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Delay of Social Search on Small-world Random Geometric Graphs

This paper studies the delay of social search by considering messages traveling between source and target individuals in small-world random geometric graphs. In particular, by considering such graphs constructed on different network domains such as, rectangular, circular and spherical network domains, an exact characterization of the average social search delay is 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 for the average social search delay are first verified by agent-based simulations, and then compared and contrasted with empirical observations in small-world experiments. It is shown 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. On the other hand, 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 consistent with experimental observations made by Travers and Milgram in 1969, as well as by others. Other somewhat surprising conclusions of the paper are that hubs have limited effect in reducing the delay of social search and the degree of social inequality existing in a society adversely affects this delay.