We saw that superclusters form intertwined systems that are separated by giant voids of almost equal size. The characteristic scale of this network can be calculated as a distance between centres of superclusters on opposite sides of void walls.

We shall determine distances between high-density regions across
the voids using the *pencil-beam* analysis of mean distances
between high-density regions, as described by EEG. The volume
under study was divided in one direction into beams. This
procedure was repeated in all three directions of coordinate axes,
therefore the total number of beams was . In each beam we
used cluster analysis to determine density maxima and derived the
mean distance between two consecutive density maxima. As a result
we obtained the mean distance between superclusters - the mean
value over voids in all beams. We used the neighbourhood radius , and two resolutions: *n* = 24 and *n* = 12. The
lower resolution was used in the case of subsamples with smaller
number of clusters as will be described below. To eliminate the
influence of the zone of avoidance we performed calculations
separately for the Northern and Southern sky. This method finds
mean distances between systems independently of the supercluster
definition given in Sect. 3. If the number of systems becomes
smaller then also the number of pencil-beams with systems detected
in them becomes smaller. In that case we performed pencil-beam
analysis with a lower resolution, these results are given in
parenthesis.

Table 2 (click here) shows the results of our calculations. Mean distances between high-density regions are given for the observed samples Acl, Asc2, and Asc4, as well as for diluted samples Ard900 and Ard580, and for random supercluster samples (first set of random samples) Rcl, Rsc2, and Rsc4 (the number of very rich superclusters in the volume under study is too small to determine distances between them using the pencil-beam method). The table shows that the distances between systems for observed samples almost do not change. This is understandable: in pencil-beams method we determine the positions of the density maxima, and the presence of clusters in low-density regions does not influence the results of this analysis. Thus the mean separation of high-density regions across voids is almost identical for all observed samples. The same occurs in the case of randomly located superclusters, only in this case distances between high-density regions are larger, and the number of detected systems is about three times smaller than in the real case (most beams cross none or only one high-density region and no distance can be derived).

We can compare the last result with the direct estimate of the characteristic distance between high-density regions using void diameters determined above. The median diameter of voids delineated by members of superclusters was about Mpc. If we add the mean size of the shortest axis of the superclusters, Mpc (EETDA, Jaaniste et al. 1997) then we have as a distance between supercluster centres across the voids a value of 120 h Mpc, close to that found using pencil-beam analysis.

We see that several tests indicate the presence of a characteristic scale of about Mpc in the distribution of rich clusters and superclusters of galaxies. This scale corresponds to the distance between superclusters across the voids. The small scatter of this distance enables us to say that the supercluster-void network is rather regular. The present paper confirms the results by EETDA based on a smaller dataset. This characteristic scale is much larger than the typical scale of voids determined by galaxies (Lindner et al. 1995), and is a manifestation of the hierarchy of the distribution of galaxies and voids. Our data suggest also that there exists no larger preferred scale in the Universe (cf. also EETDA). Thus the scale determined by the network of superclusters and voids should be the upper end of the hierarchy of the distribution of galaxies.

We shall discuss theoretical consequences of the presence of such a scale in further papers of this series (Einasto et al. 1997b; Frisch et al. 1997).

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