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Personne# Miljan Petrovic

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Community structure in graph-modeled data appears in a range of disciplines that comprise network science. Its importance relies on the influence it bears on other properties of graphs such as resilience, or prediction of missing connections. Nevertheless, research to date seems to overlook its effect on the properties of signals defined in the domain of graphs' vertices. Indeed, the framework of graph signal processing mainly focuses on local connectivity patterns reflected by the graph Laplacian, and the Laplacian's effect on signals as the graph equivalent of a differential operator. This dissertation investigates the aforementioned interplay between graph signals and the underlying community structure. We make an effort to answer questions like -- do signal's values align and in what way with the communities?; does localization of signal's energy favors a community as the vertex support? Answers to these questions should provide a clearer perspective on the relation between graph signals and networks at the level of subgraphs.This dissertation consists of two main parts. First, we investigate a particular approach -- based on modularity matrix -- of informing the graph Fourier transform about the communities without explicitly detecting them. Thereof, we show that the derived community-aware operators on signals, such as filtering or subsampling, provide a complementary view on the framework to the conventional one based on the Laplacian. Indeed, reduced signal variability within a community seems to be a valuable metric of important signal behavior, aside from its smoothness. Secondly, we explore the intricacies of a broader definition of a community -- a subgraph of any special interest, possibly identified through metadata on vertices instead of the underlying edge connectivity. Within this context, the goal is to understand the benefits of processing signals in a subgraph-restricted way. We design a new, more computationally stable type of Slepians -- bandlimited signals with energy concentrated on a subgraph. Consequently, we show that such bandlimited vectors can be successfully employed to identify a signal's oscillatory pattern localized in a subgraph of interest, by means of a modified filtering procedure. Findings from both lines of research confirm the need and benefits of a better understanding of the interaction between communities and graph signals.

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Graph spectral analysis can yield meaningful embeddings of graphs by providing insight into distributed features not directly accessible in nodal domain. Recent efforts in graph signal processing have proposed new decompositions-for example, based on wavelets and Slepians-that can be applied to filter signals defined on the graph. In this work, we take inspiration from these constructions to define a new guided spectral embedding that combines maximizing energy concentration with minimizing modified embedded distance for a given importance weighting of the nodes. We show that these optimization goals are intrinsically opposite, leading to a well-defined and stable spectral decomposition. The importance weighting allows us to put the focus on particular nodes and tune the trade-off between global and local effects. Following the derivation of our new optimization criterion, we exemplify the methodology on the C. elegans structural connectome. The results of our analyses confirm known observations on the nematode's neural network in terms of functionality and importance of cells. Compared with Laplacian embedding, the guided approach, focused on a certain class of cells (sensory neurons, interneurons, or motoneurons), provides more biological insights, such as the distinction between somatic positions of cells, and their involvement in low- or high-order processing functions.

Miljan Petrovic, Dimitri Nestor Alice Van De Ville

Joint localization of graph signals in vertex and spectral domain is achieved in Slepian vectors calculated by either maximizing energy concentration (mu) or minimizing modified embedded distance (xi) in the subgraph of interest. On the other hand, graph Laplacian is extensively used in graph signal processing as it defines graph Fourier transform (GFT) and operators such as filtering, wavelets, etc. In the context of modeling human brain as a graph, low pass (smooth over neighboring nodes) filtered graph signals represent a valuable source of information known as aligned signals. Here, we propose to define GFT and graph filtering using Slepian orthogonal basis. We explored power spectrum density estimates of random signals on Erdos-Renyi graphs and determined local discrepancies in signal behavior which cannot be accessed by the graph Laplacian, but are detected by the Slepian basis. This motivated the application of Slepian guided graph signal filtering in neuroimaging. We built a graph from diffusion-weighed brain imaging data and used blood-oxygenation-level-dependent (BOLD) time series as graph signals residing on its nodes. The dataset included recordings of 21 subjects performing a working memory task. In certain brain regions known to exhibit activity negatively correlated to performing the task, the only method capable of identifying this type of behavior in the bandlimited framework was xi-Slepian guided filtering. The localization property of the proposed approach provides significant contribution to the strength of the graph spectral analysis, as it allows inclusion of a priori knowledge of the explored graph's mesoscale structure.