The Experts below are selected from a list of 20232 Experts worldwide ranked by ideXlab platform
Jürgen Jost - One of the best experts on this subject based on the ideXlab platform.
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ollivier Ricci Curvature of directed hypergraphs
Scientific Reports, 2020Co-Authors: Marzieh Eidi, Jürgen JostAbstract:Many empirical networks incorporate higher order relations between elements and therefore are naturally modelled as, possibly directed and/or weighted, hypergraphs, rather than merely as graphs. In order to develop a systematic tool for the statistical analysis of such hypergraph, we propose a general definition of Ricci Curvature on directed hypergraphs and explore the consequences of that definition. The definition generalizes Ollivier's definition for graphs. It involves a carefully designed optimal transport problem between sets of vertices. While the definition looks somewhat complex, in the end we shall be able to express our Curvature in a very simple formula, [Formula: see text]. This formula simply counts the fraction of vertices that have to be moved by distances 0, 2 or 3 in an optimal transport plan. We can then characterize various classes of hypergraphs by their Curvature.
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Ollivier Ricci Curvature of Directed Hypergraphs
arXiv: Discrete Mathematics, 2019Co-Authors: Marzieh Eidi, Jürgen JostAbstract:We develop a definition of Ricci Curvature on directed hypergraphs and explore the consequences of that definition. The definition generalizes Ollivier's definition for graphs. It involves a carefully designed optimal transport problem between sets of vertices.
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forman Ricci Curvature for hypergraphs
arXiv: Discrete Mathematics, 2018Co-Authors: Jürgen Jost, Wilmer Leal, Guillermo Restrepo, Peter F StadlerAbstract:In contrast to graph-based models for complex networks, hypergraphs are more general structures going beyond binary relations of graphs. For graphs, statistics gauging different aspects of their structures have been devised and there is undergoing research for devising them for hypergraphs. Forman-Ricci Curvature is a statistics for graphs, which is based on Riemannian geometry, and that stresses the relational character of vertices in a network through the analysis of edges rather than vertices. In spite of the different applications of this Curvature, it has not yet been formulated for hypergraphs. Here we devise the Forman-Ricci Curvature for directed and undirected hypergraphs, where the Curvature for graphs is a particular case. We report its upper and lower bounds and the respective bounds for the graph case. The Curvature quantifies the trade-off between hyperedge(arc) size and the degree of participation of hyperedge(arc) vertices in other hyperedges(arcs). We calculated the Curvature for two large networks: Wikipedia vote network and \emph{Escherichia coli} metabolic network. In the first case the Curvature is ruled by hyperedge size, while in the second by hyperedge degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with the participation of experienced users. The Curvature values of the metabolic network allowed detecting redundant and bottle neck reactions. It is found that ADP phosphorilation is the metabolic bottle neck reaction but that the reverse reaction is not that central for the metabolism.
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Comparative analysis of two discretizations of Ricci Curvature for complex networks.
Scientific Reports, 2018Co-Authors: Areejit Samal, R. P. Sreejith, Emil Saucan, Jiao Gu, Jürgen JostAbstract:We have performed an empirical comparison of two distinct notions of discrete Ricci Curvature for graphs or networks, namely, the Forman-Ricci Curvature and Ollivier-Ricci Curvature. Importantly, these two discretizations of the Ricci Curvature were developed based on different properties of the classical smooth notion, and thus, the two notions shed light on different aspects of network structure and behavior. Nevertheless, our extensive computational analysis in a wide range of both model and real-world networks shows that the two discretizations of Ricci Curvature are highly correlated in many networks. Moreover, we show that if one considers the augmented Forman-Ricci Curvature which also accounts for the two-dimensional simplicial complexes arising in graphs, the observed correlation between the two discretizations is even higher, especially, in real networks. Besides the potential theoretical implications of these observations, the close relationship between the two discretizations has practical implications whereby Forman-Ricci Curvature can be employed in place of Ollivier-Ricci Curvature for faster computation in larger real-world networks whenever coarse analysis suffices.
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Ollivier-Ricci Curvature and the spectrum of the normalized graph Laplace operator
arXiv: Combinatorics, 2011Co-Authors: Frank Bauer, Jürgen JostAbstract:We prove the following estimate for the spectrum of the normalized Laplace operator $\Delta$ on a finite graph $G$, \begin{equation*}1- (1- k[t])^{\frac{1}{t}}\leq \lambda_1 \leq \cdots \leq \lambda_{N-1}\leq 1+ (1- k[t])^{\frac{1}{t}}, \,\forall \,\,\text{integers}\,\, t\geq 1. \end{equation*} Here $k[t]$ is a lower bound for the Ollivier-Ricci Curvature on the neighborhood graph $G[t]$, which was introduced by Bauer-Jost. In particular, when $t=1$ this is Ollivier's estimates $k\leq \lambda_1\leq \ldots \leq \lambda_{N-1}\leq 2-k$. For sufficiently large $t$ we show that, unless $G$ is bipartite, our estimates for $\lambda_1$ and $\lambda_{N-1}$ are always nontrivial and improve Ollivier's estimates for all graphs with $k\leq 0$. By definition neighborhood graphs are weighted graphs which may have loops. To understand the Ollivier-Ricci Curvature on neighborhood graphs, we generalize a sharp estimate of the Ricci Curvature given by Jost-Liu to weighted graphs with loops and relate it to the relative local frequency of triangles and loops.
Jan Maas - One of the best experts on this subject based on the ideXlab platform.
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entropic Ricci Curvature bounds for discrete interacting systems
Annals of Applied Probability, 2016Co-Authors: Max Fathi, Jan MaasAbstract:We develop a new and systematic method for proving entropic Ricci Curvature lower bounds for Markov chains on discrete sets. Using different methods, such bounds have recently been obtained in several examples (e.g., 1-dimensional birth and death chains, product chains, Bernoulli–Laplace models, and random transposition models). However, a general method to obtain discrete Ricci bounds had been lacking. Our method covers all of the examples above. In addition, we obtain new Ricci Curvature bounds for zero-range processes on the complete graph. The method is inspired by recent work of Caputo, Dai Pra and Posta on discrete functional inequalities.
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Ricci Curvature of finite markov chains via convexity of the entropy
Archive for Rational Mechanics and Analysis, 2012Co-Authors: Matthias Erbar, Jan MaasAbstract:We study a new notion of Ricci Curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci Curvature we prove discrete analogues of fundamental results by Bakry–Emery and Otto–Villani. Further, we show that Ricci Curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci Curvature lower bound for the discrete hypercube.
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Ricci Curvature of finite markov chains via convexity of the entropy
arXiv: Metric Geometry, 2011Co-Authors: Matthias Erbar, Jan MaasAbstract:We study a new notion of Ricci Curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci Curvature we prove discrete analogues of fundamental results by Bakry--Emery and Otto--Villani. Furthermore we show that Ricci Curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci Curvature lower bound for the discrete hypercube.
Geraldine Dawson - One of the best experts on this subject based on the ideXlab platform.
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measuring robustness of brain networks in autism spectrum disorder with Ricci Curvature
Scientific Reports, 2020Co-Authors: Anish K Simhal, Allen Tannenbaum, Kimberly L H Carpenter, Saad Nadeem, Joanne Kurtzberg, Allen W Song, Guillermo Sapiro, Geraldine DawsonAbstract:Ollivier-Ricci Curvature is a method for measuring the robustness of connections in a network. In this work, we use Curvature to measure changes in robustness of brain networks in children with autism spectrum disorder (ASD). In an open label clinical trials, participants with ASD were administered a single infusion of autologous umbilical cord blood and, as part of their clinical outcome measures, were imaged with diffusion MRI before and after the infusion. By using Ricci Curvature to measure changes in robustness, we quantified both local and global changes in the brain networks and their potential relationship with the infusion. Our results find changes in the Curvature of the connections between regions associated with ASD that were not detected via traditional brain network analysis.
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measuring robustness of brain networks in autism spectrum disorder with Ricci Curvature
bioRxiv, 2019Co-Authors: Anish K Simhal, Allen Tannenbaum, Kimberly L H Carpenter, Saad Nadeem, Joanne Kurtzberg, Allen W Song, Guillermo Sapiro, Geraldine DawsonAbstract:Ricci Curvature is a method for measuring the robustness of networks. In this work, we use Ricci Curvature to measure robustness of brain networks affected by autism spectrum disorder (ASD). Subjects with ASD are given a stem cell infusion and are imaged with diffusion MRI before and after the infusion. By using Ricci Curvature to measure changes in robustness, we quantify both local and global changes in the brain networks correlated with the infusion. Our results find changes in regions associated with ASD that were not detected via traditional brain network analysis.
Emil Saucan - One of the best experts on this subject based on the ideXlab platform.
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Comparative analysis of two discretizations of Ricci Curvature for complex networks.
Scientific Reports, 2018Co-Authors: Areejit Samal, R. P. Sreejith, Emil Saucan, Jiao Gu, Jürgen JostAbstract:We have performed an empirical comparison of two distinct notions of discrete Ricci Curvature for graphs or networks, namely, the Forman-Ricci Curvature and Ollivier-Ricci Curvature. Importantly, these two discretizations of the Ricci Curvature were developed based on different properties of the classical smooth notion, and thus, the two notions shed light on different aspects of network structure and behavior. Nevertheless, our extensive computational analysis in a wide range of both model and real-world networks shows that the two discretizations of Ricci Curvature are highly correlated in many networks. Moreover, we show that if one considers the augmented Forman-Ricci Curvature which also accounts for the two-dimensional simplicial complexes arising in graphs, the observed correlation between the two discretizations is even higher, especially, in real networks. Besides the potential theoretical implications of these observations, the close relationship between the two discretizations has practical implications whereby Forman-Ricci Curvature can be employed in place of Ollivier-Ricci Curvature for faster computation in larger real-world networks whenever coarse analysis suffices.
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Ricci Curvature of the internet topology
arXiv: Social and Information Networks, 2015Co-Authors: Yuyao Lin, Jie Gao, Emil SaucanAbstract:Analysis of Internet topologies has shown that the Internet topology has negative Curvature, measured by Gromov's "thin triangle condition", which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci Curvature of the Internet, defined by Ollivier, Lin, etc. Ricci Curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of Curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci Curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.
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Ricci Curvature of the Internet topology
2015 IEEE Conference on Computer Communications (INFOCOM), 2015Co-Authors: Chien-chun Ni, Xianfeng David Gu, Emil SaucanAbstract:Analysis of Internet topologies has shown that the Internet topology has negative Curvature, measured by Gromov's “thin triangle condition”, which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci Curvature of the Internet, defined by Ollivier [1], Lin et al. [2], etc. Ricci Curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of Curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci Curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.
Andrea Mondino - One of the best experts on this subject based on the ideXlab platform.
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on quotients of spaces with Ricci Curvature bounded below
Journal of Functional Analysis, 2018Co-Authors: Fernando Galazgarcia, Martin Kell, Andrea Mondino, Gerardo SosaAbstract:Abstract Let ( M , g ) be a smooth Riemannian manifold and G a compact Lie group acting on M effectively and by isometries. It is well known that a lower bound of the sectional Curvature of ( M , g ) is again a bound for the Curvature of the quotient space, which is an Alexandrov space of Curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions. The goal of the paper is to prove the corresponding stability properties for synthetic Ricci Curvature lower bounds. Specifically, we show that such stability holds for quotients of RCD ⁎ ( K , N ) -spaces, under isomorphic compact group actions and more generally under metric-measure foliations and submetries. An RCD ⁎ ( K , N ) -space is a metric measure space with an upper dimension bound N and weighted Ricci Curvature bounded below by K in a generalized sense. In particular, this shows that if ( M , g ) has Ricci Curvature bounded below by K ∈ R and dimension N, then the quotient space is an RCD ⁎ ( K , N ) -space. Additionally, we tackle the same problem for the CD / CD ⁎ and MCP Curvature-dimension conditions. We provide as well geometric applications which include: A generalization of Kobayashi's Classification Theorem of homogeneous manifolds to RCD ⁎ ( K , N ) -spaces with essential minimal dimension n ≤ N ; a structure theorem for RCD ⁎ ( K , N ) -spaces admitting actions by large (compact) groups; and geometric rigidity results for orbifolds such as Cheng's Maximal Diameter and Maximal Volume Rigidity Theorems. Finally, in two appendices we apply the methods of the paper to study quotients by isometric group actions of discrete spaces and of (super-)Ricci flows.
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sharp and rigid isoperimetric inequalities in metric measure spaces with lower Ricci Curvature bounds
Inventiones Mathematicae, 2017Co-Authors: Fabio Cavalletti, Andrea MondinoAbstract:We prove that if \((X,\mathsf {d},\mathfrak {m})\) is a metric measure space with \(\mathfrak {m}(X)=1\) having (in a synthetic sense) Ricci Curvature bounded from below by \(K>0\) and dimension bounded above by \(N\in [1,\infty )\), then the classic Levy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by Milman for any \(K\in \mathbb {R}\), \(N\ge 1\) and upper diameter bounds) holds, i.e. the isoperimetric profile function of \((X,\mathsf {d},\mathfrak {m})\) is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume \(v \in (0,1)\) and K is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume \(v \in (0,1)\) and K is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci Curvature lower bounds. Examples of spaces fitting our assumptions include measured Gromov–Hausdorff limits of Riemannian manifolds satisfying Ricci Curvature lower bounds, Alexandrov spaces with Curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci Curvature lower bounds; the result seems new even in these celebrated classes of spaces.
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sharp and rigid isoperimetric inequalities in metric measure spaces with lower Ricci Curvature bounds
arXiv: Metric Geometry, 2015Co-Authors: Fabio Cavalletti, Andrea MondinoAbstract:We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci Curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic L\'evy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by E. Milman for any $K\in \mathbb{R}$, $N\geq 1$ and upper diameter bounds) hold, i.e. the isoperimetric profile function of $(X,\mathsf{d},\mathfrak{m})$ is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume $v \in (0,1)$ and $K$ is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci Curvature lower bounds. Examples of spaces fitting our assumptions include measured Gromov-Hausdorff limits of Riemannian manifolds satisfying Ricci Curvature lower bounds and Alexandrov spaces with Curvature bounded from below; the result seems new even in these celebrated classes of spaces.
