Consider the following problem. You are given a family  G of weighted undirected sparse graphs with positive weights and a family H of functions h_0, ... , : R -> R. Define a weight of the walk as a product of the weights of its edges.  Take a tensor field F on the nodes of G and consider a new tensor field F', obtained from F by the following procedure. For a given node v, the value F'(v) of F' is obtained by taking the linear combination of the tensors F(x) over all the nodes x of G,  with the weights given in one of two ways (effectively defining two recipes for the integration procedure): (1) as h_0(shortest path distance from x to v) [singleton-H setting],  (2) as a sum over i = 0,... of h_i(sum of the weights of the i-hops walks connecting x to v). The task is to compute exactly (or approximately) F' from F in time sub-quadratic (even near-linear) in the number of nodes N. Surprisingly, efficient solutions to this class of problems have several applications across many fields of science, spanning through: fast multipole methods, graph-inputs transformers architectures in machine learning, to N-body problems in physics and quantum mechanics (computations of transition amplitudes in finite-dimensional quantum systems, discretizations of Feynman path  integral methods for general quantum systems). In this talk, we will summarize the current progress on the research to find such efficient algorithms for nontrivial families of graphs G and functions H. In particular, we will show how to address the problem of finding exact solution for d-dimensional grids, bounded (connected) treewidth graphs,  as well as approximate solutions for general graphs in setting (2) and planar graphs  in setting (1). As a by-product, we show how the research on the efficient solutions benefits from various methods across different fields of mathematics, such as: structural graph theory, Fourier analysis, the theory of structured matrices and low-displacement rank transforms, finally graph random features.