Lionel Tabourier
Abstract
The generation of random graphs using edge swaps provides a reliable method to draw uniformly random samples of sets of graphs respecting some simple constraints (e.g., degree distributions). However, in general, it is not necessarily possible to access all graphs obeying some given constraints through a classical switching procedure calling on pairs of edges. Therefore, we propose to get around this issue by generalizing this classical approach through the use of higher-order edge switches. This method, which we denote by “k-edge switching,” makes it possible to progressively improve the covered portion of a set of constrained graphs, thereby providing an increasing, asymptotically certain confidence on the statistical representativeness of the obtained sample.
- Albert, R., Jeong, H., and Barabasi, A.-L. 1999. Diameter of the world wide web. Nature 401, 130--131.Google Scholar
- Artzy-Randrup, Y. and Stone, L. 2005. Generating uniformly distributed random networks. Phys. Rev. E 72, 5, 056708.Google ScholarCross Ref
- Bansal, S., Khandelwal, S., and Meyers, L. 2008. Evolving Clustered Random Networks. Arxiv preprint cs.DM/0808.0509.Google Scholar
- Bender, E. and Canfield, E. 1978. The asymptotic number of labeled graphs with given degree sequences. J. Combin. Theory Ser. A 24, 3, 296--307.Google ScholarCross Ref
- Colbourn, C. 1977. Graph Generation. University of Waterloo, Waterloo, Ontario.Google Scholar
- Coolen, A., De Martino, A., and Annibale, A. 2009. Constrained Markovian dynamics of random graphs. J. Stat. Phys. 136, 6, 1035--1067.Google ScholarCross Ref
- Cooper, C., Dyer, M., and Greenhill, C. 2006. Sampling regular graphs and a peer-to-peer network. Comb. Probab. Comput. 16, 04, 557--593. Google ScholarDigital Library
- Eggleton, R. 1973. Graphic sequences and graphic polynomials: A report. In Infinite and Finite Sets 1, 385--392.Google Scholar
- Feder, T., Guetz, A., Mihail, M., and Saberi, A. 2006. A local switch Markov chain on given degree graphs with application in connectivity of peer-to-peer networks. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06). IEEE, Los Alamitos, CA, 69--76. Google ScholarDigital Library
- Gkantsidis, C., Mihail, M., and Zegura, E. 2003. The markov chain simulation method for generating connected power law random graphs. In Proceedings of the 5th Workshop on Algorithm Engineering and Experiments (ALENEX). SIAM, Philadelphia.Google Scholar
- Guillaume, J., Latapy, M., and Le-Blond, S. 2005. Statistical analysis of a P2P query graph based on degrees and their time-evolution. In Proceedings of the 6th International Workshop on Distributed Computing (IWDC'04). Springer, Berlin, 439--465. Google ScholarDigital Library
- Guruswami, V. 2000. Rapidly mixing markov chains: A comparison of techniques. MIT Laboratory for Computer Science. cs.washington.edu/homes/venkat/pubs/papers.html.Google Scholar
- Kannan, R., Tetali, P., and Vempala, S. 1997. Simple Markov-chain algorithms for generating bipartite graphs and tournaments. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete algorithms. SIAM, Philadelphia, 193--200. Google ScholarDigital Library
- Mahadevan, P., Krioukov, D., Fall, K., and Vahdat, A. 2006. Systematic topology analysis and generation using degree correlations. In Proceedings of the Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications (SIGCOMM'06). ACM, New York. Google ScholarDigital Library
- Miklós, I. and Podani, J. 2004. Randomization of presence-absence matrices: comments and new algorithms. Ecology Archives 85, 1, 86--92. Appendix A. http://esapubs.org/archive/ecol/E085/001/appendix-A.htm.Google ScholarCross Ref
- Milo, R., Kashtan, N., Itzkovitz, S., Newman, M., and Alon, U. 2003. On the uniform generation of random graphs with prescribed degree sequences. Arxiv preprint cond-mat/0312.028.Google Scholar
- Newman, M. 2004. Coauthorship networks and patterns of scientific collaboration. In Proc. Nat. Acad. Sciences 101, Suppl 1, 5200.Google Scholar
- Ralaivola, L., Swamidass, S., Saigo, H., and Baldi, P. 2005. Graph kernels for chemical informatics. Neural Netw. 18, 8, 1093--1110. Google ScholarDigital Library
- Rao, A., Jana, R., and Bandyopadhyay, S. 1996. A Markov chain Monte Carlo method for generating random (0, 1)-matrices with given marginals. Sankhy&abar; Indian J. Statistics, Series A, 225--242.Google Scholar
- Roberts, J. 2000. Simple methods for simulating sociomatrices with given marginal totals. Social Netw. 22, 3, 273--283.Google ScholarCross Ref
- Sinclair, A. 1993. Algorithms for Random Generation and Counting: A Markov Chain Approach. Springer, Berlin. Google ScholarDigital Library
- Stauffer, A. and Barbosa, V. 2005. A study of the edge-switching Markov-chain method for the generation of random graphs. Arxiv preprint cs.DM/0512.105.Google Scholar
- Taylor, R. 1980. Constrained switchings in graphs. Comb. Math. 8, 314--336.Google Scholar
- Taylor, R. 1982. Switchings constrained to 2-connectivity in simple graphs. SIAM J. Algebraic Discrete Meth. 3, 114.Google ScholarCross Ref
- Viger, F. and Latapy, M. 2005. Efficient and simple generation of random simple connected graphs with prescribed degree sequence. In Proceedings of the 11th International Computing and Combinatorics Conference. Springer-Verlag, Berlin, 440. Google ScholarDigital Library
Index Terms
- Generating constrained random graphs using multiple edge switches
Recommendations
Equitable coloring of random graphs
An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most one. The least positive integer k for which there exists an equitable coloring of a graph G with k colors is said to be the ...
Maximum edge-cuts in cubic graphs with large girth and in random cubic graphs
We show that for every cubic graph Gwith sufficiently large girth there exists a probability distribution on edge-cuts in Gsuch that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that Gcontains an edge-cut of size ...
Anti-Ramsey properties of random graphs
We call a coloring of the edge set of a graph G a b-bounded coloring if no color is used more than b times. We say that a subset of the edges of G is rainbow if each edge is of a different color. A graph has property A(b,H) if every b-bounded coloring ...
Comments