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Henning Meyerhenke

Publications and source records attributed to Henning Meyerhenke.

69 records · Page 4Linked to original sources
arXiv2015

Drawing Large Graphs by Multilevel Maxent-Stress Optimization

Drawing large graphs appropriately is an important step for the visual analysis of data from real-world networks. Here we present a novel multilevel algorithm to compute a graph layout with respect to a recently proposed metric that combines layout stress and entropy. As opposed to previous work, we do not solve the linear systems of the maxent-stress metric with a typical numerical solver. Instead we use a simple local iterative scheme within a multilevel approach. To accelerate local optimization, we approximate long-range forces and use shared-memory parallelism. Our experiments validate the high potential of our approach, which is particularly appealing for dynamic graphs. In comparison to the previously best maxent-stress optimizer, which is sequential, our parallel implementation is on average 30 times faster already for static graphs (and still faster if executed on one thread) while producing a comparable solution quality.

cs.DS↗
arXiv2015

Fully-dynamic Approximation of Betweenness Centrality

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 evolving networks. In previous work we proposed the first semi-dynamic algorithms that recompute an approximation of betweenness in connected graphs after batches of edge insertions. In this paper we propose the first fully-dynamic approximation algorithms (for weighted and unweighted undirected graphs that need not to be connected) with a provable guarantee on the maximum approximation error. The transfer to fully-dynamic and disconnected graphs implies additional algorithmic problems that could be of independent interest. In particular, we propose a new upper bound on the vertex diameter for weighted undirected graphs. For both weighted and unweighted graphs, we also propose the first fully-dynamic algorithms that keep track of such upper bound. In addition, we extend our former algorithm for semi-dynamic BFS to batches of both edge insertions and deletions. Using approximation, our algorithms are the first to make in-memory computation of betweenness in fully-dynamic networks with millions of edges feasible. Our experiments show that they can achieve substantial speedups compared to recomputation, up to several orders of magnitude.

cs.DS↗
arXiv2015

Structure-Preserving Sparsification of 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 by these scores. 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 100 Facebook social networks with respect to network properties including diameter, connected components, community structure, and multiple node centrality measures. Experiments with our implementations of the sparsification methods (using 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. Furthermore, the experimental results allow us to differentiate the behavior of different methods and show which method is suitable with respect to which property. Our Local Degree method is fast enough for large-scale networks and performs well across a wider range of properties than previously proposed methods.

cs.SI↗
arXiv2015

n-Level Hypergraph Partitioning

We develop a multilevel algorithm for hypergraph partitioning that contracts the vertices one at a time and thus allows very high quality. This includes a rating function that avoids nonuniform vertex weights, an efficient "semi-dynamic" hypergraph data structure, a very fast coarsening algorithm, and two new local search algorithms. One is a $k$-way hypergraph adaptation of Fiduccia-Mattheyses local search and gives high quality at reasonable cost. The other is an adaptation of size-constrained label propagation to hypergraphs. Comparisons with hMetis and PaToH indicate that the new algorithm yields better quality over several benchmark sets and has a running time that is comparable to hMetis. Using label propagation local search is several times faster than hMetis and gives better quality than PaToH for a VLSI benchmark set.

cs.DS↗
arXiv2015

Fast generation of complex networks with underlying hyperbolic geometry

Complex networks have become increasingly popular for modeling various real-world phenomena. Realistic generative network models are important in this context as they avoid privacy concerns of real data and simplify complex network research regarding data sharing, reproducibility, and scalability studies. \emph{Random hyperbolic graphs} are a well-analyzed family of geometric graphs. Previous work provided empirical and theoretical evidence that this generative graph model creates networks with non-vanishing clustering and other realistic features. However, the investigated networks in previous applied work were small, possibly due to the quadratic running time of a previous generator. In this work we provide the first generation algorithm for these networks with subquadratic running time. We prove a time complexity of $O((n^{3/2}+m) \log n)$ with high probability for the generation process. This running time is confirmed by experimental data with our implementation. The acceleration stems primarily from the reduction of pairwise distance computations through a polar quadtree, which we adapt to hyperbolic space for this purpose. In practice we improve the running time of a previous implementation by at least two orders of magnitude this way. Networks with billions of edges can now be generated in a few minutes. Finally, we evaluate the largest networks of this model published so far. Our empirical analysis shows that important features are retained over different graph densities and degree distributions.

cs.DS↗
arXiv2015

Algorithms for Mapping Parallel Processes onto Grid and Torus Architectures

Static mapping is the assignment of parallel processes to the processing elements (PEs) of a parallel system, where the assignment does not change during the application's lifetime. In our scenario we model an application's computations and their dependencies by an application graph. This graph is first partitioned into (nearly) equally sized blocks. These blocks need to communicate at block boundaries. To assign the processes to PEs, our goal is to compute a communication-efficient bijective mapping between the blocks and the PEs. This approach of partitioning followed by bijective mapping has many degrees of freedom. Thus, users and developers of parallel applications need to know more about which choices work for which application graphs and which parallel architectures. To this end, we not only develop new mapping algorithms (derived from known greedy methods). We also perform extensive experiments involving different classes of application graphs (meshes and complex networks), architectures of parallel computers (grids and tori), as well as different partitioners and mapping algorithms. Surprisingly, the quality of the partitions, unless very poor, has little influence on the quality of the mapping. More importantly, one of our new mapping algorithms always yields the best results in terms of the quality measure maximum congestion when the application graphs are complex networks. In case of meshes as application graphs, this mapping algorithm always leads in terms of maximum congestion AND maximum dilation, another common quality measure.

cs.DS↗
arXiv2015

Is Nearly-linear the same in Theory and Practice? A Case Study with a Combinatorial Laplacian Solver

Linear system solving is one of the main workhorses in applied mathematics. Recently, theoretical computer scientists have contributed sophisticated algorithms for solving linear systems with symmetric diagonally dominant matrices (a class to which Laplacian matrices belong) in provably nearly-linear time. While these algorithms are highly interesting from a theoretical perspective, there are no published results how they perform in practice. With this paper we address this gap. We provide the first implementation of the combinatorial solver by [Kelner et al., STOC 2013], which is particularly appealing for implementation due to its conceptual simplicity. The algorithm exploits that a Laplacian matrix corresponds to a graph; solving Laplacian linear systems amounts to finding an electrical flow in this graph with the help of cycles induced by a spanning tree with the low-stretch property. The results of our comprehensive experimental study are ambivalent. They confirm a nearly-linear running time, but for reasonable inputs the constant factors make the solver much slower than methods with higher asymptotic complexity. One other aspect predicted by theory is confirmed by our findings, though: Spanning trees with lower stretch indeed reduce the solver's running time. Yet, simple spanning tree algorithms perform in practice better than those with a guaranteed low stretch.

cs.DS↗
arXiv2015

Engineering Parallel Algorithms for Community Detection in Massive Networks

The amount of graph-structured data has recently experienced an enormous growth in many applications. To transform such data into useful information, fast analytics algorithms and software tools are necessary. One common graph analytics kernel is disjoint community detection (or graph clustering). Despite extensive research on heuristic solvers for this task, only few parallel codes exist, although parallelism will be necessary to scale to the data volume of real-world applications. We address the deficit in computing capability by a flexible and extensible community detection framework with shared-memory parallelism. Within this framework we design and implement efficient parallel community detection heuristics: A parallel label propagation scheme; the first large-scale parallelization of the well-known Louvain method, as well as an extension of the method adding refinement; and an ensemble scheme combining the above. In extensive experiments driven by the algorithm engineering paradigm, we identify the most successful parameters and combinations of these algorithms. We also compare our implementations with state-of-the-art competitors. The processing rate of our fastest algorithm often reaches 50M edges/second. We recommend the parallel Louvain method and our variant with refinement as both qualitatively strong and fast. Our methods are suitable for massive data sets with billions of edges.

cs.DC↗
arXiv2015

Parallel Graph Partitioning for Complex Networks

Processing large complex networks like social networks or web graphs has recently attracted considerable interest. In order to do this in parallel, we need to partition them into pieces of about equal size. Unfortunately, previous parallel graph partitioners originally developed for more regular mesh-like networks do not work well for these networks. This paper addresses this problem by parallelizing and adapting the label propagation technique originally developed for graph clustering. By introducing size constraints, label propagation becomes applicable for both the coarsening and the refinement phase of multilevel graph partitioning. We obtain very high quality by applying a highly parallel evolutionary algorithm to the coarsened graph. The resulting system is both more scalable and achieves higher quality than state-of-the-art systems like ParMetis or PT-Scotch. For large complex networks the performance differences are very big. For example, our algorithm can partition a web graph with 3.3 billion edges in less than sixteen seconds using 512 cores of a high performance cluster while producing a high quality partition -- none of the competing systems can handle this graph on our system.

cs.DC↗
arXiv2014

Approximating Betweenness Centrality in Large Evolving Networks

Betweenness centrality ranks the importance of nodes by their participation in all shortest paths of the network. Therefore computing exact betweenness values is impractical in large networks. For static networks, approximation based on randomly sampled paths has been shown to be significantly faster in practice. However, for dynamic networks, no approximation algorithm for betweenness centrality is known that improves on static recomputation. We address this deficit by proposing two incremental approximation algorithms (for weighted and unweighted connected graphs) which provide a provable guarantee on the absolute approximation error. Processing batches of edge insertions, our algorithms yield significant speedups up to a factor of $10^4$ compared to restarting the approximation. This is enabled by investing memory to store and efficiently update shortest paths. As a building block, we also propose an asymptotically faster algorithm for updating the SSSP problem in unweighted graphs. Our experimental study shows that our algorithms are the first to make in-memory computation of a betweenness ranking practical for million-edge semi-dynamic networks. Moreover, our results show that the accuracy is even better than the theoretical guarantees in terms of absolutes errors and the rank of nodes is well preserved, in particular for those with high betweenness.

cs.SI↗
arXiv2014

Partitioning Complex Networks via Size-constrained Clustering

The most commonly used method to tackle the graph partitioning problem in practice is the multilevel approach. During a coarsening phase, a multilevel graph partitioning algorithm reduces the graph size by iteratively contracting nodes and edges until the graph is small enough to be partitioned by some other algorithm. A partition of the input graph is then constructed by successively transferring the solution to the next finer graph and applying a local search algorithm to improve the current solution. In this paper, we describe a novel approach to partition graphs effectively especially if the networks have a highly irregular structure. More precisely, our algorithm provides graph coarsening by iteratively contracting size-constrained clusterings that are computed using a label propagation algorithm. The same algorithm that provides the size-constrained clusterings can also be used during uncoarsening as a fast and simple local search algorithm. Depending on the algorithm's configuration, we are able to compute partitions of very high quality outperforming all competitors, or partitions that are comparable to the best competitor in terms of quality, hMetis, while being nearly an order of magnitude faster on average. The fastest configuration partitions the largest graph available to us with 3.3 billion edges using a single machine in about ten minutes while cutting less than half of the edges than the fastest competitor, kMetis.

cs.DC↗
arXiv2014

Tree-based Coarsening and Partitioning of Complex Networks

Many applications produce massive complex networks whose analysis would benefit from parallel processing. Parallel algorithms, in turn, often require a suitable network partition. For solving optimization tasks such as graph partitioning on large networks, multilevel methods are preferred in practice. Yet, complex networks pose challenges to established multilevel algorithms, in particular to their coarsening phase. One way to specify a (recursive) coarsening of a graph is to rate its edges and then contract the edges as prioritized by the rating. In this paper we (i) define weights for the edges of a network that express the edges' importance for connectivity, (ii) compute a minimum weight spanning tree $T^m$ with respect to these weights, and (iii) rate the network edges based on the conductance values of $T^m$'s fundamental cuts. To this end, we also (iv) develop the first optimal linear-time algorithm to compute the conductance values of \emph{all} fundamental cuts of a given spanning tree. We integrate the new edge rating into a leading multilevel graph partitioner and equip the latter with a new greedy postprocessing for optimizing the maximum communication volume (MCV). Experiments on bipartitioning frequently used benchmark networks show that the postprocessing already reduces MCV by 11.3%. Our new edge rating further reduces MCV by 10.3% compared to the previously best rating with the postprocessing in place for both ratings. In total, with a modest increase in running time, our new approach reduces the MCV of complex network partitions by 20.4%.

cs.DS↗
arXiv2014

Finding all Convex Cuts of a Plane Graph in Polynomial Time

Convexity is a notion that has been defined for subsets of $\RR^n$ and for subsets of general graphs. A convex cut of a graph $G=(V, E)$ is a $2$-partition $V_1 \dot{\cup} V_2=V$ such that both $V_1$ and $V_2$ are convex, \ie shortest paths between vertices in $V_i$ never leave $V_i$, $i \in \{1, 2\}$. Finding convex cuts is $\mathcal{NP}$-hard for general graphs. To characterize convex cuts, we employ the Djokovic relation, a reflexive and symmetric relation on the edges of a graph that is based on shortest paths between the edges' end vertices. It is known for a long time that, if $G$ is bipartite and the Djokovic relation is transitive on $G$, \ie $G$ is a partial cube, then the cut-sets of $G$'s convex cuts are precisely the equivalence classes of the Djokovic relation. In particular, any edge of $G$ is contained in the cut-set of exactly one convex cut. We first characterize a class of plane graphs that we call {\em well-arranged}. These graphs are not necessarily partial cubes, but any edge of a well-arranged graph is contained in the cut-set(s) of at least one convex cut. We also present an algorithm that uses the Djokovic relation for computing all convex cuts of a (not necessarily plane) bipartite graph in $\bigO(|E|^3)$ time. Specifically, a cut-set is the cut-set of a convex cut if and only if the Djokovic relation holds for any pair of edges in the cut-set. We then characterize the cut-sets of the convex cuts of a general graph $H$ using two binary relations on edges: (i) the Djokovic relation on the edges of a subdivision of $H$, where any edge of $H$ is subdivided into exactly two edges and (ii) a relation on the edges of $H$ itself that is not the Djokovic relation. Finally, we use this characterization to present the first algorithm for finding all convex cuts of a plane graph in polynomial time.

cs.DS↗
arXiv2013

Static and Dynamic Aspects of Scientific Collaboration Networks

Collaboration networks arise when we map the connections between scientists which are formed through joint publications. These networks thus display the social structure of academia, and also allow conclusions about the structure of scientific knowledge. Using the computer science publication database DBLP, we compile relations between authors and publications as graphs and proceed with examining and quantifying collaborative relations with graph-based methods. We review standard properties of the network and rank authors and publications by centrality. Additionally, we detect communities with modularity-based clustering and compare the resulting clusters to a ground-truth based on conferences and thus topical similarity. In a second part, we are the first to combine DBLP network data with data from the Dagstuhl Seminars: We investigate whether seminars of this kind, as social and academic events designed to connect researchers, leave a visible track in the structure of the collaboration network. Our results suggest that such single events are not influential enough to change the network structure significantly. However, the network structure seems to influence a participant's decision to accept or decline an invitation.

cs.SI↗