Microscopic concavity and fluctuation bounds in a class of deposition processes

We prove fluctuation bounds for the particle current in totally asymmetric zero range processes in one dimension with nondecreasing, concave jump rates whose slope decays exponentially. Fluctuations in the characteristic directions have order of magnitude $t^{1/3}$. This is in agreement with the expectation that these systems lie in the same KPZ universality class as the asymmetric simple exclusion process. The result is via a robust argument formulated for a broad class of deposition-type processes. Besides this class of zero range processes, hypotheses of this argument have also been verified in the authors' earlier papers for the asymmetric simple exclusion and the constant rate zero range processes, and are currently under development for a bricklayers process with exponentially increasing jump rates.Comment: Improved after Referee's comments: we added explanations and changed some parts of the text. 50 pages, 1 figur

Weighted distances in scale-free preferential attachment models

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a non-negative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight.Comment: Revised version, results are unchanged. 30 pages, 1 figure. To appear in Random Structures and Algorithm

Degree distribution of shortest path trees and bias of network sampling algorithms

In this article, we explicitly derive the limiting distribution of the degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the power-law exponent of the degree distribution of this tree and compare it to the degree distribution of the original graph. We perform this analysis for the complete graph with edge weights that are powers of exponential random variables (weak disorder in the stochastic mean-field model of distance) as well as on the configuration model with edge-weights drawn according to any continuous distribution. In the latter, the focus is on settings where the degrees obey a power law, and we show that the shortest path tree again obeys a power law with the same degree power-law exponent. We also consider random r-regular graphs for large r, and show that the degree distribution of the shortest path tree is closely related to the shortest path tree for the stochastic mean field model of distance. We use our results to explain an empirically observed bias in network sampling methods. This is part of a general program initiated in previous works by Bhamidi, van der Hofstad and Hooghiemstra [7, 8, 6] of analyzing the effect of attaching random edge lengths on the geometry of random network models

Order of current variance and diffusivity in the rate one totally asymmetric zero range process

We prove that the variance of the current across a characteristic is of order t^{2/3} in a stationary constant rate totally asymmetric zero range process, and that the diffusivity has order t^{1/3}. This is a step towards proving universality of this scaling behavior in the class of one-dimensional interacting systems with one conserved quantity and concave hydrodynamic flux. The proof proceeds via couplings to show the corresponding moment bounds for a second class particle. We build on the methods developed by Balazs-Seppalainen for asymmetric simple exclusion. However, some modifications were needed to handle the larger state space. Our results translate into t^{2/3}-order of variance of the tagged particle on the characteristics of totally asymmetric simple exclusion.Comment: 23 pages; some minor typos correcte

A pedestrian's view on interacting particle systems, KPZ universality, and random matrices

These notes are based on lectures delivered by the authors at a Langeoog seminar of SFB/TR12 "Symmetries and universality in mesoscopic systems" to a mixed audience of mathematicians and theoretical physicists. After a brief outline of the basic physical concepts of equilibrium and nonequilibrium states, the one-dimensional simple exclusion process is introduced as a paradigmatic nonequilibrium interacting particle system. The stationary measure on the ring is derived and the idea of the hydrodynamic limit is sketched. We then introduce the phenomenological Kardar-Parisi-Zhang (KPZ) equation and explain the associated universality conjecture for surface fluctuations in growth models. This is followed by a detailed exposition of a seminal paper of Johansson that relates the current fluctuations of the totally asymmetric simple exclusion process (TASEP) to the Tracy-Widom distribution of random matrix theory. The implications of this result are discussed within the framework of the KPZ conjecture.Comment: 52 pages, 4 figures; to appear in J. Phys. A: Math. Theo

Weighted distances in scale-free configuration models

\u3cp\u3eIn this paper we study first-passage percolation in the configuration model with empirical degree distribution that follows a power-law with exponent τ∈ (2 , 3). We assign independent and identically distributed (i.i.d.) weights to the edges of the graph. We investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical distances. When the underlying age-dependent branching process approximating the local neighborhoods of vertices is found to produce infinitely many individuals in finite time—called explosive branching process—Baroni, Hofstad and the second author showed in Baroni et al. (J Appl Probab 54(1):146–164, 2017) that typical distances converge in distribution to a bounded random variable. The order of magnitude of typical distances remained open for the τ∈ (2 , 3) case when the underlying branching process is not explosive. We close this gap by determining the first order of magnitude of typical distances in this regime for arbitrary, not necessary continuous edge-weight distributions that produce a non-explosive age-dependent branching process with infinite mean power-law offspring distributions. This sequence tends to infinity with the amount of vertices, and, by choosing an appropriate weight distribution, can be tuned to be any growing function that is O(log log n) , where n is the number of vertices in the graph. We show that the result remains valid for the the erased configuration model as well, where we delete loops and any second and further edges between two vertices.\u3c/p\u3

First passage percolation on the Newman–Watts small world model

\u3cp\u3eThe Newman–Watts model is given by taking a cycle graph of n vertices and then adding each possible edge (Formula presented.) modn with probability ρ/n for some ρ>0 constant. In this paper we add i.i.d. exponential edge weights to this graph, and investigate typical distances in the corresponding random metric space given by the least weight paths between vertices. We show that typical distances grow as (Formyula presented.) logn for a λ>0 and determine the distribution of smaller order terms in terms of limits of branching process random variables. We prove that the number of edges along the shortest weight path follows a Central Limit Theorem, and show that in a corresponding epidemic spread model the fraction of infected vertices follows a deterministic curve with a random shift.\u3c/p\u3

Sharp bound on the truncated metric dimension of trees

A k-truncated resolving set of a graph is a subset S⊆V of its vertex set such that the vector (dk(s,v))s∈S is distinct for each vertex v∈V where dk(x,y)=min⁡{d(x,y),k+1} is the graph distance truncated at k+1. We think of elements of a k-truncated resolving set as sensors that can measure up to distance k. The k-truncated metric dimension (Tmdk) of a graph G is the minimum cardinality of a k-truncated resolving set of G. We give a sharp lower bound on Tmdk for any tree T in terms of its number of vertices |T| and the measuring radius k. Our result is that Tmdk(T)≥|T|⋅3/(k2+4k+3+1{k≡1(mod3)})+ck, disproving earlier conjectures by Frongillo et al. that suspected |T|/(⌊k2/4⌋+2k)+ck′ as general lower bound, where ck, ck′ are k-dependent constants. We provide a construction for trees with the largest number of vertices with a given Tmdk value. The proof that our optimal construction cannot be improved relies on edge-rewiring procedures of arbitrary (suboptimal) trees with arbitrary resolving sets, which reveal the structure of how small subsets of sensors measure and resolve certain areas in the tree that we call the attraction of those sensors. The notion of ‘attraction of sensors’ might be useful in other contexts beyond trees to solve related problems. We also provide an improved lower bound on Tmdk of arbitrary trees that takes into account the structural properties of the tree, in particular, the number and length of simple paths of degree-two vertices terminating in leaf vertices. This bound complements the result of the above-mentioned work of Frongillo et al., where only trees without degree-two vertices were considered, except the simple case of a single path.</p

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

When is a Scale-Free graph ultra-small?

\u3cp\u3eIn this paper we study typical distances in the configuration model, when the degrees have asymptotically infinite variance. We assume that the empirical degree distribution follows a power law with exponent τ∈ (2 , 3 ) , up to value nβn for some β \u3csub\u3en\u3c/sub\u3e≫ (log n) \u3csup\u3e-\u3c/sup\u3e \u3csup\u3eγ\u3c/sup\u3e and γ∈ (0 , 1 ). This assumption is satisfied for power law i.i.d. degrees, and also includes truncated power-law empirical degree distributions where the (possibly exponential) truncation happens at nβn. These examples are commonly observed in many real-life networks. We show that the graph distance between two uniformly chosen vertices centers around 2loglog(nβn)/|log(τ-2)|+1/(βn(3-τ)), with tight fluctuations. Thus, the graph is an ultrasmall world whenever 1 / β \u3csub\u3en\u3c/sub\u3e= o(log log n). We determine the distribution of the fluctuations around this value, in particular we prove these form a sequence of tight random variables with distributions that show log log -periodicity, and as a result it is non-converging. We describe the topology and number of shortest paths: We show that the number of shortest paths is of order nfnβn, where f \u3csub\u3en\u3c/sub\u3e∈ (0 , 1 ) is a random variable that oscillates with n. We decompose shortest paths into three segments, two ‘end-segments’ starting at each of the two uniformly chosen vertices, and a middle segment. The two end-segments of any shortest path have length loglog(nβn)/|log(τ-2)|+tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length 1 / (β \u3csub\u3en\u3c/sub\u3e(3 - τ) ) +tight, and it contains only vertices with degree at least of order n(1-fn)βn, thus all the degrees on this segment are comparable to the maximal degree. Our theorems also apply when instead of truncating the degrees, we start with a configuration model and we remove every vertex with degree at least nβn, and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees. \u3c/p\u3