Ricci curvature: An economic indicator for market fragility and systemic risk

Understanding stock market instability via graph auto-encoders

Understanding stock market instability is a key question in financial management as practitioners seek to forecast breakdowns in long-run asset co-movement patterns which expose portfolios to rapid and devastating collapses in value. These disruptions are linked to changes in the structure of market wide stock correlations which increase the risk of high volatility shocks. The structure of these co-movements can be described as a network where companies are represented by nodes while edges capture correlations between their price movements. Co-movement breakdowns then manifest as abrupt changes in the topological structure of this network. Measuring the scale of this change and learning a timely indicator of breakdowns is central in understanding both financial stability and volatility forecasting. We propose to use the edge reconstruction accuracy of a graph auto-encoder as an indicator for how homogeneous connections between assets are, which we use, based on the literature of financial network analysis, as a proxy to infer market volatility. We show, through our experiments on the Standard and Poor's index over the 2015-2022 period, that the reconstruction errors from our model correlate with volatility spikes and can be used to improve out-of-sample autoregressive modeling of volatility. Our results demonstrate that market instability can be predicted by changes in the homogeneity in connections of the financial network which expands the understanding of instability in the stock market. We discuss the implications of this graph machine learning-based volatility estimation for policy targeted at ensuring financial market stability.

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).

Attention based dynamic graph neural network for asset pricing

Recent studies suggest that networks among firms (sectors) play a vital role in asset pricing. This paper investigates these implications and develops a novel end-to-end graph neural network model for asset pricing by combining and modifying two state-of-the-art machine learning techniques. First, we apply the graph attention mechanism to learn dynamic network structures of the equity market over time and then use a recurrent convolutional neural network to diffuse and propagate firms' information into the learned networks. This novel approach allows us to model the implications of networks along with the characteristics of the dynamic comovement of asset prices. The results demonstrate the effectiveness of our proposed model in both predicting returns and improving portfolio performance. Our approach demonstrates persistent performance in different sensitivity tests and simulated data. We also show that the dynamic network learned from our proposed model captures major market events over time. Our model is highly effective in recognizing the network structure in the market and predicting equity returns and provides valuable market information to regulators and investors.

Complex network analysis techniques for the early detection of the outbreak of pandemics transmitted through air traffic

Air transport has been identified as one of the primary means whereby COVID-19 spread throughout Europe during the early stages of the pandemic. In this paper we analyse two categories of methods - dynamic network markers (DNMs) and network analysis-based methods - as potential early warning signals for detecting and anticipating COVID-19 outbreaks in Europe on the basis of accuracy regarding the daily confirmed cases. The analysis was carried out from 15 February 2020, around two weeks before the first COVID-19 cases appeared in Europe, and 1 May 2020, approximately two weeks after all the air traffic in Europe had been shut down. Daily European COVID-19 information sourced from the World Health Organization was used, whereas air traffic data from Flightradar24 has been incorporated into the analyses by means of four alternative adjacency matrices. Some DNMs have been discarded since they output multiple time series, which makes it very difficult to interpret their results. The only DNM outputting a single time series does not emulate the COVID-19 trend: it does not detect all the main peaks, which means that peak heights do not match up with the increase in the number of infected people. However, many combinations of network analysis based methods and adjacency matrices output good results (with high accuracy and 20-day advance forecasts), with only minor differences from one to another. The number of edges and the network density methods are slightly better when dynamic flight frequency information is used.

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.

Exploring robust architectures for deep artificial neural networks

The architectures of deep artificial neural networks (DANNs) are routinely studied to improve their predictive performance. However, the relationship between the architecture of a DANN and its robustness to noise and adversarial attacks is less explored, especially in computer vision applications. Here we investigate the relationship between the robustness of DANNs in a vision task and their underlying graph architectures or structures. First we explored the design space of architectures of DANNs using graph-theoretic robustness measures and transformed the graphs to DANN architectures using various image classification tasks. Then we explored the relationship between the robustness of trained DANNs against noise and adversarial attacks and their underlying architectures. We show that robustness performance of DANNs can be quantified before training using graph structural properties such as topological entropy and Olivier-Ricci curvature, with the greatest reliability for complex tasks and large DANNs. Our results can also be applied for tasks other than computer vision such as natural language processing and recommender systems.

Asim Waqas and colleagues investigated the relationship between the architectures of deep artificial neural networks (DANNs) and their robustness to noise and adversarial attacks in computer vision. The researchers found that the robustness of DANNs was highly correlated with graph-theoretic measures of entropy and curvature. This finding could help design more robust DANN architectures.

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).

A Path-Curvature Measure for Word-Based Strategy Searches in Semantic Networks

Building on a modified version of the Haantjes path-based curvature, this article offers a novel measure that considers the direction of a stream of associations in a semantic network and estimates the extent to which any single association attracts the upcoming associations to its environment—in other words, to what degree one explores that environment. We demonstrate that our measure differs from Haantjes curvature and confirm that it expresses the extent to which a stream of associations remains close to its starting point. Finally, we examine the relationship between our measure and accessibility to knowledge stored in memory. We demonstrate that a high degree of attraction facilitates the retrieval of upcoming words in the stream. By applying methods from differential geometry to semantic networks, this study contributes to our understanding of strategic search in memory.

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.

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.

Bakry-Émery Ricci Curvature Bounds for Doubly Warped Products of Weighted Spaces

We introduce a notion of doubly warped product of weighted graphs that is consistent with the doubly warped product in the Riemannian setting. We establish various discrete Bakry-Émery Ricci curvature-dimension bounds for such warped products in terms of the curvature of the constituent graphs. This requires deliberate analysis of the quadratic forms involved, prompting the introduction of some crucial notions such as curvature saturation at a vertex. In the spirit of being thorough and to provide a frame of reference, we also introduce the R 1 , R 2 -doubly warped products of smooth measure spaces and establish N -Bakry-Émery Ricci curvature (lower) bounds thereof in terms of those of the factors. At the end of these notes, we present examples and demonstrate applications of warped products with some toy models.

Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations

We analyze the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modeling the stability of fixed points in large systems defined on complex networks, such as ecosystems consisting of a large number of species that interact through a food web. We develop an exact theory for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices. This theory reveals that the nature of local interactions has a strong influence on a system's stability. We show that, in general, linear dynamical systems defined on random graphs with a prescribed degree distribution of unbounded support are unstable if they are large enough, implying a tradeoff between stability and diversity. Remarkably, in contrast to the generic case, antagonistic systems that contain only interactions of the predator-prey type can be stable in the infinite size limit. This feature for antagonistic systems is accompanied by a peculiar oscillatory behavior of the dynamical response of the system after a perturbation, when the mean degree of the graph is small enough. Moreover, for antagonistic systems we also find that there exist a dynamical phase transition and critical mean degree above which the response becomes nonoscillatory.

Geometric network analysis provides prognostic information in patients with high grade serous carcinoma of the ovary treated with immune checkpoint inhibitors

Network analysis methods can potentially quantify cancer aberrations in gene networks without introducing fitted parameters or variable selection. A new network curvature-based method is introduced to provide an integrated measure of variability within cancer gene networks. The method is applied to high-grade serous ovarian cancers (HGSOCs) to predict response to immune checkpoint inhibitors (ICIs) and to rank key genes associated with prognosis. Copy number alterations (CNAs) from targeted and whole-exome sequencing data were extracted for HGSOC patients (n = 45) treated with ICIs. CNAs at a gene level were represented on a protein–protein interaction network to define patient-specific networks with a fixed topology. A version of Ollivier–Ricci curvature was used to identify genes that play a potentially key role in response to immunotherapy and further to stratify patients at high risk of mortality. Overall survival (OS) was defined as the time from the start of ICI treatment to either death or last follow-up. Kaplan–Meier analysis with log-rank test was performed to assess OS between the high and low curvature classified groups. The network curvature analysis stratified patients at high risk of mortality with p = 0.00047 in Kaplan–Meier analysis in HGSOC patients receiving ICI. Genes with high curvature were in accordance with CNAs relevant to ovarian cancer. Network curvature using CNAs has the potential to be a novel predictor for OS in HGSOC patients treated with immunotherapy.

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.

COVID-19 non-pharmaceutical intervention portfolio effectiveness and risk communication predominance

Non-pharmaceutical interventions (NPIs) including resource allocation, risk communication, social distancing and travel restriction, are mainstream actions to control the spreading of Coronavirus disease 2019 (COVID-19) worldwide. Different countries implemented their own combinations of NPIs to prevent local epidemics and healthcare system overloaded. Portfolios, as temporal sets of NPIs have various systemic impacts on preventing cases in populations. Here, we developed a probabilistic modeling framework to evaluate the effectiveness of NPI portfolios at the macroscale. We employed a deconvolution method to back-calculate incidence of infections and estimate the effective reproduction number by using the package EpiEstim. We then evaluated the effectiveness of NPIs using ratios of the reproduction numbers and considered them individually and as a portfolio systemically. Based on estimates from Japan, we estimated time delays of symptomatic-to-confirmation and infection-to-confirmation as 7.4 and 11.4 days, respectively. These were used to correct surveillance data of other countries. Considering 50 countries, risk communication and returning to normal life were the most and least effective yielding the aggregated effectiveness of 0.11 and - 0.05 that correspond to a 22.4% and 12.2% reduction and increase in case growth. The latter is quantified by the change in reproduction number before and after intervention implementation. Countries with the optimal NPI portfolio are along an empirical Pareto frontier where mean and variance of effectiveness are maximized and minimized independently of incidence levels. Results indicate that implemented interventions, regardless of NPI portfolios, had distinct incidence reductions and a clear timing effect on infection dynamics measured by sequences of reproduction numbers. Overall, the successful suppression of the epidemic cannot work without the non-linear effect of NPI portfolios whose effectiveness optimality may relate to country-specific socio-environmental factors.

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.

Degree difference: a simple measure to characterize structural heterogeneity in complex networks

Despite the growing interest in characterizing the local geometry leading to the global topology of networks, our understanding of the local structure of complex networks, especially real-world networks, is still incomplete. Here, we analyze a simple, elegant yet underexplored measure, 'degree difference' (DD) between vertices of an edge, to understand the local network geometry. We describe the connection between DD and global assortativity of the network from both formal and conceptual perspective, and show that DD can reveal structural properties that are not obtained from other such measures in network science. Typically, edges with different DD play different structural roles and the DD distribution is an important network signature. Notably, DD is the basic unit of assortativity. We provide an explanation as to why DD can characterize structural heterogeneity in mixing patterns unlike global assortativity and local node assortativity. By analyzing synthetic and real networks, we show that DD distribution can be used to distinguish between different types of networks including those networks that cannot be easily distinguished using degree sequence and global assortativity. Moreover, we show DD to be an indicator for topological robustness of scale-free networks. Overall, DD is a local measure that is simple to define, easy to evaluate, and that reveals structural properties of networks not readily seen from other measures.

p53: 800 million years of evolution and 40 years of discovery

The evolutionarily conserved p53 protein and its cellular pathways mediate tumour suppression through an informed, regulated and integrated set of responses to environmental perturbations resulting in either cellular death or the maintenance of cellular homeostasis. The p53 and MDM2 proteins form a central hub in this pathway that receives stressful inputs via MDM2 and respond via p53 by informing and altering a great many other pathways and functions in the cell. The MDM2-p53 hub is one of the hubs most highly connected to other signalling pathways in the cell, and this may be why TP53 is the most commonly mutated gene in human cancers. Initial or truncal TP53 gene mutations (the first mutations in a stem cell) are selected for early in cancer development inectodermal and mesodermal-derived tissue-specific stem and progenitor cells and then, following additional mutations, produce tumours from those tissue types. In endodermal-derived tissue-specific stem or progenitor cells, TP53 mutations are functionally selected as late mutations transitioning the mutated cell into a malignant tumour. The order in which oncogenes or tumour suppressor genes are functionally selected for in a stem cell impacts the timing and development of a tumour.

A Note on Ricci Solitons

In this paper, we characterize trivial Ricci solitons. We observe the important role of the energy function f of a Ricci soliton (half the squared length of the potential vector field) in the charectrization of trivial Ricci solitons. We find three characterizations of connected trivial Ricci solitons by imposing different restrictions on the energy function. We also use Hessian of the potential function to characterize compact trivial Ricci solitons. Finally, we show that a solution of a Poisson equation is the energy function f of a compact Ricci soliton if and only if the Ricci soliton is trivial.

Network curvature as a hallmark of brain structural connectivity

Although brain functionality is often remarkably robust to lesions and other insults, it may be fragile when these take place in specific locations. Previous attempts to quantify robustness and fragility sought to understand how the functional connectivity of brain networks is affected by structural changes, using either model-based predictions or empirical studies of the effects of lesions. We advance a geometric viewpoint relying on a notion of network curvature, the so-called Ollivier-Ricci curvature. This approach has been proposed to assess financial market robustness and to differentiate biological networks of cancer cells from healthy ones. Here, we apply curvature-based measures to brain structural networks to identify robust and fragile brain regions in healthy subjects. We show that curvature can also be used to track changes in brain connectivity related to age and autism spectrum disorder (ASD), and we obtain results that are in agreement with previous MRI studies.

The brain can often continue to function despite lesions in many areas, but damage to particular locations may have serious effects. Here, the authors use the concept of Ollivier-Ricci curvature to investigate the robustness of brain networks.

Topological phase transitions in functional brain networks

Functional brain networks are often constructed by quantifying correlations between time series of activity of brain regions. Their topological structure includes nodes, edges, triangles, and even higher-dimensional objects. Topological data analysis (TDA) is the emerging framework to process data sets under this perspective. In parallel, topology has proven essential for understanding fundamental questions in physics. Here we report the discovery of topological phase transitions in functional brain networks by merging concepts from TDA, topology, geometry, physics, and network theory. We show that topological phase transitions occur when the Euler entropy has a singularity, which remarkably coincides with the emergence of multidimensional topological holes in the brain network. The geometric nature of the transitions can be interpreted, under certain hypotheses, as an extension of percolation to high-dimensional objects. Due to the universal character of phase transitions and noise robustness of TDA, our findings open perspectives toward establishing reliable topological and geometrical markers for group and possibly individual differences in functional brain network organization.

Community Detection on Networks with Ricci Flow

Many complex networks in the real world have community structures - groups of well-connected nodes with important functional roles. It has been well recognized that the identification of communities bears numerous practical applications. While existing approaches mainly apply statistical or graph theoretical/combinatorial methods for community detection, in this paper, we present a novel geometric approach which  enables us to borrow powerful classical geometric methods and properties. By considering networks as geometric objects and communities in a network as a geometric decomposition, we apply curvature and discrete Ricci flow, which have been used to decompose smooth manifolds with astonishing successes in mathematics, to break down communities in networks. We  tested our method on networks with ground-truth community structures, and experimentally confirmed the effectiveness of this geometric approach.

Ollivier-Ricci Curvature-Based Method to Community Detection in Complex Networks

Identification of community structures in complex network is of crucial importance for understanding the system’s function, organization, robustness and security. Here, we present a novel Ollivier-Ricci curvature (ORC) inspired approach to community identification in complex networks. We demonstrate that the intrinsic geometric underpinning of the ORC offers a natural approach to discover inherent community structures within a network based on interaction among entities. We develop an ORC-based community identification algorithm based on the idea of sequential removal of negatively curved edges symptomatic of high interactions (e.g., traffic, attraction). To illustrate and compare the performance with other community identification methods, we examine the ORC-based algorithm with stochastic block model artificial networks and real-world examples ranging from social to drug-drug interaction networks. The ORC-based algorithm is able to identify communities with either better or comparable performance accuracy and to discover finer hierarchical structures of the network. This opens new geometric avenues for analysis of complex networks dynamics.

Relationship between Entropy and Dimension of Financial Correlation-Based Network

We analyze the dimension of a financial correlation-based network and apply our analysis to characterize the complexity of the network. First, we generalize the volume-based dimension and find that it is well defined by the correlation-based network. Second, we establish the relationship between the Rényi index and the volume-based dimension. Third, we analyze the meaning of the dimensions sequence, which characterizes the level of departure from the comparison benchmark based on the randomized time series. Finally, we use real stock market data from three countries for empirical analysis. In some cases, our proposed analysis method can more accurately capture the structural differences of networks than the power law index commonly used in previous studies.

Matricial Wasserstein-1 Distance

We propose an extension of the Wasserstein 1-metric (W₁) for density matrices, matrix-valued density measures, and an unbalanced interpretation of mass transport. We use duality theory and, in particular, a "dual of the dual" formulation of W₁. This matrix analogue of the Earth Mover's Distance has several attractive features including ease of computation.