Critical phenomena in complex networks, and the emergence of dynamical abrupt transitions in the macroscopic state of the system are currently a subject of the outmost interest. We report evidence of an explosive phase synchronization in networks of chaotic units. Namely, by means of both extensive simulations of networks made up of chaotic units, and validation with an experiment of electronic circuits in a star configuration, we demonstrate the existence of a first-order transition towards synchronization of the phases of the networked units. Our findings constitute the first prove of this kind of synchronization in practice, thus opening the path to its use in real-world applications.

Trade is a fundamental pillar of economy and a form of social organization. Its empirical characterization at the worldwide scale is represented by the World Trade Web (WTW), the network built upon the trade relationships between the different countries. Several scientific studies have focused on the structural characterization of this network, as well as its dynamical properties, since we have registry of the structure of the network at different times in history. In this paper we study an abstract scenario for the development of global crises on top of the structure of connections of the WTW. Assuming a cyclic dynamics of national economies and the interaction of different countries according to the import-export balances, we are able to investigate, using a simple model of pulse-coupled oscillators, the synchronization phenomenon of crises at the worldwide scale. We focus on the level of synchronization measured by an order parameter at two different scales, one for the global system and another one for the mesoscales defined through the topology. We use the WTW network structure to simulate a network of Integrate-and-Fire oscillators for six different snapshots between years 1950 and 2000. The results reinforce the idea that globalization accelerates the global synchronization process, and the analysis at a mesoscopic level shows that this synchronization is different before and after globalization periods: after globalization, the effect of communities is almost inexistent.

The analysis of the modular structure of networks is a major challenge in complex networks theory. The validity of the modular structure obtained is essential to confront the problem of the topology-functionality relationship. Recently, several authors have worked on the limit of resolution that different community detection algorithms have, making impossible the detection of natural modules when very different topological scales coexist in the network. Existing multiresolution methods are not the panacea for solving the problem in extreme situations, and also fail. Here, we present a new hierarchical multiresolution scheme that works even when the network decomposition is very close to the resolution limit. The idea is to split the multiresolution method for optimal subgraphs of the network, focusing the analysis on each part independently. We also propose a new algorithm to speed up the computational cost of screening the mesoscale looking for the resolution parameter that best splits every subgraph. The hierarchical algorithm is able to solve a difficult benchmark proposed by [Lancichinetti & Fortunato, 2011], encouraging the further analysis of hierarchical methods based on the modularity quality function.

Singular Value Decomposition (SVD) is a technique based on linear projection theory, which has been frequently used for data analysis. It constitutes an optimal (in the sense of least squares) decomposition of a matrix in the most relevant directions of the data variance. Usually, this information is used to reduce the dimensionality of the data set in a few principal projection directions, this is called Truncated Singular Value Decomposition (TSVD). In situations where the data is continuously changing the projection might become obsolete. Since the change rate of data can be fast, it is an interesting question whether the TSVD projection of the initial data is reliable. In the case of complex networks, this scenario is particularly important when considering network growth. Here we study the reliability of the TSVD projection of growing scale free networks, monitoring its evolution at global and local scales.

Unsupervised clustering, also known as natural clustering, stands for the classification of data according to their similarities. Here we study this problem from the perspective of complex networks. Mapping the description of data similarities to graphs, we propose to extend two multiresolution modularity based algorithms to the finding of modules (clusters) in general data sets producing a multiscales' solution. We show the performance of these reported algorithms to the classification of a standard benchmark of data clustering and compare their performance.

MultiDendrograms is a Java-written application that computes agglomerative hierarchical clusterings of data. Starting from a distances (or weights) matrix, MultiDendrograms is able to calculate its dendrograms using the most common agglomerative hierarchical clustering methods. The application implements a variable-group algorithm that solves the non-uniqueness problem found in the standard pair-group algorithm. This problem arises when two or more minimum distances between different clusters are equal during the agglomerative process, because then different output clusterings are possible depending on the criterion used to break ties between distances. MultiDendrograms solves this problem implementing a variable-group algorithm that groups more than two clusters at the same time when ties occur.

Current modeling of infectious diseases allows for the study of realistic scenarios that include population heterogeneity, social structures, and mobility processes down to the individual level. The advances in the realism of epidemic description call for the explicit modeling of individual behavioral responses to the presence of disease within modeling frameworks. Here we formulate and analyze a metapopulation model that incorporates several scenarios of self-initiated behavioral changes into the mobility patterns of individuals. We find that prevalence-based travel limitations do not alter the epidemic invasion threshold. Strikingly, we observe in both synthetic and data-driven numerical simulations that when travelers decide to avoid locations with high levels of prevalence, this self-initiated behavioral change may enhance disease spreading. Our results point out that the real-time availability of information on the disease and the ensuing behavioral changes in the population may produce a negative impact on disease containment and mitigation.

Since roughly a decade ago, network science has focused among others on the problem of how the spreading of diseases depends on structural patterns. Here, we contribute to further advance our understanding of epidemic spreading processes by proposing a nonperturbative formulation of the heterogeneous mean-field approach that has been commonly used in the physics literature to deal with this kind of spreading phenomena. The nonperturbative equations we propose have no assumption about the proximity of the system to the epidemic threshold, nor any linear approximation of the dynamics. In particular, we first develop a probabilistic description at the node level of the epidemic propagation for the so-called susceptible-infected-susceptible family of models, and after we derive the corresponding heterogeneous mean-field approach. We propose to use the full extension of the approach instead of pruning the expansion to first order, which leads to a nonperturbative formulation that can be solved by fixed-point iteration, and used with reliability far away from the epidemic threshold to assess the prevalence of the epidemics. Our results are in close agreement with Monte Carlo simulations, thus enhancing the predictive power of the classical heterogeneous mean-field approach, while providing a more effective framework in terms of computational time.

Explosive collective phenomena have attracted much attention since the discovery of an explosive percolation transition. In this Letter, we demonstrate how an explosive transition shows up in the synchronization of scale-free networks by incorporating a microscopic correlation between the structural and the dynamical properties of the system. The characteristics of the explosive transition are analytically studied in a star graph reproducing the results obtained in synthetic networks. Our findings represent the first abrupt synchronization transition in complex networks and provide a deeper understanding of the microscopic roots of explosive critical phenomena.

In March 2008, the bankruptcy of Lehman Brothers marked for many the beginning of the global crisis. In an increasingly globalized world, the financial crisis spread relentlessly. Recent theories of financial fragility link globalization with economic cycles, i.e. when local crises coincide with bad credit regulation and failures in international monetary arrangements. The globalization process in recent years has been accelerated due to to the increase of international trade. Here we analyze how economic cycles can spread worldwide over the global trade network (WTW). We use the WTW network structure to simulate a network of Integrate-and-Fire oscillators for two different years, 1980 and 2000. The results reinforce the idea that globalization accelerates the global synchronization process.

We investigate the adaptation and performance of modularity-based algorithms, designed in the scope of complex networks, to analyze the mesoscopic structure of correlation matrices. Using a multiresolution analysis, we are able to describe the structure of the data in terms of clusters at different topological levels. We demonstrate the applicability of our findings in two different scenarios: to analyze the neural connectivity of the nematode Caenorhabditis elegans and to automatically classify a typical benchmark of unsupervised clustering, the Iris dataset, with considerable success.

The discovery of the important role played by the complex connectivity structure between individuals has lead to an increasing interest in the analysis of epidemic spreading in complex networks. Here we propose a discrete-time formulation of the problem of contact-based epidemic spreading, within the context of susceptible-infected-susceptible epidemic models. The proposed equations establish the relations between the probabilities of infection of individual nodes. They can be easily solved by iteration, showing an almost perfect agreement with Monte Carlo experiments throughout the whole phase diagram. This framework also allows the determination of the epidemic threshold and, unlike heterogeneous mean-field approaches, it is valid for any finite-size and weighted network.

The famous Milgram's small-world experiment revealed that there is something special in the structure of natural and man-made complex systems: without a global view of the network, a message can be routed efficiently between any pair of nodes. Our initial hypothesis is that the community structure, that provides meaningful insights on the structure and function of complex networks, is an important actor in these routing properties. To exploit the modular structure of networks we need to analyze the contribution of each node to the modules. Unfortunately, this analysis involves a huge amount of data. To reduce this problem we propose to build a map using the linear projection theory as a basis of a guided routing. First we project the matrix of contributions of each node of a given network to its modules in a plane using the Truncated Singular Value Decomposition. This two-dimensional plane reveals the structure of modules and their boundaries and we will use it as the map for navigating through the network. Considering that each node only has knowledge about its neighbors, we define a simple greedy routing algorithm to guide the communication among them. We apply our framework to the Internet Autonomous Systems (ASs) network achieving, in high percentage, close to optimal paths.

One of the most important problems in science is that of inferring knowledge from data. The most challenging issue is the unsupervised classification of patterns (observations, measurements, or feature vectors) into groups (clusters) according to their similarity. The quantification of similarity is usually performed in terms of distances or correlations between pairs. The resulting similarity matrix is a weighted complete graph. In this work we investigate the adaptation and performance of modularity-based algorithms to analyze the structure of the similarity matrix. Modularity is a quality function that allows comparing different partitions of a given graph, rewarding those partitions that are more internally cohesive than externally. In our problem cohesiveness is the representation of the similarity between members of the same group. The modularity criterion, however, has a drawback, the impossibility to find clusters below a certain size, known as the resolution limit, which depends on the topology of the graph. This is overcome by applying multi-resolution analysis. Using the multi-resolution approach for modularity-based algorithms we automatically classify typical benchmarks of unsupervised clustering with considerable success. These results open the door to the applicability of community detection algorithms in complex networks to the classification of real data sets.

The study of the sub-structure of complex networks is of major importance to relate topology and functionality. Many efforts have been devoted to the analysis of the modular structure of networks using the quality function known as modularity. However, generally speaking, the relation between topological modules and functional groups is still unknown, and depends on the semantic of the links. Sometimes, we know in advance that many connections are transitive and, as a consequence, triangles have a specific meaning. Here we propose the study of the modular structure of networks considering triangles as the building blocks of modules. The method generalizes the standard modularity and uses spectral optimization to find its maximum. We compare the partitions obtained with those resulting from the optimization of the standard modularity in several real networks. The results show that the information reported by the analysis of modules of triangles complements the information of the classical modularity analysis.

Background
Elucidating the activation pattern of molecular pathways across a given tumour type is a key challenge necessary for understanding the heterogeneity in clinical response and for developing novel more effective therapies. Gene expression signatures of molecular pathway activation derived from perturbation experiments in model systems as well as structural models of molecular interactions (''model signatures'') constitute an important resource for estimating corresponding activation levels in tumours. However, relatively few strategies for estimating pathway activity from such model signatures exist and only few studies have used activation patterns of pathways to refine molecular classifications of cancer.

Methods
Here we propose a novel network-based method for estimating pathway activation in tumours from model signatures. We find that although the pathway networks inferred from cancer expression data are highly consistent with the prior information contained in the model signatures, that they also exhibit a highly modular structure and that estimation of pathway activity is dependent on this modular structure. We apply our methodology to a panel of 438 estrogen receptor negative (ER-) and 785 estrogen receptor positive (ER+) breast cancers to infer activation patterns of important cancer related molecular pathways.

Results
We show that in ER negative basal and HER2+ breast cancer, gene expression modules reflecting T-cell helper-1 (Th1) and T-cell helper-2 (Th2) mediated immune responses play antagonistic roles as major risk factors for distant metastasis. Using Boolean interaction Cox-regression models to identify non-linear pathway combinations associated with clinical outcome, we show that simultaneous high activation of Th1 and low activation of a TGF-beta pathway module defines a subtype of particularly good prognosis and that this classification provides a better prognostic model than those based on the individual pathways. In ER+ breast cancer, we find that simultaneous high MYC and RAS activity confers significantly worse prognosis than either high MYC or high RAS activity alone. We further validate these novel prognostic classifications in independent sets of 173 ER- and 567 ER+ breast cancers.

Conclusion
We have proposed a novel method for pathway activity estimation in tumours and have shown that pathway modules antagonize or synergize to delineate novel prognostic subtypes. Specifically, our results suggest that simultaneous modulation of T-helper differentiation and TGF-beta pathways may improve clinical outcome of hormone insensitive breast cancers over treatments that target only one of these pathways.

Modular structure is pervasive in many complex networks of interactions observed in natural, social and technological sciences. Its study sheds light on the relation between the structure and function of complex systems. Generally speaking, modules are islands of highly connected nodes separated by a relatively small number of links. Every module can have contributions of links from any node in the network. The challenge is to disentangle these contributions to understand how the modular structure is built. The main problem is that the analysis of a certain partition into modules involves, in principle, as many data as number of modules times number of nodes. To confront this challenge, here we first define the contribution matrix, the mathematical object containing all the information about the partition of interest, and after, we use a Truncated Singular Value Decomposition to extract the best representation of this matrix in a plane. The analysis of this projection allow us to scrutinize the architecture of the modular structure, revealing the structure of individual modules and their interrelations.

Many epidemic processes in networks spread by stochastic contacts among their connected vertices. There are two limiting cases widely analyzed in the physics literature, the so-called contact process (CP) where the contagion is expanded at a certain rate from an infected vertex to one neighbor at a time, and the reactive process (RP) in which an infected individual effectively contacts all its neighbors to expand the epidemics. However, a more realistic scenario is obtained from the interpolation between these two cases, considering a certain number of stochastic contacts per unit time. Here we propose a discrete-time formulation of the problem of contact-based epidemic spreading. We resolve a family of models, parameterized by the number of stochastic contact trials per unit time, that range from the CP to the RP. In contrast to the common heterogeneous mean-field approach, we focus on the probability of infection of individual nodes. Using this formulation, we can construct the whole phase diagram of the different infection models and determine their critical properties.

We present a reformulation of modularity that allows the analysis of the community structure in networks of correlated data. The modularity preserves the probabilistic semantics of the original definition even when the network is directed, weighted, signed, and has self-loops. This is the most general condition one can find in the study of any network, in particular those defined from correlated data. We apply our results to a real network of correlated data between stores in the city of Lyon (France).

In this chapter we will review a recent optimization approach to the wiring connectivity in C. elegans, discussing the possible outcomes of the optimization process, its dependence on the optimization parameters, and its validation with the actual neuronal layout data. We will follow the main procedure described in the work by Chen et al. [2–4]. The results show that the current approach to optimization of neuronal layouts is still not conclusive, and then the "wiring economy principle" remains unproved.

The worldwide computing grid is essential to the LHC experiments in analysing the data collected by the detectors. Within LHCb, the computing model aims to simulate data at Tier-2 grid sites as well as non-grid resources. The reconstruction, stripping and analysis of the produced LHCb data will pimarily place at the Tier-1 centres. The computing data challenge DC06 started in May 2006 with the primary aims being to exercise the LHCb computing mod and to produce events which will be used for analyses in the forthcoming LHCb physics book. This paper gives an overview of the LHCb computing model and addresses the challenges and experiences during DC06. The management of the production of Monte Carlo data on the LCG was done using the DIRAC worklad management system which in turn uses the WLCG infrastructure and middleware. We shall report on the amount of data simulated during DC06, including the performance of the sites used. The paper will also summarise the experience gained during DC06, in particular he distribution of data to the Ter-1 sits and the access to this data.

The DIRAC system was developed in order to provide a complete solution for using the distributed computing resources of the LHCb experiment at CERN for data production and analysis. It allows a concurrent use of over 10K CPUs and 10M file replicas distributed over many tens of sites. The sites can be part of a Computing Grid such as WLCG or standalone computing clusters all integrated in a single management structure. DIRAC is a generic system with the LHCb specific functionality incorporated through a number of plug-in modules. It can be easily adapted to the needs of other communities. Special attention is paid to the resilience of the DIRAC components to allow an efficient use of non-reliable resources. The DIRAC production management components provide a framework for building highly automated data production systems including data distribution and data driven workload scheduling. In this paper we give an overview of the DIRAC system architecture and design choices. We show how different components are put together to compose an integrated data processing system including all the aspects of the LHCb experiment - from the MC production and raw data reconstruction to the final user analysis.

The structure of real complex networks is often modular, with sets of nodes more connected between them than to the rest of the network. These communities are usually reflecting a topology-functionality interplay, whose discovery is basic for the understanding of the operation of the networks. Thus, much attention has been driven to the determination of the modular structure of complex networks. Recently it has been shown that this modular organization appears at several scales of description, which may be found by a synchronization process on top of these networks. Here we make use of it for a tentative uncovering of functional groups in the neural system of the nematode C. elegans.

In agglomerative hierarchical clustering, pair-group methods suffer from a problem of non-uniqueness when two or more distances between different clusters coincide during the amalgamation process. The traditional approach for solving this drawback has been to take any arbitrary criterion in order to break ties between distances, which results in different hierarchical classifications depending on the criterion followed. In this article we propose a variable-group algorithm that consists in grouping more than two clusters at the same time when ties occur. We give a tree representation for the results of the algorithm, which we call a multidendrogram, as well as a generalization of the Lance and Williams’ formula which enables the implementation of the algorithm in a recursive way.

Modular structure is ubiquitous in real-world complex networks, and its detection is important because it gives insights into the structure–functionality relationship. The standard approach is based on the optimization of a quality function, modularity, which is a relative quality measure for the partition of a network into modules. Recently, some authors (Fortunato and Barthélemy 2007 Proc. Natl Acad. Sci. USA 104 36 and Kumpula et al 2007 Eur. Phys. J. B 56 41) have pointed out that the optimization of modularity has a fundamental drawback: the existence of a resolution limit beyond which no modular structure can be detected even though these modules might have their own entity. The reason is that several topological descriptions of the network coexist at different scales, which is, in general, a fingerprint of complex systems. Here, we propose a method that allows for multiple resolution screening of the modular structure. The method has been validated using synthetic networks, discovering the predefined structures at all scales. Its application to two real social networks allows us to find the exact splits reported in the literature, as well as the substructure beyond the actual split.

Community definitions usually focus on edges, inside and between the communities. However, the high density of edges within a community determines correlations between nodes going beyond nearest neighbors, and which are indicated by the presence of motifs. We show how motifs can be used to define general classes of nodes, including communities, by extending the mathematical expression of Newman–Girvan modularity. We construct then a general framework and apply it to some synthetic and real networks.

The ubiquity of modular structure in real-world complex networks is the focus of attention in many trials to understand the interplay between network topology and functionality. The best approaches to the identification of modular structure are based on the optimization of a quality function known as modularity. However this optimization is a hard task provided that the computational complexity of the problem is in the non-deterministic polynomial-time hard (NP-hard) class. Here we propose an exact method for reducing the size of weighted (directed and undirected) complex networks while maintaining their modularity. This size reduction allows use of heuristic algorithms that optimize modularity for a better exploration of the modularity landscape. We compare the modularity obtained in several real complex-networks by using the extremal optimization algorithm, before and after the size reduction, showing the improvement obtained. We speculate that the proposed analytical size reduction could be extended to an exact coarse graining of the network in the scope of real-space renormalization.

In this paper we apply a heuristic method based on artificial neural networks (NN) in order to trace out the efficient frontier associated to the portfolio selection problem. We consider a generalization of the standard Markowitz mean-variance model which includes cardinality and bounding constraints. These constraints ensure the investment in a given number of different assets and limit the amount of capital to be invested in each asset. We present some experimental results obtained with the NN heuristic and we compare them to those obtained with three previous heuristic methods.

The HI content and Tully-Fisher distances of 161 spiral galaxies in the region of Virgo cluster are used to gain insight into the complicated structure of this galaxy system and to constrain the role played by interactions with the hot intracluster medium on the exhaustion of the interstellar HI. We confirm previous findings that the spiral distribution is substantially more elongated toward the line-of-sight than in the plane of the sky. The filamentary structure is also reproduced in the HI deficiency distribution, which shows a central enhancement suggestive of bearing the Virgo cluster proper, extending from around 16 to 22 Mpc in radial distance, but also important enhancements in the cluster front and in a group of background galaxies. The fact that some of the spirals on the Virgo outskirts exhibit gas deficiencies as strong as those of the inner galaxies has led us to explore, by means of a dynamical model, the possibility that some peripheral objects are not newcomers. We conclude that it is not unfeasible that some of the galaxies far from the cluster, including the HI-deficient background group, have already plunged in the past into the Virgo cluster.

We conduct an investigation into the nature of and conditions within the 3D structure of the Virgo region, by adding to the analysis the HI content of its spiral population.

LHCb Calorimeters, Technical Design Report.

A new error term for dimensionality reduction, which clearly improves the quality of NLPCA neural networks, is introduced, and some illustrative examples are given. The method tries to maintain the original data structure by preserving the distances between data points.

We show that neural networks, with a suitable error function for backpropagation, can be successfully used for metric multidimensional scaling (MDS) (i.e., dimensional reduction while trying to preserve the original distances between patterns) and are in fact able to outdo the standard algebraic approach to MDS, known as classical scaling.

En este artículo estudiamos la aplicación de Redes Neuronales Artificiales a la predicción, basada únicamente en datos históricos, del futuro financiero español Bono 10, teniendo en cuenta las comisiones y la dispersión de los precios.

Given a n-dimensional sample composed of a mixture of m subsamples with different probability density functions (p.d.f.), it is possible to build a (m-1)-dimensional distribution that carries all the information about the subsample proportions in the mixture sample. This projection can be estimated without an analytical knowlegde of the p.d.f.’s of the different subsamples with the aid, for instance, of neural networks. This way, if m-1 < n it is possible to estimate the proportions of the mixture sample in a lower (m-1)-dimensional space without losing sensitivity.

In this paper we propose a new method to prevent the saturation of any set of hidden units of a multilayer neural network. This method is implemented by adding a regularization term to the standard quadratic error function, which is based on a repulsive action between pairs of patterns.

The minimization quadratic error criterion which gives rise to the backpropagation algorithm is studied using functional analysis techniques. With them, we recover easily the well-known statistical result which states that the searched global minimum is a function which assigns, to each input pattern, the expected value of its corresponding output patterns. Its application to classification tasks shows that only certain output class representations can be used to obtain the optimal Bayesian decision rule. Finally, our method permits the study of other error criterions, finding out, for instance, that absolute value errors lead to medians instead of mean values.

A new perceptron learning rule which works with multilayer neural networks made of multistate units is obtained, and the corresponding convergence theorem is proved. The definition of perceptron of maximal stability is enlarged in order to include these new multistate perceptrons, and a proof of existence and uniqueness of such optimal solutions is outlined.

The possibility of achieving optimal associative memory by means of multilayer neural networks is explored. Three original different solutions which guarantee maximal basins of attraction and storage capacity are found, and their main characteristics are outlined.

Neural network techniques for encoding-decoding processes have been developed. The net we have devised can work like a memory retrieval system in the sense of Hopfield, Feinstein and Palmer. Its behaviour for 2^R (R in N) input units has some special interesting features. In particular, the accessibilities for each initial symbol may be explicitly computed. Although thermal noise may muddle the code, we show how it can statistically rid the result of unwanted sequences while maintaining the network accuracy within a given bound.

Neural networks capable of encoding sets of patterns are analysed. Solutions are found by theoretical treatment instead of by supervised learning. The behaviour for 2^R (R in N) input units is studied and its characteristic features are discussed. The accessibilities for non-spurious patterns are calculated by analytic methods. Although thermal noise may induce wrong encoding, we show how it can rid the output of spurious sequences. Further, we compute error bounds at finite temperature.

Several theoretical and practical aspects of multilayer neural networks are studied. The main results are the following:

• Three different multilayer solutions to the problem of associative memory have been constructed, all of them sharing unlimited storage capacity, perfect recall and the removal of spurious minima and unstable states. Their retrieval power is optimal in the sense that the network's answer is selected by the maximal overlap criterion. The original contribution of these solutions has been the realization of such designs without introducing types of units different from those currently used in most neural network architectures.
• Neural network techniques for encoding-decoding processes have been developed. We have proved that it is not possible to encode arbitrary patterns with the minimal architecture, thus other non-optimal set-ups have been studied. In the simplest case of unary patterns, the accessibilities of the outputs have been calculated in two different situations: with and without thermal noise.
• A new perceptron learning rule which can be used with perceptrons made of multi-state units has been derived, and its corresponding convergence theorem has been proved. The definition of a perceptron of maximal stability has been enlarged in order to include these new multi-state perceptrons, and a proof of the existence and uniqueness of such optimal solutions has been outlined.
• The importance of the first hidden layer when multilayer neural networks with discrete activation functions are considered has been explained. As a consequence, several enhancements to the tiling algorithm have been proposed so as to increase the generalization ability of the trained nets.
• The unconstrained global minimum of the squared error criterion used in the back-propagation algorithm has been found using functional derivatives. The role played by the representation of the output patterns has been studied, showing that only certain output representations allow the achievement of the optimal Bayesian decision in classification tasks. Moreover, other error criterions have been analyzed.
• A method for the reconstruction of images from noisy data has been introduced and applied to two aerial images, showing that the results have a very competitive quality.
• Several methods based on self-supervised back-propagation have been devised for the compression of images. The new strategies admit more general applications, specifically to the diminishing of the loss of information produced by the saturation of the sigmoids.
• The performances of multi-layer feed-forward and recurrent networks have been compared when applied to time series prediction, showing that the second ones give, if proper activation functions are chosen, better predictions.