Given two sets of vectors, $A = {{a_1}, dots, {a_m}}$ and $B={{b_1},dots,{b_n}}$, our problem is to find the top-$t$ dot products, i.e., the largest $|{a_i}cdot{b_j}|$ among all possible pairs. This is a fundamental mathematical problem that appears
in numerous data applications involving similarity search, link prediction, and collaborative filtering. We propose a sampling-based approach that avoids direct computation of all $mn$ dot products. We select diamonds (i.e., four-cycles) from the weighted tripartite representation of $A$ and $B$. The probability of selecting a diamond corresponding to pair $(i,j)$ is proportional to $({a_i}cdot{b_j})^2$, amplifying the focus on the largest-magnitude entries. Experimental results indicate that diamond sampling is orders of magnitude faster than direct computation and requires far fewer samples than any competing approach. We also apply diamond sampling to the special case of maximum inner product search, and get significantly better results than the state-of-the-art hashing methods.
Community detection is expensive, and the cost generally depends at least linearly on the number of vertices in the graph. We propose working with a reduced graph that has many fewer nodes but nonetheless captures key community structure. The K-core
of a graph is the largest subgraph within which each node has at least K connections. We propose a framework that accelerates community detection by applying an expensive algorithm (modularity optimization, the Louvain method, spectral clustering, etc.) to the K-core and then using an inexpensive heuristic (such as local modularity maximization) to infer community labels for the remaining nodes. Our experiments demonstrate that the proposed framework can reduce the running time by more than 80% while preserving the quality of the solutions. Recent theoretical investigations provide support for using the K-core as a reduced representation.
Graphs are used to model interactions in a variety of contexts, and there is a growing need to quickly assess the structure of such graphs. Some of the most useful graph metrics are based on triangles, such as those measuring social cohesion. Algorit
hms to compute them can be extremely expensive, even for moderately-sized graphs with only millions of edges. Previous work has considered node and edge sampling; in contrast, we consider wedge sampling, which provides faster and more accurate approximations than competing techniques. Additionally, wedge sampling enables estimation local clustering coefficients, degree-wise clustering coefficients, uniform triangle sampling, and directed triangle counts. Our methods come with provable and practical probabilistic error estimates for all computations. We provide extensive results that show our methods are both more accurate and faster than state-of-the-art alternatives.
Graphs and networks are used to model interactions in a variety of contexts. There is a growing need to quickly assess the characteristics of a graph in order to understand its underlying structure. Some of the most useful metrics are triangle-based
and give a measure of the connectedness of mutual friends. This is often summarized in terms of clustering coefficients, which measure the likelihood that two neighbors of a node are themselves connected. Computing these measures exactly for large-scale networks is prohibitively expensive in both memory and time. However, a recent wedge sampling algorithm has proved successful in efficiently and accurately estimating clustering coefficients. In this paper, we describe how to implement this approach in MapReduce to deal with massive graphs. We show results on publicly-available networks, the largest of which is 132M nodes and 4.7B edges, as well as artificially generated networks (using the Graph500 benchmark), the largest of which has 240M nodes and 8.5B edges. We can estimate the clustering coefficient by degree bin (e.g., we use exponential binning) and the number of triangles per bin, as well as the global clustering coefficient and total number of triangles, in an average of 0.33 seconds per million edges plus overhead (approximately 225 seconds total for our configuration). The technique can also be used to study triangle statistics such as the ratio of the highest and lowest degree, and we highlight differences between social and non-social networks. To the best of our knowledge, these are the largest triangle-based graph computations published to date.
