We appeal to results from combinatorial random matrix theory to deduce that various random graph C*-algebras are asymptotically almost surely Kirchberg algebras with trivial K₁. This in particular implies that, with high probability, the stable isomorphism classes of such algebras are exhausted by variations of Cuntz algebras that we term "Cuntz polygons". These probabilistically generic algebras can be assembled into a Fraisse class whose limit structure G is consequently relevant to any K-theoretic analysis of finite graph C*-algebras. We also use computer simulations to experimentally verify the behaviour predicted by theory and to estimate the asymptotic probabilities of obtaining stable isomorphism classes represented by actual Cuntz algebras. These probabilities depend on the frequencies with which the Sylow p-subgroups of K₀ are cyclic and in some cases can be computed from existing theory. For random symmetric r-regular multigraphs, current theory can describe these frequencies for finite sets of odd primes p not dividing r-1. A novel aspect of the collected data is the observation of new heuristics outside of this case, leading to a conjecture for the asymptotic probability of these graphs yielding C*-algebras stably isomorphic to Cuntz algebras. For other models of random multigraphs including Bernoulli (di)graphs, the data also allow us to estimate and heuristically explain the (surprisingly high) asymptotic probabilities of exact isomorphism to a Cuntz algebra. Recognising the role played by Cuntz-Krieger algebras in the theory of symbolic dynamics, we also collect supplemental data to estimate (and in some cases, actually compute) the asymptotic probability of a random subshift of finite type being flow equivalent to a full shift.

BJ was supported by the GACR project 22-07833K and partially supported by the Simons Foundation Award No 663281 granted to the Institute of Mathematics of the Polish Academy of Sciences for the years 2021-2023. He is grateful to the organisers of the stimulating 'Graph Algebras' and 'Generic Structures' conferences held at the Bedlewo Centre in 2023. IK is partially supported by the Praemium Academiae of M. Markl and by Czech science foundation (GACR) under the grant GA22-00091S. This collaboration would not have been possible without the excellent working environment at the Institute of Mathematics of the Czech Academy of Sciences (RVO: 67985840). The computer code was written with some facilitation by ChatGPT. Both authors are indebted to Soren Eilers and the anonymous referee of an earlier version of this article for pointing out the connections to symbolic dynamics and for encouraging a deeper dive into the question of exact vs stable isomorphism between random graph algebras and Cuntz polygons.