The structure of a random graph at the point of the phase transition

Section: Related Results and Open Questionsmentioning

confidence: 91%

“…So we have to show that p i |C r | is small. To do this, note that B, B / (15), (13) ≤ 2p i 8E(U 0 ) ε 3/2 4 i−1 k (12), (10) ≤ 3pt…”

Section: So Let Us Now Consider Anymentioning

confidence: 99%

The order of the largest complete minor in a random graph

Fountoulakis

Kühn

Osthus

2008

“…Stepanov proved this for p I 0 [36, Theorem 31 and conjectured that it would also hold for positive p. His conjecture was proved for all fixed p by tuczak, Pittel, and Wierman [28].…”

Section: One Corollary Of Theorem 7 Is the Fact That A Random Graph Wmentioning

confidence: 95%

“…Let us, for simplicity, assume that p is bounded. Then the proof of Lemma 5 is easily modified to show that the rth term of the sum is O(n2/'(r + l)e-fr), uniformly in n and r. Thus, by dominated convergence, E n , = (f( (2) It is instructive to compare this expression with alternative formulas for the same quantity obtained in [28] by a different method:…”

Section: Rnmentioning

confidence: 99%

The birth of the giant component

Janson

Knuth

Łuczak

et al. 1993

Self Cite

354

481

Limiting distributions are derived for the sparse connected components that are present when a random graph on n vertices has approximately f n edges. In particular, we show that such a graph consists entirely of trees, unicyclic components, and bicyclic components with probability approaching cosh i= 0.9325 as n + m. The limiting probability that it consists o f trees, unicyclic components, and at most one another component is approximately 0.9957; the limiting probability that it is planar lies between 0.987 and 0.9998. When a random graph evolves and the number of edges passes 4n, its components grow in cyclic complexity according to an interesting Markov process whose asymptotic structure is derived. The probability that there never is more than a single component with more edges than vertices, throughout the evolution, approaches 5 wl18 = 0.8727. A "uniform" model of random graphs, which allows self-loops and multiple edges, is shown to lead to formulas that are substantially simpler than the analogous formulas for the classical random graphs of ErdBs and RCnyi. The notions of "excess" and "deficiency," which are significant characteristics of the generating function as well as of the graphs themselves, lead to a 233 the multigraph process, because it can generate graphs with self-loops x-x, and it can also generate multiple edges. Notice that a self-loop x-x is generated with probability 1 ln', while an edge x-y with x # y is generated with probability 2 ln' because it can occur either as ( x , y ) or ( y, x ) .The second evolution procedure, introduced by Erdos and Rknyi [12], is called the permutation model or the graph process. In this case we consider all N = ( ) possible edges x-y with x < y and introduce them in random order, with all N! permutations considered equally likely. In this model there are no self-loops or multiple edges.A multigraph M on n labeled vertices can be defined by a symmetric n X n matrix of nonnegative integers m x y , where mxy = myx is the number of undirected edges x-y in G. For purposes of analysis, we shall assign a compensation factor to M ; if m = Ez=, E:=, mxy is the total number of edges, the number of sequences ( x , , y , ) ( x 2 , y z ) . . . ( x , , y, ) that lead to M is then exactly (The factor 2" accounts for choosing either ( x , y ) or ( y , x ) ; the 2mxx in the denominator of K ( M ) compensates for the case x = y . The other factor m ! accounts for permutations of the pairs, with mxy! in K ( M ) to compensate for permutations between multiple edges.) Equation (1.2) tells us that K ( M ) is a natural weighting factor for a multigraph M , because it corresponds to the relative frequency with which M tends to occur in applications. For example, consider multigraphs on three vertices (1, 2, 3) having exactly three edges. The edges will form the cycle M , = {1-2, 2-3, 3-1) much more often than they will form three identical self-loops M2 = { 1-1, 1-1, 1-1}, when the multigraphs are generated in a uniform way. For if we consider the 36 possible sequences ( x , , y...

“…For instance, in the fully subcritical regime [p = (1 − ε)/n for ε > 0 fixed], C 1 is a tree of known (logarithmic) size and diameter. In the critical window (ε = O(n −1/3 ) the distribution of |C 1 | was determined in [1,26], and the diameter was found in [28]. See [9,19] for further information.…”

Section: Introductionmentioning

confidence: 99%

Anatomy of a young giant component in the random graph

Ding

Kim

Lubetzky

et al. 2010

We provide a complete description of the giant component of the Erdős-Rényi random graph G(n, p) as soon as it emerges from the scaling window, i.e., for p = (1 + ε)/n where ε 3 n → ∞ and ε = o(1).Our description is particularly simple for ε = o(n −1/4 ), where the giant component C 1 is contiguous with the following model (i.e., every graph property that holds with high probability for this model also holds w.h.p. for C 1 ). Let Z be normal with mean 2 3 ε 3 n and variance ε 3 n, and let K be a random 3-regular graph on 2 Z vertices. Replace each edge of K by a path, where the path lengths are i.i.d. geometric with mean 1/ε. Finally, attach an independent Poisson(1 − ε)-Galton-Watson tree to each vertex.A similar picture is obtained for larger ε = o(1), in which case the random 3-regular graph is replaced by a random graph with N k vertices of degree k for k ≥ 3, where N k has mean and variance of order ε k n.This description enables us to determine fundamental characteristics of the supercritical random graph. Namely, we can infer the asymptotics of the diameter of the giant component for any rate of decay of ε, as well as the mixing time of the random walk on C 1 .