We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

We study the performance of discrete-time consensus protocols in the presence of additive noise. When the consensus dynamic corresponds to a reversible Markov chain, we give an exact expression for a weighted version of steady-state disagreement in terms of the stationary distribution and hitting times in an underlying graph. We then show how this result can be used to characterize the noise robustness of a class of protocols for formation control in terms of the Kemeny constant of an underlying graph.

Scaling Laws for Consensus Protocols Subject to Noise

The National Coastal Shelterbelt System Construction Project (NCSSCP) was proposed to increase the afforestation area and neutralize the impact of urbanization, especially in the southeast coastal sub-region of China. In this study, we identified the spatio-temporal evolution characteristics and predicted the land use and land cover changes (LUCC) associated with this project by modeling scenarios, seeking to explore the path of sustainable development. The spatial structure was analyzed using the landscape pattern index approach and the land use transfer matrix. By coupling the Markov model and patch-generating a land-use simulation model (PLUS), different scenarios were analyzed to predict the quantity and spatial changes. According to the results, based on the current trends and due to the impact of urbanization, the forest area was predicted to decrease by 633.19 km2, whilst appearing more spatially fragmented and separated. However, with the completion of the NCSSCP target, the forest area was predicted to increase by 1666.12 km2, and the spatial structure would appear more cohesive and concentrated. From an overall perspective, the afforestation target of NCSSCP will not be completed under the present trend. It is difficult for the afforestation speed of the NCSSCP to keep up with the speed of urbanization. Therefore, giving consideration to both the afforestation speed and quality and reducing the speed of urbanization to balance the economy and ecology would be beneficial in terms of the realization of the aims of sustainable development.

/pdf/spatio-temporal-dynamic-of-the-land-use-cover-change-and-1rneklq0.pdf

Spatio-Temporal Dynamic of the Land Use/Cover Change and Scenario Simulation in the Southeast Coastal Shelterbelt System Construction Project Region of China

We consider the allocation of m balls (jobs) into n bins (servers). In the standard Two-Choice process, at each step t = 1, 2, . . . ,m we first sample two randomly chosen bins, compare their two loads and then place a ball in the least loaded bin. It is well-known that for any ≥ n, this results in a gap (difference between the maximum and average load) of log2 logn + Θ(1) (with high probability).

Balanced Allocations with the Choice of Noise

In this tutorial, we provide an extensive survey of the work on dispersion of mobile robots, introduced by Augustine and Moses Jr. [ICDCN 2018]. The problem of dispersion of k robots, initially arbitrarily placed on the nodes of an n node graph, requires the robots to autonomously move around to reach a configuration such that there are at most ⌈k/n⌉ robots on each node. Typically, the metrics used to gauge solutions to this problem are the time until dispersion is achieved and the memory requirement per robot. Although this problem was introduced recently, much work has been done on it in the setting it was originally introduced in, as well as extensions to new settings. We will provide an overview of the techniques and results until now as well as possible future work. Our presentation will be in two parts. In the first part, we will present the foundations of the dispersion problem and some fundamental results. In the second part, we discuss various extensions to different settings and recent developments.

Dispersion of Mobile Robots

We study two random processes on an n-vertex graph inspired by the internal diffusion limited aggregation (IDLA) model. These processes can also be regarded as protocols for allocating jobs in a distributed network of servers. In both processes n particles start from an arbitrary but fixed origin. Each particle performs a simple random walk until it first encounters an unoccupied vertex, at which point the vertex becomes occupied and the random walk terminates. In one of the processes, called Sequential-IDLA, a single particle moves until settling and only then does the next particle start whereas in the second process, called Parallel-IDLA, all unsettled particles move simultaneously. The second process is akin to running the first in parallel. Our main goal is to analyze the so-called dispersion time of these processes, which is the maximum number of steps performed by any of the n particles. In order to compare the two processes, we develop a coupling which shows the dispersion time of the Parallel-IDLA stochastically dominates that of the Sequential-IDLA; however, the total number of steps performed by all particles has the same distribution in both processes. This coupling also gives us that dispersion time of Parallel-IDLA is bounded in expectation by dispersion time of the Sequential-IDLA up to a multiplicative log n factor. Moreover, we derive asymptotic upper and lower bound on the dispersion time for several graph classes, such as cliques, cycles, binary trees, d-dimensional grids, hypercubes and expanders. Most of our bounds are tight up to a multiplicative constant.

pdf/the-dispersion-time-of-random-walks-on-finite-graphs-3j7ktsi9zj.pdf

The Dispersion Time of Random Walks on Finite Graphs

We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number $n$ of vertices on discrete tori and bounded degree trees, of order $\mathcal{O}(n \log \log n)$ on bounded degree expanders, and of order $\mathcal{O}(n (\log \log n)^2)$ on the Erdős-Renyi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy, and prove a dichotomy in efficiency between computing strategies for hitting and cover times.

The Power of Two Choices for Random Walks

We study the biased random walk where at each step of a random walk a "controller" can, with a certain small probability, fix the next step. This model was introduced by Azar et al. [STOC1992]; we extend their work to the time dependent setting and consider cover times of this walk. We obtain new bounds on the cover and hitting times and make progress towards resolving a conjecture of Azar et al. on maximising values of the stationary distribution. We also consider the problem of computing an optimal strategy for the controller to minimise the cover time and show that for directed graphs determining the cover time is PSPACE-complete.

Time Dependent Biased Random Walks

Twin-width is a graph width parameter recently introduced by Bonnet, Kim,
Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product
$\star$, we address the question: is the twin-width of $G\star H$ bounded by a
function of the twin-widths of $G$ and $H$ and their maximum degrees? It is
known that a bound of this type holds for strong products (Bonnet, Geniet, Kim,
Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold
for Cartesian, tensor/direct, corona, rooted, replacement, and zig-zag
products. For the lexicographical product it is known that the twin-width of
the product of two graphs is exactly the maximum of the twin-widths of the
individual graphs (Bonnet, Kim, Reinald, Thomass\'{e} & Watrigant; IPEC 2021).
In contrast, for the modular product we show that no bound can hold. In
addition, we provide examples showing many of our bounds are tight, and give
improved bounds for certain classes of graphs.

/pdf/bounds-on-the-twin-width-of-product-graphs-vqfinqx3.pdf

Bounds on the Twin-Width of Product Graphs

Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for $\textit{worst-case}$ start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker, and Tuttle in 2008) remains an open problem. 
First, we improve and tighten various bounds on the $\textit{stationary}$ cover time when $k$ random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of $\Omega((n/k) \log n)$ on the stationary cover time, holding for any $n$-vertex graph $G$ and any $1 \leq k =o(n\log n )$. Secondly, we establish the $\textit{stationary}$ cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising $\textit{worst-case}$ cover times in terms of $\textit{stationary}$ cover times and a novel, relaxed notion of mixing time for multiple walks called the $\textit{partial mixing time}$. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the $\textit{worst-case}$ cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.

pdf/multiple-random-walks-on-graphs-mixing-few-to-cover-many-2edzji16tz.pdf

Multiple Random Walks on Graphs: Mixing Few to Cover Many

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