Related papers: Remarks on power-law random graphs
We define direct sums and a corresponding notion of connectedness for graph limits. Every graph limit has a unique decomposition as a direct sum of connected components. As is well-known, graph limits may be represented by symmetric…
We study the behaviour of random labelled and unlabelled cographs with n vertices as n tends to infinity. Our main result is a novel probabilistic limit in the space of graphons.
In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results…
We introduce probability-graphons which are probability kernels that generalize graphons to the case of weighted graphs. Probability-graphons appear as the limit objects to study sequences of large weighted graphs whose distribution of…
Many natural and social systems develop complex networks, that are usually modelled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semi-circle law is known to…
The independence numbers of powers of graphs have been long studied, under several definitions of graph products, and in particular, under the strong graph product. We show that the series of independence numbers in strong powers of a fixed…
We study the detection and the reconstruction of a large very dense subgraph in a social graph with $n$ nodes and $m$ edges given as a stream of edges, when the graph follows a power law degree distribution, in the regime when $m=O(n. \log…
Subgraphs such as cliques, loops and stars form crucial connections in the topologies of real-world networks. Random graph models provide estimates for how often certain subgraphs appear, which in turn can be tested against real-world…
Degree distribution, or equivalently called degree sequence, has been commonly used to be one of most significant measures for studying a large number of complex networks with which some well-known results have been obtained. By contrast,…
We perform a massive evaluation of neural networks with architectures corresponding to random graphs of various types. We investigate various structural and numerical properties of the graphs in relation to neural network test accuracy. We…
The classical random graph model $G(n,\lambda/n)$ satisfies a `duality principle', in that removing the giant component from a supercritical instance of the model leaves (essentially) a subcritical instance. Such principles have been proved…
We consider a conditionally Poissonian random graph model where the mean degrees, `capacities', follow a power-tailed distribution with finite mean and infinite variance. Such a graph of size $N$ has a giant component which is super-small…
The $t$-e.c. and pseudo-random property are typical properties of random graphs. In this note, we study the gap between them which has not been studied well. As a main result, we give the first explicit construction of infinite families of…
In many network problems, graphs may change by the addition of nodes, or the same problem may need to be solved in multiple similar graphs. This generates inefficiency, as analyses and systems that are not transferable have to be…
For any fixed integer $R \geq 2$ we characterise the typical structure of undirected graphs with vertices $1, ..., n$ and maximum degree $R$, as $n$ tends to infinity. The information is used to prove that such graphs satisfy a labelled…
The theory of graphons has proven to be a powerful tool in many areas of graph theory. In this paper, we introduce several foundational aspects of the theory of digraphons -- asymmetric two-variable functions that arise as limits of…
Fokker-Planck equations represent a suitable description of the finite-time behavior for a large class of particle systems as the size of the population tends to infinity. Recently, the theory of graph limits has been introduced in the…
We identify the scaling limit of random intersection graphs inside their critical windows. The limit graphs vary according to the clustering regimes, and coincide with the continuum Erdos--Renyi graph in two out of the three regimes. Our…
We find that scale-free random networks are excellently modeled by a deterministic graph. This graph has a discrete degree distribution (degree is the number of connections of a vertex) which is characterized by a power-law with exponent…
In this paper we make use of graphon theory to study opinion dynamics on large undirected networks. The opinion dynamics models that we take into consideration allow for negative interactions between the individuals, i.e. competing entities…