2020-10-13 多智能体基本图论
總入度:
Δ=[a12+a13+a140000a21+a23+a240000a31+a32+a340000a41+a42+a43]\Delta=\left[\begin{array}{llll} a_{12}+a_{13}+a_{14} & 0 & 0 & 0\\ 0 & a_{21}+a_{23}+a_{24} & 0 & 0\\ 0 & 0 & a_{31}+a_{32}+a_{34} & 0\\ 0 & 0 & 0 & a_{41}+a_{42}+a_{43} \end{array}\right] Δ=?????a12?+a13?+a14?000?0a21?+a23?+a24?00?00a31?+a32?+a34?0?000a41?+a42?+a43???????
權重矩陣(aii=0a_{ii}=0aii?=0,aija_{ij}aij?表示從 jjj 到 iii 的路徑存在):
A=[0a12a13a14a210?a23a24a31a320a34a41a42a430]A=\left[\begin{array}{llll} 0 & a_{12} & a_{13} & a_{14}\\ a_{21} & 0 & -a_{23} & a_{24}\\ a_{31} & a_{32} & 0 & a_{34}\\ a_{41} & a_{42} & a_{43} & 0 \end{array}\right] A=?????0a21?a31?a41??a12?0a32?a42??a13??a23?0a43??a14?a24?a34?0??????
其中 aija_{ij}aij? 表示 jjj 指向 iii,如果存在就是1,不存在就是0。
Laplacian矩陣:
L=Δ?A=[a12+a13+a14?a12?a13?a14?a21a21+a23+a24?a23?a24?a31?a32a31+a32+a34?a34?a41?a42?a43a41+a42+a43]L=\Delta-A =\left[\begin{array}{llll} a_{12}+a_{13}+a_{14} & -a_{12} & -a_{13} & -a_{14}\\ -a_{21} & a_{21}+a_{23}+a_{24} & -a_{23} & -a_{24}\\ -a_{31} & -a_{32} & a_{31}+a_{32}+a_{34} & -a_{34}\\ -a_{41} & -a_{42} & -a_{43} & a_{41}+a_{42}+a_{43} \end{array}\right] L=Δ?A=?????a12?+a13?+a14??a21??a31??a41???a12?a21?+a23?+a24??a32??a42???a13??a23?a31?+a32?+a34??a43???a14??a24??a34?a41?+a42?+a43???????
總結
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