Tore Opsahl

Are triangles made up by strong ties?

April 17, 2009

Simmel (1923[1950]) argued that a strong social tie could not exist without being embedded among other ties. For example, two best friends would always have common friends and acquaintances.

Simmel’s argument was used as a key assumption by Granovetter (1973) when he formulated the Strength of Weak Ties theory. The theory states that weak ties in social networks (e.g., acquaintances as opposed to best friends) are more valuable than strong ties as they bring novel information to a person (ego). This assumption is based on the argument that ego’s close contacts are more likely to know each other (Simmel, 1923[1950]). This argument is illustrated in this sample network for node B. The node has two strong ties to two nodes that are connected, and two weak ties to nodes that are disconnected. Since the close contacts are connected, they are more likely to move in the same social circles as ego and each other than ego’s acquaintances. Thus, close friends’ knowledge is more likely to overlap with ego’s existing knowledge and each others’.

Following Granovetter, Burt (1992) and Coleman (1988) developed two sides of this theory. Burt developed the notion of structural holes, which states that a person benefits from being a broker among other disconnected contacts. On the contrary, Coleman argued that, although it might be beneficial to have many acquaintances when it comes to explicit knowledge, the same cannot be argued for tacit knowledge. A key factor in tacit knowledge transfer is close bonds, shared framework, and trust. These elements describe embedded relationships. Thus, ego must be embedded to optimise tacit knowledge transfer, and broker to optimise explicit knowledge transfer.

Simmel’s claim that strong social ties cannot exist without being embedded is key to this area of investigation; however, are triangles more likely to be constructed by strong ties than by weak ones?

Doreian (1969) and Opsahl and Panzarasa (2009) studied how the clustering coefficient in weighted networks changed when only considering ties with at least a certain value. As the cut-off value increased, the clustering coefficient generally decreased. However, it is not accurate to argue that strong ties is less embedded in triangles than weak ties based on such an analysis. This is due to the fact that the network structure changes when applying an increasinly restrictive cut-off as ties are removed. This implies that the networks on which the clustering coefficient is measured is not comparable.

A possible way of appropriately testing the above question is to compare the weighted clustering coefficient () to the clustering coefficient calculated on a binary network where all ties with a weight greater than 0 is set to present (). In so doing, the two coefficients are calculated on the same network structure. The only difference is that the former takes the weights of ties into consideration. This is possible due to the fact that if weights are randomly assigned in the network, the weighted clustering coefficient is equal to the binary coefficient.

If the ratio between the weighted clustering coefficient and the binary coefficient is higher than 1, it can be argued that triplets made up by strong ties are more likely to be closed than triplets made up by weak ties (). On the contrary, it can be argued that triplets made up by weak ties are more likely to be closed than those made up by strong ties if the ratio is less than 1.

Want to test it with your data?

First, you need to ensure that your data confirm to the tnet standard for weighted networks. Then you need to load it into an R session. The sample network above can be loaded using the following command:

# Load network as an object called net:
net <- cbind(
i=c(1,1,2,2,2,2,3,3,4,5,5,6),
j=c(2,3,1,3,4,5,1,2,2,2,6,5),
w=c(4,2,4,4,1,2,2,4,1,2,1,1))

Second, you need to download and install if you have not done so already for another function.

Third, by running the following commands, you will get the ratio between the weighted and binary clustering coefficient:

# Load tnet
library(tnet)

# Create a binary version of net
net.b <- dichotomise(net, GT=0)

# Calculate the clustering coefficient of the weighted network and of the binary network, and then, divided the former by the latter:
clustering_w(net, measure="am")/
clustering_w(net.b, measure="am")

Empirical tests

To empirically investigate whether strong ties are more likely to be part of closed triplets than weak ties, I have relied upon a set of publicly available datasets that I have collected.

Network		Nodes	Ties
1	Freeman’s EIES network (time 1)	48	695	0.7627	0.7702	1.0099
2	Freeman’s EIES network (time 2)	48	830	0.8131	0.8214	1.0102
3	Freeman’s EIES network (messages)	32	460	0.6386	0.7378	1.1555
4	Consulting (advice)	46	879	0.6932	0.7130	1.0285
5	Consulting (value)	46	858	0.6764	0.6852	1.0131
6	Research team (advice)	77	2228	0.6848	0.7127	1.0408
7	Research team (awareness)	77	2326	0.6785	0.6957	1.0253
8	C.elegans’ neural network	306	2345	0.1818	0.2364	1.3009
9	US airport network	500	2980	0.3514	0.4765	1.3562
10	Newman’s scientific collaboration network	16730	47594	0.3596	0.3178	0.8838

As it is possible to see from the above table, 9 of the 10 networks have a weighted clustering coefficient that is higher than the traditional clustering coefficient. This suggest that triangles generally are, in fact as suggested by Simmel (1923[1950]), made up by strong ties.

The tenth network, which shows the opposite result, is different from the other nine networks. It is a one-mode projection of a two-mode network, whereas the first nine networks are native one-mode networks. This feature could explain the observed result as the one-mode projection is constructed in a way that penalises the weight of ties within triangles, or more specifically, large fully connected cliques (Newman, 2001). See the description of the construction for more details. The next post will describe some methods for creating weighted one-mode projections of two-mode networks.

The code needed to created the above table was:

# Load tnet
library(tnet)

# Load networks
data(Freemans.EIES, Cross.Parker.Consulting, Cross.Parker.Manufacturing, celegans.n306, USairport.n500, Newman.Condmat.95.99)
net <- list()
net[[1]] <- Freemans.EIES.net.1.n48
net[[2]] <- Freemans.EIES.net.2.n48
net[[3]] <- Freemans.EIES.net.3.n32
net[[4]] <- Cross.Parker.Consulting.net.info
net[[5]] <- Cross.Parker.Consulting.net.value
net[[6]] <- Cross.Parker.Manufacturing.net.info
net[[7]] <- Cross.Parker.Manufacturing.net.aware
net[[8]] <- celegans.n306.net
net[[9]] <- USairport.n500.net
net[[10]] <- Newman.Condmat.95.99.net.1mode.wNewman

# Calculate values
output <- data.frame(CGT0=NaN, Cw=NaN, Ratio=NaN)
for(i in 1:length(net)) {
  output[i,"CGT0"] <- clustering_w(dichotomise(net[[i]]), measure="am")
  output[i,"Cw"] <- clustering_w(net[[i]], measure="am")
}
output[,"Ratio"] <- output[,"Cw"]/output[,"CGT0"]

References

Burt, R. S., 1992. Structural Holes: The Social Structure of Competition. Harvard University Press, Cambridge, MA.

Coleman, J. S., 1988. Social capital in the creation of human capital. American Journal of Sociology 94, S95-S120.

Doreian, P., 1969. A note on the detection of cliques in valued graphs. Sociometry 32 (2), 237-242.

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

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

Simmel, G., 1950. The Sociology of Georg Simmel (KH Wolff, trans.). Free Press, New York, NY.

Entry Filed under: Network thoughts. Tags: network, social network analysis, weighted networks, valued networks, strength of ties, clustering coefficient, ties, Links, edges, arcs, embeddedness, complex networks, graphs.