Tolerance analysis in scale-free social networks with varying degree exponents

There are many complex networks like World-Wide Web, internet and social networks have been reported to be scale-free. The major property of scale-free networks is their degree distributions are in power law form. Generally, the degree exponents of scale-free networks fall into the range of (2, 3). The purpose of this paper is to investigate other situations where the degree exponents may lie outside the range.

In this paper, analysis has been carried out by varying the degree exponents in the range of (0.5, 4.5). In total, 243 scenarios have been generated with varying network size of 1,000, 2,000 and 4,000, and degree exponents in the range of (0.5, 4.5) using interval of 0.05.

The following five indicators have been investigated: average density, average clustering coefficient, average path length, average diameter and average node degree. These indicators vary with the network size and degree exponent. If certain indicators do not satisfy with the user requirement using degree exponents of (2, 3), one can further increase or decrease the value with tradeoff. Results recommend that for degree exponents in (0.5, 2), 26 possible scale-free networks can be selected whereas for (3, 4.5), 41 possible scale-free networks can be selected, assuming a 100 percent deviation on the network parameters.

A tolerance analysis is given for the tradeoff and guideline is drawn to help better design of scale-free network for degree exponents in range of (0.5, 2) and (3, 4.5) using network size 1,000, 2,000 and 4,000. The methodology is applicable to any network size.