SIG-NET: Exploring the interface between SIGnal processing and NETwork science

复杂性科学在过去几年中所经历的质的进步直接与计算能力的增加有关，这使得运行大规模模拟和处理大量(经验)数据成为可能：所谓的大数据范式。提出新的方法和见解来处理、存储和提取海量数据的信息是至关重要的。这些数据集自然分为两种不同的类型。首先，从金融指标的时间演化或湍流的不规则运动到语音的波形信号，复杂的系统产生了极其复杂的单变量/多变量时间序列，其隐藏的结构需要使用快速和新颖的方法来处理和分析。其次，复杂系统相互作用模式的相互交织的体系结构自然地用图形来表示和建模-这种方法的一个范例是大脑，由建模突触连接的边连接的单个单元(神经元)建模。这些分布式处理系统通常位于有序性和随机性(所谓的复杂网络范例)之间，并有不同的风格(无定向/定向、静态/时态、单层/多层)。这两个数据集家族中的每一个都有自己的数学语料库来处理这些数据的描述和表征，即信号处理和网络科学。该项目的工作假设是，可以通过将数据集映射到另一种数学表示来检索编码或隐藏在数据集中的信息，其中信息的提取可能最终会更简单。因此，我们的目标是探索通过将时间序列映射到图中从而使用网络科学来表征信号及其潜在的动力学来提取哪些新信息：简而言之，进行图论时间序列分析。我们也对对偶问题感兴趣，即从图中提取时间序列，从而使用时间序列分析和信号处理的工具来描述、比较和分类许多类型的网络：图的信号处理。我们将考虑具体的方法(可见性算法、马尔可夫链理论、波动分析)，并将能够定义和验证描述信号的新的图论度量和描述图的新的信号论度量，以及建立数学上可靠的理论来将这两种方法联系起来。最终，我们的研究结果将在一个软件中实现，该软件的输入是一个时间序列/复杂网络，其输出是从多个角度(信号处理和图论角度)描述被研究对象的一组关键特征。然后，这些功能将提供给自动分类器，用于模式识别和数据分析。

The qualitative step forward that Complexity Science has experienced in the last years is directly related to an increase of computation capacity, enabling the possibility of running large scale simulations and handling large amounts of (empirical) data: the so called Big Data paradigm. It is fundamental to come along with new methods and insights to deal, store and extract information from large amounts of data.These datasets naturally come in two different types. First, from the time evolution of some financial indicator or the irregular motion of turbulent fluids to the waveform signal of speech, complex systems produce incredibly complicated univariate/multivariate time series, whose hidden structure should be processed and analysed using fast and novel approaches. Second, the intertwined architecture of the interaction patterns of complex systems is naturally represented and modeled in terms of graphs -a paradigmatic of this approach being the brain, modeled by single units (neurons) connected by edges that model synaptic connections. These distributed processing systems usually lay at the edge between order and randomness (the so-called complex network paradigm) and come in different flavours (undirected/directed, static/temporal, monolayer/multilayer). Each of these two families of datasets have its own mathematical corpus that deals with the description and characterisation of these data, namely signal processing and network science.The working hypothesis of this project is that information encoded or hidden in a data set can be retrieved by mapping such data set into an alternative mathematical representation, where the extraction of information may be eventually simpler. As such, we aim to explore what new information can be extracted by mapping time series into graphs and therefore using network science to characterise signals and their underlying dynamics: in short, to make graph-theoretical time series analysis. We are also interested in the dual problem, namely extracting time series from graphs and therefore using the tools of time series analysis and signal processing to describe, compare and classify networks of many kinds: a signal processing of graphs.We will consider specific methods (visibility algorithms, Markov chain theory, fluctuation analysis) and will be able to define and validate new graph-theoretical measures to describe signals and new signal-theoretic measures to describe graphs, as well as to build a mathematically sound and solid theory to relate these two approaches.Ultimately, the results of our research will be implemented in a software whose input is a time series/complex network and whose output is a set of key features which describe the object under study from several angles (both the signal processing and graph theoretic angle). These features will then feed automatic classifiers for pattern recognition and data analytics.