Differential Equations for Random Processes and Random Graphs

Probabilistic Analysis of Satisfiability Algorithms

“…The following theorem from [1] (based on Theorem 2 of [97]) is used to approximate clause flows by differential equations. Hypothesis (i) ensures that m i (j) does not change too quickly from iteration to iteration of an algorithm; hypothesis (ii) tells us what we expect the rate of change of m i (j) to be and involves functions which are calculated from a knowledge of what the algorithm is doing; and hypothesis (iii) ensures that this rate of change does not change too quickly.…”

Approximation Algorithm for Random MAX-kSAT

“…In this part of the analysis we are going to use Wormalds's theorem [9]. Wormalds's theorem provides a method for analyzing parameters of a random process using differential equations.…”

“…Our analysis relies on the method of differential equations studied by Wormald in [9]. This method have been used extensively before in the approximation (lower bounds) of the satisfiability threshold (see [2], [7] and [3]).…”

A Simple Population Protocol for Fast Robust Approximate Majority

“…This says that if the differences X t −X t−1 in a martingale with X 0 = 0 are all bounded by by 1, then Pr[X t > λ] ≤ exp(−λ 2 /2t), and by symmetry Pr[X t < −λ] ≤ exp(−λ 2 /2t); the proof is by bounding E[exp(αX t )] for a suitable choice of α. The upper bound also holds for supermartingales (as observed in [14]); the intuition is that any extra drop in X t only makes it harder to exceed λ.…”

“…Despite substantial efforts, we were unable to apply more powerful tools to this problem. Part of the reason is that we are trying to obtain exact asymptotic bounds on a system in which much of the interesting behavior occurs when particular tokens are very rare or when the behavior of the protocol is highly random (e.g., with evenly balanced numbers of x and y tokens); this (together with the fact that the corresponding systems of differential equations do not have closed-form solutions) appears to rule out arguments based on classical techniques involving reduction to a continuous process in the limit (e.g., [12,14]). Similarly, approaches based on direct computation of hitting times or eigenvalues of the resulting Markov chain would appear to require substantially more work than a direct potential function argument.…”