Random unitary matrices associated to a graph

We analyze composed quantum systems consisting of $k$ subsystems, each described by states in the $n$-dimensional Hilbert space. Interaction between subsystems can be represented by a graph, with vertices corresponding to individual subsystems and edges denoting a generic interaction, modeled by random unitary matrices of order $n^2$. The global evolution operator is represented by a unitary matrix of size $N=n^k$. We investigate statistical properties of such matrices and show that they display spectral properties characteristic to Haar random unitary matrices provided the corresponding graph is connected. Thus basing on random unitary matrices of a small size $n^2$ one can construct a fair approximation of large random unitary matrices of size $n^{k}$. Graph--structured random unitary matrices investigated here allow one to define the corresponding structured ensembles of random pure states.Comment: 13 pages, 10 figures, 1 tabl

Prisoner's dilemma in structured scale-free networks

The conventional wisdom is that scale-free networks are prone to cooperation spreading. In this paper we investigate the cooperative behaviors on the structured scale-free network. On the contrary of the conventional wisdom that scale-free networks are prone to cooperation spreading, the evolution of cooperation is inhibited on the structured scale-free network while performing the prisoner's dilemma (PD) game. Firstly, we demonstrate that neither the scale-free property nor the high clustering coefficient is responsible for the inhibition of cooperation spreading on the structured scale-free network. Then we provide one heuristic method to argue that the lack of age correlations and its associated `large-world' behavior in the structured scale-free network inhibit the spread of cooperation. The findings may help enlighten further studies on evolutionary dynamics of the PD game in scale-free networks.Comment: Definitive version accepted for publication in Journal of Physics

A Global Workspace perspective on mental disorders

Recent developments in Global Workspace theory suggest that human consciousness can suffer interpenetrating dysfunctions of mutual and reciprocal interaction with embedding environments which will have early onset and often insidiously staged developmental progression, possibly according to a cancer model. A simple rate distortion argument implies that, if an external information source is pathogenic, then sufficient exposure to it is sure to write a sufficiently accurate image of it on mind and body in a punctuated manner so as to initiate or promote simililarly progressively punctuated developmental disorder. There can, thus, be no simple, reductionist brain chemical 'bug in the program' whose 'fix' can fully correct the problem. On the contrary, the growth of an individual over the life course, and the inevitable contact with a toxic physical, social, or cultural environment, can be expected to initiate developmental problems which will become more intrusive over time, most obviously according to some damage accumulation model, but likely according to far more subtle, highly punctuated, schemes analogous to tumorigenesis. The key intervention, at the population level, is clearly to limit such exposures, a question of proper environmental sanitation, in a large sense, a matter of social justice which has long been understood to be determined almost entirely by the interactions of cultural trajectory, group power relations, and economic structure, with public policy. Intervention at the individual level appears limited to triggering or extending periods of remission, as is the case with most cancers

Entering the blackboard jungle: canonical dysfunction in conscious machines

The central paradigm of Artificial Intelligence is rapidly shifting toward biological models for both robotic devices and systems performing such critical tasks as network management and process control. Here we apply recent mathematical analysis of the necessary conditions for consciousness in humans in an attempt to gain some understanding of the likely canonical failure modes inherent to a broad class of global workspace/blackboard machines designed to emulate biological functions. Similar problems are likely to confront other possible architectures, although their mathematical description may be far less straightforward

Distributed Robust Set-Invariance for Interconnected Linear Systems

We introduce a class of distributed control policies for networks of discrete-time linear systems with polytopic additive disturbances. The objective is to restrict the network-level state and controls to user-specified polyhedral sets for all times. This problem arises in many safety-critical applications. We consider two problems. First, given a communication graph characterizing the structure of the information flow in the network, we find the optimal distributed control policy by solving a single linear program. Second, we find the sparsest communication graph required for the existence of a distributed invariance-inducing control policy. Illustrative examples, including one on platooning, are presented.Comment: 8 Pages. Submitted to American Control Conference (ACC), 201