Up to a rescaling factor for the variable x, the probability functions for CG and 7D collapse on top of each other within the entire range of x. The probability distribution P ( C 1, V ) d C 1 of the largest cluster of size C 1 for whole percolation configurations is observed to follow a single-variable function P ¯ ( x ) d x, with x ≡ C 1 / V d f * for both CG and 7D. The size distribution of bridge-free clusters in 7D displays two-scaling behavior with exponents τ = 4 and τ ′ = 1. For the size distribution, we find the behavior on the CG is governed by a modified Fisher exponent τ ′ = 1, while for leaf-free clusters in 7D, it is governed by Fisher exponent τ = 5 / 2. For the number of clusters, we numerically find the number of leaf-free and bridge-free clusters on the CG scale as ∼ ln V, while for 7D they scale as ∼ V. We also study the behavior of the number and the size distribution of leaf-free and bridge-free clusters. On the CG and in 7D, the whole, leaf-free, and bridge-free clusters all have the shortest-path volume fractal dimension d min * ≈ 1 / 3, characterizing their graph diameters. Further, we observe that while the bridge-free dimension d bf * = 1 / 3 holds for both the CG and 7D cases, the volume fractal dimensions of the leaf-free clusters are different: d lf, 7 D * = 0.669 ( 9 ) ≈ 2 / 3 and d lf, CG * = 0.3337 ( 17 ) ≈ 1 / 3. It is shown that the fraction of nonbridge (biconnected) bonds vanishes, ρ n, CG → 0, for large CGs, but converges to a finite value, ρ n, 7 D = 0.006 193 1 ( 7 ), for the 7D hypercube. Deleting branch bonds from percolation configurations produces leaf-free configurations, whereas deleting all bridge bonds leads to bridge-free configurations composed of blobs. We then classify occupied bonds into bridge bonds, which includes branch and junction bonds, and nonbridge bonds a bridge bond is a branch bond if and only if its deletion produces at least one tree. We numerically confirm that for both cases, the critical number density n ( s, V ) of clusters of size s obeys a scaling form n ( s, V ) ∼ s − τ n ̃ ( s / V d f * ) with identical volume fractal dimension d f * = 2 / 3 and exponent τ = 1 + 1 / d f * = 5 / 2. We study critical bond percolation on a seven-dimensional hypercubic lattice with periodic boundary conditions (7D) and on the complete graph (CG) of finite volume (number of vertices) V.
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