268 research outputs found
Optimal synchronization of complex networks
We study optimal synchronization in networks of heterogeneous phase
oscillators. Our main result is the derivation of a synchrony alignment
function that encodes the interplay between network structure and oscillators'
frequencies and can be readily optimized. We highlight its utility in two
general problems: constrained frequency allocation and network design. In
general, we find that synchronization is promoted by strong alignments between
frequencies and the dominant Laplacian eigenvectors, as well as a matching
between the heterogeneity of frequencies and network structure.Comment: 5 pages, 4 figure
Synchronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the synchrony alignment function
Synchronization is central to many complex systems in engineering physics
(e.g., the power-grid, Josephson junction circuits, and electro-chemical
oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms).
Despite these widespread applications---for which proper functionality depends
sensitively on the extent of synchronization---there remains a lack of
understanding for how systems evolve and adapt to enhance or inhibit
synchronization. We study how network modifications affect the synchronization
properties of network-coupled dynamical systems that have heterogeneous node
dynamics (e.g., phase oscillators with non-identical frequencies), which is
often the case for real-world systems. Our approach relies on a synchrony
alignment function (SAF) that quantifies the interplay between heterogeneity of
the network and of the oscillators and provides an objective measure for a
system's ability to synchronize. We conduct a spectral perturbation analysis of
the SAF for structural network modifications including the addition and removal
of edges, which subsequently ranks the edges according to their importance to
synchronization. Based on this analysis, we develop gradient-descent algorithms
to efficiently solve optimization problems that aim to maximize phase
synchronization via network modifications. We support these and other results
with numerical experiments.Comment: 25 pages, 6 figure
Super-resolution community detection for layer-aggregated multilayer networks
Applied network science often involves preprocessing network data before
applying a network-analysis method, and there is typically a theoretical
disconnect between these steps. For example, it is common to aggregate
time-varying network data into windows prior to analysis, and the tradeoffs of
this preprocessing are not well understood. Focusing on the problem of
detecting small communities in multilayer networks, we study the effects of
layer aggregation by developing random-matrix theory for modularity matrices
associated with layer-aggregated networks with nodes and layers, which
are drawn from an ensemble of Erd\H{o}s-R\'enyi networks. We study phase
transitions in which eigenvectors localize onto communities (allowing their
detection) and which occur for a given community provided its size surpasses a
detectability limit . When layers are aggregated via a summation, we
obtain , where is the number of
layers across which the community persists. Interestingly, if is allowed to
vary with then summation-based layer aggregation enhances small-community
detection even if the community persists across a vanishing fraction of layers,
provided that decays more slowly than . Moreover,
we find that thresholding the summation can in some cases cause to decay
exponentially, decreasing by orders of magnitude in a phenomenon we call
super-resolution community detection. That is, layer aggregation with
thresholding is a nonlinear data filter enabling detection of communities that
are otherwise too small to detect. Importantly, different thresholds generally
enhance the detectability of communities having different properties,
illustrating that community detection can be obscured if one analyzes network
data using a single threshold.Comment: 11 pages, 8 figure
Collective frequency variation in network synchronization and reverse PageRank
A wide range of natural and engineered phenomena rely on large networks of
interacting units to reach a dynamical consensus state where the system
collectively operates. Here we study the dynamics of self-organizing systems
and show that for generic directed networks the collective frequency of the
ensemble is {\it not} the same as the mean of the individuals' natural
frequencies. Specifically, we show that the collective frequency equals a
weighted average of the natural frequencies, where the weights are given by an
out-flow centrality measure that is equivalent to a reverse PageRank
centrality. Our findings uncover an intricate dependence of the collective
frequency on both the structural directedness and dynamical heterogeneity of
the network, and also reveal an unexplored connection between synchronization
and PageRank, which opens the possibility of applying PageRank optimization to
synchronization. Finally, we demonstrate the presence of collective frequency
variation in real-world networks by considering the UK and Scandinavian power
grids
Erosion of synchronization in networks of coupled oscillators
We report erosion of synchronization in networks of coupled phase
oscillators, a phenomenon where perfect phase synchronization is unattainable
in steady-state, even in the limit of infinite coupling. An analysis reveals
that the total erosion is separable into the product of terms characterizing
coupling frustration and structural heterogeneity, both of which amplify
erosion. The latter, however, can differ significantly from degree
heterogeneity. Finally, we show that erosion is marked by the reorganization of
oscillators according to their node degrees rather than their natural
frequencies.Comment: 5 pages, 4 figure
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