Related papers: Optimal co-adapted coupling for a random walk on t…

In a coalescing random walk, a set of particles make independent random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph.…

In a recent paper of Eichelsbacher and Koenig (2008) the model of ordered random walks has been considered. There it has been shown that, under certain moment conditions, one can construct a k-dimensional random walk conditioned to stay in…

Maximum partial subgraph isomorphism compares two graphs (nodes joined by edges) to find a largest common subgraph. A common use case, for graphs with labeled nodes, seeks to find instances of a \textit{query} graph with $q$ nodes in a…

This paper considers an infinite system of instantaneously coalescing rate one simple random walks on $\mathbb{Z}^2$, started from the initial condition with all sites in $\mathbb{Z}^2$ occupied. We show that the correlation functions of…

Considering the optimal alignment of two i.i.d. random sequences of length $n$, we show that when the scoring function is chosen randomly, almost surely the empirical distribution of aligned letter pairs in all optimal alignments converges…

We report on some recent developments in the search for optimal network topologies. First we review some basic concepts on spectral graph theory, including adjacency and Laplacian matrices, and paying special attention to the topological…

The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a…

Graph vertex embeddings based on random walks have become increasingly influential in recent years, showing good performance in several tasks as they efficiently transform a graph into a more computationally digestible format while…

Exact coupling of random walks is studied. Conditions for admitting a successful exact coupling are given that are necessary and in the Abelian case also sufficient. In the Abelian case, it is shown that a random walk $S$ with step-length…

We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case…

The quantum walk dynamics obey the laws of quantum mechanics with an extra locality constraint, which demands that the evolution operator is local in the sense that the walker must visit the neighboring locations before endeavoring to…

Message-passing architectures struggle to sufficiently model long-range dependencies in node and graph prediction tasks. We propose a novel approach exploiting hierarchical graph structures and adaptive random walks to address this…

We introduce a cover time problem for random walks on dynamic graphs in which the graph expands in time and the walker moves at random times. Time to cover all nodes and number of returns to original states are analyzed in resulting model.

We study Hamilton cycles and perfect matchings in a uniform attachment graph. In this random graph, vertices are added sequentially, and when a vertex $t$ is created, it makes $k$ independent and uniform choices from $\{1,\dots,t-1\}$ and…

Graph embedding based on random-walks supports effective solutions for many graph-related downstream tasks. However, the abundance of embedding literature has made it increasingly difficult to compare existing methods and to identify…

We prove that every connected cubic graph with $n$ vertices has a maximal matching of size at most $\frac{5}{12} n+ \frac{1}{2}$. This confirms the cubic case of a conjecture of Baste, F\"urst, Henning, Mohr and Rautenbach (2019) on regular…

Random walks are fundamental tools for analyzing complex networked systems, including social networks, biological systems, and communication infrastructures. While classical random walks focus on pairwise interactions, many real-world…

The performance of distributed averaging depends heavily on the underlying topology. In various fields, including compressed sensing, multi-party computation, and abstract graph theory, graphs may be expected to be free of short cycles,…

We exhibit some explicit co-adapted couplings for n-dimensional Brownian motion and all its Levy stochastic areas. In the two-dimensional case we show how to derive exact asymptotics for the coupling time under various mixed coupling…

Quantum walks are powerful kernels in quantum computing protocols that possess strong capabilities in speeding up various simulation and optimisation tasks. One striking example is given by quantum walkers evolving on glued trees for their…