Given a sequence of non-negative real numbers 0 ; 1 ; : : : which sum to 1, we consider a random graph having approximately i n vertices of degree i. In [12] the authors essentially show that if P i(i \Gamma 2) i ? 0 then the graph a.s. has a giant component, while if P i(i \Gamma 2) i ! 0 then a.s. all components in the graph are small. In this paper we analyze the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine

Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power-law degree distributions. Thus there has been a lot of recent interest in defining and studying ‘inhomogeneous ’ random graph models. One of the most studied properties of these new models is their ‘robustness’, or, equivalently, the ‘phase transition ’ as an edge density parameter is varied. For G(n, p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogenous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this already, and others can be approximated by models with

Abstract. 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 ∨ 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 → α, the problem is known to have a phase transition at αc = 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,δ) = (α−(n,δ),α+(n,δ)) such that the probability of satisfiability is greater than 1 − δ for α < α − and is less than δ for α> α+. We show that W(n,δ) = (1 − Θ(n −1/3),1 + Θ(n −1/3)), where the constants implicit in Θ depend on δ. 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 + ε)n, where ε may depend on n as long as |ε | is sufficiently small and |ε|n 1/3 is sufficiently large, we show that the probability of satisfiability decays like exp ( −Θ ( nε 3)) above the window, and goes to one like 1 − Θ ( n −1 |ε | −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.

The single source shortest path (SSSP) problem lacks parallel solutions which are fast and simultaneously work-efficient. We propose simple criteria which divide Dijkstra's sequential SSSP algorithm into a number of phases, such that the operations within a phase can be done in parallel. We give a PRAM algorithm based on these criteria and analyze its performance on random digraphs with random edge weights uniformly distributed in [0, 1]. We use

We consider how to maintain the topological order of a directed acyclic graph (DAG) in the presence of edge insertions and deletions. We present a new algorithm and, although this has marginally inferior time complexity compared with the best previously known result, we find that its simplicity leads to better performance in practice. In addition, we provide an empirical comparison against three alternatives over a large number of random DAG's. The results show our algorithm is the best for sparse graphs and, surprisingly, that an alternative with poor theoretical complexity performs marginally better on dense graphs.

There are several algorithms for producing the canonical DFA from a given NFA. While the theoretical complexities of these algorithms are known, there has not been a systematic empirical comparison between them. In this work we propose a probabilistic framework for testing the performance of automatatheoretic algorithms. We conduct a direct experimental comparison between Hopcroft 's and Brzozowski's algorithms. We show that while Hopcroft's algorithm has better overall performance, Brzozowski's algorithm performs better for "highdensity " NFA. We also consider the universality problem, which is traditionally solved explicitly via the subset construction. We propose an encoding that allows this problem to be solved symbolically via a model-checker. We compare the performance of this approach to that of the standard explicit algorithm, and show that the explicit approach performs significantly better.

We give a short proof that the largest component of the random graph G(n, 1/n) is of size approximately n 2/3. The proof gives explicit bounds for the probability that the ratio is very large or very small.

We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d ≥ 3 is fixed and n grows. We prove mean-field behavior around the critical probability pc = 1 d−1. In particular, we show that there is a scaling window of width n −1/3 around pc in which the sizes of the largest components are roughly n 2/3 and we describe their limiting joint distribution. We also show that for the subcritical regime, i.e. p = (1 − ε(n))pc where ε(n) = o(1) but ε(n)n 1/3 → ∞, the sizes of the largest components are concentrated around an explicit function of n and ε(n) which is of order o(n 2/3). In the supercritical regime, i.e. p = (1 + ε(n))pc where ε(n) = o(1) but ε(n)n1/3 → ∞, the size of the largest component is concentrated around the value 2d d−2ε(n)n and a duality principle holds: other component sizes are distributed as in the subcritical regime.

We consider how to maintain the topological order of a directed acyclic graph (DAG) in the presence of edge insertions and deletions. We present a new algorithm and obtain a marginally improved complexity result over the previously known O(||#||log||#||). In addition, we provide an empirical comparison against three existing solutions using random DAG's. The results show our algorithm to out perform the others on sparse graphs. Finally, we show how the algorithm can be extended to identify strongly connected components online.

Abstract. Let {Gn} be a sequence of finite transitive graphs with vertex degree d = d(n) and |Gn | = n. Denote by p t (v, v) the return probability after t steps of the non-backtracking random walk on Gn. We show that if p t (v, v) has quasi-random properties, then critical bond-percolation on Gn behaves as it would on a random graph. More precisely, if lim sup n