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## Description

We propose generalizations of a binary diffuse interface model for graph segmentation to the case of multiple classes. The original binary diffuse interface model adapts the Ginzburg-Landau (GL) continuum energy functional to a semi-supervised setup on graphs. The graph structure is used to encode a measure of similarity between data points. A small sample of labeled data points (semi-supervised) serves as seeds from which label information can be propagated throughout the graph structure. In this way, the problem can be posed as one of function estimation over the nodes of the graph (learning on graphs) with the GL energy providing a framework to evaluate the quality of data segmentation. We develop two multiclass generalizations, one based on a scalar representation and other based on a vector-field representation. In the scalar version, we modify the GL energy functional to remove the prejudicial effect that the order of the labelings, given by integer values, could have in the smoothing term of the diffuse interface model. In this way, the characteristics of the multiclass classification task are incorporated directly into the energy functional, with a measure of smoothness independent of label order. In the vector-field representation, a vector-valued quantity is assigned to every node in the graph, such that each of its components represents the fraction of the phase, or class, attributed to that particular data point. The vector-field representation has the advantage of leaving the Laplacian term almost unmodified in the GL functional, enabling the use of efficient numerical minimization schemes. We compare the performance of the two multiclass formulations in synthetic data as well as real benchmark sets. The experimental results demonstrate that both methods are competitive with the state-of-the-art in graph-based algorithms. Moving beyond the GL functional framework, we explore the replicator operator, an approach based on the dynamics of epidemics on graphs. We show that although this linear operator models a non-diffusive process, it is equivalent to a symmetric normalized Laplacian on a graph reweighted according to the eigenvector centrality of the adjacency matrix. The operator is able to identify nodes that are more influential and, when used in clustering applications, tends to preserve cliques. This motivates its use as an alternative method for unsupervised graph segmentation of communities in social networks. We conclude by exploring the role of feature selection and appropriate graph representations in real applications. For community detection problems, we probe different graph constructions based on the reweighting scheme of the replicator operator. We demonstrate that in these applications, our multiclass methods yield promising results.