Editor's introduction : Six themes of 'six degrees'

level, a mathematically precise definition of what constitutes a small-world network is logarithmic scaling of the average geodesic in a network. That is, a network is considered “small” if the average geodesic grows logarithmically with increasing network size (Newman, 2003: 183). Furthermore, in recent years attempts have been made at quantifying “small-world-ness” as a continuous property, thereby giving guidance to researchers as to the relative size of small-world properties for a given network (Humphries and Gurney, 2008; Uzzi and Spiro, 2005). When calculating the degree of small-world-ness, one needs to carefully choose the correct version of a random network for comparison with the network of interest; for instance, the degree of small-world-ness in a bipartite network requires comparison to a random bipartite network (see Uzzi et al., 2007: 85). Besides probing real networks for characteristics derived from formal network models, researchers have also studied the structural and dynamic properties of network models. Whenever building a network structure implies costs – and it is hard to conceive of an example where this is not the case – there is a trade-off between minimizing construction costs and guaranteeing optimal operation of processes within a given network structure (cf. Fisk, 2004). For example, this trade-off applies when strategically forming a social network, building traffic infrastructure, or growing neural networks. With regard to optimizing and sustaining operability of processes in a given network structure, several issues appear relevant: First, to the degree that network flows are costly, a process that involves fewer steps is more efficient. Networks with short geodesics are therefore ideal as they provide structural shortcuts for network flows to reach a node from any other node in the network. Second, bandwidth is an issue: if network traffic is intense, a few shortcuts in the network through which the majority of network flows must pass may either increase the risk of failure of these nodes and, as a consequence, lead to network breakdown, or it may slow down network flows considerably. Third, robustness against failure and attacks is important: Scale-free networks are highly robust against random failure or attacks – for randomly hitting one of the few highly connected nodes is improbable – but they are sensitive to targeted attacks against highly connected nodes (Albert et al., 2000, 2001; see also Bao et al., 2009). When the basic type of process under examination is not diffusion or contagion but some kind of targeted search process, the underlying network structure as such does not matter but rather the underlying structure that is actually searchable by intelligent agents. In the search framework, the structural dimension therefore relates to the question which structures are searchable by agents using local information and, from a network-design perspective, which structures match most efficiently to particular types of search algorithms. What Milgram’s letter referral studies demonstrate is exactly that people are able to find short paths in their social networks. But they also show that a substantial number of letter-referrals in the respective Editor’s Introduction xxvii intro.qxp 9/10/2012 6:42 PM Page xxvii