8,402 research outputs found
Intertwining wavelets or Multiresolution analysis on graphs through random forests
We propose a new method for performing multiscale analysis of functions
defined on the vertices of a finite connected weighted graph. Our approach
relies on a random spanning forest to downsample the set of vertices, and on
approximate solutions of Markov intertwining relation to provide a subgraph
structure and a filter bank leading to a wavelet basis of the set of functions.
Our construction involves two parameters q and q'. The first one controls the
mean number of kept vertices in the downsampling, while the second one is a
tuning parameter between space localization and frequency localization. We
provide an explicit reconstruction formula, bounds on the reconstruction
operator norm and on the error in the intertwining relation, and a Jackson-like
inequality. These bounds lead to recommend a way to choose the parameters q and
q'. We illustrate the method by numerical experiments.Comment: 39 pages, 12 figure
The Bipartite Swapping Trick on Graph Homomorphisms
We provide an upper bound to the number of graph homomorphisms from to
, where is a fixed graph with certain properties, and varies over
all -vertex, -regular graphs. This result generalizes a recently resolved
conjecture of Alon and Kahn on the number of independent sets. We build on the
work of Galvin and Tetali, who studied the number of graph homomorphisms from
to when is bipartite. We also apply our techniques to graph
colorings and stable set polytopes.Comment: 22 pages. To appear in SIAM J. Discrete Mat
Tilting theory via stable homotopy theory
We show that certain tilting results for quivers are formal consequences of
stability, and as such are part of a formal calculus available in any abstract
stable homotopy theory. Thus these results are for example valid over arbitrary
ground rings, for quasi-coherent modules on schemes, in the differential-graded
context, in stable homotopy theory and also in the equivariant, motivic or
parametrized variant thereof. In further work, we will continue developing this
calculus and obtain additional abstract tilting results. Here, we also deduce
an additional characterization of stability, based on Goodwillie's strongly
(co)cartesian n-cubes.
As applications we construct abstract Auslander-Reiten translations and
abstract Serre functors for the trivalent source and verify the relative
fractionally Calabi-Yau property. This is used to offer a new perspective on
May's axioms for monoidal, triangulated categories.Comment: minor improvements in the presentation (the definition of a strong
stable equivalence made more precise, references updated and added
Dominating sets in projective planes
We describe small dominating sets of the incidence graphs of finite
projective planes by establishing a stability result which shows that
dominating sets are strongly related to blocking and covering sets. Our main
result states that if a dominating set in a projective plane of order is
smaller than (i.e., twice the size of a Baer subplane), then
it contains either all but possibly one points of a line or all but possibly
one lines through a point. Furthermore, we completely characterize dominating
sets of size at most . In Desarguesian planes, we could rely on
strong stability results on blocking sets to show that if a dominating set is
sufficiently smaller than 3q, then it consists of the union of a blocking set
and a covering set apart from a few points and lines.Comment: 19 page
Random Forests and Networks Analysis
D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient
algorithm based on loop-erased random walks to sample uniform spanning trees
and more generally weighted trees or forests spanning a given graph. This
algorithm provides a powerful tool in analyzing structures on networks and
along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we
focused on applications of spanning rooted forests on finite graphs. The
resulting main conclusions are reviewed in this paper by collecting related
theorems, algorithms, heuristics and numerical experiments. A first
foundational part on determinantal structures and efficient sampling procedures
is followed by four main applications: 1) a random-walk-based notion of
well-distributed points in a graph 2) how to describe metastable dynamics in
finite settings by means of Markov intertwining dualities 3) coarse graining
schemes for networks and associated processes 4) wavelets-like pyramidal
algorithms for graph signals.Comment: Survey pape
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