Using the Trotter-Kato theorem we prove the convergence of the unitary dynamics generated by an increasingly singular Hamiltonian in the case of a single field coupling. The limit dynamics is a quantum stochastic evolution of Hudson-Parthasarathy type, and we establish in the process a graph limit convergence of the pre-limit Hamiltonian operators to the Chebotarev-Gregoratti-von Waldenfels Ham...

Representation of a swarm of independent robotic agents under graph-theoretic constructs allows for more formal analysis of convergence properties. We consider the local and global convergence behavior of an N -member swarm of agents in a modified consensus problem wherein the connectivity of agents is governed by probabilistic functions. The addition of a random walk control ensures Lyapunov s...

ℓ 1-graph [19, 4], a sparse graph built by reconstructing each datum with all the other data using sparse representation , has been demonstrated to be effective in clustering high dimensional data and recovering independent subspaces from which the data are drawn. It is well known that ℓ 1-norm used in ℓ 1-graph is a convex relaxation of ℓ 0-norm for enforcing the sparsity. In order to handle g...

Robust asymptotic stability for hybrid systems is considered. For this purpose, a generalized solution concept is developed. The first step is to characterize a hybrid time domain that permits an efficient description of the convergence of a sequence of solutions. Graph convergence is used. Then a generalized solution definition is given that leads to continuity with respect to initial conditio...

Since no theoretical foundations for proving the convergence of Graph Shift Algorithm have been reported, we provide a generic framework consisting of three key GS components to fit the Zangwill’s convergence theorem. We show that the sequence set generated by the GS procedures always terminates at a local maximum, or at worst, contains a subsequence which converges to a local maximum of the si...

The asymptotic distributions of the number of vertices of a given degree in random graphs, where the probabilities of edges may not be the same, are given. Using the method of Poisson convergence, distributions in a general and particular cases (complete, almost regular and bipartite graphs) are obtained.

We study the ancestral recombination graph for a pair of sites in a geographically structured population. In particular, we consider the limiting behavior of the graph, under Wright's island model, as the number of subpopulations, or demes, goes to infinity. After an instantaneous sample-size adjustment, the graph becomes identical to the two-locus graph in an unstructured population, but with ...

In this article we apply some graph-theoretic results to the study of coalescence in a structured population with migration. The graph is the pattern of migration among subpopulations, or demes, and we use the theory of random walks on graphs to characterize the ease with which ancestral lineages can traverse the habitat in a series of migration events. We identify conditions under which the co...

We address the problem of a set of agents reaching consensus by computing the average of their initial states. We propose two randomized algorithms over a directed communication graph where either a random node broadcast its value or a randomly selected pair of nodes communicate in a distributed fashion. The proposed algorithms guarantee convergence in three important definitions, namely: almos...

Simulation has indicated that distributed self-organization of autonomous swarms might be achieved through Gibbs sampler-based simulated annealing. However, the dynamic nature of the underlying graph presents significant challenges in convergence analysis. As a first step toward such analysis, convergence of the algorithm is established in this paper for two special cases: single vehicle with l...