arXiv subjects

Peter Mörters

Publications and source records attributed to Peter Mörters.

At least 37 records · Page 2Linked to original sources

arXiv2020

Competing growth processes with random growth rates and random birth times

Finding the most powerful node in a dynamic random network, the largest set in a partition-valued stochastic process, or the largest family in an evolving population at a given time, can be a very difficult problem. This is particularly the case when the underlying stochastic process has complex dependencies and the individual strength of an object has an impact that only plays out over time. We propose a novel technique to deal with such problems and show how it can be applied to a broad range of examples where it produces new insight and surprising results. The method relies on two steps: In the first step, which is highly problem dependent, the problem is embedded into continuous time so that the evolution of the sizes of objects after their individual birth times become approximately independent while we only need minimal control over the birth times themselves. Once such an embedding is achieved, the second step is to apply a Poisson limit theorem that allows a comparison of object sizes in a critical window and therefore allows a description of features of extremal objects. In this paper we prove such a versatile limit theorem, based on extreme value theory, and show how the technique can be used to study extremal behaviour in different types of preferential attachment networks with fitness, branching processes with selection and mutation, and random permutations with random cycle weights.

math.PR↗

arXiv2019

Characterising random partitions by random colouring

Let $(X_1,X_2,...)$ be a random partition of the unit interval $[0,1]$, i.e. $X_i\geq0$ and $\sum_{i\geq1} X_i=1$, and let $(\varepsilon_1,\varepsilon_2,...)$ be i.i.d. Bernoulli random variables of parameter $p \in (0,1)$. The Bernoulli convolution of the partition is the random variable $Z =\sum_{i\geq1} \varepsilon_i X_i$. The question addressed in this article is: Knowing the distribution of $Z$ for some fixed $p\in(0,1)$, what can we infer about the random partition? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter $p$ is not equal to $1/2$.

math.PR↗

arXiv2018

The age-dependent random connection model

We investigate a class of growing graphs embedded into the $d$-dimensional torus where new vertices arrive according to a Poisson process in time, are randomly placed in space and connect to existing vertices with a probability depending on time, their spatial distance and their relative ages. This simple model for a scale-free network is called the age-based spatial preferential attachment network and is based on the idea of preferential attachment with spatially induced clustering. We show that the graphs converge weakly locally to a variant of the random connection model, which we call the age-dependent random connection model. This is a natural infinite graph on a Poisson point process where points are marked by a uniformly distributed age and connected with a probability depending on their spatial distance and both ages. We use the limiting structure to investigate asymptotic degree distribution, clustering coefficients and typical edge lengths in the age-based spatial preferential attachment network.

math.PR↗

arXiv2018

Metastability of the contact process on fast evolving scale-free networks

We study the contact process in the regime of small infection rates on finite scale-free networks with stationary dynamics based on simultaneous updating of all connections of a vertex. We allow the update rates of individual vertices to increase with the strength of a vertex, leading to a fast evolution of the network. We first develop an approach for inhomogeneous networks with general kernel and then focus on two canonical cases, the factor kernel and the preferential attachment kernel. For these specific networks we identify and analyse four possible strategies how the infection can survive for a long time. We show that there is fast extinction of the infection when neither of the strategies is successful, otherwise there is slow extinction and the most successful strategy determines the asymptotics of the metastable density as the infection rate goes to zero. We identify the domains in which these strategies dominate in terms of phase diagrams for the exponent describing the decay of the metastable density.

math.PR↗

arXiv2018

A re-entrant phase transition in the survival of secondary infections on networks

We study the dynamics of secondary infections on networks, in which only the individuals currently carrying a certain primary infection are susceptible to the secondary infection. In the limit of large sparse networks, the model is mapped to a branching process spreading in a random time-sensitive environment, determined by the dynamics of the underlying primary infection. When both epidemics follow the Susceptible-Infective-Recovered model, we show that in order to survive, it is necessary for the secondary infection to evolve on a timescale that is closely matched to that of the primary infection on which it depends.

math.PR↗

arXiv2017

The shape of the emerging condensate in effective models of condensation

We consider effective models of condensation where the condensation occurs as time t goes to infinity. We provide natural conditions under which the build-up of the condensate occurs on a spatial scale of 1/t and has the universal form of a Gamma density. The exponential parameter of this density is determined only by the equation and the total mass of the condensate, while the power law parameter may in addition depend on the decay properties of the initial condition near the condensation point. We apply our results to some examples, including simple models of Bose-Einstein condensation.

math-ph↗

arXiv2017

The Semi-Infinite Asymmetric Exclusion Process: Large Deviations via Matrix Products

We study the totally asymmetric exclusion process on the positive integers with a single particle source at the origin. Liggett (1975) has shown that the long term behaviour of this process has a phase transition: If the particle production rate at the source is below a critical value, the stationary measure is a product measure, otherwise the stationary measure is spatially correlated. Following the approach of Derrida et al. (1993) it was shown by Großkinsky (2004) that these correlations can be described by means of a matrix product representation. In this paper we derive a large deviation principle with explicit rate function for the particle density in a macroscopic box based on this representation. The novel and rigorous technique we develop for this problem combines spectral theoretical and combinatorial ideas and has the potential to be applicable to other models described by matrix products.

math-ph↗

arXiv2017

Distances in scale free networks at criticality

Scale-free networks with moderate edge dependence experience a phase transition between ultrasmall and small world behaviour when the power law exponent passes the critical value of three. Moreover, there are laws of large numbers for the graph distance of two randomly chosen vertices in the giant component. When the degree distribution follows a pure power law these show the same asymptotic distances of $\frac{\log N}{\log\log N}$ at the critical value three, but in the ultrasmall regime reveal a difference of a factor two between the most-studied rank-one and preferential attachment model classes. In this paper we identify the critical window where this factor emerges. We look at models from both classes when the asymptotic proportion of vertices with degree at least~$k$ scales like $k^{-2} (\log k)^{2α+ o(1)}$ and show that for preferential attachment networks the typical distance is $\big(\frac{1}{1+α}+o(1)\big)\frac{\log N}{\log\log N}$ in probability as the number~$N$ of vertices goes to infinity. By contrast the typical distance in a rank one model with the same asymptotic degree sequence is $\big(\frac{1}{1+2α}+o(1)\big)\frac{\log N}{\log\log N}.$ As $α\to\infty$ we see the emergence of a factor two between the length of shortest paths as we approach the ultrasmall regime.

math.PR↗

arXiv2016

The largest fragment of a homogeneous fragmentation process

We show that in homogeneous fragmentation processes the largest fragment at time $t$ has size $e^{-t Φ'(\bar{p})}t^{-\frac32 (\log Φ)'(\bar{p})+o(1)},$ where $Φ$ is the Lévy exponent of the fragmentation process, and $\bar{p}$ is the unique solution of the equation $(\log Φ)'(\bar{p})=\frac1{1+\bar{p}}$. We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.

math.PR↗

arXiv2015

The spread of infections on evolving scale-free networks

We study the contact process on a class of evolving scale-free networks, where each node updates its connections at independent random times. We give a rigorous mathematical proof that there is a transition between a phase where for all infection rates the infection survives for a long time, at least exponential in the network size, and a phase where for sufficiently small infection rates extinction occurs quickly, at most like the square root of the network size. The phase transition occurs when the power-law exponent crosses the value four. This behaviour is in contrast to that of the contact process on the corresponding static model, where there is no phase transition, as well as that of a classical mean-field approximation, which has a phase transition at power-law exponent three. The new observation behind our result is that temporal variability of networks can simultaneously increase the rate at which the infection spreads in the network, and decrease the time which the infection spends in metastable states.

math.PR↗

arXiv2015

Condensation and symmetry-breaking in the zero-range process with weak site disorder

Condensation phenomena in particle systems typically occur as one of two distinct types: either as a spontaneous symmetry breaking in a homogeneous system, in which particle interactions enforce condensation in a randomly located site, or as an explicit symmetry breaking in a system with background disorder, in which particles condensate in the site of extremal disorder. In this paper we confirm a recent conjecture by Godrèche and Luck by showing, for a zero range process with weak site disorder, that there exists a phase where condensation occurs with an intermediate type of symmetry-breaking, in which particles condensate in a site randomly chosen from a range of sites favoured by disorder. We show that this type of condensation is characterised by the occurrence of a Gamma distribution in the law of the disorder at the condensation site. We further investigate fluctuations of the condensate size and confirm a phase diagram, again conjectured by Godrèche and Luck, showing the existence of phases with normal and anomalous fluctuations.

math.PR↗

arXiv2015

Spatial preferential attachment networks: Power laws and clustering coefficients

We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent $τ>2$. The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value $τ=3$. Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.

math.PR↗

arXiv2014

Unbiased shifts of Brownian motion

Let $B=(B_t)_{t\in {\mathbb{R}}}$ be a two-sided standard Brownian motion. An unbiased shift of $B$ is a random time $T$, which is a measurable function of $B$, such that $(B_{T+t}-B_T)_{t\in {\mathbb{R}}}$ is a Brownian motion independent of $B_T$. We characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of $B$. For any probability distribution $ν$ on ${\mathbb{R}}$ we construct a stopping time $T\ge0$ with the above properties such that $B_T$ has distribution $ν$. We also study moment and minimality properties of unbiased shifts. A crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing stationary diffuse random measures on ${\mathbb{R}}$. Another new result is an analogue for diffuse random measures on ${\mathbb{R}}$ of the cycle-stationarity characterisation of Palm versions of stationary simple point processes.

math.PR↗

arXiv2013

Vulnerability of robust preferential attachment networks

Scale-free networks with small power law exponent are known to be robust, meaning that their qualitative topological structure cannot be altered by random removal of even a large proportion of nodes. By contrast, it has been argued in the science literature that such networks are highly vulnerable to a targeted attack, and removing a small number of key nodes in the network will dramatically change the topological structure. Here we analyse a class of preferential attachment networks in the robust regime and prove four main results supporting this claim: After removal of an arbitrarily small proportion epsilon>0 of the oldest nodes (1) the asymptotic degree distribution has exponential instead of power law tails; (2) the largest degree in the network drops from being of the order of a power of the network size n to being just logarithmic in n; (3) the typical distances in the network increase from order log log n to order log n; and (4) the network becomes vulnerable to random removal of nodes. Importantly, all our results explicitly quantify the dependence on the proportion epsilon of removed vertices. For example, we show that the critical proportion of nodes that have to be retained for survival of the giant component undergoes a steep increase as epsilon moves away from zero, and a comparison of this result with similar ones for other networks reveals the existence of two different universality classes of robust network models. The key technique in our proofs is a local approximation of the network by a branching random walk with two killing boundaries, and an understanding of the particle genealogies in this process, which enters into estimates for the spectral radius of an associated operator.

math.PR↗

arXiv2013

Random networks with sublinear preferential attachment: The giant component

We study a dynamical random network model in which at every construction step a new vertex is introduced and attached to every existing vertex independently with a probability proportional to a concave function f of its current degree. We give a criterion for the existence of a giant component, which is both necessary and sufficient, and which becomes explicit when f is linear. Otherwise it allows the derivation of explicit necessary and sufficient conditions, which are often fairly close. We give an explicit criterion to decide whether the giant component is robust under random removal of edges. We also determine asymptotically the size of the giant component and the empirical distribution of component sizes in terms of the survival probability and size distribution of a multitype branching random walk associated with f.

math.PR↗

arXiv2012

Emergence of condensation in Kingman's model of selection and mutation

We describe the onset of condensation in the simple model for the balance between selection and mutation given by Kingman in terms of a scaling limit theorem. Loosely speaking, this shows that the wave moving towards genes of maximal fitness has the shape of a gamma distribution. We conjecture that this wave shape is a universal phenomenon that can also be found in a variety of more complex models, well beyond the genetics context, and provide some further evidence for this.

math.PR↗

arXiv2012

On the most visited sites of planar Brownian motion

Let (B_t : t > 0) be a planar Brownian motion and define gauge functions $ϕ_α(s)=log(1/s)^{-α}$ for $α>0$. If $α<1$ we show that almost surely there exists a point x in the plane such that $H^{ϕ_α}({t > 0 : B_t=x})>0$, but if $α>1$ almost surely $H^{ϕ_α} ({t > 0 : B_t=x})=0$ simultaneously for all $x\in R^2$. This resolves a longstanding open problem posed by S.,J. Taylor in 1986.

math.PR↗

arXiv2011

Typical distances in ultrasmall random networks

We show that in preferential attachment models with power-law exponent $τ\in(2,3)$ the distance between randomly chosen vertices in the giant component is asymptotically equal to $(4+o(1))\, \frac{\log\log N}{-\log (τ-2)}$, where $N$ denotes the number of nodes. This is twice the value obtained for several types of configuration models with the same power-law exponent. The extra factor reveals the different structure of typical shortest paths in preferential attachment graphs.

math.PR↗