John Lapinskas

Leslie Ann Goldberg, John Lapinskas

In this paper we prove Aldous's conjecture from 1987 that there is no backoff protocol that is stable for any positive arrival rate. The setting is a communication channel for coordinating requests for a shared resource. Each user who wants to access the resource makes a request by sending a message to the channel. The users don't have any way to communicate with each other, except by sending messages to the channel. The operation of the channel proceeds in discrete time steps. If exactly one message is sent to the channel during a time step then this message succeeds (and leaves the system). If multiple messages are sent during a time step then these messages collide. Each of the users that sent these messages therefore waits a random amount of time before re-sending. A backoff protocol is a randomised algorithm for determining how long to wait -- the waiting time is a function of how many collisions a message has had. Specifically, a backoff protocol is described by a send sequence $\overline{p} = (p_0,p_1,p_2,\ldots)$. If a message has had $k$ collisions before a time step then, with probability $p_k$, it sends during that time step, whereas with probability $1-p_k$ it is silent (waiting for later). The most famous backoff protocol is binary exponential backoff, where $p_k = 2^{-k}$. Under Kelly's model, in which the number of new messages that arrive in the system at each time step is given by a Poisson random variable with mean $λ$, Aldous proved that binary exponential backoff is unstable for any positive $λ$. He conjectured that the same is true for any backoff protocol. We prove this conjecture.

Zylan Benjert, Júlia Komjáthy, Johannes Lengler et al.

It is a fundamental question in epidemiology to estimate, model and predict the growth rate of a pandemic. Analogously, analysing the diffusion of innovation, (fake) news, memes, and rumours is of key importance in the social sciences. The resulting epidemic growth curves can be classified according to their growth rates. These have been found to range from exponential to both faster super-exponential curves and slower subexponential or polynomial curves. Previous research has lacked a unified explanatory framework capable of accommodating super-exponential, (stretched) exponential, and polynomial growth patterns within the same contact network. In this paper we propose a simple agent-based network model that can capture all these phases. We provide such a framework by modelling how transmission rates depend on spatial distance and on individuals' numbers of contacts. By comparing the growth rate of spreading processes with or without degree-dependent and/or distance-dependent contact rates through data-driven and synthetic simulations on real and modelled networks with underlying geometry, we find evidence that even a 'sublinear presence' of these causes may cause a significant slow down of the growth rate on the same underlying network. We find that the growth rate is governed by a combination of three factors: geometry, the prevalence of weak ties, and superspreaders. We confirm our results with rigorous proofs in a theoretical model, using a spatial multiscale-argument in long-range heterogeneous first passage percolation. Our results give a plausible explanation of why the consecutive waves of a single pandemic can differ in their growth even if their spreading mechanisms are similar.