The obvious way to measure this is to ask each question twice — once about your inner circle, once about strangers — and subtract.
I spent a while on that version. It does not work, for two separate reasons, and the second one is the interesting one.
Difference scores fall apart when the halves agree
The reliability of a difference between two measurements is
(alpha - r) / (1 - r)
where r is how strongly the two correlate. And the two halves of this design correlate strongly, because they are the same trait asked twice about different people.
Run the numbers for these facets. At alpha .78 and a cross-frame correlation of .70, the difference score comes out at .27. At alpha .65 it goes negative — which is not a metaphor, it means the difference contains less signal than noise.
Cronbach and Furby wrote this down in 1970, in a sentence that should be on a poster:
Investigators who ask questions regarding gain scores would ordinarily be better advised to frame their questions in other ways.
The second reason, which matters more
Even if the arithmetic worked, inner-circle-minus-strangers throws away everything in between.
And "everything in between" is where the actual question lives. Does your warmth taper gradually as people get less close, or does it fall off a cliff at the edge of some circle? Those are different descriptions of a person. A two-point measurement cannot tell them apart, because both produce the same difference.
Which of those it is is the claim. A design that cannot see it is not a cheaper version of the right instrument; it is an instrument for a different question.
So: four ordered levels — inner circle, friends, acquaintances, strangers — and instead of a subtraction, a decomposition into four pieces. How much of the trait you have overall. How steeply it changes. Whether the change accelerates or decelerates. And a fourth piece no theory predicts, which therefore makes a free per-person estimate of how noisy that respondent's answers were.
Why not a discount curve
There is a well-developed literature on social discounting that fits a hyperbolic curve to how much you will give up for someone at a given social distance, and it would be flattering to borrow its machinery.
It does not transfer. In that work, distance is a rank position among a hundred named people — a genuinely ratio-scaled quantity. My four rings are ordinal categories. Coding them 1, 2, 3, 4 rather than 1, 5, 20, 100 moves the fitted parameter by two orders of magnitude, and a number whose value is set by an arbitrary coding choice cannot be the headline. Those curves are computed and reported as diagnostics, and they are never the number at the top of the page.