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Abstract In this work, we introduce a general model for a collection of innovation processes in order to model and analyze the interaction among them. We provide theoretical results, analytically proven, and we show how the proposed model fits the behaviors observed in some real data sets (from Reddit and Gutenberg). It is worth mentioning that the given applications are only examples of the potentialities of the proposed model and related results: due to its abstractness and generality, it can be applied to many interacting innovation processes.

Abstract In the existing literature about innovation processes, the proposed models often satisfy the Heaps’ law, regarding the rate at which novelties appear, and the Zipf’s law, that states a power law behavior for the frequency distribution of the elements. However, there are empirical cases far from showing a pure power law behavior and such a deviation is mostly present for elements with high frequencies. We explain this phenomenon by means of a suitable “damping” effect in the probability of a repetition of an old element. We introduce an extremely general model, whose key element is the update function, that can be suitably chosen in order to reproduce the behaviour exhibited by the empirical data. In particular, we explicit the update function for some Twitter data sets and show great performances with respect to Heaps’ law and, above all, with respect to the fitting of the frequency-rank plots for low and high frequencies. Moreover, we also give other examples of update functions, that are able to reproduce the behaviors empirically observed in other contexts.

In neuroscience, the structural connectivity matrix of synaptic weights between neurons is one of the critical factors that determine the overall function of a network of neurons. The mechanisms of signal transduction have been intensively studied at different time and spatial scales and both the cellular and molecular levels. While a better understanding and knowledge of some basic processes of information handling by neurons has been achieved, little is known about the organization and function of complex neuronal networks. Experimental methods are now available to simultaneously monitor the electrical activity of a large number of neurons in real time. The analysis of the data related to the activities of individual neurons can become a very valuable tool for the study of the dynamics and architecture of neural networks. In particular, advances in optical imaging techniques allow us to record up to thousands of neurons nowadays. However, most of the efforts have been focused on calcium signals, that lack relevant aspects of cell activity. In recent years, progresses in the field of genetically encoded voltage indicators have shown that imaging signals could be well suited to record spiking and synaptic events from a large population of neurons. Here, we present a methodology to infer the connectivity of a population of neurons from their voltage traces. At first, putative synaptic events were detected. Then, a multi-class logistic regression was used to fit the putative events to the spiking activities and a penalization term was allowed to regulate the sparseness of the inferred network. The proposed Multi-Class Logistic Regression with L1 penalization (MCLRL) was benchmarked against data obtained from in silico network simulations. MCLRL properly inferred the connectivity of all tested networks, as indicated by the Matthew correlation coefficient (MCC). Importantly, MCLRL was accomplished to reconstruct the connectivity among subgroups of neurons sampled from the network. The robustness of MCLRL to noise was also assessed and the performances remained high (MCC>0.95) even in extremely high noise conditions (>95% noisy events). Finally, we devised a procedure to determine the optimal MCLRL regularization term, which allows us to envision its application to experimental data.

Twitter is among the most used online platforms for the political communications, due to the concision of its messages (which is particularly suitable for political slogans) and the quick diffusion of messages. Especially when the argument stimulate the emotionality of users, the content on Twitter is shared with extreme speed and thus studying the tweet sentiment if of utmost importance to predict the evolution of the discussions and the register of the relative narratives. In this article, we present a model able to reproduce the dynamics of the sentiments of tweets related to specific topics and periods and to provide a prediction of the sentiment of the future posts based on the observed past. The model is a recent variant of the Pólya urn, introduced and studied in Aletti and Crimaldi (2019, 2020), which is characterized by a "local" reinforcement, i.e. a reinforcement mechanism mainly based on the most recent observations, and by a random persistent fluctuation of the predictive mean. In particular, this latter feature is capable of capturing the trend fluctuations in the sentiment curve. While the proposed model is extremely general and may be also employed in other contexts, it has been tested on several Twitter data sets and demonstrated greater performances compared to the standard Pólya urn model. Moreover, the different performances on different data sets highlight different emotional sensitivities respect to a public event.

The development of the hyperspectral remote sensor technology allows the acquisition of images with a very detailed spectral information for each pixel. Because of this, hyperspectral images (HSI) potentially possess larger capabilities in solving many scientific and practical problems in agriculture, biomedical, ecological, geological, hydrological studies. However, their analysis requires developing specialized and fast algorithms for data processing, due the high dimensionality of the data. In this work, we propose a new semi-supervised method for multilabel segmentation of HSI that combines a suitable linear discriminant analysis, a similarity index to compare different spectra, and a random walk based model with a direct label assignment. The user-marked regions are used for the projection of the original high-dimensional feature space to a lower dimensional space, such that the class separation is maximized. This allows to retain in an automatic way the most informative features, lightening the successive computational burden. The part of the random walk is related to a combinatorial Dirichlet problem involving a weighted graph, where the nodes are the projected pixel of the original HSI, and the positive weights depend on the distances between these nodes. We then assign to each pixel of the original image a probability quantifying the likelihood that the pixel (node) belongs to some subregion. The computation of the spectral distance involves both the coordinates in a features space of a pixel and of its neighbors. The final segmentation process is therefore reduced to a suitable optimization problem coupling the probabilities from the random walker computation, and the similarity with respect the initially labeled pixels. We discuss the properties of the new method with experimental results carried on benchmark images.

Abstract Common clustering algorithms require multiple scans of all the data to achieve convergence, and this is prohibitive when large databases, with data arriving in streams, must be processed. Some algorithms to extend the popular K-means method to the analysis of streaming data are present in literature since 1998 (Bradley et al. in Scaling clustering algorithms to large databases. In: KDD. p. 9–15, 1998; O’Callaghan et al. in Streaming-data algorithms for high-quality clustering. In: Proceedings of IEEE international conference on data engineering. p. 685, 2001), based on the memorization and recursive update of a small number of summary statistics, but they either don’t take into account the specific variability of the clusters, or assume that the random vectors which are processed and grouped have uncorrelated components. Unfortunately this is not the case in many practical situations. We here propose a new algorithm to process data streams, with data having correlated components and coming from clusters with different covariance matrices. Such covariance matrices are estimated via an optimal double shrinkage method, which provides positive definite estimates even in presence of a few data points, or of data having components with small variance. This is needed to invert the matrices and compute the Mahalanobis distances that we use for the data assignment to the clusters. We also estimate the total number of clusters from the data.

In recent papers the authors introduce, study and apply a variant of the Eggenberger—Pólya urn, called the “rescaled” Pólya urn, which, for a suitable choice of the model parameters, exhibits a reinforcement mechanism mainly based on the last observations, a random persistent fluctuation of the predictive mean and the almost sure convergence of the empirical mean to a deterministic limit. In this work, motivated by some empirical evidence, we show that the multidimensional Wright—Fisher diffusion with mutation can be obtained as a suitable limit of the predictive means associated to a family of rescaled Pólya urns.

Networks and graphs offer a suitable and powerful framework for studying the spread of infection in human and animal populations. In the case of a heterogeneous population, the social contact network has a pivotal role in the analysis of directly transmitted infectious diseases. The literature presents several works where network-based models encompass realistic features (such as contacts networks or host–pathogen biological data), but analytical results are nonetheless scarce. As a significant example, in this paper, we develop a multi-group version of the epidemiological SEIR population-based model. Each group can represent a social subpopulation with the same habits or a group of geographically localized people. We consider also heterogeneity in the weighting of contacts between two groups. As a simple application, we propose a simple control algorithm in which we optimize the connection weights in order to minimize the combination between an economic cost and a social cost. Some numerical simulations are also provided.

Image segmentation is an essential but critical component in low level vision, image analysis, pattern recognition, and now in robotic systems. In addition, it is one of the most challenging tasks in image processing and determines the quality of the final results of the image analysis. Colour based segmentation could hence offer more significant extraction of information as compared to intensity or texture based segmentation. In this work, we propose a new local or global method for multi-label segmentation that combines a random walk based model with a direct label assignment computed using a suitable colour distance. Our approach is a semi-automatic image segmentation technique, since it requires user interaction for the initialisation of the segmentation process. The random walk part involves a combinatorial Dirichlet problem for a weighted graph, where the nodes are the pixel of the image, and the positive weights are related to the distances between pixels: in this work we propose a novel colour distance for computing such weights. In the random walker model we assign to each pixel of the image a probability quantifying the likelihood that the node belongs to some subregion. The computation of the colour distance is pursued by employing the coordinates in a colour space (e.g., RGB, XYZ, YCbCr) of a pixel and of the ones in its neighbourhood (e.g., in a 8–neighbourhood). The segmentation process is, therefore, reduced to an optimisation problem coupling the probabilities from the random walker approach, and the similarity with respect the labelled pixels. A further investigation involves an adaptive preprocess strategy using a regression tree for learning suitable weights to be used in the computation of the colour distance. We discuss the properties of the new method also by comparing with standard random walk and k−means approaches. The experimental results carried on the White Blood Cell (WBC) dataset and GrabCut datasets show the remarkable performance of the proposed method in comparison with state-of-the-art methods, such as normalised random walk and normalised lazy random walk, with respect to segmentation quality and computational time. Moreover, it reveals to be very robust with respect to the presence of noise and to the choice of the colourspace.

Taylor’s law quantifies the scaling properties of the fluctuations of the number of innovations occurring in open systems. Urn-based modeling schemes have already proven to be effective in modeling this complex behaviour. Here, we present analytical estimations of Taylor’s law exponents in such models, by leveraging on their representation in terms of triangular urn models. We also highlight the correspondence of these models with Poisson–Dirichlet processes and demonstrate how a non-trivial Taylor’s law exponent is a kind of universal feature in systems related to human activities. We base this result on the analysis of four collections of data generated by human activity: (i) written language (from a Gutenberg corpus); (ii) an online music website (Last.fm); (iii) Twitter hashtags; (iv) an online collaborative tagging system (Del.icio.us). While Taylor’s law observed in the last two datasets agrees with the plain model predictions, we need to introduce a generalization to fully characterize the behaviour of the first two datasets, where temporal correlations are possibly more relevant. We suggest that Taylor’s law is a fundamental complement to Zipf’s and Heaps’ laws in unveiling the complex dynamical processes underlying the evolution of systems featuring innovation.