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1. A Family of Tractable Graph Distances

   https://doi.org/10.1137/1.9781611975321.38  

   Bento, Jose; Ioannidis, Stratis (January 2018, SIAM International Conference on Data Mining)

   Important data mining problems such as nearest-neighbor search and clustering admit theoretical guarantees when restricted to objects embedded in a metric space. Graphs are ubiquitous, and clustering and classification over graphs arise in diverse areas, including, e.g., image processing and social networks. Unfortunately, popular distance scores used in these applications, that scale over large graphs, are not metrics and thus come with no guarantees. Classic graph distances such as, e.g., the chemical and the CKS distance are arguably natural and intuitive, and are indeed also metrics, but they are intractable: as such, their computation does not scale to large graphs. We define a broad family of graph distances, that includes both the chemical and the CKS distance, and prove that these are all metrics. Crucially, we show that our family includes metrics that are tractable. Moreover, we extend these distances by incorporating auxiliary node attributes, which is important in practice, while maintaining both the metric property and tractability. 

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2. Embedding Signals on Graphs with Unbalanced Diffusion Earth Mover’s Distance

   https://doi.org/10.1109/ICASSP43922.2022.9746556  

   Tong, Alexander; Huguet, Guillaume; Shung, Dennis; Natik, Amine; Kuchroo, Manik; Lajoie, Guillaume; Wolf, Guy; Krishnaswamy, Smita (April 2022, EEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   In modern relational machine learning it is common to encounter large graphs that arise via interactions or similarities between observations in many domains. Further, in many cases the target entities for analysis are actually signals on such graphs. We propose to compare and organize such datasets of graph signals by using an earth mover’s distance (EMD) with a geodesic cost over the underlying graph. Typically, EMD is computed by optimizing over the cost of transporting one probability distribution to another over an underlying metric space. However, this is inefficient when computing the EMD between many signals. Here, we propose an unbalanced graph EMD that efficiently embeds the unbalanced EMD on an underlying graph into an L1 space, whose metric we call unbalanced diffusion earth mover’s distance (UDEMD). Next, we show how this gives distances between graph signals that are robust to noise. Finally, we apply this to organizing patients based on clinical notes, embedding cells modeled as signals on a gene graph, and organizing genes modeled as signals over a large cell graph. In each case, we show that UDEMD-based embeddings find accurate distances that are highly efficient compared to other methods. 

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3. On Light Spanners, Low-treewidth Embeddings and Efficient Traversing in Minor-free Graphs

   Cohen-Addad, Vincent; Filtser, Arnold; Klein, Philip N; Le, Hung (October 2020, Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science)

   null (Ed.)

   Understanding the structure of minor-free metrics, namely shortest path metrics obtained over a weighted graph excluding a fixed minor, has been an important research direction since the fundamental work of Robertson and Seymour. A fundamental idea that helps both to understand the structural properties of these metrics and lead to strong algorithmic results is to construct a “small-complexity” graph that approximately preserves distances between pairs of points of the metric. We show the two following structural results for minor-free metrics: 1) Construction of a light subset spanner. Given a subset of vertices called terminals, and ϵ, in polynomial time we construct a sub graph that preserves all pairwise distances between terminals up to a multiplicative 1+ϵ factor, of total weight at most Oϵ(1) times the weight of the minimal Steiner tree spanning the terminals. 2) Construction of a stochastic metric embedding into low treewidth graphs with expected additive distortion ϵD. Namely, given a minor-free graph G=(V,E,w) of diameter D, and parameter ϵ, we construct a distribution D over dominating metric embeddings into treewidth-Oϵ(log n) graphs such that ∀u,v∈V, Ef∼D[dH(f(u),f(v))]≤dG(u,v)+ϵD. Our results have the following algorithmic consequences: (1) the first efficient approximation scheme for subset TSP in minor-free metrics; (2) the first approximation scheme for bounded-capacity vehicle routing in minor-free metrics; (3) the first efficient approximation scheme for bounded-capacity vehicle routing on bounded genus metrics. En route to the latter result, we design the first FPT approximation scheme for bounded-capacity vehicle routing on bounded-treewidth graphs (parameterized by the treewidth). 

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4. Graph diffusion distance: Properties and efficient computation

   https://doi.org/10.1371/journal.pone.0249624  

   Scott, C. B.; Mjolsness, Eric (April 2021, PLOS ONE)

   Oliva, Gabriele (Ed.)

   We define a new family of similarity and distance measures on graphs, and explore their theoretical properties in comparison to conventional distance metrics. These measures are defined by the solution(s) to an optimization problem which attempts find a map minimizing the discrepancy between two graph Laplacian exponential matrices, under norm-preserving and sparsity constraints. Variants of the distance metric are introduced to consider such optimized maps under sparsity constraints as well as fixed time-scaling between the two Laplacians. The objective function of this optimization is multimodal and has discontinuous slope, and is hence difficult for univariate optimizers to solve. We demonstrate a novel procedure for efficiently calculating these optima for two of our distance measure variants. We present numerical experiments demonstrating that (a) upper bounds of our distance metrics can be used to distinguish between lineages of related graphs; (b) our procedure is faster at finding the required optima, by as much as a factor of 10 3 ; and (c) the upper bounds satisfy the triangle inequality exactly under some assumptions and approximately under others. We also derive an upper bound for the distance between two graph products, in terms of the distance between the two pairs of factors. Additionally, we present several possible applications, including the construction of infinite “graph limits” by means of Cauchy sequences of graphs related to one another by our distance measure. 

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5. Path-Based Spectral Clustering: Guarantees, Robustness to Outliers, and Fast Algorithms

   Little, Anna; Maggioni, Mauro; Murphy, James M (January 2020, Journal of machine learning research)

   null (Ed.)

   We consider the problem of clustering with the longest-leg path distance (LLPD) metric, which is informative for elongated and irregularly shaped clusters. We prove finite-sample guarantees on the performance of clustering with respect to this metric when random samples are drawn from multiple intrinsically low-dimensional clusters in high-dimensional space, in the presence of a large number of highdimensional outliers. By combining these results with spectral clustering with respect to LLPD, we provide conditions under which the Laplacian eigengap statistic correctly determines the number of clusters for a large class of data sets, and prove guarantees on the labeling accuracy of the proposed algorithm. Our methods are quite general and provide performance guarantees for spectral clustering with any ultrametric. We also introduce an efficient, easy to implement approximation algorithm for the LLPD based on a multiscale analysis of adjacency graphs, which allows for the runtime of LLPD spectral clustering to be quasilinear in the number of data points. 

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