Commute Time Distance Research Articles

Graph structure reforming framework enhanced by commute time distance for graph classification

As a graph data mining task, graph classification has high academic value and wide practical application. Among them, the graph neural network-based method is one of the mainstream methods. Most graph neural networks (GNNs) follow the message passing paradigm and can be called Message Passing Neural Networks (MPNNs), achieving good results in structural data-related tasks. However, it has also been reported that these methods suffer from over-squashing and limited expressive power. In recent years, many works have proposed different solutions to these problems separately, but none has yet considered these shortcomings in a comprehensive way. After considering these several aspects comprehensively, we identify two specific defects: information loss caused by local information aggregation, and an inability to capture higher-order structures. To solve these issues, we propose a plug-and-play framework based on Commute Time Distance (CTD), in which information is propagated in commute time distance neighborhoods. By considering both local and global graph connections, the commute time distance between two nodes is evaluated with reference to the path length and the number of paths in the whole graph. Moreover, the proposed framework CTD-MPNNs (Commute Time Distance-based Message Passing Neural Networks) can capture higher-order structural information by utilizing commute paths to enhance the expressive power of GNNs. Thus, our proposed framework can propagate and aggregate messages from defined important neighbors and model more powerful GNNs. We conduct extensive experiments using various real-world graph classification benchmarks. The experimental performance demonstrates the effectiveness of our framework. Codes are released on https://github.com/Haldate-Yu/CTD-MPNNs.

Free Energy Node Embedding via Generalized Skip-gram with Negative Sampling

A widely established set of unsupervised node embedding methods can be interpreted as consisting of two distinctive steps: i) the definition of a similarity matrix based on the graph of interest followed by ii) an explicit or implicit factorization of such matrix. Inspired by this viewpoint, we propose improvements in both steps of the framework. On the one hand, we propose to encode node similarities based on the free energy distance, which interpolates between the shortest path and the commute time distances, thus, providing an additional degree of flexibility. On the other hand, we propose a matrix factorization method based on a loss function that generalizes that of the skip-gram model with negative sampling to arbitrary similarity matrices. Compared with factorizations based on the widely used <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> loss, the proposed method can better preserve node pairs associated with higher similarity scores. Moreover, it can be easily implemented using advanced automatic differentiation toolkits and computed efficiently by leveraging GPU resources. Node clustering, node classification, and link prediction experiments on real-world datasets demonstrate the effectiveness of incorporating free-energy-based similarities as well as the proposed matrix factorization compared with state-of-the-art alternatives.

Configuration spaces of hard spheres.

Hard sphere systems are often used to model simple fluids. The configuration spaces of hard spheres in a three-dimensional torus modulo various symmetry groups are comparatively simple and could provide valuable information about the nature of phase transitions. Specifically, the topological changes in the configuration space as a function of packing fraction have been conjectured to be related to the onset of first-order phase transitions. The critical configurations for 1 to 12 spheres are sampled using a Morse-theoretic approach, and they are available in an online, interactive database. Explicit triangulations are constructed for the configuration spaces of the two sphere system, and their topological and geometric properties are studied. The critical configurations are found to be associated with geometric changes to the configuration space that connect previously distant regions and reduce the configuration space diameter as measured by the commute time and diffusion distances. The number of such critical configurations around the packing fraction of the solid-liquid phase transition increases exponentially with the number of spheres, suggesting that the onset of the first-order phase transition in the thermodynamic limit is associated with a discontinuity in the configuration space diameter.

A Metric on Directed Graphs and Markov Chains Based on Hitting Probabilities

The shortest-path, commute time, and diffusion distances on undirected graphs have been widely employed in applications such as dimensionality reduction, link prediction, and trip planning. Increas...

Modelling landscape genetic connectivity of the mountain pine beetle in western Canada

The current mountain pine beetle (MPB; Dendroctonus ponderosae Hopkins, 1902) outbreak has reached more than 25 million hectares of forests in North America, affecting pine species throughout the region and substantially changing landscapes. However, landscape features that enhance or limit dispersal during the geographic expansion associated with the outbreak are poorly understood. One of the obstacles in evaluating the effects of landscape features on dispersal is the parameterization of resistance surfaces, which are often constructed based on biased expert opinion or by making assumptions in the calculation of ecological distances. In this study, we assessed the impact of four environmental variables on MPB genetic connectivity across western Canada. We optimized resistance surfaces using genetic algorithms and models of maximum likelihood population effects, based on pairwise genetic distances and ecological distances calculated using random-walk commute-time distances. Unlike other methods for the development of resistance surfaces, this approach does not make a priori assumptions about the direction or shape of the relationships between environmental features and their cost to movement. We found highest support for a composite resistance surface including elevation and climate. These results further the understanding of MPB movement during an outbreak. Additionally, we demonstrated how to use our results for management purposes.

Local commute-time guided MDS for 3D non-rigid object retrieval

In this paper, we propose a 3D non-rigid shape retrieval method based on canonical shape analysis. Our main idea is to transform the problem of non-rigid shape retrieval into a rigid shape retrieval problem via the well-known multidimensional scaling (MDS) approach and random walk on graphs. We first segment the non-rigid shape into local partitions based on its salient features. Then, we calculate a local MDS problem for each partition, where the local commute time distance is used as weighting function in order to preserve local shape details. Finally, we aggregate the set of local MDS problems as a global constrained problem. The constraint is formulated using the biharmonic function between local salient features. In contrast to MDS method, the proposed local MDS is computationally efficient, parameters free and gives isometry-invariant forms with minimum features distortion. Due to these advantageous properties, the proposed method achieved good retrieval accuracy on non-rigid shape benchmark datasets.

Accurate and Efficient Computation of Laplacian Spectral Distances and Kernels

This paper introduces the Laplacian spectral distances, as a function that resembles the usual distance map, but exhibits properties e.g. smoothness, locality, invariance to shape transformations that make them useful to processing and analysing geometric data. Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and reduce to the heat diffusion, wave, biharmonic and commute-time distances for specific filters. In particular, the smoothness of the spectral distances and the encoding of local and global shape properties depend on the convergence of the filtered eigenvalues to zero. Instead of applying a truncated spectral approximation or prolongation operators, we propose a computation of Laplacian distances and kernels through the solution of sparse linear systems. Our approach is free of user-defined parameters, overcomes the evaluation of the Laplacian spectrum and guarantees a higher approximation accuracy than previous work.

Manifold Clustering and Visualization with Commute Time Distance

Manifold Clustering and Visualization with Commute Time Distance

Trade Integration and Trade Imbalances in the European Union: A Network Perspective

We study the ever more integrated and ever more unbalanced trade relationships between European countries. To better capture the complexity of economic networks, we propose two global measures that assess the trade integration and the trade imbalances of the European countries. These measures are the network (or indirect) counterparts to traditional (or direct) measures such as the trade-to-GDP (Gross Domestic Product) and trade deficit-to-GDP ratios. Our indirect tools account for the European inter-country trade structure and follow (i) a decomposition of the global trade flow into elementary flows that highlight the long-range dependencies between exporting and importing economies and (ii) the commute-time distance for trade integration, which measures the impact of a perturbation in the economy of a country on another country, possibly through intermediate partners by domino effect. Our application addresses the impact of the launch of the Euro. We find that the indirect imbalance measures better identify the countries ultimately bearing deficits and surpluses, by neutralizing the impact of trade transit countries, such as the Netherlands. Among others, we find that ultimate surpluses of Germany are quite concentrated in only three partners. We also show that for some countries, the direct and indirect measures of trade integration diverge, thereby revealing that these countries (e.g. Greece and Portugal) trade to a smaller extent with countries considered as central in the European Union network.

Developments in the theory of randomized shortest paths with a comparison of graph node distances

There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into account the global structure of the graph. In this article, we develop the theory of one family of graph node distances, known as the randomized shortest path dissimilarity, which has its foundation in statistical physics. We show that the randomized shortest path dissimilarity can be easily computed in closed form for all pairs of nodes of a graph. Moreover, we come up with a new definition of a distance measure that we call the free energy distance. The free energy distance can be seen as an upgrade of the randomized shortest path dissimilarity as it defines a metric, in addition to which it satisfies the graph-geodetic property. The derivation and computation of the free energy distance are also straightforward. We then make a comparison between a set of generalized distances that interpolate between the shortest path distance and the commute time, or resistance distance. This comparison focuses on the applicability of the distances in graph node clustering and classification. The comparison, in general, shows that the parametrized distances perform well in the tasks. In particular, we see that the results obtained with the free energy distance are among the best in all the experiments.

Walking on Minimax Paths for k-NN Search

Link-based dissimilarity measures, such as shortest path or Euclidean commute time distance, base their distance on paths between nodes of a weighted graph. These measures are known to be better suited to data manifold with nonconvex-shaped clusters, compared to Euclidean distance, so that k-nearest neighbor (NN) search is improved in such metric spaces. In this paper we present a new link-based dissimilarity measure based on minimax paths between nodes. Two main benefits of minimax path-based dissimilarity measure are: (1) only a subset of paths is considered to make it scalable, while Euclidean commute time distance considers all possible paths; (2) it better captures nonconvex-shaped cluster structure, compared to shortest path distance. We define the total cost assigned to a path between nodes as Lp norm of intermediate costs of edges involving the path, showing that minimax path emerges from our Lp norm over paths framework. We also define minimax distance as the intermediate cost of the longest edge on the minimax path, then present a greedy algorithm to compute k smallest minimax distances between a query and N data points in O(log N + k log k) time. Numerical experiments demonstrate that our minimax k-NN algorithm reduce the search time by several orders of magnitude, compared to existing methods, while the quality of k-NN search is significantly improved over Euclidean distance.

An interactive analysis of harmonic and diffusion equations on discrete 3D shapes

Recent results in geometry processing have shown that shape segmentation, comparison, and analysis can be successfully addressed through the spectral properties of the Laplace–Beltrami operator, which is involved in the harmonic equation, the Laplacian eigenproblem, the heat diffusion equation, and the definition of spectral distances, such as the bi-harmonic, commute time, and diffusion distances. In this paper, we study the discretization and the main properties of the solutions to these equations on 3D surfaces and their applications to shape analysis. Among the main factors that influence their computation, as well as the corresponding distances, we focus our attention on the choice of different Laplacian matrices, initial boundary conditions, and input shapes. These degrees of freedom motivate our choice to address this study through the executable paper, which allows the user to perform a large set of experiments and select his/her own parameters. Finally, we represent these distances in a unified way and provide a simple procedure to generate new distances on 3D shapes.

Commute time distance transformation applied to spectral imagery and its utilization in material clustering

Spectral image analysis problems often begin by applying a transformation that generates an alternative representation of the spectral data with the intention of exposing hidden features not discernable in the original space. We introduce and demonstrate a transformation based on a Markov-chain model of a random walk on a graph via application to spectral image clustering. The random walk is quantified by a measure known as the average commute time distance (CTD), which is the average length that a random walker takes, when starting at one node, to transition to another and return to the starting node. This distance metric has the important characteristic of increasing when the number of paths between two nodes decreases and/or the lengths of those paths increase. Once a similarity graph is built on the spectral data, a transformation based on an eigendecomposition of the graph Laplacian matrix is applied that embeds the nodes of the graph into a Euclidean space with the separation between nodes equal to the square-root of the average commute time distance. This is referred to as the Commute Time Distance transformation. As an example of the utility of this data transformation, results are shown for standard clustering algorithms applied to hyperspectral data sets.

Biharmonic distance

Measuring distances between pairs of points on a 3D surface is a fundamental problem in computer graphics and geometric processing. For most applications, the important properties of a distance are that it is a metric, smooth, locally isotropic, globally “shape-aware,” isometry-invariant, insensitive to noise and small topology changes, parameter-free, and practical to compute on a discrete mesh. However, the basic methods currently popular in computer graphics (e.g., geodesic and diffusion distances) do not have these basic properties. In this article, we propose a new distance measure based on the biharmonic differential operator that has all the desired properties. This new surface distance is related to the diffusion and commute-time distances, but applies different (inverse squared) weighting to the eigenvalues of the Laplace-Beltrami operator, which provides a nice trade-off between nearly geodesic distances for small distances and global shape-awareness for large distances. The article provides theoretical and empirical analysis for a large number of meshes.

Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation

This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted and undirected graph. It is based on a Markov-chain model of random walk through the database. More precisely, we compute quantities (the average commute time, the pseudoinverse of the Laplacian matrix of the graph, etc.) that provide similarities between any pair of nodes, having the nice property of increasing when the number of paths connecting those elements increases and when the length of paths decreases. It turns out that the square root of the average commute time is a Euclidean distance and that the pseudoinverse of the Laplacian matrix is a kernel matrix (its elements are inner products closely related to commute times). A principal component analysis (PCA) of the graph is introduced for computing the subspace projection of the node vectors in a manner that preserves as much variance as possible in terms of the Euclidean commute-time distance. This graph PCA provides a nice interpretation to the Fiedler vector, widely used for graph partitioning. The model is evaluated on a collaborative-recommendation task where suggestions are made about which movies people should watch based upon what they watched in the past. Experimental results on the MovieLens database show that the Laplacian-based similarities perform well in comparison with other methods. The model, which nicely fits into the so-called statistical relational learning framework, could also be used to compute document or word similarities, and, more generally, it could be applied to machine-learning and pattern-recognition tasks involving a relational database