相关论文: Where do power laws come from?
Power law distribution seems to be an important characteristic of web graphs. Several existing web graph models generate power law graphs by adding new vertices and non-uniform edge connectivities to existing graphs. Researchers have…
We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent $\alpha+1$, where $\alpha \in (1,2)$. The limiting components are…
The spectral and localization properties of heterogeneous random graphs are determined by the resolvent distributional equations, which have so far resisted an analytic treatment. We solve analytically the resolvent equations of random…
Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of "low-degree dependencies'' such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We…
We propose a simple random process inducing various types of random graphs and the scale free random graphs among others. The model is of a threshold nature and differs from the preferential attachment approach discussed in the literature…
We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$…
The study of domination in graphs has led to a variety of domination problems studied in the literature. Most of these follow the following general framework: Given a graph $G$ and an integer $k$, decide if there is a set $S$ of $k$…
We study the realizability of scale free-networks with a given degree sequence, showing that the fraction of realizable sequences undergoes two first-order transitions at the values 0 and 2 of the power-law exponent. We substantiate this…
We consider a variant of so called power-law random graph. A sequence of expected degrees corresponds to a power-law degree distribution with finite mean and infinite variance. In previous works the asymptotic picture with number of nodes…
Let $\mathbb{G}^{D}$ be the set of graphs $G(V,\, E)$ with $\left|V\right|=n$, and the degree sequence equal to $D=(d_{1},\, d_{2},\,\dots,\, d_{n})$. In addition, for $\frac{1}{2}<a<1$, we define the set of graphs with an almost given…
Network data appear in a number of applications, such as online social networks and biological networks, and there is growing interest in both developing models for networks as well as studying the properties of such data. Since individual…
Specify a randomized algorithm that, given a very large graph or network, extracts a random subgraph. What can we learn about the input graph from a single subsample? We derive laws of large numbers for the sampler output, by relating…
We study limits of convergent sequences of string graphs, that is, graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We…
An $r$-uniform hypergraph $H$ consists of a set of vertices $V$ and a set of edges whose elements are $r$-subsets of $V$. We define a hypertree to be a connected hypergraph which contains no cycles. A hypertree spans a hypergraph $H$ if it…
Random network models generated using sparse exchangeable graphs have provided a mechanism to study a wide variety of complex real-life networks. In particular, these models help with investigating power-law properties of degree…
There are typically several nonisomorphic graphs having a given degree sequence, and for any two degree sequence terms it is often possible to find a realization in which the corresponding vertices are adjacent and one in which they are…
We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary length. The implications of our results to the structural and dynamical properties of networks…
Recurrence networks are a novel tool of nonlinear time series analysis allowing the characterisation of higher-order geometric properties of complex dynamical systems based on recurrences in phase space, which are a fundamental concept in…
Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $[n]:=\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $m/n\leq 1$, with high probability the…
The power-law behavior is ubiquitous in a majority of real-world networks, and it was shown to have a strong effect on various combinatorial, structural, and dynamical properties of graphs. For example, it has been shown that in real-life…