Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

A nonuniform popularity-similarity optimization (nPSO) model to efficiently generate realistic complex networks with communities

The hidden metric space behind complex network topologies is a fervid topic in current network science and the hyperbolic space is one of the most studied, because it seems associated to the structural organization of many real complex systems. The Popularity-Similarity-Optimization (PSO) model simulates how random geometric graphs grow in the hyperbolic space, reproducing strong clustering and scale-free degree distribution, however it misses to reproduce an important feature of real complex networks, which is the community organization. The Geometrical-Preferential-Attachment (GPA) model was recently developed to confer to the PSO also a community structure, which is obtained by forcing different angular regions of the hyperbolic disk to have variable level of attractiveness. However, the number and size of the communities cannot be explicitly controlled in the GPA, which is a clear limitation for real applications. Here, we introduce the nonuniform PSO (nPSO) model that, differently from GPA, forces heterogeneous angular node attractiveness by sampling the angular coordinates from a tailored nonuniform probability distribution, for instance a mixture of Gaussians. The nPSO differs from GPA in other three aspects: it allows to explicitly fix the number and size of communities; it allows to tune their mixing property through the network temperature; it is efficient to generate networks with high clustering. After several tests we propose the nPSO as a valid and efficient model to generate networks with communities in the hyperbolic space, which can be adopted as a realistic benchmark for different tasks such as community detection and link prediction

Learning Combinatorial Node Labeling Algorithms

We present a graph neural network to learn graph coloring heuristics using reinforcement learning. Our learned deterministic heuristics give better solutions than classical degree-based greedy heuristics and only take seconds to evaluate on graphs with tens of thousands of vertices. As our approach is based on policy-gradients, it also learns a probabilistic policy as well. These probabilistic policies outperform all greedy coloring baselines and a machine learning baseline. Our approach generalizes several previous machine-learning frameworks, which applied to problems like minimum vertex cover. We also demonstrate that our approach outperforms two greedy heuristics on minimum vertex cover

The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201