Curvature Of Graph Research Articles

Graph Curvature Flow-Based Masked Attention.

Graph neural networks (GNNs) have revolutionized drug discovery in chemistry and biology, enhancing efficiency and reducing resource demands. However, classical GNNs often struggle to capture long-range dependencies due to challenges like oversmoothing and oversquashing. Graph Transformers address these issues by employing global self-attention mechanisms that allow direct information exchange between any pair of nodes, enabling the modeling of long-range interactions. Despite this, Graph Transformers often face difficulties in capturing the nuanced structural information on graphs. To overcome these challenges, we introduce the CurvFlow-Transformer, a novel graph Transformer model incorporating a curvature flow-based masked attention mechanism. By leveraging a topologically enhanced mask matrix, the attention layer can effectively detect subtle structural differences within graphs, balancing the focus between global mutual information and local structural details of molecules. The CurvFlow-Transformer demonstrates superior performance on the MoleculeNet data set, surpassing several state-of-the-art models across various tasks. Moreover, the model provides unique insights into the relationship between molecular structure and chemical properties by analyzing the attention heat coefficients of individual atoms.

Graph Information Vanishing Phenomenon in Implicit Graph Neural Networks

Graph neural networks (GNNs) have been highly successful in graph representation learning. The goal of GNNs is to enrich node representations by aggregating information from neighboring nodes. Much work has attempted to improve the quality of aggregation by introducing a variety of graph information with representational capabilities. The class of GNNs that improves the quality of aggregation by encoding graph information with representational capabilities into the weights of neighboring nodes through different learnable transformation structures (LTSs) are referred to as implicit GNNs. However, we argue that LTSs only transform graph information into the weights of neighboring nodes in the direction that minimizes the loss function during the learning process and does not actually utilize the effective properties of graph information, a phenomenon that we refer to as graph information vanishing (GIV). To validate this point, we perform thousands of experiments on seven node classification benchmark datasets. We first replace the graph information utilized by five implicit GNNs with random values and surprisingly observe that the variation range of accuracies is less than ± 0.3%. Then, we quantitatively characterize the similarity of the weights generated from graph information and random values by cosine similarity, and the cosine similarities are greater than 0.99. The empirical experiments show that graph information is equivalent to initializing the input of LTSs. We believe that graph information as an additional supervised signal to constrain the training of GNNs can effectively solve GIV. Here, we propose GinfoNN, which utilizes both labels and discrete graph curvature as supervised signals to jointly constrain the training of the model. The experimental results show that the classification accuracies of GinfoNN improve by two percentage points over baselines on large and dense datasets.

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.

Forman-Ricci curvature of tournaments

tournaments are a type of directed graph which have been used to study the geometry of classical flag manifolds. We became interested in this type of graphs because the combinatorial properties of tournaments can be used to study geometric properties of the flag manifolds. [21]introduced the Forman-Ricci curvature for directed and undirected hypergraphs and obtained the curvature for graphs as a particular case. In this work we present the basic ideas about the Forman- Ricci curvature for directed graphs, characterize the parabolic tournaments in terms of Forman-Ricci curvature and calculate the Forman-Ricci curvature for any tournament.

Ollivier Curvature of Random Geometric Graphs Converges to Ricci Curvature of Their Riemannian Manifolds

Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curvature and their smooth counterparts remain lurking and moot. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth space. Here we prove that in proper settings the Ollivier–Ricci curvature of random geometric graphs in Riemannian manifolds converges to the Ricci curvature of the manifold. This is the first rigorous result linking curvature of random graphs to curvature of smooth spaces. Our results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities. Here the scaling of the average degree, as a function of the graph size, can range from nearly logarithmic to nearly linear.

A cosine rule-based discrete sectional curvature for graphs

Abstract How does one generalize differential geometric constructs such as curvature of a manifold to the discrete world of graphs and other combinatorial structures? This problem carries significant importance for analysing models of discrete spacetime in quantum gravity; inferring network geometry in network science; and manifold learning in data science. The key contribution of this article is to introduce and validate a new estimator of discrete sectional curvature for random graphs with low metric-distortion. The latter are constructed via a specific graph sprinkling method on different manifolds with constant sectional curvature. We define a notion of metric distortion, which quantifies how well the graph metric approximates the metric of the underlying manifold. We show how graph sprinkling algorithms can be refined to produce hard annulus random geometric graphs with minimal metric distortion. We construct random geometric graphs for spheres, hyperbolic and Euclidean planes; upon which we validate our curvature estimator. Numerical analysis reveals that the error of the estimated curvature diminishes as the mean metric distortion goes to zero, thus demonstrating convergence of the estimate. We also perform comparisons to other existing discrete curvature measures. Finally, we demonstrate two practical applications: (i) estimation of the earth’s radius using geographical data; and (ii) sectional curvature distributions of self-similar fractals.

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

BackgroundExtracting 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.ResultsWe 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.ConclusionsWe 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.

Ricci curvature, Bruhat graphs and Coxeter groups

We consider the discrete Ricci curvature for graphs as defined by Schmuckenschläger [Convex geometric analysis, MSRI Publications, 1998] and compute its value for Bruhat graphs associated to finite Coxeter groups. To do so we work with the geometric realization of a finite Coxeter group and a classical result obtained by Dyer in [Compositio Math. 78 (1991), pp. 185–191]. As an application we obtain a bound for the spectral gap of the Bruhat graph of any finite Coxeter group and an isoperimetric inequality for them. Our proofs are case-free.

A New Transport Distance and Its Associated Ricci Curvature of Hypergraphs

Abstract The coarse Ricci curvature of graphs introduced by Ollivier as well as its modification by Lin–Lu– Yau have been studied from various aspects. In this paper, we propose a new transport distance appropriate for hypergraphs and study a generalization of Lin–Lu–Yau type curvature of hypergraphs. As an application, we derive a Bonnet–Myers type estimate for hypergraphs under a lower Ricci curvature bound associated with our transport distance. We remark that our transport distance is new even for graphs and worthy of further study.

Curvature graph neural network

Graph neural networks (GNNs) have achieved great success in many graph-based tasks. Much work is dedicated to empowering GNNs with adaptive locality ability, which enables the measurement of the importance of neighboring nodes to the target node by a node-specific mechanism. However, the current node-specific mechanisms are deficient in distinguishing the importance of nodes in the topology structure. We believe that the structural importance of neighboring nodes is closely related to their importance in aggregation. In this paper, we introduce discrete graph curvature (the Ricci curvature) to quantify the strength of the structural connection of pairwise nodes. We propose a curvature graph neural network (CGNN), which effectively improves the adaptive locality ability of GNNs by leveraging the structural properties of graph curvature. To improve the adaptability of curvature on various datasets, we explicitly transform curvature into the weights of neighboring nodes by the necessary negative curvature processing module and curvature normalization module. Then, we conduct numerous experiments on various synthetic and real-world datasets. The experimental results on synthetic datasets show that CGNN effectively exploits the topology structure information and that the performance is significantly improved. CGNN outperforms the baselines on 5 dense node classification benchmark datasets. This study provides a deepened understanding of how to utilize advanced topology information and assign the importance of neighboring nodes from the perspective of graph curvature and encourages bridging the gap between graph theory and neural networks. The source code is available at https://github.com/GeoX-Lab/CGNN.

Emergent Einstein Equation in p-adic Conformal Field Theory Tensor Networks.

We take the tensor network describing explicit p-adic conformal field theory partition functions proposed in [L.-Y. Hung et al., J. High Energy Phys. 04 (2019) 170JHEPFG1029-847910.1007/JHEP04(2019)170], and consider boundary conditions of the network describing a deformed Bruhat-Tits (BT) tree geometry. We demonstrate that this geometry satisfies an emergent graph Einstein equation in a unique way that is consistent with the bulk effective matter action encoding the same correlation function as the tensor network, at least in the perturbative limit order by order away from the pure BT tree. Moreover, the (perturbative) definition of the graph curvature in the mathematics [Y. Lin and S.-T. Yau, Tohoku Math. J. 63, 605 (2011)TOMJAM0040-873510.2748/tmj/1325886283; Y. Ollivier, J. Funct. Anal. 256, 810 (2009)JFUAAW0022-123610.1016/j.jfa.2008.11.001] and physics [S. S. Gubser et al., J. High Energy Phys. 06 (2017) 157JHEPFG1029-847910.1007/JHEP06(2017)157] literature naturally emerges from the consistency requirements of the emergent Einstein equation. This could provide new insights into the understanding of gravitational dynamics potentially encoded in more general tensor networks.

Ricci-flat Graphs with Girth Four

Lin-Lu-Yau introduced a notion of Ricci curvature for graphs and obtained a complete classification for all Ricci-flat graphs with girth at least five. In this paper, we characterize all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.

Interacting thermofield doubles and critical behavior in random regular graphs

We discuss numerically the non-perturbative effects in exponential random graphs which are analogue of eigenvalue instantons in matrix models. The phase structure of exponential random graphs with chemical potential for 4-cycles and degree preserving constraint is clarified. The first order phase transition at critical value of chemical potential for 4-cycles into bipartite phase with a formation of fixed number of bipartite clusters is found for ensemble of random regular graphs (RRG). We consider the similar phase transition in combinatorial quantum gravity based of the Ollivier graph curvature for RRG supplemented with hard-core constraint and show that a order of a phase transition and the structure of emerging phase depend on a vertex degree d in RRG. For d = 3 the bipartite closed ribbon emerges at bipartite phase while for d > 3 the ensemble of isolated or weakly interacting hypercubes supplemented with the bipartite closed ribbon gets emerged at the first order phase transition with a clear-cut hysteresis. If the additional connectedness condition is imposed the bipartite phase gets identified as the closed chain of weakly coupled hypercubes. Since the ground state of isolated hypercube is the thermofield double (TFD) we suggest that the dual holographic picture involves multiboundary wormholes. Treating RRG as a model of a Hilbert space for a interacting many-body system we discuss the patterns of the Hilbert space fragmentation at the phase transition. We also briefly comment on a possible relation of the found phase transition to the problem of holographic interpretation of a partial deconfinement transition in the gauge theories.

KBER: A kernel bandwidth estimate using the Ricci curvature

The choice of a bandwidth can affect dramatically the accuracy of classification methods relying on it. Recently, a large number of methods to choose the bandwidth have been developed. This article presents a simple method called KBER to estimate the bandwidth using the average Ricci curvature of graphs. The Radial Basis Function kernel (RBF) has been chosen in our work for its simplicity and its popularity in this kind of research; it is also possible to apply our method to any kernel with the same kind of parameter.

Ricci curvature, graphs and eigenvalues

We express the discrete Ricci curvature of a graph as the minimal eigenvalue of a family of matrices, one for each vertex of a graph whose entries depend on the local adjacency structure of the graph. Using this method we compute or bound the Ricci curvature of Cayley graphs of finite Coxeter groups and affine Weyl groups. As an application we obtain an isoperimetric inequality that holds for all Cayley graphs of finite Coxeter groups.

Ollivier-Ricci curvature convergence in random geometric graphs

Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional definition of curvature of a Riemannian manifold. Here we show that Ollivier curvature of random geometric graphs in any Riemannian manifold converges in the continuum limit to Ricci curvature of the underlying manifold, but only if the definition of Ollivier graph curvature is properly generalized to apply to mesoscopic graph neighborhoods. This result establishes the first rigorous link between a definition of curvature applicable to networks and a traditional definition of curvature of smooth spaces.

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.

FORMAN–RICCI CURVATURE FOR HYPERGRAPHS

Hypergraphs serve as models of complex networks that capture more general structures than binary relations. For graphs, a wide array of statistics has been devised to gauge different aspects of their structures. Hypergraphs lack behind in this respect. The Forman–Ricci curvature is a statistics for graphs based on Riemannian geometry, which stresses the relational character of vertices in a network by focusing on the edges rather than on the vertices. Despite many successful applications of this measure to graphs, Forman–Ricci curvature has not been introduced for hypergraphs. Here, we define the Forman–Ricci curvature for directed and undirected hypergraphs such that the curvature for graphs is recovered as a special case. It quantifies the trade-off between hyperedge (arc) size and the degree of participation of hyperedge (arc) vertices in other hyperedges (arcs). Here, we determine upper and lower bounds for Forman–Ricci curvature both for hypergraphs in general and for graphs in particular. The measure is then applied to two large networks: the Wikipedia vote network and the metabolic network of the bacterium Escherichia coli. In the first case, the curvature is governed by the size of the hyperedges, while in the second example, it is dominated by the hyperedge degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with the participation of experienced users. The curvature values of the metabolic network allowed detecting redundant and bottle neck reactions. It is found that ADP phosphorylation is the metabolic bottle neck reaction but that the reverse reaction is not similarly central for the metabolism. Furthermore, we show the utility of the Forman–Ricci curvature for quantification of assortativity in hypergraphs and illustrate the idea by investigating three metabolic networks.

Edge-based analysis of networks: curvatures of graphs and hypergraphs

The relations, rather than the elements, constitute the structure of networks. We therefore develop a systematic approach to the analysis of networks, modelled as graphs or hypergraphs, that is based on structural properties of (hyper)edges, instead of vertices. For that purpose, we utilize so-called network curvatures. These curvatures quantify the local structural properties of (hyper)edges, that is, how, and how well, they are connected to others. In the case of directed networks, they assess the input they receive and the output they produce, and relations between them. With those tools, we can investigate biological networks. As examples, we apply our methods here to protein–protein interaction, transcriptional regulatory and metabolic networks.

The Graph Curvature Calculator and the Curvatures of Cubic Graphs

We classify all cubic graphs with either non-negative Ollivier-Ricci curvature or non-negative Bakry-Émery curvature everywhere. We show in both curvature notions that the non-negatively curved graphs are the prism graphs and the Möbius ladders. As a consequence of the classification result we show that non-negatively curved cubic expanders do not exist. We also introduce the Graph Curvature Calculator, an online tool developed for calculating the curvature of graphs under several variants of the curvature notions that we use in the classification.