The degree-restricted random process is far from uniform
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866908484545544192 |
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| author | Molloy, Michael Surya, Erlang Warnke, Lutz |
| author_facet | Molloy, Michael Surya, Erlang Warnke, Lutz |
| contents | The degree-restricted random process is a natural algorithmic model for generating graphs with degree sequence D_n=(d_1, \ldots, d_n): starting with an empty n-vertex graph, it sequentially adds new random edges so that the degree of each vertex v_i remains at most d_i. Wormald conjectured in 1999 that, for d-regular degree sequences D_n, the final graph of this process is similar to a uniform random d-regular graph.
In this paper we show that, for degree sequences D_n that are not nearly regular, the final graph of the degree-restricted random process differs substantially from a uniform random graph with degree sequence D_n. The combinatorial proof technique is our main conceptual contribution: we adapt the switching method to the degree-restricted process, demonstrating that this enumeration technique can also be used to analyze stochastic processes (rather than just uniform random models, as before). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_00835 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The degree-restricted random process is far from uniform Molloy, Michael Surya, Erlang Warnke, Lutz Combinatorics Discrete Mathematics Probability 05C80, 60C05, 68W20, 60G99, 90B15 The degree-restricted random process is a natural algorithmic model for generating graphs with degree sequence D_n=(d_1, \ldots, d_n): starting with an empty n-vertex graph, it sequentially adds new random edges so that the degree of each vertex v_i remains at most d_i. Wormald conjectured in 1999 that, for d-regular degree sequences D_n, the final graph of this process is similar to a uniform random d-regular graph. In this paper we show that, for degree sequences D_n that are not nearly regular, the final graph of the degree-restricted random process differs substantially from a uniform random graph with degree sequence D_n. The combinatorial proof technique is our main conceptual contribution: we adapt the switching method to the degree-restricted process, demonstrating that this enumeration technique can also be used to analyze stochastic processes (rather than just uniform random models, as before). |
| title | The degree-restricted random process is far from uniform |
| topic | Combinatorics Discrete Mathematics Probability 05C80, 60C05, 68W20, 60G99, 90B15 |
| url | https://arxiv.org/abs/2211.00835 |