Ricci curvature and the stream of thought.

The characterization of complex networks with tools originating in geometry, for instance through the statistics of so-called Ricci curvatures, is a well established tool of network science. Various types of such Ricci curvatures capture different aspects of network geometry. In the present work, we investigate Bakry–Émery–Ricci curvature, a notion of discrete Ricci curvature that has been studied much in geometry, but so far has not been applied to networks. We explore on standard classes of artificial networks as well as on selected empirical ones to what the statistics of that curvature are similar to or different from that of other curvatures, how it is correlated to other important network measures, and what it tells us about the underlying network. We observe that most vertices typically have negative curvature. Furthermore, the curvature distributions are different for different types of model networks. We observe a high positive correlation between Bakry–Émery–Ricci and both Forman–Ricci and Ollivier–Ricci curvature, and in particular with the augmented version of Forman–Ricci curvature while comparing them for both model and real-world networks. We investigate the correlation of Bakry–Émery–Ricci curvature with degree, clustering coefficient and vertex centrality measures. Also, we investigate the importance of vertices with highly negative curvature values to maintain communication in the network. Additionally, for Forman–Ricci, Augmented Forman–Ricci and Ollivier–Ricci curvature, we compare the robustness of the networks by comparing the sum of the incident edges and the minimum of the incident edges as vertex measures and find that the sum identifies vertices that are important for maintaining the connectivity of the network. The computational time for Bakry–Émery–Ricci curvature is shorter than that required for Ollivier–Ricci curvature but higher than for Augmented Forman–Ricci curvature. We therefore conclude that for empirical network analysis, the latter is the tool of choice.

PointGS: Bridging and fusing geometric and semantic space for 3D point cloud analysis

Discrete Ricci curvatures for directed networks

Graph neural networks(GNNs) have been demonstrated to depend on whether the node effective information is sufficiently passing. Discrete curvature (Ricci curvature) is used to study graph connectivity and information propagation efficiency with a geometric perspective, and has been raised in recent years to explore the efficient message-passing structure of GNNs. However, most empirical studies are based on directly observed graph structures or heuristic topological assumptions, and lack in-depth exploration of underlying optimal information transport structures for downstream tasks. We suggest that graph curvature optimization is more in-depth and essential than directly rewiring or learning for graph structure with richer message-passing characterization and better information transport interpretability. From both graph geometry and information theory perspectives, we propose the novel Discrete Curvature Graph Information Bottleneck (CurvGIB) framework to optimize the information transport structure and learn better node representations simultaneously. CurvGIB advances the Variational Information Bottleneck (VIB) principle for Ricci curvature optimization to learn the optimal information transport pattern for specific downstream tasks. The learned Ricci curvature is used to refine the optimal transport structure of the graph, and the node representation is fully and efficiently learned. Moreover, for the computational complexity of Ricci curvature differentiation, we combine Ricci flow and VIB to deduce a curvature optimization approximation to form a tractable IB objective function. Extensive experiments on various datasets demonstrate the superior effectiveness and interpretability of CurvGIB.

We study the eigenvalues of the connection Laplacian on a graph with an orthogonal group or unitary group signature. We establish higher order Buser type inequalities, i.e., we provide upper bounds for eigenvalues in terms of Cheeger constants in the case of nonnegative Ricci curvature. In this process, we discuss the concepts of Cheeger type constants and a discrete Ricci curvature for connection Laplacians and study their properties systematically. The Cheeger constants are defined as mixtures of the expansion rate of the underlying graph and the frustration index of the signature. The discrete curvature, which can be computed efficiently via solving semidefinite programming problems, has a characterization by the heat semigroup for functions combined with a heat semigroup for vector fields on the graph.

Current unbiased approaches to mine the large amounts of patient-level data on mutations, structural variations and gene expression result in an unwieldy amount of interactions and correlations, which cannot be parsed to identify disease drivers. Here we present an approach to encode mutational and structural variant data at a patient level in a semantic association space. This approach transforms the presence of a mutation (or other feature) in each patient into the quantitative semantic association score of the corresponding gene and the phenotype of interest, which we have trained on all publicly available literature using word-embedding neural networks. Using data from The Cancer Genome Atlas (TCGA), we encoded the mutation or structural variant status (incl. copy number, fusion and chromothripsis) of all patients in the Lung Adenocarcinoma and Mesothelioma cohorts into our semantic space. For each cancer, we first defined the set of genes that are most associated to it according to the literature. To project each patient into this semantic space, we next determined if each patient had a mutation in the genes representing the disease semantic vector (e.g. NSCLC). For TCGA data we only counted non-Silent mutations and represented them as a binary number for each gene, i.e. 0 if the patient had no mutations in that gene and 1 if the patient had a non-Silent mutation in the gene. Each patient was then encoded in a binary vector with each member corresponding to a gene from the disease semantic vector. For example, lung adenocarcinoma was associated to 1,367 genes in our semantic space. A lung adenocarcinoma patient’s vector would them be composed of 1,367 binary numbers dictating if the gene is mutated or not in that patient. We then multiply these binary vectors with the semantic disease vector to obtain the patient’s projection in the disease space, which in effect replaces the binary number with the Semantic Association Score between the gene and the disease. Contrary to clustering patient samples by their mutation or structural variant data alone, our projected patient vectors clustered patients together into 22 groups with high patient-to-patient similarity. These clusters recapitulate canonical knowledge about the disease, e.g. Lung Adenocarcinoma patients form clusters that include EGFR-driven and KRAS-driven cohorts. We also see novel groups of patients driven by genes such as MET, STK11 and MALAT1. These clusters can be further stratified by their survival status and other clinical features. We validated our approach with a non-TCGA Mesothelioma cohort, revealing similarities in patient stratification regardless of the data source. This approach represents a dramatic shift in patient segmentation, delivering real-time grouping of patients and biomarker identification, which can accelerate clinical trial design and therapeutic development strategy. Citation Format: Enrique Garcia-Rivera, Aaron S. Mansfield, Karthik Murugadoss, Murali Aravamudan. Patient segmentation using machine-learning based literature and genomic data synthesis uncovers novel cohorts of NSCLC and mesothelioma patients [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2019; 2019 Mar 29-Apr 3; Atlanta, GA. Philadelphia (PA): AACR; Cancer Res 2019;79(13 Suppl):Abstract nr 2452.

Current unbiased approaches to mine the large amounts of patient-level data on mutations, structural variations and gene expression result in an unwieldy amount of interactions and correlations, which cannot be parsed to identify disease drivers. Here we present an approach to encode mutational and structural variant data at a patient level in a semantic association space. This approach transforms the presence of a mutation (or other feature) in each patient into the quantitative semantic association score of the corresponding gene and the phenotype of interest, which we have trained on all publicly available literature using word-embedding neural networks. Using data from The Cancer Genome Atlas (TCGA), we encoded the mutation or structural variant status (incl. copy number, fusion and chromothripsis) of all patients in the Lung Adenocarcinoma and Mesothelioma cohorts into our semantic space. For each cancer, we first defined the set of genes that are most associated to it according to the literature. To project each patient into this semantic space, we next determined if each patient had a mutation in the genes representing the disease semantic vector (e.g. NSCLC). For TCGA data we only counted non-Silent mutations and represented them as a binary number for each gene, i.e. 0 if the patient had no mutations in that gene and 1 if the patient had a non-Silent mutation in the gene. Each patient was then encoded in a binary vector with each member corresponding to a gene from the disease semantic vector. For example, lung adenocarcinoma was associated to 1,367 genes in our semantic space. A lung adenocarcinoma patient’s vector would them be composed of 1,367 binary numbers dictating if the gene is mutated or not in that patient. We then multiply these binary vectors with the semantic disease vector to obtain the patient’s projection in the disease space, which in effect replaces the binary number with the Semantic Association Score between the gene and the disease. Contrary to clustering patient samples by their mutation or structural variant data alone, our projected patient vectors clustered patients together into 22 groups with high patient-to-patient similarity. These clusters recapitulate canonical knowledge about the disease, e.g. Lung Adenocarcinoma patients form clusters that include EGFR-driven and KRAS-driven cohorts. We also see novel groups of patients driven by genes such as MET, STK11 and MALAT1. These clusters can be further stratified by their survival status and other clinical features. We validated our approach with a non-TCGA Mesothelioma cohort, revealing similarities in patient stratification regardless of the data source. This approach represents a dramatic shift in patient segmentation, delivering real-time grouping of patients and biomarker identification, which can accelerate clinical trial design and therapeutic development strategy. Citation Format: Enrique Garcia-Rivera, Aaron S. Mansfield, Karthik Murugadoss, Murali Aravamudan. Patient segmentation using machine-learning based literature and genomic data synthesis uncovers novel cohorts of NSCLC and mesothelioma patients [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2019; 2019 Mar 29-Apr 3; Atlanta, GA. Philadelphia (PA): AACR; Cancer Res 2019;79(13 Suppl):Abstract nr 2452.

In order to improve the efficiency of spatial information service composition and enhance its stability, this paper proposes to introduce the spatial information service combination network into the geometric space and use Ricci curvature to study the local structure of spatial information service chain and explore the way of connecting to the spatial information service network, and analyse the Ricci curvature and its distribution and excavate the network structure and analyse its meaning. It is found that negative curvature nodes are more heavily loaded and have more service invocations. They are prefer to be distributed over the edge of the graph structure composed of spatial information service chains, and more inclined to the intersection of local structures of the network, the positive curvature on the contrary. And the proportion of negative curvature of this service network is large, which indicates that there is strong connectivity and recall between these different service chains.

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi-linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingartens formula, the Ricci curvature, and the Codazzi-Mainardi equations. For matrix analogues of embedded surfaces, we define discrete curvatures and Euler characteristics, and a non-commutative Gauss-Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and explicit examples are provided. Furthermore, we illustrate the fact that techniques from differential geometry can carry over to matrix analogues by proving that a bound on the discrete Gauss curvature implies a bound on the eigenvalues of the discrete Laplace operator.

The widespread use of deep learning in processing point cloud data promotes the development of neural networks designed for point clouds. Point-based methods are increasingly becoming the mainstream in point cloud neural networks due to their high efficiency and performance. However, most of these methods struggle to balance both the geometric and semantic space of the point cloud, which usually leads to unclear local feature aggregation in geometric space and poor global feature extraction in semantic space. To address these two defects, we propose a bilateral feature fusion module capable of combining geometric and semantic data from the point cloud to enhance local feature extraction. In addition, we propose an offset vector attention module for better extraction of global features from point clouds. We provide specific ablation studies and visualizations in the article to validate our key modules. Experimental results show that the proposed method performs superior in both point cloud classification and segmentation tasks.

The damage of the novel Coronavirus disease (COVID-19) is reaching an unprecedented scale. There are numerous classical epidemiology models trying to quantify epidemiology metrics. To forecast epidemics, classical approaches usually need parameter estimations, such as the contagion rate or the basic reproduction number. Here, we propose a data-driven, parameter-free, geometric approach to access the emergence of a pandemic state by studying the Forman–Ricci and Ollivier–Ricci network curvatures. Discrete Ollivier–Ricci curvature has been used successfully to forecast risk in financial networks and we suggest that those results can provide analogous results for COVID-19 epidemic time-series. We first compute both curvatures in a toy-model of epidemic time-series with delays, which allows us to create epidemic networks. We also compared our results to classical network metrics. By doing so, we are able to verify that the Ollivier–Ricci and Forman–Ricci curvatures can be a parameter-free estimate for identifying a pandemic state in the simulated epidemic. On this basis, we then compute both Forman–Ricci and Ollivier–Ricci curvatures for real epidemic networks built from COVID-19 epidemic time-series available at the World Health Organization (WHO). This approach allows us to detect early warning signs of the emergence of the pandemic. The advantage of our method lies in providing an early geometrical data marker for the pandemic state, regardless of parameter estimation and stochastic modelling. This work opens the possibility of using discrete geometry to study epidemic networks.

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.

In this paper, we consider the problem of approximately aligning/matching two graphs. Given two graphs \(G_{1}=(V_{1},E_{1})\) and \(G_{2}=(V_{2},E_{2})\), the objective is to map nodes \(u, v \in G_1\) to nodes \(u',v'\in G_2\) such that when u, v have an edge in \(G_1\), very likely their corresponding nodes \(u', v'\) in \(G_2\) are connected as well. This problem with subgraph isomorphism as a special case has extra challenges when we consider matching complex networks exhibiting the small world phenomena. In this work, we propose to use ‘Ricci flow metric’, to define the distance between two nodes in a network. This is then used to define similarity of a pair of nodes in two networks respectively, which is the crucial step of network alignment. Specifically, the Ricci curvature of an edge describes intuitively how well the local neighborhood is connected. The graph Ricci flow uniformizes discrete Ricci curvature and induces a Ricci flow metric that is insensitive to node/edge insertions and deletions. With the new metric, we can map a node in \(G_1\) to a node in \(G_2\) whose distance vector to only a few preselected landmarks is the most similar. The robustness of the graph metric makes it outperform other methods when tested on various complex graph models and real world network data sets (Emails, Internet, and protein interaction networks) (The source code of computing Ricci curvature and Ricci flow metric are available: https://github.com/saibalmars/GraphRicciCurvature).

The systemic stability of a stock market is one of the core issues in the financial field. The market can be regarded as a complex network whose nodes are stocks connected by edges that signify their correlation strength. Since the market is a strongly nonlinear system, it is difficult to measure the macroscopic stability and depict market fluctuations in time. In this article, we use a geometric measure derived from discrete Ricci curvature to capture the higher-order nonlinear architecture of financial networks. In order to confirm the effectiveness of our method, we use it to analyze the CSI 300 constituents of China’s stock market from 2005 to 2020 and the systemic stability of the market is quantified through the network’s Ricci-type curvatures. Furthermore, we use a hybrid model to analyze the curvature time series and predict the future trends of the market accurately. As far as we know, this is the first article to apply Ricci curvature to forecast the systemic stability of China’s stock market, and our results show that Ricci curvature has good explanatory power for the market stability and can be a good indicator to judge the future risk and volatility of China’s stock market.

Soft sets are efficient mathematical structures to model systems in multiple relations. Since a soft set is basically set system, it is possible to endow them with a proper distance function to obtain a metric space. By this embedding, we propose a discretization of the Ricci curvatures that stresses the relational character of universe elements in a soft set through the analysis of parameters rather than the elements themselves. The Forman and Ollivier-type Ricci curvatures we propose here quantifies the trade-off between parameter size and the cardinality of participation of parameterized universe elements in other parameters. Such discretizations of the Ricci curvature have already been applied to complex systems; however, it has not yet been formulated for soft sets. In this study, our main question is whether the defined geometric concept determines statistics for soft set models. Two examples are discussed for the answer to this question. The first example Ricci on soft sets model of occupational accidents occurred in Turkey in 2013–2014 is compared with the Wasserstein distance of the curvature distributions. The second example is the use of Ricci curvatures as an indicator in the soft sets model of a financial system while the system is in stress. These real world examples show that discrete Ricci curvatures for soft sets offer effective statistics.