162 research outputs found
Synchronisation Properties of Trees in the Kuramoto Model
We consider the Kuramoto model of coupled oscillators, specifically the case
of tree networks, for which we prove a simple closed-form expression for the
critical coupling. For several classes of tree, and for both uniform and
Gaussian vertex frequency distributions, we provide tight closed form bounds
and empirical expressions for the expected value of the critical coupling. We
also provide several bounds on the expected value of the critical coupling for
all trees. Finally, we show that for a given set of vertex frequencies, there
is a rearrangement of oscillator frequencies for which the critical coupling is
bounded by the spread of frequencies.Comment: 21 pages, 19 Figure
Spectral Network Principle for Frequency Synchronization in Repulsive Laser Networks
Network synchronization of lasers is critical for reaching high-power levels
and for effective optical computing. Yet, the role of network topology for the
frequency synchronization of lasers is not well understood. Here, we report our
significant progress toward solving this critical problem for networks of
heterogeneous laser model oscillators with repulsive coupling. We discover a
general approximate principle for predicting the onset of frequency
synchronization from the spectral knowledge of a complex matrix representing a
combination of the signless Laplacian induced by repulsive coupling and a
matrix associated with intrinsic frequency detuning. We show that the gap
between the two smallest eigenvalues of the complex matrix generally controls
the coupling threshold for frequency synchronization. In stark contrast with
Laplacian networks, we demonstrate that local rings and all-to-all networks
prevent frequency synchronization, whereas full bipartite networks have optimal
synchronization properties. Beyond laser models, we show that, with a few
exceptions, the spectral principle can be applied to repulsive Kuramoto
networks. Our results may provide guidelines for optimal designs of scalable
laser networks capable of achieving reliable synchronization
Critical phenomena in complex networks
The combination of the compactness of networks, featuring small diameters,
and their complex architectures results in a variety of critical effects
dramatically different from those in cooperative systems on lattices. In the
last few years, researchers have made important steps toward understanding the
qualitatively new critical phenomena in complex networks. We review the
results, concepts, and methods of this rapidly developing field. Here we mostly
consider two closely related classes of these critical phenomena, namely
structural phase transitions in the network architectures and transitions in
cooperative models on networks as substrates. We also discuss systems where a
network and interacting agents on it influence each other. We overview a wide
range of critical phenomena in equilibrium and growing networks including the
birth of the giant connected component, percolation, k-core percolation,
phenomena near epidemic thresholds, condensation transitions, critical
phenomena in spin models placed on networks, synchronization, and
self-organized criticality effects in interacting systems on networks. We also
discuss strong finite size effects in these systems and highlight open problems
and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references,
extende
On the typical and atypical solutions to the Kuramoto equations
The Kuramoto model is a dynamical system that models the interaction of
coupled oscillators. There has been much work to effectively bound the number
of equilibria to the Kuramoto model for a given network. By formulating the
Kuramoto equations as a system of algebraic equations, we first relate the
complex root count of the Kuramoto equations to the combinatorics of the
underlying network by showing that the complex root count is generically equal
to the normalized volume of the corresponding adjacency polytope of the
network. We then give explicit algebraic conditions under which this bound is
strict and show that there are conditions where the Kuramoto equations have
infinitely many equilibria.Comment: 28 page
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