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

Nilomber Roy, M. M. B. Sheraj, M. K. Hassan

TL;DR

The paper studies bond percolation on mediation-driven attachment (MDA) networks under a generalized best-of-$M$ Achlioptas process. It shows that the critical point $t_c$ and the associated exponents $(α,β,γ)$ depend weakly on the degree exponent $ω$ but strongly on the choice parameter $M$, with $M=2$ producing a non-explosive continuous transition and $M=3,4$ yielding explosively sharp but continuous transitions due to an entropic powder-keg mechanism. Finite-size scaling extracts $ u$, $α$, $β$, and $γ$, revealing that $β/ν$ decreases while $α/ν$ and $γ/ν$ increase with $M$, indicating different universality classes across $M$ and $m$; the Rushbrooke inequality $α+2β+γ\,=\,2$ is satisfied. The work highlights how topology and competitive link occupation jointly shape percolation, with practical implications for robustness and controllability in scale-free-like networks and for understanding entropy-driven explosive transitions.

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 $ω$ and the choice parameter $M$ on the critical point $t_c$ and the critical exponents ($β,α,γ$), which define universality classes and obey the Rushbrooke inequality $α+ 2β+ γ\geq 2$. 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 $ω$ 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.

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

TL;DR

The paper studies bond percolation on mediation-driven attachment (MDA) networks under a generalized best-of- Achlioptas process. It shows that the critical point and the associated exponents depend weakly on the degree exponent but strongly on the choice parameter , with producing a non-explosive continuous transition and yielding explosively sharp but continuous transitions due to an entropic powder-keg mechanism. Finite-size scaling extracts , , , and , revealing that decreases while and increase with , indicating different universality classes across and ; the Rushbrooke inequality is satisfied. The work highlights how topology and competitive link occupation jointly shape percolation, with practical implications for robustness and controllability in scale-free-like networks and for understanding entropy-driven explosive transitions.

Abstract

We investigate bond percolation on mediation-driven attachment (MDA) networks under the generalized Achlioptas process, where candidate bonds are sampled and the one that minimizes the resulting cluster size is selected the best-of- 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 and the choice parameter on the critical point and the critical exponents (), which define universality classes and obey the Rushbrooke inequality . Using entropy, the order parameter, and their derivatives (representing specific heat and susceptibility respectively), we show that both and the universality class depend only weakly on but strongly on , while the Rushbrooke inequality remains valid throughout. For , the order parameter varies continuously without a clear order-disorder transition. By contrast, and display explosive percolation that still corresponds to a continuous phase transition, with producing a significantly sharper and clearer order-disorder transition. This sharpening is traced to an enhanced powder-keg effect at larger , underscoring the entropic origin of explosive percolation.

Paper Structure

This paper contains 12 sections, 18 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Network Topology
  3. Power of choice in percolation
  4. Achlioptas process
  5. Order--disorder transition
  6. Entropic origins of powder keg effect
  7. Finite-size scaling and Critical exponents
  8. Critical exponent of susceptibility
  9. Critical exponent of specific heat
  10. Critical exponent of order parameter
  11. Universality
  12. Summary

Figures (6)

  • Figure 1: We show a mineature of MDA network in (a). Next, plot (b) shows $\ln P(k)$ vs $\ln k$ for $m=50$, yielding a straight line with slope $2.9358$, representing the degree exponent.
  • Figure 2: Plots of relative entropy and relative order parameter for (a) $m=50,M=3$ (b) $m=50,M=4$ (c) $m=100,M=3$, (d) $m=100, M=4$ (e) $m=200,M=3$ and for (f) $m=200,M=4$.
  • Figure 3: Plots of relative entropy and relative order parameter for (a) $m=50,M=2$ (b) $m=100,M=2$ (c) $m=200,M=2$ and (d) presents the corresponding result for ER network with $M = 2$, shown for comparison.
  • Figure 4: Plots of susceptibility $\chi$ versus relative link density $t$ for different network sizes are shown in (a) and (d) for $M=3$ with $m=50$ and $M=4$ with $m=200$. In the inset we show plots of $\log(\chi_h)$ versus $\log(N)$ and find straight lines whose slopes give an estimate of $\gamma/\nu$. In (b) and (e) we plot $\chi(t,N)N^{-\gamma/\nu}$ versus $t-t_c(N)$ find that all the peaks of the respective plot of (a) and (d) collapse at $t=t_c(N)$. The quality of peak collapse suggests how good are the estimated values of $\gamma/\nu$. In the inset of (b) and (e) we show plots of $\log(t-t_c)$ versus $\log(N)$, the slopes of the resulting straight lines give an estimate of $1/\nu$. When we plot $\chi(t,N)N^{-\gamma/\nu}$ versus $(t-t_c)N^{1/\nu}$ we find an excellent data collapse which proves that the values of $\gamma/\nu$ and $1/\nu$ are as good as the theoretical values.
  • Figure 5: Plots of specific heat $C(t,N)$ versus $t$ for different network sizes are shown in (a) and (c) for $M=3$ with $m=50$ and $M=4$ with $m=200$. In the inset we show plots of $\log(C_h)$ versus $\log(N)$ and find straight lines whose slopes give an estimate of $\alpha/\nu$. We used the $1/\nu$ which was previously found from susceptibility for corresponding $M$ and $m$. Then in (b) and (d) we plot $C(t,N)N^{-\alpha/\nu}$ versus $(t-t_c)N^{1/\nu}$ and obtain an excellent data collapse revealing that the values of $\alpha/\nu$ and $1/\nu$ are the best we can get numerically.
  • ...and 1 more figures