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Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton Cliques in Hyperbolic Random Graphs. Algorithmica 2018: 23242344
Most complex real world networks display scalefree features. This characteristic motivated the study of numerous random graph models with a powerlaw degree distribution. There is, however, no established and simple model which also has a high clustering of vertices as typically observed in real data. Hyperbolic random graphs bridge this gap. This natural model has recently been introduced by Krioukov et al. and has shown theoretically and empirically to fulfill all typical properties of real world networks, including powerlaw degree distribution and high clustering. We study cliques in hyperbolic random graphs \(G\) and present new results on the expected number of \(k\)cliques \(E[K_k]\) and the size of the largest clique \(\omega(G)\). We observe that there is a phase transition at powerlaw exponent \(\beta = 3\). More precisely, for \(\beta\)\(\in\)\((2,3)\) we prove \(E[K_k] = \) \(n^{k(3\beta)/2} \Theta(k)^{k}\) and \(\omega(G) = \) \(\Theta\)\((n^{(3\beta)/2})\), while for \(\beta \geq 3\) we prove \(E[K_k]=n \Theta(k)^{k}\) and \(\omega(G)=\Theta(\log(n)/ \log\log n)\). Furthermore, we show that for \(\beta \geq 3\), cliques in hyperbolic random graphs can be computed in time \(O(n)\). If the underlying geometry is known, cliques can be found with worstcase runtime \(O(m n^{2.5})\) for all values of \(\beta\).

Friedrich, Tobias; Krohmer, Anton On the diameter of hyperbolic random graphs. SIAM Journal on Discrete Mathematics 2018: 13141334
Large realworld networks are typically scalefree. Recent research has shown that such graphs are described best in a geometric space. More precisely, the internet can be mapped to a hyperbolic space such that geometric greedy routing is close to optimal (Boguñá, Papadopoulos, and Krioukov. Nature Communications, 1:62, 2010). This observation has pushed the interest in hyperbolic networks as a natural model for scalefree networks. Hyperbolic random graphs follow a power law degree distribution with controllable exponent \(\beta\) and show high clustering (Gugelmann, Panagiotou, and Peter. ICALP, pp. 573–585, 2012). For understanding the structure of the resulting graphs and for analyzing the behavior of network algorithms, the next question is bounding the size of the diameter. The only known explicit bound is \(O(\)\((\log n)\)\(^{32/((3  \beta)(5  \beta))+1})\)(Kiwi and Mitsche. ANALCO, pp. 26–39, 2015). We present two much simpler proofs for an improved upper bound of \(O((\log n)\)\(^{2/(3  \beta)})\) and a lower bound of \(\Omega(\log n)\). If \(\beta > 3\), we show that the latter bound is tight by proving an upper bound of \(O(\log n)\) for the diameter.

Bläsius, Thomas; Freiberger, Cedric; Friedrich, Tobias; Katzmann, Maximilian; MontenegroRetana, Felix; Thieffry, Marianne Efficient Shortest Paths in ScaleFree Networks with Underlying Hyperbolic Geometry. International Colloquium on Automata, Languages, and Programming (ICALP) 2018: 20:120:14
A common way to accelerate shortest path algorithms on graphs is the use of a bidirectional search, which simultaneously explores the graph from the start and the destination. It has been observed recently that this strategy performs particularly well on scalefree realworld networks. Such networks typically have a heterogeneous degree distribution (e.g., a powerlaw distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is \(\tilde{O}(n\)\(^{2  1/ \alpha}+\)\(n^{1/(2\alpha)}\)\(+ \delta_{\max})\) with high probability, where \(\alpha\)\(\in\)\((0.5, 1)\) controls the powerlaw exponent of the degree distribution, and \(\delta_{\max}\) is the maximum degree. This bound is sublinear, improving the obvious worstcase linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.

Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton Hyperbolic Random Graphs: Separators and Treewidth. European Symposium on Algorithms (ESA) 2016: 15:115:16
When designing and analyzing algorithms, one can obtain better and more realistic results for practical instances by assuming a certain probability distribution on the input. The worstcase runtime is then replaced by the expected runtime or by bounds that hold with high probability (whp), i.e., with probability \(1  O(1/n)\), on the random input. Hyperbolic random graphs can be used to model complex realworld networks as they share many important properties such as a small diameter, a large clustering coefficient, and a powerlaw degreedistribution. Divide and conquer is an important algorithmic design principle that works particularly well if the instance admits small separators. We show that hyperbolic random graphs in fact have comparatively small separators. More precisely, we show that a hyperbolic random graph can be expected to have a balanced separator hierarchy with separators of size \(O(\sqrt{n^{(3\beta)}})\), \(O(\log n)\), and \(O(1)\) if \(2 < \beta < 3\), \(\beta = 3\) and \(3 < \beta\), respectively (\(\beta\) is the powerlaw exponent). We infer that these graphs have whp a treewidth of \(O(\sqrt{n^{(3  \beta)}})\), \(O(\log^{2}n)\), and \(O(\log n)\), respectively. For \(2 < \beta < 3\), this matches a known lower bound. For the more realistic (but harder to analyze) binomial model, we still prove a sublinear bound on the treewidth. To demonstrate the usefulness of our results, we apply them to obtain fast matching algorithms and an approximation scheme for Independent Set.

Friedrich, Tobias; Krohmer, Anton On the Diameter of Hyperbolic Random Graphs. International Colloquium on Automata, Languages and Programming (ICALP) 2015: 614625
Large realworld networks are typically scalefree. Recent research has shown that such graphs are described best in a geometric space. More precisely, the internet can be mapped to a hyperbolic space such that geometric greedy routing performs close to optimal (Boguna, Papadopoulos, and Krioukov. Nature Communications, 1:62, 2010). This observation pushed the interest in hyperbolic networks as a natural model for scalefree networks. Hyperbolic random graphs follow a powerlaw degree distribution with controllable exponent \(\beta\) and show high clustering (Gugelmann, Panagiotou, and Peter. ICALP, pp. 573585, 2012). For understanding the structure of the resulting graphs and for analyzing the behavior of network algorithms, the next question is bounding the size of the diameter. The only known explicit bound is \(O((\log n)\)\(^{32/((3\beta)(5\beta))})\) (Kiwi and Mitsche. ANALCO, pp. 2639, 2015). We present two much simpler proofs for an improved upper bound of \(O((\log n)\)\(^{2/(3\beta)})\) and a lower bound of \(\Omega(\log n)\). If the average degree is bounded from above by some constant, we show that the latter bound is tight by proving an upper bound of \(O(\log n)\).

Friedrich, Tobias; Krohmer, Anton Cliques in Hyperbolic Random Graphs. International Conference on Computer Communications (INFOCOM) 2015: 15441552
Most complex realworld networks display scalefree features. This motivated the study of numerous random graph models with a powerlaw degree distribution. There is, however, no established and simple model which also has a high clustering of vertices as typically observed in real data. Hyperbolic random graphs bridge this gap. This natural model has recently been introduced by Papadopoulos, Krioukov, Boguna, Vahdat (INFOCOM, pp. 29732981, 2010) and has shown theoretically and empirically to fulfill all typical properties of realworld networks, including powerlaw degree distribution and high clustering. We study cliques in hyperbolic random graphs \(G\) and present new results on the expected number of \(k\)cliques \(E[K_k]\) and the size of the largest clique \(\omega(G)\). We observe that there is a phase transition at powerlaw exponent \(\gamma = 3\). More precisely, for \(\gamma\) \(\in\) \((2,3)\) we prove \(E[K_k] = \)\(n^{k(3\gamma)/2}\)\(\Theta(k)^{k}\) and \(\omega(G) = \)\(\Theta(\)\(n^{(3\gamma)/2})\) while for \(\gamma \ge 3\) we prove \(E[K_k] = n \Theta(k)^{k}\) and \(\omega(G) = \Theta(\log(n)/\log \log n)\). We empirically compare the \(\omega(G)\) value of several scalefree random graph models with realworld networks. Our experiments show that the \(\omega(G)\)predictions by hyperbolic random graphs are much closer to the data than other scalefree random graph models.