Anton Krohmer

Cliques in hyperbolic random graphs

Most complex real-world networks display scale-free features. This motivated the study of numerous random graph models with a power-law 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. 2973–2981, 2010) and has shown theoretically and empirically to fulfill all typical properties of real-world networks, including power-law 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 ω(G). We observe that there is a phase transition at power-law exponent γ = 3. More precisely, for γ e (2,3) we prove E[K k ] = nk(3-γ)/2 Θ(k)−k and ω(G) = Θ(n(3-γ)/2) while for γ  3 we prove E[K k ] = nΘ(k)−k and ω(G) = Θ(log(n)/log log n). We empirically compare the ω(G) value of several scale-free random graph models with real-world networks. Our experiments show that the ω(G)-predictions by hyperbolic random graphs are much closer to the data than other scale-free random graph models.

Parameterized Clique on Scale-Free Networks

Finding cliques in graphs is a classical problem which is in general NP-hard and parameterized intractable. However, in typical applications like social networks or protein-protein interaction networks, the considered graphs are scale-free, i.e., their degree sequence follows a power law. Their specific structure can be algorithmically exploited and makes it possible to solve clique much more efficiently. We prove that on inhomogeneous random graphs with n nodes and power law exponent γ, cliques of size k can be found in time \(\mathcal{O}(n^2)\) for γ ≥ 3 and in time \(\mathcal{O}(n\, \exp(k^4))\) for 2 < γ < 3.

Efficient algorithms for extracting biological key pathways with global constraints

The integrated analysis of data of different types and with various interdependencies is one of the major challenges in computational biology. Recently, we developed KeyPathwayMiner, a method that combines biological networks modeled as graphs with disease-specific genetic expression data gained from a set of cases (patients, cell lines, tissues, etc.). We aimed for finding all maximal connected sub-graphs where all nodes but

Hyperbolic Random Graphs: Separators and Treewidth

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RatFish: a file sharing protocol provably secure against rational users

are expressed in all cases but at most

Towards a Systematic Evaluation of Generative Network Models

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On the Diameter of Hyperbolic Random Graphs

, i.e. key pathways. Thereby, we combined biological networks with OMICS data, instead of analyzing these data sets in isolation. Here we present an alternative approach that avoids a certain bias towards hub nodes: We now aim for extracting all maximal connected sub-networks where all but at most

Bounds on the satisfiability threshold for power law distributed random SAT

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Efficient Embedding of Scale-Free Graphs in the Hyperbolic Plane.

nodes are expressed in all cases but in total (!) at most

Efficient key pathway mining: combining networks and OMICS data

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