Delay of Social Search on Small-world Graphs ⋆

This paper introduces an analytical framework for two small-world network models, and studies the delay of targeted social search by considering messages traveling between source and target individuals in these networks. In particular, by considering graphs constructed on different network domains, such as rectangular, circular and spherical network domains, analytical solutions for the average social search delay and the delay distribution are obtained as a function of source-target separation, distribution of the number of long-range connections and geometrical properties of network domains. Derived analytical formulas are first verified by agent-based simulations, and then compared with empirical observations in small-world experiments. Our formulas indicate that individuals tend to communicate with one another only through their short-range contacts, and the average social search delay rises linearly, when the separation between the source and target is small. As this separation increases, long-range connections are more commonly used, and the average social search delay rapidly saturates to a constant value and stays almost the same for all large values of the separation. These results are qualitatively consistent with experimental observations made by Travers and Milgram in 1969 as well as by others. Moreover, analytical distributions predicted by our models for the delay of social search are also compared with corresponding empirical distributions, and good statistical matches between them are observed. Other somewhat surprising conclusions of the paper are that hubs have limited effect in reducing the delay of social search and variation in node degree distribution adversely affects this delay.