11,646 research outputs found
Evolution of network structure by temporal learning
We study the effect of learning dynamics on network topology. A network of
discrete dynamical systems is considered for this purpose and the coupling
strengths are made to evolve according to a temporal learning rule that is
based on the paradigm of spike-time-dependent plasticity. This incorporates
necessary competition between different edges. The final network we obtain is
robust and has a broad degree distribution.Comment: revised manuscript in communicatio
From the Jordan product to Riemannian geometries on classical and quantum states
The Jordan product on the self-adjoint part of a finite-dimensional
-algebra is shown to give rise to Riemannian metric
tensors on suitable manifolds of states on , and the covariant
derivative, the geodesics, the Riemann tensor, and the sectional curvature of
all these metric tensors are explicitly computed. In particular, it is proved
that the Fisher--Rao metric tensor is recovered in the Abelian case, that the
Fubini--Study metric tensor is recovered when we consider pure states on the
algebra of linear operators on a finite-dimensional
Hilbert space , and that the Bures--Helstrom metric tensors is
recovered when we consider faithful states on .
Moreover, an alternative derivation of these Riemannian metric tensors in terms
of the GNS construction associated to a state is presented. In the case of pure
and faithful states on , this alternative geometrical
description clarifies the analogy between the Fubini--Study and the
Bures--Helstrom metric tensor.Comment: 32 pages. Minor improvements. References added. Comments are welcome
Synchronization in discrete-time networks with general pairwise coupling
We consider complete synchronization of identical maps coupled through a
general interaction function and in a general network topology where the edges
may be directed and may carry both positive and negative weights. We define
mixed transverse exponents and derive sufficient conditions for local complete
synchronization. The general non-diffusive coupling scheme can lead to new
synchronous behavior, in networks of identical units, that cannot be produced
by single units in isolation. In particular, we show that synchronous chaos can
emerge in networks of simple units. Conversely, in networks of chaotic units
simple synchronous dynamics can emerge; that is, chaos can be suppressed
through synchrony
Synchronization of networks with prescribed degree distributions
We show that the degree distributions of graphs do not suffice to
characterize the synchronization of systems evolving on them. We prove that,
for any given degree sequence satisfying certain conditions, there exists a
connected graph having that degree sequence for which the first nontrivial
eigenvalue of the graph Laplacian is arbitrarily close to zero. Consequently,
complex dynamical systems defined on such graphs have poor synchronization
properties. The result holds under quite mild assumptions, and shows that there
exists classes of random, scale-free, regular, small-world, and other common
network architectures which impede synchronization. The proof is based on a
construction that also serves as an algorithm for building non-synchronizing
networks having a prescribed degree distribution.Comment: v2: A new theorem and a numerical example added. To appear in IEEE
Trans. Circuits and Systems I: Fundamental Theory and Application
Symbolic Synchronization and the Detection of Global Properties of Coupled Dynamics from Local Information
We study coupled dynamics on networks using symbolic dynamics. The symbolic
dynamics is defined by dividing the state space into a small number of regions
(typically 2), and considering the relative frequencies of the transitions
between those regions. It turns out that the global qualitative properties of
the coupled dynamics can be classified into three different phases based on the
synchronization of the variables and the homogeneity of the symbolic dynamics.
Of particular interest is the {\it homogeneous unsynchronized phase} where the
coupled dynamics is in a chaotic unsynchronized state, but exhibits (almost)
identical symbolic dynamics at all the nodes in the network. We refer to this
dynamical behaviour as {\it symbolic synchronization}. In this phase, the local
symbolic dynamics of any arbitrarily selected node reflects global properties
of the coupled dynamics, such as qualitative behaviour of the largest Lyapunov
exponent and phase synchronization. This phase depends mainly on the network
architecture, and only to a smaller extent on the local chaotic dynamical
function. We present results for two model dynamics, iterations of the
one-dimensional logistic map and the two-dimensional H\'enon map, as local
dynamical function.Comment: 21 pages, 7 figure
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