increase your collision rate

Luck looks random up close and structural from far away. The people who keep getting lucky are usually the ones who put themselves in the path of the most good chances. You can't summon a specific opportunity, but you can raise how often one comes within reach. The cleanest way I have to think about it is the same equation chemists use for how often molecules hit each other.

\[ R = n \, \sigma \, v \]

R is your rate of useful encounters. n is the density of useful people around you, sigma is your cross section, meaning how visible and reachable you are (what you publish, ship, and show up to), and v is how fast you move through that space. you can't choose who you run into, but you control sigma, and R rises with it.

A collision isn't the same as value. Each one can form a bond, and bonds come in different strengths. A quick hello is weak. A year of working side by side is strong. Weak bonds are cheap and bring you news from parts of the world you aren't in. Strong bonds are expensive, and they're the only ones that produce the big things -- a cofounder, a lead investor, or a job nobody posted. So there's a real tradeoff between many shallow bonds and a few deep ones.

To make the tradeoff concrete, model it. Say you have a fixed time budget T, in hours per year. Each relationship takes d hours to build, so you can afford N = T/d of them. A bond's strength grows with time but levels off, which the term (1 minus e to the minus d over tau) captures, where tau is the time it takes to form a real bond. The payoff from a bond rises faster than its strength, because trust pays in jumps, so call the payoff (bond strength) raised to a power p, where the exponent p is at least 1 and says how steeply trust pays off.

\[ \begin{aligned} W(d) &= \underbrace{\tfrac{T}{d}}_{\text{how many}} \times \underbrace{\big(1 - e^{-d/\tau}\big)^{p}}_{\text{payoff per bond}} \\ \text{set } \tfrac{dW}{dd} = 0,\ \ u \equiv \tfrac{d}{\tau}: \quad & p\,u\,e^{-u} = 1 - e^{-u} \\ &\Rightarrow\; e^{u} = p\,u + 1 \\ p = 2: \quad & e^{u} = 2u + 1 \;\Rightarrow\; u^{*} \approx 1.26 \\ \tau = 10\text{ h},\ T = 600\text{ h}: \quad & d^{*} = 1.26\,\tau \approx 12.6\text{ h}, \quad N \approx 48 \end{aligned} \]

W is total value, how many relationships (T/d) times the payoff of each. setting the slope dW/dd to zero finds the top of the curve. u = d/tau is depth in bond-times, u-star (found numerically) is the u that maximizes W, the best depth is d-star = u-star × tau ≈ 12.6 h, and that allows N = T/d-star ≈ 48 relationships. the best depth is finite, neither zero nor infinite.

payoff exponent p 2.0 bonding time τ 10 h

best depth 12.6 h, about 48 relationships a year (budget T = 600 h)

drag p toward 1, where you only need information, and the peak slides toward shallow and many. push p up, where you need real trust, and it moves to deep and few.

So your surface area decides how many chances show up, and your depth decides how many turn into something real. Most people get one of the two wrong. They either meet almost no one, or they meet everyone and never build anything that lasts.