Related papers: A continuum limit for dense spatial networks

We define a class of continuous graded graphs similar to the graph of Gelfand--Tsetlin patterns, and describe the set of all ergodic central measures of discrete type on the path spaces of such graphs. The main observation is that an…

Motivated in part by various sequences of graphs growing under random rules (like internet models), convergent sequences of dense graphs and their limits were introduced by Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi and by Lov\'asz and…

Inferring time-varying graph structures from high-dimensional nodal observations is a fundamental problem arising in neuroscience, finance, climatology, and beyond. Two intrinsic challenges govern this problem: maintaining the…

We introduce a broad class of equations that are described by a graph, which includes many well-studied systems. For these, we show that the number of solutions (or the dimension of the solution set) can be bounded by studying certain…

We study the problem of deterministic approximate counting of matchings and independent sets in graphs of bounded connective constant. More generally, we consider the problem of evaluating the partition functions of the monomer-dimer model…

We propose a new method for spatio-temporal forecasting on arbitrarily distributed points. Assuming that the observed system follows an unknown partial differential equation, we derive a continuous-time model for the dynamics of the data…

Multiscale modelling aims to systematically construct macroscale models of materials with fine microscale structure. However, macroscale boundary conditions are typically not systematically derived, but rely on heuristic arguments,…

Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…

The goal of the paper is to lay the foundation for the qualitative analogue of the classical, quantitative sparse graph limit theory. In the first part of the paper we introduce the qualitative analogues of the Benjamini-Schramm and…

When solving partial differential equations (PDEs), classical numerical methods often require fine mesh grids and small time stepping to meet stability, consistency, and convergence conditions, leading to high computational cost. Recently,…

The capacity of a graph is defined as the rate of exponential growth of independent sets in the strong powers of the graph. In the strong power an edge connects two sequences if at each position their letters are equal or adjacent. We…

Betweenness is a well-known centrality measure that ranks the nodes of a network according to their participation in shortest paths. Since an exact computation is prohibitive in large networks, several approximation algorithms have been…

One of the most influential results in neural network theory is the universal approximation theorem [1, 2, 3] which states that continuous functions can be approximated to within arbitrary accuracy by single-hidden-layer feedforward neural…

We are interested in exploring interacting particle systems that can be seen as microscopic models for a particular structure of coupled transport flux arising when different populations are jointly evolving. The scenarios we have in mind…

In the dense graph limit theory, the topology of the set of graphs is defined by the distribution of the subgraphs spanned by finite number of random vertices. Vera T. S\'os proposed a question that if we consider only the number of edges…

While previous works have shown that an overwhelming number of scale-free networks are sparse, there still exist some real-world networks including social networks, urban networks, information networks, which are by observation dense. In…

Empirical networks are often globally sparse, with a small average number of connections per node, when compared to the total size of the network. However, this sparsity tends not to be homogeneous, and networks can also be locally dense,…

In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…

Preferential attachment graphs are random graphs designed to mimic properties of typical real world networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already…

Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have…