4,646 research outputs found
Strong Structural Controllability of Systems on Colored Graphs
This paper deals with structural controllability of leader-follower networks.
The system matrix defining the network dynamics is a pattern matrix in which a
priori given entries are equal to zero, while the remaining entries take
nonzero values. The network is called strongly structurally controllable if for
all choices of real values for the nonzero entries in the pattern matrix, the
system is controllable in the classical sense. In this paper we introduce a
more general notion of strong structural controllability which deals with the
situation that given nonzero entries in the system's pattern matrix are
constrained to take identical nonzero values. The constraint of identical
nonzero entries can be caused by symmetry considerations or physical
constraints on the network. The aim of this paper is to establish graph
theoretic conditions for this more general property of strong structural
controllability.Comment: 13 page
Spin Foams and Noncommutative Geometry
We extend the formalism of embedded spin networks and spin foams to include
topological data that encode the underlying three-manifold or four-manifold as
a branched cover. These data are expressed as monodromies, in a way similar to
the encoding of the gravitational field via holonomies. We then describe
convolution algebras of spin networks and spin foams, based on the different
ways in which the same topology can be realized as a branched covering via
covering moves, and on possible composition operations on spin foams. We
illustrate the case of the groupoid algebra of the equivalence relation
determined by covering moves and a 2-semigroupoid algebra arising from a
2-category of spin foams with composition operations corresponding to a fibered
product of the branched coverings and the gluing of cobordisms. The spin foam
amplitudes then give rise to dynamical flows on these algebras, and the
existence of low temperature equilibrium states of Gibbs form is related to
questions on the existence of topological invariants of embedded graphs and
embedded two-complexes with given properties. We end by sketching a possible
approach to combining the spin network and spin foam formalism with matter
within the framework of spectral triples in noncommutative geometry.Comment: 48 pages LaTeX, 30 PDF figure
Synchronization in random networks with given expected degree sequences
Synchronization in random networks with given expected degree sequences is studied. We also investigate in details the synchronization in networks whose topology is described by classical random graphs, power-law random graphs and hybrid graphs when N goes to infinity. In particular, we show that random graphs almost surely synchronize. We also show that adding small number of global edges to a local graph makes the corresponding hybrid graph to synchroniz
Fundamental Limits of Cloud and Cache-Aided Interference Management with Multi-Antenna Edge Nodes
In fog-aided cellular systems, content delivery latency can be minimized by
jointly optimizing edge caching and transmission strategies. In order to
account for the cache capacity limitations at the Edge Nodes (ENs),
transmission generally involves both fronthaul transfer from a cloud processor
with access to the content library to the ENs, as well as wireless delivery
from the ENs to the users. In this paper, the resulting problem is studied from
an information-theoretic viewpoint by making the following practically relevant
assumptions: 1) the ENs have multiple antennas; 2) only uncoded fractional
caching is allowed; 3) the fronthaul links are used to send fractions of
contents; and 4) the ENs are constrained to use one-shot linear precoding on
the wireless channel. Assuming offline proactive caching and focusing on a high
signal-to-noise ratio (SNR) latency metric, the optimal information-theoretic
performance is investigated under both serial and pipelined fronthaul-edge
transmission modes. The analysis characterizes the minimum high-SNR latency in
terms of Normalized Delivery Time (NDT) for worst-case users' demands. The
characterization is exact for a subset of system parameters, and is generally
optimal within a multiplicative factor of 3/2 for the serial case and of 2 for
the pipelined case. The results bring insights into the optimal interplay
between edge and cloud processing in fog-aided wireless networks as a function
of system resources, including the number of antennas at the ENs, the ENs'
cache capacity and the fronthaul capacity.Comment: 34 pages, 15 figures, submitte
Sampling and Reconstruction of Sparse Signals on Circulant Graphs - An Introduction to Graph-FRI
With the objective of employing graphs toward a more generalized theory of
signal processing, we present a novel sampling framework for (wavelet-)sparse
signals defined on circulant graphs which extends basic properties of Finite
Rate of Innovation (FRI) theory to the graph domain, and can be applied to
arbitrary graphs via suitable approximation schemes. At its core, the
introduced Graph-FRI-framework states that any K-sparse signal on the vertices
of a circulant graph can be perfectly reconstructed from its
dimensionality-reduced representation in the graph spectral domain, the Graph
Fourier Transform (GFT), of minimum size 2K. By leveraging the recently
developed theory of e-splines and e-spline wavelets on graphs, one can
decompose this graph spectral transformation into the multiresolution low-pass
filtering operation with a graph e-spline filter, and subsequent transformation
to the spectral graph domain; this allows to infer a distinct sampling pattern,
and, ultimately, the structure of an associated coarsened graph, which
preserves essential properties of the original, including circularity and,
where applicable, the graph generating set.Comment: To appear in Appl. Comput. Harmon. Anal. (2017
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