Detecting and Characterizing Small Dense Bipartite-like Subgraphs by the Bipartiteness Ratio Measure

We study the problem of finding and characterizing subgraphs with small \textit{bipartiteness ratio}. We give a bicriteria approximation algorithm \verb|SwpDB| such that if there exists a subset $S$ of volume at most $k$ and bipartiteness ratio $\theta$, then for any $0<\epsilon<1/2$, it finds a set $S'$ of volume at most $2k^{1+\epsilon}$ and bipartiteness ratio at most $4\sqrt{\theta/\epsilon}$. By combining a truncation operation, we give a local algorithm \verb|LocDB|, which has asymptotically the same approximation guarantee as the algorithm \verb|SwpDB| on both the volume and bipartiteness ratio of the output set, and runs in time $O(\epsilon^2\theta^{-2}k^{1+\epsilon}\ln^3k)$, independent of the size of the graph. Finally, we give a spectral characterization of the small dense bipartite-like subgraphs by using the $k$th \textit{largest} eigenvalue of the Laplacian of the graph.Comment: 17 pages; ISAAC 201

Detecting and characterizing small dense bipartite-like subgraphs by the bipartiteness ratio measure

We study the problem of finding and characterizing subgraphs with small bipartiteness ratio. We give a bicriteria approximation algorithm SwpDB such that if there exists a subset S of volume at most k and bipartiteness ratio &theta;, then for any 0 &lt; &Epsilon; &lt; 1/2, it finds a set S&prime; of volume at most 2k1+&Epsilon; and bipartiteness ratio at most 4&radic;&theta;/&Epsilon;. By combining a truncation operation, we give a local algorithm LocDB, which has asymptotically the same approximation guarantee as the algorithm SwpDB on both the volume and bipartiteness ratio of the output set, and runs in time O(&Epsilon;2 &theta;-2 k 1+&Epsilon; ln 3 k), independent of the size of the graph. Finally, we give a spectral characterization of the small dense bipartite-like subgraphs by using the kth largest eigenvalue of the Laplacian of the graph. &copy; 2013 Springer-Verlag.We study the problem of finding and characterizing subgraphs with small bipartiteness ratio. We give a bicriteria approximation algorithm SwpDB such that if there exists a subset S of volume at most k and bipartiteness ratio &theta;, then for any 0 &lt; &Epsilon; &lt; 1/2, it finds a set S&prime; of volume at most 2k1+&Epsilon; and bipartiteness ratio at most 4&radic;&theta;/&Epsilon;. By combining a truncation operation, we give a local algorithm LocDB, which has asymptotically the same approximation guarantee as the algorithm SwpDB on both the volume and bipartiteness ratio of the output set, and runs in time O(&Epsilon;2 &theta;-2 k 1+&Epsilon; ln 3 k), independent of the size of the graph. Finally, we give a spectral characterization of the small dense bipartite-like subgraphs by using the kth largest eigenvalue of the Laplacian of the graph. &copy; 2013 Springer-Verlag

On learning the structure of clusters in graphs

Graph clustering is a fundamental problem in unsupervised learning, with numerous applications in computer science and in analysing real-world data. In many real-world applications, we find that the clusters have a significant high-level structure. This is often overlooked in the design and analysis of graph clustering algorithms which make strong simplifying assumptions about the structure of the graph. This thesis addresses the natural question of whether the structure of clusters can be learned efficiently and describes four new algorithmic results for learning such structure in graphs and hypergraphs. The first part of the thesis studies the classical spectral clustering algorithm, and presents a tighter analysis on its performance. This result explains why it works under a much weaker and more natural condition than the ones studied in the literature, and helps to close the gap between the theoretical guarantees of the spectral clustering algorithm and its excellent empirical performance. The second part of the thesis builds on the theoretical guarantees of the previous part and shows that, when the clusters of the underlying graph have certain structures, spectral clustering with fewer than k eigenvectors is able to produce better output than classical spectral clustering in which k eigenvectors are employed, where k is the number of clusters. This presents the first work that discusses and analyses the performance of spectral clustering with fewer than k eigenvectors, and shows that general structures of clusters can be learned with spectral methods. The third part of the thesis considers efficient learning of the structure of clusters with local algorithms, whose runtime depends only on the size of the target clusters and is independent of the underlying input graph. While the objective of classical local clustering algorithms is to find a cluster which is sparsely connected to the rest of the graph, this part of the thesis presents a local algorithm that finds a pair of clusters which are densely connected to each other. This result demonstrates that certain structures of clusters can be learned efficiently in the local setting, even in the massive graphs which are ubiquitous in real-world applications. The final part of the thesis studies the problem of learning densely connected clusters in hypergraphs. The developed algorithm is based on a new heat diffusion process, whose analysis extends a sequence of recent work on the spectral theory of hypergraphs. It allows the structure of clusters to be learned in datasets modelling higher-order relations of objects and can be applied to efficiently analyse many complex datasets occurring in practice. All of the presented theoretical results are further extensively evaluated on both synthetic and real-word datasets of different domains, including image classification and segmentation, migration networks, co-authorship networks, and natural language processing. These experimental results demonstrate that the newly developed algorithms are practical, effective, and immediately applicable for learning the structure of clusters in real-world data