Abstract. We survey recent trends in practical algorithms for balanced graph partitioning, point to applications and discuss future research directions.

We introduce NetworKit, an open-source software package for analyzing the structure of large complex networks. Appropriate algorithmic solutions are required to handle increasingly common large graph data sets containing up to billions of connections. We describe the methodology applied to develop scalable solutions to network analysis problems, including techniques like parallelization, heuristics for computationally expensive problems, efficient data structures, and modular software architecture. Our goal for the software is to package results of our algorithm engineering efforts and put them into the hands of domain experts. NetworKit is implemented as a hybrid combining the kernels written in C++ with a Python front end, enabling integration into the Python ecosystem of tested tools for data analysis and scientific computing. The package provides a wide range of functionality (including common and novel analytics algorithms and graph generators) and does so via a convenient interface. In an experimental comparison with related software, NetworKit shows the best performance on a range of typical analysis tasks.

Finding central nodes is a fundamental problem in network analysis. Betweenness centrality is a well-known measure which quantifies the importance of a node based on the fraction of shortest paths going though it. Due to the dynamic nature of many today's networks, algorithms that quickly update centrality scores have become a necessity. For betweenness, several dynamic algorithms have been proposed over the years, targeting different update types (incremental-and decremental-only, fully-dynamic). In this paper we introduce a new dynamic algorithm for updating betweenness centrality after an edge insertion or an edge weight decrease. Our method is a combination of two independent contributions: a faster algorithm for updating pairwise distances as well as number of shortest paths, and a faster algorithm for updating dependencies. Whereas the worst-case running time of our algorithm is the same as recomputation, our techniques considerably reduce the number of operations performed by existing dynamic betweenness algorithms. Our experimental evaluation on a variety of real-world networks reveals that our approach is significantly faster than the current state-of-the-art dynamic algorithms, approximately by one order of magnitude on average. ACM Subject Classification G.2.2 Graph Theory

The GraphBLAS standard (GraphBlas.org) is being developed to bring the potential of matrix-based graph algorithms to the broadest possible audience. Mathematically, the GraphBLAS defines a core set of matrix-based graph operations that can be used to implement a wide class of graph algorithms in a wide range of programming environments. This paper provides an introduction to the mathematics of the GraphBLAS. Graphs represent connections between vertices with edges. Matrices can represent a wide range of graphs using adjacency matrices or incidence matrices. Adjacency matrices are often easier to analyze while incidence matrices are often better for representing data. Fortunately, the two are easily connected by matrix multiplication. A key feature of matrix mathematics is that a very small number of matrix operations can be used to manipulate a very wide range of graphs. This composability of a small number of operations is the foundation of the GraphBLAS. A standard such as the GraphBLAS can only be effective if it has low performance overhead. Performance measurements of prototype GraphBLAS implementations indicate that the overhead is low.

We develop a multilevel algorithm for hypergraph partitioning that contracts the vertices one at a time. Using several caching and lazy-evaluation techniques during coarsening and refinement, we reduce the running time by up to two-orders of magnitude compared to a naive n-level algorithm that would be adequate for ordinary graph partitioning. The overall performance is even better than the widely used hMetis hypergraph partitioner that uses a classical multilevel algorithm with few levels. Aided by a portfolio-based approach to initial partitioning and adaptive budgeting of imbalance within recursive bipartitioning, we achieve very high quality. We assembled a large benchmark set with 310 hypergraphs stemming from application areas such VLSI, SAT solving, social networks, and scientific computing. We achieve significantly smaller cuts than hMetis and PaToH, while being faster than hMetis. Considerably larger improvements are observed for some instance classes like social networks, for bipartitioning, and for partitions with an allowed imbalance of 10%. The algorithm presented in this work forms the basis of our hypergraph partitioning framework KaHyPar (Karlsruhe Hypergraph Partitioning).