claims and their exact SIRSN hypotheses need their cited papers; no visual judgment or success claim is made here.

Short English statement

Construct a technically valid SIRSN whose appearance also matches the intended qualitative road-network benchmark. The source defines no objective realism test, so known scale-invariant constructions are only partial progress.

Sources

• Local UnsolvedMath registry, permanent ID $9700028$, exact source code catalogue entry, "A visually realistic SIRSN." Proposer David Aldous.
• Aldous, Open problems, page sirsn.html, linked paper Open Problem 28; index https://www.stat.berkeley.edu/~aldous/Research/OP/index.html. Recorded access 2026-07-29; HTML text with linked-primary-document extraction.
• Supplied PARTIAL-PROGRESS summary names Kendall (2017), Kahn (2016), Kendall (2020), and Blanc--Curien--Kahn (2024/2025) for Poisson-line models and refinements. Complete titles, publication details, and exact axiom scope are not supplied.