955 research outputs found
Graph Laplacians and their convergence on random neighborhood graphs
Given a sample from a probability measure with support on a submanifold in
Euclidean space one can construct a neighborhood graph which can be seen as an
approximation of the submanifold. The graph Laplacian of such a graph is used
in several machine learning methods like semi-supervised learning,
dimensionality reduction and clustering. In this paper we determine the
pointwise limit of three different graph Laplacians used in the literature as
the sample size increases and the neighborhood size approaches zero. We show
that for a uniform measure on the submanifold all graph Laplacians have the
same limit up to constants. However in the case of a non-uniform measure on the
submanifold only the so called random walk graph Laplacian converges to the
weighted Laplace-Beltrami operator.Comment: Improved presentation, typos corrected, to appear in JML
Hypothesis Testing For Network Data in Functional Neuroimaging
In recent years, it has become common practice in neuroscience to use
networks to summarize relational information in a set of measurements,
typically assumed to be reflective of either functional or structural
relationships between regions of interest in the brain. One of the most basic
tasks of interest in the analysis of such data is the testing of hypotheses, in
answer to questions such as "Is there a difference between the networks of
these two groups of subjects?" In the classical setting, where the unit of
interest is a scalar or a vector, such questions are answered through the use
of familiar two-sample testing strategies. Networks, however, are not Euclidean
objects, and hence classical methods do not directly apply. We address this
challenge by drawing on concepts and techniques from geometry, and
high-dimensional statistical inference. Our work is based on a precise
geometric characterization of the space of graph Laplacian matrices and a
nonparametric notion of averaging due to Fr\'echet. We motivate and illustrate
our resulting methodologies for testing in the context of networks derived from
functional neuroimaging data on human subjects from the 1000 Functional
Connectomes Project. In particular, we show that this global test is more
statistical powerful, than a mass-univariate approach. In addition, we have
also provided a method for visualizing the individual contribution of each edge
to the overall test statistic.Comment: 34 pages. 5 figure
On the Connectivity of Unions of Random Graphs
Graph-theoretic tools and techniques have seen wide use in the multi-agent
systems literature, and the unpredictable nature of some multi-agent
communications has been successfully modeled using random communication graphs.
Across both network control and network optimization, a common assumption is
that the union of agents' communication graphs is connected across any finite
interval of some prescribed length, and some convergence results explicitly
depend upon this length. Despite the prevalence of this assumption and the
prevalence of random graphs in studying multi-agent systems, to the best of our
knowledge, there has not been a study dedicated to determining how many random
graphs must be in a union before it is connected. To address this point, this
paper solves two related problems. The first bounds the number of random graphs
required in a union before its expected algebraic connectivity exceeds the
minimum needed for connectedness. The second bounds the probability that a
union of random graphs is connected. The random graph model used is the
Erd\H{o}s-R\'enyi model, and, in solving these problems, we also bound the
expectation and variance of the algebraic connectivity of unions of such
graphs. Numerical results for several use cases are given to supplement the
theoretical developments made.Comment: 16 pages, 3 tables; accepted to 2017 IEEE Conference on Decision and
Control (CDC
Unbounded quantum graphs with unbounded boundary conditions
We consider metric graphs with a uniform lower bound on the edge lengths but
no further restrictions. We discuss how to describe every local self-adjoint
Laplace operator on such graphs by boundary conditions in the vertices given by
projections and self-adjoint operators. We then characterize the lower bounded
self-adjoint Laplacians and determine their associated quadratic form in terms
of the operator families encoding the boundary conditions
On occurrence of spectral edges for periodic operators inside the Brillouin zone
The article discusses the following frequently arising question on the
spectral structure of periodic operators of mathematical physics (e.g.,
Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can
obtain the correct spectrum by using the values of the quasimomentum running
over the boundary of the (reduced) Brillouin zone only, rather than the whole
zone? Or, do the edges of the spectrum occur necessarily at the set of
``corner'' high symmetry points? This is known to be true in 1D, while no
apparent reasons exist for this to be happening in higher dimensions. In many
practical cases, though, this appears to be correct, which sometimes leads to
the claims that this is always true. There seems to be no definite answer in
the literature, and one encounters different opinions about this problem in the
community.
In this paper, starting with simple discrete graph operators, we construct a
variety of convincing multiply-periodic examples showing that the spectral
edges might occur deeply inside the Brillouin zone. On the other hand, it is
also shown that in a ``generic'' case, the situation of spectral edges
appearing at high symmetry points is stable under small perturbations. This
explains to some degree why in many (maybe even most) practical cases the
statement still holds.Comment: 25 pages, 10 EPS figures. Typos corrected and a reference added in
the new versio
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