@article{FriedrichKatzmannKrohmer2018,
  author    = {Friedrich, Tobias and Katzmann, Maximilian and Krohmer, Anton},
  title     = {Unbounded Discrepancy of Deterministic Random Walks on Grids},
  series = {SIAM journal on discrete mathematics},
  volume    = {32},
  journal   = {SIAM journal on discrete mathematics},
  number    = {4},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {0895-4801},
  doi       = {10.1137/17M1131088},
  pages     = {2441 -- 2452},
  year      = {2018},
  abstract  = {Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs called the rotor-router model. In this model, instead of distributing tokens randomly, each vertex serves its neighbors in a fixed deterministic order. For most setups, both processes behave in a remarkably similar way: Starting with the same initial configuration, the number of tokens in the rotor-router model deviates only slightly from the expected number of tokens on the corresponding vertex in the random walk model. The maximal difference over all vertices and all times is called single vertex discrepancy. Cooper and Spencer [Combin. Probab. Comput., 15 (2006), pp. 815-822] showed that on Z(d), the single vertex discrepancy is only a constant c(d). Other authors also determined the precise value of c(d) for d = 1, 2. All of these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph Z(d). We show that this assumption is crucial by proving that, otherwise, the single vertex discrepancy can become arbitrarily large. For all dimensions d >= 1 and arbitrary discrepancies l >= 0, we construct configurations that reach a discrepancy of at least l.},
  language  = {en}
}
@phdthesis{Krohmer2016,
  author    = {Krohmer, Anton},
  title     = {Structures \& algorithms in hyperbolic random graphs},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-395974},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xii, 102},
  year      = {2016},
  abstract  = {Complex networks are ubiquitous in nature and society. They appear in vastly different domains, for instance as social networks, biological interactions or communication networks. Yet in spite of their different origins, these networks share many structural characteristics. For instance, their degree distribution typically follows a power law. This means that the fraction of vertices of degree k is proportional to k^(-β) for some constant β; making these networks highly inhomogeneous. Furthermore, they also typically have high clustering, meaning that links between two nodes are more likely to appear if they have a neighbor in common. To mathematically study the behavior of such networks, they are often modeled as random graphs. Many of the popular models like inhomogeneous random graphs or Preferential Attachment excel at producing a power law degree distribution. Clustering, on the other hand, is in these models either not present or artificially enforced. Hyperbolic random graphs bridge this gap by assuming an underlying geometry to the graph: Each vertex is assigned coordinates in the hyperbolic plane, and two vertices are connected if they are nearby. Clustering then emerges as a natural consequence: Two nodes joined by an edge are close by and therefore have many neighbors in common. On the other hand, the exponential expansion of space in the hyperbolic plane naturally produces a power law degree sequence. Due to the hyperbolic geometry, however, rigorous mathematical treatment of this model can quickly become mathematically challenging. In this thesis, we improve upon the understanding of hyperbolic random graphs by studying its structural and algorithmical properties. Our main contribution is threefold. First, we analyze the emergence of cliques in this model. We find that whenever the power law exponent β is 2 < β < 3, there exists a clique of polynomial size in n. On the other hand, for β >= 3, the size of the largest clique is logarithmic; which severely contrasts previous models with a constant size clique in this case. We also provide efficient algorithms for finding cliques if the hyperbolic node coordinates are known. Second, we analyze the diameter, i. e., the longest shortest path in the graph. We find that it is of order O(polylog(n)) if 2 < β < 3 and O(logn) if β > 3. To complement these findings, we also show that the diameter is of order at least Ω(logn). Third, we provide an algorithm for embedding a real-world graph into the hyperbolic plane using only its graph structure. To ensure good quality of the embedding, we perform extensive computational experiments on generated hyperbolic random graphs. Further, as a proof of concept, we embed the Amazon product recommendation network and observe that products from the same category are mapped close together.},
  language  = {en}
}