Interplay of choice and topology in percolation on mediation-driven attachment networks

Abstract

We investigate bond percolation on mediation-driven attachment (MDA) networks under the generalized Achlioptas process, where $M>1$ candidate bonds are sampled and the one that minimizes the resulting cluster size is selected the best-of-$M$ rule. This framework offers a systematic approach to investigate how network topology and choice mechanisms jointly shape percolation behavior. We analyze the effects of the degree exponent $\omega$ and the choice parameter $M$ on the critical point $t_c$ and the critical exponents ($\beta,\alpha,\gamma$), which define universality classes and obey the Rushbrooke inequality $\alpha + 2\beta + \gamma \geq 2$. Using entropy, the order parameter, and their derivatives (representing specific heat and susceptibility respectively), we show that both $t_c$ and the universality class depend only weakly on $\omega$ but strongly on $M$, while the Rushbrooke inequality remains valid throughout. For $M=2$, the order parameter varies continuously without a clear order-disorder transition. By contrast, $M=3$ and $M=4$ display explosive percolation that still corresponds to a continuous phase transition, with $M=4$ producing a significantly sharper and clearer order-disorder transition. This sharpening is traced to an enhanced powder-keg effect at larger $M$, underscoring the entropic origin of explosive percolation.