Community Detection on Networks with Ricci Flow

ORCO: Ollivier-Ricci Curvature-Omics-an unsupervised method for analyzing robustness in biological systems

MOTIVATION

Although recent advanced sequencing technologies have improved the resolution of genomic and proteomic data to better characterize molecular phenotypes, efficient computational tools to analyze and interpret large-scale omic data are still needed.

RESULTS

To address this, we have developed a network-based bioinformatic tool called Ollivier-Ricci curvature for omics (ORCO). ORCO incorporates omics data and a network describing biological relationships between the genes or proteins and computes Ollivier-Ricci curvature (ORC) values for individual interactions. ORC is an edge-based measure that assesses network robustness. It captures functional cooperation in gene signaling using a consistent information-passing measure, which can help investigators identify therapeutic targets and key regulatory modules in biological systems. ORC has identified novel insights in multiple cancer types using genomic data and in neurodevelopmental disorders using brain imaging data. This tool is applicable to any data that can be represented as a network.

AVAILABILITY AND IMPLEMENTATION

ORCO is an open-source Python package and is publicly available on GitHub at https://github.com/aksimhal/ORC-Omics.

Parallel median consensus clustering in complex networks

We develop an algorithm that finds the consensus among many different clustering solutions of a graph. We formulate the problem as a median set partitioning problem and propose a greedy optimization technique. Unlike other approaches that find median set partitions, our algorithm takes graph structure into account and finds a comparable quality solution much faster than the other approaches. For graphs with known communities, our consensus partition captures the actual community structure more accurately than alternative approaches. To make it applicable to large graphs, we remove sequential dependencies from our algorithm and design a parallel algorithm. Our parallel algorithm achieves 35x speedup when utilizing 64 processing cores for large real-world graphs representing mass cytometry data from single-cell experiments.

ORCO: Ollivier-Ricci Curvature-Omics - an unsupervised method for analyzing robustness in biological systems

Although recent advanced sequencing technologies have improved the resolution of genomic and proteomic data to better characterize molecular phenotypes, efficient computational tools to analyze and interpret the large-scale omic data are still needed. To address this, we have developed a network-based bioinformatic tool called Ollivier-Ricci curvature-omics (ORCO). ORCO incorporates gene interaction information with omic data into a biological network, and computes Ollivier-Ricci curvature (ORC) values for individual interactions. ORC, an edge-based measure, indicates network robustness and captures global gene signaling changes in functional cooperation using a consistent information passing measure, thereby helping identify therapeutic targets and regulatory modules in biological systems. This tool can be applicable to any data that can be represented as a network. ORCO is an open-source Python package and publicly available on GitHub at https://github.com/aksimhal/ORC-Omics.

Deep learning as Ricci flow

Deep neural networks (DNNs) are powerful tools for approximating the distribution of complex data. It is known that data passing through a trained DNN classifier undergoes a series of geometric and topological simplifications. While some progress has been made toward understanding these transformations in neural networks with smooth activation functions, an understanding in the more general setting of non-smooth activation functions, such as the rectified linear unit (ReLU), which tend to perform better, is required. Here we propose that the geometric transformations performed by DNNs during classification tasks have parallels to those expected under Hamilton's Ricci flow-a tool from differential geometry that evolves a manifold by smoothing its curvature, in order to identify its topology. To illustrate this idea, we present a computational framework to quantify the geometric changes that occur as data passes through successive layers of a DNN, and use this framework to motivate a notion of 'global Ricci network flow' that can be used to assess a DNN's ability to disentangle complex data geometries to solve classification problems. By training more than 1500 DNN classifiers of different widths and depths on synthetic and real-world data, we show that the strength of global Ricci network flow-like behaviour correlates with accuracy for well-trained DNNs, independently of depth, width and data set. Our findings motivate the use of tools from differential and discrete geometry to the problem of explainability in deep learning.

Charting cellular differentiation trajectories with Ricci flow

Complex biological processes, such as cellular differentiation, require intricate rewiring of intra-cellular signalling networks. Previous characterisations revealed a raised network entropy underlies less differentiated and malignant cell states. A connection between entropy and Ricci curvature led to applications of discrete curvatures to biological networks. However, predicting dynamic biological network rewiring remains an open problem. Here we apply Ricci curvature and Ricci flow to biological network rewiring. By investigating the relationship between network entropy and Forman-Ricci curvature, theoretically and empirically on single-cell RNA-sequencing data, we demonstrate that the two measures do not always positively correlate, as previously suggested, and provide complementary rather than interchangeable information. We next employ Ricci flow to derive network rewiring trajectories from stem cells to differentiated cells, accurately predicting true intermediate time points in gene expression time courses. In summary, we present a differential geometry toolkit for understanding dynamic network rewiring during cellular differentiation and cancer.

A community partitioning algorithm for cyberspace

Community partitioning is an effective technique for cyberspace mapping. However, existing community partitioning algorithm only uses the topological structure of the network to divide the community and disregards factors such as real hierarchy, overlap, and directionality of information transmission between communities in cyberspace. Consequently, the traditional community division algorithm is not suitable for dividing cyberspace resources effectively. Based on cyberspace community structure characteristics, this study introduces an algorithm that combines an improved local fitness maximization (LFM) algorithm with the PageRank (PR) algorithm for community partitioning on cyberspace resources, called PR-LFM. First, seed nodes are determined using degree centrality, followed by local community expansion. Nodes belonging to multiple communities undergo further partitioning so that they are retained in the community where they are most important, thus preserving the community's original structure. The experimental data demonstrate good results in the resource division of cyberspace.

Deeper Exploiting Graph Structure Information by Discrete Ricci Curvature in a Graph Transformer

Graph-structured data, operating as an abstraction of data containing nodes and interactions between nodes, is pervasive in the real world. There are numerous ways dedicated to extract graph structure information explicitly or implicitly, but whether it has been adequately exploited remains an unanswered question. This work goes deeper by heuristically incorporating a geometric descriptor, the discrete Ricci curvature (DRC), in order to uncover more graph structure information. We present a curvature-based topology-aware graph transformer, termed Curvphormer. This work expands the expressiveness by using a more illuminating geometric descriptor to quantify the connections within graphs in modern models and to extract the desired structure information, such as the inherent community structure in graphs with homogeneous information. We conduct extensive experiments on a variety of scaled datasets, including PCQM4M-LSC, ZINC, and MolHIV, and obtain a remarkable performance gain on various graph-level tasks and fine-tuned tasks.

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

Introduction

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.

Methods

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).

Results

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.

Discussion

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.

PWN: enhanced random walk on a warped network for disease target prioritization

BACKGROUND

Extracting meaningful information from unbiased high-throughput data has been a challenge in diverse areas. Specifically, in the early stages of drug discovery, a considerable amount of data was generated to understand disease biology when identifying disease targets. Several random walk-based approaches have been applied to solve this problem, but they still have limitations. Therefore, we suggest a new method that enhances the effectiveness of high-throughput data analysis with random walks.

RESULTS

We developed a new random walk-based algorithm named prioritization with a warped network (PWN), which employs a warped network to achieve enhanced performance. Network warping is based on both internal and external features: graph curvature and prior knowledge.

CONCLUSIONS

We showed that these compositive features synergistically increased the resulting performance when applied to random walk algorithms, which led to PWN consistently achieving the best performance among several other known methods. Furthermore, we performed subsequent experiments to analyze the characteristics of PWN.

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.

Coarse Graining on Financial Correlation Networks

Community structure detection is an important and valuable task in financial network studies as it forms the basis of many statistical applications such as prediction, risk analysis, and recommendation. Financial networks have a natural multi-grained structure that leads to different community structures at different levels. However, few studies pay attention to these multi-part features of financial networks. In this study, we present a geometric coarse graining method based on Voronoi regions of a financial network. Rather than studying the dense structure of the network, we perform our analysis on the triangular maximally filtering of a financial network. Such filtered topology emerges as an efficient approach because it keeps local clustering coefficients steady and it underlies the network geometry. Moreover, in order to capture changes in coarse grains geometry throughout a financial stress, we study Haantjes curvatures of paths that are the farthest from the center in each of the Voronoi regions. We performed our analysis on a network representation comprising the stock market indices BIST (Borsa Istanbul), FTSE100 (London Stock Exchange), and Nasdaq-100 Index (NASDAQ), across three financial crisis periods. Our results indicate that there are remarkable changes in the geometry of coarse grains.

Graph Ricci curvatures reveal atypical functional connectivity in autism spectrum disorder

While standard graph-theoretic measures have been widely used to characterize atypical resting-state functional connectivity in autism spectrum disorder (ASD), geometry-inspired network measures have not been applied. In this study, we apply Forman-Ricci and Ollivier-Ricci curvatures to compare networks of ASD and typically developing individuals (N = 1112) from the Autism Brain Imaging Data Exchange I (ABIDE-I) dataset. We find brain-wide and region-specific ASD-related differences for both Forman-Ricci and Ollivier-Ricci curvatures, with region-specific differences concentrated in Default Mode, Somatomotor and Ventral Attention networks for Forman-Ricci curvature. We use meta-analysis decoding to demonstrate that brain regions with curvature differences are associated to those cognitive domains known to be impaired in ASD. Further, we show that brain regions with curvature differences overlap with those brain regions whose non-invasive stimulation improves ASD-related symptoms. These results suggest the utility of graph Ricci curvatures in characterizing atypical connectivity of clinically relevant regions in ASD and other neurodevelopmental disorders.

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.

Understanding the Variability in Graph Data Sets through Statistical Modeling on the Stiefel Manifold

Network analysis provides a rich framework to model complex phenomena, such as human brain connectivity. It has proven efficient to understand their natural properties and design predictive models. In this paper, we study the variability within groups of networks, i.e., the structure of connection similarities and differences across a set of networks. We propose a statistical framework to model these variations based on manifold-valued latent factors. Each network adjacency matrix is decomposed as a weighted sum of matrix patterns with rank one. Each pattern is described as a random perturbation of a dictionary element. As a hierarchical statistical model, it enables the analysis of heterogeneous populations of adjacency matrices using mixtures. Our framework can also be used to infer the weight of missing edges. We estimate the parameters of the model using an Expectation-Maximization-based algorithm. Experimenting on synthetic data, we show that the algorithm is able to accurately estimate the latent structure in both low and high dimensions. We apply our model on a large data set of functional brain connectivity matrices from the UK Biobank. Our results suggest that the proposed model accurately describes the complex variability in the data set with a small number of degrees of freedom.

Ollivier Persistent Ricci Curvature-Based Machine Learning for the Protein-Ligand Binding Affinity Prediction

Efficient molecular featurization is one of the major issues for machine learning models in drug design. Here, we propose a persistent Ricci curvature (PRC), in particular, Ollivier PRC (OPRC), for the molecular featurization and feature engineering, for the first time. The filtration process proposed in the persistent homology is employed to generate a series of nested molecular graphs. Persistence and variation of Ollivier Ricci curvatures on these nested graphs are defined as OPRC. Moreover, persistent attributes, which are statistical and combinatorial properties of OPRCs during the filtration process, are used as molecular descriptors and further combined with machine learning models, in particular, gradient boosting tree (GBT). Our OPRC-GBT model is used in the prediction of the protein-ligand binding affinity, which is one of the key steps in drug design. Based on three of the most commonly used data sets from the well-established protein-ligand binding databank, that is, PDBbind, we intensively test our model and compare with existing models. It has been found that our model can achieve the state-of-the-art results and has advantages over traditional molecular descriptors.

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.

Physics approach to the variable-mass optimal-transport problem

Optimal transport (OT) has become a discipline by itself that offers solutions to a wide range of theoretical problems in probability and mathematics with applications in several applied fields such as imaging sciences, machine learning, and in data sciences in general. The traditional OT problem suffers from a severe limitation: its balance condition imposes that the two distributions to be compared be normalized and have the same total mass. However, it is important for many applications to be able to relax this constraint and allow for mass creation and/or destruction. This is true, for example, in all problems requiring partial matching. In this paper, we propose an approach to solving a generalized version of the OT problem, which we refer to as the discrete variable-mass optimal-transport (VMOT) problem, using techniques adapted from statistical physics. Our first contribution is to fully describe this formalism, including all the proofs of its main claims. In particular, we derive a strongly concave effective free-energy function that captures the constraints of the VMOT problem at a finite temperature. From its maximum we derive a weak distance (i.e., a divergence) between possibly unbalanced distribution functions. The temperature-dependent OT distance decreases monotonically to the standard variable-mass OT distance, providing a robust framework for temperature annealing. Our second contribution is to show that the implementation of this formalism has the same properties as the regularized OT algorithms in time complexity, making it a competitive approach to solving the VMOT problem. We illustrate applications of the framework to the problem of partial two- and three-dimensional shape-matching problems.