CHARMNET - Characterising Models for Networks

网络已成为表示和分析复杂数据集的有用工具。这些数据集出现在许多背景下--例如，生物网络被用来代表细胞内代理人的相互作用，社会网络代表个人或社会实体之间的互动，例如引用其他网站的网站，贸易网络反映国家之间的贸易关系。由于它们所代表的数据的复杂性，网络给分析带来了相当大的障碍。通常，独立观测的标准统计框架不再适用--网络被用来准确地表示数据，因为它们往往不是相互独立的。虽然每个网络本身都可以被视为观测，但通常不存在对整个网络的独立观测。要理解网络，可以使用概率模型。然后，可以使用应用概率中的工具来研究由这些模型生成的网络的行为。即使是相对简单的模型也会在分析中带来挑战，更现实的复杂模型往往无法进行严格的数学处理。因此，根据感兴趣的网络行为，用更简单的模型来近似复杂的模型可能是合理的。评估这种近似值的误差对于确定近似值是否合适至关重要。该项目将派生出与公共基础过程相关的网络模型的特征。然后，这一共同的基础过程将允许通过比较模型的特征来比较模型。在这种比较的基础上，可以通过首先使用SIMPLER模型来获得检验统计量在零假设下的分布，然后考虑近似误差，从而得到近似检验程序。在实践中，对于给定的数据集，模型将与数据相适应。这种拟合过程引入了一些变异性，这本身就会导致与模型的一些偏差。使用理论统计和应用概率中的工具，可以使用明确的误差项再次评估这些偏差。该项目将利用这样的观察结果，即评估这种近似误差的方法很好地适用于分析所谓的图形神经网络，图形神经网络正在成为人工智能中的一种工具。因此，该项目将在概率和人工智能之间产生一种新的联系，这将引发应用于网络分析之外的想法。结果将应用于三个公开可用的网络集：蛋白质-蛋白质相互作用网络、政治博客网络和世界贸易网络。选择这些网络是因为它们带来的挑战：到目前为止，还没有被普遍接受的蛋白质-蛋白质相互作用网络模型；此外，这些网络背后的数据包含大量错误。政治博客数据被用作基准；已经为这些网络提出了几个模型，我们的方法将允许对它们进行定量比较。世界贸易网络是加权的、定向的、动态的和空间的，因此说明了我们的方法将能够处理的复杂性。

Networks have emerged as useful tool to represent and analyse complex data sets. These data sets appear in many contexts - for example, biological networks are used to represent the interplay of agents within a cell, social networks represent interactions between individuals or social entities such as websites referring to other websites, trade networks reflect trade relationships between countries. Due to the complexity of the data which they represent, networks pose considerable obstacles for analysis. Typically the standard statistical framework of independent observations no longer applies - networks are used to represent the data precisely because they are often not independent of each other. While each network itself can be viewed as an observation, usually there are no independent observations of the whole network available. To understand networks, probabilistic models can be employed. The behaviour of networks which are generated from such models can then be studied with tools from applied probability. Even relatively simple models provide challenges in their analysis, with more realistic complex models often out of reach of a rigorous mathematical treatment. Hence, depending on the network behaviour of interest, it may be reasonable to approximate a complex model with a simpler model. Assessing the error in such an approximation is crucial to determine whether the approximation is suitable. This project will derive characterisations of network models which relate to a common underlying process. This common underlying process will then allow to compare models through comparing their characterisations. Based on such comparisons, approximate test procedures can be derived by first using the simpler model to obtain the distribution of the test statistic under the null hypothesis and then taking the approximation error into account. In practice, for a given data set, a model would be fitted to the data. This fitting process introduces some variability which in itself will result in some deviations from the model. Using tools from theoretical statistics as well as applied probability, these deviations can again be assessed, with an explicit error term. The project will exploit the observation that the method for assessing this approximation error is well adapted to analyse so-called graph neural networks, which are emerging as a tool in Artificial Intelligence. Thus the project will yield a new connection between Probability and Artificial Intelligence which will spark ideas beyond the application to network analysis.The results will be applied to three network sets which are publicly available: protein-protein interaction networks, political blog networks, and World Trade networks. These networks are chosen because of the challenges they pose: there is to date no generally accepted model for protein-protein interaction network; moreover, the data underlying these networks contain a large amount of errors. Political blog data are used as a benchmark; several models have been proposed for these networks, and our approach will allow to compare them quantitatively. World Trade networks are weighted, directed, dynamic and spatial, and thus illustrate the complexity which our approach will be able to tackle.