In order to compare the spatial distribution of clusters and superclusters with random distributions and to investigate the influence of the selection effects on the number density of clusters we generated two sets of randomly located superclusters.
In both sets the number of clusters was the same as in the observed catalogue (approximately 1300), they occupy the same volume, and are combined into superclusters that have multiplicity distribution similar to that of real superclusters. First we generated supercluster centres, and then supercluster members around each centre. The radius of superclusters was chosen in accordance with observations: 10 h Mpc and 20 h Mpc, for poor and rich superclusters, respectively (EETDA, Jaaniste et al. 1997). Clusters outside the sphere of radius, , and near the Galactic plane, , were excluded. The selection effects were taken into account in two different ways.
In the first set of random catalogues centres of superclusters and locations of isolated clusters were generated using a censored random distribution: in order to avoid overlapping of the superclusters a minimum distance of 40 h Mpc was chosen between these centres and a minimum distance of 24 h Mpc was chosen between clusters that did not belong to the superclusters, as in the case of real isolated clusters. This set of random supercluster catalogues was generated without taking into account detailed selection effects (i.e. clusters were absent from the zone of avoidance but the changes in the mean density of clusters in distance and in galactic latitude were ignored).
In the second set of random catalogues we took into account the selection functions derived from the observed sample of all clusters. The locations of the supercluster centres and isolated clusters were generated completely randomly, the number of members for each supercluster was generated according to both the multiplicity function of real superclusters, and to the selection functions which determined the probability to find a cluster at a given galactic latitude and distance from the observer. The number of clusters that were generated but not included to the catalogue due to selection effects gives us an estimate of the real number of clusters in the volume under study (see the previous section). The final multiplicity of superclusters was determined using the clustering algorithm and a neighbourhood radius of 24 h Mpc.
To check the validity of the selection function procedures we calculated the density distribution of clusters for both sets of random supercluster catalogues. As expected, the dependence of the cluster density on the galactic latitude and distance was similar to the observed one in the case of the second set of random catalogues, while in the case of the first set this dependence was much weaker. The multiplicity distribution of superclusters for the second set of models was almost identical to the multiplicity distribution in the real catalogue. We shall discuss the influence of differences in the selection function on the tests applied in the present paper in corresponding sections.