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The phase transition in inhomogeneous random graphs
A very general model of an inhomogeneous random graph with (conditional) independence between the edges is introduced, which scales so that the number of edges is linear in thenumber of vertices.
The Diameter of a Scale-Free Random Graph
We consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number m of earlier vertices, where each earlier vertex is chosen with probability
The degree sequence of a scale‐free random graph process
Here the authors obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3.9±0.1 is obtained.
The phase transition in inhomogeneous random graphs
The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the
Directed scale-free graphs
A model for directed scale-free graphs that grow with preferential attachment depending in a natural way on the in- and out-degrees is introduced, reproducing observed properties of the worldwide web.
Mathematical results on scale‐free random graphs
There has been much interest in studying large-scale real-world networks and attempting to model their properties using random graphs, and the work in this field falls very roughly into the following categories.
The critical probability for random Voronoi percolation in the plane is 1/2
We study percolation in the following random environment: let Z be a Poisson process of constant intensity on ℝ2, and form the Voronoi tessellation of ℝ2 with respect to Z. Colour each Voronoi cell
A polynomial of graphs on surfaces
ribbon graphs, i.e., graphs realized as disks (vertices) joined together by strips (edges) glued to their boundaries, corresponding to neighbourhoods of graphs embedded into surfaces. We construct a
Spanning Subgraphs of Random Graphs
  • O. Riordan
  • Mathematics
    Combinatorics, Probability and Computing
  • 1 March 2000
A question of Bollobás is answered by showing that, as d → ∞, Gp almost surely has a spanning subgraph isomorphic to H, which implies that the number of d-cubes in G ∈ [Gscr ](n, M) is asymptotically normally distributed for M in a certain range.
Metrics for sparse graphs
This paper deals mainly with graphs with $o(n^2)$ but $\omega(n)$ edges: a companion paper [arXiv:0812.2656] will discuss the (more problematic still) case of {\em extremely sparse} graphs, with O( n) edges.