*2.2. Ecological Network Analysis*

The dependence and control degrees of one sector to other sectors can present the system's ecological hierarchy structure. The dependence degree means the ability of one sector receives urban solid waste from other sectors, while the control degree denotes the ability of one sector delivers urban solid waste to other sectors. The sum of all sectors' dependence (or control) degrees is equal to 1. To reflect how the variations in solid waste flow of a sector influence the USWS's ecological hierarchy structure, indexes (i.e., pulling force weight and driving force weight) in the ecological control analysis method are used to detect the sectoral dependence and control degrees as follows [31]:

$$\mathbf{Y} = \operatorname{diag}(\mathbf{T}) \ast \mathbf{N} \tag{9}$$

$$\mathbf{ID} = \mathbf{Y} - \mathbf{D} = y\_{ij} - f\_{ij} \tag{10}$$

$$w\_i = \sum\_{j=1}^{n} y\_{ij} / \sum\_{i=1}^{n} \sum\_{j=1}^{n} y\_{ij} \tag{11}$$

$$w\_{j} = \sum\_{i=1}^{n} y\_{ij} / \sum\_{i=1}^{n} \sum\_{j=1}^{n} y\_{ij} \tag{12}$$

where **Y** is the sectoral contribution weight, *yij* is the integral flow from sector *i* to *j*, **ID** is the indirect flows of solid waste of sectors, *w<sup>j</sup>* is the pulling force weight (PFW) of sector *j*, indicating the ability of sectors *j* receives solid waste from other sectors and *w<sup>i</sup>* is the driving force weight (DFW) of sector *i*, meaning the ability of sector *i* delivers solid waste to other sectors. The difference between PFW and DFW indicates the role one sector plays in the waste flow chain.

Ecological utility analysis can be utilized to reveal the interconnection among various sectors in the USWS. The dimensionless direct utility matrix **D** examines the mutual benefit, and the integral utility intensity matrix **U** contains all solid waste interflows pathway. **D** and **U** can be calculated based on Equations (13) and (14) [32,33]:

$$\mathbf{D} = [d\_{\mathrm{ij}}] = (f\_{\mathrm{ij}} - f\_{\mathrm{ji}}) / T\_{\mathrm{i}} \tag{13}$$

$$\mathbf{U} = (\mathbf{D})^0 + (\mathbf{D})^1 + (\mathbf{D})^2 + \dots \\ (\mathbf{D})^\infty = (\mathbf{I} - \mathbf{D})^{-1} \tag{14}$$

Transforming **U** to sign(**U**) (including sign**U**(+) and sign**U**(−)) judges the integral ecological relationships between pairwise sectors. Relationships include: (i) exploitation (+, −) means sector *i* exploits *j*, indicating sector *i* receive wastes from *j* (the same applies to (−, +)); (ii) competition (−, −) means the relationship is harmful to both sectors; (iii) mutualism (+, +) means the relationship is beneficial to both sectors; (iv) neutralized (0, 0) means there is no impact on each other. Three indexes are employed to assess the comprehensive properties of the USWS:

$$SI = \sum\_{i=1}^{n} \sum\_{j=1}^{n} \mu\_{ij} \tag{15}$$

$$MI = \text{sign}\mathcal{U}(+)/\text{sign}\mathcal{U}(-) \tag{16}$$

$$R = \frac{N(+,+) + N(-,-)}{N} \tag{17}$$

where sign*U*(+) and sign*U*(−) are the number of positive and negative signs in **U**; *N*(+, +) and *N*(−, −) are the amounts of mutualism and competition relationships and *N* is the total number of all relationships. Synergism index (*SI*) and mutualism index (*MI*) assess fitness and symbiosis of the USWS [34]. When *MI* > 1 and *SI* > 0, the USWS is mutualistic. Otherwise, the USWS requires to be modified.
