arXiv subjects

Henning Meyerhenke

Publications and source records attributed to Henning Meyerhenke.

At least 55 records · Page 3Linked to original sources

arXiv2018

Topology-induced Enhancement of Mappings

In this paper we propose a new method to enhance a mapping $μ(\cdot)$ of a parallel application's computational tasks to the processing elements (PEs) of a parallel computer. The idea behind our method \mswap is to enhance such a mapping by drawing on the observation that many topologies take the form of a partial cube. This class of graphs includes all rectangular and cubic meshes, any such torus with even extensions in each dimension, all hypercubes, and all trees. Following previous work, we represent the parallel application and the parallel computer by graphs $G_a = (V_a, E_a)$ and $G_p = (V_p, E_p)$. $G_p$ being a partial cube allows us to label its vertices, the PEs, by bitvectors such that the cost of exchanging one unit of information between two vertices $u_p$ and $v_p$ of $G_p$ amounts to the Hamming distance between the labels of $u_p$ and $v_p$. By transferring these bitvectors from $V_p$ to $V_a$ via $μ^{-1}(\cdot)$ and extending them to be unique on $V_a$, we can enhance $μ(\cdot)$ by swapping labels of $V_a$ in a new way. Pairs of swapped labels are local \wrt the PEs, but not \wrt $G_a$. Moreover, permutations of the bitvectors' entries give rise to a plethora of hierarchies on the PEs. Through these hierarchies we turn \mswap into a hierarchical method for improving $μ(\cdot)$ that is complementary to state-of-the-art methods for computing $μ(\cdot)$ in the first place. In our experiments we use \mswap to enhance mappings of complex networks onto rectangular meshes and tori with 256 and 512 nodes, as well as hypercubes with 256 nodes. It turns out that common quality measures of mappings derived from state-of-the-art algorithms can be improved considerably.

cs.DC↗

arXiv2018

Updating Dynamic Random Hyperbolic Graphs in Sublinear Time

Generative network models play an important role in algorithm development, scaling studies, network analysis, and realistic system benchmarks for graph data sets. A complex network model gaining considerable popularity builds random hyperbolic graphs, generated by distributing points within a disk in the hyperbolic plane and then adding edges between points with a probability depending on their hyperbolic distance. We present a dynamic extension to model gradual network change, while preserving at each step the point position probabilities. To process the dynamic changes efficiently, we formalize the concept of a probabilistic neighborhood: Let $P$ be a set of $n$ points in Euclidean or hyperbolic space, $q$ a query point, $\operatorname{dist}$ a distance metric, and $f : \mathbb{R}^+ \rightarrow [0,1]$ a monotonically decreasing function. Then, the probabilistic neighborhood $N(q, f)$ of $q$ with respect to $f$ is a random subset of $P$ and each point $p \in P$ belongs to $N(q,f)$ with probability $f(\operatorname{dist}(p,q))$. We present a fast, sublinear-time query algorithm to sample probabilistic neighborhoods from planar point sets. For certain distributions of planar $P$, we prove that our algorithm answers a query in $O((|N(q,f)| + \sqrt{n})\log n)$ time with high probability. This enables us to process a node movement in random hyperbolic graphs in sublinear time, resulting in a speedup of about one order of magnitude in practice compared to the fastest previous approach. Apart from that, our query algorithm is also applicable to Euclidean geometry, making it of independent interest for other sampling or probabilistic spreading scenarios.

cs.DS↗

arXiv2018

Many-to-many Correspondences between Partitions: Introducing a Cut-based Approach

Let $\mathcal{P}$ and $\mathcal{P}'$ be finite partitions of the set $V$. Finding good correspondences between the parts of $\mathcal{P}$ and those of $\mathcal{P}'$ is helpful in classification, pattern recognition, and network analysis. Unlike common similarity measures for partitions that yield only a single value, we provide specifics on how $\mathcal{P}$ and $\mathcal{P'}$ correspond to each other. To this end, we first define natural collections of best correspondences under three constraints \cone, \ctwo, and \cthree. In case of \cone, the best correspondences form a minimum cut basis of a certain bipartite graph, whereas the other two lead to minimum cut bases of $\mathcal{P}$ \wrt $\mathcal{P}'$. We also introduce a constraint, \cfour, which tightens \cthree; both are useful for finding consensus partitions. We then develop branch-and-bound algorithms for finding minimum $P_s$-$P_t$ cuts of $\mathcal{P}$ and thus $\vert \mathcal{P} \vert -1$ best correspondences under \ctwo, \cthree, and \cfour, respectively. In a case study, we use the correspondences to gain insight into a community detection algorithm. The results suggest, among others, that only very minor losses in the quality of the correspondences occur if the branch-and-bound algorithm is restricted to its greedy core. Thus, even for graphs with more than half a million nodes and hundreds of communities, we can find hundreds of best or almost best correspondences in less than a minute.

cs.DM↗

arXiv2017

Computing Top-k Closeness Centrality in Fully-dynamic Graphs

Closeness is a widely-studied centrality measure. Since it requires all pairwise distances, computing closeness for all nodes is infeasible for large real-world networks. However, for many applications, it is only necessary to find the k most central nodes and not all closeness values. Prior work has shown that computing the top-k nodes with highest closeness can be done much faster than computing closeness for all nodes in real-world networks. However, for networks that evolve over time, no dynamic top-k closeness algorithm exists that improves on static recomputation. In this paper, we present several techniques that allow us to efficiently compute the k nodes with highest (harmonic) closeness after an edge insertion or an edge deletion. Our algorithms use information obtained during earlier computations to omit unnecessary work. However, they do not require asymptotically more memory than the static algorithms (i. e., linear in the number of nodes). We propose separate algorithms for complex networks (which exhibit the small-world property) and networks with large diameter such as street networks, and we compare them against static recomputation on a variety of real-world networks. On many instances, our dynamic algorithms are two orders of magnitude faster than recomputation; on some large graphs, we even reach average speedups between $10^3$ and $10^4$.

cs.DS↗

arXiv2017

Maxent-Stress Optimization of 3D Biomolecular Models

Knowing a biomolecule's structure is inherently linked to and a prerequisite for any detailed understanding of its function. Significant effort has gone into developing technologies for structural characterization. These technologies do not directly provide 3D structures; instead they typically yield noisy and erroneous distance information between specific entities such as atoms or residues, which have to be translated into consistent 3D models. Here we present an approach for this translation process based on maxent-stress optimization. Our new approach extends the original graph drawing method for the new application's specifics by introducing additional constraints and confidence values as well as algorithmic components. Extensive experiments demonstrate that our approach infers structural models (i. e., sensible 3D coordinates for the molecule's atoms) that correspond well to the distance information, can handle noisy and error-prone data, and is considerably faster than established tools. Our results promise to allow domain scientists nearly-interactive structural modeling based on distance constraints.

cs.DS↗

arXiv2017

Shared Memory Parallel Subgraph Enumeration

The subgraph enumeration problem asks us to find all subgraphs of a target graph that are isomorphic to a given pattern graph. Determining whether even one such isomorphic subgraph exists is NP-complete---and therefore finding all such subgraphs (if they exist) is a time-consuming task. Subgraph enumeration has applications in many fields, including biochemistry and social networks, and interestingly the fastest algorithms for solving the problem for biochemical inputs are sequential. Since they depend on depth-first tree traversal, an efficient parallelization is far from trivial. Nevertheless, since important applications produce data sets with increasing difficulty, parallelism seems beneficial. We thus present here a shared-memory parallelization of the state-of-the-art subgraph enumeration algorithms RI and RI-DS (a variant of RI for dense graphs) by Bonnici et al. [BMC Bioinformatics, 2013]. Our strategy uses work stealing and our implementation demonstrates a significant speedup on real-world biochemical data---despite a highly irregular data access pattern. We also improve RI-DS by pruning the search space better; this further improves the empirical running times compared to the already highly tuned RI-DS.

cs.DC↗

arXiv2017

Computing top-k Closeness Centrality Faster in Unweighted Graphs

Given a connected graph $G=(V,E)$, the closeness centrality of a vertex $v$ is defined as $\frac{n-1}{\sum_{w \in V} d(v,w)}$. This measure is widely used in the analysis of real-world complex networks, and the problem of selecting the $k$ most central vertices has been deeply analysed in the last decade. However, this problem is computationally not easy, especially for large networks: in the first part of the paper, we prove that it is not solvable in time $Ø(|E|^{2-ε})$ on directed graphs, for any constant $ε>0$, under reasonable complexity assumptions. Furthermore, we propose a new algorithm for selecting the $k$ most central nodes in a graph: we experimentally show that this algorithm improves significantly both the textbook algorithm, which is based on computing the distance between all pairs of vertices, and the state of the art. For example, we are able to compute the top $k$ nodes in few dozens of seconds in real-world networks with millions of nodes and edges. Finally, as a case study, we compute the $10$ most central actors in the IMDB collaboration network, where two actors are linked if they played together in a movie, and in the Wikipedia citation network, which contains a directed edge from a page $p$ to a page $q$ if $p$ contains a link to $q$.

cs.DS↗

arXiv2017

Faster Betweenness Centrality Updates in Evolving Networks

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.

cs.DS↗

arXiv2017

Generating realistic scaled complex networks

Research on generative models is a central project in the emerging field of network science, and it studies how statistical patterns found in real networks could be generated by formal rules. Output from these generative models is then the basis for designing and evaluating computational methods on networks, and for verification and simulation studies. During the last two decades, a variety of models has been proposed with an ultimate goal of achieving comprehensive realism for the generated networks. In this study, we (a) introduce a new generator, termed ReCoN; (b) explore how ReCoN and some existing models can be fitted to an original network to produce a structurally similar replica, (c) use ReCoN to produce networks much larger than the original exemplar, and finally (d) discuss open problems and promising research directions. In a comparative experimental study, we find that ReCoN is often superior to many other state-of-the-art network generation methods. We argue that ReCoN is a scalable and effective tool for modeling a given network while preserving important properties at both micro- and macroscopic scales, and for scaling the exemplar data by orders of magnitude in size.

cs.SI↗

arXiv2016

Querying Probabilistic Neighborhoods in Spatial Data Sets Efficiently

$\newcommand{\dist}{\operatorname{dist}}$ In this paper we define the notion of a probabilistic neighborhood in spatial data: Let a set $P$ of $n$ points in $\mathbb{R}^d$, a query point $q \in \mathbb{R}^d$, a distance metric $\dist$, and a monotonically decreasing function $f : \mathbb{R}^+ \rightarrow [0,1]$ be given. Then a point $p \in P$ belongs to the probabilistic neighborhood $N(q, f)$ of $q$ with respect to $f$ with probability $f(\dist(p,q))$. We envision applications in facility location, sensor networks, and other scenarios where a connection between two entities becomes less likely with increasing distance. A straightforward query algorithm would determine a probabilistic neighborhood in $Θ(n\cdot d)$ time by probing each point in $P$. To answer the query in sublinear time for the planar case, we augment a quadtree suitably and design a corresponding query algorithm. Our theoretical analysis shows that -- for certain distributions of planar $P$ -- our algorithm answers a query in $O((|N(q,f)| + \sqrt{n})\log n)$ time with high probability (whp). This matches up to a logarithmic factor the cost induced by quadtree-based algorithms for deterministic queries and is asymptotically faster than the straightforward approach whenever $|N(q,f)| \in o(n / \log n)$. As practical proofs of concept we use two applications, one in the Euclidean and one in the hyperbolic plane. In particular, our results yield the first generator for random hyperbolic graphs with arbitrary temperatures in subquadratic time. Moreover, our experimental data show the usefulness of our algorithm even if the point distribution is unknown or not uniform: The running time savings over the pairwise probing approach constitute at least one order of magnitude already for a modest number of points and queries.

cs.DS↗

arXiv2016

Mathematical Foundations of the GraphBLAS

The GraphBLAS standard (GraphBlas.org) is being developed to bring the potential of matrix based graph algorithms to the broadest possible audience. Mathematically the Graph- BLAS 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 mul- tiplication. 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 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.

cs.MS↗

arXiv2016

Generating massive complex networks with hyperbolic geometry faster in practice

Generative network models play an important role in algorithm development, scaling studies, network analysis, and realistic system benchmarks for graph data sets. The commonly used graph-based benchmark model R-MAT has some drawbacks concerning realism and the scaling behavior of network properties. A complex network model gaining considerable popularity builds random hyperbolic graphs, generated by distributing points within a disk in the hyperbolic plane and then adding edges between points whose hyperbolic distance is below a threshold. We present in this paper a fast generation algorithm for such graphs. Our experiments show that our new generator achieves speedup factors of 3-60 over the best previous implementation. One billion edges can now be generated in under one minute on a shared-memory workstation. Furthermore, we present a dynamic extension to model gradual network change, while preserving at each step the point position probabilities.

cs.DS↗

arXiv2016

Better partitions of protein graphs for subsystem quantum chemistry

Determining the interaction strength between proteins and small molecules is key to analyzing their biological function. Quantum-mechanical calculations such as \emph{Density Functional Theory} (DFT) give accurate and theoretically well-founded results. With common implementations the running time of DFT calculations increases quadratically with molecule size. Thus, numerous subsystem-based approaches have been developed to accelerate quantum-chemical calculations. These approaches partition the protein into different fragments, which are treated separately. Interactions between different fragments are approximated and introduce inaccuracies in the calculated interaction energies. To minimize these inaccuracies, we represent the amino acids and their interactions as a weighted graph in order to apply graph partitioning. None of the existing graph partitioning work can be directly used, though, due to the unique constraints in partitioning such protein graphs. We therefore present and evaluate several algorithms, partially building upon established concepts, but adapted to handle the new constraints. For the special case of partitioning a protein along the main chain, we also present an efficient dynamic programming algorithm that yields provably optimal results. In the general scenario our algorithms usually improve the previous approach significantly and take at most a few seconds.

cs.DS↗

arXiv2016

Structure-Preserving Sparsification Methods for Social Networks

Sparsification reduces the size of networks while preserving structural and statistical properties of interest. Various sparsifying algorithms have been proposed in different contexts. We contribute the first systematic conceptual and experimental comparison of \textit{edge sparsification} methods on a diverse set of network properties. It is shown that they can be understood as methods for rating edges by importance and then filtering globally or locally by these scores. We show that applying a local filtering technique improves the preservation of all kinds of properties. In addition, we propose a new sparsification method (\textit{Local Degree}) which preserves edges leading to local hub nodes. All methods are evaluated on a set of social networks from Facebook, Google+, Twitter and LiveJournal with respect to network properties including diameter, connected components, community structure, multiple node centrality measures and the behavior of epidemic simulations. In order to assess the preservation of the community structure, we also include experiments on synthetically generated networks with ground truth communities. Experiments with our implementations of the sparsification methods (included in the open-source network analysis tool suite NetworKit) show that many network properties can be preserved down to about 20\% of the original set of edges for sparse graphs with a reasonable density. The experimental results allow us to differentiate the behavior of different methods and show which method is suitable with respect to which property. While our Local Degree method is best for preserving connectivity and short distances, other newly introduced local variants are best for preserving the community structure.

cs.SI↗

arXiv2016

An Empirical Comparison of Big Graph Frameworks in the Context of Network Analysis

Complex networks are relational data sets commonly represented as graphs. The analysis of their intricate structure is relevant to many areas of science and commerce, and data sets may reach sizes that require distributed storage and processing. We describe and compare programming models for distributed computing with a focus on graph algorithms for large-scale complex network analysis. Four frameworks - GraphLab, Apache Giraph, Giraph++ and Apache Flink - are used to implement algorithms for the representative problems Connected Components, Community Detection, PageRank and Clustering Coefficients. The implementations are executed on a computer cluster to evaluate the frameworks' suitability in practice and to compare their performance to that of the single-machine, shared-memory parallel network analysis package NetworKit. Out of the distributed frameworks, GraphLab and Apache Giraph generally show the best performance. In our experiments a cluster of eight computers running Apache Giraph enables the analysis of a network with about 2 billion edges, which is too large for a single machine of the same type. However, for networks that fit into memory of one machine, the performance of the shared-memory parallel implementation is far better than the distributed ones. The study provides experimental evidence for selecting the appropriate framework depending on the task and data volume.

cs.DC↗

arXiv2015

NetworKit: A Tool Suite for Large-scale Complex Network Analysis

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.

cs.SI↗

arXiv2015

k-way Hypergraph Partitioning via n-Level Recursive Bisection

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).

cs.DS↗

arXiv2015

Approximating Betweenness Centrality in Fully-dynamic Networks

Betweenness is a well-known centrality measure that ranks the nodes of a network according to their participation in shortest paths. Since an exact computation is prohibitive in large networks, several approximation algorithms have been proposed. Besides that, recent years have seen the publication of dynamic algorithms for efficient recomputation of betweenness in networks that change over time. In this paper we propose the first betweenness centrality approximation algorithms with a provable guarantee on the maximum approximation error for dynamic networks. Several new intermediate algorithmic results contribute to the respective approximation algorithms: (i) new upper bounds on the vertex diameter, (ii) the first fully-dynamic algorithm for updating an approximation of the vertex diameter in undirected graphs, and (iii) an algorithm with lower time complexity for updating single-source shortest paths in unweighted graphs after a batch of edge actions. Using approximation, our algorithms are the first to make in-memory computation of betweenness in dynamic networks with millions of edges feasible. Our experiments show that our algorithms can achieve substantial speedups compared to recomputation, up to several orders of magnitude. Moreover, the approximation accuracy is usually significantly better than the theoretical guarantee in terms of absolute error. More importantly, for reasonably small approximation error thresholds, the rank of nodes is well preserved, in particular for nodes with high betweenness.

cs.DS↗