Modern social networks frequently encompass multiple distinct types of connectivity information; for instance, explicitly acknowledged friend relationships might complement behavioral measures that link users according to their actions or interests. One way to represent these networks is as multi-layer graphs, where each layer contains a unique set of edges over the same underlying vertices (users). Edges in different layers typically have related but distinct semantics; depending on the application multiple layers might be used to reduce noise through averaging, to perform multifaceted analyses, or a combination of the two. However, it is not obvious how to extend standard graph analysis techniques to the multi-layer setting in a flexible way. In this paper we develop latent variable models and methods for mining multi-layer networks for connectivity patterns based on noisy data.
The literature on single layer networks is large, with contributions coming from many different fields. There are many results on structural and spectral properties of a single-layer network, including community detection @cite_17 , random walk return times @cite_21 , and percolation theory results @cite_27 . Diffusion or infection models have also been studied in the context of complex networks (see @cite_26 , for instance).
@cite_27 @cite_21 @cite_26 @cite_17
2179290474 2026313208 2780544323 2169404653
This paper introduces the use of single-layer and deep convolutional networks for remote sensing data analysis. Direct application to multi- and hyperspectral imagery of supervised (shallow or deep) convolutional networks is very challenging given the high input data dimensionality and the relatively small amount of available labeled data. Therefore, we propose the use of greedy layerwise unsupervised pretraining coupled with a highly efficient algorithm for unsupervised learning of sparse features. The algorithm is rooted on sparse representations and enforces both population and lifetime sparsity of the extracted features, simultaneously. We successfully illustrate the expressive power of the extracted representations in several scenarios: classification of aerial scenes, as well as land-use classification in very high resolution or land-cover classification from multi- and hyperspectral images. The proposed algorithm clearly outperforms standard principal component analysis (PCA) and its kernel counterpart (kPCA), as well as current state-of-the-art algorithms of aerial classification, while being extremely computationally efficient at learning representations of data. Results show that single-layer convolutional networks can extract powerful discriminative features only when the receptive field accounts for neighboring pixels and are preferred when the classification requires high resolution and detailed results. However, deep architectures significantly outperform single-layer variants, capturing increasing levels of abstraction and complexity throughout the feature hierarchy. Many complex networks in nature have directed links, a property that affects the network’s navigability and large-scale topology. Here we study the percolation properties of such directed scale-free networks with correlated in and out degree distributions. We derive a phase diagram that indicates the existence of three regimes, determined by the values of the degree exponents. In the first regime we regain the known directed percolation mean field exponents. In contrast, the second and third regimes are characterized by anomalous exponents, which we calculate analytically. In the third regime the network is resilient to random dilution, i.e., the percolation threshold is pc!1. Recent studies have shown that the performance of single-image super-resolution methods can be significantly boosted by using deep convolutional neural networks. In this study, we present a novel single-image super-resolution method by introducing dense skip connections in a very deep network. In the proposed network, the feature maps of each layer are propagated into all subsequent layers, providing an effective way to combine the low-level features and high-level features to boost the reconstruction performance. In addition, the dense skip connections in the network enable short paths to be built directly from the output to each layer, alleviating the vanishing-gradient problem of very deep networks. Moreover, deconvolution layers are integrated into the network to learn the upsampling filters and to speedup the reconstruction process. Further, the proposed method substantially reduces the number of parameters, enhancing the computational efficiency. We evaluate the proposed method using images from four benchmark datasets and set a new state of the art. There has been a quickly growing interest in networks, since they can represent the structure of a wide class of complex systems occurring from the level of cells to society. Data obtained on real networks show that the corresponding graphs exhibit unexpected nontrivial properties, e.g., anomalous degree distributions, diameter, spreading phenomena, clustering coefficient, and correlations [1]. Very recently great attention has been paid to the local structural units of networks. Small and well defined subgraphs have been introduced as ‘‘motifs’’ [2]. Their distribution and clustering properties [2,3] can be used to interpret global features as well. Somewhat larger units, made up of vertices that are more densely connected to each other than to the rest of the network, are often referred to as communities [4], and have been considered to be the essential structural units of real networks. They have no obvious definition, and most of the recent methods for their identification rely on dividing the network into smaller pieces. The biggest drawback of these methods is that they do not allow for overlapping communities, although overlaps are generally assumed to be crucial features of communities. In this Letter we lay down the fundamentals of a kind of percolation phenomenon on graphs, which can also be used as an effective and deterministic method for uniquely identifying overlapping communities in large real networks [5]. Meanwhile, the various aspects of the classical Erdo ýs