| Summary: | Free-forming or casual groups are groups in which individuals are in face-to-face interactions and are free to maintain or terminate contact with one another, such as clusters of people at a cocktail party, play groups in a children’s playground or shopping groups in a mall. Stochastic models of casual groups assume that group sizes are the products of natural processes by which groups acquire and lose members. The size distributions predicted by these models have been the object of controversy since their derivation in the 1960s because of the neglect of fluctuations around the mean values of random variables that characterize a collection of groups. Here, we check the validity of these mean-field approximations using an exact stochastic simulation algorithm to study the processes of the acquisition and loss of group members. In addition, we consider the situation where the appeal of a group of size <i>i</i> to isolates is proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>i</mi><mi>α</mi></msup></semantics></math></inline-formula>. We find that, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>, the mean-field approximation fits the equilibrium simulation results very well, even for a relatively small population size <i>N</i>. However, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, this approximation scheme fails to provide a coherent description of the distribution of group sizes. We find a discontinuous phase transition at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>α</mi><mi>c</mi></msub><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> that separates the regime where the variance of the group size does not depend on <i>N</i> from the regime where it grows linearly with <i>N</i>. In the latter regime, the system is composed of a single large group that coexists with a large number of isolates. Hence, the same underlying acquisition-and-loss process can explain the existence of small, temporary casual groups and of large, stable social groups.
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