But spectral clustering remains a frequently used baseline, and a serious contender to state-of-the-art graphembeddingmethods,e.g.[20,11,28,22].

Spectral clustering algorithms provide approximate solutions to hard optimization problems that formulate graph partitioning in terms of the graph conductance. It is well understood that the quality of these approximate solutions is negatively affected by a possibly significant gap between the conductance and the second eigenvalue of the graph. In this paper we show that for \textbf{any} graph $G$, there exists a `spectral maximizer' graph $H$ which is cut-similar to $G$, but has eigenvalues that are near the theoretical limit implied by the cut structure of $G$. Applying then spectral clustering on $H$ has the potential to produce improved cuts that also exist in $G$ due to the cut similarity. This leads to the second contribution of this work: we describe a practical spectral modification algorithm that raises the eigenvalues of the input graph, while preserving its cuts. Combined with spectral clustering on the modified graph, this yields demonstrably improved cuts.

This section collects a number of required notions from spectral graph theory and puts spectral modification in perspective with important recent discoveries that inspire it.

We thank the reviewers for their nice and helpful comments. Our modification algorithm is intended as a proof of concept of what we view as a general framework. There have been several recent works on graph embedding methods. Our experimental examples are well known and appreciated in spectral graph theory. We can report that the unsupervised versions of recent graph embedding methods (e.g.

Spectral clustering algorithms provide approximate solutions to hard optimization problems that formulate graph partitioning in terms of the graph conductance. It is well understood that the quality of these approximate solutions is negatively affected by a possibly significant gap between the conductance and the second eigenvalue of the graph. In this paper we show that for \textbf{any} graph G, there exists a spectral maximizer' graph H which is cut-similar to G, but has eigenvalues that are near the theoretical limit implied by the cut structure of G . Applying then spectral clustering on H has the potential to produce improved cuts that also exist in G due to the cut similarity. This leads to the second contribution of this work: we describe a practical spectral modification algorithm that raises the eigenvalues of the input graph, while preserving its cuts. Combined with spectral clustering on the modified graph, this yields demonstrably improved cuts.

Spectral clustering algorithms provide approximate solutions to hard optimization problems that formulate graph partitioning in terms of the graph conductance. It is well understood that the quality of these approximate solutions is negatively affected by a possibly significant gap between the conductance and the second eigenvalue of the graph. In this paper we show that for \textbf{any} graph $G$, there exists a spectral maximizer' graph $H$ which is cut-similar to $G$, but has eigenvalues that are near the theoretical limit implied by the cut structure of $G$. Applying then spectral clustering on $H$ has the potential to produce improved cuts that also exist in $G$ due to the cut similarity. This leads to the second contribution of this work: we describe a practical spectral modification algorithm that raises the eigenvalues of the input graph, while preserving its cuts.