Risk Resonance

Resonance is a compelling component of the study of mechanical systems, as it helps use identify "special situations" where a monotonic perspective on parameter dependence fails, perhaps un-expectedly.

Despite it's clear importance in mechanics, there is little discussion of the importance of resonance in probabilistic systems, as far as I can tell. This seems something worth correcting.

First, we won't be able to find a perfect parallel -- mechanical resonances depend on forcing, momentum, rotational effects, and superposition, which we won't find in simple probabilistic models. But if we widen our views a little, we can still make some progress. Classical resonance appears because of two competing effects in a system -- a natural resonance frequency and a forcing frequency. When these two frequencies are out of alignment, oscillation amplitudes respond monotonely; a faster forcing frequency leads to a smaller amplitude response. But in ranges where the frequencies coincide, we see a peak in the amplitude of oscillations, which we call resonance.

One analog for this in a random system could be how the probability of an outcome responds when it depends on two different competing paths. The simplest such example I can think of is a 4-state continuous-time Markov chain with two possible outcomes. Here's a diagram.

1 3 2 4 b c a d

We start at state 1, and ask, "What's the probability of ending up in state 4, rather than state 3?" A simple calculation shows the probability of ending up in 4 is \[\frac{cd}{(a+c)(b+d)}.\] This probability responds monotonely to each of the parameters. Increasing rate $c$ or $d$ increases the chance of reaching state $4$. Increasing rate $a$ or $b$ decreases the chance of reaching state $4$.

However, if the rates are related to each other, the relationship may become more complicated. For instance, suppose that $a$ and $d$ are always equal, so that the chance of reaching state 4 is actually \[\frac{ca}{(a+c)(b+a)}.\] Now, if we start with $a=0$, there is no chance of getting to state 4. Increasing $a$ a little increases the probability of reaching state $4$. But as $a$ get's really big, the chance of getting to state 4 again goes back to zero. Turns out, we most likely to get to state 4 if $a = \sqrt{bc}$!

Produced by GNUPLOT 4.4 patchlevel 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 2 4 6 8 10 12 14 Probability of reaching state 4 a

This is risk resonance. It's a situation where some factor in a system couples pathways in a way that can make an outcome respond non-monotonely. Another example is the case where two 1-directional random walks collide.

a a a a a a b b b b b b

The probability of reaching the red state when we start at the upper left corner is \[ \frac{6 a^2 b^2}{(a+b)^4} \] which is maximized when $a=b$.

Produced by GNUPLOT 4.4 patchlevel 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.01 0.1 1 10 100 Hitting Probability Relative transition rate (a/b) The take-home? When 2 or more transitions in a Markov process are dependent on a common factor, outcomes may not respond monotonely to factor perturbations -- changes that reduce risk in one setting may increase risk in a separate setting.