The jamming constant of uniform random graphs

Bermolen, Paola; Jonckheere, Matthieu; Moyal, Pascal

doi:10.1016/j.spa.2016.10.005

Cited by 19 publications

(32 citation statements)

References 23 publications

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“…In contrast to the Euclidean space, rsa on the crg(c, α d ) model is analytically solvable, even at later times when the filled space becomes more dense (large c). To do so, we will extend the mean-field techniques recently developed for analyzing rsa on random graph models [4,6,13,32]. The main goal of these works was to find greedy independent sets (or colorings) of large random networks.…”

Section: Clustered Random Graphsmentioning

confidence: 99%

Corrected Mean-Field Model for Random Sequential Adsorption on Random Geometric Graphs

2018

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A notorious problem in mathematics and physics is to create a solvable model for random sequential adsorption of non-overlapping congruent spheres in the d-dimensional Euclidean space with d 2. Spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres. Due to spatial correlations, characterizing the fraction of accepted spheres remains largely intractable. We study this fraction by taking a novel approach that compares random sequential adsorption in Euclidean space to the nearest-neighbor blocking on a sequence of clustered random graphs. This random network model can be thought of as a corrected mean-field model for the interaction graph between the attempted spheres. Using functional limit theorems, we characterize the fraction of accepted spheres and its fluctuations.

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Section: Clustered Random Graphsmentioning

confidence: 99%

Corrected Mean-Field Model for Random Sequential Adsorption on Random Geometric Graphs

2018

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“…Assumption (2.1) can however be considered a reasonable approximation for many systems, and it is for example satisfied by ER random graphs. We refer the reader to [2] for a study of scaling limits in infinite dimension that applies to a larger class of problems.…”

Section: Fluid Limit and Diffusion Approximation For Homogeneous Graphsmentioning

confidence: 99%

Scaling Limits and Generic Bounds for Exploration Processes

2017

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We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become explored. Given an initial number of vertices $N$ growing to infinity, we study statistical properties of the proportion of explored nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the \emph{jamming constant}, through a diffusion approximation for the exploration process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e. random sequential adsorption. As opposed to homogeneous random graphs, these do not allow for a reduction in dimensionality. Instead we build on a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and obtain generic bounds: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, we give two trajectorial interpretations of our bounds by constructing two coupled processes that have the same fluid limits.Comment: 25 pages, 3 figure

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“…The problem of finding the size of a greedy independent set in a more general random graph with given degrees was first studied in a recent preprint by Bermolen, Jonckheere and Moyal . Their approach is superficially similar to ours, in that they construct the graph and the independent set simultaneously, but their analysis is significantly more complicated.…”

Section: Introductionmentioning

confidence: 95%

“…There is often also a discrete version, typically taking place on a regular lattice, where an object arrives and selects a location on the lattice uniformly at random, and then inhibits later objects from occupying neighbouring points. Bermolen, Jonckheere and Moyal and Finch , §5.3] also list a number of other application areas, e.g., to linguistics, sociology and computer science, again with further references.…”

Section: Introductionmentioning

confidence: 99%

The Greedy Independent Set in a Random Graph with Given Degrees

Brightwell

Janson

Luczak

2017

Random Struct Algorithms

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Abstract. We analyse the size of an independent set in a random graph on n vertices with specified vertex degrees, constructed via a simple greedy algorithm: order the vertices arbitrarily, and, for each vertex in turn, place it in the independent set unless it is adjacent to some vertex already chosen. We find the limit of the expected proportion of vertices in the greedy independent set as n → ∞, expressed as an integral whose upper limit is defined implicitly, valid whenever the second moment of a random vertex degree is uniformly bounded. We further show that the random proportion of vertices in the independent set converges to the jamming constant as n → ∞. The results hold under weaker assumptions in a random multigraph with given degrees constructed via the configuration model.

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The jamming constant of uniform random graphs

Cited by 19 publications

References 23 publications

Corrected Mean-Field Model for Random Sequential Adsorption on Random Geometric Graphs

Corrected Mean-Field Model for Random Sequential Adsorption on Random Geometric Graphs

Scaling Limits and Generic Bounds for Exploration Processes

The Greedy Independent Set in a Random Graph with Given Degrees

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