Complex networks for random lasers and photonic computing

We present a theoretical study of a random laser (RL) formed by a scale-free photonic network (PN) with dissipative photon tunneling between microcavities (MCs) hosting two-level quantum systems (TLSs). Using the Gorini-Kossakowski-Sudarshan-Lindblad framework, we identify two distinct lasing regimes. The first is independent of the tunneling parameter and emerges when the steady-state frequency of the nth mode is resonant with the TLS transition. This resembles lasing in uncoupled MCs and relies on population inversion. The second scenario arises from the nonreciprocal nature of dissipative tunneling and enables lasing of nonresonant modes even at vanishingly small population inversion, owing to energy redistribution across the network. In the latter case, interference effects along graph paths induce node-dependent frequency shifts and nonreciprocal photon transmission. We further propose a renormalization-group hypothesis for the PN to explain the observed spectral features. We analyze the phase structure of photonic modes in the scale-free RL and discuss its relevance for photonic Ising machines operating under the minimum power dissipation principle.