Degree Distribution and Quality in Complex Engineered Systems

Although the intuitive notion of hubs suggests that they constitute a set that is both highly connected and small, the set does not necessarily need to be small. For instance, Strogatz ([5], p. 274) referred to hubs as simply highly connected components, even if they constitute the majority of the components in the system.

Here we assume that a system has, at least, one hub unless its degree distribution gives positive probability weight to only one degree ko (i.e., all components have the same degree, ko ). In such degree distributions, we cannot identify hubs because we cannot split the degree distribution at any point.

The degree distributions of these systems were based on directed graphs that represent calling relationships among subroutines within each of these systems [3]. The convention used by [3] with respect to the directionality of the link between two nodes is reversed from the one used here. Our convention is similar to the one used in previous work that models software architectures using DSM representations [13,34,47] by which the direction of the link follows the direction of the “function call” (i.e., in-degree counts the number of incoming function calls). Such a difference in convention does not affect our analysis.

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The lack of significance in correlation between QD and skewness can be explained by the following thought experiment. Assume a network with non-directed dependencies and a certain (right-skewed) degree distribution. In addition, choose a D to define QD . How are both skewness and QD likely to change if a new node is added to the network? If the newly added node has a degree that is smaller than the average mean, then skewness will decrease and QD will decrease (or will stay at zero if the original QD  = 0). If the newly added node has degree greater than the mean degree but smaller than D, then skewness will increase whereas QD will decrease (or will stay equal to zero if the original QD  = 0). Finally, if the newly added node has degree greater than or equal to D, then skewness will increase and QD will also increase. Hence, the actual relationship between skewness and QD is an empirical question, which in our dataset does not exhibit a significant correlation for a wide range of D.

Because the variables are measured using very different unit scales, and to facilitate the interpretation of the estimated coefficients, we standardize each variable to its z-score before entering it into the regressions [67]. The z-score of a variable is obtained by subtracting its mean value and dividing it by its standard deviation.

To see this, note that equalizing to zero the partial derivative of yis with respect to Q0.15,in in Model 1 yields the z-score of Q0.15,in  = 2.105, which minimizes yis , all else constant. To obtain the value of 9.0%, multiply such a z-score by the standard deviation and add its mean value, 2.105 × 0.032 +  0.023 = 0.09.