{"id":36125,"date":"2022-07-28T22:45:07","date_gmt":"2022-07-28T14:45:07","guid":{"rendered":"https:\/\/swarma.org\/?p=36125"},"modified":"2022-07-28T22:45:07","modified_gmt":"2022-07-28T14:45:07","slug":"%e7%8e%9b%e4%b8%bd%e5%a5%b3%e7%8e%8b%e5%a4%a7%e5%ad%a6-ginestra-bianconi%ef%bc%9a%e9%ab%98%e9%98%b6%e7%bd%91%e7%bb%9c%e7%9a%84%e6%8b%93%e6%89%91%e7%bb%93%e6%9e%84%e4%b8%8e%e5%8a%a8%e5%8a%9b%e5%ad%a6","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=36125","title":{"rendered":"\u739b\u4e3d\u5973\u738b\u5927\u5b66 Ginestra Bianconi\uff1a\u9ad8\u9636\u7f51\u7edc\u7684\u62d3\u6251\u7ed3\u6784\u4e0e\u52a8\u529b\u5b66"},"content":{"rendered":"
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\u57286\u670828\u65e5\u7684\u9ad8\u9636\u7f51\u7edc\u8bfb\u4e66\u4f1a<\/a>\u7b2c\u4e00\u671f\uff0c\u82f1\u56fd\u739b\u4e3d\u5973\u738b\u5927\u5b66\u5e94\u7528\u6570\u5b66\u6559\u6388 Ginestra Bianconi \u7cfb\u7edf\u6027\u4ecb\u7ecd\u4e86\u9ad8\u9636\u7f51\u7edc\u7684\u4e3b\u8981\u7ed3\u6784\u7ec4\u5408\u3001\u62d3\u6251\u548c\u51e0\u4f55\u7ed3\u6784\uff0c\u4ee5\u53ca\u9ad8\u9636\u7f51\u7edc\u62d3\u6251\u548c\u52a8\u529b\u5b66\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u3002\u672c\u6587\u662f\u6b64\u6b21\u62a5\u544a\u7684\u6574\u7406\u3002\u9ad8\u9636\u7f51\u7edc\u8bfb\u4e66\u4f1a<\/a>\u7531\u7535\u5b50\u79d1\u6280\u5927\u5b66\u5415\u7433\u5a9b\u8001\u5e08\u7b49\u53d1\u8d77\uff0c\u56f4\u7ed5\u9ad8\u9636\u4ea4\u4e92\u7f51\u7edc\u7684\u57fa\u672c\u6982\u5ff5\u3001\u6a21\u578b\u3001\u65b9\u6cd5\u4e0e\u5e94\u7528\u7b49\u7814\u7a76\u8fdb\u884c\u7814\u8ba8\uff0c\u8be6\u60c5\u89c1\u6587\u672b\uff0c\u6b22\u8fce\u611f\u5174\u8da3\u7684\u670b\u53cb\u53c2\u4e0e\u3002<\/strong><\/section>\n
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Ginestra Bianconi<\/span> <\/span>| \u8bb2\u8005<\/strong><\/span><\/span><\/p>\n

\u9ad8\u723d<\/span> <\/span>| \u6574\u7406<\/strong><\/span><\/strong><\/span><\/strong><\/p>\n

\u5218\u6ce2\u3001\u4efb\u6653\u9f99<\/span> <\/span>| \u6821\u5ba1<\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/p>\n

\u9093\u4e00\u96ea<\/span> <\/span>| <\/strong><\/span>\u7f16\u8f91<\/strong><\/p>\n


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\u4e2a\u4eba\u4e3b\u9875\uff1ahttps:\/\/www.qmul.ac.uk\/maths\/profiles\/bianconig.html<\/span><\/p>\n

\u53c2\u8003\u4e66\u7c4d\uff1aHigher order network: An Introduction to Simplicial Complexes<\/span><\/section>\n
\u8bba\u6587\u94fe\u63a5\uff1ahttps:\/\/pattern.swarma.org\/outlink?target=http:\/\/www.cambridge.org\/core_title\/gb\/546920<\/span><\/section>\n<\/section>\n<\/section>\n<\/blockquote>\n

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\u9ad8\u9636\u7f51\u7edc<\/span>\uff08Higher-order network\uff09<\/span>\u662f\u63cf\u8ff0\u4e24\u4e2a\u6216\u591a\u4e2a\u8282\u70b9\u4e4b\u95f4\u76f8\u4e92\u4f5c\u7528\u7684\u7f51\u7edc\u7ed3\u6784\uff0c\u4e0e\u7f51\u7edc\u51e0\u4f55\u3001\u7ec4\u5408\u4e0e\u7edf\u8ba1\u6027\u8d28\u3001\u7f51\u7edc\u62d3\u6251\u7d27\u5bc6\u5173\u8054\uff0c\u5e38\u7528\u8d85\u56fe<\/span>\uff08hypergraph\uff09<\/span>\u548c\u5355\u7eaf\u590d\u5f62<\/span>\uff08simplicial complex\uff09<\/span>\u6765\u8868\u793a\u3002\u63a5\u4e0b\u6765\u4ece\u6982\u5ff5\u5165\u624b\uff0c\u4e86\u89e3\u9ad8\u9636\u7f51\u7edc\u7ed3\u6784\u3002<\/span><\/section>\n
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\u00b7 \u8d85\u8fb9\uff08Hyperedges\uff09<\/span><\/strong>\uff1a\u662f\u5177\u6709\u76f8\u4e92\u4f5c\u7528\u7684\u8282\u70b9\u96c6\u5408\u3002\u5982\u56fe1\uff0c\u4e24\u4e2a\u8282\u70b9\u8fde\u63a5\u4e00\u6761\u8fde\u8fb9\uff0c\u5f62\u62102-\u8d85\u8fb9\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0cm\u4e2a\u8282\u70b9\u53ef\u4ee5\u5f97\u5230m-\u8d85\u8fb9\uff0c\u8868\u793am\u4e2a\u8282\u70b9\u4e4b\u95f4\u7684\u4ea4\u4e92\u3002<\/span><\/section>\n
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\u00b7 <\/span><\/strong>\u8d85\u56fe\uff08Hypergraphs\uff09<\/strong>\uff1a\u8d85\u56feG\u662f\u7531n\u4e2a\u8282\u70b9\u7684\u96c6\u5408V\u548c\u8d85\u8fb9\u7684\u96c6\u5408E\u5b9a\u4e49\uff0c\u5176\u4e2d(m+1)-\u8d85\u8fb9\u8868\u793am+1\u4e2a\u8282\u70b9\u7684\u96c6\u5408\uff0c\u5229\u7528\u8d85\u8fb9\u6765\u63cf\u8ff0\u8282\u70b9\u95f4\u7684\u591a\u4f53\u4ea4\u4e92\u3002\u6240\u4ee5\uff0c\u6bcf\u4e2a\u7531\u8282\u70b9\u5b50\u96c6\u5f62\u6210\u7684\u8d85\u8fb9\u53ef\u4ee5\u5c5e\u4e8e\u4e5f\u53ef\u4ee5\u4e0d\u5c5e\u4e8e\u8d85\u56fe\u3002<\/span><\/section>\n

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\u00b7 <\/span><\/strong>\u5355\u7eaf\u5f62\uff08Simplex\uff09<\/strong>\uff1a\u4e5f\u53ef\u79f0\u4e3a\u5355\u5f62\uff0c\u4e00\u4e2ad\u7ef4\u5355\u7eaf\u5f62<\/span>\uff08\u4e5f\u8868\u793a\u4e3ad-\u5355\u5f62\uff09<\/span>\u662f\u7531\u4e00\u7ec4(d+1)\u76f8\u4e92\u4f5c\u7528\u7684\u8282\u70b9\u5f62\u6210\u7684\u3002\u5b83\u63cf\u8ff0\u4e86\u8282\u70b9\u95f4\u7684\u591a\u4f53\u4ea4\u4e92\uff0c\u5e76\u5141\u8bb8\u5355\u5f62\u7684\u62d3\u6251\u548c\u51e0\u4f55\u89e3\u91ca\u3002\u4f8b\u5982\uff0c\u8282\u70b9\u662f0-\u5355\u7eaf\u5f62\uff0c\u94fe\u63a5\u662f1-\u5355\u7eaf\u5f62\uff0c\u4e09\u89d2\u5f62\u662f2-\u5355\u7eaf\u5f62\uff0c\u56db\u9762\u4f53\u662f3-\u5355\u7eaf\u5f62\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002<\/span><\/section>\n

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\u00b7 \u5355\u5f62\u7684\u9762\uff08Faces\uff09<\/span><\/strong>\uff1ad\u7ef4\u5355\u5f62\u7684\u9762\u662f\u7531\u5355\u5f62\u7684\u8282\u70b9\u7684\u9002\u5f53\u5b50\u96c6\u6784\u6210\u7684\u5355\u5f62\u3002\u5728\u56fe3\u4e2d\uff0c\u523b\u753b\u4e86\u56db\u9762\u4f53\u7684\u9762\u30023\u7ef4\u5355\u7eaf\u5f62\u7684\u9762\u5305\u542b\uff1a4\u4e2a 0-\u5355\u7eaf\u5f62\uff0c6\u4e2a 1-\u5355\u7eaf\u5f62\uff0c4\u4e2a 2-\u5355\u7eaf\u5f62\u3002<\/span><\/section>\n

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\u00b7 \u5355\u7eaf\u590d\u5f62\uff08Simplicial complex\uff09<\/span><\/strong>\uff1a\u53c8\u79f0\u590d\u5f62\uff0c\u662f\u4e00\u4e2a\u7531\u5355\u7eaf\u5f62\u7ec4\u6210\u7684\u96c6\u5408\uff0c\u6bcf\u4e2a\u5355\u7eaf\u5f62\u7684\u9762\u90fd\u662f\u6b64\u96c6\u5408\u4e2d\u7684\u5143\u7d20\uff0c\u5176\u7ef4\u5ea6\u4e3a\u8be5\u96c6\u5408\u4e2d\u6700\u5927\u7ef4\u5ea6\u5355\u7eaf\u5f62\u7684\u7ef4\u5ea6\u3002<\/span><\/section>\n
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\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5355\u7eaf\u590d\u5f62\u4e0e\u8d85\u56fe\u7684\u533a\u522b\u5728\u4e8e\u5355\u7eaf\u590d\u5f62\u5305\u542b\u7ed9\u5b9a\u5355\u7eaf\u5f62\u7684\u6240\u6709\u5b50\u96c6\u3002<\/strong><\/span><\/section>\n
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\u00b7 \u7ef4\u9762\uff08Facets\uff09<\/span><\/strong>\uff1a\u7ef4\u9762\u662f\u5355\u7eaf\u590d\u5f62\u4e2d\u7684\u5355\u5f62\uff0c\u5b83\u4e0d\u662f\u4efb\u4f55\u5176\u4ed6\u5355\u7eaf\u5f62\u7684\u9762\u3002\u56e0\u6b64\uff0c\u5355\u7eaf\u590d\u5f62\u5b8c\u5168\u7531\u5b83\u7684\u7ef4\u9762\u7684\u5e8f\u5217\u51b3\u5b9a\u3002<\/span><\/section>\n

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\u00b7 \u7eaf\u5355\u7eaf\u590d\u5f62\uff08Pure simplicial complex\uff09<\/span><\/strong>\uff1a\u4e00\u4e2a\u7eafd\u7ef4\u5355\u7eaf\u590d\u5f62\u662f\u7531\u4e00\u7ec4d\u7ef4\u5355\u5f62\u53ca\u5176\u9762\u6784\u6210\u7684\u3002\u56e0\u6b64\uff0c\u7eafd\u7ef4\u5355\u7eaf\u590d\u5f62\u53ea\u5141\u8bb8d\u7ef4\u5355\u5f62\u4f5c\u4e3a\u9762\u3002\u8fd9\u610f\u5473\u7740\u7eafd\u7ef4\u5355\u7eaf\u5f62\u590d\u5f62\u662f\u7531d\u7ef4\u5355\u7eaf\u5f62\u6cbf\u7740\u5b83\u4eec\u7684\u8868\u9762\u7c98\u63a5\u800c\u5f62\u6210\u7684\u3002<\/span><\/section>\n

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\u00b7 \u5355\u5143\u590d\u5f62\uff08Cell complex\uff09<\/span><\/strong>\uff1a\u5176\u5177\u6709\u4e24\u4e2a\u6027\u8d28\uff1a(a) \u5b83\u7531\u4e00\u7ec4\u95ed\u5305\u5355\u5143\u7ec4\u6210\uff0c\u8fd9\u610f\u5473\u7740\u6bcf\u4e2a\u5355\u5143\u90fd\u88ab\u4e00\u4e2a\u6709\u9650\u7684\u5f00\u653e\u5355\u5143\u8054\u5408\u8986\u76d6\uff1b(b) \u7ed9\u5b9a\u5355\u5143\u590d\u5f62\u7684\u4e24\u4e2a\u5355\u5143\uff0c\u5b83\u4eec\u7684\u4ea4\u96c6\u5c5e\u4e8e\u5355\u5143\u590d\u5f62\uff0c\u6216\u8005\u5b83\u4eec\u7684\u4ea4\u96c6\u662f\u7a7a\u96c6\u3002<\/span><\/section>\n

<\/span><\/section>\n

\u00b7 \u5355\u7eaf\u590d\u5f62\u9aa8\u67b6\uff08Simplicial complex skeleton\uff09<\/span><\/strong>\uff1ad\u7ef4\u5355\u7eaf\u590d\u5f62K\u4e2d\u6240\u6709\u5c0f\u4e8e\u7b49\u4e8e\u7ef4\u5ea6r\u7684\u5355\u7eaf\u6027\u7684\u96c6\u5408\uff0c\u79f0\u4e3a\u5355\u7eaf\u590d\u5f62K\u7684r\u7ef4\u9aa8\u67b6\uff0c\u8bb0\u4e3aKr<\/sup><\/span>\u3002\u5982\u679c\u53ea\u8003\u8651\u4e00\u4e2a\u5355\u7eaf\u590d\u5f62\u7f51\u7edc\u4e2d\u7684\u8282\u70b9\u548c\u8fde\u8fb9\uff0c\u5373\u4e3a\u8be5\u5355\u7eaf\u590d\u5f62\u7f51\u7edc\u76841\u7ef4\u9aa8\u67b6\u3002<\/span><\/p>\n


<\/span><\/p>\n

\u00b7 \u56e2\u590d\u5f62\uff08Clique complex\uff09<\/span><\/strong>\uff1a\u5047\u5b9a\u4e8c\u5143\u7f51\u7edc\u4e2d\u7684\u6bcf\u4e2a\u56e2\u90fd\u662f\u5355\u7eaf\u5f62\uff0c\u5c31\u53ef\u4ee5\u4ece\u4e8c\u5143\u7f51\u7edc\u4e2d\u751f\u6210\u5355\u7eaf\u590d\u5f62\u7f51\u7edc\u3002\u5982\u679c\u7ed9\u51fa\u4e00\u4e2a\u53ea\u6709\u8282\u70b9\u548c\u8fde\u8fb9\u7ec4\u6210\u7684\u4e8c\u5143\u7f51\u7edc\uff0c\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u5176\u4e2d\u7684\u6240\u6709\u56e2\uff0c\u628a\u6bcf\u4e2a\u56e2\u4f5c\u4e3a\u5355\u7eaf\u5f62\u4f7f\u7f51\u7edc\u8f6c\u6362\u4e3a\u4e00\u4e2a\u56e2\u590d\u5f62\u7f51\u7edc\u3002<\/span><\/p>\n


<\/span><\/strong><\/p>\n

\u00b7 \u5e7f\u4e49\u5ea6\uff08Generalized degrees\uff09<\/span><\/strong>\uff1a\u4e00\u4e2am\u7ef4\u5355\u7eaf\u5f62<\/span>\u03b1<\/span>\u7684\u5e7f\u4e49\u5ea6kd, m<\/sub>(\u03b1<\/span>)\u8868\u793a\u5173\u8054\u5230\u8be5\u5355\u7eaf\u5f62\u03b1<\/span>\u7684d\u7ef4\u5355\u7eaf\u5f62\u7684\u6570\u91cf\u3002<\/span><\/p>\n


<\/section>\n
<\/section>\n
\u56fe4 \u5e7f\u4e49\u5ea6\u793a\u4f8b\u56fe<\/span><\/section>\n

<\/span><\/section>\n
2. \u7f51\u7edc\u62d3\u6251\u4e0e\u51e0\u4f55<\/span><\/strong><\/span>
<\/section>\n

<\/section>\n
\u590d\u6742\u7cfb\u7edf\u7684\u7f51\u7edc\u51e0\u4f55\u662f\u7f51\u7edc\u6f14\u5316\u7684\u5148\u9a8c\u524d\u63d0\u8fd8\u662f\u7f51\u7edc\u52a8\u529b\u5b66\u7684\u6d8c\u73b0\u73b0\u8c61\uff1f\u5728\u6d8c\u73b0\u51e0\u4f55\u7684\u6846\u67b6\u4e0b\uff0c\u5177\u6709\u51e0\u4f55\u7ed3\u6784\u7684\u7f51\u7edc\u662f\u7531\u7eaf\u7cb9\u7ec4\u5408\u7684\u975e\u5e73\u8861\u52a8\u529b\u5b66<\/strong>\u751f\u6210\u7684\uff0c\u5373\u72ec\u7acb\u4e8e\u7f51\u7edc\u51e0\u4f55\u7ed3\u6784\u7684\u975e\u5e73\u8861\u52a8\u529b\u5b66\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u5177\u6709\u504f\u597d\u7684\u7f51\u7edc\u51e0\u4f55\uff08Network Geometry with Flavor \uff0cNGF\uff09<\/span><\/strong>\uff1a\u662f\u4e00\u4e2a\u901a\u7528\u7684\u6570\u5b66\u6846\u67b6\uff0c\u7528\u4e8e\u589e\u957f\u663e\u793a\u6d8c\u73b0\u7684\u53cc\u66f2\u7f51\u7edc\u51e0\u4f55\u7684\u7b80\u5355\u590d\u5408\u4f53\u3002\u53cd\u6620\u4e86\u6d8c\u73b0\u53cc\u66f2\u51e0\u4f55\u4e2d\u53ef\u76f4\u63a5\u63cf\u8ff0\u7684\u4e30\u5bcc\u7684\u76f8\u4e92\u4f5c\u7528\u3002<\/span><\/section>\n


<\/p>\n

Slim Triangle Conditiond\uff08<\/span>\u03b4<\/span>–<\/span>slim \u4e09\u89d2\u5f62\u6027\u8d28\uff09<\/span><\/strong> \uff1a\u5982\u679c\u5b58\u5728\u4e00\u4e2a\u03b4<\/span>>0\uff0c\u4f7f\u5f97\u5bf9\u4e8e\u4efb\u610f\u7684\u4e09\u4e2a\u8282\u70b9r\uff0cs\uff0cq\uff0c\u4e24\u4e24\u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\u4e3aPrs<\/sub>\uff0cPsq<\/sub>\uff0cPrq<\/sub>\uff0c\u4efb\u610f\u4e24\u4e2a\u6700\u77ed\u8def\u5f84\u03b4<\/span>-\u90bb\u57df\u7684\u5e76\u96c6\uff08\u4f8b\u5982\uff0c<\/sub><\/span>\uff09\u90fd\u5305\u62ec\u53e6\u4e00\u6761\u8def\u5f84\u4e2d\u7684\u8282\u70b9(P<\/span>rq<\/sub> )\u3002\u03b4<\/span>-\u90bb\u57df\u662f\u6307\u5230\u4e00\u6761\u8def\u5f84\u4e2d\u6240\u6709\u8282\u70b9\u7684\u8ddd\u79bb\u90fd\u5c0f\u4e8e\u7b49\u4e8e\u03b4<\/span>\u7684\u8282\u70b9\u548c\u8fde\u8fb9\u7ec4\u6210\u7684\u96c6\u5408\u3002<\/span><\/p>\n


<\/span><\/p>\n

\u00b7 \u683c\u7f57\u83ab\u592b\u53cc\u66f2\u6027\uff08Gromov <\/span><\/strong>\u03b4<\/span>-hyperbolicity\uff09<\/span><\/strong>\uff1a\u5982\u679c\u4e00\u4e2a\u7f51\u7edc\u670d\u4ece\u03b4<\/span>-slim \u4e09\u89d2\u5f62\u6027\u8d28\uff0c\u5219\u79f0\u8fd9\u4e2a\u7f51\u7edc\u662f\u03b4<\/span>-\u53cc\u66f2\u7f51\u7edc\u3002<\/span><\/p>\n


<\/span><\/p>\n

\u56fe5(a)\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u4e09\u8282\u70b9\u7f51\u7edc\u793a\u610f\u56fe\uff0c\u8fd9\u4e09\u4e2a\u8282\u70b9\u7ec4\u6210\u7684\u4e09\u89d2\u5f62\u6ee1\u8db3\u03b4<\/span>-slim \u4e09\u89d2\u5f62\u6027\u8d28\uff0c\u56e0\u6b64\u8fd9\u4e2a\u7f51\u7edc\u53ef\u4ee5\u88ab\u79f0\u4f5c\u4e3a\u03b4<\/span>-\u53cc\u66f2\u7f51\u7edc\u3002\u56fe5(b)\u662f\u4e00\u4e2a\u6811\u72b6\u7f51\u7edc\uff0c\u5728\u56fe\u4e2d\u4e09\u4e2a\u6a59\u8272\u8282\u70b9\u6ee1\u8db3\u03b4<\/span>-slim \u4e09\u89d2\u5f62\u6027\u8d28\uff0c\u5e76\u4e14\u4e09\u6761\u6700\u77ed\u8def\u5f84\u4ea4\u4e8e\u84dd\u8272\u8282\u70b9\uff0c\u6240\u4ee5\u03b4<\/span>=0\uff1b\u56fe5(c)\u662f\u4e00\u4e2aNGF\u7f51\u7edc\uff0c\u5176\u4e2d\u7684\u4e09\u4e2a\u6a59\u8272\u8282\u70b9\u6ee1\u8db3\u03b4<\/span>-slim \u4e09\u89d2\u5f62\u6027\u8d28\uff0c\u5e76\u4e14\u4e09\u6761\u6700\u77ed\u8def\u5f84\u4e4b\u95f4\u7684\u8ddd\u79bb\u6700\u591a\u4e3a1\uff0c\u56e0\u6b64\u6b64\u03b4<\/span>=1\u3002<\/span><\/p>\n


<\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe5 Gromov\u53cc\u66f2\u7387\u793a\u4f8b\u56fe<\/span><\/section>\n

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 <\/span><\/section>\n

\n
\n
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<\/svg><\/section>\n<\/section>\n<\/section>\n
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\u4e8c\u3001\u9ad8\u9636\u52a8\u529b\u5b66<\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

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<\/section>\n<\/section>\n<\/section>\n


<\/p>\n<\/section>\n<\/h3>\n

1. \u4ee3\u6570\u62d3\u6251<\/span><\/strong><\/span><\/h3>\n

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\u00b7 \u8d1d\u8482\u6570\uff08Betti numbers\uff09<\/span><\/strong><\/span><\/section>\n

<\/span><\/section>\n
\u8d1d\u8482\u6570\u8868\u793a\u5355\u7eaf\u590d\u5f62\u7684m\u7ef4\u6d1e\u7684\u6570\u91cf\uff0c\u4e5f\u7b49\u4e8e\u540c\u8c03\u7fa4\u7684\u79e9\u3002\u5176\u4e2d\u540c\u8c03\u7fa4\u662f\u7a7a\u95f4\u8868\u793a\u4e2d\u7684m-\u5faa\u73af\u94fe\u7684\u540c\u8c03\u7c7b\uff0c\u5b83\u4eec\u4e0e\u4e0d\u662f(m+1)-\u94fe\u8fb9\u754c\u7684\u5faa\u73af\u94fe\u6709\u5173\u3002<\/span><\/section>\n

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<\/section>\n

\u56fe6 \u8d1d\u8482\u6570\u793a\u4f8b\u56fe<\/span><\/p>\n

\uff08\u56fe\u4e2d\u8868\u793a\u4e00\u4e2a\u70b9\u3001\u4e00\u4e2a\u5706\u3001\u4e00\u4e2a\u7403\u548c\u4e00\u4e2a\u73af\u9762\u53ca\u5176\u5bf9\u5e94\u7684Betti\u6570\uff09<\/span><\/p>\n


<\/span><\/section>\n
\u00b7 \u5355\u5f62\u65b9\u5411\uff08Orientation of the simplex\uff09<\/span><\/strong><\/section>\n

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<\/section>\n

<\/section>\n
\u4e00\u4e2am\u7ef4\u5b9a\u5411\u5355\u7eaf\u5f62\u662f\u4e00\u4e2am+1\u4e2a\u8282\u70b9\u7684\u96c6\u5408\uff0c\u5b83\u4e0e\u4e00\u4e2a\u65b9\u5411\u76f8\u5173\u8054\uff0c\u5176\u4e2d\u03b4(\u03c0)\u8868\u793a\u7f6e\u6362\u7684\u5947\u5076\u6027\u3002<\/span><\/section>\n

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\u00b7 \u8fb9\u754c\u7b97\u5b50\uff08Boundary operator\uff09<\/span><\/strong><\/span><\/section>\n

<\/span><\/section>\n
\u8fb9\u754c\u6620\u5c04\u662f\u4e00\u4e2a\u7ebf\u6027\u7b97\u5b50\uff0c \u7531\u4e0b\u5f0f\u7ed9\u51fa\uff0c\u662f\u7531\u5355\u7eaf\u590d\u5f62\u7684\u6bcf\u4e2am-\u5355\u5f62\u4e0a\u7684\u4f5c\u7528\u51b3\u5b9a\u7684\u3002<\/span><\/section>\n

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<\/section>\n

<\/section>\n
\u8fb9\u754c\u7b97\u5b50\u5177\u6709\u8fb9\u754c\u7684\u8fb9\u754c\u4e3a\u7a7a<\/strong>\u7684\u6027\u8d28\uff0c \u8fd9\u4e2a\u6027\u8d28\u76f4\u63a5\u6765\u81ea\u8fb9\u754c\u7684\u5b9a\u4e49\u3002<\/span><\/section>\n
<\/span><\/section>\n

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\u00b7 \u62d3\u6251\u805a\u7c7b\uff08Topological clustering\uff09<\/span><\/strong><\/span><\/section>\n

<\/span><\/section>\n
\u7531\u8282\u70b9\u7684\u6240\u6709\u90bb\u57df\u53ca\u5176\u8fde\u63a5\u96c6\u5408\u6240\u5f62\u6210\u7684\u56e2\u5355\u7eaf\u590d\u5f62\u3002<\/span><\/section>\n

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 <\/span><\/section>\n
2. \u62d3\u6251\u4fe1\u53f7\uff0c\u970d\u5947\u62c9\u666e\u62c9\u65af\u7b97\u5b50\u548c\u72c4\u62c9\u514b\u7b97\u5b50<\/span><\/strong><\/span><\/section>\n

<\/span><\/section>\n
\u00b7 \u62d3\u6251\u4fe1\u53f7\uff08Topological signals\uff09<\/span><\/strong><\/span><\/section>\n

<\/span><\/section>\n
\u5355\u7eaf\u590d\u5f62\u548c\u7f51\u7edc\u4e0d\u4ec5\u53ef\u4ee5\u652f\u6301\u5b9a\u4e49\u5728\u8282\u70b9\u4e0a\u7684\u52a8\u6001\u53d8\u91cf<\/span>\uff08\u4fe1\u53f7\uff09<\/span>\uff0c\u800c\u4e14\u53ef\u4ee5\u652f\u6301\u5b9a\u4e49\u5728\u9ad8\u9636\u5355\u7eaf\u5f62\u4e0a\u7684\u52a8\u6001\u53d8\u91cf<\/span>\uff08\u4fe1\u53f7\uff09<\/span>\uff0c\u8fd9\u4e9b\u4fe1\u53f7\u79f0\u4e3a\u62d3\u6251\u4fe1\u53f7\u3002\u62d3\u6251\u4fe1\u53f7\u5305\u62ec\uff1a\u534f\u4f5c\u7f51\u7edc\u4e2d\u7684\u5f15\u6587\uff1b\u7ed9\u5b9a\u4f4d\u7f6e\u7684\u98ce\u901f\uff1b\u6d77\u6d0b\u7ed9\u5b9a\u4f4d\u7f6e\u7684\u6d77\u6d41\uff1b\u751f\u7269\u8fd0\u8f93\u7f51\u7edc\u4e2d\u7684\u6d41\u91cf\uff1b\u7a81\u89e6\u4fe1\u53f7\uff1b\u5927\u8111\u4e2d\u7684\u8fb9\u7f18\u4fe1\u53f7\u7b49\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u5173\u8054\u77e9\u9635\u7684\u62c9\u666e\u62c9\u65af\u77e9\u9635\uff08Graph Laplacian in terms of the incidence matrix\uff09<\/span><\/strong>\u8868\u793a\u4e3a\uff1a<\/span><\/section>\n
<\/section>\n

<\/section>\n
\u9ad8\u9636Laplacians\u53ef\u4ee5\u7528\u5173\u8054\u77e9\u9635\u5b9a\u4e49\u4e3a\uff1a<\/span><\/section>\n

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\u970d\u5947\u5206\u89e3\uff08Hodge decomposition\uff09\uff1a<\/span><\/strong>\u5b9a\u7406\u8868\u793a\u4e3a\u66f2\u9762\u4e0a\u4efb\u610f\u4e00\u4e2a\u5149\u6ed1\u5207\u5411\u91cf\u573a\uff0c\u53ef\u4ee5\u88ab\u552f\u4e00\u5730\u5206\u89e3\u4e3a\u4e09\u4e2a\u5411\u91cf\u573a\uff1a<\/span>\u68af\u5ea6\u573a\u3001\u6563\u5ea6\u573a\u548c\u8c03\u548c\u573a\u3002<\/span>\u970d\u5947\u5206\u89e3\u7ecf\u5e38\u88ab\u7528\u4e8e\u5149\u6ed1\u5316\u4e00\u4e2a\u77e2\u91cf\u573a\uff0c\u5c06\u4e00\u4e2a\u4e0d\u53ef\u79ef\u77e2\u91cf\u573a\u53d8\u5f97\u53ef\u79ef\u3002<\/span><\/section>\n

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\u5728\u7f51\u7edc\u4e2d\uff0c\u7528\u62d3\u6251\u65cb\u91cf<\/span>\uff08Topological spinor\uff09<\/span>\u63cf\u8ff0\u7f51\u7edc\u62d3\u6251\u4fe1\u53f7\u7684\u52a8\u6001\u72b6\u6001\uff0c\u662f\u4e00\u4e2a\u75310-\u534f\u94fe\u548c1-\u534f\u94fe\u6784\u6210\u7684\u5757\u7ed3\u6784\u5411\u91cf\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u7f51\u7edc\u4e0a\u7684Dirac\u7b97\u5b50<\/span>\uff08Topological Dirac operator on a network\uff09<\/span>\u5b9a\u4e49\uff1a<\/span><\/section>\n

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\u76f8\u5e94\u5730\uff0c\u53ef\u4ee5\u63a8\u5e7f\u5230\u9ad8\u7ef4\u5355\u5f62\u3002<\/span><\/section>\n

<\/section>\n
3. \u7f51\u7edc\u4e0a\u7684Kuramoto\u6a21\u578b\uff08\u540c\u6b65\uff09<\/span><\/strong><\/span><\/section>\n

<\/section>\n
\u4e0d\u540c\u9891\u7387\u7684\u632f\u8361\u5668\u8026\u5408\u5f3a\u5ea6\u8db3\u591f\u5927\u65f6\uff0c\u4f1a\u51fa\u73b0\u540c\u6b65\u73b0\u8c61\u3002\u5728\u751f\u6d3b\u4e2d\u7684\u540c\u6b65\u73b0\u8c61\u6709\u5927\u8111\u52a8\u529b\u5b66\uff0c\u8424\u706b\u866b\u7684\u540c\u65f6\u95ea\u70c1\u7b49\u73b0\u8c61\u3002<\/span><\/section>\n
 <\/span><\/section>\n
1975 \u5e74 Yoshiki Kuramoto \u63d0\u51fa\u7684Kuramoto\u6a21\u578b\u662f\u7814\u7a76\u540c\u6b65\u7684\u4e00\u4e2a\u91cd\u8981\u53c2\u8003\u6a21\u578b. \u5982\u4e0b\u7ecf\u5178\u7684Kuramoto\u6a21\u578b\u63cf\u8ff0\u4e86\u4e0e\u8282\u70b9\uff0c\u53730-\u5355\u7eaf\u5f62\uff0c\u76f8\u5173\u7684\u76f8\u4f4d\u7684\u540c\u6b65\u53d8\u5316\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe7  Kuramoto\u6a21\u578b<\/span><\/section>\n
 <\/span><\/section>\n
\u540c\u6b65\u76f8\u53d8\u7531\u5168\u5c40\u5e8f\u53c2\u91cfR\u523b\u753b\uff0c\u5f53\u8026\u5408\u5e38\u6570\u03c3\u8f83\u5c0f\u65f6\uff0c\u76f8\u4f4d\u52a8\u529b\u5b66\u4e0d\u76f8\u5e72\uff1b\u5f53\u8026\u5408\u5e38\u6570\u03c3\u5927\u4e8e\u540c\u6b65\u9608\u503c\u03c3c\u65f6\uff0c\u53d1\u751f\u76f8\u53d8\u3002\u4e0b\u56fe\u662f\u7531\u8282\u70b9 (0-\u5355\u7eaf\u5f62)\u3001\u94fe (1-\u5355\u7eaf\u5f62) \u548c\u4e09\u89d2\u5f62 (2-\u5355\u7eaf\u5f62\uff0c\u6a59\u8272\u9634\u5f71\u90e8\u5206) \u7ec4\u6210\u7684\u5355\u7eaf\u590d\u5f62\u3002<\/span><\/section>\n

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<\/section>\n
 <\/span><\/section>\n
\u5355\u7eaf\u590d\u5f62\u4e0d\u4ec5\u53ef\u4ee5\u523b\u753b\u8282\u70b9 (0-\u5355\u7eaf\u5f62) \u7684\u52a8\u529b\u5b66\uff0c\u4e5f\u53ef\u4ee5\u523b\u753b\u9ad8\u9636\u5355\u7eaf\u5f62\u7684\u52a8\u529b\u5b66\u3002<\/span><\/strong><\/span><\/section>\n
 <\/span><\/section>\n
\u7528\u5173\u8054\u77e9\u9635\u8868\u793a\u6807\u51c6 Kuramoto\u6a21\u578b\uff1a<\/span><\/section>\n

<\/span><\/section>\n
\u5728\u4e00\u4e2a\u7f51\u7edc\u6216\u5355\u590d\u5f62\u9aa8\u67b6\u4e2d\uff0c\u4e0e\u8282\u70b9\u76f8\u5173\u7684\u76f8\u4f4d\u03b8\u6ee1\u8db3\u5982\u4e0b\u65b9\u7a0b\uff0c\u76f8\u4f4d\u03b8\u548c\u81ea\u7136\u9891\u7387\u03c9\u5982\u4e0b\uff1a<\/span><\/section>\n
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\u5bf9\u4e8e\u9ad8\u9636Kuramoto\u6a21\u578b\uff0c\u53ef\u4ee5\u5c06\u4e00\u4e2a\u632f\u8361\u901a\u91cf\u5173\u8054\u5230\u6bcf\u4e00\u4e2a\u94fe\u8def\uff0c\u5373\u5c06\u76f8\u4f4d\u5173\u8054\u5230\u6bcf\u4e2am\u7ef4\u5355\u7eaf\u5f62\u4e0a\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u6240\u4ee5\u7528\u7528\u5173\u8054\u77e9\u9635\u8868\u793a\u9ad8\u9636 Kuramoto \u6a21\u578b\u6765\u8868\u793a\u5355\u7eaf\u7684\u540c\u6b65\uff1a<\/span><\/section>\n

<\/span><\/section>\n
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\u975e\u81ea\u9002\u5e94\u8026\u5408\u5e8f\u53c2\u91cf\u4e0b\uff0c\u62d3\u6251\u540c\u6b65\u662f\u8fde\u7eed\u7684\uff0c\u4e14\u03c3c=0\u3002<\/span><\/section>\n

<\/section>\n
4. Dirac\u540c\u6b65<\/span><\/strong><\/span><\/section>\n

<\/section>\n
Dirac\u540c\u6b65\u5229\u7528Dirac\u7b97\u5b50\u5728\u5c40\u90e8\u548c\u62d3\u6251\u4e0a\u8026\u5408\u4e0d\u540c\u7ef4\u5ea6\u7684\u62d3\u6251\u4fe1\u53f7\u3002Dirac\u540c\u6b65\u5177\u6709\u70ed\u529b\u5b66\u65f6\u6ede\u73af\u3002\u5176\u5e8f\u53c2\u91cf\u5305\u62ec\u8282\u70b9\u4fe1\u53f7\u548c\u94fe\u8def\u4fe1\u53f7<\/span>\uff08\u6295\u5f71\u5728\u8282\u70b9\u4e0a\uff09<\/span>\u7684\u7ebf\u6027\u7ec4\u5408\u3002<\/span><\/section>\n
 <\/span><\/section>\n
Dirac\u540c\u6b65\u5bfc\u81f4\u8282\u5f8b\u76f8\u4f4d\u7684\u51fa\u73b0\uff0c\u5728\u65cb\u8f6c\u6846\u67b6\u4e2d\u5e8f\u53c2\u91cf\u81ea\u53d1\u5730\u83b7\u5f97\u4e00\u4e2a\u52a8\u529b\u5b66\u76f8\u4f4d\uff0c\u5373\u5728\u672c\u5f81\u76f8\u4f4d\u5e73\u5747\u4e3a\u96f6\u7684\u6846\u67b6\u4e2d\u3002\u5176\u8282\u5f8b\u9636\u6bb5\u63ed\u793a\u4e86\u5927\u8111\u8282\u5f8b\u51fa\u73b0\u7684\u62d3\u6251\u673a\u5236\u3002<\/span><\/section>\n

<\/section>\n
5. \u5176\u4ed6\u52a8\u529b\u5b66<\/span><\/strong><\/span><\/section>\n

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\u5c40\u90e8\u6d41\u884c\u75c5\u7684\u4f20\u64ad\u5f71\u54cd\uff0c\u53ef\u4ee5\u7528\u65f6\u95f4\u8d85\u56fe\u6765\u6a21\u62df\uff0c\u9608\u503c\u6548\u5e94\u662f\u5bfc\u81f4\u975e\u7ebf\u6027\u4f20\u67d3\u6838\u7684\u91cd\u8981\u56e0\u7d20\u3002<\/span><\/section>\n

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\u591a\u91cd\u8d85\u56fe\u7531\u5c42\u6784\u6210\uff0c\u6bcf\u4e2a\u5c42\u6355\u6349\u7ed9\u5b9a\u9636\u7684\u76f8\u4e92\u4f5c\u7528\uff0c\u5176\u4e2d\u5305\u542b\u5408\u4f5c\u6548\u5e94\u7684\u9ad8\u9636\u6e17\u6d41\u95ee\u9898\u662f\u4e0d\u8fde\u7eed\u7684\u3002<\/span><\/section>\n

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 <\/span><\/section>\n

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<\/svg><\/section>\n<\/section>\n<\/section>\n
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\u603b\u7ed3<\/span><\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n


<\/p>\n<\/section>\n<\/h3>\n

\u5728\u672c\u6b21\u8bb2\u5ea7\u4e2d\uff0c\u4e3b\u8bb2\u4eba\u5bf9\u9ad8\u9636\u7f51\u7edc\u7684\u57fa\u672c\u6982\u5ff5\u8fdb\u884c\u4e86\u8be6\u7ec6\u7684\u8bb2\u89e3\uff0c\u5e76\u6307\u51fa\u5355\u7eaf\u590d\u5f62\u662f\u4e00\u4e2a\u975e\u5e38\u4e00\u822c\u7684\u6570\u5b66\u6846\u67b6\u3002\u5355\u7eaf\u590d\u5f62\u9664\u4e86\u53ef\u4ee5\u5bf9\u9ad8\u9636\u7f51\u7edc\u8fdb\u884c\u5b8c\u6574\u7684\u62d3\u6251\u5206\u6790\u5916\uff0c\u8fd8\u6709\u6355\u6349\u590d\u6742\u7cfb\u7edf\u7684\u591a\u4f53\u76f8\u4e92\u4f5c\u7528\uff0c\u63ed\u793a\u5355\u7eaf\u62d3\u6251\u548c\u5355\u7eaf\u51e0\u4f55\u5728\u52a8\u529b\u5b66\u91cd\u8981\u4f5c\u7528\u7684\u4f18\u70b9\uff0c\u4e3a\u542c\u8005\u5c55\u793a\u4e86\u9ad8\u9636\u7f51\u7edc\u7684\u6574\u4f53\u89c6\u56fe\u3002\u4ece\u793e\u4f1a\u8fc7\u7a0b\u5230\u795e\u7ecf\u79d1\u5b66\u7684\u4f17\u591a\u590d\u6742\u7cfb\u7edf\u5b9e\u4f8b\u4e0a\uff0c\u9ad8\u9636\u62d3\u6251\u5206\u6790\u5177\u6709\u5f88\u5927\u6f5c\u529b\uff0c\u9ad8\u9636\u7f51\u7edc\u52a8\u529b\u5b66\u5df2\u7ecf\u6210\u4e3a\u76ee\u524d\u5feb\u901f\u53d1\u5c55\u7684\u7814\u7a76\u65b9\u5411\u3002<\/span><\/section>\n

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\u8bb2\u8005\u4ecb\u7ecd<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n

Ginestra Bianconi\uff0c<\/span><\/strong>\u82f1\u56fd\u4f26\u6566\u739b\u4e3d\u5973\u738b\u5927\u5b66\u5e94\u7528\u6570\u5b66\u6559\u6388\uff0c\u827e\u4f26\u00b7\u56fe\u7075\u7814\u7a76\u6240\u7684\u827e\u4f26\u00b7\u56fe\u7075\u7814\u7a76\u5458\u3002\u5979\u7684\u7814\u7a76\u9886\u57df\u662f\u7edf\u8ba1\u529b\u5b66\u548c\u7f51\u7edc\u79d1\u5b66\uff0c\u5177\u4f53\u5305\u62ec\u7f51\u7edc\u7406\u8bba\u53ca\u5176\u8de8\u5b66\u79d1\u5e94\u7528\u3002\u5979\u662f\u7f51\u7edc\u5efa\u6a21\u7684\u4e13\u5bb6\uff0c\u5305\u62ec\u589e\u957f\u7f51\u7edc\u6a21\u578b\u548c\u5747\u8861\u7f51\u7edc\u7cfb\u7efc\u3002\u540c\u65f6\u4e5f\u51fa\u7248\u4e86\u5173\u4e8e\u7f51\u7edc\u52a8\u6001\u8fc7\u7a0b\u7684\u91cd\u8981\u8457\u4f5c\uff0c\u7279\u522b\u662f\u6e17\u6d41\u7814\u7a76\u3002\u6700\u8fd1\u51e0\u5e74\uff0c\u5979\u4e00\u76f4\u4e13\u6ce8\u4e8e\u591a\u5c42\u7f51\u7edc\u3001\u5355\u7eaf\u590d\u5f62\u3001\u7f51\u7edc\u51e0\u4f55\u548c\u62d3\u6251\u3002\u5979\u662f Multilayer Networks: Structure and Function\uff08\u725b\u6d25\u5927\u5b66\u51fa\u7248\u793e\u51fa\u7248\uff09 \u548c Higher-order networks: An introduction to simplicial complexes \uff08\u5251\u6865\u5927\u5b66\u51fa\u7248\u793e\u51fa\u7248\uff09 \u4e24\u672c\u4e66\u7684\u4f5c\u8005\u3002<\/span><\/span><\/section>\n
\u76ee\u524d\uff0c\u5979\u662f JPhys Complex \u671f\u520a\u7684\u4e3b\u7f16\uff0cPlosOne \u548c Scientific Reports \u7684\u7f16\u8f91\u30022020\u5e74\uff0c\u5979\u88ab NetSci \u5b66\u4f1a\u6388\u4e88 Network Scientific Fellowships\uff0c\u4ee5\u8868\u5f70\u5bf9\u7f51\u7edc\u79d1\u5b66\u7684\u57fa\u7840\u6027\u8d21\u732e\uff0c\u7279\u522b\u662f\u5728\u590d\u6742\u7f51\u7edc\u4e2d\u73bb\u8272-\u7231\u56e0\u65af\u5766\u51dd\u805a\u7684\u63d0\u51fa\uff0c\u4ee5\u53ca\u5728\u591a\u5c42\u7f51\u7edc\u7ed3\u6784\u548c\u52a8\u529b\u5b66\u65b9\u9762\u7684\u8fdb\u5c55\u3002<\/span><\/section>\n
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\u9ad8\u9636\u7f51\u7edc\u8bfb\u4e66\u4f1a\u542f\u52a8<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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\u968f\u7740\u5bf9\u73b0\u5b9e\u4e16\u754c\u7684\u63a2\u7d22\u4e0d\u65ad\u6df1\u5165\uff0c\u4eba\u4eec\u53d1\u73b0\u5728\u8bb8\u591a\u771f\u5b9e\u7684\u590d\u6742\u7cfb\u7edf\u4e2d\uff0c\u7ec4\u6210\u7cfb\u7edf\u7684\u4e2a\u4f53\u4e4b\u95f4\u4e0d\u4ec5\u5b58\u5728\u4e8c\u5143\u4ea4\u4e92\u5173\u7cfb\uff0c\u4e5f\u5e7f\u6cdb\u5b58\u5728\u591a\u4e2a\u4f53\u540c\u65f6\uff08\u6216\u4ee5\u7279\u5b9a\u987a\u5e8f\uff09\u8fdb\u884c\u4ea4\u4e92\uff0c\u5373\u9ad8\u9636\u4ea4\u4e92\u73b0\u8c61\u3002\u4e3a\u6b64\uff0c\u7814\u7a76\u4eba\u5458\u5206\u522b\u53d1\u5c55\u51fa\u4e86\u57fa\u4e8e\u8d85\u56fe\u3001\u5355\u7eaf\u590d\u5f62\u3001\u4f9d\u8d56\u5173\u7cfb\u7b49\u7684\u7f51\u7edc\u9ad8\u9636\u8868\u793a\u6a21\u578b\uff0c\u4e3a\u590d\u6742\u7f51\u7edc\u5206\u6790\u548c\u7814\u7a76\u63d0\u4f9b\u4e86\u65b0\u7684\u601d\u8def\u3002\u4e3a\u4e86\u4fc3\u8fdb\u6b64\u9886\u57df\u7684\u4ea4\u6d41\u4e0e\u5408\u4f5c\uff0c\u6211\u4eec\u53d1\u8d77\u4e86\u3010\u9ad8\u9636\u7f51\u7edc\u8bfb\u4e66\u4f1a<\/a>\u3011\u3002<\/span><\/p>\n


<\/p>\n

\u96c6\u667a\u4ff1\u4e50\u90e8\u8bfb\u4e66\u4f1a\u662f\u9762\u5411\u5e7f\u5927\u79d1\u7814\u5de5\u4f5c\u8005\u7684\u7cfb\u5217\u8bba\u6587\u7814\u8bfb\u6d3b\u52a8\uff0c\u5176\u76ee\u7684\u662f\u5171\u540c\u6df1\u5165\u5b66\u4e60\u63a2\u8ba8\u67d0\u4e2a\u79d1\u5b66\u8bae\u9898\uff0c\u6fc0\u53d1\u79d1\u7814\u7075\u611f\uff0c\u4fc3\u8fdb\u79d1\u7814\u5408\u4f5c\u3002\u3010\u9ad8\u9636\u7f51\u7edc\u8bfb\u4e66\u4f1a<\/a>\u3011\u7531\u7535\u5b50\u79d1\u6280\u5927\u5b66\u5415\u7433\u5a9b\u8001\u5e08\u3001\u4efb\u6653\u9f99\u8001\u5e08\u53ca\u4e2d\u56fd\u5730\u8d28\u5927\u5b66\uff08\u5317\u4eac\uff09\u7ba1\u9752\u8001\u5e08\u8054\u5408\u53d1\u8d77\uff0c\u6bcf\u5468\u5206\u4eab\u65f6\u95f4\u4e3a\u6bcf\u5468\u56db 19:30-21:30 \u8fdb\u884c\uff0c\u9884\u8ba1\u6301\u7eed 10-12 \u5468\u3002\u8fd9\u5176\u95f4\uff0c\u6211\u4eec\u5c06\u56f4\u7ed5\u9ad8\u9636\u4ea4\u4e92\u7f51\u7edc\u7684\u57fa\u672c\u6982\u5ff5\u3001\u6a21\u578b\u3001\u65b9\u6cd5\u4e0e\u5e94\u7528\u7b49\u7814\u7a76\u8fdb\u884c\u7814\u8ba8\uff0c\u672c\u6b21\u8bfb\u4e66\u4f1a\u5206\u4eab\u4f1a\u6309\u7167\u300c\u57fa\u7840\u7406\u8bba\u300d+\u300c\u6df1\u5165\u7406\u8bba\u300d+\u300c\u6848\u4f8b\u7814\u8ba8\u300d\u7684\u6a21\u5f0f\u5c55\u5f00\u3002<\/span><\/p>\n


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\u8be6\u60c5\u8bf7\u89c1\uff1a<\/span><\/p>\n

\u63a2\u7d22\u590d\u6742\u7cfb\u7edf\u9ad8\u9636\u4ea4\u4e92\u7684\u5965\u79d8 | \u9ad8\u9636\u7f51\u7edc\u8bfb\u4e66\u4f1a\u542f\u52a8<\/strong><\/a><\/p>\n


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\u63a8\u8350\u9605\u8bfb<\/span><\/strong><\/strong><\/p>\n