Dorothea Wagner

On Modularity Clustering

Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, particularly in the complex systems literature, although its properties are not well understood. We study the problem of finding clusterings with maximum modularity, thus providing theoretical foundations for past and present work based on this measure. More precisely, we prove the conjectured hardness of maximizing modularity both in the general case and with the restriction to cuts and give an Integer Linear Programming formulation. This is complemented by first insights into the behavior and performance of the commonly applied greedy agglomerative approach.

Engineering Route Planning Algorithms

Algorithms for route planning in transportation networks have recently undergone a rapid development, leading to methods that are up to three million times faster than Dijkstras algorithm. We give an overview of the techniques enabling this development and point out frontiers of ongoing research on more challenging variants of the problem that include dynamically changing networks, time-dependent routing, and flexible objective functions.

Experiments on Graph Clustering Algorithms

A promising approach to graph clustering is based on the intuitive notion of intra-cluster density vs. inter-cluster sparsity. While both formalizations and algorithms focusing on particular aspects of this rather vague concept have been proposed no conclusive argument on their appropriateness has been given. As a first step towards understanding the consequences of particular con- ceptions, we conducted an experimental evaluation of graph clustering approaches. By combining proven techniques from graph partitioning and geometric clustering, we also introduce a new approach that compares favorably.

Analysis and visualization of social networks

Social network analysis is a subdiscipline of the social sciences using graph-theoretic concepts to understand and explain social structure.We describe the main issues in social network analysis. General principles are laid out for visualizing network data in a way that conveys structural information relevant to specific research questions. Based on these innovative graph drawing techniques integrating the analysis and visualization of social networks are introduced.

Finding, counting and listing all triangles in large graphs, an experimental study

In the past, the fundamental graph problem of triangle counting and listing has been studied intensively from a theoretical point of view. Recently, triangle counting has also become a widely used tool in network analysis. Due to the very large size of networks like the Internet, WWW or social networks, the efficiency of algorithms for triangle counting and listing is an important issue. The main intention of this work is to evaluate the practicability of triangle counting and listing in very large graphs with various degree distributions. We give a surprisingly simple enhancement of a well known algorithm that performs best, and makes triangle listing and counting in huge networks feasible. This paper is a condensed presentation of [SW05].

Route Planning in Transportation Networks

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.

Approximating Clustering Coefficient and Transitivity

Since its introduction in the year 1998 by Watts and Strogatz, the clustering coecient has become a frequently used tool for analyzing graphs. In 2002 the transitivity was proposed by Newman, Watts and Strogatz as an alternative to the clustering coecient. As many networks considered in complex systems are huge, the ecient computation of such network parameters is crucial. Several algorithms with polynomial running time can be derived from results known in graph theory. The main contribution of this work is a new fast approximation algorithm for the weighted clustering coecient which also gives very ecient approximation algorithms for the clustering coecient and the transitivity. We namely present an algorithm with running time in O(1) for the clustering coecient, respectively with running time in O(n) for the transitivity. By an experimental study we demonstrate the performance of the proposed algorithms on real-world data as well as on generated graphs. Moreover we give a simple graph generator algorithm that works according to the preferential attachment rule but also generates graphs with adjustable clustering coecient.

Combining hierarchical and goal-directed speed-up techniques for dijkstra's algorithm

In recent years, highly effective hierarchical and goal-directed speed-up techniques for routing in large road networks have been developed. This article makes a systematic study of combinations of such techniques. These combinations turn out to give the best results in many scenarios, including graphs for unit disk graphs, grid networks, and time-expanded timetables. Besides these quantitative results, we obtain general insights for successful combinations.

Efficient models for timetable information in public transportation systems

We consider two approaches that model timetable information in public transportation systems as shortest-path problems in weighted graphs. In the time-expanded approach, every event at a station, e.g., the departure of a train, is modeled as a node in the graph, while in the time-dependent approach the graph contains only one node per station. Both approaches have been recently considered for (a simplified version of) the earliest arrival problem, but little is known about their relative performance. Thus far, there are only theoretical arguments in favor of the time-dependent approach. In this paper, we provide the first extensive experimental comparison of the two approaches. Using several real-world data sets, we evaluate the performance of the basic models and of several new extensions towards realistic modeling. Furthermore, new insights on solving bicriteria optimization problems in both models are presented. The time-expanded approach turns out to be more robust for modeling more complex scenarios, whereas the time-dependent approach shows a clearly better performance.

On finding graph clusterings with maximum modularity

Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, and in particular in the complex systems literature, although its properties are not well understood. We study the problem of finding clusterings with maximum modularity, thus providing theoretical foundations for past and present work based on this measure. More precisely, we prove the conjectured hardness of maximizing modularity both in the general case and with the restriction to cuts, and give an Integer Linear Programming formulation. This is complemented by first insights into the behavior and performance of the commonly applied greedy agglomaration approach.