• Corpus ID: 119324444

An Independent Process Approximation to Sparse Random Graphs with a Prescribed Number of Edges and Triangles

@article{Desalvo2015AnIP,
  title={An Independent Process Approximation to Sparse Random Graphs with a Prescribed Number of Edges and Triangles},
  author={Stephen Desalvo and Michaela Puck Rombach},
  journal={arXiv: Probability},
  year={2015}
}
We prove a $pre$-$asymptotic$ bound on the total variation distance between the uniform distribution over two types of undirected graphs with $n$ nodes. One distribution places a prescribed number of $k_T$ triangles and $k_S$ edges not involved in a triangle independently and uniformly over all possibilities, and the other is the uniform distribution over simple graphs with exactly $k_T$ triangles and $k_S$ edges not involved in a triangle. As a corollary, for $k_S = o(n)$ and $k_T = o(n)$ as… 

Figures from this paper

References

SHOWING 1-10 OF 25 REFERENCES
On the absolute constants in the Berry-Esseen-type inequalities
By a modification of the method that was applied in (Korolev and Shevtsova, 2010), here the inequalities $\Delta_n\leq0.3328(\beta_3+0.429)/\sqrt{n}$ and $\Delta_n\leq0.33554(\beta_3+0.415)/\sqrt{n}$
A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs
Tuning clustering in random networks with arbitrary degree distributions.
TLDR
A generator of random networks where both the degree-dependent clustering coefficient and the degree distribution are tunable and an universal relation among clustering and degree-degree correlations for all networks is unveiled.
Poisson Approximation for Dependent Trials
by a Poisson distribution and a derivation of a bound on the distance between the distribution of W and the Poisson distribution with mean E(W ). This new method is based on previous work by C. Stein
Degree and clustering coefficient in sparse random intersection graphs
We establish asymptotic vertex degree distribution and examine its relation to the clustering coecient in two popular random intersection graph models of Godehardt and Jaworski (2001). For sparse
Random graphs with clustering.
  • M. Newman
  • Computer Science
    Physical review letters
  • 2009
TLDR
It is shown how standard random-graph models can be generalized to incorporate clustering and give exact solutions for various properties of the resulting networks, including sizes of network components, size of the giant component if there is one, position of the phase transition at which the giant components forms, and position ofThe phase transition for percolation on the network.
Mathematical results on scale‐free random graphs
TLDR
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.
Random geometric graphs.
TLDR
An analytical expression for the cluster coefficient is derived, which shows that the graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality are distinctly different from standard random graphs, even for infinite dimensionality.
Two moments su ce for Poisson approx-imations: the Chen-Stein method
Convergence to the Poisson distribution, for the number of occurrences of dependent events, can often be established by computing only first and second moments, but not higher ones. This remarkable
Highly clustered scale-free networks.
  • K. Klemm, V. Eguíluz
  • Computer Science
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2002
TLDR
The model shows stylized features of real-world networks: power-law distribution of degree, linear preferential attachment of new links, and a negative correlation between the age of a node and its link attachment rate.
...
1
2
3
...