The IOTA Whitepaper in Section 3 does not seem to define k. Is k the number of verifications required by a transaction as noted earlier in the Whitepaper or something else? If it's the former I do not understand. Can someone people please help me understand?
Stability of the system, and cutsets
Let L(t) be the total number of tips in the system at time t. One expects that thestochastic process L(t) remains stable. More precisely, one expects the process to be positive recurrent, see Sections 4.4 and 6.5 of  for formal definitions. In particular, positive recurrence implies that the limit of P[L(t) =k] as t → ∞ should exist and be positive for all k ≥1. Intuitively, we expect that L(t) should fluctuate around a constant value, and not escape to infinity. If L(t) were to escape to infinity, many unapproved transactions would be left behind.
This is the simplest approach. One may also study similar systems where transactions must approve k other transactions for a general k ≥ 2, or have an entirely different set of rules.