Social networking has been a feature of human society. From the early hunter-gatherer tribes, medieval guilds, the twentieth century workplaces, up to online entities like Facebook and Twitter, it is difficult to think of a time or place where all people did not belong to at least one cooperative group. It follows that social network formation has been studied extensively in the past decades and will continue to be a popular area of research. Past research has primarily confined itself to considering cases in which new members are introduced into the networks by making a constant number of connections to those who are already present in the networks. Our study aims to fill the glaring gap in the variety of network formation modelling. Most notably, we want to consider scenarios in which the number of connections new members make to those already present in the networks is determined by chance. More specifically, the number of connections made to existing members when a new one is introduced into the network is characterized by a positive integer-valued random variable. The objective of the study is to determine the distribution of degree of a node in this kind of social networks. It is determined that the node degree distribution is a mixture of geometric distributions. Three numerical examples are provided in the study to demonstrate the validity of our findings.
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