Tanja Hartmann

Clustering Evolving Networks

Roughly speaking, clustering evolving networks aims at detecting structurally dense subgroups in networks that evolve over time. This implies that the subgroups we seek for also evolve, which results in many additional tasks compared to clustering static networks. We discuss these additional tasks and difficulties resulting thereof and present an overview on current approaches to solve these problems. We focus on clustering approaches in online scenarios, i.e., approaches that incrementally use structural information from previous time steps in order to incorporate temporal smoothness or to achieve low running time. Moreover, we describe a collection of real world networks and generators for synthetic data that are often used for evaluation.

Fully-dynamic hierarchical graph clustering using cut trees

Algorithms or target functions for graph clustering rarely admit quality guarantees or optimal results in general. However, a hierarchical clustering algorithm by Flake et al., which is based on minimum s-t-cuts whose sink sides are of minimum size, yields such a provable guarantee. We introduce a new degree of freedom to this method by allowing arbitrary minimum s-t-cuts and show that this unrestricted algorithm is complete, i.e., any clustering hierarchy based on minimum s-t-cuts can be found by choosing the right cuts. This allows for a more comprehensive analysis of a graphs structure. Additionally, we present a dynamic version of the unrestricted approach which employs this new degree of freedom to maintain a hierarchy of clusterings fulfilling this quality guarantee and effectively avoid changing the clusterings.

Dynamic Graph Clustering Using Minimum-Cut Trees

Algorithms or target functions for graph clustering rarely admit quality guarantees or optimal results in general. Based on properties of minimum-cut trees, a clustering algorithm by Flake et al. does however yield such a provable guarantee. We show that the structure of minimum-s-t-cuts in a graph allows for an efficient dynamic update of minimum-cut trees, and present a dynamic graph clustering algorithm that maintains a clustering fulfilling this quality quarantee, and that effectively avoids changing the clustering. Experiments on real-world dynamic graphs complement our theoretical results.

Simultaneous Embeddability of Two Partitions

We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region that contains exactly those points that belong to the elements in the block and that is bounded by a simple closed curve. We establish three main classes of simultaneous embeddability weak, strong, and full embeddability that differ by increasingly strict well-formedness conditions on how different block regions are allowed to intersect. We show that these simultaneous embeddability classes are closely related to different planarity concepts of hypergraphs. For each embeddability class we give a full characterization. We show that i every pair of partitions has a weak simultaneous embedding, ii it is NP-complete to decide the existence of a strong simultaneous embedding, and iii the existence of a full simultaneous embedding can be tested in linear time.

Cubic Augmentation of Planar Graphs

In this paper we study the problem of augmenting a planar graph such that it becomes 3-regular and remains planar. We show that it is NP-hard to decide whether such an augmentation exists. On the other hand, we give an efficient algorithm for the variant of the problem where the input graph has a fixed planar (topological) embedding that has to be preserved by the augmentation. We further generalize this algorithm to test efficiently whether a 3-regular planar augmentation exists that additionally makes the input graph connected or biconnected. If the input graph should become even triconnected, we show that the existence of a 3-regular planar augmentation is again NP-hard to decide.

Hierarchies of predominantly connected communities

We consider communities whose vertices are predominantly connected, i.e., the vertices in each community are stronger connected to other community members of the same community than to vertices outside the community. Flake et al. introduced a hierarchical clustering algorithm that finds predominantly connected communities of different coarseness depending on an input parameter. We present a simple and efficient method for constructing a clustering hierarchy according to Flake et al. that supersedes the necessity of choosing feasible parameter values and guarantees the completeness of the resulting hierarchy, i.e., the hierarchy contains all clusterings that can be constructed by the original algorithm for any parameter value. However, predominantly connected communities are not organized in a single hierarchy. Thus, we further develop a framework that, after precomputing at most 2(n−1) maximum flows, admits a linear time construction of a clustering Ω(S) of predominantly connected communities that contains a given community S and is maximum in the sense that any further clustering of predominantly connected communities that also contains S is hierarchically nested in Ω(S). We further generalize this construction yielding a clustering with similar properties for k given communities in O(kn) time. This admits the analysis of a networks structure with respect to various communities in different hierarchies.

Identifikation von Clustern in Graphen

ZusammenfassungAlgorithm Engineering für Graphclustern beinhaltet mehr als die Entwicklung gut funktionierender Algorithmen für konkrete Anwendungen oder Datensätze. Es geht vielmehr um den systematischen Entwurf von Algorithmen für formal sauber gefasste Probleme und deren Analyse und Evaluation unter Betrachtung angemessener Qualitätsmaße. Die Wahl eines Qualitätsmaßes und eine dementsprechend saubere Formulierung eines Optimierungsproblems ist bereits für das intuitiv nahe liegende Paradigma eines starken Zusammenhangs innerhalb der Cluster gegenüber einem schwachen Zusammenhang zwischen den Clustern eine Herausforderung. Umso bedeutender ist der Erkenntnisgewinn, der aus der Methodik des Algorithm Engineering für Graphclustern erzielt werden kann. Viele Aspekte, die in diesem Artikel nur kurz angerissen werden, sind in der Arbeit [9] ausführlich beschrieben.

Fast and Simple Fully-Dynamic Cut Tree Construction

A cut tree of an undirected weighted graph G = (V,E) encodes a minimum s-t-cut for each vertex pair {s,t} ⊆ V and can be iteratively constructed by n − 1 maximum flow computations. They solve the multiterminal network flow problem, which asks for the all-pairs maximum flow values in a network and at the same time they represent n − 1 non-crossing, linearly independent cuts that constitute a minimum cut basis of G. Hence, cut trees are resident in at least two fundamental fields of network analysis and graph theory, which emphasizes their importance for many applications. In this work we present a fully-dynamic algorithm that efficiently maintains a cut tree for a changing graph. The algorithm is easy to implement and has a high potential for saving cut computations under the assumption that a local change in the underlying graph does rarely affect the global cut structure. We document the good practicability of our approach in a brief experiment on real world data.