Computing distances and finding shortest paths in massive real-world networks is a fundamental algorithmic task in network analysis. There are two main approaches to solving this task. On one end are traversal-based algorithms like bidirectional breadth-first search (BiBFS), which have no preprocessing step but are slow on individual distance inquiries. On the other end are indexing-based approaches, which create and maintain a large index. This allows for answering individual inquiries very fast; however, index creation is prohibitively expensive. We seek to bridge these two extremes: quickly answer distance inquiries without the need for costly preprocessing.

Graphs are a fundamental data structure used to represent relationships in domains as diverse as the social sciences, bioinformatics, cybersecurity, the Internet, and more. One of the central observations in network science is that real-world graphs are globally sparse, yet contain numerous "pockets" of high edge density. A fundamental task in graph mining is to discover these dense subgraphs. Most common formulations of the problem involve finding a single (or a few) "optimally" dense subsets. But in most real applications, one does not care for the optimality. Instead, we want to find a large collection of dense subsets that covers a significant fraction of the input graph. We give a mathematical formulation of this problem, using a new definition of regularly triangle-rich (RTR) families. These families capture the notion of dense subgraphs that contain many triangles and have degrees comparable to the subgraph size. We design a provable algorithm, RTRExtractor, that can discover RTR families that approximately cover any RTR set. The algorithm is efficient and is inspired by recent results that use triangle counts for community testing and clustering. We show that RTRExtractor has excellent behavior on a large variety of real-world datasets. It is able to process graphs with hundreds of millions of edges within minutes. Across many datasets, RTRExtractor achieves high coverage using high edge density datasets. For example, the output covers a quarter of the vertices with subgraphs of edge density more than (say) 0.5, for datasets with 10M+ edges. We show an example of how the output of RTRExtractor correlates with meaningful sets of similar vertices in a citation network, demonstrating the utility of RTRExtractor for unsupervised graph discovery tasks.

Consider property testing on bounded degree graphs and let ∊ > 0 denote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP 2013) asserts that all properties of planar graphs (more generally hyperfinite) are testable with query complexity only depending on ∊. Recent advances in testing minor-freeness have proven that all additive and monotone properties of planar graphs can be tested in poly(∊–1) queries. Some properties falling outside this class, such as Hamiltonicity, also have a similar complexity for planar graphs. Motivated by these results, we ask: can all properties of planar graphs can be tested in poly(∊–1) queries? Is there a uniform query complexity upper bound for all planar properties, and what is the “hardest” such property to test? We discover a surprisingly clean and optimal answer. Any property of bounded degree planar graphs can be tested in exp(O(∊–2)) queries. Moreover, there is a matching lower bound, up to constant factors in the exponent. The natural property of testing isomorphism to a fixed graph requires exp(Ω(∊–2)) queries, thereby showing that (up to polynomial dependencies) isomorphism to an explicit fixed graph is the hardest property of planar graphs. The upper bound is a straightforward adaptation of the Newman-Sohler analysis that tracks dependencies on ∊ more carefully. The main technical contribution is the lower bound construction, which is achieved by a special family of planar graphs that are all mutually far from each other. We can also apply our techniques to get analogous results for bounded treewidth graphs. We prove that all properties of bounded treewidth graphs can be tested in exp(O(∊–1 log ∊–1)) queries. Moreover, testing isomorphism to a fixed forest requires exp(Ω(∊–1)) queries.

Data traffic sequences from two campus FDDI rings, an Ethernet, two entry/exit points of the NSFNET, and sub-sequences belonging to popular TCP port numbers on one of the FDDI rings indicate that appropriately differenced time-series generated from these traces can be modeled as auto-regressive moving average (ARMA) processes. The variates of the ARMA filter are, however, non-Gaussian. A sequence of steps leading through (i) parameter estimation, (ii) generating the distribution of the variates, (iii) forecasting tail percentiles, and (iv) synthetic generation of non-negative integer sequences is presented. The data indicates that parameter estimates drift slowly with time and may need to be re-computed periodically for accurate forecasts. The forecasting algorithm, has potential application in dynamic resource allocation. The synthetic traffic generation algorithm may be used in simulation studies of resource management algorithms.