Related papers: Typical distances in ultrasmall random networks
Suppose in a graph $G$ vertices can be either red or blue. Let $k$ be odd. At each time step, each vertex $v$ in $G$ polls $k$ random neighbours and takes the majority colour. If it doesn't have $k$ neighbours, it simply polls all of them,…
Preferential attachment models are a common class of graph models which have been used to explain why power-law distributions appear in the degree sequences of real network data. One of the things they lack, however, is higher-order network…
We study the distribution of diameters d of Erd\"os-R\'enyi random graphs with average connectivity c. The diameter d is the maximum among all shortest distances between pairs of nodes in a graph and an important quantity for all dynamic…
The classical preferential attachment model is sensitive to the choice of the initial configuration of the network. As the number of initial nodes and their degree grow, so does the time needed for an equilibrium degree distribution to be…
We consider the survival probability $f(t)$ of a random walk with a constant hopping rate $w$ on a host lattice of fractal dimension $d$ and spectral dimension $d_s\le 2$, with spatially correlated traps. The traps form a sublattice with…
In order to better understand dynamical functions on amounts of natural and man-made complex systems, lots of researchers from a wide range of disciplines, covering statistic physics, mathematics, theoretical computer science, and so on,…
We study random digraphs on sequences of expanders with bounded average degree {which converge locally in probability}. We prove that the threshold for the existence of a giant strongly connected component, as well as the asymptotic…
We study the average nearest neighbor degree $a(k)$ of vertices with degree $k$. In many real-world networks with power-law degree distribution $a(k)$ falls off in $k$, a property ascribed to the constraint that any two vertices are…
Numerous works have been proposed to generate random graphs preserving the same properties as real-life large scale networks. However, many real networks are better represented by hypergraphs. Few models for generating random hypergraphs…
In the on-line nearest-neighbour graph (ONG), each point after the first in a sequence of points in R^d is joined by an edge to its nearest-neighbour amongst those points that precede it in the sequence. We study the large-sample asymptotic…
One of the best-known models in network science is preferential attachment. In this model, the probability of attaching to a node depends on the degree of all nodes in the population, and thus depends on global information. In many…
We study the component structure of the random graph $G=G_{n,m,d}$. Here $d=O(1)$ and $G$ is sampled uniformly from ${\mathcal G}_{n,m,d}$, the set of graphs with vertex set $[n]$, $m$ edges and maximum degree at most $d$. If $m=\mu n/2$…
We determine the asymptotic size of the largest component in the $2$-type binomial random graph $G(\mathbf{n},P)$ near criticality using a refined branching process approach. In $G(\mathbf{n},P)$ every vertex has one of two types, the…
It is shown that in a subcritical random graph with given vertex degrees satisfying a power law degree distribution with exponent $\gamma>3$, the largest component is of order $n^{1/(\gamma-1)}$. More precisely, the order of the largest…
We consider the preferential attachment model. This is a growing random graph such that at each step a new vertex is added and forms $m$ connections. The neighbors of the new vertex are chosen at random with probability proportional to…
We study the optimal distance in networks, $\ell_{\scriptsize opt}$, defined as the length of the path minimizing the total weight, in the presence of disorder. Disorder is introduced by assigning random weights to the links or nodes. For…
In this work we consider a growing random graph sequence where a new vertex is less likely to join to an existing vertex with high degree and more likely to join to a vertex with low degree. In contrast to the well studied…
We consider degree-biased random walkers whose probability to move from a node to one of its neighbors of degree $k$ is proportional to $k^{\alpha}$, where $\alpha$ is a tuning parameter. We study both numerically and analytically three…
For a graph $G=(V,E)$, let $\tau(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $\tau(G) \leq…
Analytic solution for the average path length in a large class of random graphs is found. We apply the approach to classical random graphs of Erd\"{o}s and R\'{e}nyi (ER) and to scale-free networks of Barab\'{a}si and Albert (BA). In both…