### Abstract

We introduce a minimal generative model for densifying networks in which a new node attaches to a randomly selected target node and also to each of its neighbors with probability p. The networks that emerge from this copying mechanism are sparse for p<12 and dense (average degree increasing with number of nodes N) for p≥12. The behavior in the dense regime is especially rich; for example, individual network realizations that are built by copying are disparate and not self-averaging. Further, there is an infinite sequence of structural anomalies at p=23, 34, 45, etc., where the N dependences of the number of triangles (3-cliques), 4-cliques, undergo phase transitions. When linking to second neighbors of the target can occur, the probability that the resulting graph is complete - all nodes are connected - is nonzero as N→∞.

Original language | English |
---|---|

Article number | 218301 |

Journal | Physical review letters |

Volume | 117 |

Issue number | 21 |

DOIs | |

Publication status | Published - 16 Nov 2016 |

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*Physical review letters*,

*117*(21), [218301]. https://doi.org/10.1103/PhysRevLett.117.218301

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*Physical review letters*, vol. 117, no. 21, 218301. https://doi.org/10.1103/PhysRevLett.117.218301

**Structural transitions in densifying networks.** / Lambiotte, R.; Krapivsky, P. L.; Bhat, U.; Redner, S.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Structural transitions in densifying networks

AU - Lambiotte, R.

AU - Krapivsky, P. L.

AU - Bhat, U.

AU - Redner, S.

PY - 2016/11/16

Y1 - 2016/11/16

N2 - We introduce a minimal generative model for densifying networks in which a new node attaches to a randomly selected target node and also to each of its neighbors with probability p. The networks that emerge from this copying mechanism are sparse for p<12 and dense (average degree increasing with number of nodes N) for p≥12. The behavior in the dense regime is especially rich; for example, individual network realizations that are built by copying are disparate and not self-averaging. Further, there is an infinite sequence of structural anomalies at p=23, 34, 45, etc., where the N dependences of the number of triangles (3-cliques), 4-cliques, undergo phase transitions. When linking to second neighbors of the target can occur, the probability that the resulting graph is complete - all nodes are connected - is nonzero as N→∞.

AB - We introduce a minimal generative model for densifying networks in which a new node attaches to a randomly selected target node and also to each of its neighbors with probability p. The networks that emerge from this copying mechanism are sparse for p<12 and dense (average degree increasing with number of nodes N) for p≥12. The behavior in the dense regime is especially rich; for example, individual network realizations that are built by copying are disparate and not self-averaging. Further, there is an infinite sequence of structural anomalies at p=23, 34, 45, etc., where the N dependences of the number of triangles (3-cliques), 4-cliques, undergo phase transitions. When linking to second neighbors of the target can occur, the probability that the resulting graph is complete - all nodes are connected - is nonzero as N→∞.

UR - http://www.scopus.com/inward/record.url?scp=84995948317&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.117.218301

DO - 10.1103/PhysRevLett.117.218301

M3 - Article

VL - 117

JO - Physical review letters

JF - Physical review letters

SN - 0031-9007

IS - 21

M1 - 218301

ER -