4,417 research outputs found

### Entropy rate of higher-dimensional cellular automata

We introduce the entropy rate of multidimensional cellular automata. This
number is invariant under shift-commuting isomorphisms; as opposed to the
entropy of such CA, it is always finite. The invariance property and the
finiteness of the entropy rate result from basic results about the entropy of
partitions of multidimensional cellular automata. We prove several results that
show that entropy rate of 2-dimensional automata preserve similar properties of
the entropy of one dimensional cellular automata.
In particular we establish an inequality which involves the entropy rate, the
radius of the cellular automaton and the entropy of the d-dimensional shift. We
also compute the entropy rate of permutative bi-dimensional cellular automata
and show that the finite value of the entropy rate (like the standard entropy
of for one-dimensional CA) depends on the number of permutative sites.
Finally we define the topological entropy rate and prove that it is an
invariant for topological shift-commuting conjugacy and establish some
relations between topological and measure-theoretic entropy rates

### Taylor series expansions for the entropy rate of Hidden Markov Processes

Finding the entropy rate of Hidden Markov Processes is an active research
topic, of both theoretical and practical importance. A recently used approach
is studying the asymptotic behavior of the entropy rate in various regimes. In
this paper we generalize and prove a previous conjecture relating the entropy
rate to entropies of finite systems. Building on our new theorems, we establish
series expansions for the entropy rate in two different regimes. We also study
the radius of convergence of the two series expansions

### Entropy rate calculations of algebraic measures

Let $K = \{0,1,...,q-1\}$. We use a special class of translation invariant
measures on $K^\mathbb{Z}$ called algebraic measures to study the entropy rate
of a hidden Markov processes. Under some irreducibility assumptions of the
Markov transition matrix we derive exact formulas for the entropy rate of a
general $q$ state hidden Markov process derived from a Markov source corrupted
by a specific noise model. We obtain upper bounds on the error when using an
approximation to the formulas and numerically compute the entropy rates of two
and three state hidden Markov models

### Entropy Rate of Diffusion Processes on Complex Networks

The concept of entropy rate for a dynamical process on a graph is introduced.
We study diffusion processes where the node degrees are used as a local
information by the random walkers. We describe analitically and numerically how
the degree heterogeneity and correlations affect the diffusion entropy rate. In
addition, the entropy rate is used to characterize complex networks from the
real world. Our results point out how to design optimal diffusion processes
that maximize the entropy for a given network structure, providing a new
theoretical tool with applications to social, technological and communication
networks.Comment: 4 pages (APS format), 3 figures, 1 tabl

### Analyticity of Entropy Rate of Hidden Markov Chains

We prove that under mild positivity assumptions the entropy rate of a hidden
Markov chain varies analytically as a function of the underlying Markov chain
parameters. A general principle to determine the domain of analyticity is
stated. An example is given to estimate the radius of convergence for the
entropy rate. We then show that the positivity assumptions can be relaxed, and
examples are given for the relaxed conditions. We study a special class of
hidden Markov chains in more detail: binary hidden Markov chains with an
unambiguous symbol, and we give necessary and sufficient conditions for
analyticity of the entropy rate for this case. Finally, we show that under the
positivity assumptions the hidden Markov chain {\em itself} varies
analytically, in a strong sense, as a function of the underlying Markov chain
parameters.Comment: The title has been changed. The new main theorem now combines the old
main theorem and the remark following the old main theorem. A new section is
added as an introduction to complex analysis. General principle and an
example to determine the domain of analyticity of entropy rate have been
added. Relaxed conditions for analyticity of entropy rate and the
corresponding examples are added. The section about binary markov chain
corrupted by binary symmetric noise is taken out (to be part of another
paper

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