Reciprocity -- arc or dyad?

Post date: Jan 25, 2015 9:10:34 PM

What's the difference between arc-based reciprocity and dyad-based reciprocity? An arc is an ordered pair (x,y) where x sends a tie to y. An arc (x,y) is said to be reciprocated if the arc (y,x) is also present. Now, for the formulas below, let R be the number of reciprocated arcs and U be the number of unreciprocated arcs.

Arc Reciprocity

Of all outgoing arcs, what proportion are reciprocated?

• • Reciprocity = R/(R+U)

Dyad Reciprocity

Of all dyads in which there there is at least one arc (i.e., either x-->y or y-->x), what proportion are symmetric? i.e., what proportion have both x-->y and y-->x?

• • Reciprocity = R/(R+2U)

It should be noted that reciprocity is affected by density: the more ties you have, the more reciprocity. In the limit, if the network is 100% dense, every tie is reciprocated. A nice feature of arc reciprocity is that its expected value equals density if there is no particular tendency toward reciprocation. So, if arc reciprocity is much higher than density, you know there is some force (norms?) pushing for reciprocation.