Invariant Measures, Hausdorff Dimension and Dimension Drop of some Harmonic Measures on Galton-Watson Trees

We consider infinite Galton-Watson trees without leaves together with i.i.d.~random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, to build explicit invariant measures. We apply this result, together with the ergodic theory on Galton-Watson trees developed in \cite{LPP95}, to the computation of Hausdorff dimensions of harmonic measures in two cases. The first one is the harmonic measure of the (transient) $\lambda$-biased random walk on Galton-Watson trees, for which the invariant measure and the dimension were not explicitly known. The second case is a model of random walk on a Galton-Watson trees with random lengths for which we compute the dimensions of the harmonic measure and show dimension drop phenomenon for the natural metric on the boundary and another metric that depends on the random lengths.Comment: 37 pages, 5 figure

FABilT – finding answers in a billion triples

This submission presents the application of two coupled systems to the Billion Triples Challenge. The first system (Watson) provides the infrastructure which allows the second one (PowerAqua) to pose natural language queries to the billion triple datasets. Watson is a gateway to the Semantic Web: it crawls and indexes semantic data online to provide a variety of access mechanisms for human users and applications.We show here how we indexed most of the datasets provided for the challenge, thus obtaining an infrastructure (comprising web services, API, web interface, etc.) which supports the exploration of these datasets and makes them available to any Watson-based application. PowerAqua is an open domain question answering system which allows users to pose natural language queries to large scale collections of heterogeneous semantic data. In this paper, we discuss the issues we faced in configuring PowerAqua and Watson for the challenge and report on our results. The system composed of Watson and PowerAqua, and applied to the Billion Triples Challenge, is called FABilT