Contours in the $r_{\mathrm{v}}-\overline{{\delta }_{\mathrm{v}}}$ space for the full observed (left) and simulated (right) void samples. The voids are divided into three populations according to their λ_v values, following the binning strategy described in Sect. 3.2.2. It can be seen that the λ_v subdivision is primarily driven by the $\overline{{\delta }_{\mathrm{v}}}$ values: voids with λ_v ∈ (5, ∞) (orange) exhibit positive $\overline{{\delta }_{\mathrm{v}}}$ values, voids with λ_v ∈ ( − ∞, −5] (purple) show negative $\overline{{\delta }_{\mathrm{v}}}$, and voids in the range λ_v ∈ ( − 5, 5] (green) tend to have $\overline{{\delta }_{\mathrm{v}}}\sim 0$, forming a transition zone between the two populations. Negative-λ_v voids are also more likely to have larger r_v values compared to positive-λ_v voids.