Guided Graph Spectral Embedding: Application to the C. elegans Connectome

TL;DR

This paper introduces a new guided spectral embedding method for graph analysis, applied to the C. elegans connectome, which emphasizes specific nodes and balances global and local features, yielding biologically meaningful insights.

Contribution

The work presents a novel spectral embedding technique combining energy concentration and distance minimization, tailored for targeted node importance in graph structures.

Findings

Enhanced biological insights over Laplacian embedding

Effective focus on specific neuron types in C. elegans

Stable spectral decomposition achieved through opposite optimization goals

Abstract

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-e.g., 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 these optimization goals are intrinsically opposite, leading to a well-defined and stable spectral decomposition. The importance weighting allows 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 and its linear approximation, 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 to Laplacian embedding, the guided approach, focused on a certain class of cells (sensory, inter- and motoneurons), provides more biological insights, such as the distinction between somatic positions of cells, and their involvement in low or high order processing functions.