### Citations

2385 | On generalized graphs,
- Bollobas
- 1965
(Show Context)
Citation Context ... G n;p and G n;m when the expected degree is a constant and power-law graphs. For sparse random graphs diameter results of such precision were known earlier only for regular graphs of constant degree =-=[4]-=-. Weaker results (to within a constant factor) were known for diameter of sparse G n;p [5] and random `expected-degree' power law graphs [10]. Our results have several applications. We make it possibl... |

746 | Branching Processes,
- Athreya, Ney
- 1972
(Show Context)
Citation Context ...xed degree sequence D satisfying * D ! * can be understood in terms of the p.g.f. of the limiting residual distribution _ = _ * . The p.g.f. also plays a key role in the theory of branching processes =-=[2]-=-. 33sWe begin by noting that the k'th derivative of the p.g.f. is given by (k) _ (z) = E h X _ \Deltas(X _ \Gammas1) \Deltas\Deltas\Deltas(X _ \Gammask + 1) \Deltasz X_\Gamma k i = E h (X _ ) k \Delta... |

706 |
Foundations of Modern Probability
- Kallenberg
- 2002
(Show Context)
Citation Context ..._ [ffl 1 +ffl 2 ] : This theorem allow us to derive a large deviation inequality regarding the growth rate of BFS neighborhoods. The proof is adapted from the upper bound proof of Cramer's Theorem in =-=[9]-=-. 19sLemma 3.12 Let U be an endpoint arrangement satisfying * U 1 \Gamma ! * and m = jU j ! 1, where * has residual distribution _ = _ * . Let R ` U be the initial endpoint queue for an iteration of C... |

506 | A critical point for random graphs with a given degree sequence. Random Structures and Algorithms
- Molloy, Reed
- 1995
(Show Context)
Citation Context ...et. 2.3 Asymptotics. In the previous section, we defined the random graph G D for any degree sequence D. We seek to study such random graphs asymptotically. Several authors, including Molloy and Reed =-=[12, 13]-=-, and Aiello, Chung, and Lu [1], have accomplished this by creating infinite sequences D 1 ; D 2 ; : : : of degree sequences, and examining limits as in lim n!1 P[G Dn satisfies A]: (Here D n is a deg... |

406 | A random graph model for massive graphs
- Aiello, Chung, et al.
- 2000
(Show Context)
Citation Context ...) where (i) k = i(i \Gammas1) \Deltas\Deltas\Deltas(i \Gammask + 1). Definition 2.2 The probability generating function (p.g.f.) of X _ is the function _ (z) = E[z X_ ] = 1 X i=0 z i _(i) (5) for z 2 =-=[0; 1]-=-. In general, all generic random variables are assumed to be independent unless certain dependencies are made explicit. For any integer n * 0 and any distribution _, we let X n X _ (6) denote the sum ... |

197 | The size of the giant component of a random graph with a given degree sequence
- Molloy, Reed
- 1998
(Show Context)
Citation Context ...et. 2.3 Asymptotics. In the previous section, we defined the random graph G D for any degree sequence D. We seek to study such random graphs asymptotically. Several authors, including Molloy and Reed =-=[12, 13]-=-, and Aiello, Chung, and Lu [1], have accomplished this by creating infinite sequences D 1 ; D 2 ; : : : of degree sequences, and examining limits as in lim n!1 P[G Dn satisfies A]: (Here D n is a deg... |

128 | Sudden emergence of a giant k-core in a random graph - Pittel, Spencer, et al. - 1996 |

116 |
Asymptotic enumeration by degree sequence of graphs of high degreeā,
- McKay, Wormald
- 1990
(Show Context)
Citation Context ...obability under the configuration model, then conditioning on simplicity produces a uniformly random simple graph with degree sequence D. The following fact follows from a result of McKay and Wormald =-=[11]-=- Fact 2.1 If the maximum degree of a degree sequence is o(n 1=3 ) and the average degree is \Theta (1), then a random configuration produces a simple graph with constant probability. If a degree seque... |

37 | The cores of random hypergraphs with a given degree sequence, Random Structures Algorithms 25
- Cooper
(Show Context)
Citation Context ...scribe the CM (or configuration model) algorithm for generating a random graph with a fixed degree sequence as well the CM ^-core algorithm from [7, 8] (other results are known for the ^-core problem =-=[14, 6]-=-, but our results depend mainly on the treatment in [8]). In this section, we also introduce an important tool in our analysis, the CM breadth-first search algorithm, and analyze its key properties in... |

16 |
Cores and Connectivity in Sparse Random Graphs. The
- Fernholz, Ramachandran
- 2004
(Show Context)
Citation Context ...ondition does not hold; in particular, we achieve bounds on large deviations for neighborhood sizes of a breadth-first search in a random graph. Our actual result on diameter uses our earlier results =-=[7, 8]-=- on ^-core (especially 2-core) to develop a method of identifying and bounding the worst-case distance between two connected vertices. For instance, in a random graph that has a constant fraction of v... |