School of Communication and Electronic Engineering, East China Normal University, Shanghai 200241, China.
School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China.
Chaos. 2019 Nov;29(11):113106. doi: 10.1063/1.5121401.
Deep learning has taken part in the competition since not long ago to learn and identify phase transitions in physical systems such as many-body quantum systems, whose underlying lattice structures are generally regular as they are in Euclidean space. Real networks have complex structural features that play a significant role in dynamics in them, and thus the structural and dynamical information of complex networks cannot be directly learned by existing neural network models. Here, we propose a novel and effective framework to learn the epidemic threshold in complex networks by combining the structural and dynamical information into the learning procedure. Considering the strong performance of learning in Euclidean space, the Convolutional Neural Network (CNN) is used, and, with the help of "confusion scheme," we can identify precisely the outbreak threshold of epidemic dynamics. To represent the high-dimensional network data set in Euclidean space for CNN, we reduce the dimensionality of a network by using graph representation learning algorithms and discretize the embedded space to convert it into an imagelike structure. We then creatively merge the nodal dynamical states with the structural embedding by multichannel images. In this manner, the proposed model can draw the conclusion from both structural and dynamical information. A large number of simulations show a great performance in both synthetic and empirical network data sets. Our end to end machine learning framework is robust and universally applicable to complex networks with arbitrary size and topology.
深度学习最近开始参与到学习和识别物理系统（如多体量子系统）相变的竞争中，这些物理系统的底层晶格结构通常是规则的，就像在欧几里得空间中一样。实际网络具有复杂的结构特征，这些特征在网络动力学中起着重要作用，因此现有的神经网络模型无法直接学习复杂网络的结构和动力学信息。在这里，我们提出了一个新颖而有效的框架，通过将结构和动力学信息结合到学习过程中，来学习复杂网络中的流行病阈值。考虑到在欧几里得空间中学习的强大性能，我们使用卷积神经网络（CNN），并借助“混淆方案”，可以精确地识别流行病动力学的爆发阈值。为了将高维网络数据集转换为 CNN 中的欧几里得空间，我们使用图表示学习算法来降低网络的维度，并将嵌入空间离散化，将其转换为类似图像的结构。然后，我们通过多通道图像创造性地将节点动力学状态与结构嵌入合并。通过这种方式，所提出的模型可以从结构和动力学信息中得出结论。大量的模拟表明，该模型在合成和经验网络数据集上都具有出色的性能。我们的端到端机器学习框架是稳健的，并且可以普遍适用于具有任意大小和拓扑结构的复杂网络。
