On the robustness of centrality measures against link weight quantization in social networks

Abstract

In social network analysis, individuals are represented as nodes in a graph, social ties among them are represented as links, and the strength of the social ties can be expressed as link weights. However, in social network analyses where the strength of a social tie is expressed as a link weight, the link weight may be quantized to take only a few discrete values. In this paper, expressing a continuous value of social tie strength as a few discrete value is referred to as link weight quantization, and we study the effects of link weight quantization on centrality measures through simulations and experiments utilizing network generation models that generate synthetic social networks and real social network datasets. Our results show that (1) the effects of link weight quantization on the centrality measures are not significant when determining the most important node in a graph, (2) conversely, a 5–8 quantization level is needed to determine other important nodes, and (3) graphs with a highly skewed degree distribution or with a high correlation between node degree and link weights are robust against link weight quantization.

Appendix: quantization errors of linear and logarithmic quantization

We derive the quantization errors of linear and logarithmic quantization. Without loss of generality, we assume link weights are distributed over [1, m]. As a representative skewed distribution, we assume link weights follow a truncated Pareto distribution with probability density function defined as following equation.

$$\begin{aligned} p(x) = {\left\{ \begin{array}{ll} \frac{\alpha x^{-\alpha -1}}{1-m^{-\alpha }} &{} (1 \le x \le m)\\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

(4)

Let n be the quantization level. Then, the expected value of the error of linear quantization is given by

$$\sum _{k=1}^n \int _{1+\frac{(m-1)}{n}(k-1)}^{1+\frac{(m-1)}{n}k}p(x) (1+\frac{m-1}{n}k-x) dx,$$

(5)

and the expected value of the error of logarithmic quantization is given by

$$\sum _{k=1}^n \int _{m^{\frac{k-1}{n}}}^{m^{\frac{k}{n}}}p(x) (m^{\frac{k}{n}}-x) dx.$$

(6)

The relation between quantization level n and quantization error is shown in Fig. 16. The relation between the parameter $\alpha$ and quantization errors is shown in Fig. 17.

Fig. 16

Relation between quantization level n and expected value of quantization error (parameter $\alpha =3$, maximum link weight $m=10$)

Fig. 17

Relation between parameter $\alpha$ and expected value of quantization error (quantization level $n=10$, maximum link weight $m=10$)

These figures show that when the link weight distribution is the truncated Pareto distribution, which is a skewed distribution, quantization error of logarithmic quantization is smaller than that of linear quantization.