504 research outputs found
Directed cycles and related structures in random graphs: II--Dynamic properties
We study directed random graphs (random graphs whose edges are directed) as
they evolve in discrete time by the addition of nodes and edges. For two
distinct evolution strategies, one that forces the graph to a condition of near
acyclicity at all times and another that allows the appearance of nontrivial
directed cycles, we provide analytic and simulation results related to the
distributions of degrees. Within the latter strategy, in particular, we
investigate the appearance and behavior of the strong components that were our
subject in the first part of this study.Comment: submitted to Physica
The Scaling Window of the 2-SAT Transition
We consider the random 2-satisfiability problem, in which each instance is a
formula that is the conjunction of m clauses of the form (x or y), chosen
uniformly at random from among all 2-clauses on n Boolean variables and their
negations. As m and n tend to infinity in the ratio m/n --> alpha, the problem
is known to have a phase transition at alpha_c = 1, below which the probability
that the formula is satisfiable tends to one and above which it tends to zero.
We determine the finite-size scaling about this transition, namely the scaling
of the maximal window W(n,delta) = (alpha_-(n,delta),alpha_+(n,delta)) such
that the probability of satisfiability is greater than 1-delta for alpha <
alpha_- and is less than delta for alpha > alpha_+. We show that
W(n,delta)=(1-Theta(n^{-1/3}),1+Theta(n^{-1/3})), where the constants implicit
in Theta depend on delta. We also determine the rates at which the probability
of satisfiability approaches one and zero at the boundaries of the window.
Namely, for m=(1+epsilon)n, where epsilon may depend on n as long as |epsilon|
is sufficiently small and |epsilon|*n^(1/3) is sufficiently large, we show that
the probability of satisfiability decays like exp(-Theta(n*epsilon^3)) above
the window, and goes to one like 1-Theta(1/(n*|epsilon|^3)) below the window.
We prove these results by defining an order parameter for the transition and
establishing its scaling behavior in n both inside and outside the window.
Using this order parameter, we prove that the 2-SAT phase transition is
continuous with an order parameter critical exponent of 1. We also determine
the values of two other critical exponents, showing that the exponents of 2-SAT
are identical to those of the random graph.Comment: 57 pages. This version updates some reference
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