An Experimental Study of the Coloring Problem on Human Subject Networks

ion of many human and organizational problems in which it is desirable or necessary to distinguish one_s behavior from that of neighboring parties. As a specific scenario, consider the problem faced by faculty members scheduling departmental events—recurring classes, one-time seminars, exams, and so on—in a limited number of available rooms. We can view the events to be scheduled as the vertices in a network,with an edge connecting any pair of events that temporally overlap, even partially. Clearly, two such events must be assigned to different rooms or Bcolors,[ thus yielding a natural graph coloring problem. Furthermore, even when there is a centralized firstcome, first-serve sign-up sheet for rooms, this mechanism is simply the starting point for the negotiation of a solution, and the problem is still solved in a largely distributed fashion by the participants: Facultymembers routinely query the current holder of a room whether they might be able to switch to a different room, whether their event will really require their entire time slot, and the like. Other coloring-like problems arise in a variety of social activities (such as selecting a cell phone ringtone that differs from those of family members, friends, and colleagues); technological coordination Eselecting a channel unused by nearby parties in a wireless communication network (13, 14)^; and individual differentiation within an organization (developing an expertise not duplicated by others nearby). Graph coloring also generalizes many traditional problems in logistics and operations research (12). The coloring problem was chosen for both its simplicity of description and its contrast to other distributed network optimization problems. Unlike the well-studied studied navigation or shortest-paths problem, optimal coloring is notoriously intractable from the viewpoint of even centralized computation (12, 15). In fact, even weak approximations (in which many more colors than the chromatic number are permitted) are known to be equally difficult (16, 17). We report here on the findings from two extensive experimental sessions held in January 2006 with 55 University of Pennsylvania undergraduate students (18). Subjects were given a series of coloring experiments in which the network had one of six topologies, each chosen according to recently proposed models of network formation (Fig. 1 and Table 1). Three of these six begin with a simple cycle and then add a varying number of randomly chosen chords while preserving a chromatic number of two. These Bsmall worlds[ networks (9, 19) are intended to model the mixture of local connectivity (as induced by geography) with longdistance connectivity (as induced by travel or chance meetings) often found in social and other networks. The fourth cycle-based network adopted a more engineered or hierarchical structure, with two distinguished individuals having inordinately high connectivity. The fifth and sixth networks were generated according to the well-studied preferential attachment University of Pennsylvania, Department of Computer and Information Science, 3330 Walnut Street, Philadelphia, PA 19104, USA. *To whom correspondence should be addressed. E-mail: mkearns@cis.upenn. edu Fig. 1. Network topologies with sample colorings found by subjects. From left to right and top to bottom: simple cycle, 5-chord cycle, 20-chord cycle, leader cycle, and preferential attachment with two and three links initially added to each new vertex. 11 AUGUST 2006 VOL 313 SCIENCE www.sciencemag. org 824 REPORTS