Exact formulae for the Lovasz Theta Function of sparse Circulant Graphs

The Lovasz theta function has attracted a lot of attention for its connection with diverse issues, such as communicating without errors and computing large cliques in graphs. Indeed this function enjoys the remarkable property of being computable in polynomial time, despite being sandwitched between clique and chromatic number, two well known hard to compute quantities. In this paper I provide a closed formula for the Lovasz function of a specific class of sparse circulant graphs thus generalizing Lovasz results on cycle graphs (circulant graphs of degree 2)

The Power of Pivoting for Exact Clique Counting

Clique counting is a fundamental task in network analysis, and even the simplest setting of $3$-cliques (triangles) has been the center of much recent research. Getting the count of $k$-cliques for larger $k$ is algorithmically challenging, due to the exponential blowup in the search space of large cliques. But a number of recent applications (especially for community detection or clustering) use larger clique counts. Moreover, one often desires \textit{local} counts, the number of $k$-cliques per vertex/edge. Our main result is Pivoter, an algorithm that exactly counts the number of $k$-cliques, \textit{for all values of $k$}. It is surprisingly effective in practice, and is able to get clique counts of graphs that were beyond the reach of previous work. For example, Pivoter gets all clique counts in a social network with a 100M edges within two hours on a commodity machine. Previous parallel algorithms do not terminate in days. Pivoter can also feasibly get local per-vertex and per-edge $k$-clique counts (for all $k$) for many public data sets with tens of millions of edges. To the best of our knowledge, this is the first algorithm that achieves such results. The main insight is the construction of a Succinct Clique Tree (SCT) that stores a compressed unique representation of all cliques in an input graph. It is built using a technique called \textit{pivoting}, a classic approach by Bron-Kerbosch to reduce the recursion tree of backtracking algorithms for maximal cliques. Remarkably, the SCT can be built without actually enumerating all cliques, and provides a succinct data structure from which exact clique statistics ($k$-clique counts, local counts) can be read off efficiently.Comment: 10 pages, WSDM 202

VoG: Summarizing and Understanding Large Graphs

How can we succinctly describe a million-node graph with a few simple sentences? How can we measure the "importance" of a set of discovered subgraphs in a large graph? These are exactly the problems we focus on. Our main ideas are to construct a "vocabulary" of subgraph-types that often occur in real graphs (e.g., stars, cliques, chains), and from a set of subgraphs, find the most succinct description of a graph in terms of this vocabulary. We measure success in a well-founded way by means of the Minimum Description Length (MDL) principle: a subgraph is included in the summary if it decreases the total description length of the graph. Our contributions are three-fold: (a) formulation: we provide a principled encoding scheme to choose vocabulary subgraphs; (b) algorithm: we develop \method, an efficient method to minimize the description cost, and (c) applicability: we report experimental results on multi-million-edge real graphs, including Flickr and the Notre Dame web graph.Comment: SIAM International Conference on Data Mining (SDM) 201