Graph embedding: time and space

大型网络分析的进步使许多领域受益，从通过模拟两个IP地址正常通信来识别入侵者的网络安全，到神经科学，大脑连接的差异可能是精神分裂症的迹象。这样的分析通常涉及第一步，其中网络数据被表示为称为嵌入的低维向量空间，在该空间中行为相似的节点是“接近”的。本项目将专注于产生正确地表示大型动态网络的嵌入。这涉及获取动态网络随时间的“快照”，然后计算每个快照的嵌入表示，使得节点的社区随着时间的推移被正确地定位。这是一个潜在位置模型的例子，该模型生成随机网络，在该网络中，同一社区的节点将表现相似并共享潜在位置。由此，我们可以考虑一个动态随机区块模型，其中社区和潜在位置将随着时间的推移而移动(或保持不变)。这个项目的目标是找到一种方法，当网络渐近变大时，能够完美地恢复动态随机块模型中每个节点的社区成员资格。如果实现，这将是一个非常重要的结果，对于跨许多应用程序域执行正确的分析具有简单的影响。

Advances in the analysis of large networks have benefitted very many fields, from cyber-security in the identification of intruders by modelling when two IP addresses should normally communicate, to neuroscience in which differences in brain connectivity could be an indication of schizophrenia. Analyses like these often involve a first step where the network data is represented as a lower-dimensional vector space known as an embedding, in which nodes that behave similarly are ""close"".This project will focus on producing embeddings that correctly represent large dynamic networks. This involves taking ""snapshots"" of the dynamic network over time and then calculating an embedding representation of each snapshot such that communities of nodes are positioned correctly over time.A simple method of modelling a network with community structure is the stochastic block model. This is an example of a latent position model which generates random networks where nodes of the same community will behave similarly and share latent positions. From this, we can consider a dynamic stochastic block model where communities and latent positions will move (or stay static) over time. The goal of this project is then to discover a method that can perfectly recover the community membership of each node in a dynamic stochastic block model as the network becomes asymptotically large. If achieved, this will be an incredibly important result with simple implications for the correct analysis to perform across many application domains.