100,346 research outputs found
Causal graph dynamics
We extend the theory of Cellular Automata to arbitrary, time-varying graphs.
In other words we formalize, and prove theorems about, the intuitive idea of a
labelled graph which evolves in time - but under the natural constraint that
information can only ever be transmitted at a bounded speed, with respect to
the distance given by the graph. The notion of translation-invariance is also
generalized. The definition we provide for these "causal graph dynamics" is
simple and axiomatic. The theorems we provide also show that it is robust. For
instance, causal graph dynamics are stable under composition and under
restriction to radius one. In the finite case some fundamental facts of
Cellular Automata theory carry through: causal graph dynamics admit a
characterization as continuous functions, and they are stable under inversion.
The provided examples suggest a wide range of applications of this mathematical
object, from complex systems science to theoretical physics. KEYWORDS:
Dynamical networks, Boolean networks, Generative networks automata, Cayley
cellular automata, Graph Automata, Graph rewriting automata, Parallel graph
transformations, Amalgamated graph transformations, Time-varying graphs, Regge
calculus, Local, No-signalling.Comment: 25 pages, 9 figures, LaTeX, v2: Minor presentation improvements, v3:
Typos corrected, figure adde
Quantum Causal Graph Dynamics
Consider a graph having quantum systems lying at each node. Suppose that the
whole thing evolves in discrete time steps, according to a global, unitary
causal operator. By causal we mean that information can only propagate at a
bounded speed, with respect to the distance given by the graph. Suppose,
moreover, that the graph itself is subject to the evolution, and may be driven
to be in a quantum superposition of graphs---in accordance to the superposition
principle. We show that these unitary causal operators must decompose as a
finite-depth circuit of local unitary gates. This unifies a result on Quantum
Cellular Automata with another on Reversible Causal Graph Dynamics. Along the
way we formalize a notion of causality which is valid in the context of quantum
superpositions of time-varying graphs, and has a number of good properties.
Keywords: Quantum Lattice Gas Automata, Block-representation,
Curtis-Hedlund-Lyndon, No-signalling, Localizability, Quantum Gravity, Quantum
Graphity, Causal Dynamical Triangulations, Spin Networks, Dynamical networks,
Graph Rewriting.Comment: 8 pages, 1 figur
Causal connectivity of evolved neural networks during behavior
To show how causal interactions in neural dynamics are modulated by behavior, it is valuable to analyze these interactions without perturbing or lesioning the neural mechanism. This paper proposes a method, based on a graph-theoretic extension of vector autoregressive modeling and 'Granger causality,' for characterizing causal interactions generated within intact neural mechanisms. This method, called 'causal connectivity analysis' is illustrated via model neural networks optimized for controlling target fixation in a simulated head-eye system, in which the structure of the environment can be experimentally varied. Causal connectivity analysis of this model yields novel insights into neural mechanisms underlying sensorimotor coordination. In contrast to networks supporting comparatively simple behavior, networks supporting rich adaptive behavior show a higher density of causal interactions, as well as a stronger causal flow from sensory inputs to motor outputs. They also show different arrangements of 'causal sources' and 'causal sinks': nodes that differentially affect, or are affected by, the remainder of the network. Finally, analysis of causal connectivity can predict the functional consequences of network lesions. These results suggest that causal connectivity analysis may have useful applications in the analysis of neural dynamics
An example of the stochastic dynamics of a causal set
An example of a discrete pregeometry on a microscopic scale is introduced.
The model is a directed dyadic acyclic graph. This is the particular case of a
causal set. The particles in this model must be self-organized repetitive
structures. The dynamics of this model is a stochastic sequential growth
dynamics. New vertexes are added one by one. The probability of this addition
depends on the structure of existed graph. The particular case of the dynamics
is considered. The numerical simulation provides some symptoms of
self-organization.Comment: 7 pages, 5 figures, work presented at the conference "Foundations of
Probability and Physics-6" (FPP6) held on 12-15 June 2011 at the Linnaeus
University, V\"axj\"o, Swede
Network Cosmology
Prediction and control of the dynamics of complex networks is a central
problem in network science. Structural and dynamical similarities of different
real networks suggest that some universal laws might accurately describe the
dynamics of these networks, albeit the nature and common origin of such laws
remain elusive. Here we show that the causal network representing the
large-scale structure of spacetime in our accelerating universe is a power-law
graph with strong clustering, similar to many complex networks such as the
Internet, social, or biological networks. We prove that this structural
similarity is a consequence of the asymptotic equivalence between the
large-scale growth dynamics of complex networks and causal networks. This
equivalence suggests that unexpectedly similar laws govern the dynamics of
complex networks and spacetime in the universe, with implications to network
science and cosmology
Topological reversibility and causality in feed-forward networks
Systems whose organization displays causal asymmetry constraints, from
evolutionary trees to river basins or transport networks, can be often
described in terms of directed paths (causal flows) on a discrete state space.
Such a set of paths defines a feed-forward, acyclic network. A key problem
associated with these systems involves characterizing their intrinsic degree of
path reversibility: given an end node in the graph, what is the uncertainty of
recovering the process backwards until the origin? Here we propose a novel
concept, \textit{topological reversibility}, which rigorously weigths such
uncertainty in path dependency quantified as the minimum amount of information
required to successfully revert a causal path. Within the proposed framework we
also analytically characterize limit cases for both topologically reversible
and maximally entropic structures. The relevance of these measures within the
context of evolutionary dynamics is highlighted.Comment: 9 pages, 3 figure
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