## 3 Preliminaries
### Inter-Core Crosstalk  
Since MCF tightly encapsulates the spatial channel into each core, the effect of inter-core crosstalk on degrading the transmission signal should be considered in the relevant research. There are two ways to reduce this crosstalk. One is to keep the cores at an appropriate distance, which can be achieved by properly designing MCF. The other designs an effective spectrum allocation method to be studied in this paper. When allocating FSs to the current connection request, the crosstalk resulting from the possible FSs of other requests needs to be considered. If the crosstalk value of the allocated FSs on the selected transmission path is greater than the given crosstalk threshold, the crosstalk will seriously affect the recovery of the receiver signal and cause the current request to be blocked. Equation (2) is used to calculate the mean inter-core crosstalk (XT) [1], which serves as the basis for calculating the crosstalk value for each request and is used widely in existing studies. \[h=2k^{2}r/(\beta cp) \tag{1}\]  
Figure 5: Node architecture of core switching with modification to [17]. (Reprinted with permission from Ref. [17]. 2015, [PERSON]). \[XT=\frac{n-n\times exp[-(n+1)\times 2 hL]}{1+n\times exp[-(n-1)\times 2 hL]} \tag{2}\]  
where \(h\) in Equation (1) represents the mean increase in inter-core crosstalk per fiber length unit, and the parameters \(k,r,\beta,c_{p}\) are all related to MCF, denoting the coupling coefficient, bend radius, propagation constant, and core-pitch, respectively. We assume that these parameters cannot be changed. In Equation (2), \(XT\), \(n\), and \(L\) denote the crosstalk, the number of adjacent cores, and the length of each fiber, respectively. For the seven-core fiber shown in Figure 2c, \(n\) equals 6 for the center core 7 but equals 3 for the other cores. Therefore, core 7 is most seriously affected by crosstalk from adjacent cores. From the above equations, we can see that inter-core crosstalk is affected by the number of adjacent cores and the fiber length is disregarded in the algorithm design. Therefore, the main influencing factor is the first one, especially since crosstalk occurs only when the same FS is occupied on adjacent cores. Figure 6 illustrates the inter-core crosstalk calculation using the seven-core fiber in Figure 2c as an example. For the FS, \(XT_{k,l}^{f}\) represents the number of crosstalk FSs at link \(l\) on the selected path \(p\) for the connection request \(r_{k}\), and \(XT_{k}^{f}=\sum\limits_{l\in p}XT_{k,l}^{f}\) denotes the number of crosstalk FSs for \(r_{k}\). Cores \(1\) and \(2\), and cores \(2\) and \(3\) are adjacent. Therefore, there are crosstalk FSs between them. Cores \(1\) and \(3\) are not adjacent, so there is no crosstalk between them. The numbers \(1\) through \(9\) in the upper part of each core are the FS index. For request \(4\), the selected path length is \(3\), i.e., it occupies links \(1\), \(2\), and \(3\). For FS \(5\) on core \(2\) of link \(1\), request \(3\) and request \(5\) occupy the same FS on cores \(1\) and \(3\) that are adjacent to core \(2\), so request \(4\) has crosstalk at FS \(5\) and \(XT_{4,1}^{5}=2\). Analogously, for FS \(5\) on link \(2\), there is a crosstalk FS for request \(4\), which results from request \(5\), i.e., \(XT_{4,2}^{5}=1\). There is no request occupying FS \(5\) on core \(1\) and core \(3\) of link \(3\), then \(XT_{4,3}^{5}=0\). Therefore, the crosstalk value of request \(4\) at FS \(5\) equals \(XT_{4}^{5}=XT_{4,1}^{5}+XT_{4,2}^{5}+XT_{4,3}^{5}=2+1+0=3\). Similarly, the crosstalk value of request \(4\) at FS \(6\) is \(XT_{6}^{6}=XT_{6,1}^{6}+XT_{4,2}^{6}+XT_{4,3}^{6}=1+1+0=2\). The total crosstalk value of request \(4\) equals \(XT_{4}^{5}+XT_{4}^{5}=3+2=5\). When the crosstalk value of each request is greater than the crosstalk threshold, the network cannot provide transport service to the request.