3,392 research outputs found
Persistent Homology Over Directed Acyclic Graphs
We define persistent homology groups over any set of spaces which have
inclusions defined so that the corresponding directed graph between the spaces
is acyclic, as well as along any subgraph of this directed graph. This method
simultaneously generalizes standard persistent homology, zigzag persistence and
multidimensional persistence to arbitrary directed acyclic graphs, and it also
allows the study of more general families of topological spaces or point-cloud
data. We give an algorithm to compute the persistent homology groups
simultaneously for all subgraphs which contain a single source and a single
sink in arithmetic operations, where is the number of vertices in
the graph. We then demonstrate as an application of these tools a method to
overlay two distinct filtrations of the same underlying space, which allows us
to detect the most significant barcodes using considerably fewer points than
standard persistence.Comment: Revised versio
On cohomology theory of (di)graphs
To a digraph with a choice of certain integral basis, we construct a CW
complex, whose integral singular cohomology is canonically isomorphic to the
path cohomology of the digraph as introduced in \cite{GLMY}. The homotopy type
of the CW complex turns out to be independent of the choice of basis. After a
very brief discussion of functoriality, this construction immediately implies
some of the expected but perhaps combinatorially subtle properties of the
digraph cohomology and homotopy proved very recently \cite{GLMY2}. Furthermore,
one gets a very simple expected formula for the cup product of forms on the
digraph. On the other hand, we present an approach of using sheaf theory to
reformulate (di)graph cohomologies. The investigation of the path cohomology
from this framework, leads to a subtle version of Poincare lemma for digraphs,
which follows from the construction of the CW complex.Comment: 17 page
1-Safe Petri nets and special cube complexes: equivalence and applications
Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net
unfolds into an event structure . By a result of Thiagarajan
(1996 and 2002), these unfoldings are exactly the trace regular event
structures. Thiagarajan (1996 and 2002) conjectured that regular event
structures correspond exactly to trace regular event structures. In a recent
paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on
the striking bijection between domains of event structures, median graphs, and
CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018) we
proved that Thiagarajan's conjecture is true for regular event structures whose
domains are principal filters of universal covers of (virtually) finite special
cube complexes.
In the current paper, we prove the converse: to any finite 1-safe Petri net
one can associate a finite special cube complex such that the
domain of the event structure (obtained as the unfolding of
) is a principal filter of the universal cover of .
This establishes a bijection between 1-safe Petri nets and finite special cube
complexes and provides a combinatorial characterization of trace regular event
structures.
Using this bijection and techniques from graph theory and geometry (MSO
theory of graphs, bounded treewidth, and bounded hyperbolicity) we disprove yet
another conjecture by Thiagarajan (from the paper with S. Yang from 2014) that
the monadic second order logic of a 1-safe Petri net is decidable if and only
if its unfolding is grid-free.
Our counterexample is the trace regular event structure
which arises from a virtually special square complex . The domain of
is grid-free (because it is hyperbolic), but the MSO
theory of the event structure is undecidable
Nice labeling problem for event structures: a counterexample
In this note, we present a counterexample to a conjecture of Rozoy and
Thiagarajan from 1991 (called also the nice labeling problem) asserting that
any (coherent) event structure with finite degree admits a labeling with a
finite number of labels, or equivalently, that there exists a function such that an event structure with degree
admits a labeling with at most labels. Our counterexample is based on
the Burling's construction from 1965 of 3-dimensional box hypergraphs with
clique number 2 and arbitrarily large chromatic numbers and the bijection
between domains of event structures and median graphs established by
Barth\'elemy and Constantin in 1993
The Vietoris-Rips complexes of a circle
Given a metric space X and a distance threshold r>0, the Vietoris-Rips
simplicial complex has as its simplices the finite subsets of X of diameter
less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian
manifold and r is sufficiently small, then the Vietoris-Rips complex is
homotopy equivalent to the original manifold. Little is known about the
behavior of Vietoris-Rips complexes for larger values of r, even though these
complexes arise naturally in applications using persistent homology. We show
that as r increases, the Vietoris-Rips complex of the circle obtains the
homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ...,
until finally it is contractible. As our main tool we introduce a directed
graph invariant, the winding fraction, which in some sense is dual to the
circular chromatic number. Using the winding fraction we classify the homotopy
types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset
of the circle, and we study the expected homotopy type of the Vietoris-Rips
complex of a uniformly random sample from the circle. Moreover, we show that as
the distance parameter increases, the ambient Cech complex of the circle also
obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the
7-sphere, ..., until finally it is contractible.Comment: Final versio
Rearrangement Groups of Fractals
We construct rearrangement groups for edge replacement systems, an infinite
class of groups that generalize Richard Thompson's groups F, T, and V .
Rearrangement groups act by piecewise-defined homeomorphisms on many
self-similar topological spaces, among them the Vicsek fractal and many Julia
sets. We show that every rearrangement group acts properly on a locally finite
CAT(0) cubical complex, and we use this action to prove that certain
rearrangement groups are of type F infinity.Comment: 48 pages, 37 figure
Complexes of not -connected graphs
Complexes of (not) connected graphs, hypergraphs and their homology appear in
the construction of knot invariants given by V. Vassiliev. In this paper we
study the complexes of not -connected -hypergraphs on vertices. We
show that the complex of not -connected graphs has the homotopy type of a
wedge of spheres of dimension . This answers one of the
questions raised by Vassiliev in connection with knot invariants. For this case
the -action on the homology of the complex is also determined. For
complexes of not -connected -hypergraphs we provide a formula for the
generating function of the Euler characteristic, and we introduce certain
lattices of graphs that encode their topology. We also present partial results
for some other cases. In particular, we show that the complex of not
-connected graphs is Alexander dual to the complex of partial matchings
of the complete graph. For not -connected graphs we provide a formula
for the generating function of the Euler characteristic
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