{"id":2254,"date":"2024-10-09T09:09:36","date_gmt":"2024-10-09T01:09:36","guid":{"rendered":"https:\/\/www.gnn.club\/?p=2254"},"modified":"2024-10-10T14:42:02","modified_gmt":"2024-10-10T06:42:02","slug":"graph-convolutional-network%ef%bc%88gcn%ef%bc%89","status":"publish","type":"post","link":"http:\/\/gnn.club\/?p=2254","title":{"rendered":"Graph Convolutional Network\uff08GCN\uff09"},"content":{"rendered":"\n

\u5148\u6765\u770b\u4e00\u4e0b\u4e4b\u524d\u8bb2\u89e3\u7684\u6734\u7d20\u56fe\u795e\u7ecf\u7f51\u7edc\uff0c\u5982\u56fe1\u6240\u793a\u3002<\/p>\n\n\n

\n
\"\"
\u56fe1 \u6734\u7d20\u56fe\u795e\u7ecf\u7f51\u7edc<\/figcaption><\/figure><\/div>\n\n\n

\u56fe1\u5de6\u4e0a\u89d2\u65b9\u6846\u90e8\u5206\u53ef\u4ee5\u770b\u4f5c\u56fe\u795e\u7ecf\u7f51\u7edc\u7684\u521d\u59cb\u72b6\u6001\u3002\u4ee51\u53f7\u8282\u70b9\u4e3a\u4f8b\uff0c\u5728\u56fe\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u4fe1\u606f\u7684\u4f20\u9012\u662f\u5148\u6c47\u805a\u4e00\u53f7\u8282\u70b9\u7684\u90bb\u5c45\u8282\u70b9\u4fe1\u606f\uff0c\u5f97\u5230\u6c47\u805a\u540e\u7684\u65b0\u5411\u91cf\uff0c\u8fd9\u4e2a\u5411\u91cf\u53ef\u4ee5\u770b\u4f5c\u56fe\u795e\u7ecf\u7f51\u7edc\u7b2c\u4e00\u5c42\u7684\u8f93\u5165\u4fe1\u606f\\(H^{(l)}\\)\u3002\u7136\u540e\\(H^{(l)}\\) \u7ecf\u8fc7\u4e00\u4e2a MLP \u7684\u6620\u5c04, \u5f97\u5230\u4e00\u4e2a\u65b0\u7684\u8f93\u51fa\u5411\u91cf\\(H^{(l+1)}\\) , \u8fd9\u4e2a\u5411\u91cf\u5219\u4f5c\u4e3a\u7b2c\u4e8c\u5c42\u56fe\u795e\u7ecf\u7f51\u7edc\u7684\u8f93\u5165\u4fe1\u606f, \u4f9d\u6b21\u7c7b\u63a8, \u53ef\u4ee5\u5b9a\u4e49\u51fa\u4e00\u4e2a\u591a\u5c42\u7684\u56fe\u795e\u7ecf\u7f51\u7edc\u3002\u5c42\u4e0e\u5c42\u4e4b\u95f4\u7684\u4fe1\u606f\u4f20\u9012\u516c\u5f0f\u53ef\u4ee5\u5199\u4f5c:<\/p>\n\n\n

\\(H^{(l+1)}=\\sigma(H^{(l)}W^{(l)})\\)<\/p>\n\n\n

\u800c\u56fe\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u7684\u8ba1\u7b97\u516c\u5f0f\u5219\u4e3a\uff1a<\/p>\n\n\n

\\(H^{l+1}=\\sigma(\\tilde{D}^{-\\frac12}\\tilde{AD}^{-\\frac12}H(l)W(l))\\)<\/p>\n\n\n

\u4e24\u8005\u6700\u4e3b\u8981\u7684\u533a\u522b\u5c31\u662f\u56fe\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u6bd4\u56fe\u795e\u7ecf\u7f51\u7edc\u591a\u4e86\u4e00\u4e2a\\(\\tilde{\\boldsymbol{D}}^{-\\frac12}\\tilde{\\boldsymbol{AD}}^{-\\frac12}\\)\u3002\u5176\u4e2d\uff0c\\(\\tilde{A}=A+I_N\\)\uff0c \\(A\\) \u5c31\u662f\u56fe\u7684\u90bb\u63a5\u77e9\u9635\uff0c\\(I_{N}\\)\u662f\u4e00\u4e2a\u5168\u4e00\u7684\u5bf9\u89d2\u77e9\u9635\uff0c\u5373\uff1a<\/p>\n\n\n

\\(\\begin{aligned}&\\boldsymbol{A}=\\begin{pmatrix}0.&1.&1.&0.\\\\1.&0.&1.&0.\\\\1.&1.&0.&1.\\\\0.&0.&1.&0.\\end{pmatrix}&\\boldsymbol{I}_N=\\begin{pmatrix}1.&0.&0.&0.\\\\0.&1.&0.&0.\\\\0.&0.&1.&0.\\\\0.&0.&0.&1.\\end{pmatrix}\\\\&\\tilde{\\boldsymbol{A}}=\\boldsymbol{A}+\\boldsymbol{I}_N=\\begin{pmatrix}1.&1.&1.&0.\\\\1.&1.&1.&0.\\\\1.&1.&1.&1.\\\\0.&0.&1.&1.\\end{pmatrix}\\end{aligned}\\)<\/p>\n\n\n

\u90bb\u63a5\u77e9\u9635 \\(A\\)\u8868\u793a\u7684\u662f\u8282\u70b9\u4e0e\u8282\u70b9\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u5168\u4e00\u7684\u5bf9\u89d2\u77e9\u9635\\(I_{N}\\)\u8868\u793a\u7684\u662f\u8282\u70b9\u81ea\u8eab\uff0c\u6240\u4ee5\\(\\tilde{A}=A+I_N\\)\u8868\u793a\u8003\u8651\u4e86\u8282\u70b9\u81ea\u8eab\u4fe1\u606f\u7684\u90bb\u63a5\u77e9\u9635\u3002<\/p>\n\n\n\n

\u5148\u5c06\\(A\\)\u77e9\u9635\u52a0\u5165\u56fe\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u7684\u8ba1\u7b97\u516c\u5f0f\u4e2d\uff0c\u5f97\u5230\uff1a<\/p>\n\n\n

\\(H^{(l+1)}=\\sigma(\\tilde{A}H^{(l)}W^{(l)})\\)<\/p>\n\n\n

\u5176\u4e2d\uff0c\\(H^{(l)}\\)\u7684\u503c\u662f\u56fe1\u4e2d\u7684\u8282\u70b9\u521d\u59cb\u5316\u5411\u91cf\uff1a<\/p>\n\n\n

\\(\\tilde{\\boldsymbol{A}}\\boldsymbol{H}^{(l)}=\\begin{pmatrix}1.&1.&1.&0.\\\\1.&1.&1.&0.\\\\1.&1.&1.&1.\\\\0.&0.&1.&1.\\end{pmatrix}*\\begin{pmatrix}1.&0.&0.\\\\0.&1.&0.\\\\0.&0.&1.\\\\0.&1.&1.\\end{pmatrix}=\\begin{pmatrix}1.&1+1.&1.\\\\1.&1.&1.\\\\1.&1+1.&1+1.\\\\0.&1.&1+1.\\end{pmatrix}\\)<\/p>\n\n\n

\u4e0a\u5f0f\u8868\u793a\u6bcf\u4e2a\u8282\u70b9\u5411\u91cf\u8981\u540c\u65f6\u8003\u8651\u81ea\u8eab\u8282\u70b9\u548c\u5b83\u76f8\u90bb\u8282\u70b9\u7684\u4fe1\u606f\u3002\\(\\tilde{D}\\)\u662f\u5ea6\u77e9\u9635\uff0c\u8868\u793a\u7684\u662f\u6bcf\u4e2a\u8282\u70b9\u7684\u5ea6\u6570\u3002\u4f8b\u5982\uff0c\u56fe1\u4e2d\u7684\u4e00\u53f7\u548c\u4e8c\u53f7\u8282\u70b9\uff0c\u8003\u8651\u5176\u81ea\u8fde\u63a5\uff0c\u5219\u5ea6\u6570\u7b49\u4e8e3\uff1b\u540c\u7406\uff0c\u4e09\u53f7\u8282\u70b9\u5ea6\u6570\u4e3a4\uff0c\u800c\u56db\u53f7\u8282\u70b9\u5ea6\u6570\u4e3a2\uff0c\u56e0\u6b64\u5f97\u5230\u4e0b\u8ff0\u5ea6\u77e9\u9635\u3002<\/p>\n\n\n

\\(\\tilde{\\boldsymbol{D}}=\\begin{pmatrix}3.&0.&0.&0.\\\\0.&3.&0.&0.\\\\0.&0.&4.&0.\\\\0.&0.&0.&2.\\end{pmatrix}\\)<\/p>\n\n\n

\\(\\tilde{D}^{-1}\\)\u8868\u793a\u5c06\u5ea6\u77e9\u9635\u4e2d\u7684\u5143\u7d20\u53d6\u5012\u6570\uff0c\u5f97\u5230\uff1a<\/p>\n\n\n

\\(\\tilde{\\boldsymbol{D}}^{-1}=\\begin{pmatrix}1\/3.&0.&0.&0.\\\\0.&1\/3.&0.&0.\\\\0.&0.&1\/4.&0.\\\\0.&0.&0.&1\/2.\\end{pmatrix}\\)<\/p>\n\n\n

\\(\\tilde{D}^{-1}(\\tilde{A}H^{(l)})\\)\u76f8\u5f53\u4e8e\u5bf9\\(\\tilde{A}H^{(l)}\\)\u7ed3\u679c\u7684\u884c\u8fdb\u884c\u5f52\u4e00\u5316\u64cd\u4f5c\uff0c\u5b83\u6709\u52a9\u4e8e\u907f\u514d\u7531\u4e8e\u56fe\u4e2d\u8282\u70b9\u5ea6\u7684\u4e0d\u5747\u5300\u5206\u5e03\u5f15\u8d77\u7684\u68af\u5ea6\u7206\u70b8\u6216\u6d88\u5931\u95ee\u9898\u3002\u786e\u4fdd\u4e86\u6240\u6709\u8282\u70b9\u7684\u7279\u5f81\u8d21\u732e\u5728\u805a\u5408\u65f6\u88ab\u9002\u5f53\u5730\u7f29\u653e\uff0c\u4ece\u800c\u63d0\u9ad8\u4e86\u5b66\u4e60\u7684\u7a33\u5b9a\u6027\u548c\u6548\u679c\u3002\u516c\u5f0f\u8ba1\u7b97\u5982\u4e0b\uff1a<\/p>\n\n\n

\\(\\tilde{\\boldsymbol{D}}^{-1(}\\tilde{A}\\boldsymbol{H}^{(l)})=\\begin{pmatrix}1\/3.&0.&0.&0.\\\\0.&1\/3.&0.&0.\\\\0.&0.&1\/4.&0.\\\\0.&0.&0.&1\/2.\\end{pmatrix}*\\begin{pmatrix}1.&1+1.&1.\\\\1.&1.&1.\\\\1.&1+1.&1+1.\\\\0.&1.&1+1.\\end{pmatrix}=\\begin{pmatrix}1\/3*1.&1\/3*(1+1).&1\/3*1.\\\\1\/3*1.&1\/3*1.&1\/3*1.\\\\1\/4*1.&1\/4*(1+1).&1\/4*(1+1).\\\\1\/2*0.&1\/2*1.&1\/4*(1+1).\\end{pmatrix}\\)<\/p>\n\n\n

\u540c\u7406\uff0c\\((\\tilde{A}H^{(l)})\\tilde{D}^{-1}\\)\u76f8\u5f53\u4e8e\u5bf9\\(\\tilde{A}H^{(l)}\\)\u7ed3\u679c\u7684\u5217\u505a\u5f52\u4e00\u5316\u3002\u6700\u540e\uff0c\u5f0f\\(H^{l+1}=\\sigma(\\tilde{D}^{-\\frac12}\\tilde{AD}^{-\\frac12}H(l)W(l))\\)\u4e2d\u5de6\u4e58\u548c\u53f3\u4e58\u7684\u77e9\u9635\u4f7f\u7528\u7684\u662f\\(\\tilde{D}^{-\\frac12}\\)\u800c\u4e0d\u662f\\(\\boldsymbol{\\tilde{D}}^{-1}\\)\u7684\u539f\u56e0\u662f\uff1a\u5f53\u5bf9\u884c\u5217\u5143\u7d20\u5404\u505a\u4e00\u6b21\u5f52\u4e00\u5316\u540e\uff0c\u76f8\u5f53\u4e8e\u5bf9\u8282\u70b9\u5411\u91cf\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u505a\u4e86\u4e24\u6b21\u5f52\u4e00\u5316\uff0c\u4e5f\u5c31\u662f\u591a\u505a\u4e86\u4e00\u6b21\uff0c\u56e0\u6b64\u8fd9\u91cc\u4f7f\u7528\u505a\u5f52\u4e00\u5316\u7684\u77e9\u9635\u7684\u662f\\(\\tilde{D}^{-\\frac12}\\)\u3002\u6700\u7ec8\uff0c\u56fe\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u7684\u8ba1\u7b97\u516c\u5f0f\u5219\u4e3a\uff1a<\/p>\n\n\n

\\(\\boldsymbol{H}^{l+1}=\\sigma(\\tilde{\\boldsymbol{D}}^{-\\frac12}\\tilde{\\boldsymbol{AD}}^{-\\frac12}\\boldsymbol{H}(l)\\boldsymbol{W}(l))\\)<\/p>\n\n\n

\u5c3d\u7ba1\u56fe\u5377\u79ef\u7f51\u7edc\uff08GCN\uff09\u548c\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\uff08CNN\uff09\u90fd\u662f\u201c\u5377\u79ef\u201d\u6982\u5ff5\u7684\u6269\u5c55\uff0c\u4f46\u5b83\u4eec\u5728\u5904\u7406\u6570\u636e\u7684\u65b9\u5f0f\u548c\u7ed3\u6784\u4e0a\u6709\u5f88\u5927\u7684\u4e0d\u540c\u3002\u5728CNN\u4e2d\uff0c\u5377\u79ef\u662f\u5b9a\u4e49\u5728\u89c4\u5219\u7684\u7f51\u683c\u6570\u636e\uff08\u5982\u56fe\u50cf\uff09\u4e0a\u7684\u3002\u8fd9\u79cd\u5377\u79ef\u64cd\u4f5c\u901a\u5e38\u6d89\u53ca\u5230\u56fa\u5b9a\u5927\u5c0f\u7684\u6ee4\u6ce2\u5668\uff08\u5377\u79ef\u6838\uff09\u5728\u8f93\u5165\u6570\u636e\u4e0a\u6ed1\u52a8\uff0c\u63d0\u53d6\u5c40\u90e8\u7279\u5f81\u3002\u5728GCN\u4e2d\uff0c\u5377\u79ef\u64cd\u4f5c\u662f\u5728\u56fe\u7ed3\u6784\u4e0a\u5b9a\u4e49\u7684\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u9700\u8981\u8003\u8651\u56fe\u7684\u62d3\u6251\u7ed3\u6784\u3002GCN\u4e2d\u7684\u201c\u5377\u79ef\u201d\u662f\u901a\u8fc7\u805a\u5408\u4e00\u4e2a\u8282\u70b9\u53ca\u5176\u90bb\u5c45\u7684\u7279\u5f81\u6765\u5b9e\u73b0\u7684\uff0c\u8fd9\u79cd\u805a\u5408\u8003\u8651\u4e86\u56fe\u7684\u8fde\u63a5\u6027\u800c\u975e\u7a7a\u95f4\u7684\u4e34\u8fd1\u6027\u3002\u66f4\u91cd\u8981\u7684\u662f\u5728\u805a\u5408\u90bb\u5c45\u8282\u70b9\u4fe1\u606f\u65f6\uff0c\u5f15\u5165\u4e86\u81ea\u8eab\u8282\u70b9\u7684\u4fe1\u606f\uff0c\u5e76\u901a\u8fc7\u8282\u70b9\u7684\u5ea6\u5bf9\u805a\u5408\u7684\u7279\u5f81\u8fdb\u884c\u5f52\u4e00\u5316\u3002\u8fd9\u79cd\u5f52\u4e00\u5316\u5904\u7406\u6709\u52a9\u4e8e\u907f\u514d\u8282\u70b9\u5ea6\u7684\u5927\u5c0f\u5bf9\u7279\u5f81\u805a\u5408\u9020\u6210\u8fc7\u5927\u7684\u5f71\u54cd\uff0c\u4f7f\u5f97\u7279\u5f81\u8868\u793a\u66f4\u52a0\u5e73\u6ed1\u548c\u7a33\u5b9a\u3002\u603b\u7ed3\u6765\u8bf4\uff0c\u5c3d\u7ba1GCN\u4e2d\u7684\u201c\u5377\u79ef\u201d\u4e0e\u4f20\u7edf\u7684CNN\u5728\u6280\u672f\u7ec6\u8282\u4e0a\u6709\u6240\u4e0d\u540c\uff0c\u4f46\u5176\u6838\u5fc3\u601d\u60f3\u90fd\u662f\u5229\u7528\u5c40\u90e8\u8fde\u63a5\u7ed3\u6784\u4fe1\u606f\u8fdb\u884c\u7279\u5f81\u5b66\u4e60\u548c\u805a\u5408\uff0c\u8fd9\u4e5f\u662f\u4e3a\u4ec0\u4e48\u5c06\u8fd9\u79cd\u64cd\u4f5c\u79f0\u4e3a\u201c\u5377\u79ef\u201d\u7684\u539f\u56e0\u3002 <\/p>\n\n\n\n

\u53e6\u5916\uff0cGCN\u66f4\u9002\u7528\u4e8eTransductive\u4efb\u52a1\uff0c\u56e0\u4e3aGCN\u5728\u66f4\u65b0\u8282\u70b9\u5411\u91cf\u65f6\uff0c\u5229\u7528\u5230\u4e86\u57fa\u4e8e\u6574\u4e2a\u90bb\u63a5\u77e9\u9635\u4fe1\u606f\u7684\u62c9\u666e\u62c9\u65af\u77e9\u9635\\((\\tilde{D}^{-\\frac{1}{2}}\\tilde{AD}^{-\\frac{1}{2}})\\)\uff0c\u4ece\u8fd9\u79cd\u610f\u4e49\u4e0a\u8bb2\u5b83\u805a\u5408\u90bb\u5c45\u7279\u5f81\u7684\u65f6\u5019\uff0c\u8bad\u7ec3\u51fa\u6765\u7684\u6743\u91cd\\(W\\)\u662f\u8003\u8651\u4e86\u6574\u4e2a\u56fe\u7ed3\u6784\u7684\uff0c\u5982\u679c\u56fe\u7ed3\u6784\u6539\u53d8\uff0c\u90a3\u4e48\u5c31\u4e0d\u518d\u9002\u7528\u3002\u56e0\u6b64GCN\u5728\u5904\u7406Inductive\u4efb\u52a1\u65f6\u8868\u73b0\u4e0d\u4f73\u3002<\/p>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

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