Obscure results and open problems - some of my favorites - Things I would like some smart young person to work on

Sep 18, 2026

David Aldous presents a set of probability and modeling questions inspired by obscure results and open problems. The talk highlights routed planar networks, including scale-invariance and comparisons between real-world subnetwork topologies and explicit scale-invariant models, while asking about efficient enumeration and canonical one-parameter families of distributions. A second topic explores fringe-tree combinatorics in a phylogenetic random tree model, including whether the model reproduces observed fringe behavior and matches data-driven clade proportions.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages36

Reading Time1 h 31 min

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Obscure results and open problems: some of my favorites

Topic 1: Probability distributions on routed planar networks

Open problems about “routed networks”

Topic 2: Combinatorics of fringe trees in a phylogenetic model

Obscure results and open problems: some of my favorites. David Aldous 28 January 2025 David Aldous Obscure results and open problems: some of my favorites. Sub-title: Things I would like some smart young person to work on. Probability distributions on routed planar networks. Combinatorics of fringe trees in a phylogenetic model. Nearest neighbor of nearest neighbor, on the complete network. Are you smarter than an AI? The Nearest Unvisited Vertex walk. Find a qualitatively realistic model for a subway network. A layer model for random DAGs David Aldous Obscure results and open problems: some of my favorites. Topic 1: Probability distributions on routed planar networks. q q q q q q q 7 points in a window. David Aldous Obscure results and open problems: some of my favorites. Scale-invariance means: doing this within a randomly positioned window, the statistics of the subnetwork observed don't depend on the scale, i.e. don't depend on whether the side length is 10 km or 100 km. David Aldous Obscure results and open problems: some of my favorites. As undergraduate project we have looked at real-world subnetwork topologies (for k = 4 vertices, roughly at corners of a square) . David Aldous Obscure results and open problems: some of my favorites. and listed all 71 topologies on 4 vertices – different conventions from usual planar graph theory. Could compare distributions over these topologies in real-world spatial networks and in models of spatial networks. David Aldous Obscure results and open problems: some of my favorites. Open problems about “routed networks” Efficient naming (coding) of the different topologies? Enumeration? [cf. Catherine Greenhill talk Thursday] Is there a (mathematically) canonical one-parameter family of probability distributions on these topologies? (In our small data study) empirical frequencies of topologies are very non-uniform. Can one make a model where some of the real-world-rare topologies don’t occur? We have 2 explicit models of scale-invariant networks; calculate/simulate the distribution for these models. References: [https://escholarship.org/content/qt1g44k4dk/qt1g44k4dk.pdf](https://escholarship.org/content/qt1g44k4dk/qt1g44k4dk.pdf) (the 71 topologies) . [https://arxiv.org/abs/1204.0817](https://arxiv.org/abs/1204.0817) and [https://arxiv.org/abs/2407.07887](https://arxiv.org/abs/2407.07887)[ ](https://arxiv.org/abs/2407.07887)(scale-invariant network models) . [All links are live on these slides posted on my home page] David Aldous Obscure results and open problems: some of my favorites. Topic 2: Combinatorics of fringe trees in a phylogenetic model. 26 DAVID J. ALDOUS AND SVANTE JANSON Bottoa:vilaAdlgorausm showiOnsorgesueneltcnfop77earprolems: some of my favorites. Broad ongoing project with Svante Janson and Boris Pittel: average age ≈ 76. We have a model for a random phylogenetic tree on n leaves (Svante talk tomorrow) Has zero parameters, so gives explicit numerical predictions. Model designed to mimic the (very uneven) way that large splits occur in real phylogenetic trees. But does it happen to also mimic the fringe behavior? David Aldous Obscure results and open problems: some of my favorites. THE CRITICAL BETA-SPLITTING RANDOM TREE III 27 Figure 7 . Proportions of leaves in clades of a given shape, for each shape with 2 − 6 leaves in the fringe tree. The top number is from our model, the bottom number [···] from our small data set. David Aldous Obscure results and open problems: some of my favorites.

FAQs

What does “scale-invariance” mean in the routed planar network discussion?

Scale-invariance means the observed subnetwork statistics in a randomly positioned window do not depend on the window’s scale, such as whether the side length is 10 km or 100 km.

What open questions are raised for routed network topologies?

The talk asks about efficient coding/enumeration of different topologies, the existence of a canonical one-parameter family of distributions, and how to model rare real-world topologies that do not appear in limited data.

What is the purpose of the phylogenetic fringe-tree model in Topic 2?

The model for a random phylogenetic tree has no parameters and gives explicit numerical predictions designed to mimic the uneven way large splits occur; the open question is whether it also mimics the fringe behavior seen in data.

Sep 18, 2026

Uploaded by Felix Montgomery