We propose the new EI-clustering method for random networks. Regarding the underlying graph of a random network, EI-clustering is an advanced statistical tool for community detection and based on the estimation of the extremal index (EI) associated with each node. The EI metric is estimated by samples of indices of the node influences. The latter quantities are determined by the PageRank and a Max-Linear Model. The EI values of both models are estimated by a blocks estimator for each node which is considered as the root of a Thorny Branching Tree. Generations of descendant nodes related to the root node of the tree are used as blocks. The reciprocal of the EI value indicates the average number of influential nodes per generation containing at least one influential node. In the context of random graphs the EI metric indicates the ability of a randomly selected node to attract highly ranked nodes in its orbit. Looking at the changing shape of a plot of the EI metric versus the node number, the node communities are detected. The EI-clustering method is compared with the conductance measure regarding the data set of a real Web graph.