Spectral and Dynamical Properties in Classes of Sparse Networks with Mesoscopic Inhomogeneities

We study structure, eigenvalue spectra and diffusion dynamics in a wide class of networks with subgraphs (modules) at mesoscopic scale. The networks are grown within the model with three parameters controlling the number of modules, their internal structure as scale-free and correlated subgraphs, and the topology of connecting network. Within the exhaustive spectral analysis for both the adjacency matrix and the normalized Laplacian matrix we identify the spectral properties which characterize the mesoscopic structure of sparse cyclic graphs and trees. The minimally connected nodes, clustering, and the average connectivity affect the central part of the spectrum. The number of distinct modules leads to an extra peak at the lower part of the Laplacian spectrum in cyclic graphs. Such a peak does not occur in the case of topologically distinct tree-subgraphs connected on a tree. Whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs. We also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue equal 2 of the Laplacian. We corroborate the results with simulations of the random walk on several types of networks. Our results for the distribution of return-time of the walk to the origin (autocorrelator) agree well with recent analytical solution for trees, and it appear to be independent on their mesoscopic and global structure. For the cyclic graphs we find new results with twice larger stretching exponent of the tail of the distribution, which is virtually independent on the size of cycles. The modularity and clustering contribute to a power-law decay at short return times

Generalized Quantum Theory: Overview and Latest Developments

The main formal structures of Generalized Quantum Theory are summarized. Recent progress has sharpened some of the concepts, in particular the notion of an observable, the action of an observable on states (putting more emphasis on the role of proposition observables), and the concept of generalized entanglement. Furthermore, the active role of the observer in the structure of observables and the partitioning of systems is emphasized.Comment: 14 pages, update in reference

Geometric representations for minimalist grammars

We reformulate minimalist grammars as partial functions on term algebras for strings and trees. Using filler/role bindings and tensor product representations, we construct homomorphisms for these data structures into geometric vector spaces. We prove that the structure-building functions as well as simple processors for minimalist languages can be realized by piecewise linear operators in representation space. We also propose harmony, i.e. the distance of an intermediate processing step from the final well-formed state in representation space, as a measure of processing complexity. Finally, we illustrate our findings by means of two particular arithmetic and fractal representations.Comment: 43 pages, 4 figure

Epistemic Entanglement due to Non-Generating Partitions of Classical Dynamical Systems

Quantum entanglement relies on the fact that pure quantum states are dispersive and often inseparable. Since pure classical states are dispersion-free they are always separable and cannot be entangled. However, entanglement is possible for epistemic, dispersive classical states. We show how such epistemic entanglement arises for epistemic states of classical dynamical systems based on phase space partitions that are not generating. We compute epistemically entangled states for two coupled harmonic oscillators.Comment: 13 pages, no figures; International Journal of Theoretical Physics, 201

A dynamical systems perspective on the relationship between symbolic and non-symbolic computation

It has been claimed that connectionist (artificial neural network) models of language processing, which do not appear to employ “rules”, are doing something different in kind from classical symbol processing models, which treat “rules” as atoms (e.g., McClelland and Patterson in Trends Cogn Sci 6(11):465–472, 2002). This claim is hard to assess in the absence of careful, formal comparisons between the two approaches. This paper formally investigates the symbol-processing properties of simple dynamical systems called affine dynamical automata, which are close relatives of several recurrent connectionist models of language processing (e.g., Elman in Cogn Sci 14:179–211, 1990). In line with related work (Moore in Theor Comput Sci 201:99–136, 1998; Siegelmann in Neural networks and analog computation: beyond the Turing limit. Birkhäuser, Boston, 1999), the analysis shows that affine dynamical automata exhibit a range of symbol processing behaviors, some of which can be mirrored by various Turing machine devices, and others of which cannot be. On the assumption that the Turing machine framework is a good way to formalize the “computation” part of our understanding of classical symbol processing, this finding supports the view that there is a fundamental “incompatibility” between connectionist and classical models (see Fodor and Pylyshyn 1988; Smolensky in Behav Brain Sci 11(1):1–74, 1988; beim Graben in Mind Matter 2(2):29--51,2004b). Given the empirical successes of connectionist models, the more general, super-Turing framework is a preferable vantage point from which to consider cognitive phenomena. This vantage may give us insight into ill-formed as well as well-formed language behavior and shed light on important structural properties of learning processes

Unifying syntactic theory and sentence processing difficulty through a connectionist minimalist parser

Syntactic theory provides a rich array of representational assumptions about linguistic knowledge and processes. Such detailed and independently motivated constraints on grammatical knowledge ought to play a role in sentence comprehension. However most grammar-based explanations of processing difficulty in the literature have attempted to use grammatical representations and processes per se to explain processing difficulty. They did not take into account that the description of higher cognition in mind and brain encompasses two levels: on the one hand, at the macrolevel, symbolic computation is performed, and on the other hand, at the microlevel, computation is achieved through processes within a dynamical system. One critical question is therefore how linguistic theory and dynamical systems can be unified to provide an explanation for processing effects. Here, we present such a unification for a particular account to syntactic theory: namely a parser for Stabler's Minimalist Grammars, in the framework of Smolensky's Integrated Connectionist/Symbolic architectures. In simulations we demonstrate that the connectionist minimalist parser produces predictions which mirror global empirical findings from psycholinguistic research