Random geometric graphs.

@article{Dall2002RandomGG,
  title={Random geometric graphs.},
  author={Jesper Dall and Michael Christensen},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2002},
  volume={66 1 Pt 2},
  pages={
          016121
        }
}
We analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size of the largest cluster. We derive an analytical expression for the cluster coefficient, which shows that the graphs are distinctly different from standard random graphs, even for infinite dimensionality. Insights relevant for graph bipartitioning are included… 
Degree Correlations in Random Geometric Graphs
  • A. Antonioni, M. Tomassini
  • Mathematics, Computer Science
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2012
TLDR
New results for the two-point degree correlation function in terms of the clustering coefficient of the graphs for two-dimensional space in particular, with extensions to arbitrary finite dimensions are presented.
Topological properties of the one dimensional exponential random geometric graph
TLDR
It is shown that the asymptotic properties of a graph with a truncated exponential distribution can be obtained using the exponential random geometric graph.
Hamiltonicity of graphs perturbed by a random geometric graph
We study Hamiltonicity in graphs obtained as the union of a deterministic n-vertex graph H with linear degrees and a d-dimensional random geometric graph Gd(n, r), for any d ≥ 1. We obtain an
Joint large deviation result for empirical measures of the coloured random geometric graphs
We prove joint large deviation principle for the empirical pair measure and empirical locality measure of the near intermediate coloured random geometric graph models on n points picked uniformly in
Submitted to the Annals of Applied Probability HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS By
We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose. We also show
Scale-invariant geometric random graphs.
TLDR
A class of growing geometric random graphs that are invariant under rescaling of space and time are introduced and analysis reveals a dichotomy between scale-free and Poisson distributions of in- and out-degree.
A Geometric Preferential Attachment Model of Networks II
We study a random graph G n that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power law degree distribution where the
Connectivity of inhomogeneous random graphs
TLDR
The results generalize the classical result for G(n, p), when p=clogn/n and show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined.
Percolation, Connectivity, Coverage and Colouring of Random Geometric Graphs
In this review paper, we shall discuss some recent results concerning several models of random geometric graphs, including the Gilbert disc model G r , the k-nearest neighbour model G k nn and the
...
...

References

SHOWING 1-10 OF 60 REFERENCES
A Critical Point for Random Graphs with a Given Degree Sequence
TLDR
It is shown that if Σ i(i - 2)λi > 0, then such graphs almost surely have a giant component, while if λ0, λ1… which sum to 1, then almost surely all components in such graphs are small.
Are randomly grown graphs really random?
TLDR
It is concluded that grown graphs, however randomly they are constructed, are fundamentally different from their static random graph counterparts.
Extremal Optimization of Graph Partitioning at the Percolation Threshold
The benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem. The performance of this new method, called extremal
Percolation properties of random ellipses.
  • Xia, Thorpe
  • Mathematics
    Physical review. A, General physics
  • 1988
TLDR
The percolation concentration for identical ellipses with aspect ratio b/a in a two-dimensional plane where both the centers and orientations are random is computed and an interpolation formula appears superior to all these effective-medium theories.
The Size of the Giant Component of a Random Graph with a Given Degree Sequence
TLDR
The size of the giant component in the former case, and the structure of the graph formed by deleting that component is analyzed, which is basically that of a random graph with n′=n−∣C∣ vertices, and with λ′in′ of them of degree i.
On the evolution of random graphs
(n) k edges have equal probabilities to be chosen as the next one . We shall 2 study the "evolution" of such a random graph if N is increased . In this investigation we endeavour to find what is the
Systematic derivation of percolation thresholds in continuum systems.
  • Alon, Drory, Balberg
  • Physics
    Physical review. A, Atomic, molecular, and optical physics
  • 1990
TLDR
Analytic expressions for the critical percolation density in the continuum are derived using the direct-connectedness expansion method and the results obtained are in excellent agreement with Monte Carlo simulations.
Small worlds
TLDR
This paper considers some particular instances of small world models, and rigorously investigates the distribution of their inter‐point network distances, framed in terms of approximations, whose accuracy increases with the size of the network.
Scaling and percolation in the small-world network model.
  • M. Newman, D. Watts
  • Mathematics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1999
TLDR
There is one nontrivial length-scale in the small-world network model of Watts and Strogatz, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit.
Statistical mechanics of complex networks
TLDR
A simple model based on these two principles was able to reproduce the power-law degree distribution of real networks, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network.
...
...