Related papers: The diameter of KPKVB random graphs
We introduce a new family of models for growing networks. In these networks new edges are attached preferentially to vertices with higher number of connections, and new vertices are created by already existing ones, inheriting part of their…
We study the diameter of $C_1$, the largest component of the Erd\H{o}s-R\'enyi random graph $G(n,p)$ in the emerging supercritical phase, i.e., for $p = \frac{1+\epsilon}n$ where $\epsilon^3 n \to \infty$ and $\epsilon=o(1)$. This parameter…
A network is formed using the $N$ sites of an one-dimensional lattice in the shape of a ring as nodes and each node with the initial degree $k_{in}=2$. $N$ links are then introduced to this network, each link starts from a distinct node,…
Complex networks in different areas exhibit degree distributions with heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential…
Complex networks have been studied extensively due to their relevance to many real systems as diverse as the World-Wide-Web (WWW), the Internet, energy landscapes, biological and social networks…
The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…
Scale-free networks are characterized by a degree distribution with power-law behavior and have been shown to arise in many areas, ranging from the World Wide Web to transportation or social networks. Degree distributions of observed…
Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have…
We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs between two competing metrics, and show that this family is equivalent to a family of preferential attachment…
We give a combinatorial proof of the following theorem. Let $G$ be any finite group acting transitively on a set of cardinality $n$. If $S \subseteq G$ is a random set of size $k$, with $k \geq (\log n)^{1+\varepsilon}$ for some…
We establish a general formula for the maximum size of finite length block codes with minimum pairwise distance no less than $d$. The achievability argument involves an iterative construction of a set of radius-$d$ balls, each centered at a…
Markov width of a graph is a graph invariant defined as the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables. We show that a graph has Markov width at most four if and only if it…
Random graphs with power-law degrees can model scale-free networks as sparse topologies with strong degree heterogeneity. Mathematical analysis of such random graphs proved successful in explaining scale-free network properties such as…
The $\textit{volume density}$ of a hyperbolic link $K$ is defined to be the ratio of the hyperbolic volume of $K$ to the crossing number of $K$. We show that there are sequences of non-alternating links with volume density approaching…
We study a deterministic scale-free network recently proposed by Barab\'{a}si, Ravasz and Vicsek. We find that there are two types of nodes: the hub and rim nodes, which form a bipartite structure of the network. We first derive the exact…
We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.
We address the problem of constructing large undirected circulant networks with given degree and diameter. First we discuss the theoretical upper bounds and their asymptotics, and then we describe and implement a computer-based method to…
The information regarding the structure of a single protein is encoded in the network of interacting amino acids. Considering each protein as a weighted and unweighted network of amino acids we have analyzed a total of forty nine protein…
The size and number of maximum matchings in a network have found a large variety of applications in many fields. As a ubiquitous property of diverse real systems, power-law degree distribution was shown to have a profound influence on size…
We introduce a two-dimensional growth model where every new site is located, at a distance $r$ from the barycenter of the pre-existing graph, according to the probability law $1/r^{2+\alpha_G} (\alpha_G \ge 0)$, and is attached to (only)…