Adaptability and “Intermediate Phase” in Randomly Connected Networks

We present a simple model that enables us to analytically characterize a floppy to rigid transition and an associated self-adaptive intermediate phase in a random bond network. In this intermediate phase, the network adapts itself to lower the stress due to constraints. Our simulations verify this picture. We use these insights to identify applications of these ideas in computational problems such as vertex cover and $K$-satisfiability.

Figure 1

Example of a network with $N_1=11$, $N_3=5$, so that $x_3=0.3125$. It has $N_{11}=1$, $N_{13}=9$, $N_{33}=3$, $N_3^{(0)}=1$, $N_3^{(1)}=3$, $N_3^{(2)}=0$, and $N_3^{(3)}=1$ (see text). The two subgraphs on the right are rigid; the one on the left has three internal degrees of freedom: the rotations around the $3\mathrm{\text{−}}3$ bonds. There is no stress in the network.

Figure 2

Comparison of analytical calculations (solid lines) with numerics (symbols). The diamonds represent the number of floppy modes per atom; the circles (dots) the fraction of the network in the percolating rigid (stressed) cluster. The dashed line is the Maxwell (i.e., constraint counting) estimate for the number of floppy modes. The numerics are performed on networks of $4\times {10}^4$ atoms.

Figure 3

Energy (dashed line), entropy (dotted line), and free energy (solid line) as a function of the parameter $a$ for different $x_3$; from top to bottom, $x_3=0.48$ (floppy phase), $x_3=0.5$ (intermediate phase), $x_3=0.53$ (rigid phase). The circle indicates the rigidity transition varying $a$ at fixed $x_3$; the diamond indicates the free energy minimum; both coincide in the intermediate phase. In this figure and throughout the text, $T=5$.

Figure 4

Comparison between MC simulations (circles for the parameter $a$, diamonds for the number of floppy modes per atom) and analytical results (dotted lines and solid lines, respectively, correspond to one bond and two bond correlation level). The two dot-dashed vertical lines indicate the location of the two phase transitions. The dashed line is the Maxwell estimate for the number of floppy modes. MC simulations involve $2\times {10}^4\mathrm{atoms}$, for ${10}^6$ MC steps.

Figure 5

Diamonds (circles): probability that a rigid (stressed) cluster percolates the entire sample, as a function of $x_3$. The runs were made with $N=2\times {10}^4\mathrm{atoms}$, for ${10}^6$ MC steps. Solid and dashed lines are guides for the eyes; dot-dashed vertical lines indicate the location of the two phase transitions.