15,155 research outputs found
On the stability of the Kuramoto model of coupled nonlinear oscillators
We provide an analysis of the classic Kuramoto model of coupled nonlinear
oscillators that goes beyond the existing results for all-to-all networks of
identical oscillators. Our work is applicable to oscillator networks of
arbitrary interconnection topology with uncertain natural frequencies. Using
tools from spectral graph theory and control theory, we prove that for
couplings above a critical value, the synchronized state is locally
asymptotically stable, resulting in convergence of all phase differences to a
constant value, both in the case of identical natural frequencies as well as
uncertain ones. We further explain the behavior of the system as the number of
oscillators grows to infinity.Comment: 8 Pages. An earlier version appeared in the proceedings of the
American Control Conference, Boston, MA, June 200
Dynamics over Signed Networks
A signed network is a network with each link associated with a positive or
negative sign. Models for nodes interacting over such signed networks, where
two different types of interactions take place along the positive and negative
links, respectively, arise from various biological, social, political, and
economic systems. As modifications to the conventional DeGroot dynamics for
positive links, two basic types of negative interactions along negative links,
namely the opposing rule and the repelling rule, have been proposed and studied
in the literature. This paper reviews a few fundamental convergence results for
such dynamics over deterministic or random signed networks under a unified
algebraic-graphical method. We show that a systematic tool of studying node
state evolution over signed networks can be obtained utilizing generalized
Perron-Frobenius theory, graph theory, and elementary algebraic recursions.Comment: In press, SIAM Revie
Robustness of large-scale stochastic matrices to localized perturbations
Upper bounds are derived on the total variation distance between the
invariant distributions of two stochastic matrices differing on a subset W of
rows. Such bounds depend on three parameters: the mixing time and the minimal
expected hitting time on W for the Markov chain associated to one of the
matrices; and the escape time from W for the Markov chain associated to the
other matrix. These results, obtained through coupling techniques, prove
particularly useful in scenarios where W is a small subset of the state space,
even if the difference between the two matrices is not small in any norm.
Several applications to large-scale network problems are discussed, including
robustness of Google's PageRank algorithm, distributed averaging and consensus
algorithms, and interacting particle systems.Comment: 12 pages, 4 figure
Generalized Opinion Dynamics from Local Optimization Rules
We study generalizations of the Hegselmann-Krause (HK) model for opinion
dynamics, incorporating features and parameters that are natural components of
observed social systems. The first generalization is one where the strength of
influence depends on the distance of the agents' opinions. Under this setup, we
identify conditions under which the opinions converge in finite time, and
provide a qualitative characterization of the equilibrium. We interpret the HK
model opinion update rule as a quadratic cost-minimization rule. This enables a
second generalization: a family of update rules which possess different
equilibrium properties. Subsequently, we investigate models in which a external
force can behave strategically to modulate/influence user updates. We consider
cases where this external force can introduce additional agents and cases where
they can modify the cost structures for other agents. We describe and analyze
some strategies through which such modulation may be possible in an
order-optimal manner. Our simulations demonstrate that generalized dynamics
differ qualitatively and quantitatively from traditional HK dynamics.Comment: 20 pages, under revie
Tight Bounds for Consensus Systems Convergence
We analyze the asymptotic convergence of all infinite products of matrices
taken in a given finite set, by looking only at finite or periodic products. It
is known that when the matrices of the set have a common nonincreasing
polyhedral norm, all infinite products converge to zero if and only if all
infinite periodic products with period smaller than a certain value converge to
zero, and bounds exist on that value.
We provide a stronger bound holding for both polyhedral norms and polyhedral
seminorms. In the latter case, the matrix products do not necessarily converge
to 0, but all trajectories of the associated system converge to a common
invariant space. We prove our bound to be tight, in the sense that for any
polyhedral seminorm, there is a set of matrices such that not all infinite
products converge, but every periodic product with period smaller than our
bound does converge.
Our technique is based on an analysis of the combinatorial structure of the
face lattice of the unit ball of the nonincreasing seminorm. The bound we
obtain is equal to half the size of the largest antichain in this lattice.
Explicitly evaluating this quantity may be challenging in some cases. We
therefore link our problem with the Sperner property: the property that, for
some graded posets, -- in this case the face lattice of the unit ball -- the
size of the largest antichain is equal to the size of the largest rank level.
We show that some sets of matrices with invariant polyhedral seminorms lead
to posets that do not have that Sperner property. However, this property holds
for the polyhedron obtained when treating sets of stochastic matrices, and our
bound can then be easily evaluated in that case. In particular, we show that
for the dimension of the space , our bound is smaller than the
previously known bound by a multiplicative factor of
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