167 research outputs found
Hardy-Muckenhoupt Bounds for Laplacian Eigenvalues
We present two graph quantities Psi(G,S) and Psi_2(G) which give constant factor estimates to the Dirichlet and Neumann eigenvalues, lambda(G,S) and lambda_2(G), respectively. Our techniques make use of a discrete Hardy-type inequality due to Muckenhoupt
Graph Embedding Techniques for Bounding Condition Numbers of Incomplete Factor Preconditioning
We extend graph embedding techniques for bounding the spectral condition number of preconditioned systems involving symmetric, irreducibly diagonally dominant M-matrices to systems where the preconditioner is not diagonally dominant. In particular, this allows us to bound the spectral condition number when the preconditioner is based on an incomplete factorization. We provide a review of previous techniques, describe our extension, and give examples both of a bound for a model problem, and of ways in which our techniques give intuitive way of looking at incomplete factor preconditioners
Optimal Scale-Free Small-World Graphs with Minimum Scaling of Cover Time
The cover time of random walks on a graph has found wide practical
applications in different fields of computer science, such as crawling and
searching on the World Wide Web and query processing in sensor networks, with
the application effects dependent on the behavior of cover time: the smaller
the cover time, the better the application performance. It was proved that over
all graphs with nodes, complete graphs have the minimum cover time . However, complete graphs cannot mimic real-world networks with small
average degree and scale-free small-world properties, for which the cover time
has not been examined carefully, and its behavior is still not well understood.
In this paper, we first experimentally evaluate the cover time for various
real-world networks with scale-free small-world properties, which scales as
. To better understand the behavior of the cover time for real-world
networks, we then study the cover time of three scale-free small-world model
networks by using the connection between cover time and resistance diameter.
For all the three networks, their cover time also behaves as . This
work indicates that sparse networks with scale-free and small-world topology
are favorable architectures with optimal scaling of cover time. Our results
deepen understanding the behavior of cover time in real-world networks with
scale-free small-world structure, and have potential implications in the design
of efficient algorithms related to cover time
On Weighted Graph Sparsification by Linear Sketching
A seminal work of [Ahn-Guha-McGregor, PODS'12] showed that one can compute a
cut sparsifier of an unweighted undirected graph by taking a near-linear number
of linear measurements on the graph. Subsequent works also studied computing
other graph sparsifiers using linear sketching, and obtained near-linear upper
bounds for spectral sparsifiers [Kapralov-Lee-Musco-Musco-Sidford, FOCS'14] and
first non-trivial upper bounds for spanners [Filtser-Kapralov-Nouri, SODA'21].
All these linear sketching algorithms, however, only work on unweighted graphs.
In this paper, we initiate the study of weighted graph sparsification by
linear sketching by investigating a natural class of linear sketches that we
call incidence sketches, in which each measurement is a linear combination of
the weights of edges incident on a single vertex. Our results are:
1. Weighted cut sparsification: We give an algorithm that computes a -cut sparsifier using linear
measurements, which is nearly optimal.
2. Weighted spectral sparsification: We give an algorithm that computes a -spectral sparsifier using
linear measurements. Complementing our algorithm, we then prove a superlinear
lower bound of measurements for computing some
-spectral sparsifier using incidence sketches.
3. Weighted spanner computation: We focus on graphs whose largest/smallest
edge weights differ by an factor, and prove that, for incidence
sketches, the upper bounds obtained by~[Filtser-Kapralov-Nouri, SODA'21] are
optimal up to an factor
Synchronization and Transient Stability in Power Networks and Non-Uniform Kuramoto Oscillators
Motivated by recent interest for multi-agent systems and smart power grid
architectures, we discuss the synchronization problem for the network-reduced
model of a power system with non-trivial transfer conductances. Our key insight
is to exploit the relationship between the power network model and a
first-order model of coupled oscillators. Assuming overdamped generators
(possibly due to local excitation controllers), a singular perturbation
analysis shows the equivalence between the classic swing equations and a
non-uniform Kuramoto model. Here, non-uniform Kuramoto oscillators are
characterized by multiple time constants, non-homogeneous coupling, and
non-uniform phase shifts. Extending methods from transient stability,
synchronization theory, and consensus protocols, we establish sufficient
conditions for synchronization of non-uniform Kuramoto oscillators. These
conditions reduce to and improve upon previously-available tests for the
standard Kuramoto model. Combining our singular perturbation and Kuramoto
analyses, we derive concise and purely algebraic conditions that relate
synchronization and transient stability of a power network to the underlying
system parameters and initial conditions
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