The scaling limit of a critical random directed graph
We consider the random directed graph ⃗G(n,p) with vertex set {1,2,...,n} in which each of the n(n − 1) possible directed edges is present indepen- dently with probability p. We are interested in the strongly connected com- ponents of this directed graph. A phase transition for the emergence of a gi...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Institute of Mathematical Statistics
2023
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| _version_ | 1826309865065152512 |
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| author | Goldschmidt, C Stephenson, R |
| author_facet | Goldschmidt, C Stephenson, R |
| author_sort | Goldschmidt, C |
| collection | OXFORD |
| description | We consider the random directed graph ⃗G(n,p) with vertex set {1,2,...,n}
in which each of the n(n − 1) possible directed edges is present indepen-
dently with probability p. We are interested in the strongly connected com-
ponents of this directed graph. A phase transition for the emergence of a
giant strongly connected component is known to occur at p = 1/n, with
critical window p = 1/n + λn−4/3 for λ ∈ R. We show that, within this
critical window, the strongly connected components of ⃗G(n,p), ranked in
decreasing order of size and rescaled by n−1/3, converge in distribution to a
sequence (C1,C2,...) of finite strongly connected directed multigraphs with
edge lengths which are either 3-regular or loops. The convergence occurs in
the sense of an ℓ1 sequence metric for which two directed multigraphs are
close if there are compatible isomorphisms between their vertex and edge
sets which roughly preserve the edge lengths. Our proofs rely on a depth-first
exploration of the graph which enables us to relate the strongly connected
components to a particular spanning forest of the undirected Erd˝os–Rényi
random graph G(n,p), whose scaling limit is well understood. We show that
the limiting sequence (C1,C2,...) contains only finitely many components
which are not loops. If we ignore the edge lengths, any fixed finite sequence
of 3-regular strongly connected directed multigraphs occurs with positive
probability. |
| first_indexed | 2024-03-07T07:42:05Z |
| format | Journal article |
| id | oxford-uuid:2fd40076-182e-4e26-aa55-6b17ef8683f1 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:42:05Z |
| publishDate | 2023 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | oxford-uuid:2fd40076-182e-4e26-aa55-6b17ef8683f12023-05-04T13:42:41ZThe scaling limit of a critical random directed graphJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2fd40076-182e-4e26-aa55-6b17ef8683f1EnglishSymplectic ElementsInstitute of Mathematical Statistics2023Goldschmidt, CStephenson, RWe consider the random directed graph ⃗G(n,p) with vertex set {1,2,...,n} in which each of the n(n − 1) possible directed edges is present indepen- dently with probability p. We are interested in the strongly connected com- ponents of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at p = 1/n, with critical window p = 1/n + λn−4/3 for λ ∈ R. We show that, within this critical window, the strongly connected components of ⃗G(n,p), ranked in decreasing order of size and rescaled by n−1/3, converge in distribution to a sequence (C1,C2,...) of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs in the sense of an ℓ1 sequence metric for which two directed multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge lengths. Our proofs rely on a depth-first exploration of the graph which enables us to relate the strongly connected components to a particular spanning forest of the undirected Erd˝os–Rényi random graph G(n,p), whose scaling limit is well understood. We show that the limiting sequence (C1,C2,...) contains only finitely many components which are not loops. If we ignore the edge lengths, any fixed finite sequence of 3-regular strongly connected directed multigraphs occurs with positive probability. |
| spellingShingle | Goldschmidt, C Stephenson, R The scaling limit of a critical random directed graph |
| title | The scaling limit of a critical random directed graph |
| title_full | The scaling limit of a critical random directed graph |
| title_fullStr | The scaling limit of a critical random directed graph |
| title_full_unstemmed | The scaling limit of a critical random directed graph |
| title_short | The scaling limit of a critical random directed graph |
| title_sort | scaling limit of a critical random directed graph |
| work_keys_str_mv | AT goldschmidtc thescalinglimitofacriticalrandomdirectedgraph AT stephensonr thescalinglimitofacriticalrandomdirectedgraph AT goldschmidtc scalinglimitofacriticalrandomdirectedgraph AT stephensonr scalinglimitofacriticalrandomdirectedgraph |