相关论文: On The Modified Newman-Watts Small World and Its R…

Since its recent introduction, the small-world effect has been identified in several important real-world systems. Frequently, it is a consequence of the existence of a few long-range connections, which dominate the original regular…

The metric dimension (MD) of a graph is a combinatorial notion capturing the minimum number of landmark nodes needed to distinguish every pair of nodes in the graph based on graph distance. We study how much the MD can increase if we add a…

Using both numerical simulations and scaling arguments, we study the behavior of a random walker on a one-dimensional small-world network. For the properties we study, we find that the random walk obeys a characteristic scaling form. These…

In this paper we consider the simple random walk on $\mathbb{Z}^d$, $d \geq 3$, conditioned to stay in a large domain $D_N$ of typical diameter $N$. Considering the range up to time $t_N \geq N^{2+\delta}$ for some $\delta > 0$, we…

We study graphs obtained by successive creation and destruction of edges into small neighborhoods of the vertices. Starting with a circle graph of large diameter we obtain small world graphs with logarithmic diameter, high clustering…

We study analytically a simple random walk model on a one-dimensional lattice, where at each time step the walker resets to the maximum of the already visited positions (to the rightmost visited site) with a probability $r$, and with…

A network is said to have the properties of a small world if a suitably defined average distance between any two nodes is proportional to the logarithm of the number of nodes, $N$. In this paper, we present a novel derivation of the…

Let $P$ be a path graph of $n$ vertices embedded in a metric space. We consider the problem of adding a new edge to $P$ to minimize the radius of the resulting graph. Previously, a similar problem for minimizing the diameter of the graph…

We show that the total variation mixing time of the simple random walk on the giant component of supercritical Erdos-Renyi graphs is log^2 n. This statement was only recently proved, independently, by Fountoulakis and Reed. Our proof…

The connection between coined and continuous-time quantum walk models has been addressed in a number of papers. In most of those studies, the continuous-time model is derived from coined quantum walks by employing dimensional reduction and…

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…

Mixing properties of discrete-time quantum walks on two-dimensional grids with torus-like boundary conditions are analyzed, focusing on their connection to the complexity of the corresponding abstract search algorithm. In particular, an…

We study spreading processes in temporal graphs, i. e., graphs whose connections change over time. These processes naturally model real-world phenomena such as infectious diseases or information flows. More precisely, we investigate how…

This paper investigates the addition of random edges to arbitrary dense graphs; in particular, we determine the number of random edges required to ensure various monotone properties including the appearance of a fixed size clique, small…

Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We…

The dynamics of the avalanche width in the evolution model is described using a random walk picture. In this approach the critical exponents for avalanche distribution, $\tau$, and avalanche average time, $\gamma$, are found to be the same…

Consider a random walk in random environment on a supercritical Galton--Watson tree, and let $\tau_n$ be the hitting time of generation $n$. The paper presents a large deviation principle for $\tau_n/n$, both in quenched and annealed cases.…

We study the growing patterns in the rotor-router model formed by adding $N$ walkers at the center of a $L \times L$ two-dimensional square lattice, starting with a periodic background of arrows, and relaxing to a stable configuration. The…

We consider dynamical percolation on the complete graph $K_n$, where each edge refreshes its state at rate $\mu \ll 1/n$, and is then declared open with probability $p = \lambda/n$ where $\lambda > 1$. We study a random walk on this…

Consider a randomly-oriented two dimensional Manhattan lattice where each horizontal line and each vertical line is assigned, once and for all, a random direction by flipping independent and identically distributed coins. A deterministic…