On Large-Scale Graph Generation with Validation of Diverse Triangle Statistics at Edges and Vertices

Researchers developing implementations of distributed graph analytic algorithms require graph generators that yield graphs sharing the challenging characteristics of real-world graphs (small-world, scale-free, heavy-tailed degree distribution) with efficiently calculable ground-truth solutions to the desired output. Reproducibility for current generators used in benchmarking are somewhat lacking in this respect due to their randomness: the output of a desired graph analytic can only be compared to expected values and not exact ground truth. Nonstochastic Kronecker product graphs meet these design criteria for several graph analytics. Here we show that many flavors of triangle participation can be cheaply calculated while generating a Kronecker product graph. Given two medium-sized scale-free graphs with adjacency matrices $A$ and $B$, their Kronecker product graph has adjacency matrix $C = A \otimes B$. Such graphs are highly compressible: $|{\cal E}|$ edges are represented in ${\cal O}(|{\cal E}|^{1/2})$ memory and can be built in a distributed setting from small data structures, making them easy to share in compressed form. Many interesting graph calculations have worst-case complexity bounds ${\cal O}(|{\cal E}|^p)$ and often these are reduced to ${\cal O}(|{\cal E}|^{p/2})$ for Kronecker product graphs, when a Kronecker formula can be derived yielding the sought calculation on $C$ in terms of related calculations on $A$ and $B$. We focus on deriving formulas for triangle participation at vertices, ${\bf t}_C$, a vector storing the number of triangles that every vertex is involved in, and triangle participation at edges, $\Delta_C$, a sparse matrix storing the number of triangles at every edge.Comment: 10 pages, 7 figures, IEEE IPDPS Graph Algorithms Building Block

An Ensemble Framework for Detecting Community Changes in Dynamic Networks

Dynamic networks, especially those representing social networks, undergo constant evolution of their community structure over time. Nodes can migrate between different communities, communities can split into multiple new communities, communities can merge together, etc. In order to represent dynamic networks with evolving communities it is essential to use a dynamic model rather than a static one. Here we use a dynamic stochastic block model where the underlying block model is different at different times. In order to represent the structural changes expressed by this dynamic model the network will be split into discrete time segments and a clustering algorithm will assign block memberships for each segment. In this paper we show that using an ensemble of clustering assignments accommodates for the variance in scalable clustering algorithms and produces superior results in terms of pairwise-precision and pairwise-recall. We also demonstrate that the dynamic clustering produced by the ensemble can be visualized as a flowchart which encapsulates the community evolution succinctly.Comment: 6 pages, under submission to HPEC Graph Challeng

Multilevel Aggregation Methods for Small-World Graphs with Application to Random-Walk Ranking

We describe multilevel aggregation in the specific context of using Markov chains to rank the nodes of graphs. More generally, aggregation is a graph coarsening technique that has a wide range of possible uses regarding information retrieval applications. Aggregation successfully generates efficient multilevel methods for solving nonsingular linear systems and various eigenproblems from discretized partial differential equations, which tend to involve mesh-like graphs. Our primary goal is to extend the applicability of aggregation to similar problems on small-world graphs, with a secondary goal of developing these methods for eventual applicability towards many other tasks such as using the information in the hierarchies for node clustering or pattern recognition. The nature of small-world graphs makes it difficult for many coarsening approaches to obtain useful hierarchies that have complexity on the order of the number of edges in the original graph while retaining the relevant properties of the original graph. Here, for a set of synthetic graphs with the small-world property, we show how multilevel hierarchies formed with non-overlapping strength-based aggregation have optimal or near optimal complexity. We also provide an example of how these hierarchies are employed to accelerate convergence of methods that calculate the stationary probability vector of large, sparse, irreducible, slowly-mixing Markov chains on such small-world graphs. The stationary probability vector of a Markov chain allows one to rank the nodes in a graph based on the likelihood that a long random walk visits each node. These ranking approaches have a wide range of applications including information retrieval and web ranking, performance modeling of computer and communication systems, analysis of social networks, dependability and security analysis, and analysis of biological systems

The Treatment of Recurrent Abdominal Pain in Children: A Controlled Comparison of Cognitive-Behavioral Family Intervention and Standard Pediatric Care

This study describes the results of a controlled clinical trial involving 44 7- to 14-year-old children with recurrent abdominal pain who were randomly allocated to either cognitive-behavioral family intervention (CBFI) or standard pediatric care (SPC). Both treatment conditions resulted in significant improvements on measures of pain intensity and pain behavior. However, the children receiving CBFI had a higher rate of complete elimination of pain, lower levels of relapse at 6- and 12-month follow-up, and lower levels of interference with their activities as a result of pain and parents reported a higher level of satisfaction with the treatment than children receiving SPC. After controlling for pretreatment levels of pain, children's active self-coping and mothers' caregiving strategies were significant independent predictors of pain behavior at posttreatment

A CSO Search for ll-C3_3H+^+: Detection in the Orion Bar PDR

The results of a Caltech Submillimeter Observatory (CSO) search for $l$-C$_3$H$^+$, first detected by Pety et al. (2012) in observations toward the Horsehead photodissociation region (PDR), are presented. A total of 39 sources were observed in the 1 mm window. Evidence of emission from $l$-C$_3$H$^+$ is found in only a single source - the Orion Bar PDR region, which shows a rotational temperature of 178(13) K and a column density of 7(2) x $10^{11}$ cm$^{-2}$. In the remaining sources, upper limits of ~10$^{11} - 10^{13}$ cm$^{-2}$ are found. These results are discussed in the context of guiding future observational searches for this species.Comment: 9 pages, 8 figures, 4 table