## 2 The Resilient Recovery of the SN Based on Outer Synchronization  
### Supply Network Model  
The supply network (SN) is also known as the supply chain network. The SN can be simplified to a single supply chain when there is only one chain in it. In the supply chain, raw materials, intermediate materials, and finished goods are procured exclusively. A simple supply chain has been described previously by a set of differential equations [31; 32; 33]. This model can express the complex behavior of the supply chain perfectly. However, it cannot express complex network structures, meaning it is only suitable for a single-chain structure. With the increase of supply objects and purchase objects, the supply chain structure is gradually upgraded to a network structure. The primary goal of the SN is to satisfy customer demands at the lowest cost. From the perspective of structure, SN can be viewed as a collection of nodes \(V=\{v_{i}\}\) (retailers, distributors, and manufacturers) and arcs \(E=\{e_{ij}\}\) (the set of relations between the enterprise nodes). SN can be defined as a set of self-organizing agents that interact through a series of links. Therefore, SN can be expressed as \(G=(V,E)\). Since there isn't a central, authoritative organizing node, the SN is a self-organizing system. A dynamic SN is formed when each node is treated as a dynamic system. Each node has the behavior of ordering goods, production, and sales, whether it is a retailer, distributor, or manufacturer. In view of these studies [31; 32; 33], the status of each node in the SN can be described as  
\[x_{i}(t)=\left[x_{i1}(t),x_{i2}(t),x_{i3}(t)\right]^{T} \tag{1}\]  
The status of each enterprise can be expressed as Equation (2). The parameters are expressed in Table 1.  
\[ \begin{array}{rcl}F_{i}(x_{i}(t),\alpha,\varphi_{i})&=&\left(\begin{array}[] {c}\dot{x}_{i1}(t)\\ \dot{x}_{i2}(t)\\ \dot{x}_{i3}(t)\end{array}\right)&=&\left(\begin{array}{c}m(x_{i2}(t)-x_{i1 }(t))\\ rx_{i1}(t)-x_{i1}(t)x_{i3}(t)-x_{i2}(t)\\ x_{i1}(t)x_{i2}(t)-bx_{i3}(t)\end{array}\right)&\\ &=&\left(\begin{array}{c}0\\ -x_{i1}(t)x_{i3}(t)-x_{i2}(t)\\ x_{i1}(t)x_{i2}(t)\end{array}\right)+&\left(\begin{array}{cccc}x_{i2}(t)-x_ {i1}(t)&0&0\\ 0&0&x_{i1}(t)\\ 0&x_{i3}(t)&0\end{array}\right)&\left(\begin{array}{c}m\\ -b\\ r\end{array}\right)\end{array}  \tag{2}\]  
Figure 1: The framework of this study. The variation of demand quantity \(\dot{x}_{i1}(t)\) relates to the dissatisfaction \(m\) in the preceding period. The variation of the supply quantity \(\dot{x}_{i2}(t)\) is influenced by the information distortion \(r\) and the supply quantities \(x_{i2}\) in the preceding period. It also needs to take into account both the demand quantities \(x_{i1}\) and the inventory \(x_{i3}\). The variation of inventory \(\dot{x}_{i3}(t)\) not only relates to the demand quantities \(x_{i1}\) and the supply quantities \(x_{i2}\), but also depends on safety stock [34]. The state of each enterprise can be seen as a process, including order-taking, product-making, and goods distribution; similarly, the status of each node is affected by its neighboring nodes [35; 36; 37]. This is expressed by Equation (3). \[\dot{x}_{i}(t)=F_{i}(x_{i}(t),\alpha,\varphi_{i})+\sum_{j\in N_{i}}g_{ij} \Gamma(x_{j}(t)-x_{i}(t)) \tag{3}\] \[\left( \begin{array}{c}\dot{x}_{i1}(t)\\ \dot{x}_{i2}(t)\\ \dot{x}_{i3}(t)\end{array} \right)=\left( \begin{array}{c}m(x_{i2}(t)-x_{i1}( t))\\ rx_{i1}(t)-x_{i1}(t)x_{i3}(t)-x_{i2}(t)\\ x_{i1}(t)x_{i2}(t)-bx_{i3}(t)\end{array} \right)+\sum_{j\in N_{i}}g_{ij}\left( \begin{array}{c}c_{1}\\ c_{2}\\ c_{3}\end{array} \right)\left( \begin{array}{c}x_{j1}(t)-x_{i1}(t)\\ x_{j2}(t)-x_{i2}(t)\\ x_{j3}(t)-x_{3}(t)\end{array} \right)\]  
In Equation (3), the first item represents the state of node \(v_{i}\) itself, and the second item represents the impact on the state of node \(v_{i}\) by the neighboring nodes with a supply relationship. The state of each node is mainly adjusted by its own demand quantities \(x_{i1}\), supply quantities \(x_{i2}\), and inventory \(x_{i3}\).