Tore Opsahl

Thesis: 1.1 Weighted networks

A major limitation of many methods used for studying large-scale networks stems from the fact that the strength of ties is not taken into account. Granovetter (1973) argued that the strength of a social tie is a function of its duration, emotional intensity, intimacy, and exchange of services. For non-social networks, the strength often reflects the function performed by the ties, e.g. carbon flow (mg/m²/day) between species in food webs (Luczkowich et al., 2003; Nordlund, 2007) or the number of synapses and gap junctions in a neural networks (Watts and Strogatz, 1998). In infrastructure and information networks, variations in the strength of a tie depend on the flow of information, energy, people, and goods along that tie (Barrat et al., 2004; Guimera et al., 2005; Pastor-Satorras and Vespignani, 2004). The strength of a tie is generally operationalised into a weight that is attached to the tie, thereby creating a weighted network (Wasserman and Faust, 1994).

Exploring the information that weights hold allows us to further our understanding of networks. In social networks, strong ties are often found among socially embedded individuals (Granovetter, 1973). In fact, Simmel (1950) argued that a strong tie cannot exist without other indirect ties (weak and strong). Strong ties facilitate change in the face of uncertainty (Krackhardt, 1992), reinforce obligations, expectations, and social norms (Coleman, 1988), and promote the transfer of complex and tacit knowledge by sustaining individuals’ motivation to assist one another (Hansen, 1999; Reagans and McEvily, 2003). Strong ties also aid communication through, for example, the development of relationship-specific heuristics (Uzzi, 1997). In the behaviour of non-social networks, such as technological and transportation ones, strong ties also play crucial roles. For instance, in the network of routers on the Internet they form part of the large backbones that provide national or inter-continental connectivity (Pastor-Satorras and Vespignani, 2004), whereas in airport networks they represent major international or trans-oceanic routes (Barrat et al., 2004).

One of Granovetter’s (1973) major findings is the idea that novel and explicit information is more likely to flow to individuals through weak ties than through strong ones. An individual’s friends tend to move in the same circles, and therefore are likely to receive the same information that the individual already possesses. Conversely, acquaintances are likely to know people that the individual does not, and thus receive more novel information. If this information is explicit or codified, it can easily be transferred from the acquaintance to the individual (Levin and Cross, 2004). In light of these findings, the measures that scholars typically apply to study networks should be sensitive to tie strength and capture the difference between strong and weak ties. This will ensure that the full richness of the data is retained.

However, even though ties in many empirical networks have naturally a strength associated with them, most network measures can only be applied to binary networks, i.e. networks where ties are either present or absent (Scott, 2000; Wasserman and Faust, 1994). Therefore, researchers must convert weighted networks into binary ones. A common way of doing this is to dichotomise a weighted network into a series of binary ones by using a set of cutoffs. A tie is set to present if its weight is higher than the cut-off, and removed otherwise. As Figure 1 shows, a weighted network (a) can create a range of binary networks (b-d) depending on the cut-off. This figure illustrate how the subjectivity of the researcher’s choice of the cut-off value will produce biases that may invalidate subsequent analysis.

A better way to analyse weighted networks would be to redefine and generalise current methods to explicitly take weights into account. Currently, only a small set of measures have been generalised (Freeman et al., 1991; Newman, 2001c). For example, Newman (2001c) generalised Freeman’s (1978) closeness measure by applying a method from computer science to define the distances among nodes (Dijkstra, 1959). The generalised methods would aid the analysis by removing the bias resulting from the subjective choice of the cut-off.