The acquaintance time of (percolated) random geometric graphs

In this paper, we study the acquaintance time \AC(G) defined for a connected graph $G$. We focus on \G(n,r,p), a random subgraph of a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from $[0,1]^2$, and two vertices are adjacent with probability $p$ if the Euclidean distance between them is at most $r$. We present asymptotic results for the acquaintance time of \G(n,r,p) for a wide range of $p=p(n)$ and $r=r(n)$. In particular, we show that with high probability \AC(G) = \Theta(r^{-2}) for G \in \G(n,r,1), the "ordinary" random geometric graph, provided that $\pi n r^2 - \ln n \to \infty$ (that is, above the connectivity threshold). For the percolated random geometric graph G \in \G(n,r,p), we show that with high probability \AC(G) = \Theta(r^{-2} p^{-1} \ln n), provided that p n r^2 \geq n^{1/2+\eps} and p < 1-\eps for some \eps>0

Do narcissism and emotional intelligence win us friends? Modeling dynamics of peer popularity using inferential network analysis

This research investigated effects of narcissism and emotional intelligence (EI) on popularity in social networks. In a longitudinal field study we examined the dynamics of popularity in 15 peer groups in two waves (N=273).We measured narcissism, ability EI, explicit and implicit self-esteem. In addition, we measured popularity at zero acquaintance and three months later. We analyzed the data using inferential network analysis (temporal exponential random graph modeling, TERGM) accounting for self-organizing network forces. People high in narcissism were popular, but increased less in popularity over time than people lower in narcissism. In contrast, emotionally intelligent people increased more in popularity over time than less emotionally intelligent people. The effects held when we controlled for explicit and implicit self-esteem. These results suggest that narcissism is rather disadvantageous and that EI is rather advantageous for long-term popularity

Acquaintance time of random graphs near connectivity threshold

Benjamini, Shinkar, and Tsur stated the following conjecture on the acquaintance time: asymptotically almost surely $AC(G) \le p^{-1} \log^{O(1)} n$ for a random graph $G \in G(n,p)$, provided that $G$ is connected. Recently, Kinnersley, Mitsche, and the second author made a major step towards this conjecture by showing that asymptotically almost surely $AC(G) = O(\log n / p)$, provided that $G$ has a Hamiltonian cycle. In this paper, we finish the task by showing that the conjecture holds in the strongest possible sense, that is, it holds right at the time the random graph process creates a connected graph. Moreover, we generalize and investigate the problem for random hypergraphs

Cost-efficient vaccination protocols for network epidemiology

We investigate methods to vaccinate contact networks -- i.e. removing nodes in such a way that disease spreading is hindered as much as possible -- with respect to their cost-efficiency. Any real implementation of such protocols would come with costs related both to the vaccination itself, and gathering of information about the network. Disregarding this, we argue, would lead to erroneous evaluation of vaccination protocols. We use the susceptible-infected-recovered model -- the generic model for diseases making patients immune upon recovery -- as our disease-spreading scenario, and analyze outbreaks on both empirical and model networks. For different relative costs, different protocols dominate. For high vaccination costs and low costs of gathering information, the so-called acquaintance vaccination is the most cost efficient. For other parameter values, protocols designed for query-efficient identification of the network's largest degrees are most efficient