## Abstract

Consider a random preferential attachment model G(p) for network evolution that allows both node and edge arrivals. Starting with an arbitrary nonempty graph G0, at each time step, there are two possible events: with probability p > 0 a new node arrives and a new edge is added between the new node and an existing node, and with probability 1 - p a new edge is added between two existing nodes. In both cases, the involved existing nodes are chosen at random according to preferential attachment, i.e., with probability proportional to their degree. G(p) is known to generate power law networks, i.e., the fraction of nodes with degree k is proportional to k^{-β}. Here β = (4 - p)/(2 - p) is in the range (2, 3]. Denoting the number of nodes of degree k at time t by m_{k} _{t}, we significantly improve some long-standing results. In particular, we show that m_{k} _{t} is concentrated around its mean with a deviation of O(t), which is independent of k. We also tightly bound the expectation E [m_{kt}] with an additive error of O(1/k), which is independent of t. These new bounds allow us to tightly estimate m_{k,t} for a considerably larger k values than before. This, in turn, enables us to estimate other important quantities, e.g., the size of the k-rich club, namely, the set of all nodes with a degree at least k. Finally, we introduce a new generalized model, G(pt,rt,qt), which extends G(p) by allowing also time-varying probabilities for node and edge arrivals, as well as the formation of new components. We show that the extended model can produce power law networks with any exponent β in the range (1,∞). Furthermore, the concentration bounds established for m_{k,t} in G(p) also apply in G(pt,rt,qt).

Original language | English |
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Title of host publication | KDD 2017 - Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining |

Publisher | Association for Computing Machinery |

Pages | 45-53 |

Number of pages | 9 |

ISBN (Electronic) | 9781450348874 |

DOIs | |

State | Published - 13 Aug 2017 |

Externally published | Yes |

Event | 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2017 - Halifax, Canada Duration: 13 Aug 2017 → 17 Aug 2017 |

### Publication series

Name | Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining |
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Volume | Part F129685 |

### Conference

Conference | 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2017 |
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Country/Territory | Canada |

City | Halifax |

Period | 13/08/17 → 17/08/17 |

### Bibliographical note

Publisher Copyright:© 2017 ACM.

## Keywords

- Degree bounds
- Power law
- Preferential attachment