Tore Opsahl

Defining Weighted Networks

Ties in many empirical networks have naturally a strength associated with them. For example, in social networks, some contacts are friends, whereas others are simply acquaintances. Granovetter (1973, pg. 1361) 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) 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).

The way tie strength is recorded as weights (operationalisation of tie strength) can introduce subjective biases in the analysis of weighted networks that might invalidate the analysis. Therefore, a method that retains elements from the research question and setting should be applied as it affects the outcomes of weighted network measures.

A precise definition of tie strength often exists in non-social networks (e.g. the number of synapses and gap junctions), making the operationalisation of tie strength into weights straight forward. Conversely, in social networks, this tend not to be the case. Most social networks are collected through survey instruments (Wasserman and Faust, 1994). For example, a friendship network could be collected by asking people to designate others from a list or roaster as “know this person’s name”; “acquaintace”, or “friend”. By collecting social relations in such a way, two issues exist.

First, what is considered an “acquaintance” or a “friend” can differ considerably from one person to another. This bias is usually referred to as the informant inaccuracy bias (Bernard et al., 1984). A possible way of overcoming this bias is to design questions that are targeting the elements of a social tie (duration, emotional intensity, intimacy, and exchange of services; Granovetter, 1973). A question that does this is:

Please indicate how often you have turned to this person for information or advice on work-related topics in the past three months.

with the ordinal scale: 0, Do not know this person; 1, Never; 2, Seldom; 3, Sometimes; 4, Often; 5, Very Often. (This question was used by Cross and Parker, 2004, to collect an advice network.)

Second, by using an ordinal scale (associating 0, 1, 2, 3, 4, and 5 to the six levels of relationship), the differences among the levels might be misrepresented. In addition, with the ordinal scale used in the above question, answers are subjected to the inevitable bias that comes from the different ways in which different people assess duration. One possible way to overcome the scale issues is to design a scale that is closer to a ratio scale and reflects duration more accurately. For example, a better scale for the above question could be: 0, Do not know this person/Never; 1, Once; 3, Monthly; 6, Bi-weekly; 12, Weekly (Opsahl et al., 2010; Opsahl and Panzarasa, 2009).

Researchers should carefully design questions and, in turn, the scale of weights, to overcome these issues. Marsden and Campbell (1984) conducted a comparative analysis of Granovetter’s (1973) four criteria for defining tie weights. They found that emotional intensity was a better indicator of strength of friendship than the other three criteria. Nevertheless, I believe that researchers should choose the appropriate measures of tie strength depending on the nature of the nodes and ties and, more generally, on the context of the research setting. In turn, such a scale is likely to yield a network dataset that is richer in information, more robust against potential inaccuracies emanating from subjective judgments, and more suitable to investigations that rely on weighted network measures.

References

Barrat, A., Barthelemy, M., Pastor-Satorras, R., Vespignani, A., 2004. The architecture of complex weighted networks. Proceedings of the National Academy of Sciences 101 (11), 3747-3752. arXiv:cond-mat/0311416

Bernard, H. R., Killworth, P. D., Kronenfeld, D., Sailer, L. D., 1984. The problem of informant accuracy: the validity of retrospective data. Annual Review of Anthropology 13, 495-517.

Cross, R., Parker, A., 2004. The Hidden Power of Social Networks. Harvard Business School Press, Boston, MA.

Granovetter, M., 1973. The strength of weak ties. American Journal of Sociology 78, 1360-1380.

Guimera, R., Mossa, S., Turtschi, A., Amaral, L. A. N., 2005. The worldwide air transportation network: Anomalous centrality, community structure, and cities’ global roles. Proceedings of the National Academy of Sciences 102, 7794-7799. arXiv:cond-mat/0312535

Luczkowich, J. J., Borgatti, S. P., Johnson, J. C., Everett, M. G., 2003. Defining and measuring trophic role similarity in food webs using regular equivalence. Journal of Theoretical Biology 220, 303321.

Marsden, P.V., Campbell, K.E., 1984. Measuring Tie Strength. Social Forces 63, 482-501.

Nordlund, C., 2007. Identifying regular blocks in valued networks: A heuristic applied to the St. Marks carbon flow data, and international trade in cereal products. Social Networks 29 (1), 59-69.

Opsahl, T., Agneessens, F., Skvoretz, J., 2010. Node centrality in weighted networks: Generalizing degree and shortest paths. Social Networks 32 (3), 245-251.

Opsahl, T., Panzarasa, P., 2009. Clustering in weighted networks. Social Networks 31 (2), 155-163.

Pastor-Satorras, R., Vespignani, A., 2004. Evolution and Structure of the Internet. Cambridge University Press, New York, NY.

Wasserman, S., Faust, K., 1994. Social Network Analysis. Cambridge University Press, Cambridge, MA.

Watts, D. J., Strogatz, S. H., 1998. Collective dynamics of “small-world” networks. Nature 393, 440-442.