Random graphs with arbitrary degree distributions and their applications.

@article{Newman2000RandomGW,
  title={Random graphs with arbitrary degree distributions and their applications.},
  author={Mark E. J. Newman and Steven H. Strogatz and Duncan J. Watts},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2000},
  volume={64 2 Pt 2},
  pages={
          026118
        }
}
Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact… 

Approximating sparse graphs: The random overlapping communities model

A simple, easy to sample, random graph model that captures the limiting spectra of many sequences of interest, including the sequence of hypercube graphs, and it is shown that the model is an effective approximation for individual graphs.

Directed Random Graphs with given Degree Distributions

An algorithm to construct in- and out-degree sequences from samples of i.i.d. observations from F and G that with high probability will be graphical, that is, from which a simple directed graph can be drawn is proposed.

The average distances in random graphs with given expected degrees

  • F. ChungLinyuan Lu
  • Mathematics, Computer Science
    Proceedings of the National Academy of Sciences of the United States of America
  • 2002
It is shown that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n/log d́, where d̃ is the weighted average of the sum of squares of the expected degrees.

Localization and Universality of Eigenvectors in Directed Random Graphs.

A general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links is presented and it is shown that in the high connectivity limit the distribution of theright eigen vector components is solely determined by the degree distribution.

Properties of dense partially random graphs.

  • S. Risau-Gusman
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2004
This work studies the properties of random graphs where for each vertex a neighborhood has been previously defined and obtains analytically the distribution of eigenvalues of the corresponding adjacency matrices and applies the results to the calculation of the mixing rate and synchronizability threshold.

Distances in power-law random graphs

This work will study two random graph models, the configuration model and the preferential attachment model, which will have power-law degree sequences when the number of vertices tends to infinity, and some new results will be presented.

The Average Distance in a Random Graph with Given Expected Degrees

It is shown that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n/ logd̃ where d̃ is the weighted average of the sum of squares of the expected degrees.

Connected Graphs with a Given Degree Sequence: Efficient Sampling, Correlations, Community Detection and Robustness

It is argued that keeping graphs in one connected piece, or component, is key for many applications where complex graphs are assumed to be connected either by definition or by construction.

Techniques for analyzing dynamic random graph models of web-like networks: An overview

The focus has primarily been on dynamic random graph models that attempt to account for the observed statistical properties of web-like networks through certain dynamic processes guided by simple stochastic rules.

Random graphs with arbitrary clustering and their applications.

This paper extends the analytical solutions to the percolation properties of random networks with homogeneous clustering to investigate networks that contain clusters whose nodes are not degree equivalent, including multilayer networks.
...

References

SHOWING 1-10 OF 66 REFERENCES

A random graph model for massive graphs

A random graph model is proposed which is a special case of sparse random graphs with given degree sequences which involves only a small number of parameters, called logsize and log-log growth rate, which capture some universal characteristics of massive graphs.

The Size of the Giant Component of a Random Graph with a Given Degree Sequence

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.

The Web as a Graph: Measurements, Models, and Methods

This paper describes two algorithms that operate on the Web graph, addressing problems from Web search and automatic community discovery, and proposes a new family of random graph models that point to a rich new sub-field of the study of random graphs, and raises questions about the analysis of graph algorithms on the Internet.

A Critical Point for Random Graphs with a Given Degree Sequence

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.

A portion of the well-known collaboration graph

An on-going project in which a portion of this graph—in particular, a list of all people with small Erdős numbers—is made available in electronic form is reported on.

ON THE STRENGTH OF CONNECTEDNESS OF A RANDOM GRAPH

If G is an arbitrary non-complete graph, let cp(G) denote the least number к such that by deleting к appropriately chosen vertices from G (i. e. deleting the к points in question and all edges

On the strength of connectedness of a random graph

If G is an arbitrary non-complete graph, let cZ1(G) denote the least number k such that by deleting k appropriately chosen vertices from G (i. e. deleting the k points in question and all edges

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

Exact solution of site and bond percolation on small-world networks.

  • C. MooreM. Newman
  • Mathematics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 2000
An exact solution of the model for both site and bond percolation is given, including the position of thePercolation transition at which epidemic behavior sets in, the values of the critical exponents governing this transition, the mean and variance of the distribution of cluster sizes below the transition, and the size of the giant component (epidemic) above the transition.

Emergence of scaling in random networks

A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
...