7 research outputs found
Extremal problems on shadows and hypercuts in simplicial complexes
Let be an -vertex forest. We say that an edge is in the
shadow of if contains a cycle. It is easy to see that if
is "almost a tree", that is, it has edges, then at least
edges are in its shadow and this is tight.
Equivalently, the largest number of edges an -vertex cut can have is
. These notions have natural analogs in higher
-dimensional simplicial complexes, graphs being the case . The results
in dimension turn out to be remarkably different from the case in graphs.
In particular the corresponding bounds depend on the underlying field of
coefficients. We find the (tight) analogous theorems for . We construct
-dimensional "-almost-hypertrees" (defined below) with an empty
shadow. We also show that the shadow of an "-almost-hypertree"
cannot be empty, and its least possible density is . In
addition we construct very large hyperforests with a shadow that is empty over
every field.
For even, we construct -dimensional -almost-hypertree whose shadow has density .
Finally, we mention several intriguing open questions
The Bulk and the Extremes of Minimal Spanning Acycles and Persistence Diagrams of Random Complexes
Frieze showed that the expected weight of the minimum spanning tree (MST) of
the uniformly weighted graph converges to . Recently, this result was
extended to a uniformly weighted simplicial complex, where the role of the MST
is played by its higher-dimensional analogue -- the Minimum Spanning Acycle
(MSA). In this work, we go beyond and look at the histogram of the weights in
this random MSA -- both in the bulk and in the extremes. In particular, we
focus on the `incomplete' setting, where one has access only to a fraction of
the potential face weights. Our first result is that the empirical distribution
of the MSA weights asymptotically converges to a measure based on the shadow --
the complement of graph components in higher dimensions. As far as we know,
this result is the first to explore the connection between the MSA weights and
the shadow. Our second result is that the extremal weights converge to an
inhomogeneous Poisson point process. A interesting consequence of our two
results is that we can also state the distribution of the death times in the
persistence diagram corresponding to the above weighted complex, a result of
interest in applied topology.Comment: 15 pages, 5 figures, Corrected Typo