Graphs possess exotic features like variable size and absence of natural ordering of the nodes that make them difficult to analyze and compare. To circumvent this problem and learn on graphs, graph feature representation is required. Main difficulties with feature extraction lie in the trade-off between expressiveness, consistency and efficiency, i.e. the capacity to extract features that represent the structural information of the graph while being deformation-consistent and isomorphism-invariant. While state-of-the-art methods enhance expressiveness with powerful graph neural-networks, we propose to leverage natural spectral properties of graphs to study a simple graph feature: the graph Laplacian spectrum (GLS). We analyze the representational power of this object that satisfies both isomorphism-invariance, expressiveness and deformation-consistency. In particular, we propose a theoretical analysis based on graph perturbation to understand what kind of comparison between graphs we do when comparing GLS. To do so, we derive bounds for the distance between GLS that are related to the divergence to isomorphism, a standard computationally expensive graph divergence. Finally, we experiment GLS as graph representation through consistency tests and classification tasks, and show that it is a strong graph feature representation baseline.
We study theoretically the consistency the Laplacian spectrum and use it as whole-graph embeddding This paper forcuses on the laplacian spectrum of a graph as means to generate a representation to be used to compare graphs and classify them. This work proposed to use Graph Laplacian spectrum to learn graph representation.