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More on quasi-random graphs, subgraph counts and graph limits
We study some properties of graphs (or, rather, graph sequences) defined by
demanding that the number of subgraphs of a given type, with vertices in
subsets of given sizes, approximatively equals the number expected in a random
graph. It has been shown by several authors that several such conditions are
quasi-random, but that there are exceptions. In order to understand this
better, we investigate some new properties of this type. We show that these
properties too are quasi-random, at least in some cases; however, there are
also cases that are left as open problems, and we discuss why the proofs fail
in these cases.
The proofs are based on the theory of graph limits; and on the method and
results developed by Janson (2011), this translates the combinatorial problem
to an analytic problem, which then is translated to an algebraic problem.Comment: 35 page
Identifiability of Points and Rigidity of Hypergraphs under Algebraic Constraints
Identifiability of data is one of the fundamental problems in data science.
Mathematically it is often formulated as the identifiability of points
satisfying a given set of algebraic relations. A key question then is to
identify sufficient conditions for observations to guarantee the
identifiability of the points.
This paper proposes a new general framework for capturing the identifiability
problem when a set of algebraic relations has a combinatorial structure and
develops tools to analyze the impact of the underlying combinatorics on the
local or global identifiability of points. Our framework is built on the
language of graph rigidity, where the measurements are Euclidean distances
between two points, but applicable in the generality of hypergraphs with
arbitrary algebraic measurements. We establish necessary and sufficient
(hyper)graph theoretical conditions for identifiability by exploiting
techniques from graph rigidity theory and algebraic geometry of secant
varieties
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