Machine learning techniques have emerged as potential alternatives to traditional physics-based modeling and partial differential equation solvers. Among these machine techniques, Graph Neural Networks (GNNs) simulate physics via graph models; GNNs embed relevant physical features into data structures, perform message passing within the graphs, produce new attributes based on system’s relations...

We study the spectral convergence of graph Laplacians to Laplace-Beltrami operator when kernelized affinity matrix is constructed from N random samples on a d-dimensional manifold in an ambient Euclidean space. By analyzing Dirichlet form and constructing candidate approximate eigenfunctions via convolution with heat kernel, we prove eigen-convergence rates as increases. The best eigenvalue rat...

The dominating set concept in graphs has been used in many applications. In large graphs finding the minimum dominating set is difficult. The minimum dominating set problem in graphs seek a set D of minimum number of vertices such that each vertex of the graph is either in D or adjacent to a vertex in D. In a graph on n nodes if there is a single node of degree n-1 then that single node forms a...

This paper considers a distributed gossip approach for finding a Nash equilibrium in networked games on graphs. In such games a player’s cost function may be affected by the actions of any subset of players. An interference graph is employed to illustrate the partially-coupled cost functions and the asymmetric information requirements. For a given interference graph, network communication betwe...

Distributed consensus computation over random graph processes is considered. The random graph process is defined as a sequence of random variables which take values from the set of all possible digraphs over the node set. At each time step, every node updates its state based on a Bernoulli trial, independent in time and among different nodes: either averaging among the neighbor set generated by...

This work considers the convergence rate of multi-agent systems with discrete-time single-integrator dynamics and undirected interaction topologies. In recent work it has been proven that in case of lattice interaction topologies the convergence rate can be bounded away from zero, independent of the network size, using asymmetric weightings that give the interaction graph a preferred communicat...