1. Problem session
This (below) was an attempt to record the problem session (Tuesday, November 1st), the way it proceeded. Your additions and comments are welcome. Dan

Problem 1.1.
[Shavitt] Need ways to compute geodesics on spaces where curvature is variable (using embedded graphs). 
Problem 1.15.
[Krioukov] Why measure it? (What are the realworld applications?)
Remark. [Weinberger] Wasn’t that your result, that the internet has curvature $1$ and dimension $2$?

Remark. [Krioukov] So, yes, we need more examples like that, showing the significance of curvature.


Problem 1.2.
[Holmes] Find (different) notions of distance on the space of graphs, or perhaps measures on this space. In particular: how far is a given graph from being a tree?
Remark. [Holmes] perhaps need to look at nondense graphs; I have the impression that the problem is solved for dense ones.

Remark. [Gao] How about treewidth?

Remark. [Sullivan] Treewidth is not good enough. There is a relation to treelength.


Problem 1.25.
[Saniee] Is curvature related to clustering?
Remark. [Baryshnikov] What is meant by clustering?


Problem 1.35.
[Mahoney] For random graphs, is there an idea for a degree of heterogeneity that could indicate hyperbolicity
Remark. [Weinberger] Similar to the question of frequency of hyperbolic groups among finitely generated/presented groups: could ask –
* What is a set of conditions ensuring hyperbolicity of an ensemble of random graphs (i.e., the set of hyperbolic graphs in the ensemble has probability one)?
* (Mahoney) important to emphasize randomness.


Problem 1.4.
[Krioukov(?)] What would be meant by a graph having *positive* curvature? (Grids don’t have it; trees don’t; how about spheres?)
Remark. [Baryshnikov, Weinberger] That depends on a choice of scale.


Problem 1.45.
[Krioukov/Jonckheere] What precise notions of negative curvature imply congestion?
Remark. [Weinberger] What price are you willing to pay to avoid congestion?

Remark. [Bonahon/Saniee] *Where* does congestion occur? (something analogous to convex core?)


Problem 1.5.
What are the data required to define congestion?
Remark. [Holmes] Do weights on the network or some such affect this?


Problem 1.55.
[Baryshnikov] Is there a scale at which the structure of a (real) network “looks like” a manifold? (e.g., Krioukov’s result)
Remark. [Krioukov] This relates to the (previous) questions about how to define hyperbolicity or curvature for a single finite graph.

Cite this as: AimPL: Geometry of large networks, available at http://aimpl.org/largenetworks.