CuSP

Graph analytics systems must analyze graphs with billions of vertices and edges which require several terabytes of storage. Distributed-memory clusters are often used for analyzing such large graphs since the main memory of a single machine is usually restricted to a few hundreds of gigabytes. This requires partitioning the graph among the machines in the cluster. Existing graph analytics systems use a built-in partitioner that incorporates a particular partitioning policy, but the best policy is dependent on the algorithm, input graph, and platform. Therefore, built-in partitioners are not sufficiently flexible. Stand-alone graph partitioners are available, but they too implement only a few policies. CuSP is a fast streaming edge partitioning framework which permits users to specify the desired partitioning policy at a high level of abstraction and quickly generates highquality graph partitions. For example, it can partition wdc12, the largest publicly available web-crawl graph with 4 billion vertices and 129 billion edges, in under 2 minutes for clusters with 128 machines. Our experiments show that it can produce quality partitions 6× faster on average than the state-of-theart stand-alone partitioner in the literature while supporting a wider range of partitioning policies.

BGRAP: Balanced GRAph Partitioning Algorithm for Large Graphs

The definition of effective strategies for graph partitioning is a major challenge in distributed environments since an effective graph partitioning allows to considerably improve the performance of large graph data analytics computations. In this paper, we propose a multi-objective and scalable Balanced GRAph Partitioning (\algo) algorithm, based on Label Propagation (LP) approach, to produce balanced graph partitions. \algo defines a new efficient initialization procedure and different objective functions to deal with either vertex or edge balance constraints while considering edge direction in graphs. \algo is implemented of top of the open source distributed graph processing system Giraph. The experiments are performed on various graphs with different structures and sizes (going up to 50.6M vertices and 1.9B edges) while varying the number of partitions. We evaluate \algo using several quality measures and the computation time. The results show that \algo (i) provides a good balance while reducing the cuts between the different computed partitions (ii) reduces the global computation time, compared to LP-based algorithms.

A theory of spectral partitions of metric graphs

We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815–838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic—rather than numerical—results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45–72, 2005), Helffer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:101–138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

MetaCell: analysis of single-cell RNA-seq data using K-nn graph partitions

scRNA-seq profiles each represent a highly partial sample of mRNA molecules from a unique cell that can never be resampled, and robust analysis must separate the sampling effect from biological variance. We describe a methodology for partitioning scRNA-seq datasets into metacells: disjoint and homogenous groups of profiles that could have been resampled from the same cell. Unlike clustering analysis, our algorithm specializes at obtaining granular as opposed to maximal groups. We show how to use metacells as building blocks for complex quantitative transcriptional maps while avoiding data smoothing. Our algorithms are implemented in the MetaCell R/C++ software package.

Correspondence between Multilevel Graph Partitions and Tree Decompositions

We present a mapping between rooted tree decompositions and node separator based multilevel graph partitions. Significant research into both tree decompositions and graph partitions exists. We hope that our result allows for an easier knowledge transfer between the two research avenues.

Spanning Properties of Yao and 𝜃-Graphs in the Presence of Constraints

We present improved upper bounds on the spanning ratio of constrained [Formula: see text]-graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text], and adds an edge to the closest vertex in each cone. Constrained Yao-graphs have the additional property that no edge properly intersects any of the given line segment constraints. Constrained [Formula: see text]-graphs are similar to constrained Yao-graphs, but use a different method to determine the closest vertex. We present tight bounds on the spanning ratio of a large family of constrained [Formula: see text]-graphs. We show that constrained [Formula: see text]-graphs with [Formula: see text] ([Formula: see text] and integer) cones have a tight spanning ratio of [Formula: see text], where [Formula: see text] is [Formula: see text]. We also present improved upper bounds on the spanning ratio of the other families of constrained [Formula: see text]-graphs. These bounds match the current upper bounds in the unconstrained setting. We also show that constrained Yao-graphs with an even number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text] and constrained Yao-graphs with an odd number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text]. As is the case with constrained [Formula: see text]-graphs, these bounds match the current upper bounds in the unconstrained setting, which implies that like in the unconstrained setting using more cones can make the spanning ratio worse.