2,416 research outputs found
Higher-dimensional models of networks
Networks are often studied as graphs, where the vertices stand for entities
in the world and the edges stand for connections between them. While relatively
easy to study, graphs are often inadequate for modeling real-world situations,
especially those that include contexts of more than two entities. For these
situations, one typically uses hypergraphs or simplicial complexes.
In this paper, we provide a precise framework in which graphs, hypergraphs,
simplicial complexes, and many other categories, all of which model higher
graphs, can be studied side-by-side. We show how to transform a hypergraph into
its nearest simplicial analogue, for example. Our framework includes many new
categories as well, such as one that models broadcasting networks. We give
several examples and applications of these ideas
Visualizing Sensor Network Coverage with Location Uncertainty
We present an interactive visualization system for exploring the coverage in
sensor networks with uncertain sensor locations. We consider a simple case of
uncertainty where the location of each sensor is confined to a discrete number
of points sampled uniformly at random from a region with a fixed radius.
Employing techniques from topological data analysis, we model and visualize
network coverage by quantifying the uncertainty defined on its simplicial
complex representations. We demonstrate the capabilities and effectiveness of
our tool via the exploration of randomly distributed sensor networks
Investigating The Algebraic Structure of Dihomotopy Types
This presentation is the sequel of a paper published in GETCO'00 proceedings
where a research program to construct an appropriate algebraic setting for the
study of deformations of higher dimensional automata was sketched. This paper
focuses precisely on detailing some of its aspects. The main idea is that the
category of homotopy types can be embedded in a new category of dihomotopy
types, the embedding being realized by the Globe functor. In this latter
category, isomorphism classes of objects are exactly higher dimensional
automata up to deformations leaving invariant their computer scientific
properties as presence or not of deadlocks (or everything similar or related).
Some hints to study the algebraic structure of dihomotopy types are given, in
particular a rule to decide whether a statement/notion concerning dihomotopy
types is or not the lifting of another statement/notion concerning homotopy
types. This rule does not enable to guess what is the lifting of a given
notion/statement, it only enables to make the verification, once the lifting
has been found.Comment: 28 pages ; LaTeX2e + 4 figures ; Expository paper ; Minor typos
corrections ; To appear in GETCO'01 proceeding
Weighted simplicial complex reconstruction from mobile laser scanning using sensor topology
We propose a new method for the reconstruction of simplicial complexes
(combining points, edges and triangles) from 3D point clouds from Mobile Laser
Scanning (MLS). Our method uses the inherent topology of the MLS sensor to
define a spatial adjacency relationship between points. We then investigate
each possible connexion between adjacent points, weighted according to its
distance to the sensor, and filter them by searching collinear structures in
the scene, or structures perpendicular to the laser beams. Next, we create and
filter triangles for each triplet of self-connected edges and according to
their local planarity. We compare our results to an unweighted simplicial
complex reconstruction.Comment: 8 pages, 11 figures, CFPT 2018. arXiv admin note: substantial text
overlap with arXiv:1802.0748
Globular realization and cubical underlying homotopy type of time flow of process algebra
We construct a small realization as flow of every precubical set (modeling
for example a process algebra). The realization is small in the sense that the
construction does not make use of any cofibrant replacement functor and of any
transfinite construction. In particular, if the precubical set is finite, then
the corresponding flow has a finite globular decomposition. Two applications
are given. The first one presents a realization functor from precubical sets to
globular complexes which is characterized up to a natural S-homotopy. The
second one proves that, for such flows, the underlying homotopy type is
naturally isomorphic to the homotopy type of the standard cubical complex
associated with the precubical set.Comment: 31 pages, 1 figure, LaTeX2
Forman's Ricci curvature - From networks to hypernetworks
Networks and their higher order generalizations, such as hypernetworks or
multiplex networks are ever more popular models in the applied sciences.
However, methods developed for the study of their structural properties go
little beyond the common name and the heavy reliance of combinatorial tools. We
show that, in fact, a geometric unifying approach is possible, by viewing them
as polyhedral complexes endowed with a simple, yet, the powerful notion of
curvature - the Forman Ricci curvature. We systematically explore some aspects
related to the modeling of weighted and directed hypernetworks and present
expressive and natural choices involved in their definitions. A benefit of this
approach is a simple method of structure-preserving embedding of hypernetworks
in Euclidean N-space. Furthermore, we introduce a simple and efficient manner
of computing the well established Ollivier-Ricci curvature of a hypernetwork.Comment: to appear: Complex Networks '18 (oral presentation
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