For both phases, there is one major network comprising thousands of nodes and edges. Around the major network, there are a few small outlying networks consisting of only a few nodes. As shown in

Figure 1, the carbon trading network can be considered a complex network due to its complexities in both network structure and evolution. In this section, we propose an integrated method to analyze the network. First, time windows (as defined in

Section 3.4) were introduced to make the network time-varying. Through these time windows, the evolutionary features of the network can be observed, including, for example, the growth of the network nodes and edges. Second, metric indicators (defined in

Section 3.2) are used to measure the structural features of the network. The evolution of those features is also analyzed with the time windows. Finally, the scale-free feature of the carbon trading network is tested by both linear regression and segmented linear regression (as mentioned in

Section 3.3).

As shown in

Table 2, the label of a time window, namely

${w}_{i}$ (or

${W}_{i}$), corresponds to a specific time range. For example,

${w}_{33}$ represents the time range of 1 January 2005 to 1 May 2008 of Phase I, and

${W}_{1}$ represents the time range of 1 January 2008 to 1 July 2008 of Phase II. For ease of comparison, the labels for the time windows of Phase I and II are combined in all following figures. In the combined labels, the time window

${w}_{33}$ of Phase I and

${W}_{1}$ of Phase II share the same label because they have 4 months of overlap.

#### 4.1. Growth Features of the Carbon Trading Network

The size of the EU carbon market can be measured by the number of nodes and edges. The overall sizes of the networks are reported in

Table 3. In Phase I, 4605 firms built 7803 trading relationships with each other. It may be noted that 106 firms are missing compared to the number of firms listed in

Table 1; this is because those firms did not trade any allowances with other firms—in other words, they did not build any trading relationships in the EU ETS. In Phase II, the number of firms grew to 6644, which built 19,393 trading relationships with each other—a striking increase from Phase I.

Figure 2 reports the number of nodes and edges observed during each time window. As is shown, the number of nodes and edges increased significantly during both phases. During the first year of the Phase I market (label

${w}_{6}$), carbon trading was a new concept for firms, and the first surrender of allowances had not yet been executed, so the number of firms and trading relationships increased very slowly. From then on, as the pressure of allowance surrenders increased, firms gradually entered the trading market and the size of the Phase I market increased continuously. Additionally, as previously mentioned, the Phase I market was a trial market, and it was closed on May 2008. However, it can be observed from

Figure 2 that the Phase I market still displayed an increasing tendency at the end of the market.

The Phase II market is the successor of the Phase I market. In Phase II, the market size increased rapidly at the beginning. By the first surrender of Phase II (label ${W}_{11}$), the Phase II market size was already equivalent to that of the final Phase I market. After ${W}_{11}$, the number of firms plateaued, but the trading relationships continued to increase at a rapid pace.

Figure 3 illustrates the growth rate of the nodes and edges, in which the extremum values are marked (red points and textured columns). For both nodes and edges, the extremum value of growth appears in the same periods, namely,

${w}_{3}$,

${w}_{6}$,

${w}_{10}$,

${w}_{18}$,

${w}_{23}$,

${w}_{31}$, and

${w}_{35}$ for Phase I, and

${W}_{3}$,

${W}_{11}$,

${W}_{18}$,

${W}_{23}$,

${W}_{30}$,

${W}_{35}$,

${W}_{43}$,

${W}_{47}$, and

${W}_{47}$ for Phase II.

Generally, the extremum values appear twice in each year: once around May, when firms are required to surrender allowances, and once around December, when most active future contracts expire. These are the periods when there are high levels of activity in carbon trading, and it can be seen that the size of the carbon trading network increased rapidly at those times. However, the growth rates of nodes and edges decreased significantly outside of those periods. The growth rate decreased to 5% after the first surrender (label ${w}_{11}$ of Phase I or label ${W}_{11}$ of Phase II), and decreased even further to 1% after the second surrender (label ${w}_{23}$ of Phase I or label ${W}_{23}$ of Phase II). These decreases in growth rates imply that the scale of the markets became increasingly stable. It is also notable that the nodes and edges grew rapidly in the last 2 months of Phase I (label ${w}_{34}$ and ${w}_{35}$), when the Phase I market was about to be closed. Once the Phase I market was closed, the value of the Phase I allowances would disappear; hence, lots of firms swarmed into the market to sell their remaining allowances as fast as they could.

Figure 4 illustrates the network densities of the carbon trading network. As is shown, network density decreased rapidly at the beginning of both phases, but then the decrease became steadier over time. Specifically, network density converged at the level of 0.1% after the second surrender of Phase I (label

${w}_{23}$), as well as following the first surrender of Phase II (label

${W}_{11}$), thus proving that the market structures were stabilizing.

It can also be noticed that in both phases, the levels of network density were very low, even when the market structures were stable. For example, in Phase I, there were 4605 firms in the time window ${w}_{35}$. According to Equation (1), these firms represented a potential for more than 21 million relationships. However, only 19,393 relationships were built, representing a very low network density value of 0.074%. It can therefore be concluded that the connection of the network was far from complete.

#### 4.3. Scale-Free Features of the Carbon Trading Network

In complex network theory, if the distribution of a network satisfies the power law (see Equation (6)), then the network is said to be a scale-free network. To test whether carbon trading is a scale-free network, we apply both linear and segmented linear modeling to the trading data from both phases. The time windows of ${w}_{27}$ and ${W}_{27}$ are chosen for modeling the networks because the market is more stable at the end of each period.

Figure 7 illustrates the degree distribution of both networks. As is shown, the probability density of degree

$P\left(k\right)$ decreases sharply towards zero as degree

$k$ grows. This reveals that most firms in the EU ETS have only few trading partners, but a few firms have a vast number of trading partners. Notably, the density curve of

${w}_{27}$ is steeper than the density curve of

${W}_{27}$. In the time window of

${w}_{27}$, about 61% of firms had only one trading partner, while about 4.5% of firms had more than 10 trading partners. By contrast, in the time window of

${W}_{27}$, only 42% of firms had only one trading partner, but about 10% of firms had more than 10 trading partners.

When logarithms are taken of both

$P\left(k\right)$ and

$k$, their relationship becomes linear, as shown in

Figure 8. As an essential characteristic of power law, the linear relationship between

$\mathrm{log}P\left(k\right)$ and

$\mathrm{log}k$ can be modeled by the following model:

where

$\alpha $ and

$\beta $ respectively indicate the scaling and intercepting exponents. The linear least square method has been commonly used in previous research to estimate the exponents. Therefore, we separately estimate the exponents for the two networks of

${w}_{27}$ and

${W}_{27}$. As shown in

Table 4, the estimated values of

$\alpha $ are both significant at a 1% level for both datasets, and the goodness of fit

${R}^{2}$ values are both above 90%. However, the residuals analysis shows that the linear model of Equation (10) is invalid (

Table 4) for both networks. For the

${w}_{27}$ network, the insignificance of Shapiro-Wilk implies that the residuals obey the normal distribution, but the significance of Durbin-Watson implies the residuals exhibit heteroscedasticity. Therefore, the linear model is invalid for

${w}_{27}$. For Phase II, the model is also invalid because the residuals exhibit heteroscedasticity.

Since the linear model is invalid, we try to model the data using the broken power law. By performing logarithmic transformations on Equations (7) and (8), segmented linear models can be built as follows:

In the segmented linear model, ${\alpha}_{1}$, ${\alpha}_{2}$, and $\mathsf{\gamma}=\mathrm{ln}{k}_{0}$ are the exponents which can be estimated by segmented linear regression. ${k}_{0}={e}^{\gamma}$ is the threshold of the broken power law.

As shown in

Table 5, for both networks, all the estimated exponents were significant at a 1% level, and the goodness of fit (

${R}^{2}$) values are higher than 90%. The residuals analysis shows that both the Shapiro-Wilk test and the Durbin-Watson test of the residuals are insignificant at a 5% level, which indicates that the broken power law model is valid. As shown in

Figure 9, the form of the broken power law was similar for the two networks. Therefore, while we only discuss the results of

${w}_{27}$, the same conclusions can be drawn for the other.

In the ${w}_{27}$ network, the estimated value of the exponent $\gamma $ is 3.24, so the threshold of the broken power law is about 25. If the nodes degree is less than 25, then the degree distribution obeys a power law of which the exponent is 2.10; otherwise, the degree distribution follows another power law of which the exponent is 1.18. According to Equation (6), the density function of the former power law is steeper than the latter, which brings a higher heterogeneity to the network.

By applying the broken power law model to each time window, we illustrate the changes in the exponents and validity of the broken power law model (

Figure 10). First, the broken power law distribution is invalid around the periods when firms and trading relationships increased rapidly (e.g.,

${w}_{10}$ and

${w}_{18}$ for Phase I or

${W}_{11}$ and

${W}_{18}$ for Phase II). Second, for the firms whose degree was below the threshold, the heterogeneity in trading relationships is higher and stable. Third, for the firms whose degree is above the threshold, the heterogeneity in trading relationships is lower and unstable. Fourth, the degree distribution of the overall carbon trading network can be approximately described as the first part of the broken power law distribution (the power law in Equation (7)), because more than 95% of the firms had trading partners below the threshold of the broken power-law (see the colored columns in

Figure 10). Compared to the approximated power law model, the broken power law model is flatter in the lower tail of the distribution, which leads to a long tail effect on the data.