## Section: New Results

### Graph Signal Processing and Machine Learning

Participants : Paulo Gonçalves, Esteban Bautista Ruiz, Mikhail Tsitsvero, Sarah de Nigris.

#### Analytic signal in many dimensions

In a series of two articles [30] and [54] (in collaboration with P. Borgnat), we extended analytic signal to the multidimensional case. First we showed how to obtain separate phase-shifted components and how to combine them into instantaneous amplitude and phase. Secondly we defined the proper hypercomplex analytic signal as a holomorphic hypercomplex function on the boundary of polydisk in the hypercomplex space. Next it was shown that the correct phase-shifted components can be obtained by positive frequency restriction of the Scheffers-Fourier transform based on the commutative and associative algebra generated by the set of elliptic hypercomplex numbers. Moreover we demonstrated that for d > 2 there is no corresponding Clifford-Fourier transform that allows to recover phase-shifted components correctly. Finally the euclidean-domain construction of instantaneous amplitude was extended to manifold and manifold-like graphs and point clouds.

#### BGP Zombies: an Analysis of Beacons Stuck Routes

Joint work with Romain Fontugne (IIJ Research Lab, Japan) and Patrice Abry (CNRS, Physics Lab of ENS de Lyon) [25].

Network operators use the Border Gateway Protocol (BGP) to control the global visibility of their networks. When withdrawing an IP prefix from the Internet, an origin network sends BGP withdraw messages, which are expected to propagate to all BGP routers that hold an entry for that address space in their routing table. Yet network operators occasionally report issues where routers maintain routes to IP prefixes withdrawn by their origin network. We refer to this problem as BGP zombies and characterize their appearance using RIS BGP beacons, a set of prefixes withdrawn every four hours at predetermined times. Across the 27 monitored beacon prefixes, we observe usually more than one zombie outbreak per day. But their presence is highly volatile, on average a monitored peer misses 1.8% withdraws for an IPv4 beacon (2.7% for IPv6).
We also discovered that BGP zombies can propagate to other ASes, for example, zombies in a transit network are inevitably affect- ing its customer networks. **We employ a graph-based semi-supervised machine learning technique to estimate the scope of zombies propagation**, and found that most of the observed zombie outbreaks are small (i.e. on average 10% of monitored ASes for IPv4 and 17% for IPv6). We also report some large zombie outbreaks with almost all monitored ASes affected.

#### Design of graph filters and filterbanks

Book chapter [43], co-authored with Nicolas Tremblay (CNRS, UGA Gipsa-Lab) and Pierre Borgnat (CNRS, Physics Lab, ENS de Lyon).

Basic operations in graph signal processing consist in processing signals indexed on graphs either by filtering them or by changing their domain of representation, in order to better extract or analyze the important information they contain. The aim of this chapter is to review general concepts underlying such filters and representations of graph signals. We first recall the different Graph Fourier Transforms that have been developed in the literature, and show how to introduce a notion of frequency analysis for graph signals by looking at their variations. Then, we move to the introduction of graph filters, that are defined like the classical equivalent for 1D signals or 2D images, as linear systems which operate on each frequency band of a signal. Some examples of filters and of their implementations are given. Finally, as alternate representations of graph signals, we focus on multiscale transforms that are defined from filters. Continuous multiscale transforms such as spectral wavelets on graphs are reviewed, as well as the versatile approaches of filterbanks on graphs. Several variants of graph filterbanks are discussed, for structured as well as arbitrary graphs, with a focus on the central point of the choice of the decimation or aggregation operators.