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Some social networks, such as LinkedIn and ResearchGate, allow user endorsements for specific skills. From the number and quality of the endorsements received, an authority score can be assigned to each profile. In Perez-Roses et al (Proceedings of INNOV 2015: the fourth international conference on communications, computation, networks and technologies, Barcelona, pp. 68---73. http://www.thinkmind.org/index.php?view=instancei Comput Commun 73:200---210. http://dx.doi.org/10.1016/j.comcom.2015.08.018, 2016), an authority score computation method was proposed, which takes into account the relations existing among different skills. The method described in Perez-Roses et al (Proceedings of INNOV 2015: the fourth international conference on communications, computation, networks and technologies, Barcelona, pp 68---73. http://www.thinkmind.org/index.php?view=instancei Comput Commun 73:200---210. http://dx.doi.org/10.1016/j.comcom.2015.08.018, 2016) is based on enriching the digraph of endorsements corresponding to a specific skill, and then applying a ranking method suitable for weighted digraphs, such as PageRank. In this paper we take the method of Perez-Roses et al (Proceedings of INNOV 2015: the fourth international conference on communications, computation, networks and technologies, Barcelona, pp 68---73. http://www.thinkmind.org/index.php?view=instancei Comput Commun 73:200---210. http://dx.doi.org/10.1016/j.comcom.2015.08.018, 2016) to the limit, by successive application of the enrichment step, and we study the mathematical properties of the endorsement digraphs resulting from that process. In particular, we prove that the endorsements converge to some values between 0 and 1, and they reach the value 1 only in some specific circumstances. This allows the use of the limit values as input to the ranking algorithm.