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A recurrent neural network (RNN) is a universal approximator of dynamical systems, whose performance often depends on sensitive hyperparameters. Tuning of such hyperparameters may be difficult and, typically, based on a trial-and-error approach. In this work, we adopt a graph-based framework to interpret and characterize the internal RNN dynamics. Through this insight, we are able to design a principled unsupervised method to derive configurations with maximized performances, in terms of prediction error and memory capacity. In particular, we propose to model time series of neurons activations with the recently introduced horizontal visibility graphs, whose topological properties reflect important dynamical features of the underlying dynamic system. Successively, each graph becomes a layer of a larger structure, called multiplex. We show that topological properties of such a multiplex reflect important features of RNN dynamics and are used to guide the tuning procedure. To validate the proposed method, we consider a class of RNNs called echo state networks. We perform experiments and discuss results on several benchmarks and real-world dataset of call data records.

We investigate the scaling of the cross-correlations calculated for two-variable time series containing vertex properties in the context of complex networks. Time series of such observables are obtained by means of stationary, unbiased random walks. We consider three vertex properties that provide, respectively, short, medium, and long-range information regarding the topological role of vertices in a given network. In order to reveal the relation between these quantities, we applied the multifractal cross-correlation analysis technique, which provides information about the nonlinear effects in coupling of time series. We show that the considered network models are characterized by unique multifractal properties of the cross-correlation. In particular, it is possible to distinguish between Erd\"{o}s-R\'{e}nyi, Barab\'{a}si-Albert, and Watts-Strogatz networks on the basis of fractal cross-correlation. Moreover, the analysis of protein contact networks reveals characteristics shared with both scale-free and small-world models., Comment: Revised version. Previous title "Exploring complex networks by means of two-variable time series of vertex observables"

One-class classifiers offer valuable tools to assess the presence of outliers in data. In this paper, we propose a design methodology for one-class classifiers based on entropic spanning graphs. Our approach takes into account the possibility to process also non-numeric data by means of an embedding procedure. The spanning graph is learned on the embedded input data and the outcoming partition of vertices defines the classifier. The final partition is derived by exploiting a criterion based on mutual information minimization. Here, we compute the mutual information by using a convenient formulation provided in terms of the $\alpha$-Jensen difference. Once training is completed, in order to associate a confidence level with the classifier decision, a graph-based fuzzy model is constructed. The fuzzification process is based only on topological information of the vertices of the entropic spanning graph. As such, the proposed one-class classifier is suitable also for data characterized by complex geometric structures. We provide experiments on well-known benchmarks containing both feature vectors and labeled graphs. In addition, we apply the method to the protein solubility recognition problem by considering several representations for the input samples. Experimental results demonstrate the effectiveness and versatility of the proposed method with respect to other state-of-the-art approaches., Comment: Extended and revised version of the paper "One-Class Classification Through Mutual Information Minimization" presented at the 2016 IEEE IJCNN, Vancouver, Canada

It is a widely accepted fact that the computational capability of recurrent neural networks is maximized on the so-called "edge of criticality". Once the network operates in this configuration, it performs efficiently on a specific application both in terms of (i) low prediction error and (ii) high short-term memory capacity. Since the behavior of recurrent networks is strongly influenced by the particular input signal driving the dynamics, a universal, application-independent method for determining the edge of criticality is still missing. In this paper, we aim at addressing this issue by proposing a theoretically motivated, unsupervised method based on Fisher information for determining the edge of criticality in recurrent neural networks. It is proven that Fisher information is maximized for (finite-size) systems operating in such critical regions. However, Fisher information is notoriously difficult to compute and either requires the probability density function or the conditional dependence of the system states with respect to the model parameters. The paper takes advantage of a recently-developed non-parametric estimator of the Fisher information matrix and provides a method to determine the critical region of echo state networks, a particular class of recurrent networks. The considered control parameters, which indirectly affect the echo state network performance, are explored to identify those configurations lying on the edge of criticality and, as such, maximizing Fisher information and computational performance. Experimental results on benchmarks and real-world data demonstrate the effectiveness of the proposed method.

In this paper, we elaborate over the well-known interpretability issue in echo state networks. The idea is to investigate the dynamics of reservoir neurons with time-series analysis techniques taken from research on complex systems. Notably, we analyze time-series of neuron activations with Recurrence Plots (RPs) and Recurrence Quantification Analysis (RQA), which permit to visualize and characterize high-dimensional dynamical systems. We show that this approach is useful in a number of ways. First, the two-dimensional representation offered by RPs provides a way for visualizing the high-dimensional dynamics of a reservoir. Our results suggest that, if the network is stable, reservoir and input denote similar line patterns in the respective RPs. Conversely, the more unstable the ESN, the more the RP of the reservoir presents instability patterns. As a second result, we show that the $\mathrm{L_{max}}$ measure is highly correlated with the well-established maximal local Lyapunov exponent. This suggests that complexity measures based on RP diagonal lines distribution provide a valuable tool to quantify the degree of network stability. Finally, our analysis shows that all RQA measures fluctuate on the proximity of the so-called edge of stability, where an ESN typically achieves maximum computational capability. We verify that the determination of the edge of stability provided by such RQA measures is more accurate than two well-known criteria based on the Jacobian matrix of the reservoir. Therefore, we claim that RPs and RQA-based analyses can be used as valuable tools to design an effective network given a specific problem., Comment: Revised version. 24 pages; 12 figures

In this paper, we propose a novel data-driven approach for removing trends (detrending) from nonstationary, fractal and multifractal time series. We consider real-valued time series relative to measurements of an underlying dynamical system that evolves through time. We assume that such a dynamical process is predictable to a certain degree by means of a class of recurrent networks called Echo State Network (ESN), which are capable to model a generic dynamical process. In order to isolate the superimposed (multi)fractal component of interest, we define a data-driven filter by leveraging on the ESN prediction capability to identify the trend component of a given input time series. Specifically, the (estimated) trend is removed from the original time series and the residual signal is analyzed with the multifractal detrended fluctuation analysis procedure to verify the correctness of the detrending procedure. In order to demonstrate the effectiveness of the proposed technique, we consider several synthetic time series consisting of different types of trends and fractal noise components with known characteristics. We also process a real-world dataset, the sunspot time series, which is well-known for its multifractal features and has recently gained attention in the complex systems field. Results demonstrate the validity and generality of the proposed detrending method based on ESNs., Comment: Revised version

In this paper, we analyze 48 signals of rest tremor velocity related to 12 distinct subjects affected by Parkinson's disease. The subjects belong to two different groups, formed by four and eight subjects with, respectively, high- and low-amplitude rest tremors. Each subject is tested in four settings, given by combining the use of deep brain stimulation and L-DOPA medication. We develop two main feature-based representations of such signals, which are obtained by considering (i) the long-term correlations and multifractal properties, and (ii) the power spectra. The feature-based representations are initially utilized for the purpose of characterizing the subjects under different settings. In agreement with previous studies, we show that deep brain stimulation does not significantly characterize neither of the two groups, regardless of the adopted representation. On the other hand, the medication effect yields statistically significant differences in both high- and low-amplitude tremor groups. We successively test several different instances of the two feature-based representations of the signals in the setting of supervised classification and (nonlinear) feature transformation. We consider three different classification problems, involving the recognition of (i) the presence of medication, (ii) the use of deep brain stimulation, and (iii) the membership to the high- and low-amplitude tremor groups. Classification results show that the use of medication can be discriminated with higher accuracy, considering many of the feature-based representations. Notably, we show that the best results are obtained with a parsimonious, two-dimensional representation encoding the long-term correlations and multifractal character of the signals., Comment: 10 pages, 41 references

In this paper, we study long-term correlations and multifractal properties elaborated from time series of three-phase current signals coming from an industrial electric arc furnace plant. Implicit sinusoidal trends are suitably detected by considering the scaling of the fluctuation functions. Time series are then filtered via a Fourier-based analysis, removing hence such strong periodicities. In the filtered time series we detected long-term, positive correlations. The presence of positive correlations is in agreement with the typical V--I characteristic (hysteresis) of the electric arc furnace, providing thus a sound physical justification for the memory effects found in the current time series. The multifractal signature is strong enough in the filtered time series to be effectively classified as multifractal.

In this paper we present a generative model for protein contact networks. The soundness of the proposed model is investigated by focusing primarily on mesoscopic properties elaborated from the spectra of the graph Laplacian. To complement the analysis, we study also classical topological descriptors, such as statistics of the shortest paths and the important feature of modularity. Our experiments show that the proposed model results in a considerable improvement with respect to two suitably chosen generative mechanisms, mimicking with better approximation real protein contact networks in terms of diffusion properties elaborated from the Laplacian spectra. However, as well as the other considered models, it does not reproduce with sufficient accuracy the shortest paths structure. To compensate this drawback, we designed a second step involving a targeted edge reconfiguration process. The ensemble of reconfigured networks denotes improvements that are statistically significant. As a byproduct of our study, we demonstrate that modularity, a well-known property of proteins, does not entirely explain the actual network architecture characterizing protein contact networks. In fact, we conclude that modularity, intended as a quantification of an underlying community structure, should be considered as an emergent property of the structural organization of proteins. Interestingly, such a property is suitably optimized in protein contact networks together with the feature of path efficiency., Comment: 18 pages, 67 references

Power losses reduction is one of the main targets for any electrical energy distribution company. In this paper, we face the problem of joint optimization of both topology and network parameters in a real smart grid. We consider a portion of the Italian electric distribution network managed by the ACEA Distribuzione S.p.A. located in Rome. We perform both the power factor correction (PFC) for tuning the generators and the distributed feeder reconfiguration (DFR) to set the state of the breakers. This joint optimization problem is faced considering a suitable objective function and by adopting genetic algorithms as global optimization strategy. We analyze admissible network configurations, showing that some of these violate constraints on current and voltage at branches and nodes. Such violations depend only on pure topological properties of the configurations. We perform tests by feeding the simulation environment with real data concerning hourly samples of dissipated and generated active and reactive power values of the ACEA smart grid. Results show that removing the configurations violating the electrical constraints from the solution space leads to interesting improvements in terms of power loss reduction. To conclude, we provide also an electrical interpretation of the phenomenon using graph-based pattern analysis techniques., Comment: 35 pages, 38 references

This paper presents the SPARE C++ library, an open source software tool conceived to build pattern recognition and soft computing systems. The library follows the requirement of the generality: most of the implemented algorithms are able to process user-defined input data types transparently, such as labeled graphs and sequences of objects, as well as standard numeric vectors. Here we present a high-level picture of the SPARE library characteristics, focusing instead on the specific practical possibility of constructing pattern recognition systems for different input data types. In particular, as a proof of concept, we discuss two application instances involving clustering of real-valued multidimensional sequences and classification of labeled graphs., Comment: Home page: https://sourceforge.net/p/libspare/home/Spare/

The multifractal detrended fluctuation analysis of time series is able to reveal the presence of long-range correlations and, at the same time, to characterize the self-similarity of the series. The rich information derivable from the characteristic exponents and the multifractal spectrum can be further analyzed to discover important insights about the underlying dynamical process. In this paper, we employ multifractal analysis techniques in the study of protein contact networks. To this end, initially a network is mapped to three different time series, each of which is generated by a stationary unbiased random walk. To capture the peculiarities of the networks at different levels, we accordingly consider three observables at each vertex: the degree, the clustering coefficient, and the closeness centrality. To compare the results with suitable references, we consider also instances of three well-known network models and two typical time series with pure monofractal and multifractal properties. The first result of notable interest is that time series associated to proteins contact networks exhibit long-range correlations (strong persistence), which are consistent with signals in-between the typical monofractal and multifractal behavior. Successively, a suitable embedding of the multifractal spectra allows to focus on ensemble properties, which in turn gives us the possibility to make further observations regarding the considered networks. In particular, we highlight the different role that small and large fluctuations of the considered observables play in the characterization of the network topology.

We propose a multi-agent algorithm able to automatically discover relevant regularities in a given dataset, determining at the same time the set of configurations of the adopted parametric dissimilarity measure yielding compact and separated clusters. Each agent operates independently by performing a Markovian random walk on a suitable weighted graph representation of the input dataset. Such a weighted graph representation is induced by the specific parameter configuration of the dissimilarity measure adopted by the agent, which searches and takes decisions autonomously for one cluster at a time. Results show that the algorithm is able to discover parameter configurations that yield a consistent and interpretable collection of clusters. Moreover, we demonstrate that our algorithm shows comparable performances with other similar state-of-the-art algorithms when facing specific clustering problems.

In this paper, we study the structure and dynamical properties of protein contact networks with respect to other biological networks, together with simulated archetypal models acting as probes. We consider both classical topological descriptors, such as the modularity and statistics of the shortest paths, and different interpretations in terms of diffusion provided by the discrete heat kernel, which is elaborated from the normalized graph Laplacians. A principal component analysis shows high discrimination among the network types, either by considering the topological and heat kernel based vector characterizations. Furthermore, a canonical correlation analysis demonstrates the strong agreement among those two characterizations, providing thus an important justification in terms of interpretability for the heat kernel. Finally, and most importantly, the focused analysis of the heat kernel provides a way to yield insights on the fact that proteins have to satisfy specific structural design constraints that the other considered networks do not need to obey. Notably, the heat trace decay of an ensemble of varying-size proteins denotes subdiffusion, a peculiar property of proteins.

Representing patterns as labeled graphs is becoming increasingly common in the broad field of computational intelligence. Accordingly, a wide repertoire of pattern recognition tools, such as classifiers and knowledge discovery procedures, are nowadays available and tested for various datasets of labeled graphs. However, the design of effective learning procedures operating in the space of labeled graphs is still a challenging problem, especially from the computational complexity viewpoint. In this paper, we present a major improvement of a general-purpose classifier for graphs, which is conceived on an interplay between dissimilarity representation, clustering, information-theoretic techniques, and evolutionary optimization algorithms. The improvement focuses on a specific key subroutine devised to compress the input data. We prove different theorems which are fundamental to the setting of the parameters controlling such a compression operation. We demonstrate the effectiveness of the resulting classifier by benchmarking the developed variants on well-known datasets of labeled graphs, considering as distinct performance indicators the classification accuracy, computing time, and parsimony in terms of structural complexity of the synthesized classification models. The results show state-of-the-art standards in terms of test set accuracy and a considerable speed-up for what concerns the computing time., Comment: Revised version

We evaluate a version of the recently-proposed classification system named Optimized Dissimilarity Space Embedding (ODSE) that operates in the input space of sequences of generic objects. The ODSE system has been originally presented as a classification system for patterns represented as labeled graphs. However, since ODSE is founded on the dissimilarity space representation of the input data, the classifier can be easily adapted to any input domain where it is possible to define a meaningful dissimilarity measure. Here we demonstrate the effectiveness of the ODSE classifier for sequences by considering an application dealing with the recognition of the solubility degree of the Escherichia coli proteome. Solubility, or analogously aggregation propensity, is an important property of protein molecules, which is intimately related to the mechanisms underlying the chemico-physical process of folding. Each protein of our dataset is initially associated with a solubility degree and it is represented as a sequence of symbols, denoting the 20 amino acid residues. The herein obtained computational results, which we stress that have been achieved with no context-dependent tuning of the ODSE system, confirm the validity and generality of the ODSE-based approach for structured data classification., Comment: 10 pages, 49 references

This paper deals with the relations among structural, topological, and chemical properties of the E.Coli proteome from the vantage point of the solubility/aggregation propensity of proteins. Each E.Coli protein is initially represented according to its known folded 3D shape. This step consists in representing the available E.Coli proteins in terms of graphs. We first analyze those graphs by considering pure topological characterizations, i.e., by analyzing the mass fractal dimension and the distribution underlying both shortest paths and vertex degrees. Results confirm the general architectural principles of proteins. Successively, we focus on the statistical properties of a representation of such graphs in terms of vectors composed of several numerical features, which we extracted from their structural representation. We found that protein size is the main discriminator for the solubility, while however there are other factors that help explaining the solubility degree. We finally analyze such data through a novel one-class classifier, with the aim of discriminating among very and poorly soluble proteins. Results are encouraging and consolidate the potential of pattern recognition techniques when employed to describe complex biological systems., Comment: To appear in Current Bioinformatics, Bentham Science

The one-class classification problem is a well-known research endeavor in pattern recognition. The problem is also known under different names, such as outlier and novelty/anomaly detection. The core of the problem consists in modeling and recognizing patterns belonging only to a so-called target class. All other patterns are termed non-target, and therefore they should be recognized as such. In this paper, we propose a novel one-class classification system that is based on an interplay of different techniques. Primarily, we follow a dissimilarity representation based approach; we embed the input data into the dissimilarity space by means of an appropriate parametric dissimilarity measure. This step allows us to process virtually any type of data. The dissimilarity vectors are then represented through a weighted Euclidean graphs, which we use to (i) determine the entropy of the data distribution in the dissimilarity space, and at the same time (ii) derive effective decision regions that are modeled as clusters of vertices. Since the dissimilarity measure for the input data is parametric, we optimize its parameters by means of a global optimization scheme, which considers both mesoscopic and structural characteristics of the data represented through the graphs. The proposed one-class classifier is designed to provide both hard (Boolean) and soft decisions about the recognition of test patterns, allowing an accurate description of the classification process. We evaluate the performance of the system on different benchmarking datasets, containing either feature-based or structured patterns. Experimental results demonstrate the effectiveness of the proposed technique., Comment: To appear in IEEE-TNNLS

This paper builds upon the fundamental work of Niwa et al. [34], which provides the unique possibility to analyze the relative aggregation/folding propensity of the elements of the entire Escherichia coli (E. coli) proteome in a cell-free standardized microenvironment. The hardness of the problem comes from the superposition between the driving forces of intra- and inter-molecule interactions and it is mirrored by the evidences of shift from folding to aggregation phenotypes by single-point mutations [10]. Here we apply several state-of-the-art classification methods coming from the field of structural pattern recognition, with the aim to compare different representations of the same proteins gathered from the Niwa et al. data base; such representations include sequences and labeled (contact) graphs enriched with chemico-physical attributes. By this comparison, we are able to identify also some interesting general properties of proteins. Notably, (i) we suggest a threshold around 250 residues discriminating "easily foldable" from "hardly foldable" molecules consistent with other independent experiments, and (ii) we highlight the relevance of contact graph spectra for folding behavior discrimination and characterization of the E. coli solubility data. The soundness of the experimental results presented in this paper is proved by the statistically relevant relationships discovered among the chemico-physical description of proteins and the developed cost matrix of substitution used in the various discrimination systems., Comment: 17 pages, 3 figures, 46 references

Researches in granular modeling produced a variety of mathematical models, such as intervals, (higher-order) fuzzy sets, rough sets, and shadowed sets, which are all suitable to characterize the so-called information granules. Modeling of the input data uncertainty is recognized as a crucial aspect in information granulation. Moreover, the uncertainty is a well-studied concept in many mathematical settings, such as those of probability theory, fuzzy set theory, and possibility theory. This fact suggests that an appropriate quantification of the uncertainty expressed by the information granule model could be used to define an invariant property, to be exploited in practical situations of information granulation. In this perspective, a procedure of information granulation is effective if the uncertainty conveyed by the synthesized information granule is in a monotonically increasing relation with the uncertainty of the input data. In this paper, we present a data granulation framework that elaborates over the principles of uncertainty introduced by Klir. Being the uncertainty a mesoscopic descriptor of systems and data, it is possible to apply such principles regardless of the input data type and the specific mathematical setting adopted for the information granules. The proposed framework is conceived (i) to offer a guideline for the synthesis of information granules and (ii) to build a groundwork to compare and quantitatively judge over different data granulation procedures. To provide a suitable case study, we introduce a new data granulation technique based on the minimum sum of distances, which is designed to generate type-2 fuzzy sets. We analyze the procedure by performing different experiments on two distinct data types: feature vectors and labeled graphs. Results show that the uncertainty of the input data is suitably conveyed by the generated type-2 fuzzy set models., Comment: 16 pages, 9 figures, 52 references