9 research outputs found
Infinite Excess Entropy Processes with Countable-State Generators
We present two examples of finite-alphabet, infinite excess entropy processes
generated by invariant hidden Markov models (HMMs) with countable state sets.
The first, simpler example is not ergodic, but the second is. It appears these
are the first constructions of processes of this type. Previous examples of
infinite excess entropy processes over finite alphabets admit only invariant
HMM presentations with uncountable state sets.Comment: 13 pages, 3 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/ieepcsg.ht
Structural Information in Two-Dimensional Patterns: Entropy Convergence and Excess Entropy
We develop information-theoretic measures of spatial structure and pattern in
more than one dimension. As is well known, the entropy density of a
two-dimensional configuration can be efficiently and accurately estimated via a
converging sequence of conditional entropies. We show that the manner in which
these conditional entropies converge to their asymptotic value serves as a
measure of global correlation and structure for spatial systems in any
dimension. We compare and contrast entropy-convergence with mutual-information
and structure-factor techniques for quantifying and detecting spatial
structure.Comment: 11 pages, 5 figures,
http://www.santafe.edu/projects/CompMech/papers/2dnnn.htm
Thermodynamic Depth of Causal States: When Paddling around in Occam's Pool Shallowness Is a Virtue
Thermodynamic depth is an appealing but flawed structural complexity measure.
It depends on a set of macroscopic states for a system, but neither its
original introduction by Lloyd and Pagels nor any follow-up work has considered
how to select these states. Depth, therefore, is at root arbitrary.
Computational mechanics, an alternative approach to structural complexity,
provides a definition for a system's minimal, necessary causal states and a
procedure for finding them. We show that the rate of increase in thermodynamic
depth, or {\it dive}, is the system's reverse-time Shannon entropy rate, and so
depth only measures degrees of macroscopic randomness, not structure. To fix
this we redefine the depth in terms of the causal state
representation----machines---and show that this representation gives
the minimum dive consistent with accurate prediction. Thus, -machines
are optimally shallow.Comment: 11 pages, 9 figures, RevTe
Dynamical Phase Transitions in Graph Cellular Automata
Discrete dynamical systems can exhibit complex behaviour from the iterative
application of straightforward local rules. A famous example are cellular
automata whose global dynamics are notoriously challenging to analyze. To
address this, we relax the regular connectivity grid of cellular automata to a
random graph, which gives the class of graph cellular automata. Using the
dynamical cavity method (DCM) and its backtracking version (BDCM), we show that
this relaxation allows us to derive asymptotically exact analytical results on
the global dynamics of these systems on sparse random graphs. Concretely, we
showcase the results on a specific subclass of graph cellular automata with
``conforming non-conformist'' update rules, which exhibit dynamics akin to
opinion formation. Such rules update a node's state according to the majority
within their own neighbourhood. In cases where the majority leads only by a
small margin over the minority, nodes may exhibit non-conformist behaviour.
Instead of following the majority, they either maintain their own state, switch
it, or follow the minority. For configurations with different initial biases
towards one state we identify sharp dynamical phase transitions in terms of the
convergence speed and attractor types. From the perspective of opinion dynamics
this answers when consensus will emerge and when two opinions coexist almost
indefinitely.Comment: 15 page