Coordination in a Social Network Bertil Hatt ∗ ´ EconomiX , FranceTelecom† Extended Abstract — February 14, 2007 We consider a large population; each agent i has relatives j, and the set of ties draws a graph, or social network. Such a simplifying vision enlightens many new properties (Albert and Barab´ asi 2001, Newman 2003). The number of i’s relatives is his degree di . We consider a service S that presents: • a fixed subscription or entry cost c, which can be monetary or attention (possibly monetized though ads), it can be filling forms or understanding efforts, both reduced by a good design; • a utility a for each of his relatives subscribed too, dSi in total; this can be format or features, structural management hence the total utility: ui = a dSi − c. This is inspired from Arthur (1989), but in a local fashion. Marginal costs can be subtracted from a; intrinsic utility of the product can reduce c: however, our idea is to focus on social network externalities. An agent would subscribe if his expected utility is positive, hence if he knows enough subscribers: dSi ≥ δ := ac . Subscription depends on i’s anticipation of success among his relatives; e. g. a convention of no adoption is a sub-optimal Nash-equilibrium. If two relatives have agreed to subscribe, though they have mutual interest in keeping their word, one can fail to its promise if he discovers his other relatives are not subscribing. Coordination is key, beyond one’s direct acquaintances. We then consider how often users with many contacts establish ties with well-connected agents too: degreehomophily, or assortativity. Many studied complex networks proved to be “client-server” structured, with a local influential element being directly interacting with many single-ended agents: these are dis-assortative network. On the contrary, social network are often assortative, with agents having many ties being linked to similarly social agents. This can be due to homophily on a social, assertive skill; such networks can also appear if ties are established between members of a group — affiliation networks, such as common board members: the ∗ Univ. † FT

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size difference between boards explain more assortativity then having some members of several boards know each other (Girvan and Newman 2002). Not all social networks are assortative though, often because of asymmetric availability constraints; hierarchies are generally dis-assortative networks. By introducing in our model a cross-distribution function, that measures assortativity, we can define a function φ that ties the degree of an agent with the quality level he expects from a service to adopt it: • in a dis-assortative network, φ rapidly reaches a roof and many poorly designed services have no chances; • in an assortative network, φ is very high in cores or very inter-connected agents: a service might be relevant for them — and possibly be bettered for less-savvy users, or grow as an asymmetric communication mean. As agents are uncertain about who among their relatives will adopt, the possibility to unsubscribe (at a cost) can encourage agents to subscribe; risk-averse agents are even more sensitive to that. However, clustering should resolve a significant share of that local information issue. Next, we consider competition. Entry is generally possible, but made more difficult by the incumbent, whether the new service is trying to differentiate by a cheaper, easier to use service or a better product, with more features. Simultaneous entry of several equivalent services is a pure convention game; if the services are different, coexistence is only possible under a set of network conditions — and the more compatible are the services, the more likely they coexist. In either case, the collective assumption that a service can convince a significant share of the market is needed. Finally, we consider the role of homophily, independent of the degree: for a relevant service, the quality demands are lower, and the adoption is easier in the proper niche. —℘— This model could suit communication services, like phone, fax, e-mail, conferencing tools, instant messaging (IM) and more specifically social network services (SNS). Chat rooms, forums and virtual words have similar communication features, and a similar viral diffusion, but their point is more often to meet new people; however, once adopted, ties established on-line constitute a user-base, with similar positive externalities and “stickiness”. We offer two arguments in favor of testable, ramping offers. First one considers risk aversion: agents overestimate the cost of using a service, especially if consequential. Second one considers coordination: an incremental evolution is adopted by a larger share of the population — avoiding to be trapped in niches like a

significant, specialized service would be. Step-by-step ramping offers the possibility for every user to adopt the relevant (compatible) service. Our result could apply to many services whose utility is non exclusively based on relatives’ choice, but with decisive social network effects: fashion and cultural goods, but also all standard- driven technology —software— come to mind. Complex goods demanding expert assistance can be preferred when help is provided on a gift-exchange basis. Other non-digital services can be understood with this model: caf´es and bars in Europe are mostly for enjoyable conversation; conferences are preferred when attended with like-minded experts; night-clubs are increasingly appreciated when joining a large group.

References Albert, R´ eka and Albert-L´ aszl´ o Barab´ asi, “Statistical Mechanics of Complex Networks,” June 2001. arXiv. Arthur, W. Brian, “Competing Technologies, Increasing Returns, and Lock-In by Historical Events,” The Economic Journal, March 1989, 99 (384), 161– 131. Girvan, Michelle and Mark E. J. Newman, “Community structure in social and biological networks,” in “Proceedings of the National Academy of Science” Lawrence A. Sherpp June 2002. Newman, Mark E. J., “The Structure and Function of Complex Networks,” March 2003. arXiv.

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