Metric dimension for random graphs

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$

Vertex-pursuit in random directed acyclic graphs

We examine a dynamic model for the disruption of information flow in hierarchical social networks by considering the vertex-pursuit game Seepage played in directed acyclic graphs (DAGs). In Seepage, agents attempt to block the movement of an intruder who moves downward from the source node to a sink. The minimum number of such agents required to block the intruder is called the green number. We propose a generalized stochastic model for DAGs with given expected total degree sequence. Seepage and the green number is analyzed in stochastic DAGs in both the cases of a regular and power law degree sequence. For each such sequence, we give asymptotic bounds (and in certain instances, precise values) for the green number

Clique colouring of binomial random graphs

A clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colours in such a colouring is the clique chromatic number. In this paper, we study the asymptotic behaviour of the clique chromatic number of the random graph G(n,p) for a wide range of edge-probabilities p=p(n). We see that the typical clique chromatic number, as a function of the average degree, forms an intriguing step function

A spatial web graph model with local influence regions

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SimpleHypergraphs.jl—novel software framework for modelling and analysis of hypergraphs

Hypergraphs are natural generalization of graphs in which a single (hyper)edge can connect any number of vertices. As a result, hypergraphs are suitable and useful to model many important networks and processes. Typical applications are related to social data analysis and include situations such as exchanging emails with several recipients, reviewing products on social platforms, or analyzing security vulnerabilities of information networks. In many situations, using hypergraphs instead of classical graphs allows us to better capture and analyze dependencies within the network. In this paper, we propose a new library, named SimpleHypergraphs.jl, designed for efficient hypegraph analysis. The library exploits the Julia language flexibility and direct support for distributed computing in order to bring a new quality for simulating and analyzing processes represented as hypergraphs. In order to show how the library can be used we study two case studies based on the Yelp dataset. Results are promising and confirm the ability of hypergraphs to provide more insight than standard graph-based approaches