From Random Lines to Metric Spaces

Consider an improper Poisson line process, marked by positive speeds so as to satisfy a scale-invariance property (actually, scale-equivariance). The line process can be characterized by its intensity measure, which belongs to a one-parameter family if scale and Euclidean invariance are required. This paper investigates a proposal by Aldous, namely that the line process could be used to produce a scale-invariant random spatial network (SIRSN) by means of connecting up points using paths which follow segments from the line process at the stipulated speeds. It is shown that this does indeed produce a scale-invariant network, under suitable conditions on the parameter; indeed that this produces a parameter-dependent random geodesic metric for d-dimensional space ($d\geq2$), where geodesics are given by minimum-time paths. Moreover in the planar case it is shown that the resulting geodesic metric space has an almost-everywhere-unique-geodesic property, that geodesics are locally of finite mean length, and that if an independent Poisson point process is connected up by such geodesics then the resulting network places finite length in each compact region. It is an open question whether the result is a SIRSN (in Aldous' sense; so placing finite mean length in each compact region), but it may be called a pre-SIRSN.Comment: Version 1: 46 pages, 10 figures Version 2: 47 pages, 10 figures (various typos and stylistic amendments, added dedication to Burkholder, added references concerning Lipschitz property and Sobolev space

Optimal relay location and power allocation for low SNR broadcast relay channels

We consider the broadcast relay channel (BRC), where a single source transmits to multiple destinations with the help of a relay, in the limit of a large bandwidth. We address the problem of optimal relay positioning and power allocations at source and relay, to maximize the multicast rate from source to all destinations. To solve such a network planning problem, we develop a three-faceted approach based on an underlying information theoretic model, computational geometric aspects, and network optimization tools. Firstly, assuming superposition coding and frequency division between the source and the relay, the information theoretic framework yields a hypergraph model of the wideband BRC, which captures the dependency of achievable rate-tuples on the network topology. As the relay position varies, so does the set of hyperarcs constituting the hypergraph, rendering the combinatorial nature of optimization problem. We show that the convex hull C of all nodes in the 2-D plane can be divided into disjoint regions corresponding to distinct hyperarcs sets. These sets are obtained by superimposing all k-th order Voronoi tessellation of C. We propose an easy and efficient algorithm to compute all hyperarc sets, and prove they are polynomially bounded. Using the switched hypergraph approach, we model the original problem as a continuous yet non-convex network optimization program. Ultimately, availing on the techniques of geometric programming and $p$-norm surrogate approximation, we derive a good convex approximation. We provide a detailed characterization of the problem for collinearly located destinations, and then give a generalization for arbitrarily located destinations. Finally, we show strong gains for the optimal relay positioning compared to seemingly interesting positions.Comment: In Proceedings of INFOCOM 201

Biologically inspired distributed machine cognition: a new formal approach to hyperparallel computation

The irresistable march toward multiple-core chip technology presents currently intractable pdrogramming challenges. High level mental processes in many animals, and their analogs for social structures, appear similarly massively parallel, and recent mathematical models addressing them may be adaptable to the multi-core programming problem

On a registration-based approach to sensor network localization

We consider a registration-based approach for localizing sensor networks from range measurements. This is based on the assumption that one can find overlapping cliques spanning the network. That is, for each sensor, one can identify geometric neighbors for which all inter-sensor ranges are known. Such cliques can be efficiently localized using multidimensional scaling. However, since each clique is localized in some local coordinate system, we are required to register them in a global coordinate system. In other words, our approach is based on transforming the localization problem into a problem of registration. In this context, the main contributions are as follows. First, we describe an efficient method for partitioning the network into overlapping cliques. Second, we study the problem of registering the localized cliques, and formulate a necessary rigidity condition for uniquely recovering the global sensor coordinates. In particular, we present a method for efficiently testing rigidity, and a proposal for augmenting the partitioned network to enforce rigidity. A recently proposed semidefinite relaxation of global registration is used for registering the cliques. We present simulation results on random and structured sensor networks to demonstrate that the proposed method compares favourably with state-of-the-art methods in terms of run-time, accuracy, and scalability

Institutional Cognition

We generalize a recent mathematical analysis of Bernard Baars' model of human consciousness to explore analogous, but far more complicated, phenomena of institutional cognition. Individual consciousness is limited to a single, tunable, giant component of interacting cogntivie modules, instantiating a Global Workspace. Human institutions, by contrast, seem able to multitask, supporting several such giant components simultaneously, although their behavior remains constrained to a topology generated by cultural context and by the path-dependence inherent to organizational history. Surprisingly, such multitasking, while clearly limiting the phenomenon of inattentional blindness, does not eliminate it. This suggests that organizations (or machines) explicitly designed along these principles, while highly efficient at certain sets of tasks, would still be subject to analogs of the subtle failure patterns explored in Wallace (2005b, 2006). We compare and contrast our results with recent work on collective efficacy and collective consciousness

Machine Hyperconsciousness

Individual animal consciousness appears limited to a single giant component of interacting cognitive modules, instantiating a shifting, highly tunable, Global Workspace. Human institutions, by contrast, can support several, often many, such giant components simultaneously, although they generally function far more slowly than the minds of the individuals who compose them. Machines having multiple global workspaces -- hyperconscious machines -- should, however, be able to operate at the few hundred milliseconds characteistic of individual consciousness. Such multitasking -- machine or institutional -- while clearly limiting the phenomenon of inattentional blindness, does not eliminate it, and introduces characteristic failure modes involving the distortion of information sent between global workspaces. This suggests that machines explicitly designed along these principles, while highly efficient at certain sets of tasks, remain subject to canonical and idiosyncratic failure patterns analogous to, but more complicated than, those explored in Wallace (2006a). By contrast, institutions, facing similar challenges, are usually deeply embedded in a highly stabilizing cultural matrix of law, custom, and tradition which has evolved over many centuries. Parallel development of analogous engineering strategies, directed toward ensuring an 'ethical' device, would seem requisite to the sucessful application of any form of hyperconscious machine technology