## AbstractWe continue our study of online prediction of the labelling of a graph. We show a fundamental limitation of Laplacian-based algorithms: if the graph has a large diameter then the number of mistakes made by such algorithms may be proportional to the square root of the number of vertices, even when tackling simple problems. We overcome this drawback by means of an efficient algorithm which achieves a logarithmic mistake bound. It is based on the notion of a spine, a path graph which provides a linear embedding of the original graph. In practice, graphs may exhibit cluster structure; thus in the last part, we present a modified algorithm which achieves the “best of both worlds”: it performs well locally in the presence of cluster structure, and globally on large diameter graphs.
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