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This paper gives a short overview of a model checking tool for real-time systems. The modeling language are timed automata extended with concepts for modular modeling. The tool provides reachability analysis and refinement checking, both implemented using the data structure BDD. Good variable orderings for the BDDs are computed from the modular structure of the model and an estimate of the BDD size. This leads to a significant performance improvement compared to the tool RED and the BDD-based version of Kronos.
In this paper we analyze the efficiency of binary decision diagrams (BDDs) and clock difference diagrams (CDDs) in the verification of timed automata. Therefore we present analytical and empirical complexity results for three communication protocols. The contributions of the analyses are: Firstly, they show that BDDs and CDDs of polynomial size exist for the reachability sets of the three protocols. This is the first evidence that CDDs can grow only polynomially for models with non-trivial state space explosion. Secondly, they show that CDD-based tools, which currently use at least exponential space for two of the protocols, will only find polynomial-size CDDs if they use better variable orders, as the BDD-based tool Rabbit does. Finally, they give insight into the dependency of the BDD and CDD size on properties of the model, in particular the number of automata and the magnitude of the clock values.
We introduce an energy model whose minimum energy drawings reveal the clusters of the drawn graph. Here a cluster is a set of nodes with many internal edges and few edges to nodes outside the set. The drawings of the best-known force and energy models do not clearly show clusters for graphs whose diameter is small relative to the number of nodes. We formally characterize the minimum energy drawings of our energy model. This characterization shows in what sense the drawings separate clusters, and how the distance of separated clusters to the other nodes can be interpreted
Modularity is a widely used quality measure for graph clusterings. Its exact maximization is prohibitively expensive for large graphs. Popular heuristics progressively merge clusters starting from singletons (coarsening), and optionally improve the resulting clustering by moving vertices between clusters (refinement). This paper experimentally compares existing and new heuristics of this type with respect to their effectiveness (achieved modularity) and runtime. For coarsening, it turns out that the most widely used criterion for merging clusters (modularity increase) is outperformed by other simple criteria, and that a recent multi-step algorithm is no improvement over simple single-step coarsening for these criteria. For refinement, a new multi-level algorithm produces significantly better clusterings than conventional single-level algorithms. A comparison with published benchmark results and algorithm implementations shows that combinations of coarsening and multi-level refinement are competitive with the best algorithms in the literature.
Two natural and widely used representations for the community structure of networks are clusterings,which partition the vertex set into disjoint subsets, and layouts, which assign the vertices to positions in a metric space. This paper unifies prominent characterizations of layout quality and clustering quality, by showing that energy models of pairwise attraction and repulsion subsume Newman and Girvan's modularity measure. Layouts with optimal energy are relaxations of, and are thus consistent with, clusterings with optimal modularity, which isof practical relevance because both representations are complementary and often used together.