Ollivier-Ricci Curvature Research Articles

Augmentations of Forman’s Ricci curvature and their applications in community detection

Abstract The notion of curvature on graphs has recently gained traction in the networks community, with the Ollivier–Ricci curvature (ORC) in particular being used for several tasks in network analysis, such as community detection. In this work, we choose a different approach and study augmentations of the discretization of the Ricci curvature proposed by Forman (AFRC). We empirically and theoretically investigate its relation to the ORC and the un-augmented Forman–Ricci curvature. In particular, we provide evidence that the AFRC frequently gives sufficient insight into the structure of a network to be used for community detection, and therefore provides a computationally cheaper alternative to previous ORC-based methods. Our novel AFRC-based community detection algorithm is competitive with an ORC-based approach.

Bakry–Émery–Ricci curvature: an alternative network geometry measure in the expanding toolbox of graph Ricci curvatures

Abstract 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.

Multi-scale geometric network analysis identifies melanoma immunotherapy response gene modules

Melanoma response to immune-modulating therapy remains incompletely characterized at the molecular level. In this study, we assess melanoma immunotherapy response using a multi-scale network approach to identify gene modules with coordinated gene expression in response to treatment. Using gene expression data of melanoma before and after treatment with nivolumab, we modeled gene expression changes in a correlation network and measured a key network geometric property, dynamic Ollivier-Ricci curvature, to distinguish critical edges within the network and reveal multi-scale treatment-response gene communities. Analysis identified six distinct gene modules corresponding to sets of genes interacting in response to immunotherapy. One module alone, overlapping with the nuclear factor kappa-B pathway (NFkB), was associated with improved patient survival and a positive clinical response to immunotherapy. This analysis demonstrates the usefulness of dynamic Ollivier-Ricci curvature as a general method for identifying information-sharing gene modules in cancer.

Curvature-enhanced graph convolutional network for biomolecular interaction prediction

Geometric deep learning has demonstrated a great potential in non-Euclidean data analysis. The incorporation of geometric insights into learning architecture is vital to its success. Here we propose a curvature-enhanced graph convolutional network (CGCN) for biomolecular interaction prediction. Our CGCN employs Ollivier-Ricci curvature (ORC) to characterize network local geometric properties and enhance the learning capability of GCNs. More specifically, ORCs are evaluated based on the local topology from node neighborhoods, and further incorporated into the weight function for the feature aggregation in message-passing procedure. Our CGCN model is extensively validated on fourteen real-world bimolecular interaction networks and analyzed in details using a series of well-designed simulated data. It has been found that our CGCN can achieve the state-of-the-art results. It outperforms all existing models, as far as we know, in thirteen out of the fourteen real-world datasets and ranks as the second in the rest one. The results from the simulated data show that our CGCN model is superior to the traditional GCN models regardless of the positive-to-negative-curvature ratios, network densities, and network sizes (when larger than 500).

Gene interaction network analysis in multiple myeloma detects complex immune dysregulation associated with shorter survival

The plasma cell cancer multiple myeloma (MM) varies significantly in genomic characteristics, response to therapy, and long-term prognosis. To investigate global interactions in MM, we combined a known protein interaction network with a large clinically annotated MM dataset. We hypothesized that an unbiased network analysis method based on large-scale similarities in gene expression, copy number aberration, and protein interactions may provide novel biological insights. Applying a novel measure of network robustness, Ollivier-Ricci Curvature, we examined patterns in the RNA-Seq gene expression and CNA data and how they relate to clinical outcomes. Hierarchical clustering using ORC differentiated high-risk subtypes with low progression free survival. Differential gene expression analysis defined 118 genes with significantly aberrant expression. These genes, while not previously associated with MM, were associated with DNA repair, apoptosis, and the immune system. Univariate analysis identified 8/118 to be prognostic genes; all associated with the immune system. A network topology analysis identified both hub and bridge genes which connect known genes of biological significance of MM. Taken together, gene interaction network analysis in MM uses a novel method of global assessment to demonstrate complex immune dysregulation associated with shorter survival.

Network Modeling of Apoptosis-Related Genes Reveal a High-Risk Cluster in Multiple Myeloma

The plasma cell cancer multiple myeloma (MM) varies significantly in genomic characteristics, response to therapy, and long-term prognosis. Recent studies have shown the complexity of biological systems associated with poor outcomes in MM. One hallmark of cancer pathogenesis, applicable to MM, is the dysregulation of apoptotic cell death. Understanding the relationship between known drivers of MM and critical pathways in the apoptosis gene network is key towards identifying novel therapeutic targets. Recently, there has been a surge of interest in developing network-based approaches to study gene interactions and use machine learning tools to further elucidate useful biological factors. It is well-known that genes operate as network systems; however, interpretable computational methods are still limited. In this work, we combine a set of network analysis tools to provide novel biological insights into how the apoptosis network is dysregulated in MM. We identify a set of genes associated with poor outcomes in MM using both univariate and network-based approaches. Genomic and clinical data, including RNA-sequencing (RNA-seq) and copy number alteration (CNA), for 659 subjects were obtained from the Multiple Myeloma Research Foundation's CoMMpass (IA19) database. The apoptosis gene set is defined using the apoptosis hallmark pathway provided by the gene set enrichment analysis tool. The interactions between the given list of genes are provided via MetaCore. Univariate gene modeling is done using a Cox proportional hazards model. To determine the relationship between genes, we used Ollivier-Ricci curvature (ORC), a measure of network robustness. The ORC value between two genes incorporates information about both the two genes in question and their local neighborhoods, a concept critical for understanding the role a neighborhood of genes has on a given connection. Once edge information is computed, we used a uniform manifold approximation and projection (UMAP) to reduce the complexity of the dataset. The resulting projection is clustered using k-means clustering, and survival analysis of the given clusters is done using the Kaplan-Meier (KM) method. To understand which features are most informative, we used a random forest to predict the cluster labels using the RNA-seq data. After five-fold cross-validation, feature importance was computed using random forest's permutation importance methodology. All statistical significance testing was corrected for multiple comparisons via the Benjamini-Hochberg false discovery rate method with an alpha set to 0.05. Using RNA-seq data of genes associated with apoptosis, we identified nineteen apoptosis genes associated with poor prognosis in MM. Some of these genes, such as ANXA1 and WEE1 are known to have an effect in MM, while others, such as AVPR1A have yet to be studied extensively for their role in MM. When examining CNA data associated with apoptosis, a similar pattern emerged - nineteen genes were associated with poor prognosis. However, between the two key gene sets found by RNA-seq and CNA, only FAS appeared in both lists. FAS, Fas Cell Surface Death Receptor, is critical for triggering apoptosis. Half of the 37 genes identified using RNA-seq and CNA data are directly connected to the TP53 gene. While TP53 mutations are not a marker of the presence of MM itself, TP53 mutations have been shown to be associated with poorer outcomes. After reducing the complexity of the Ollivier-Ricci curvature analysis of the RNA-seq apoptosis network using UMAPs, the data revealed five clusters with KM curves for progression-free survival were significant (p<0.001). To identify the genes most impactful for separating these clusters, we trained a random forest model on the RNA-seq - note, it was the edge robustness values that were clustered - and evaluated its performance for predicted class membership. The model had an overall accuracy of approximately 78% with F1 scores for each class ranging from 61% to 87%. The top three features associated with the model include CTNNB1, RELA, and LEF1. Of note, CTNNB1 and RELA are directly interacted with the TP53 gene. Because these three genes discriminate between all five clusters, their expression levels may play a mediating effect on MM outcomes. Further investigation with external datasets is needed to validate these results.

On computing discretized Ricci curvatures of graphs: Local algorithms and (localized) fine-grained reductions

Characterizing shapes of high-dimensional objects via Ricci curvatures plays a critical role in many research areas in mathematics and physics. However, even though several discretizations of Ricci curvatures for discrete combinatorial objects such as networks have been proposed and studied by mathematicians, the computational complexity aspects of these discretizations have escaped the attention of theoretical computer scientists to a large extent. In this paper, we study one such discretization, namely the Ollivier-Ricci curvature, from the perspective of efficient computation by fine-grained reductions and local query-based algorithms. Our main contributions are the following.▹We relate our curvature computation problem to minimum weight perfect matching problem on complete bipartite graphs via fine-grained reduction.▹We formalize the computational aspects of the curvature computation problems in suitable frameworks so that they can be studied by researchers in local algorithms.▹We provide the first known lower and upper bounds on queries for query-based algorithms for the curvature computation problems in our local algorithms framework. En route, we also illustrate a localized version of our fine-grained reduction. We believe that our results bring forth an intriguing set of research questions, motivated both in theory and practice, regarding designing efficient algorithms for curvatures of geometrical objects.

Geometric graph neural networks on multi-omics data to predict cancer survival outcomes

The advance of sequencing technologies has enabled a thorough molecular characterization of the genome in human cancers. To improve patient prognosis predictions and subsequent treatment strategies, it is imperative to develop advanced computational methods to analyze large-scale, high-dimensional genomic data. However, traditional machine learning methods face a challenge in handling the high-dimensional, low-sample size problem that is shown in most genomic data sets. To address this, our group has developed geometric network analysis techniques on multi-omics data in connection with prior biological knowledge derived from protein-protein interactions (PPIs) or pathways. Geometric features obtained from the genomic network, such as Ollivier-Ricci curvature and the invariant measure of the associated Markov chain, have been shown to be predictive of survival outcomes in various cancers. In this study, we propose a novel supervised deep learning method called geometric graph neural network (GGNN) that incorporates such geometric features into deep learning for enhanced predictive power and interpretability. More specifically, we utilize a state-of-the-art graph neural network with sparse connections between the hidden layers based on known biology of the PPI network and pathway information. Geometric features along with multi-omics data are then incorporated into the corresponding layers. The proposed approach utilizes a local–global principle in such a manner that highly predictive features are selected at the front layers and fed directly to the last layer for multivariable Cox proportional-hazards regression modeling. The method was applied to multi-omics data from the CoMMpass study of multiple myeloma and ten major cancers in The Cancer Genome Atlas (TCGA). In most experiments, our method showed superior predictive performance compared to other alternative methods.

Discrete Ricci curvatures capture age-related changes in human brain functional connectivity networks.

Geometry-inspired notions of discrete Ricci curvature have been successfully used as markers of disrupted brain connectivity in neuropsychiatric disorders, but their ability to characterize age-related changes in functional connectivity is unexplored. We apply Forman-Ricci curvature and Ollivier-Ricci curvature to compare functional connectivity networks of healthy young and older subjects from the Max Planck Institute Leipzig Study for Mind-Body-Emotion Interactions (MPI-LEMON) dataset (N = 225). We found that both Forman-Ricci curvature and Ollivier-Ricci curvature can capture whole-brain and region-level age-related differences in functional connectivity. Meta-analysis decoding demonstrated that those brain regions with age-related curvature differences were associated with cognitive domains known to manifest age-related changes-movement, affective processing, and somatosensory processing. Moreover, the curvature values of some brain regions showing age-related differences exhibited correlations with behavioral scores of affective processing. Finally, we found an overlap between brain regions showing age-related curvature differences and those brain regions whose non-invasive stimulation resulted in improved movement performance in older adults. Our results suggest that both Forman-Ricci curvature and Ollivier-Ricci curvature correctly identify brain regions that are known to be functionally or clinically relevant. Our results add to a growing body of evidence demonstrating the sensitivity of discrete Ricci curvature measures to changes in the organization of functional connectivity networks, both in health and disease.

Abstract 2061: Protein network analysis uncovers a poor-survival subtype in multiple myeloma

Abstract Multiple myeloma (MM) prognosis incorporates a variety of metrics including treatment received, clinical factors, and genomic characteristics, with several specific genomic features predicting for shorter progression free survival (PFS). No study to date has integrated genomic data from a systems view, incorporating interactions between biomarkers in a network. We use a geometric network analysis that integrates complex interactions to characterize patterns of biological behavior not captured by individual genomic events. The methodology is mathematically well-defined and has no fitting parameters. We hypothesized that such a systems mathematical approach applied to gene interaction networks may delineate biologically relevant MM subtypes and potential new therapeutic targets. We overlaid RNA-Seq and copy number alteration data from the MMRF CoMMpass study (IA19) on a gene interactome derived from the Human Protein Reference Database using a novel graph metric of network robustness — Ollivier-Ricci curvature (ORC). Results were clustered, with the optimal number determined via silhouette score. Survival analysis for PFS was performed employing Kaplan-Meier and log-rank tests. A differential gene expression analysis between high and low risk groups was conducted. Differences in scalar ORC between the low-risk and high-risk groups were examined and contextualized using a pathway analysis. Pathway analysis was performed using the Broad Institute’s Gene Set Enrichment Analysis tool and the pathways used are from the hallmark gene set from the Human MSigDB collection. The dataset included 659 patients and the incorporated protein-protein interactions resulted in a network with 8,468 nodes and 33,695 edges. The ORC analysis discovered 6 clusters, with specific genomic features being associated with clusters predicting for long [hyperdiploidy, t(11:14)], and short [t(4;14), MAF/MAFB translocations] PFS. A differential gene expression analysis comparing the high risk and low risk groups identified 118 key genes. These genes were associated with various pathways both known and unknown to be associated with multiple myeloma, including mitotic spindle, DNA repair, inflammatory response, and the P53 pathways. Further scalar curvature analysis showed differences in the apoptosis, TGF beta signaling, and other signaling pathways. In summary, we applied the geometric network analysis tool ORC to multi-omics data in MM represented as biological networks to identify individuals at high risk of short PFS and relevant biological correlates. Decreased robustness of signaling near immune-related genes was associated with shorter survival, highlighting the plausible utility of using these methods to uncover new biological insights. Citation Format: Anish K. Simhal, Kylee H. Maclachlan, Rena Elkin, Jiening Zhu, Saad Z. Usmani, Jonathan J. Keats, Larry Norton, Joseph O. Deasy, Jung Hun Oh, Allen Tannenbaum. Protein network analysis uncovers a poor-survival subtype in multiple myeloma [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2023; Part 1 (Regular and Invited Abstracts); 2023 Apr 14-19; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2023;83(7_Suppl):Abstract nr 2061.

Abstract 5367: Deep neural networks using protein-protein network information predict multiple myeloma survival

Abstract The modern development of sequencing technologies provides a comprehensive molecular portrait of human cancers. There is a strong need to develop methods to not only improve patient prognosis predictions but also to understand the driving factors for treatment. However, the high-dimension, low-sample size nature of the genomic data poses challenges for typical machine learning algorithms. The systematic understanding of genes with respect to a network (protein-protein interaction (PPI) network) is a way to handle the limit and the nonparametric analysis of geometric properties such as Ollivier-Ricci curvature and associated invariant measure developed by our group have proven to be successful for the prediction of survival in multiple cancers. In this work, we propose a novel supervised deep learning approach combining the aforementioned geometric methods, which benefit from the flexibility provided by deep learning techniques while still preserving much of the interpretability of the geometric analysis. We take advantage of a state-of-the-art graph neural network approach. Sparse connections between layers were inspired by the known biology of the PPI network from the Human Protein Reference Database (HPRD) and pathway information from the Kyoto Encyclopedia of Genes and Genomes (KEGG) database, supplemented with geometric network features which are fed into the network in corresponding layers. The prediction is based on a local-global principle, where highly predictive features are selected from early layers of the network and fed directly to the final layer to produce a multivariable Cox regression. We applied our method to RNA-Seq gene expression data from the CoMMpass study of multiple myeloma (MM). More specifically, 657 patients in the data set were randomly divided into training, validation and set-aside testing sets by a ratio of 6:2:2. We obtained an average C-index 0.66 of the prediction in the testing set from a 10-fold data split. Dichotomizing the testing set by its mean value to define high-risk vs. low-risk yielded a significant p-value of the log-rank test in the set-aside data (p-value =3e-4). We observed that geometric protein network information not only improved the outcome prediction (vs. 6% worse without geometric feature inputs), but was also more robust to fold splitting. From our model, we identified WEE1, CENPE and CENPF as top genes driving survival differences (higher expression of WEE1 increased risk and lower negative curvature between CENPE and CENPF increased risk). WEE1 is a cell cycle-related gene that regulates DNA repair and CENPE and CENPF are components of a fibrous layer of mitotic kinetochores, which have been indicated in the literature to be related to the prognosis as well as possible targets for treatment. While it is therefore logical that these genes would be implicated in the natural history of MM, they were identified entirely on the basis of network analysis. Citation Format: Jiening Zhu, Jung Hun Oh, Anish K. Simhal, Rena Elkin, Larry Norton, Joseph O. Deasy, Allen R. Tannenbaum. Deep neural networks using protein-protein network information predict multiple myeloma survival. [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2023; Part 1 (Regular and Invited Abstracts); 2023 Apr 14-19; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2023;83(7_Suppl):Abstract nr 5367.

Geometric Network Analysis Defines Poor-Prognosis Subtypes in Multiple Myeloma

The prognosis for patients with multiple myeloma (MM) is dependent on a variety of measures including treatment received, clinical factors, and genomic characteristics, with several specific genomic features predicting for shorter progression free survival (PFS). Studies have been conducted to identify biomarkers using gene expression profiles and copy number variation (CNV) data. However, no study to date has integrated genomic data from a systems view, incorporating interactions between biomarkers in a network. We apply an innovative geometric network analysis that integrates complex interactions to characterize patterns of biological behavior not captured by individual genomic events. The methodology is mathematically well-defined and has no fitting parameters. We hypothesized that such a systems mathematical approach applied to gene interaction networks may delineate biologically relevant MM subtypes and potential new therapeutic targets. We overlaid RNA-Seq and CNV data from the MMRF CoMMpass dataset (IA19) on a gene interactome derived from the Human Protein Reference Database (HPRD) using a novel graph metric of network robustness - Ollivier-Ricci curvature (ORC). ORC considers both local and global connectivity in assessing the robustness of each pathway characterized by feedback loops in a network. Hierarchical agglomerative clustering of the resulting ORC matrix computed for each omics-dataset was performed to identify subtypes based on the patterns of curvature, with the optimal number of clusters determined by silhouette score. Survival analysis for progression free survival (PFS) was performed employing Kaplan-Meier and log-rank tests. A differential gene expression analysis between high and low risk groups, as identified by the gene expression data, was then carried out using raw read counts with DESeq2. Further downstream bioinformatic analysis was conducted using MetaCore software. Both RNA-Seq and CNV data were available in 659 patients. Gene product interactions were determined by protein-protein interactions derived from HPRD, resulting in a network with 8,468 nodes and 33,695 edges. The ORC analysis using RNA-Seq data resulted in 6 clusters, and Kaplan-Meier analysis showed statistical significance in PFS with log-rank p=0.0016. Clusters predicting the best and worst PFS were associated with specific genomic features including CCND1 and MAF/MAFB translocations, respectively (each p<0.05, Fisher's exact test). Considering the central role of CCND1 in MM biology, we repeated our analysis with t(11;14) patients excluded, with the resulting clustering remaining significant for predicting PFS. The ORC analysis of CNV data on HPRD resulted in 8 clusters, and Kaplan-Meier analysis showed statistical significance in PFS with log-rank p=0.0082. Genomic features associated with the clusters included hyperdiploidy and CCND1 (longest PFS), and MAF-translocations and APOBEC-activity (shortest PFS) (each p<0.05, Fisher's). The worst PFS cluster had a higher prevalence of chromothripsis (p<0.0001, Fisher's) than other clusters, known to be an independent predictor for MM survival. Considering the impact of hyperdiploidy on CNV in MM, analyzing hyperploid and non-hyperdiploid patients separately was still significant for PFS prediction. After DESeq2 analysis to perform differential gene expression analysis between the high and low risk groups, the Benjamini-Hochberg method was applied to adjust the resulting p-values, producing false discovery rate (FDR). A list of 118 genes derived with FDR < 0.05 and |fold change| > 3.5 was input into MetaCore to identify biological correlates associated with the risk of PFS. Key implicated biological processes included cell cycle progression, inflammation and immune response. In this study, we applied the geometric network analysis tool ORC to multi-omics data in MM represented as biological networks to identify individuals at high risk of short PFS and relevant biological correlates. One hundred and eighteen key genes associated with MM prognosis were identified. Decreased robustness of signaling near immune-related genes was associated with shorter survival. For both RNAseq and CNV-based data, patient clusters were associated with known predictors of PFS, and remained predictive when removing specific genomic features, indicating that our methods are providing new biological insights. Figure 1View largeDownload PPTFigure 1View largeDownload PPT Close modal

Changes in the geometry and robustness of diffusion tensor imaging networks: Secondary analysis from a randomized controlled trial of young autistic children receiving an umbilical cord blood infusion.

Diffusion tensor imaging (DTI) has been used as an outcome measure in clinical trials for several psychiatric disorders but has rarely been explored in autism clinical trials. This is despite a large body of research suggesting altered white matter structure in autistic individuals. The current study is a secondary analysis of changes in white matter connectivity from a double-blind placebo-control trial of a single intravenous cord blood infusion in 2-7-year-old autistic children (1). Both clinical assessments and DTI were collected at baseline and 6 months after infusion. This study used two measures of white matter connectivity: change in node-to-node connectivity as measured through DTI streamlines and a novel measure of feedback network connectivity, Ollivier-Ricci curvature (ORC). ORC is a network measure which considers both local and global connectivity to assess the robustness of any given pathway. Using both the streamline and ORC analyses, we found reorganization of white matter pathways in predominantly frontal and temporal brain networks in autistic children who received umbilical cord blood treatment versus those who received a placebo. By looking at changes in network robustness, this study examined not only the direct, physical changes in connectivity, but changes with respect to the whole brain network. Together, these results suggest the use of DTI and ORC should be further explored as a potential biomarker in future autism clinical trials. These results, however, should not be interpreted as evidence for the efficacy of cord blood for improving clinical outcomes in autism. This paper presents a secondary analysis using data from a clinical trial that was prospectively registered with ClinicalTrials.gov(NCT02847182).

Inferring functional communities from partially observed biological networks exploiting geometric topology and side information

Cellular biological networks represent the molecular interactions that shape function of living cells. Uncovering the organization of a biological network requires efficient and accurate algorithms to determine the components, termed communities, underlying specific processes. Detecting functional communities is challenging because reconstructed biological networks are always incomplete due to technical bias and biological complexity, and the evaluation of putative communities is further complicated by a lack of known ground truth. To address these challenges, we developed a geometric-based detection framework based on Ollivier-Ricci curvature to exploit information about network topology to perform community detection from partially observed biological networks. We further improved this approach by integrating knowledge of gene function, termed side information, into the Ollivier-Ricci curvature algorithm to aid in community detection. This approach identified essential conserved and varied biological communities from partially observed Arabidopsis protein interaction datasets better than the previously used methods. We show that Ollivier-Ricci curvature with side information identified an expanded auxin community to include an important protein stability complex, the Cop9 signalosome, consistent with previous reported links to auxin response and root development. The results show that community detection based on Ollivier-Ricci curvature with side information can uncover novel components and novel communities in biological networks, providing novel insight into the organization and function of complex networks.

Applying Ollivier-Ricci curvature to indicate the mismatch of travel demand and supply in urban transit network

Urban transit networks need to be upgraded in accordance with urban development. While methods have been studied to design an optimal transit network given the locations of stations, these methods focus on the whole network as the optimization object. However, the strategy to improve parts of an existing transit network based on the gap between travel demand and infrastructure supply is rarely investigated. Different parts of the same network should not have the same priority to be upgraded. An evaluation method is required for this purpose to decide which parts of a transit network should be upgraded first. We argue that Ollivier-Ricci curvature, a concept from differential geometry, can serve as the evaluation method. The basic hypothesis is that Ollivier-Ricci curvature indicates the theoretical attractiveness of the fastest path between any two nodes, thereby capturing the internal structure of the transit network. Meanwhile, travel demand and behavior can be approximated by urban mobility big data. By comparing the theoretical attractiveness and the flow volume between a path between two stations, the traits of lines with mismatched supply and demand are identified, and macro strategies are suggested. A case study in Beijing demonstrates how the proposed method works out followed by a comparison with traditional betweenness centrality analysis.

Understanding Changes in the Topology and Geometry of Financial Market Correlations during a Market Crash.

In econophysics, the achievements of information filtering methods over the past 20 years, such as the minimal spanning tree (MST) by Mantegna and the planar maximally filtered graph (PMFG) by Tumminello et al., should be celebrated. Here, we show how one can systematically improve upon this paradigm along two separate directions. First, we used topological data analysis (TDA) to extend the notions of nodes and links in networks to faces, tetrahedrons, or k-simplices in simplicial complexes. Second, we used the Ollivier-Ricci curvature (ORC) to acquire geometric information that cannot be provided by simple information filtering. In this sense, MSTs and PMFGs are but first steps to revealing the topological backbones of financial networks. This is something that TDA can elucidate more fully, following which the ORC can help us flesh out the geometry of financial networks. We applied these two approaches to a recent stock market crash in Taiwan and found that, beyond fusions and fissions, other non-fusion/fission processes such as cavitation, annihilation, rupture, healing, and puncture might also be important. We also successfully identified neck regions that emerged during the crash, based on their negative ORCs, and performed a case study on one such neck region.

Unfolding the multiscale structure of networks with dynamical Ollivier-Ricci curvature

Describing networks geometrically through low-dimensional latent metric spaces has helped design efficient learning algorithms, unveil network symmetries and study dynamical network processes. However, latent space embeddings are limited to specific classes of networks because incompatible metric spaces generally result in information loss. Here, we study arbitrary networks geometrically by defining a dynamic edge curvature measuring the similarity between pairs of dynamical network processes seeded at nearby nodes. We show that the evolution of the curvature distribution exhibits gaps at characteristic timescales indicating bottleneck-edges that limit information spreading. Importantly, curvature gaps are robust to large fluctuations in node degrees, encoding communities until the phase transition of detectability, where spectral and node-clustering methods fail. Using this insight, we derive geometric modularity to find multiscale communities based on deviations from constant network curvature in generative and real-world networks, significantly outperforming most previous methods. Our work suggests using network geometry for studying and controlling the structure of and information spreading on networks.

Inner-outer curvatures, Ollivier-Ricci curvature and volume growth of graphs

We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger inner-outer curvature growth than a model graph, then it has faster volume growth too. We also study the relationhips of volume growth with other kind of curvatures, such as the Ollivier-Ricci curvature.

Enhanced Forman curvature and its relation to Ollivier curvature

Recent advances in emergent geometry and discretized approaches to quantum gravity have relied upon the notion of a discrete measure of graph curvature. We focus on the two main measures that have been studied, the so-called Ollivier-Ricci and Forman-Ricci curvatures. These two approaches have a very different origin, and both have advantages and disadvantages. In this work we study the relationship between the two measures for a class of graphs that are important in quantum gravity applications. We discover that under a specific set of circumstances they are equivalent, opening up the possibility of replacing the more fundamental Ollivier-Ricci curvature by the computationally more accessible Forman-Ricci curvature in certain applications to models of emergent spacetime and quantum gravity.

Network geometry and market instability.

The complexity of financial markets arise from the strategic interactions among agents trading stocks, which manifest in the form of vibrant correlation patterns among stock prices. Over the past few decades, complex financial markets have often been represented as networks whose interacting pairs of nodes are stocks, connected by edges that signify the correlation strengths. However, we often have interactions that occur in groups of three or more nodes, and these cannot be described simply by pairwise interactions but we also need to take the relations between these interactions into account. Only recently, researchers have started devoting attention to the higher-order architecture of complex financial systems, that can significantly enhance our ability to estimate systemic risk as well as measure the robustness of financial systems in terms of market efficiency. Geometry-inspired network measures, such as the Ollivier–Ricci curvature and Forman–Ricci curvature, can be used to capture the network fragility and continuously monitor financial dynamics. Here, we explore the utility of such discrete Ricci curvatures in characterizing the structure of financial systems, and further, evaluate them as generic indicators of the market instability. For this purpose, we examine the daily returns from a set of stocks comprising the USA S&P-500 and the Japanese Nikkei-225 over a 32-year period, and monitor the changes in the edge-centric network curvatures. We find that the different geometric measures capture well the system-level features of the market and hence we can distinguish between the normal or ‘business-as-usual’ periods and all the major market crashes. This can be very useful in strategic designing of financial systems and regulating the markets in order to tackle financial instabilities.