Structures in supercritical scale-free percolation

Scale-free percolation is a percolation model on $\mathbb{Z}^d$ which can be used to model real-world networks. We prove bounds for the graph distance in the regime where vertices have infinite degrees. We fully characterize transience vs. recurrence for dimension 1 and 2 and give sufficient conditions for transience in dimension 3 and higher. Finally, we show the existence of a hierarchical structure for parameters where vertices have degrees with infinite variance and obtain bounds on the cluster density.Comment: Revised Definition 2.5 and an argument in Section 6, results are unchanged. Correction of minor typos. 29 pages, 7 figure

Comma Selection Outperforms Plus Selection on OneMax with Randomly Planted Optima

It is an ongoing debate whether and how comma selection in evolutionary algorithms helps to escape local optima. We propose a new benchmark function to investigate the benefits of comma selection: OneMax with randomly planted local optima, generated by frozen noise. We show that comma selection (the $(1,\lambda)$ EA) is faster than plus selection (the $(1+\lambda)$ EA) on this benchmark, in a fixed-target scenario, and for offspring population sizes $\lambda$ for which both algorithms behave differently. For certain parameters, the $(1,\lambda)$ EA finds the target in $\Theta(n \ln n)$ evaluations, with high probability (w.h.p.), while the $(1+\lambda)$ EA) w.h.p. requires almost $\Theta((n\ln n)^2)$ evaluations. We further show that the advantage of comma selection is not arbitrarily large: w.h.p. comma selection outperforms plus selection at most by a factor of $O(n \ln n)$ for most reasonable parameter choices. We develop novel methods for analysing frozen noise and give powerful and general fixed-target results with tail bounds that are of independent interest.Comment: An extended abstract will be published at GECCO 202

Not all interventions are equal for the height of the second peak

In this paper we conduct a simulation study of the spread of an epidemic like COVID-19 with temporary immunity on finite spatial and non-spatial network models. In particular, we assume that an epidemic spreads stochastically on a scale-free network and that each infected individual in the network gains a temporary immunity after its infectious period is over. After the temporary immunity period is over, the individual becomes susceptible to the virus again. When the underlying contact network is embedded in Euclidean geometry, we model three different intervention strategies that aim to control the spread of the epidemic: social distancing, restrictions on travel, and restrictions on maximal number of social contacts per node. Our first finding is that on a finite network, a long enough average immunity period leads to extinction of the pandemic after the first peak, analogous to the concept of "herd immunity". For each model, there is a critical average immunity length $L_c$ above which this happens. Our second finding is that all three interventions manage to flatten the first peak (the travel restrictions most efficiently), as well as decrease the critical immunity length $L_c$, but elongate the epidemic. However, when the average immunity length $L$ is shorter than $L_c$, the price for the flattened first peak is often a high second peak: for limiting the maximal number of contacts, the second peak can be as high as 1/3 of the first peak, and twice as high as it would be without intervention. Thirdly, interventions introduce oscillations into the system and the time to reach equilibrium is, for almost all scenarios, much longer. We conclude that network-based epidemic models can show a variety of behaviors that are not captured by the continuous compartmental models.Comment: 17 pages text, 27 figures of which 22 in the appendi

Distance evolutions in growing preferential attachment graphs

We study the evolution of the graph distance and weighted distance between two fixed vertices in dynamically growing random graph models. More precisely, we consider preferential attachment models with power-law exponent $\tau\in(2,3)$, sample two vertices $u_t,v_t$ uniformly at random when the graph has $t$ vertices, and study the evolution of the graph distance between these two fixed vertices as the surrounding graph grows. This yields a discrete-time stochastic process in $t'\geq t$, called the distance evolution. We show that there is a tight strip around the function $4\frac{\log\log(t)-\log(\log(t'/t)\vee1)}{|\log(\tau-2)|}\vee 2$ that the distance evolution never leaves with high probability as $t$ tends to infinity. We extend our results to weighted distances, where every edge is equipped with an i.i.d. copy of a non-negative random variable $L$.Comment: 42 pages, 4 figures. Revised version with corrected typos and more elaborate proofs. Includes correction of an error in Theorem 2.5 that required a shift of indices in the summatio

Increasing efficacy of contact-tracing applications by user referrals and stricter quarantining.

We study the effects of two mechanisms which increase the efficacy of contact-tracing applications (CTAs) such as the mobile phone contact-tracing applications that have been used during the COVID-19 epidemic. The first mechanism is the introduction of user referrals. We compare four scenarios for the uptake of CTAs-(1) the p% of individuals that use the CTA are chosen randomly, (2) a smaller initial set of randomly-chosen users each refer a contact to use the CTA, achieving p% in total, (3) a small initial set of randomly-chosen users each refer around half of their contacts to use the CTA, achieving p% in total, and (4) for comparison, an idealised scenario in which the p% of the population that uses the CTA is the p% with the most contacts. Using agent-based epidemiological models incorporating a geometric space, we find that, even when the uptake percentage p% is small, CTAs are an effective tool for mitigating the spread of the epidemic in all scenarios. Moreover, user referrals significantly improve efficacy. In addition, it turns out that user referrals reduce the quarantine load. The second mechanism for increasing the efficacy of CTAs is tuning the severity of quarantine measures. Our modelling shows that using CTAs with mild quarantine measures is effective in reducing the maximum hospital load and the number of people who become ill, but leads to a relatively high quarantine load, which may cause economic disruption. Fortunately, under stricter quarantine measures, the advantages are maintained but the quarantine load is reduced. Our models incorporate geometric inhomogeneous random graphs to study the effects of the presence of super-spreaders and of the absence of long-distant contacts (e.g., through travel restrictions) on our conclusions

Structures in supercritical scale-free percolation

\u3cp\u3eScale-free percolation is a percolation model on Z\u3csup\u3ed\u3c/sup\u3e which can be used to model real-world networks. We prove bounds for the graph distance in the regime where vertices have infinite degrees. We fully characterize transience versus recurrence for dimension 1 and 2 and give sufficient conditions for transience in dimension 3 and higher. Finally, we show the existence of a hierarchical structure for parameters where vertices have degrees with infinite variance and obtain bounds on the cluster density.\u3c/p\u3

Simulation data for "Increasing efficacy of contact-tracing applications by user referrals and stricter quarantining"

This is the data that is generated by our simulations for the paper "Increasing efficacy of contact-tracing applications by user referrals and stricter quarantining". All figures are made with these data sets. This upload also contains the graphs on top of which the epidemics are simulated