611 research outputs found
Density of Range Capturing Hypergraphs
For a finite set of points in the plane, a set in the plane, and a
positive integer , we say that a -element subset of is captured
by if there is a homothetic copy of such that ,
i.e., contains exactly elements from . A -uniform -capturing
hypergraph has a vertex set and a hyperedge set consisting
of all -element subsets of captured by . In case when and
is convex these graphs are planar graphs, known as convex distance function
Delaunay graphs.
In this paper we prove that for any , any , and any convex
compact set , the number of hyperedges in is at most , where is the number of -element
subsets of that can be separated from the rest of with a straight line.
In particular, this bound is independent of and indeed the bound is tight
for all "round" sets and point sets in general position with respect to
.
This refines a general result of Buzaglo, Pinchasi and Rote stating that
every pseudodisc topological hypergraph with vertex set has
hyperedges of size or less.Comment: new version with a tight result and shorter proo
What is good mathematics?
Some personal thoughts and opinions on what ``good quality mathematics'' is,
and whether one should try to define this term rigorously. As a case study, the
story of Szemer\'edi's theorem is presented.Comment: 12 pages, no figures. To appear, Bull. Amer. Math. So
The Data Big Bang and the Expanding Digital Universe: High-Dimensional, Complex and Massive Data Sets in an Inflationary Epoch
Recent and forthcoming advances in instrumentation, and giant new surveys,
are creating astronomical data sets that are not amenable to the methods of
analysis familiar to astronomers. Traditional methods are often inadequate not
merely because of the size in bytes of the data sets, but also because of the
complexity of modern data sets. Mathematical limitations of familiar algorithms
and techniques in dealing with such data sets create a critical need for new
paradigms for the representation, analysis and scientific visualization (as
opposed to illustrative visualization) of heterogeneous, multiresolution data
across application domains. Some of the problems presented by the new data sets
have been addressed by other disciplines such as applied mathematics,
statistics and machine learning and have been utilized by other sciences such
as space-based geosciences. Unfortunately, valuable results pertaining to these
problems are mostly to be found only in publications outside of astronomy. Here
we offer brief overviews of a number of concepts, techniques and developments,
some "old" and some new. These are generally unknown to most of the
astronomical community, but are vital to the analysis and visualization of
complex datasets and images. In order for astronomers to take advantage of the
richness and complexity of the new era of data, and to be able to identify,
adopt, and apply new solutions, the astronomical community needs a certain
degree of awareness and understanding of the new concepts. One of the goals of
this paper is to help bridge the gap between applied mathematics, artificial
intelligence and computer science on the one side and astronomy on the other.Comment: 24 pages, 8 Figures, 1 Table. Accepted for publication: "Advances in
Astronomy, special issue "Robotic Astronomy
Clustering of tag-induced sub-graphs in complex networks
We study the behavior of the clustering coefficient in tagged networks. The
rich variety of tags associated with the nodes in the studied systems provide
additional information about the entities represented by the nodes which can be
important for practical applications like searching in the networks. Here we
examine how the clustering coefficient changes when narrowing the network to a
sub-graph marked by a given tag, and how does it correlate with various other
properties of the sub-graph. Another interesting question addressed in the
paper is how the clustering coefficient of the individual nodes is affected by
the tags on the node. We believe these sort of analysis help acquiring a more
complete description of the structure of large complex systems
Exact solutions for diluted spin glasses and optimization problems
We study the low temperature properties of p-spin glass models with finite
connectivity and of some optimization problems. Using a one-step functional
replica symmetry breaking Ansatz we can solve exactly the saddle-point
equations for graphs with uniform connectivity. The resulting ground state
energy is in perfect agreement with numerical simulations. For fluctuating
connectivity graphs, the same Ansatz can be used in a variational way: For
p-spin models (known as p-XOR-SAT in computer science) it provides the exact
configurational entropy together with the dynamical and static critical
connectivities (for p=3, \gamma_d=0.818 and \gamma_s=0.918 resp.), whereas for
hard optimization problems like 3-SAT or Bicoloring it provides new upper
bounds for their critical thresholds (\gamma_c^{var}=4.396 and
\gamma_c^{var}=2.149 resp.).Comment: 4 pages, 1 figure, accepted for publication in PR
Hypergraphs Demonstrate Anastomoses During Divergent Integration
Complex networks can be used to analyze structures and systems in the embryo.
Not only can we characterize growth and the emergence of form, but also
differentiation. The process of differentiation from precursor cell populations
to distinct functional tissues is of particular interest. These phenomena can
be captured using a hypergraph consisting of nodes represented by cell type
categories and arranged as a directed cyclic graph (lineage hypergraph) and a
complex network (spatial hypergraph). The lineage hypergraph models the
developmental process as an n-ary tree, which can model two or more descendent
categories per division event. A lineage tree based on the mosaic development
of the nematode C. elegans (2-ary tree), is used to capture this process. Each
round of divisions produces a new set of categories that allow for exchange of
cells between types. An example from single-cell morphogenesis based on the
cyanobacterial species Nostoc punctiforme (multiple discontinuous 2-ary tree)
is also used to demonstrate the flexibility of this method. This model allows
for new structures to emerge (such as a connectome) while also demonstrating
how precursor categories are maintained for purposes such as dedifferentiation
or other forms of cell fate plasticity. To understand this process of divergent
integration, we analyze the directed hypergraph and categorical models, in
addition to considering the role of network fistulas (spaces that conjoin two
functional modules) and spatial restriction.Comment: 21 pages, 8 figure
Persistent Dirac of Path and Hypergraph
This work introduces the development of path Dirac and hypergraph Dirac
operators, along with an exploration of their persistence. These operators
excel in distinguishing between harmonic and non-harmonic spectra, offering
valuable insights into the subcomplexes within these structures. The paper
showcases the functionality of these operators through a series of examples in
various contexts. An important facet of this research involves examining the
operators' sensitivity to filtration, emphasizing their capacity to adapt to
topological changes. The paper also explores a significant application of
persistent path Dirac and persistent hypergraph Dirac in the field of molecular
science, specifically in the analysis of molecular structures. The study
introduces strict preorders derived from molecular structures, which generate
graphs and digraphs with intricate path structures. The depth of information
within these path complexes reflects the complexity of different preorder
classes influenced by molecular structures. This characteristic underscores the
effectiveness of these tools in the realm of topological data analysis.Comment: typos corrected, figures fixe
Hyperedge bundling : A practical solution to spurious interactions in MEG/EEG source connectivity analyses
Inter-areal functional connectivity (FC), neuronal synchronization in particular, is thought to constitute a key systems-level mechanism for coordination of neuronal processing and communication between brain regions. Evidence to support this hypothesis has been gained largely using invasive electrophysiological approaches. In humans, neuronal activity can be non-invasively recorded only with magneto-and electroencephalography (MEG/EEG), which have been used to assess FC networks with high temporal resolution and whole-scalp coverage. However, even in source-reconstructed MEG/EEG data, signal mixing, or "source leakage", is a significant confounder for FC analyses and network localization. Signal mixing leads to two distinct kinds of false-positive observations: artificial interactions (AI) caused directly by mixing and spurious interactions (SI) arising indirectly from the spread of signals from true interacting sources to nearby false loci. To date, several interaction metrics have been developed to solve the AI problem, but the SI problem has remained largely intractable in MEG/EEG all-to-all source connectivity studies. Here, we advance a novel approach for correcting SIs in FC analyses using source-reconstructed MEG/EEG data. Our approach is to bundle observed FC connections into hyperedges by their adjacency in signal mixing. Using realistic simulations, we show here that bundling yields hyperedges with good separability of true positives and little loss in the true positive rate. Hyperedge bundling thus significantly decreases graph noise by minimizing the false-positive to true-positive ratio. Finally, we demonstrate the advantage of edge bundling in the visualization of large-scale cortical networks with real MEG data. We propose that hypergraphs yielded by bundling represent well the set of true cortical interactions that are detectable and dissociable in MEG/EEG connectivity analysis.Peer reviewe
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