![\"\"](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/03\/20250308221453752.png\")
\u8003\u8651\u56fe1\u4e24\u8df3\uff082-hop\uff09\u8303\u56f4\u5185\u8ba1\u7b97\u56fe\u7684\u51e0\u4e2a\u7279\u6b8a\u60c5\u51b5\uff1a<\/p>\n\n\n\n
\uff081\uff09\u8282\u70b91\u548c\u8282\u70b94\uff0c\u5177\u6709\u76f8\u540c\u5ea6\u6570\uff0c\u4f46\u5230\u5176\u4e24\u8df3\uff082-hop\uff09\u90bb\u5c45\u7684\u4fe1\u606f\u4e0a\u5ea6\u6570\u4e0d\u540c\uff0c\u5bfc\u81f4\u8ba1\u7b97\u56fe\u7ed3\u6784\u4e0d\u540c\uff0c\u56e0\u6b64\u53ef\u4ee5\u533a\u5206\u8fd9\u4e24\u4e2a\u8282\u70b9\uff0c\u5982\u56fe2\u6240\u793a\u3002<\/p>\n\n\n
![\"\"](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/03\/20250308221543786.png\")
\uff082\uff09\u8282\u70b91\u548c\u8282\u70b95\uff0c\u56e0\u5176\u81ea\u8eab\u5ea6\u6570\u4e0d\u540c\u800c\u5177\u6709\u4e0d\u540c\u7684\u5c40\u90e8\u7ed3\u6784\u4fe1\u606f\uff0c\u5982\u56fe3\u6240\u793a\u3002<\/p>\n\n\n
![\"\"](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/03\/20250308221628190.png\")
\uff083\uff09\u8282\u70b91\u548c\u8282\u70b92\uff0c\u5177\u6709\u76f8\u540c\u5ea6\u6570\uff0c\u4e5f\u5177\u6709\u76f8\u540c\u7684\u90bb\u5c45\u7ed3\u6784\uff0c\u5728\u4e24\u8df3\u90bb\u5c45\u4e0a\uff0c\u90fd\u5177\u6709\u76f8\u540c\u7684\u5c40\u90e8\u7ed3\u6784\uff0c\u5982\u56fe4\u6240\u793a\u3002<\/p>\n\n\n
![\"\"](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/03\/20250308221925820.png\")
\u5728\u56fe4\u7684\u60c5\u51b5\u4e0b\uff0c\u5047\u8bbe\u6240\u6709\u8282\u70b9\u7684\u521d\u59cb\u5316\u7279\u5f81\u5411\u91cf\u76f8\u540c\uff0c\u4e14\u805a\u5408\u8282\u70b9\u4fe1\u606f\u65f6\u4ec5\u8003\u8651\u4e24\u8df3\u8303\u56f4\u5185\u7684\u4fe1\u606f\uff0c\u90a3\u4e48\u8282\u70b91\u548c\u8282\u70b92\u5c06\u88ab\u5d4c\u5165\u5230\u540c\u4e00\u4e2a\u8868\u793a\u5411\u91cf\uff0cGNN\u65e0\u6cd5\u533a\u5206\u8fd9\u4e24\u4e2a\u8282\u70b9\u3002\u76f8\u6bd4\u4e4b\u4e0b\u8282\u70b93\u30014\u30015\u7531\u4e8e\u8ba1\u7b97\u56fe\u4e0d\u540c\uff0c\u7406\u8bba\u4e0a\u662f\u53ef\u80fd\u7531GNN\u533a\u5206\u5f00\u7684\u3002GNN\u6a21\u578b\u5e94\u8be5\u6709\u5c06\u4e0d\u540c\u7684\u8ba1\u7b97\u56fe\u6620\u5c04\u5230\u4e0d\u540c\u7684\u8282\u70b9\u5411\u91cf\u7684\u80fd\u529b\u3002\u60f3\u8981\u533a\u5206\u4e0d\u540c\u7684\u8ba1\u7b97\u56fe\uff0c\u805a\u5408\u7684\u65b9\u5f0f\u662f\u5173\u952e\uff0c\u805a\u5408\u51fd\u6570\u8868\u8fbe\u80fd\u529b\u8d8a\u5f3a\uff0cGNN\u8868\u8fbe\u80fd\u529b\u8d8a\u5f3a\u3002<\/p>\n\n\n\n
\u7136\u800c\uff0c\u4e0d\u7ba1\u662f\u6c42\u548c\u3001\u6c42\u5e73\u5747\uff0c\u8fd8\u662f\u6c42\u6700\u5927\u503c\u8fd9\u4e9b\u5e7f\u6cdb\u88ab\u4f7f\u7528\u7684\u805a\u5408\u65b9\u5f0f\uff0c\u90fd\u4e0d\u662f\u6700\u4f73\u7684\u9009\u62e9\uff0c\u5982\u56fe5\u6240\u793a\uff0c\u4e24\u79cd\u56fe\u6570\u636e\u7684\u8ba1\u7b97\u56fe\u662f\u4e0d\u540c\u7684\u3002\u4f46\u9488\u5bf9\u56fe5 (a)\u7684\u521d\u59cb\u5316\u8282\u70b9\u5411\u91cf\uff0c\u6c42\u5e73\u5747\u548c\u6c42\u6700\u5927\u503c\u7684\u805a\u5408\u65b9\u5f0f\u90fd\u4e0d\u80fd\u6709\u6548\u7684\u533a\u5206\u51fa\u4e0d\u540c\u7684\u8ba1\u7b97\u56fe\u3002\u7c7b\u4f3c\u7684\uff0c\u5bf9\u4e8e\u56fe5(b)\u7684\u521d\u59cb\u5316\u8282\u70b9\u5411\u91cf\uff0c\u6c42\u548c\u7684\u805a\u5408\u65b9\u5f0f\u4e5f\u4e0d\u80fd\u6709\u6548\u533a\u5206\u8fd9\u4e24\u79cd\u8ba1\u7b97\u56fe\u3002\u8fd9\u662f\u56e0\u4e3a\u56fe\u901a\u8fc7\u805a\u5408\u8282\u70b9\u5411\u91cf\u540e\u5374\u5f97\u5230\u4e86\u76f8\u540c\u7684\u65b0\u7684\u8282\u70b9\u5411\u91cf\u3002<\/p>\n\n\n
![\"\"](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/03\/20250308222037476.png\")
\u4ece\u6570\u5b66\u7684\u89d2\u5ea6\u4e0a\u6765\u8bb2\uff0c\u5e0c\u671bGNN\u7684\u805a\u5408\u51fd\u6570\u662f\u4e00\u79cd\u5355\u5c04\u51fd\u6570\uff08Injective Function\uff09\uff0c\u5373\u53ef\u628a\u4e0d\u540c\u7684\u5143\u7d20\u6620\u5c04\u6210\u4e0d\u540c\u8f93\u51fa\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5e0c\u671bGNN\u7684\u805a\u5408\u51fd\u6570\u80fd\u628a\u4e0d\u540c\u7684\u8ba1\u7b97\u56fe\u6620\u5c04\u751f\u6210\u4e0d\u540c\u7684\u8282\u70b9\u5411\u91cf\u3002\u663e\u7136\uff0c\u6c42\u548c\u51fd\u6570\u3001\u6c42\u5e73\u5747\u51fd\u6570\u3001\u6c42\u6700\u5927\u503c\u51fd\u6570\u90fd\u4e0d\u662f\u5355\u5c04\u51fd\u6570\u3002\u6839\u636eXu et al. \u5728\u8bba\u6587Graph Wavelet Neural Network\u4e2d\u5f97\u5230\u5b9a\u7406\u53ef\u77e5\uff1a\u4efb\u4e00\u5355\u5c04\u51fd\u6570\u90fd\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n
\\(\\phi(\\sum_{x\\in S}f(x))\\)<\/p>\n\n\n
\u5176\u4e2d\uff0c\\(\\phi\\)\u548c\\(\\mathrm{f}\\)\u8868\u793a\u67d0\u79cd\u975e\u7ebf\u6027\u51fd\u6570\uff0c\\(\\mathrm{S}\\)\u662f\u8f93\u5165\u4fe1\u606f\\(\\mathrm{x}\\)\u7684\u96c6\u5408\u3002\u95ee\u9898\u7684\u5173\u952e\u662f\u8fd9\u4e2a\u975e\u7ebf\u6027\u51fd\u6570\\(\\phi\\)\u548c\\(\\mathrm{f}\\)\u8be5\u600e\u4e48\u6c42\u89e3\uff1f\u6b64\u65f6\u6df1\u5ea6\u5b66\u4e60\u7684\u9b45\u529b\u6765\u4e86\uff0c\u5bf9\u4e8e\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\u800c\u8a00\uff0c\u7406\u8bba\u4e0a\u5b83\u53ef\u4ee5\u62df\u5408\u4efb\u610f\u4efb\u4f55\u8fde\u7eed\u51fd\u6570\uff0c\u6240\u4ee5\u65e2\u7136\u4e0d\u6e05\u695a\u8981\u6c42\u89e3\u7684\u51fd\u6570\\(\\phi\\)\u548c\\(\\mathrm{f}\\)\u662f\u4ec0\u4e48\uff0c\u53ef\u4ee5\u76f4\u63a5\u901a\u8fc7\u795e\u7ecf\u7f51\u7edc\u8ba9\u5b83\u81ea\u5df1\u5b66\uff0c\u901a\u8fc7\u5b9a\u4e49\u635f\u5931\uff0c\u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u66f4\u65b0\u7f51\u7edc\u7684\u53c2\u6570\u5c31\u597d\u3002\u6b64\u65f6\uff0c\u8fd9\u4e2a\u5355\u5c04\u51fd\u6570\u7684\u5f62\u5f0f\u53d8\u6210\u4e86\uff1a<\/p>\n\n\n
\\(\\mathrm{MLP}_\\phi(\\sum_{xinS}\\mathrm{MLP}_f(x))\\)<\/p>\n\n\n
\u4f7f\u7528\u4e0a\u5f0f\u7684\u51fd\u6570\u4f5c\u4e3a\u805a\u5408\u51fd\u6570\u7684\u56fe\u795e\u7ecf\u7f51\u7edc\u88ab\u79f0\u4e3aGraph Isomorphism Network\uff08GIN\uff09\u3002\u5728GIN\u7684\u5b9e\u9645\u5b9e\u73b0\u4e2d\uff0c\\(\\phi\\)\u548c\\(\\mathrm{f}\\)\u4e24\u4e2a\u975e\u7ebf\u6027\u51fd\u6570\u662f\u901a\u8fc7\u4e00\u4e2aMLP\u5b9e\u73b0\u7684\uff0c\u56e0\u4e3aMLP\u672c\u8eab\u53ef\u4ee5\u8868\u793a\u51fd\u6570\u7684\u7ec4\u5408\u3002\u5728\u7b2c\u4e00\u6b21\u8fed\u4ee3\u4e2d\uff0c\u5982\u679c\u8f93\u5165\u7279\u5f81\u662f\u72ec\u70ed\u7f16\u7801\u5f62\u5f0f\uff0c\u5728\u6c42\u548c\u4e4b\u524d\u4e0d\u9700\u8981MLP\uff0c\u56e0\u4e3a\u72ec\u70ed\u7f16\u7801\u6c42\u548c\u672c\u8eab\u662f\u5355\u5c04\u7684\uff0c\u6700\u540e\uff0c\u4f7f\u7528\u4e00\u4e2a\u53ef\u5b66\u4e60\u53c2\u6570\\(\\epsilon\\)\uff08\u4e5f\u53ef\u4ee5\u662f\u4e00\u4e2a\u56fa\u5b9a\u7684\u7f29\u653e\u56e0\u5b50\uff09\u6765\u51b3\u5b9a\u805a\u5408\u8282\u70b9\u81ea\u8eab\u5411\u91cf\u65f6\u7684\u6bd4\u91cd\u3002\u5b8c\u6574\u7684GIN\u5c42\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n\n\n
\\(h_{\\nu}^{(k)}=\\mathrm{MLP}^{(k)}((1+\\epsilon^{(k)})\\cdot h_{\\nu}^{(k-1)}+\\sum_{u\\in N_{(v)}}h_{u}^{(k-1)})\\)<\/p>\n\n\n
\u5728\u4e0a\u5f0f\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\\(\\nu\\)\u7684\u65b0\u8868\u793a\\(h_{v}^{(k)}\\)\u5728\u7b2c\\(\\mathrm{k}\\)\u8f6e\u8fed\u4ee3\u4e2d\u662f\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u8ba1\u7b97\u5f97\u51fa\u7684\uff1a\u9996\u5148\u8fdb\u884c\u90bb\u5c45\u4fe1\u606f\u7684\u805a\u5408\uff0c\u5bf9\u8282\u70b9\\(\\nu\\)\u7684\u6240\u6709\u90bb\u5c45\u8282\u70b9\\(\\mathrm{u}\\)\u7684\u5f53\u524d\u8868\u793a\\(h_{u}^{(k-1)}\\)\u8fdb\u884c\u6c42\u548c\u805a\u5408\uff0c\u8fd9\u8868\u793a\u4e3a\\(\\sum_{u\\in N(\\nu)}h_{u}^{(k-1)}\\)\uff0c\u5176\u4e2d\\(N(\\nu)\\)\u662f\u8282\u70b9\\(\\nu\\)\u7684\u90bb\u5c45\u8282\u70b9\u96c6\u5408\u3002\u7136\u540e\u52a0\u4e0a\u8282\u70b9\\(\\nu\\)\u81ea\u8eab\u5728\u4e0a\u4e00\u8f6e\u8fed\u4ee3\u4e2d\u7684\u4fe1\u606f\\(h_{v}^{(k-1)}\\)\uff0c\u5e76\u5c06\u5176\u4e58\u4ee5\u4e00\u4e2a\u6743\u91cd\\(1+\\epsilon^{(k)}\\)\u3002\u8fd9\u91cc\u7684\\(\\epsilon^{(k)}\\)\u662f\u5728\u7b2c\\(\\mathrm{k}\\)\u8f6e\u8fed\u4ee3\u4e2d\u4e00\u4e2a\u53ef\u5b66\u4e60\u7684\u53c2\u6570\uff0c\u5b83\u5141\u8bb8\u6a21\u578b\u8c03\u6574\u81ea\u8eab\u8282\u70b9\u4fe1\u606f\u4e0e\u90bb\u5c45\u4fe1\u606f\u7684\u76f8\u5bf9\u91cd\u8981\u6027\u3002\u6700\u540e\uff0c\u5c06\u4e0a\u8ff0\u4e24\u90e8\u5206\u7684\u548c\u4f5c\u4e3a\u8f93\u5165\u4f20\u9012\u7ed9\u4e00\u4e2a\u591a\u5c42\u611f\u77e5\u5668\\(MLP^{(k)}\\)\uff0c\u8fd9\u662f\u4e00\u4e2a\u53ef\u5b66\u4e60\u7684\u975e\u7ebf\u6027\u51fd\u6570\uff0c\u7528\u6765\u751f\u6210\u8282\u70b9\\(\\nu\\)\u7684\u65b0\u8868\u793a \u3002\u5982\u679c\u8f93\u5165\u7279\u5f81\u4e0d\u662f\u72ec\u70ed\u7f16\u7801\u5f62\u5f0f\uff0c\u5b58\u5728\u91cd\u590d\u7684\u7279\u5f81\u5411\u91cf\u8868\u793a\uff0cGIN\u7684\u5f62\u5f0f\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n\n\n
\\(h_{\\nu}^{(k)}=\\mathrm{MLP}^{(k)}((1+\\epsilon^{(k)})\\cdot h_{\\nu}^{(k-1)}+\\sum_{u\\in N_{(\\nu)}}\\mathrm{MLP}(h_{u}^{(k-1)}))\\)<\/p>\n\n\n
\u4ece\u4e0a\u5f0f\u53ef\u4ee5\u770b\u51fa\uff0c\u5982\u679c\u8f93\u5165\u7279\u5f81\u53ef\u80fd\u5b58\u5728\u91cd\u590d\u7684\u7279\u5f81\u5411\u91cf\u8868\u793a\uff0c\u90a3\u4e48\u53ea\u9700\u8981\u5728\u805a\u5408\u5b8c\u8282\u70b9\\(\\nu\\)\u7684\u90bb\u5c45\u4fe1\u606f\u4e4b\u540e\uff0c\u5728\u4e0e\u8282\u70b9\\(\\nu\\)\u81ea\u8eab\u4fe1\u606f\u6c42\u548c\u4e4b\u524d\uff0c\u5bf9\u6bcf\u4e2a\u90bb\u5c45\u8282\u70b9\u7684\u8868\u793a\u5148\u8fdb\u884c\u4e00\u4e2aMLP\u53d8\u6362\u5373\u53ef\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u5728\u8bb2\u89e3Graph Isomorphism Network\uff08GIN\uff09\u4e4b\u524d\uff0c\u6709\u5fc5\u8981\u5148\u5b66\u4e60\u4e86\u89e3\u4e00\u4e0bGNN\u6a21\u578b\u7684\u8868\u8fbe\u80fd […]<\/p>\n","protected":false},"author":1,"featured_media":2859,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[23,26],"tags":[22],"class_list":["post-2858","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-23","category-26","tag-22"],"_links":{"self":[{"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/2858","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2858"}],"version-history":[{"count":3,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/2858\/revisions"}],"predecessor-version":[{"id":2866,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/2858\/revisions\/2866"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/media\/2859"}],"wp:attachment":[{"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2858"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2858"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2858"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}