Transitive Reduction Research Articles

One Edge at a Time: A Novel Approach Towards Efficient Transitive Reduction Computation on DAGs

Given a directed acyclic graph ( ${DAG}$ ) $G$ , $G$ ’s transitive reduction ( TR ) $G^{tr}$ is the unique ${DAG}$ satisfying that $G^{tr}$ has the minimum number of edges and has the same transitive closure ( ${TC}$ ) as $G$ . ${TR}$ computation has been extensively studied during the past decades and was used in many applications, where the main problem is how to compute ${TR}$ efficiently for large graphs. However, existing approaches have either large space complexity or higher time complexity, which makes them cannot compute ${TR}$ efficiently on large dense graphs. We propose a novel approach for ${TR}$ computation, which takes every single edge as the basic processing unit, and utilizes existing reachability algorithms to test whether it is redundant or not. In this way, we avoid the costly graph traversal operation of existing approaches. We identify the performance bottleneck and propose a set of heuristics to sort edges, such that to reduce the average processing cost of each edge. We show by experimental results that our approach works much better than all the existing approaches, and can be faster than the state-of-the-art approach by more than two orders of magnitude on large dense graphs.

Accelerating reachability query processing based on $$\varvec{DAG}$$ DAG reduction

Answering reachability queries is one of the fundamental graph operations. The existing approaches build indexes and answer reachability queries on a directed acyclic graph (DAG) $$G$$ , which is constructed by coalescing each strongly connected component of the given directed graph $$\mathcal {G}$$ into a node of $$G$$ . Considering that $$G$$ can still be large to be processed efficiently, there are studies to further reduce $$G$$ to a smaller graph. However, these approaches suffer from either inefficiency in answering reachability queries, or cannot scale to large graphs. In this paper, we study DAG reduction to accelerate reachability query processing, which reduces the size of $$G$$ by computing transitive reduction (TR) followed by computing equivalence reduction (ER). For TR, we propose a bottom-up algorithm, namely buTR, which removes from $$G$$ all redundant edges to get the unique smallest DAG $$G^{t}$$ satisfying that $$G^{t}$$ has the same transitive closure as that of $$G$$ . For ER, we propose a divide-and-conquer algorithm, namely linear-ER. Given the result $$G^{t}$$ of TR, linear-ER gets a smaller DAG $$G^{\varepsilon }$$ in linear time based on equivalence relationship between nodes in $$G$$ . Our DAG reduction approaches (TR and ER) significantly improve the cost of time and space and can be scaled to large graphs. Based on the result of DAG reduction, we further propose a graph decomposition-based algorithm to efficiently answer reachability queries. We confirm the efficiency of our approaches by extensive experimental studies for TR, ER, and reachability query processing using 20 real datasets. The complete source code is available for download at https://pan.baidu.com/s/1skHBXXN .

Transitive reduction of citation networks

In many complex networks, the vertices are ordered in time, and edges represent causal connections. We propose methods of analysing such directed acyclic graphs taking into account the constraints of causality and highlighting the causal structure. We illustrate our approach using citation networks formed from academic papers, patents and US Supreme Court verdicts. We show how transitive reduction (TR) reveals fundamental differences in the citation practices of different areas, how it highlights particularly interesting work, and how it can correct for the effect that the age of a document has on its citation count. Finally, we transitively reduce null models of citation networks with similar degree distributions and show the difference in degree distributions after TR to illustrate the lack of causal structure in such models.

Reconstruction of large-scale regulatory networks based on perturbation graphs and transitive reduction: improved methods and their evaluation

BackgroundThe data-driven inference of intracellular networks is one of the key challenges of computational and systems biology. As suggested by recent works, a simple yet effective approach for reconstructing regulatory networks comprises the following two steps. First, the observed effects induced by directed perturbations are collected in a signed and directed perturbation graph (PG). In a second step, Transitive Reduction (TR) is used to identify and eliminate those edges in the PG that can be explained by paths and are therefore likely to reflect indirect effects.ResultsIn this work we introduce novel variants for PG generation and TR, leading to significantly improved performances. The key modifications concern: (i) use of novel statistical criteria for deriving a high-quality PG from experimental data; (ii) the application of local TR which allows only short paths to explain (and remove) a given edge; and (iii) a novel strategy to rank the edges with respect to their confidence. To compare the new methods with existing ones we not only apply them to a recent DREAM network inference challenge but also to a novel and unprecedented synthetic compendium consisting of 30 5000-gene networks simulated with varying biological and measurement error variances resulting in a total of 270 datasets. The benchmarks clearly demonstrate the superior reconstruction performance of the novel PG and TR variants compared to existing approaches. Moreover, the benchmark enabled us to draw some general conclusions. For example, it turns out that local TR restricted to paths with a length of only two is often sufficient or even favorable. We also demonstrate that considering edge weights is highly beneficial for TR whereas consideration of edge signs is of minor importance. We explain these observations from a graph-theoretical perspective and discuss the consequences with respect to a greatly reduced computational demand to conduct TR. Finally, as a realistic application scenario, we use our framework for inferring gene interactions in yeast based on a library of gene expression data measured in mutants with single knockouts of transcription factors. The reconstructed network shows a significant enrichment of known interactions, especially within the 100 most confident (and for experimental validation most relevant) edges.ConclusionsThis paper presents several major achievements. The novel methods introduced herein can be seen as state of the art for inference techniques relying on perturbation graphs and transitive reduction. Another key result of the study is the generation of a new and unprecedented large-scale in silico benchmark dataset accounting for different noise levels and providing a solid basis for unbiased testing of network inference methodologies. Finally, applying our approach to Saccharomyces cerevisiae suggested several new gene interactions with high confidence awaiting experimental validation.