2018 [ to top ]

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  Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton; Laue, Sören Efficient Embedding of Scale-Free Graphs in the Hyperbolic Plane. Transactions on Networking 2018
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scale-free graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require at least a quadratic runtime. Our algorithm achieves quasilinear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics like Log-likelihood and greedy routing our algorithm discovers embeddings that are very close to the ground truth.
  @article{DBLP:journals/ton/BlasiusFKL18, abstract = {Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scale-free graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require at least a quadratic runtime. Our algorithm achieves quasilinear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics like Log-likelihood and greedy routing our algorithm discovers embeddings that are very close to the ground truth.}, author = {Bläsius, Thomas and Friedrich, Tobias and Krohmer, Anton and Laue, Sören}, booktitle = {Transactions on Networking}, title = {Efficient Embedding of Scale-Free Graphs in the Hyperbolic Plane}, year = 2018 }
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  Bringmann, Karl; Friedrich, Tobias; Krohmer, Anton De-anonymization of Heterogeneous Random Graphs in Quasilinear Time. Algorithmica 2018: 1-31
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  There are hundreds of online social networks with altogether billions of users. Many such networks publicly release structural information, with all personal information removed. Empirical studies have shown, however, that this provides a false sense of privacy - it is possible to identify almost all users that appear in two such anonymized network as long as a few initial mappings are known. We analyze this problem theoretically by reconciling two versions of an artificial power-law network arising from independent subsampling of vertices and edges. We present a new algorithm that identifies most vertices and makes no wrong identifications with high probability. The number of vertices matched is shown to be asymptotically optimal. For an n-vertex graph, our algorithm uses \(n^\varepsilon\) seed nodes (for an arbitrarily small \(\varepsilon\)) and runs in quasilinear time. This improves previous theoretical results which need \(\Theta(n)\) seed nodes and have runtimes of order \(n^{1+\Omega(1)}\). Additionally, the applicability of our algorithm is studied experimentally on different networks.
  @article{DBLP:journals/algorithmica/BringmannFK18, abstract = {There are hundreds of online social networks with altogether billions of users. Many such networks publicly release structural information, with all personal information removed. Empirical studies have shown, however, that this provides a false sense of privacy - it is possible to identify almost all users that appear in two such anonymized network as long as a few initial mappings are known. We analyze this problem theoretically by reconciling two versions of an artificial power-law network arising from independent subsampling of vertices and edges. We present a new algorithm that identifies most vertices and makes no wrong identifications with high probability. The number of vertices matched is shown to be asymptotically optimal. For an n-vertex graph, our algorithm uses \(n^\varepsilon\) seed nodes (for an arbitrarily small \(\varepsilon\)) and runs in quasilinear time. This improves previous theoretical results which need \(\Theta(n)\) seed nodes and have runtimes of order \(n^{1+\Omega(1)}\). Additionally, the applicability of our algorithm is studied experimentally on different networks.}, author = {Bringmann, Karl and Friedrich, Tobias and Krohmer, Anton}, journal = {Algorithmica}, pages = {1-31}, title = {De-anonymization of Heterogeneous Random Graphs in Quasilinear Time}, year = 2018 }
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  Bläsius, Thomas; Friedrich, Tobias; Katzmann, Maximilian; Krohmer, Anton Hyperbolic Embeddings for Near-Optimal Greedy Routing. Algorithm Engineering and Experiments (ALENEX) 2018: 199-208
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Greedy routing computes paths between nodes in a network by successively moving to the neighbor closest to the target with respect to coordinates given by an embedding into some metric space. Its advantage is that only local information is used for routing decisions. We present different algorithms for generating graph embeddings into the hyperbolic plane that are well suited for greedy routing. In particular our embeddings guarantee that greedy routing always succeeds in reaching the target and we try to minimize the lengths of the resulting greedy paths. We evaluate our algorithm on multiple generated and real wold networks. For networks that are generally assumed to have a hidden underlying hyperbolic geometry, such as the Internet graph, we achieve near-optimal results, i.e., the resulting greedy paths are only slightly longer than the corresponding shortest paths. In the case of the Internet graph, they are only \(6\%\) longer when using our best algorithm, which greatly improves upon the previous best known embedding, whose creation required substantial manual intervention.
  @inproceedings{btkk2018hyperbolic, abstract = {Greedy routing computes paths between nodes in a network by successively moving to the neighbor closest to the target with respect to coordinates given by an embedding into some metric space. Its advantage is that only local information is used for routing decisions. We present different algorithms for generating graph embeddings into the hyperbolic plane that are well suited for greedy routing. In particular our embeddings guarantee that greedy routing always succeeds in reaching the target and we try to minimize the lengths of the resulting greedy paths. We evaluate our algorithm on multiple generated and real wold networks. For networks that are generally assumed to have a hidden underlying hyperbolic geometry, such as the Internet graph, we achieve near-optimal results, i.e., the resulting greedy paths are only slightly longer than the corresponding shortest paths. In the case of the Internet graph, they are only \(6\%\) longer when using our best algorithm, which greatly improves upon the previous best known embedding, whose creation required substantial manual intervention.}, author = {Bläsius, Thomas and Friedrich, Tobias and Katzmann, Maximilian and Krohmer, Anton}, booktitle = {Algorithm Engineering and Experiments (ALENEX)}, pages = {199-208}, title = {Hyperbolic Embeddings for Near-Optimal Greedy Routing}, year = 2018 }

2017 [ to top ]

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  Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton Cliques in Hyperbolic Random Graphs. Algorithmica 2017: 1-21
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Most complex real world networks display scale-free features. This characteristic 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 hyprg 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 \(\omega(G)\). We observe that there is a phase transition at power-law 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 \(\beta\geq3\) 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 worst-case runtime \(O(m \cdot n^{2.5})\) for all values of \(\beta\).
  @article{blasius2017cliques, abstract = {Most complex real world networks display scale-free features. This characteristic 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 hyprg 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 \(\omega(G)\). We observe that there is a phase transition at power-law 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 \(\beta\geq3\) 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 worst-case runtime \(O(m \cdot n^{2.5})\) for all values of \(\beta\).}, author = {Bläsius, Thomas and Friedrich, Tobias and Krohmer, Anton}, journal = {Algorithmica}, pages = {1-21}, title = {Cliques in Hyperbolic Random Graphs}, year = 2017 }
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  Friedrich, Tobias; Krohmer, Anton; Rothenberger, Ralf; Sutton, Andrew M. Phase Transitions for Scale-Free SAT Formulas. Association for the Advancement of Artificial Intelligence (AAAI) 2017: 3893-3899
  [ Abstract ] [ BibTeX ] [ URL ] [ Download ]
   
  Recently, a number of non-uniform random satisfiability models have been proposed that are closer to practical satisfiability problems in some characteristics. In contrast to uniform random Boolean formulas, scale-free formulas have a variable occurrence distribution that follows a power law. It has been conjectured that such a distribution is a more accurate model for some industrial instances than the uniform random model. Though it seems that there is already an awareness of a threshold phenomenon in such models, there is still a complete picture lacking. In contrast to the uniform model, the critical density threshold does not lie at a single point, but instead exhibits a functional dependency on the power-law exponent. For scale-free formulas with clauses of length \(k = 2\), we give a lower bound on the phase transition threshold as a function of the scaling parameter. We also perform computational studies that suggest our bound is tight and investigate the critical density for formulas with higher clause lengths. Similar to the uniform model, on formulas with \(k \geq 3\), we find that the phase transition regime corresponds to a set of formulas that are difficult to solve by backtracking search.
  @inproceedings{DBLP:conf/aaai/0001KRS17, abstract = {Recently, a number of non-uniform random satisfiability models have been proposed that are closer to practical satisfiability problems in some characteristics. In contrast to uniform random Boolean formulas, scale-free formulas have a variable occurrence distribution that follows a power law. It has been conjectured that such a distribution is a more accurate model for some industrial instances than the uniform random model. Though it seems that there is already an awareness of a threshold phenomenon in such models, there is still a complete picture lacking. In contrast to the uniform model, the critical density threshold does not lie at a single point, but instead exhibits a functional dependency on the power-law exponent. For scale-free formulas with clauses of length \(k = 2\), we give a lower bound on the phase transition threshold as a function of the scaling parameter. We also perform computational studies that suggest our bound is tight and investigate the critical density for formulas with higher clause lengths. Similar to the uniform model, on formulas with \(k \geq 3\), we find that the phase transition regime corresponds to a set of formulas that are difficult to solve by backtracking search.}, author = {Friedrich, Tobias and Krohmer, Anton and Rothenberger, Ralf and Sutton, Andrew M.}, booktitle = {Association for the Advancement of Artificial Intelligence (AAAI)}, pages = {3893-3899}, publisher = {AAAI Press}, title = {Phase Transitions for Scale-Free SAT Formulas}, year = 2017 }
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  Friedrich, Tobias; Krohmer, Anton; Rothenberger, Ralf; Sauerwald, Thomas; Sutton, Andrew M. Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT. European Symposium on Algorithms (ESA) 2017: 37:1-37:15
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The average-case analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, nearly all theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a scale-free distribution of the variables, which results in distributions closer to industrial SAT instances. We study random \(k\)-SAT on \(n\) variables, \(m=\Theta(n)\) clauses, and a power law distribution on the variable occurrences with exponent \(\beta\). We observe a satisfiability threshold at \(\beta=(2k-1)/(k-1)\). This threshold is tight in the sense that instances with \(beta < (2k-1)/(k-1)-\varepsilon\) for any constant \(\varepsilon>0\) are unsatisfiable with high probability (w.h.p.). For \(\beta\ge(2k-1)/(k-1)+\varepsilon\), the picture is reminiscent of the uniform case: instances are satisfiable w.h.p. for sufficiently small constant clause-variable ratios \(m/n\); they are unsatisfiable above a ratio \(m/n\) that depends on \(\beta\).
  @inproceedings{friedrich2017bounds, abstract = {Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The average-case analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, nearly all theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a scale-free distribution of the variables, which results in distributions closer to industrial SAT instances. We study random \(k\)-SAT on \(n\) variables, \(m=\Theta(n)\) clauses, and a power law distribution on the variable occurrences with exponent \(\beta\). We observe a satisfiability threshold at \(\beta=(2k-1)/(k-1)\). This threshold is tight in the sense that instances with \(\beta < (2k-1)/(k-1)-\varepsilon\) for any constant \(\varepsilon>0\) are unsatisfiable with high probability (w.h.p.). For \(\beta\ge(2k-1)/(k-1)+\varepsilon\), the picture is reminiscent of the uniform case: instances are satisfiable w.h.p. for sufficiently small constant clause-variable ratios \(m/n\); they are unsatisfiable above a ratio \(m/n\) that depends on \(\beta\).}, author = {Friedrich, Tobias and Krohmer, Anton and Rothenberger, Ralf and Sauerwald, Thomas and Sutton, Andrew M.}, booktitle = {European Symposium on Algorithms (ESA)}, month = {sep}, pages = {37:1-37:15}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik}, series = {LIPIcs}, title = {Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT}, volume = 87, year = 2017 }

2016 [ to top ]

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  Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton Hyperbolic Random Graphs: Separators and Treewidth. European Symposium on Algorithms (ESA) 2016: 15:1-15:16
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  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 worst-case run-time is then replaced by the expected run-time 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 real-world networks as they share many important properties such as a small diameter, a large clustering coefficient, and a power-law degree-distribution. 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 power-law 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.
  @inproceedings{DBLP:conf/esa/BlasiusFK16, abstract = {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 worst-case run-time is then replaced by the expected run-time 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 real-world networks as they share many important properties such as a small diameter, a large clustering coefficient, and a power-law degree-distribution. 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 power-law 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.}, author = {Bläsius, Thomas and Friedrich, Tobias and Krohmer, Anton}, booktitle = {European Symposium on Algorithms (ESA)}, pages = {15:1-15:16}, title = {Hyperbolic Random Graphs: Separators and Treewidth}, year = 2016 }
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  Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton; Laue, Sören Efficient Embedding of Scale-Free Graphs in the Hyperbolic Plane. European Symposium on Algorithms (ESA) 2016: 16:1-16:18

  EATCS Best Paper Award
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scale-free graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require a runtime of \(\Omega(n^{2})\). Our algorithm achieves quasilinear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics like Log-likelihood and greedy routing our algorithm discovers embeddings that are very close to the ground truth.
  @inproceedings{DBLP:conf/esa/BlasiusFKL16, abstract = {Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scale-free graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require a runtime of \(\Omega(n^{2})\). Our algorithm achieves quasilinear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics like Log-likelihood and greedy routing our algorithm discovers embeddings that are very close to the ground truth.}, author = {Bläsius, Thomas and Friedrich, Tobias and Krohmer, Anton and Laue, Sören}, booktitle = {European Symposium on Algorithms (ESA)}, note = {EATCS Best Paper Award}, pages = {16:1-16:18}, title = {Efficient Embedding of Scale-Free Graphs in the Hyperbolic Plane}, year = 2016 }

2015 [ to top ]

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  Friedrich, Tobias; Krohmer, Anton Parameterized clique on inhomogeneous random graphs. Discrete Applied Mathematics 2015: 130-138
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Finding cliques in graphs is a classical problem which is in general NP-hard and parameterized intractable. In typical applications like social networks or biological networks, however, 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 \(\beta\), cliques of size \(k\) can be found in time \(O(n)\) for \(\beta \ge 3\) and in time \(O(ne^{k^4})\) for \(2 < \beta < 3\).
  @article{DBLP:journals/dam/FriedrichK15, abstract = {Finding cliques in graphs is a classical problem which is in general NP-hard and parameterized intractable. In typical applications like social networks or biological networks, however, 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 \(\beta\), cliques of size \(k\) can be found in time \(O(n)\) for \(\beta \ge 3\) and in time \(O(ne^{k^4})\) for \(2 < \beta < 3\).}, author = {Friedrich, Tobias and Krohmer, Anton}, journal = {Discrete Applied Mathematics}, pages = {130-138}, title = {Parameterized clique on inhomogeneous random graphs}, volume = 184, year = 2015 }
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  Friedrich, Tobias; Krohmer, Anton On the Diameter of Hyperbolic Random Graphs. International Colloquium on Automata, Languages and Programming (ICALP) 2015: 614-625
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Large real-world networks are typically scale-free. 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 scale-free 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 \(O((\log n)^32/((3-\beta)(5-\beta)}))\) (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 \(\Omega(\log n)\).
  @inproceedings{DBLP:conf/icalp/FriedrichK15, abstract = {Large real-world networks are typically scale-free. 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 scale-free 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 \(O((\log n)^{32/((3-\beta)(5-\beta)}))\) (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 \(\Omega(\log n)\).}, author = {Friedrich, Tobias and Krohmer, Anton}, booktitle = {International Colloquium on Automata, Languages and Programming (ICALP)}, pages = {614-625}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, title = {On the Diameter of Hyperbolic Random Graphs}, volume = 9135, year = 2015 }
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  Friedrich, Tobias; Krohmer, Anton Cliques in hyperbolic random graphs. International Conference on Computer Communications (INFOCOM) 2015: 1544-1552
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  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 \(\omega(G)\). We observe that there is a phase transition at power-law 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 scale-free random graph models with real-world networks. Our experiments show that the \(\omega(G)\)-predictions by hyperbolic random graphs are much closer to the data than other scale-free random graph models.
  @inproceedings{DBLP:conf/infocom/FriedrichK15, abstract = {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 \(\omega(G)\). We observe that there is a phase transition at power-law 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 scale-free random graph models with real-world networks. Our experiments show that the \(\omega(G)\)-predictions by hyperbolic random graphs are much closer to the data than other scale-free random graph models.}, author = {Friedrich, Tobias and Krohmer, Anton}, booktitle = {International Conference on Computer Communications (INFOCOM)}, pages = {1544-1552}, publisher = {{IEEE}}, title = {Cliques in hyperbolic random graphs}, year = 2015 }
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  Friedrich, Tobias; Katzmann, Maximilian; Krohmer, Anton Unbounded Discrepancy of Deterministic Random Walks on Grids. International Symposium on Algorithms and Computation (ISAAC) 2015: 212-222
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs, called 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 remarkably similar: 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 (2006) showed that on \(\mathbb{Z}^d\) 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 these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph \(\mathbb{Z}^d\). We show that this assumption is crucial by proving that otherwise the single vertex discrepancy can become arbitrarily large. For all dimensions \(d \ge 1\) and arbitrary discrepancies \(\ell \ge 0\), we construct configurations that reach a discrepancy of at least \(\ell\).
  @inproceedings{DBLP:conf/isaac/FriedrichKK15, abstract = {Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs, called 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 remarkably similar: 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 (2006) showed that on \(\mathbb{Z}^d\) 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 these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph \(\mathbb{Z}^d\). We show that this assumption is crucial by proving that otherwise the single vertex discrepancy can become arbitrarily large. For all dimensions \(d \ge 1\) and arbitrary discrepancies \(\ell \ge 0\), we construct configurations that reach a discrepancy of at least \(\ell\).}, author = {Friedrich, Tobias and Katzmann, Maximilian and Krohmer, Anton}, booktitle = {International Symposium on Algorithms and Computation (ISAAC)}, pages = {212-222}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, title = {Unbounded Discrepancy of Deterministic Random Walks on Grids}, volume = 9472, year = 2015 }

2014 [ to top ]

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  Bringmann, Karl; Friedrich, Tobias; Krohmer, Anton De-anonymization of Heterogeneous Random Graphs in Quasilinear Time. European Symposium on Algorithms (ESA) 2014: 197-208
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  There are hundreds of online social networks with billions of users in total. Many such networks publicly release structural information, with all personal information removed. Empirical studies have shown, however, that this provides a false sense of privacy - it is possible to identify almost all users that appear in two such anonymized network as long as a few initial mappings are known. We analyze this problem theoretically by reconciling two versions of an artificial power-law network arising from independent subsampling of vertices and edges. We present a new algorithm that identifies most vertices and makes no wrong identifications with high probability. The number of vertices matched is shown to be asymptotically optimal. For an \(n\)-vertex graph, our algorithm uses \(n^\epsilon\) seed nodes (for an arbitrarily small \(\epsilon\)) and runs in quasilinear time. This improves previous theoretical results which need \(\Theta(n)\) seed nodes and have runtimes of order \(n^{1 + \Omega(1)}\). Additionally, the applicability of our algorithm is studied experimentally on different networks.
  @inproceedings{DBLP:conf/esa/BringmannFK14, abstract = {There are hundreds of online social networks with billions of users in total. Many such networks publicly release structural information, with all personal information removed. Empirical studies have shown, however, that this provides a false sense of privacy - it is possible to identify almost all users that appear in two such anonymized network as long as a few initial mappings are known. We analyze this problem theoretically by reconciling two versions of an artificial power-law network arising from independent subsampling of vertices and edges. We present a new algorithm that identifies most vertices and makes no wrong identifications with high probability. The number of vertices matched is shown to be asymptotically optimal. For an \(n\)-vertex graph, our algorithm uses \(n^\epsilon\) seed nodes (for an arbitrarily small \(\epsilon\)) and runs in quasilinear time. This improves previous theoretical results which need \(\Theta(n)\) seed nodes and have runtimes of order \(n^{1 + \Omega(1)}\). Additionally, the applicability of our algorithm is studied experimentally on different networks.}, author = {Bringmann, Karl and Friedrich, Tobias and Krohmer, Anton}, booktitle = {European Symposium on Algorithms (ESA)}, pages = {197-208}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, title = {De-anonymization of Heterogeneous Random Graphs in Quasilinear Time}, volume = 8737, year = 2014 }

2012 [ to top ]

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  Alcaraz, Nicolas; Friedrich, Tobias; Kötzing, Timo; Krohmer, Anton; Müller, Joachim; Pauling, Josch; Baumbach, Jan Efficient Key Pathway Mining: Combining Networks and OMICS Data. Integrative Biology 2012: 756-764
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Systems biology has emerged over the last decade. Driven by the advances in sophisticated measurement technology the research community generated huge molecular biology data sets. This comprises rather static data on the interplay of biological entities, for instance protein-protein interaction network data, as well as quite dynamic data collected for studying the behavior of individual cells or tissues in accordance to changing environmental conditions, such as DNA microarrays or RNA sequencing. Here we bring the two different data types together in order to gain higher level knowledge. We introduce a significantly improved version of the KeyPathwayMiner software framework. Given a biological network modelled as graph and a set of expression studies, KeyPathwayMiner efficiently finds and visualizes connected sub-networks where most components are expressed in most cases. It finds all maximal connected sub-networks where all nodes but k exceptions are expressed in all experimental studies but at least l exceptions. We demonstrate the power of the new approach by comparing it to similar approaches with gene expression data previously used to study Huntington's disease. In addition, we demonstrate KeyPathwayMiner's flexibility and applicability to non-array data by analyzing genome-scale DNA methylation profiles from colorectal tumor cancer patients. KeyPathwayMiner release 2 is available as a Cytoscape plugin and online at http://keypathwayminer.mpi-inf.mpg.de.
  @article{alcaraz2012efficient, abstract = {Systems biology has emerged over the last decade. Driven by the advances in sophisticated measurement technology the research community generated huge molecular biology data sets. This comprises rather static data on the interplay of biological entities, for instance protein-protein interaction network data, as well as quite dynamic data collected for studying the behavior of individual cells or tissues in accordance to changing environmental conditions, such as DNA microarrays or RNA sequencing. Here we bring the two different data types together in order to gain higher level knowledge. We introduce a significantly improved version of the KeyPathwayMiner software framework. Given a biological network modelled as graph and a set of expression studies, KeyPathwayMiner efficiently finds and visualizes connected sub-networks where most components are expressed in most cases. It finds all maximal connected sub-networks where all nodes but k exceptions are expressed in all experimental studies but at least l exceptions. We demonstrate the power of the new approach by comparing it to similar approaches with gene expression data previously used to study Huntington's disease. In addition, we demonstrate KeyPathwayMiner's flexibility and applicability to non-array data by analyzing genome-scale DNA methylation profiles from colorectal tumor cancer patients. KeyPathwayMiner release 2 is available as a Cytoscape plugin and online at http://keypathwayminer.mpi-inf.mpg.de.}, author = {Alcaraz, Nicolas and Friedrich, Tobias and Kötzing, Timo and Krohmer, Anton and Müller, Joachim and Pauling, Josch and Baumbach, Jan}, journal = {Integrative Biology}, number = 7, pages = {756-764}, title = {Efficient Key Pathway Mining: Combining Networks and OMICS Data}, volume = 4, year = 2012 }
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  Baumbach, Jan; Friedrich, Tobias; Kötzing, Timo; Krohmer, Anton; Müller, Joachim; Pauling, Josch Efficient Algorithms for Extracting Biological Key Pathways with Global Constraints. Genetic and Evolutionary Computation Conference (GECCO) 2012: 169-176
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  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 K are expressed in all cases but at most L, 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 K nodes are expressed in all cases but in total (!) at most L, i.e. accumulated over all cases and all nodes in a solution. We call this strategy GLONE (global node exceptions); the previous problem we call INES (individual node exceptions). Since finding GLONE-components is computationally hard, we developed an Ant Colony Optimization algorithm and implemented it with the KeyPathwayMiner Cytoscape framework as an alternative to the INES algorithms. KeyPathwayMiner 3.0 now offers both the INES and the GLONE algorithms. It is available as plugin from Cytoscape and online at http://keypathwayminer.mpi-inf.mpg.de.
  @inproceedings{DBLP:conf/gecco/BaumbachFKKMP12, abstract = {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 K are expressed in all cases but at most L, 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 K nodes are expressed in all cases but in total (!) at most L, i.e. accumulated over all cases and all nodes in a solution. We call this strategy GLONE (global node exceptions); the previous problem we call INES (individual node exceptions). Since finding GLONE-components is computationally hard, we developed an Ant Colony Optimization algorithm and implemented it with the KeyPathwayMiner Cytoscape framework as an alternative to the INES algorithms. KeyPathwayMiner 3.0 now offers both the INES and the GLONE algorithms. It is available as plugin from Cytoscape and online at http://keypathwayminer.mpi-inf.mpg.de.}, author = {Baumbach, Jan and Friedrich, Tobias and Kötzing, Timo and Krohmer, Anton and Müller, Joachim and Pauling, Josch}, booktitle = {Genetic and Evolutionary Computation Conference (GECCO)}, pages = {169-176}, title = {Efficient Algorithms for Extracting Biological Key Pathways with Global Constraints}, year = 2012 }
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  Friedrich, Tobias; Krohmer, Anton Parameterized Clique on Scale-Free Networks. International Symposium on Algorithms and Computation (ISAAC) 2012: 659-668
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  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 \(\gamma\), cliques of size \(k\) can be found in time \(O(n^2)\) for \(\gamma \ge 3\) and in time \(O(n, exp(k^4))\) for \(2<\gamma <3\).
  @inproceedings{DBLP:conf/isaac/FriedrichK12, abstract = {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 \(\gamma\), cliques of size \(k\) can be found in time \(O(n^2)\) for \(\gamma \ge 3\) and in time \(O(n\, exp(k^4))\) for \(2<\gamma <3\).}, author = {Friedrich, Tobias and Krohmer, Anton}, booktitle = {International Symposium on Algorithms and Computation (ISAAC)}, pages = {659-668}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, title = {Parameterized Clique on Scale-Free Networks}, volume = 7676, year = 2012 }

2010 [ to top ]

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  Backes, Michael; Ciobotaru, Oana; Krohmer, Anton RatFish: A File Sharing Protocol Provably Secure against Rational Users. European Symposium on Research in Computer Security (ESORICS) 2010: 607-625
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  The proliferation of P2P computing has recently been propelled by popular applications, most notably file sharing protocols such as BitTorrent. These protocols provide remarkable efficiency and scalability, as well as adaptivity to dynamic situations. However, none of them is secure against attacks from rational users, i.e., users that misuse the protocol if doing so increases their benefits (e.g., reduces download time or amount of upload capacity). We propose a rigorous model of rational file sharing for both seeders and leechers, building upon the concept of rational cryptography. We design a novel file sharing protocol called RatFish, and we formally prove that no rational party has an incentive to deviate from RatFish; i.e., RatFish constitutes a Nash equilibrium. Compared to existing file sharing protocols such as BitTorrent, the tracker in RatFish is assigned additional tasks while the communication overhead of a RatFish client is kept to a minimum. We demonstrate by means of a prototype implementation that RatFish is practical and efficient.
  @inproceedings{DBLP:conf/esorics/BackesCK10, abstract = {The proliferation of P2P computing has recently been propelled by popular applications, most notably file sharing protocols such as BitTorrent. These protocols provide remarkable efficiency and scalability, as well as adaptivity to dynamic situations. However, none of them is secure against attacks from rational users, i.e., users that misuse the protocol if doing so increases their benefits (e.g., reduces download time or amount of upload capacity). We propose a rigorous model of rational file sharing for both seeders and leechers, building upon the concept of rational cryptography. We design a novel file sharing protocol called RatFish, and we formally prove that no rational party has an incentive to deviate from RatFish; i.e., RatFish constitutes a Nash equilibrium. Compared to existing file sharing protocols such as BitTorrent, the tracker in RatFish is assigned additional tasks while the communication overhead of a RatFish client is kept to a minimum. We demonstrate by means of a prototype implementation that RatFish is practical and efficient.}, author = {Backes, Michael and Ciobotaru, Oana and Krohmer, Anton}, booktitle = {European Symposium on Research in Computer Security (ESORICS)}, pages = {607-625}, title = {RatFish: A File Sharing Protocol Provably Secure against Rational Users}, year = 2010 }