Related papers: Subgraphs in random networks
Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases -…
With a simple attack and repair evolution model, we investigate the stability and structural changes of the Erdos-Renyi random graphs (RG) and Barabasi-Albert scale-free (SF) networks. We introduce a new quantity, invulnerability I(s), to…
Scale-free networks with moderate edge dependence experience a phase transition between ultrasmall and small world behaviour when the power law exponent passes the critical value of three. Moreover, there are laws of large numbers for the…
In this paper we calculate the average number of cliques in random scale-free networks. We consider first the hidden variable ensemble and subsequently the Molloy Reed ensemble. In both cases we find that cliques, i.e. fully connected…
A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on…
Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with…
Using each node's degree as a proxy for its importance, the topological hierarchy of a complex network is introduced and quantified. We propose a simple dynamical process used to construct networks which are either maximally or minimally…
Detecting the dimensionality of graphs is a central topic in machine learning. While the problem has been tackled empirically as well as theoretically, existing methods have several drawbacks. On the one hand, empirical tools are…
Given connected graph $H$ which is not a star, we show that the number of copies of $H$ in a dense uniformly random regular graph is asymptotically Gaussian, which was not known even for $H$ being a triangle. This addresses a question of…
We consider a model of large regulatory gene expression networks where the thresholds activating the sigmoidal interactions between genes and the signs of these interactions are shuffled randomly. Such an approach allows for a qualitative…
A uniformly random graph on $n$ vertices with a fixed degree sequence, obeying a $\gamma$ subpower law, is studied. It is shown that, for $\gamma>3$, in a subcritical phase with high probability the largest component size does not exceed…
We investigate a randomly evolving process of subgraphs in an underlying host graph using the spectral theory of semigroups related to the Tsetlin library and hyperplane arrangements. Starting with some initial subgraph, at each iteration,…
The degrees are a classical and relevant way to study the topology of a network. They can be used to assess the goodness-of-fit for a given random graph model. In this paper we introduce goodness-of-fit tests for two classes of models.…
We introduce a growing network model---the copying model---in which a new node attaches to a randomly selected target node and, in addition, independently to each of the neighbors of the target with copying probability $p$. When…
Random key graphs were introduced to study various properties of the Eschenauer-Gligor key predistribution scheme for wireless sensor networks (WSNs). Recently this class of random graphs has received much attention in contexts as diverse…
We study the statistical behavior under random sequential renormalization(RSR) of several network models including Erd"os R'enyi (ER) graphs, scale-free networks and an annealed model (AM) related to ER graphs. In RSR the network is locally…
Generalised degrees provide a natural bridge between local and global topological properties of networks. We define the generalised degree to be the number of neighbours of a node within one and two steps respectively. Tailored random graph…
How do real graphs evolve over time? What are ``normal'' growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large…
Duplication graphs are graphs that grow by duplication of existing vertices, and are important models of biological networks, including protein-protein interaction networks and gene regulatory networks. Three models of graph growth are…
There is increasing evidence that dense networks occur in on-line social networks, recommendation networks and in the brain. In addition to being dense, these networks are often also scale-free, i.e. their degree distributions follow…