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Networks play an ubiquitous role in computer science and real-world applications, offering multiple kind of information that can be retrieved with adequate methods. With the continuous growing in the amount of data available, networks are becoming larger day by day. Consequently, the tasks that were easily achievable on smaller networks, often becomes impractical on huge amount of data, either due to the high computational cost or due to the impracticality to visualise corresponding data. Using distinctive node features to group large amount of connected data into a limited number of clusters, hence represented by a representative per cluster, proves to be a valuable approach. The resulting contracted graphs are more manageable in size and can reveal previously hidden characteristics of the original networks. Furthermore, in many real-world use cases, a definition of cluster is intrinsic with the data, eventually obtained with the injection of some expert knowledge represent by a categorical function. Clusters then results in set of connected vertices taking the same values in a finite set C. In the recent literature, Lombardi and Onofri proposed a novel, fast, and easily parallelisable approach under the name of $\gamma$-contraction to contract a graph given a categorical function. In this work, we formally define such approach by providing a rigorous mathematical definition of the problem, which, to the best of our knowledge, was missing in the existing literature. Specifically, we explore the variadic nature of the contraction operation and use it to introduce the weaker version of the colour contraction, under the name of $\beta$-contraction, that the algorithmic solution exploits. We finally dive into the details of the algorithm and we provide a full assesment on its convergence complexity relying on two constructive proofs that deeply unveil its mode of operation.

The growth potential of a crypto(currency) project can be measured by the use cases spurred by the underlying technology. However, these projects are usually distributed, with a weak feedback schemes. Hence, a metric that is widely used as a proxy for their healthiness is the number of transactions and related volumes. Nevertheless, such a metric can be subject to manipulation (the crypto market being an unregulated one magnifies such a risk). To address the cited gap we design a comprehensive methodology to process large cryptocurrency transaction graphs that, after clustering user addresses of interest, derives a compact representation of the network that highlights clusters interactions. To show the viability of our solution, we bring forward a use case centered on Polkadot, which has gained significant attention in the digital currency landscape due to its pioneering approach to interoperability and scalability. However, little is known about how many and to what extent its wide range of enabled use cases have been adopted by end-users so far. The answer to this type of question means mapping Polkadot (or any analyzed crypto project) on a palette that ranges from a thriving ecosystem to a speculative coin without compelling use cases. Our findings demonstrate that crypto exchanges exert considerable influence on the Polkadot network, owning nearly 40% of all addresses in the ledger and absorbing at least 80% of all transactions. In addition, the high volume of inter-exchange transactions (> 20%) underscores the strong interconnections among just a couple of prominent exchanges, prompting further investigations into the behavior of these actors to uncover potential unethical activities, such as wash trading. These results, while characterized by a high level of scalability and adaptability, are at the same time immune from the drawbacks of currently used metrics.

In this paper, we aim at developing new methods to join machine learning techniques and macroscopic differential models for vehicular traffic estimation and forecast. It is well known that data-driven and model-driven approaches have (sometimes complementary) advantages and drawbacks. We consider here a dataset with flux and velocity data of vehicles moving on a highway, collected by fixed sensors and classified by lane and by class of vehicle. By means of a machine learning model based on an LSTM recursive neural network, we extrapolate two important pieces of information: 1) if congestion is appearing under the sensor, and 2) the total amount of vehicles which is going to pass under the sensor in the next future (30 min). These pieces of information are then used to improve the accuracy of an LWR-based first-order multi-class model describing the dynamics of traffic flow between sensors. The first piece of information is used to invert the (concave) fundamental diagram, thus recovering the density of vehicles from the flux data, and then inject directly the density datum in the model. This allows one to better approximate the dynamics between sensors, especially if an accident happens in a not monitored stretch of the road. The second piece of information is used instead as boundary conditions for the equations underlying the traffic model, to better reconstruct the total amount of vehicles on the road at any future time. Some examples motivated by real scenarios will be discussed. Real data are provided by the Italian motorway company Autovie Venete S.p.A., Comment: 38 pages

Individual tracking of museum visitors based on portable radio beacons, an asset for behavioural analyses and comfort/performance improvements, is seeing increasing diffusion. Conceptually, this approach enables room-level localisation based on a network of small antennas (thus, without invasive modification of the existent structures). The antennas measure the intensity (RSSi) of self-advertising signals broadcasted by beacons individually assigned to the visitors. The signal intensity provides a proxy for the distance to the antennas and thus indicative positioning. However, RSSi signals are well-known to be noisy, even in ideal conditions (high antenna density, absence of obstacles, absence of crowd, ...). In this contribution, we present a method to perform accurate RSSi-based visitor tracking when the density of antennas is relatively low, e.g. due to technical constraints imposed by historic buildings. We combine an ensemble of "simple" localisers, trained based on ground-truth, with an encoding of the museum topology in terms of a total-coloured graph. This turns the localisation problem into a cascade process, from large to small scales, in space and in time. Our use case is visitors tracking in Galleria Borghese, Rome (Italy), for which our method manages >96% localisation accuracy, significantly improving on our previous work (J. Comput. Sci. 101357, 2021).

We present an all-around study of the visitors flow in crowded museums: a combination of Lagrangian field measurements and statistical analyses enable us to create stochastic digital-twins of the guests dynamics, unlocking comfort- and safety-driven optimizations. Our case study is the Galleria Borghese museum in Rome (Italy), in which we performed a real-life data acquisition campaign. We specifically employ a Lagrangian IoT-based visitor tracking system based on Raspberry Pi receivers, displaced in fixed positions throughout the museum rooms, and on portable Bluetooth Low Energy beacons handed over to the visitors. Thanks to two algorithms: a sliding window-based statistical analysis and an MLP neural network, we filter the beacons RSSI and accurately reconstruct visitor trajectories at room-scale. Via a clustering analysis, hinged on an original Wasserstein-like trajectory-space metric, we analyze the visitor paths to get behavioral insights, including the most common flow patterns. On these bases, we build the transition matrix describing, in probability, the room-scale visitor flows. Such a matrix is the cornerstone of a stochastic model capable of generating visitor trajectories in silico. We conclude by employing the simulator to increase the number of daily visitors while respecting numerous logistic and safety constraints. This is possible thanks to optimized ticketing and new entrance/exit management.

The object of this paper is to transform a set of local chain complexes to a single global complex using an equivalence relation of congruence of cells, solving topologically the numerical inaccuracies of floating-point arithmetics. While computing the space arrangement generated by a collection of cellular complexes, one may start from independently and efficiently computing the intersection of each single input 2-cell with the others. The topology of these intersections is codified within a set of (0-2)-dimensional chain complexes. The target of this paper is to merge the local chains by using the equivalence relations of {\epsilon}-congruence between 0-, 1-, and 2-cells (elementary chains). In particular, we reduce the block-diagonal coboundary matrices [\Delta_0] and [\Delta_1], used as matrix accumulators of the local coboundary chains, to the global matrices [\delta_0] and [\delta_1], representative of congruence topology, i.e., of congruence quotients between all 0-,1-,2-cells, via elementary algebraic operations on their columns. This algorithm is codified using the Julia porting of the SuiteSparse:GraphBLAS implementation of the GraphBLAS standard, conceived to efficiently compute algorithms on large graphs using linear algebra and sparse matrices [1, 2]., Comment: to submit

We tackle the issue of measuring and analyzing the visitors' dynamics in crowded museums. We propose an IoT-based system -- supported by artificial intelligence models -- to reconstruct the visitors' trajectories throughout the museum spaces. Thanks to this tool, we are able to gather wide ensembles of visitors' trajectories, allowing useful insights for the facility management and the preservation of the art pieces. Our contribution comes with one successful use case: the Galleria Borghese in Rome, Italy., Comment: 6 pages, 11 figures, International Conference on Metrology for Archaeology and Cultural Heritage

The goal of this paper is to introduce a new method in computer-aided geometry of solid modeling. We put forth a novel algebraic technique to evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with regularized operators of union, intersection, and difference, i.e., any CSG tree. The result is obtained in three steps: first, by computing an independent set of generators for the d-space partition induced by the input; then, by reducing the solid expression to an equivalent logical formula between Boolean terms made by zeros and ones; and, finally, by evaluating this expression using bitwise operators. This method is implemented in Julia using sparse arrays. The computational evaluation of every possible solid expression, usually denoted as CSG (Constructive Solid Geometry), is reduced to an equivalent logical expression of a finite set algebra over the cells of a space partition, and solved by native bitwise operators., Comment: revised version submitted to Computer-Aided Geometric Design