Spectral embedding for dynamic networks with stability guarantees

• URL: http://arxiv.org/abs/2106.01282v1
• Date: Wed, 2 Jun 2021 16:43:43 GMT
• Title: Spectral embedding for dynamic networks with stability guarantees
• Abstract summary: We consider the problem of embedding a dynamic network vector to obtain time-evolving representations of each node. These properties are defined formally within a generic dynamic latent position model. We show how this model can be recast as a multilayer random dot product graph.
• Score: 7.575860121202432
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the problem of embedding a dynamic network, to obtain time-evolving vector representations of each node, which can then be used to describe the changes in behaviour of a single node, one or more communities, or the entire graph. Given this open-ended remit, we wish to guarantee stability in the spatio-temporal positioning of the nodes: assigning the same position, up to noise, to nodes behaving similarly at a given time (cross-sectional stability) and a constant position, up to noise, to a single node behaving similarly across different times (longitudinal stability). These properties are defined formally within a generic dynamic latent position model. By showing how this model can be recast as a multilayer random dot product graph, we demonstrate that unfolded adjacency spectral embedding satisfies both stability conditions, allowing, for example, spatio-temporal clustering under the dynamic stochastic block model. We also show how alternative methods, such as omnibus, independent or time-averaged spectral embedding, lack one or the other form of stability.

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