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We show that the homology of ordered configuration spaces of finite trees with loops is torsion free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete generating set for all homology groups of configuration spaces of trees with loops and the first homology group of configuration spaces of general finite graphs. An important technique in the paper is the identification of the $E^1$-page and differentials of Mayer-Vietoris spectral sequences for configuration spaces., Comment: 21 pages, 7 figures. Revised Proposition 2.3, other minor changes

We present a new computing package Flagser, designed to construct the directed flag complex of a finite directed graph, and compute persistent homology for flexibly defined filtrations on the graph and the resulting complex. The persistent homology computation part of Flagser is based on the program Ripser [Bau18a], but is optimized specifically for large computations. The construction of the directed flag complex is done in a way that allows easy parallelization by arbitrarily many cores. Flagser also has the option of working with undirected graphs. For homology computations Flagser has an Approximate option, which shortens compute time with remarkable accuracy. We demonstrate the power of Flagser by applying it to the construction of the directed flag complex of digital reconstructions of brain microcircuitry by the Blue Brain Project and several other examples. In some instances we perform computation of homology. For a more complete performance analysis, we also apply Flagser to some other data collections. In all cases the hardware used in the computation, the use of memory and the compute time are recorded., Comment: 14 Pages, 12 figures

With the spread of Internet of Things (IoT) technologies, assets have acquired communication, processing and sensing capabilities. In response, the field of Asset Management has moved from fleet-wide failure models to individualised asset prognostics. Individualised models are seldom truly distributed, and often fail to capitalise the processing power of the asset fleet. This leads to hardly scalable machine learning centralised models that often must find a compromise between accuracy and computational power. In order to overcome this, we present a novel theoretical approach to collaborative prognostics within the Social Internet of Things. We introduce the concept of Social Asset Networks, defined as networks of cooperating assets with sensing, communicating and computing capabilities. In the proposed approach, the information obtained from the medium by means of sensors is synthesised into a Health Indicator, which determines the state of the asset. The Health Indicator of each asset evolves according to an equation determined by a triplet of parameters. Assets are given the form of the equation but they are not aware of their parametric values. To obtain these values, assets use the equation in order to perform a non-linear least squares fit of their Health Indicator data. Using these estimated parameters, they are interconnected to a subset of collaborating assets by means of a similarity metric. We show how by simply interchanging their estimates, networked assets are able to precisely determine their Health Indicator dynamics and reduce maintenance costs. This is done in real time, with no centralised library, and without the need for extensive historical data. We compare Social Asset Networks with the typical self-learning and fleet-wide approaches, and show that Social Asset Networks have a faster convergence and lower cost. This study serves as a conceptual proof for the potential of collaborative prognostics for solving maintenance problems, and can be used to justify the implementation of such a system in a real industrial fleet.