15,666 research outputs found
Distances and Isomorphism between Networks and the Stability of Network Invariants
We develop the theoretical foundations of a network distance that has
recently been applied to various subfields of topological data analysis, namely
persistent homology and hierarchical clustering. While this network distance
has previously appeared in the context of finite networks, we extend the
setting to that of compact networks. The main challenge in this new setting is
the lack of an easy notion of sampling from compact networks; we solve this
problem in the process of obtaining our results. The generality of our setting
means that we automatically establish results for exotic objects such as
directed metric spaces and Finsler manifolds. We identify readily computable
network invariants and establish their quantitative stability under this
network distance. We also discuss the computational complexity involved in
precisely computing this distance, and develop easily-computable lower bounds
by using the identified invariants. By constructing a wide range of explicit
examples, we show that these lower bounds are effective in distinguishing
between networks. Finally, we provide a simple algorithm that computes a lower
bound on the distance between two networks in polynomial time and illustrate
our metric and invariant constructions on a database of random networks and a
database of simulated hippocampal networks
Cluster Complexes via Semi-Invariants
We define and study virtual representation spaces having both positive and
negative dimensions at the vertices of a quiver without oriented cycles. We
consider the natural semi-invariants on these spaces which we call virtual
semi-invariants and prove that they satisfy the three basic theorems: the First
Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition
Theorem. In the special case of Dynkin quivers with n vertices this gives the
fundamental interrelationship between supports of the semi-invariants and the
Tilting Triangulation of the (n-1)-sphere.Comment: 34 page
Psi-floor diagrams and a Caporaso-Harris type recursion
Floor diagrams are combinatorial objects which organize the count of tropical
plane curves satisfying point conditions. In this paper we introduce Psi-floor
diagrams which count tropical curves satisfying not only point conditions but
also conditions given by Psi-classes (together with points). We then generalize
our definition to relative Psi-floor diagrams and prove a Caporaso-Harris type
formula for the corresponding numbers. This formula is shown to coincide with
the classical Caporaso-Harris formula for relative plane descendant
Gromov-Witten invariants. As a consequence, we can conclude that in our case
relative descendant Gromov-Witten invariants equal their tropical counterparts.Comment: minor changes to match the published versio
Novikov-Shubin invariants for arbitrary group actions and their positivity
We extend the notion of Novikov-Shubin invariant for free G-CW-complexes of
finite type to spaces with arbitrary G-actions and prove some statements about
their positivity. In particular we apply this to classifying spaces of discrete
groups.Comment: 18 pages, metadata change
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