Publications

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It is challenging to obtain scalable HPC performance on real applications, especially for data science applications with irregular memory access and computation patterns. To drive co-design efforts in architecture, system, and application design, we are developing miniapps representative of data science workloads. These in turn stress the state of the art in Graph BLAS-like Graph Algorithm Building Blocks (GABB). In this work, we outline a Graph BLAS-like, linear algebra based approach to miniTri, one such miniapp. We describe a task-based prototype implementation and give initial scalability results.

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A fundamental challenge for supercomputer architecture is that processors cannot be fed data from DRAM as fast as CPUs can consume it. Therefore, many applications are memory-bandwidth bound. As the number of cores per chip increases, and traditional DDR DRAM speeds stagnate, the problem is only getting worse. A variety of non-DDR 3D memory technologies (Wide I/O 2, HBM) offer higher bandwidth and lower power by stacking DRAM chips on the processor or nearby on a silicon interposer. However, such a packaging scheme cannot contain sufficient memory capacity for a node. It seems likely that future systems will require at least two levels of main memory: high-bandwidth, low-power memory near the processor and low-bandwidth high-capacity memory further away. This near memory will probably not have significantly faster latency than the far memory. This, combined with the large size of the near memory (multiple GB) and power constraints, may make it difficult to treat it as a standard cache. In this paper, we explore some of the design space for a user-controlled multi-level main memory. We present algorithms designed for the heterogeneous bandwidth, using streaming to exploit data locality. We consider algorithms for the fundamental application of sorting. Our algorithms asymptotically reduce memory-block transfers under certain architectural parameter settings. We use and extend Sandia National Laboratories' SST simulation capability to demonstrate the relationship between increased bandwidth and improved algorithmic performance. Memory access counts from simulations corroborate predicted performance. This co-design effort suggests implementing two-level main memory systems may improve memory performance in fundamental applications.

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We present a new distributed model for graph computations motivated by limited information sharing. Two or more independent entities have collected large social graphs. They wish to compute the result of running graph algorithms on the entire set of relationships. Because the information is sensitive or economically valuable, they do not wish to simply combine the information in a single location. We consider two models for computing the solution to graph algorithms in this setting: 1) limited-sharing: the two entities can share only a poly logarithmic size subgraph, 2) low-trust: the entities must not reveal any information beyond the query answer, assuming they are all honest but curious. We believe this model captures realistic constraints on cooperating autonomous data centres' have results for both models for s-t connectivity, one of the simplest graph problems that requires global information in the worst case. In the limited-sharing model, our results exploit social network structure. Standard communication complexity gives polynomial lower bounds on s-t connectivity for general graphs. However, if the graph for each data centre has a giant component and these giant components intersect, then we can overcome this lower bound, computing-t connectivity while exchanging O(log 2 n) bits for a constant number of data centers. We can also test the assumption that the giant components overlap using O(log 2 n) bits provided the (unknown) overlap is sufficiently large. The second result is in the low trust model. We give a secure multi-party computation (MPC) algorithm that 1) does not make cryptographic assumptions when there are 3 or more entities, and 2) is efficient, especially when compared to the usual garbled circuit approach. The entities learn only the yes/no answer. No party learns anything about the others' graph, not even node names. This algorithm does not require any special graph structure. This secure MPC result for s-t connectivity is one of the first that involves a few parties computing on large inputs, instead of many parties computing on a few local values.

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We present new algorithms for a distributed model for graph computations motivated by limited information sharing we first discussed in [20]. Two or more independent entities have collected large social graphs. They wish to compute the result of running graph algorithms on the entire set of relationships. Because the information is sensitive or economically valuable, they do not wish to simply combine the information in a single location. We consider two models for computing the solution to graph algorithms in this setting: 1) limited-sharing: the two entities can share only a polylogarithmic size subgraph; 2) low-trust: the entities must not reveal any information beyond the query answer, assuming they are all honest but curious. We believe this model captures realistic constraints on cooperating autonomous data centers. We have algorithms in both setting for s - t connectivity in both models. We also give an algorithm in the low-communication model for finding a planted clique. This is an anomaly- detection problem, finding a subgraph that is larger and denser than expected. For both the low- communication algorithms, we exploit structural properties of social networks to prove perfor- mance bounds better than what is possible for general graphs. For s - t connectivity, we use known properties. For planted clique, we propose a new property: bounded number of triangles per node. This property is based upon evidence from the social science literature. We found that classic examples of social networks do not have the bounded-triangles property. This is because many social networks contain elements that are non-human, such as accounts for a business, or other automated accounts. We describe some initial attempts to distinguish human nodes from automated nodes in social networks based only on topological properties.

Listing all triangles is a fundamental graph operation. Triangles can have important interpretations in real-world graphs, especially social and other interaction networks. Despite the lack of provably efficient (linear, or slightly super linear) worst-case algorithms for this problem, practitioners run simple, efficient heuristics to find all triangles in graphs with millions of vertices. How are these heuristics exploiting the structure of these special graphs to provide major speedups in running time? We study one of the most prevalent algorithms used by practitioners. A trivial algorithm enumerates all paths of length 2, and checks if each such path is incident to a triangle. A good heuristic is to enumerate only those paths of length 2 in which the middle vertex has the lowest degree. It is easily implemented and is empirically known to give remarkable speedups over the trivial algorithm. We study the behavior of this algorithm over graphs with heavy-tailed degree distributions, a defining feature of real-world graphs. The erased configuration model (ECM) efficiently generates a graph with asymptotically (almost) any desired degree sequence. We show that the expected running time of this algorithm over the distribution of graphs created by the ECM is controlled by the l4/3-norm of the degree sequence. Norms of the degree sequence are a measure of the heaviness of the tail, and it is precisely this feature that allows non trivial speedups of simple triangle enumeration algorithms. As a corollary of our main theorem, we prove expected linear-time performance for degree sequences following a power law with exponent α ≥ 7/3, and non trivial speedup whenever α ∈ (2, 3).

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This report summarizes the work performed under the project (3z(BStatitically significant relational data mining.(3y (BThe goal of the project was to add more statistical rigor to the fairly ad hoc area of data mining on graphs. Our goal was to develop better algorithms and better ways to evaluate algorithm quality. We concetrated on algorithms for community detection, approximate pattern matching, and graph similarity measures. Approximate pattern matching involves finding an instance of a relatively small pattern, expressed with tolerance, in a large graph of data observed with uncertainty. This report gathers the abstracts and references for the eight refereed publications that have appeared as part of this work. We then archive three pieces of research that have not yet been published. The first is theoretical and experimental evidence that a popular statistical measure for comparison of community assignments favors over-resolved communities over approximations to a ground truth. The second are statistically motivated methods for measuring the quality of an approximate match of a small pattern in a large graph. The third is a new probabilistic random graph model. Statisticians favor these models for graph analysis. The new local structure graph model overcomes some of the issues with popular models such as exponential random graph models and latent variable models.

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