Concurrency and Commitment: Network Scheduling and Its Consequences for Diffusion¹

¹For comments on an earlier version of this paper I am indebted to Peter Bearman, Randall Collins, Jim Ennis, Duncan Watts, and two anonymous reviewers. For their suggestions on the model, I am grateful to Ann Mische, Freda Lynn, and participants in my fall 2002 Social Dynamics class at Harvard University. For programming assistance, I thank Cheri Minton.

ABSTRACT

Network ties are thought to be concurrent—one can “have” many friends at once, for instance—but their concrete enactment is largely serial and episodic, guided by priorities that steer a person from one encounter to the next. Further, dyadic encounters require that two people be simultaneously available to interact, creating the need for coordinated scheduling. Here I study the consequences of scheduling for network diffusion, using a computer simulation that interposes a scheduling process between a pre-existing network and instances of contagion. The pace and extent of diffusion are shown to depend upon the interaction of network topology, contagion rule (on first-contact versus at some threshold), and whether actors try to remedy past scheduling imperfections. Scheduling turns central actors into diffusion bottlenecks, but can also trigger early adoption by giving actors false readings on the status of their network alters. The implications of scheduling extend beyond diffusion, to other outcomes such as decision-making, as well as to network evolution.

Keywords:

• networks
• scheduling
• simulation
• diffusion

Notes

²Of course, there are ways of communicating that do not require that two people meet face-to-face, including various forms of “asynchronous” communication such as email and old-fashioned letter-writing, neither of which require that two people are even simultaneously available (as, say, a phone call would). These are beyond the purview of the current paper, though point to important avenues for future research, as discussed later.

³Winship (1992) acknowledges as much in a later paper, in which he uses graph coloring theory to analyze the scheduling of group gatherings such as colloquia.

⁴Most does not mean all, of course. See, for instance, Zerubavel (1985) on monastery schedules. A more precise formulation is that institutions and organizations impose cyclical routines, within the context of which decentralized scheduling occurs.

⁵The program was written in the programming language C.

⁶This is not to deny the fact of chance encounters, though I do not consider them here.

⁷As remediation is currently implemented, it turns people into perfect record-keepers, who care as much about what happened, or did not happen, a long time ago as about what happened more recently. An alternative would be to add memory decay, allowing people to assign more weight to recent encounters.

⁸In a network of 100 actors, this makes for a density of .101. The networks were constrained to remain connected, so that everyone is “reachable” by everyone else.

⁹Generally, a “biased” network is one in which ties are formed non-randomly, such as according to the principles of reciprocity and closure. The earliest research on biased networks was by Rapoport (e.g., 1953); more recent extensions can be found in the work of Skvoretz (e.g., 1985; 1990; Skvoretz, Fararo, & Agneessens, 2004).

¹⁰The diagrams were made using NetDraw, available with Ucinet 6 (Borgatti, Everett, & Freeman, 1999).

¹¹The biased distribution is, loosely speaking, “scale-free,” though it is not well approximated by the standard power law function p _k  ∼ k ^γ (Strogatz, 2001), because the decay—the drop-off in frequency of higher degree—is somewhat too rapid.

¹²I bypass the question of whether the precise degree of centralization or clustering in the biased networks “grown” in the manner described above matches that found in empirical networks. Undoubtedly it depends upon the network in question and on how ties are defined. Future research may manipulate these structural properties in a more gradated fashion, perhaps using the procedure employed by Robins, Pattison, and Woolcock (2005).

¹³This assumes that actors with many ties also serve as “bridges” between otherwise-distant network regions. In support of this, the Pearson correlation between “degree centrality” (i.e., degree) and “betweenness centrality”—the latter a measure of the extent to which ego is a link in the shortest chain between each pair of individuals—is .919.

¹⁴An alternative would be to define memory as a decay function of the recent past. In referring to this as memory I do not mean to suggest that people have no recollection of non-recent events—this would be inconsistent with my implementation of remediation—but that adoption decisions are mainly affected by recent experiences, on the basis of which one infers what one's friends are doing now.

¹⁵The effect is not visually striking but is statistically significant throughout.

¹⁶This effect, and that related to remediation described in the last paragraph, does not necessarily take hold at the instant that thresholds become non-trivial. As the exact transition point depends on a number of considerations, and as it is not sharp in any case (i.e., it does not amount to an abrupt “phase transition”), I do not dwell upon it here.

¹⁷Relevant here is Dodds, Muhamad, and Watts (2003) finding that messages forwarded (via e-mail) to central individuals are particularly unlikely to reach their intended target in the allotted time, perhaps because, pressed for time, those people either fail to read or fail to pass along the messages in a timely manner.

¹For comments on an earlier version of this paper I am indebted to Peter Bearman, Randall Collins, Jim Ennis, Duncan Watts, and two anonymous reviewers. For their suggestions on the model, I am grateful to Ann Mische, Freda Lynn, and participants in my fall 2002 Social Dynamics class at Harvard University. For programming assistance, I thank Cheri Minton.