Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space

This paper focuses on devising graph signal processing tools for the treatment of data defined on the edges of a graph. We first show that conventional tools from graph signal processing may not be suitable for the analysis of such signals. More specifically, we discuss how the underlying notion of a `smooth signal' inherited from (the typically considered variants of) the graph Laplacian are not suitable when dealing with edge signals that encode a notion of flow. To overcome this limitation we introduce a class of filters based on the Edge-Laplacian, a special case of the Hodge-Laplacian for simplicial complexes of order one. We demonstrate how this Edge-Laplacian leads to low-pass filters that enforce (approximate) flow-conservation in the processed signals. Moreover, we show how these new filters can be combined with more classical Laplacian-based processing methods on the line-graph. Finally, we illustrate the developed tools by denoising synthetic traffic flows on the London street network.Comment: 5 pages, 2 figur

Network Inference from Consensus Dynamics

We consider the problem of identifying the topology of a weighted, undirected network $\mathcal G$ from observing snapshots of multiple independent consensus dynamics. Specifically, we observe the opinion profiles of a group of agents for a set of $M$ independent topics and our goal is to recover the precise relationships between the agents, as specified by the unknown network $\mathcal G$. In order to overcome the under-determinacy of the problem at hand, we leverage concepts from spectral graph theory and convex optimization to unveil the underlying network structure. More precisely, we formulate the network inference problem as a convex optimization that seeks to endow the network with certain desired properties -- such as sparsity -- while being consistent with the spectral information extracted from the observed opinions. This is complemented with theoretical results proving consistency as the number $M$ of topics grows large. We further illustrate our method by numerical experiments, which showcase the effectiveness of the technique in recovering synthetic and real-world networks.Comment: Will be presented at the 2017 IEEE Conference on Decision and Control (CDC

Spectral partitioning of time-varying networks with unobserved edges

We discuss a variant of `blind' community detection, in which we aim to partition an unobserved network from the observation of a (dynamical) graph signal defined on the network. We consider a scenario where our observed graph signals are obtained by filtering white noise input, and the underlying network is different for every observation. In this fashion, the filtered graph signals can be interpreted as defined on a time-varying network. We model each of the underlying network realizations as generated by an independent draw from a latent stochastic blockmodel (SBM). To infer the partition of the latent SBM, we propose a simple spectral algorithm for which we provide a theoretical analysis and establish consistency guarantees for the recovery. We illustrate our results using numerical experiments on synthetic and real data, highlighting the efficacy of our approach.Comment: 5 pages, 2 figure