Tricritical Point in Heterogeneous $k$-Core Percolation

$k$-core percolation is an extension of the concept of classical percolation and is particularly relevant to understanding the resilience of complex networks under random damage. A new analytical formalism has been recently proposed to deal with heterogeneous $k$-cores, where each vertex is assigned a local threshold $k_i$. In this Letter we identify a binary mixture of heterogeneous $k$-cores which exhibits a tricritical point. We investigate the new scaling scenario and calculate the relevant critical exponents, by analytical and computational methods, for Erdős-Rényi networks and 2D square lattices.

Figure 1

Phase diagram of the $\mathbf{k}=(2,3)$ mixture, showing the total mass of the percolating HKC cluster at different compositions $r$, for ER networks with $z_1=10$. The TCP at $r=1/2$ separates a line of first order transitions (dashed) from the second order line (solid). The masses of the 2-rich core and the 3-rich core in the giant HKC are also shown (dotted and dash-dotted lines in the shaded area, respectively). The inset shows the phase diagram in the ($r$, $p$) space.

Figure 2

Rescaling of the corona mass $C_{23}$ and the mean corona cluster size ${\Xi }_{23}$ at the TCP on ER networks. The data range in size from $N=2^9$ to $N=2^{18}$ via successive doublings. We find the exponent ratios $\beta /\nu =0.34(5)$ and $\gamma /\nu =0.39(4)$ from the scaling of $C_{23}$ and ${\Xi }_{23}$ at $p_t$ (insets) and show the data collapse achieved with those exponents (main panels $N=2^{14}\dots 2^{18}$; ▷, $\odot$, ★, $\oplus$, □, respectively).

Figure 3

Phase diagram for the $\mathbf{k}=(2,3)$ lattice model showing the threshold density $p$ against composition $r$ for several sizes $L=64$(□), 128(◇), 256(○), 384($\otimes$), 512(×), 768(★), 1024(▷). The arrows indicate the location of the TCP for each $L$. The inset shows the Binder cumulant $U_4(p_c)$ for various sizes, indicating the narrowing of the tricritical region for increasing size. On the right are sample configurations at the threshold density in the continuous transition region (top), at the TCP (center), and in the discontinuous region (bottom); in each case the critical HKC cluster is in lighter color (red).