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tokens: giant component fractions imply the fundamental difference in the pathway topology between Snap100 and Lévy fractals; Snap100 has more efficient pathways to connect all galaxies at a shorter linking length than Lévy fractals. Very likely, this is due to the filamentary structures in Snap100, lacking in Lévy fractals. The middle panel of Fig. 2 shows the diameters. We can again see clear separations of Snap100 from the Lévy walk models. Snap100 reaches the maximum diameter, 300, at the linking length, 1.1 h −1 Mpc, while Lévy walk models reach the maximum diameters around 200 for linking lengths near 2.0 h −1 Mpc. Even for 100 Lévy walk realizations, none of the Lévy walk models can match the diameter measurements of Snap100. Hence, both the size of the giant component and diameter are network measures that discriminate the Lévy walk topologies from the Illustris simulation, despite the data sets being constructed to have matching abundance and two-point correlation statistics. The linking length for maximum diameter is related to the inflection point of the growth curve of giant component fractions; the rate of growth of the giant component decreases after reaching the maximum diameter. This transitioning feature occurs due to the ‘saturation’ of connecting edges. At first (i.e. small linking length values), increasing the linking length results in adding new vertices and increasing the size of the connected network components. However, once the largest diameter is reached, increasing the linking length tends to form new pathways within the existing structure between more far-flung members and only slowly increases the overall size of the connected structure. Therefore, the diameter is maximized at this critical scale, transitioning from ‘growing phase’ to ‘saturating phase’. The previous percolation studies are closely related to this maximum diameter scale, though they have not measured these specific diameters. If the system size is infinite, the diameter measurements transit from finite values to an infinity near this scale. The bottom panel of Fig. 2 shows transitivity results. Again, none of Lévy walk models mimics the transitivity curve of Snap100. We note that the statistical variances of transitivity measurements are much smaller than the other measurements as shown in Fig. 2 , since a single realization of the network is statistically large enough for counting triangles. Hence, the difference of transitivities between Snap100 and Lévy fractals also suggests that Snap100 is topologically very different from Lévy fractals. An interesting feature is the difference of convexities between Snap100 (concave or ‘cup’) and Lévy fractals (convex or ‘hat’). The transitivities of Snap100 are high for small linking lengths, then decrease to a minimum transitivity at 0.4 h −1 Mpc as the linking length increases. After this, the transitivities slowly increase to 0.8. This transitivity trend of Snap100 is related to the transition between the