Three inflows

Chapter 3 described a relationship’s shared order $x$ with a drive $\mu = J – L – c$. Chapter 5 dealt with the leak $L$. This chapter takes the inflow $J$ apart. The model and the evidence suggest that it has three components, which work in different ways:

$$J = J_{\text{novelty}} + J_{\text{meaning}} + J_{\text{repair}} .$$

Novelty is the stream of new, learnable shared experience of Chapters 2 and 4. Meaning is what the two people make of their experience together: the stories, goals and rituals that turn a sequence of events into a shared life. Repair is what happens after conflict, the speed and reliability with which the two return to their usual closeness. The first two raise the drive. The third, as this chapter will show, protects the state itself, and matters most exactly when the drive is weakest.

Novelty: expanding the self

Self-expansion and the learning model of Chapter 2 describe the same thing at two levels. Early in a relationship, each partner is a large source of learnable novelty: a new person, new friends, new places, a new way of seeing. The self grows quickly, and the prediction errors are large. As the partners come to know each other, the growth slows. The experiments of Chapter 3, in which couples who did novel and arousing activities together reported larger gains in relationship quality (Aron et al. 2000), suggest that the growth can be restarted by finding new things to learn together, so that the expansion is again associated with the partner.

Meaning: turning surprise into a shared model

Novelty alone is not enough. A life of constant new experiences, none of which connects to the others, would have high surprisal and little shared structure. What turns experience into order is the work of making sense of it together.

In the language of Chapter 4, meaning compresses. A shared story, a private joke, a ritual or a common goal is a model that both partners hold, which makes the relationship’s core more predictable (lower surprisal) while the content of their life can stay novel (high epistemic value). Meaning is how novelty becomes structure instead of noise.

Repair: speed matters most near the edge

The third inflow is different in kind. Conflict is not a failure of a relationship; every relationship has it. What matters, the evidence suggests, is what happens next.

The model of Chapter 3 explains why repair matters so much, and when it matters most. Near any stable state, a system returns from a small disturbance at a rate set by the slope of its dynamics. For the equation $dx/dt = \mu x + x^3 – x^5$, that rate can be calculated exactly, and it falls as the drive approaches the tipping point.

from math import sqrt

print("   mu     order x*   recovery rate   recovery time")
for mu in (0.20, 0.0, -0.10, -0.20, -0.24, -0.249):
    x2 = (1 + sqrt(1 + 4 * mu)) / 2          # the ordered state: x*^2
    rate = -(mu + 3 * x2 - 5 * x2 ** 2)       # minus the slope of mu*x + x^3 - x^5 at x*
    print(f"{mu:+.3f}   {sqrt(x2):8.3f}   {rate:13.3f}   {1 / rate:13.1f}")
#    mu     order x*   recovery rate   recovery time
# +0.200      1.082           3.142             0.3
# +0.000      1.000           2.000             0.5
# -0.100      0.942           1.375             0.7
# -0.200      0.851           0.647             1.5
# -0.240      0.775           0.240             4.2
# -0.249      0.729           0.067            14.9

Far from the edge, the system bounces back in a fraction of a time unit. As $\mu$ falls towards $-0.25$, recovery takes longer and longer, and at the tipping point it would take forever. This is the critical slowing down of Chapter 3, and it has been observed in human emotion.

Slower recovery has a dangerous consequence. If the next conflict arrives before the couple has recovered from the last, the disturbances add up. Repair, in the model, is an extra restoring force after a conflict: an apology, a touch, a joke, a conversation that brings the state back towards order faster than it would return on its own.

import random
from math import sqrt

mu = -0.15                                   # inside the zone where order and disorder are both possible
x_hi = sqrt((1 + sqrt(1 + 4 * mu)) / 2)      # the ordered state
x_lo = sqrt((1 - sqrt(1 + 4 * mu)) / 2)      # the tipping point below it

def couple(repair, seed, days=5 * 365, dt=0.1):
    """One couple over five years. Conflicts arrive about once a month and knock x down by 0.10-0.30.
    repair = extra pull back towards x_hi after a conflict (0 = none). Returns True if the couple collapsed."""
    rng = random.Random(seed)
    x = x_hi
    for step in range(int(days / dt)):
        if rng.random() < dt / 30:
            x -= rng.uniform(0.10, 0.30)
        x += dt * (mu * x + x ** 3 - x ** 5 + repair * (x_hi - x))
        if x < x_lo:
            return True
    return False

print(f"ordered state {x_hi:.3f}, tipping point {x_lo:.3f}")
print("repair strength   couples that collapsed within 5 years (of 200)")
for repair in (0.0, 0.5, 1.0, 2.0):
    n = sum(couple(repair, seed) for seed in range(200))
    print(f"{repair:15.1f}   {n:5d}")
# ordered state 0.903, tipping point 0.429
# repair strength   couples that collapsed within 5 years (of 200)
#             0.0      86
#             0.5      11
#             1.0       3
#             2.0       0

With the same drive and the same conflicts, couples who do not repair collapse in 86 of 200 runs; modest repair cuts this to 11, and stronger repair to 3 or none. The conflicts did not change. What changed was whether they could accumulate.

Three inflows, three roles

The practical reading is a kind of triage. A couple who quarrel rarely and recover quickly can afford to put their effort into novelty and meaning. A couple whose small quarrels now take days to heal should put repair first: agree on how to signal “I want to stop this”, and practise returning quickly, before planning anything adventurous. The next chapter turns these ideas into four weeks of practice.