51 research outputs found
Evasion Paths in Mobile Sensor Networks
Suppose that ball-shaped sensors wander in a bounded domain. A sensor doesn't
know its location but does know when it overlaps a nearby sensor. We say that
an evasion path exists in this sensor network if a moving intruder can avoid
detection. In "Coordinate-free coverage in sensor networks with controlled
boundaries via homology", Vin deSilva and Robert Ghrist give a necessary
condition, depending only on the time-varying connectivity data of the sensors,
for an evasion path to exist. Using zigzag persistent homology, we provide an
equivalent condition that moreover can be computed in a streaming fashion.
However, no method with time-varying connectivity data as input can give
necessary and sufficient conditions for the existence of an evasion path.
Indeed, we show that the existence of an evasion path depends not only on the
fibrewise homotopy type of the region covered by sensors but also on its
embedding in spacetime. For planar sensors that also measure weak rotation and
distance information, we provide necessary and sufficient conditions for the
existence of an evasion path
Construction of the generalized Cech complex
In this paper, we introduce an algorithm which constructs the generalized
Cech complex. The generalized Cech complex represents the topology of a
wireless network whose cells are different in size. This complex is often used
in many application to locate the boundary holes or to save energy consumption
in wireless networks. The complexity of a construction of the Cech complex to
analyze the coverage structure is found to be a polynomial time
Computing the -coverage of a wireless network
Coverage is one of the main quality of service of a wirelessnetwork.
-coverage, that is to be covered simultaneously by network nodes, is
synonym of reliability and numerous applicationssuch as multiple site MIMO
features, or handovers. We introduce here anew algorithm for computing the
-coverage of a wirelessnetwork. Our method is based on the observation that
-coverage canbe interpreted as layers of -coverage, or simply
coverage. Weuse simplicial homology to compute the network's topology and
areduction algorithm to indentify the layers of -coverage. Weprovide figures
and simulation results to illustrate our algorithm.Comment: Valuetools 2019, Mar 2019, Palma de Mallorca, Spain. 2019. arXiv
admin note: text overlap with arXiv:1802.0844
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
Positive Alexander Duality for Pursuit and Evasion
Considered is a class of pursuit-evasion games, in which an evader tries to
avoid detection. Such games can be formulated as the search for sections to the
complement of a coverage region in a Euclidean space over a timeline. Prior
results give homological criteria for evasion in the general case that are not
necessary and sufficient. This paper provides a necessary and sufficient
positive cohomological criterion for evasion in a general case. The principal
tools are (1) a refinement of the Cech cohomology of a coverage region with a
positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore
homology of the coverage gaps with a positive cone encoding time orientation,
and (3) a positive variant of Alexander Duality. Positive cohomology decomposes
as the global sections of a sheaf of local positive cohomology over the time
axis; we show how this decomposition makes positive cohomology computable as a
linear program.Comment: 19 pages, 6 figures; improvements made throughout: e.g. positive
(co)homology generalized to arbitrary degrees; Positive Alexander Duality
generalized from homological degrees 0,1; Morse and smoothness conditions
generalized; illustrations of positive homology added. minor corrections in
proofs, notation, organization, and language made throughout. variant of
Borel-Moore homology now use
- …