{"id":62725,"date":"2025-07-26T17:02:00","date_gmt":"2025-07-26T09:02:00","guid":{"rendered":"https:\/\/swarma.org\/?p=62725"},"modified":"2025-07-26T17:02:00","modified_gmt":"2025-07-26T09:02:00","slug":"%e6%8e%a2%e7%b4%a2%e7%bd%91%e7%bb%9c%e6%b7%b1%e5%b1%82%e7%bb%93%e6%9e%84%e7%9a%84%e5%af%b9%e7%a7%b0%e6%80%a7%ef%bc%9a%e6%b1%82%e8%a7%a3%e7%bd%91%e7%bb%9c%e9%ab%98%e9%98%b6%e6%b4%9e%e6%96%b0%e6%96%b9","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=62725","title":{"rendered":"\u63a2\u7d22\u7f51\u7edc\u6df1\u5c42\u7ed3\u6784\u7684\u5bf9\u79f0\u6027\uff1a\u6c42\u89e3\u7f51\u7edc\u9ad8\u9636\u6d1e\u65b0\u65b9\u6cd5"},"content":{"rendered":"
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\u8fd1\u671f\u7531\u53f2\u5b9a\u534e\u6559\u6388\u3001\u9648\u5173\u8363\u6559\u6388\u548c\u8d75\u9633\u535a\u58eb\u751f\u5171\u540c\u64b0\u5199\u7684\u8bba\u6587\u53d1\u5e03\u5728Arxiv\u4e0a\uff0c\u8bba\u6587\u57fa\u4e8e\u90bb\u5c45\u5b50\u7f51\u7edc\u63d0\u51fa\u4e86\u540c\u8c03H\u2096-\u6838\u5206\u89e3\u6c42\u7f51\u7edc\u9ad8\u9636\u6d1e\u7684\u65b0\u65b9\u6cd5\u3002<\/span><\/span><\/strong><\/p>\n


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\u5173\u952e\u8bcd\uff1a\u90bb\u5c45\u5b50\u7f51\u7edc\uff0c\u793a\u6027\u6570\uff0c\u8fb9\u754c\u77e9\u9635\uff0c\u8d1d\u8482\u6570\uff0cH(k)-\u6838\u5206\u89e3\uff0c\u540c\u8c03\u8ba1\u7b97\u3002<\/span><\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

\"\u56fe\u7247\"<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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\u8bba\u6587\u9898\u76ee\uff1aNode-neighbor subnetworks and Hk-core decomposition <\/span><\/span><\/p>\n

\u8bba\u6587\u5730\u5740\uff1ahttps:\/\/arxiv.org\/abs\/2507.04948<\/span><\/span><\/p>\n<\/blockquote>\n


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\u8d8a\u6765\u8d8a\u591a\u7684\u7814\u7a76\u503e\u5411\u4e8e\u8ba4\u4e3a\uff0c\u5728\u591a\u4e2a\u65f6\u7a7a\u5c3a\u5ea6\u4e0b\u7a33\u5b9a\u91cd\u73b0\u7684\u62d3\u6251\u7279\u5f81\u53ef\u80fd\u66f4\u80fd\u53cd\u6620\u539f\u59cb\u6570\u636e\u7684\u771f\u5b9e\u7ed3\u6784\uff0c\u800c\u90a3\u4e9b\u4ec5\u5728\u5355\u4e00\u5c3a\u5ea6\u6216\u77ed\u65f6\u6bb5\u51fa\u73b0\u7684\u7279\u5f81\u5219\u5f88\u53ef\u80fd\u662f\u7531\u91c7\u6837\u8bef\u5dee\u6216\u566a\u58f0\u5f15\u8d77\u7684\u5076\u7136\u73b0\u8c61\u3002\u8fd9\u4fc3\u4f7f\u6211\u4eec\u501f\u52a9\u4ee3\u6570\u62d3\u6251\u4e2d\u7684<\/span><\/span>\u540c\u8c03<\/span><\/span><\/strong>\u7406\u8bba\u548c\u65b9\u6cd5\u53bb\u523b\u753b\u4e0d\u540c\u7ef4\u5ea6\u7a7a\u95f4\u4e2d\u591a\u6837\u56fe\u5f62\u6240\u5305\u542b\u7684\u5404\u79cd\u4fe1\u606f\u3002<\/span><\/span><\/span><\/p>\n


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\u968f\u7740\u9ad8\u7ef4\u6570\u636e\u7684\u6d8c\u73b0\uff0c\u62d3\u6251\u6570\u636e\u5206\u6790<\/span> (TDA) <\/span>\u6210\u4e3a\u76f4\u63a5\u53cd\u6620\u6570\u636e\u79d1\u5b66\u3001\u7f51\u7edc\u79d1\u5b66\u3001\u8ba1\u7b97\u62d3\u6251\u5b66\u4e4b\u95f4\u4ea4\u53c9\u79d1\u5b66\u8054\u7cfb\u7684\u91cd\u8981\u5de5\u5177\u548c\u7814\u7a76\u65b9\u6cd5\u3002\u4f5c\u4e3aTDA\u7684\u6838\u5fc3\u5de5\u5177\uff0c\u6301\u7eed\u540c\u8c03\u901a\u8fc7\u6784\u5efa\u70b9\u4e91\u6570\u636e\u7684\u53c2\u6570\u5316\u8fc7\u6ee4\u8fc7\u7a0b\uff0c\u7cfb\u7edf\u5730\u53bb\u8ffd\u8e2a\u62d3\u6251\u7279\u5f81\u7684\u6f14\u5316\u89c4\u5f8b\u3002\u8be5\u65b9\u6cd5\u9996\u5148\u57fa\u4e8e\u8282\u70b9\u8ddd\u79bb\u6784\u5efa\u9012\u589e\u7684\u5355\u7eaf\u590d\u5f62\u5e8f\u5217\uff0c\u7136\u540e\u8ba1\u7b97\u6bcf\u4e2a\u6ee4\u5b50\u5c3a\u5ea6\u4e0b\u7684\u540c\u8c03\u7fa4\u7ed3\u6784\uff0c\u6700\u7ec8\u83b7\u5f97\u5404\u7ef4\u8d1d\u8482\u6570\u7684\u751f\u706d\u533a\u95f4\u6761\u5f62\u7801<\/span> (barcode) <\/span>\u53ca\u5176\u6301\u7eed\u7279\u5f81\u3002\u7279\u522b\u503c\u5f97\u6307\u51fa\u7684\u662f\uff0c\u867d\u7136\u6301\u7eed\u540c\u8c03\u80fd\u6709\u6548\u8bc6\u522b\u6570\u636e\u4e2d\u7a33\u5b9a\u7684\u62d3\u6251\u7279\u5f81<\/span> (\u5982\u6301\u7eed\u5b58\u5728\u7684\u540c\u8c03\u73af)<\/span>\uff0c\u4f46\u662f\u7cbe\u786e\u8ba1\u7b97\u8fd9\u4e9b\u7279\u5f81\u5bf9\u5e94\u7684\u751f\u6210\u5143<\/span> (\u5373\u6784\u6210\u62d3\u6251\u7279\u5f81\u7684\u5355\u7eaf\u5f62\u7ec4\u5408)<\/span> \u4ecd\u9762\u4e34\u8ba1\u7b97\u4e0a\u7684\u5de8\u5927\u6311\u6218\u3002<\/span><\/span><\/p>\n


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\u4e3a\u653b\u514b\u5927\u89c4\u6a21\u7f51\u7edc\u9ad8\u9636\u6d1e\u7684\u8ba1\u7b97\u96be\u9898\uff0c\u8bba\u6587\u4f5c\u8005\u5229\u7528\u8282\u70b9\u90bb\u5c45\u5b50\u7f51\u7edc\u4e09\u6570\u7ec4<\/span> (\u90bb\u5c45\u6570\u3001\u8d1d\u8482\u6570\u3001\u793a\u6027\u6570) <\/span>\u7684\u4fe1\u606f\u6765\u7814\u7a76\u539f\u59cb\u7f51\u7edc\u7684<\/span><\/span>H(k)<\/span><\/span><\/em>-\u6838\u5206\u89e3\uff0c\u5e76\u5229\u7528\u5206\u89e3\u4fe1\u606f\u53ca\u8282\u70b9\u65b0\u6307\u6570\u53bb\u7b80\u5316\u6d1e\u7684\u8ba1\u7b97\u5e76\u63d0\u4f9b\u5bfb\u627e\u6700\u9ad8\u9636\u6d1e\u7684\u6709\u6548\u65b9\u6cd5\u3002<\/span><\/span><\/span><\/p>\n


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\u661f\u56fe\u7ed3\u6784\u53ef\u7528\u90bb\u63a5\u77e9\u9635\u7684\u884c\u548c\u6765\u8868\u793a\u8282\u70b9\u5ea6\u3002\u5708\u7ed3\u6784\u53ef\u770b\u4f5c\u662f\u4e00\u7c7b\u7279\u6b8a\u7684\u7ec4\u5408\uff0c\u5b83\u4eec\u5728\u6570\u5b66\u4e0a\u6784\u6210\u4e00\u4e2a\u201c\u5708\u7a7a\u95f4\u201d\uff1b\u800c\u7531\u9ad8\u4e00\u9636\u7ed3\u6784\u8fb9\u754c\u6784\u6210\u7684\u5708\u79f0\u4e3a\u201c\u6070\u5f53\u5708\u201d\uff0c\u7ec4\u6210\u201c\u50cf\u7a7a\u95f4\u201d\u3002\u524d\u8005\u5305\u542b\u540e\u8005\uff0c\u4e24\u8005\u7684\u201c\u5dee\u201d\u6784\u6210\u201c\u540c\u8c03\u7fa4\u201d\u3002\u4e3a\u4e86\u8ba1\u7b97\u8fd9\u4e9b\u7ed3\u6784\uff0c\u901a\u5e38\u4f7f\u7528\u8fb9\u754c\u77e9\u9635\u8fdb\u884c\u8868\u793a\u548c\u8fd0\u7b97\u3002\u901a\u8fc7\u8fb9\u754c\u77e9\u9635\u7684\u79e9r(k)\u53ef\u4ee5\u8ba1\u7b97\u7f51\u7edc\u7b2c\u4e8c\u4e2a\u62d3\u6251\u4e0d\u53d8\u91cf\u2014\u2014<\/span><\/span>k<\/span><\/span><\/em>-\u9636<\/span><\/span>\u8d1d\u8482\u6570<\/span>\u03b2(k)=m(k)-r(k)-r(k+1)\uff0c\u5176\u4e2d,\u03b2(0)\u662f\u8fde\u901a\u5206\u652f\u6570\uff0c\u03b2(k)\u662f<\/span><\/span><\/strong>k<\/span><\/span><\/em>–<\/span>\u9636\u540c\u8c03\u7fa4\u7684\u79e9\u6216\u6d1e\u7684\u4e2a\u6570\u3002<\/span><\/span><\/span><\/p>\n


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\u9996\u5148\u5f15\u8fdb\u8282\u70b9\u7684\u90bb\u5c45\u5b50\u7f51\u7edc\u3002\u8fd9\u79cd\u5b50\u7f51\u7edc\u867d\u7136\u89c4\u6a21\u5c0f\uff0c\u4f46\u5b83\u662f\u6574\u4e2a\u7f51\u7edc\u7684\u7f29\u5f71\u3002\u5728\u5b50\u7f51\u7edc\u4e2d\u9664\u4e86\u6709\u90bb\u5c45\u6570\u3001\u793a\u6027\u6570\u548c\u8d1d\u8482\u6570\u5916\uff0c\u56e0\u5176\u89c4\u6a21\u5c0f\u8fd8\u53ef\u80fd\u6709\u7279\u6b8a\u7684\u8282\u70b9\uff0c\u5982\u5ea6\u4e3a0\u7684\u5b64\u7acb\u70b9\u548c\u5ea6\u4e3a\u90bb\u5c45\u6570\u51cf1\u7684\u4e2d\u5fc3\u70b9\u3002\u90bb\u5c45\u5b50\u7f51\u7edc\u7684\u4e2d\u5fc3\u70b9\u548c\u8d1d\u8482\u6570\u6709\u4e0b\u9762\u5173\u7cfb\uff1a\u82e5\u8282\u70b9<\/span><\/span>i<\/span><\/span><\/em>\u7684\u90bb\u5c45\u5b50\u7f51\u7edc\u6709\u4e2d\u5fc3\u70b9\uff0c\u5219\u90bb\u5c45\u5b50\u7f51\u7edc\u8d1d\u8482\u6570\u4e3a(1,0,0,\u2026)<\/span><\/span>(i)<\/span><\/span><\/em>\u3002<\/span><\/span><\/span><\/p>\n


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

\u7136\u540e\u5b9a\u4e49\u4e00\u4e2a\u8282\u70b9\u7684\u65b0\u6307\u6570\uff0c\u4e3a\u8be5\u8282\u70b9\u4e09\u7c7b\u53c2\u6570\u7684\u96c6\u5408\uff1a\u90bb\u5c45\u6570\u3001\u8d1d\u8482\u6570\u4ee5\u53ca\u5c40\u90e8\u7ed3\u6784\u793a\u6027\u6570\uff0c\u8bb0\u4f5c(<\/span><\/span>n\u1d62<\/span><\/span><\/em>, (\u03b2<\/span><\/span>(k)<\/span><\/span><\/em>)<\/span><\/span>\u1d62<\/span><\/span><\/em>, <\/span><\/span>\u03c7\u1d62<\/span><\/span><\/em>)\u3002\u5176\u4e2d\u5c40\u90e8\u7ed3\u6784\u793a\u6027\u6570\u662f\u901a\u8fc7\u8be5\u8282\u70b9\u8fde\u63a5\u90bb\u5c45\u5b50\u7f51\u7edc\u7ec4\u6210\u7684\u7f51\u7edc\u793a\u6027\u6570\uff0c\u03c7(i)=1-m(0)+m(1)-m(2)…\u3002\u82e5\u539f\u59cb\u7f51\u7edc\u6700\u5927\u7684\u975e\u96f6\u8d1d\u8482\u6570\u662f<\/span><\/span>k<\/span><\/span><\/em>\u9636\u7684\uff0c\u5219\u8282\u70b9\u5b50\u7f51\u7edc\u6700\u5927\u7684\u975e\u96f6\u8d1d\u8482\u6570\u4e0d\u4f1a\u8d85<\/span><\/span>k <\/span><\/span><\/em>\u9636\u3002\u5229\u7528\u8fd9\u4e2a\u547d\u9898\uff0c\u5728\u8ba1\u7b97\u8282\u70b9\u6307\u6570\u65f6\u53ea\u9700\u622a\u65ad\u8ba1\u7b97\u5230k\u9636\u8d1d\u8482\u6570\u6216k+1\u9636\u5355\u7eaf\u5f62\uff0c\u4ece\u800c\u964d\u4f4e\u8ba1\u7b97\u590d\u6742\u5ea6\u3002<\/span><\/span><\/span><\/p>\n


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


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

<\/span><\/h3>\n

\n
\n
\n
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\n

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

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

\u540c\u8c03H\u2096-\u6838\u5206\u89e3<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

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

<\/span><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/span><\/strong><\/span><\/h3>\n

\u6587\u7ae0\u63d0\u51fa\u7f51\u7edc\u7684<\/span><\/span>H(k)<\/span><\/span><\/em>-\u6838\u5206\u89e3\u65b9\u6cd5\uff0c\u5176\u6838\u5fc3\u601d\u60f3\u662f\u901a\u8fc7\u9010\u6b65\u5220\u9664\u7279\u5b9a\u8282\u70b9\u548c\u8fb9\u6765\u7b80\u5316\u7f51\u7edc\u7ed3\u6784\uff0c\u540c\u65f6\u4fdd\u7559\u5173\u952e\u62d3\u6251\u7279\u5f81\uff0c\u4ece\u800c\u63ed\u793a\u7f51\u7edc\u7684\u540c\u8c03\u7ed3\u6784\u3002\u4e0e\u4f20\u7edf\u7684\u8282\u70b9\u5ea6\u7684<\/span><\/span>k<\/span><\/span><\/em>-\u6838\u5206\u89e3\u7c7b\u4f3c\uff0c<\/span><\/span>H(k)<\/span><\/span><\/em>-\u6838\u5206\u89e3\u7684\u6574\u4f53\u6846\u67b6\u5982\u4e0b\uff1a\u4ece\u539f\u59cb\u7f51\u7edc\u51fa\u53d1\uff0c\u4f9d\u6b21\u8ba1\u7b97\u5404\u9636\u6838\u7684\u4e09\u5143\u7ec4\u53c2\u6570(<\/span><\/span>m(k)<\/span><\/span><\/em>, <\/span><\/span>r(k)<\/span><\/span><\/em>, <\/span><\/span>b(k)<\/span><\/span><\/em>)\uff0c\u6700\u540e\u57fa\u4e8e\u8282\u70b9\u65b0\u6307\u6570(<\/span><\/span>n\u1d62<\/span><\/span><\/em>, (<\/span><\/span>b(k)<\/span><\/span><\/em>)<\/span><\/span>\u1d62<\/span><\/span><\/em>, <\/span><\/span>\u03c7\u1d62<\/span><\/span><\/em>)\u6307\u5bfc\u5220\u9664\u64cd\u4f5c\u3002\u5206\u5c42\u5220\u9664\u7b56\u7565\u5982\u4e0b\uff1a<\/span><\/span><\/span><\/p>\n


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

\uff081\uff09<\/span><\/span>H<\/span><\/span><\/em>(1)-\u6838\u6784\u5efa<\/span> (\u5220\u70b9)<\/span>\uff1a<\/span><\/span><\/span><\/p>\n

\u5220\u9664\u8d1d\u8482\u6570\u4e3a(1,0,…,0)\u7684\u8282\u70b9\uff0c\u4f18\u5148\u5904\u7406\u4f4e\u5ea6\u8282\u70b9\u548c\u4e2d\u5fc3\u70b9\uff0c\u4fdd\u6301\u7f51\u7edc\u793a\u6027\u6570\u548c\u8d1d\u8482\u6570\u4e0d\u53d8\u3002<\/span><\/span><\/p>\n


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

\uff082\uff09<\/span><\/span>H<\/span><\/span><\/em>(2)-\u6838\u6784\u5efa<\/span> (\u5220\u70b9\u8fb9)<\/span>\uff1a<\/span><\/span><\/span><\/p>\n

\u5220\u9664\u5bfc\u81f4<\/span><\/span>\u03b2<\/span><\/span><\/em>(1)\u53d8\u5316\u7684\u8282\u70b9(\u8d1d\u8482\u6570\u4e3a(<\/span><\/span>\u03b2<\/span><\/span><\/em>(0)>1,0,…,0))\uff0c\u540c\u6b65\u6e05\u7406\u65b0\u4ea7\u751f\u7684(1,0,…,0)\u8282\u70b9\uff0c\u5fc5\u8981\u65f6\u5220\u9664\u7279\u5b9a\u8fb9\u4f7f<\/span><\/span>\u03b2<\/span><\/span><\/em>(1)\u964d\u4e3a0\u3002<\/span><\/span><\/span><\/p>\n


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

\uff083\uff09<\/span><\/span>H<\/span><\/span><\/em>(3)\u81f3\u9ad8\u9636<\/span><\/span>H(k)<\/span><\/span><\/em> -\u6838\u6784\u5efa<\/span> (\u5220\u9664\u70b9\u8fb9\u548c\u5b50\u7f51\u7edc)<\/span>\uff1a<\/span><\/span><\/span><\/p>\n

\u5220\u9664<\/span><\/span>H<\/span><\/span><\/em>(2)-\u6838\u4e2d\u793a\u6027\u6570\u4e3a\u6b63\u503c\u7684\u70b9\u8fb9\uff0c\u53ef\u4ee5\u5f97\u5230<\/span><\/span>H<\/span><\/span><\/em>(3)-\u6838\u3002\u4fdd\u7559\u542b\u975e\u96f6\u9ad8\u9636\u8d1d\u8482\u6570\u7684\u5b50\u7f51\u7edc\u53ef\u5f97\u5230\u9ad8\u9636\u6838\uff0c\u4f46\u662f\u8981\u786e\u4fdd\u5220\u9664\u64cd\u4f5c\u4e0d\u5f71\u54cd\u66f4\u9ad8\u9636\u62d3\u6251\u7279\u5f81\u3002<\/span><\/span><\/span><\/p>\n


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

\u5173\u952e\u70b9\uff1a\u8282\u70b9\u5220\u9664\u8981\u9075\u5faa\u4e00\u5b9a\u7684\u62d3\u6251\u7ea6\u675f\uff0c\u6bcf\u6b65\u64cd\u4f5c\u540e\u52a8\u6001\u66f4\u65b0\u7f51\u7edc\u53c2\u6570\uff0c\u901a\u8fc7\u5c40\u90e8\u7279\u5f81<\/span> (\u90bb\u5c45\u5b50\u7f51\u7edc) <\/span>\u6307\u5bfc\u5168\u5c40\u7b80\u5316\u3002<\/span><\/span><\/p>\n


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

\u8be5\u65b9\u6cd5\u901a\u8fc7\u5206\u5c42\u5220\u9664\u7b56\u7565\uff0c\u5728\u964d\u4f4e\u7f51\u7edc\u89c4\u6a21\u7684\u540c\u65f6\uff0c\u7cfb\u7edf\u6027\u5730\u4fdd\u7559\u5404\u9636\u62d3\u6251\u7279\u5f81\uff0c\u4e3a\u9ad8\u6548\u5206\u6790\u9ad8\u9636\u6d1e\u7ed3\u6784\u63d0\u4f9b\u65b0\u9014\u5f84\u3002\u800c\u9ad8\u9636\u6d1e\u6240\u5728\u7684\u5b50\u7f51\u7edc\u662f\u7f51\u7edc\u6df1\u5c42\u4e2d\u7684\u5bf9\u79f0\u7ed3\u6784\uff0c\u89c1\u4e0b\u8282\u5b9e\u4f8b\u3002\u5b9e\u9645\u8ba1\u7b97\u65f6\uff0c\u5229\u7528\u5ea6\u4fe1\u606f\u548c\u4e2d\u5fc3\u6027\u6307\u6807\u4f18\u5316\u5220\u9664\u987a\u5e8f\uff0c\u53ef\u63d0\u9ad8\u8ba1\u7b97\u6548\u7387\u3002<\/span><\/span><\/p>\n


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


<\/span><\/strong><\/h3>\n

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

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

\u771f\u5b9e\u7f51\u7edc\u6d4b\u8bd5<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

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

<\/span><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


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

\u5728\u7ebf\u866b\u795e\u7ecf\u7f51\u7edc\u4e2d\u8fdb\u884c\u540c\u8c03<\/span><\/span>H(k)<\/span><\/span><\/em>-\u6838\u5206\u89e3\u3002\u539f\u59cb\u7f51\u7edc\u5305\u62ec297\u4e2a\u8282\u70b9\u548c2148\u6761\u8fb9\u3002\u6309\u7167\u4e0a\u8ff0\u6d41\u7a0b\u8fdb\u884c\u9010\u6b65\u5206\u89e3\u8ba1\u7b97\uff0c\u53ef\u4ee5\u5f97\u5230\u7f51\u7edc<\/span><\/span>H<\/span><\/span><\/em>(3)-\u6838\u7684\u4e09\u6570\u7ec4\u3002\u57fa\u4e8e\u5d4c\u5957\u7279\u70b9\uff0c\u7528\u8282\u70b9\u65b0\u6307\u6570\u53bb\u5bfb\u627e\u7f51\u7edc\u76844\u4e2a3\u9636\u6d1e\u3002<\/span><\/span><\/span><\/p>\n


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

<\/section>\n

\u56fe1 \u7f51\u7edc\u6700\u5c0f 3 \u9636\u6d1e<\/span><\/span><\/p>\n

<\/section>\n

\u56fe2 \u7ebf\u866b\u7f51\u7edc\u7684H(3)-\u6838(\u7565\u56fe)<\/span><\/span><\/p>\n

\u540c\u7406\uff0c\u5728\u732b\u8111\u76ae\u5c42\u7f51\u7edc\u4e2d\uff0c\u901a\u8fc7\u8ba1\u7b97\u53ef\u5f97\u5230\u5404\u9636\u6838\u53ca\u7f51\u7edc\u76842\u4e2a3\u9636\u6d1e\u3002\u5148\u5220\u4e2d\u5fc3\u70b9\uff0c\u518d\u7b97\u8d1d\u8482\u6570\uff0c\u6700\u540e\u7b97\u5230<\/span><\/span>k<\/span><\/span><\/em>\u622a\u5c3e\uff0c\u8fd9\u65b9\u6cd5\u663e\u8457\u51cf\u5c11\u4e86\u8ba1\u7b97\u91cf\u3002\u8be6\u7ec6\u7684\u8ba1\u7b97\u8fc7\u7a0b\u53ef\u53c2\u8003\u539f\u6587\u3002<\/span><\/span><\/span><\/p>\n


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

<\/section>\n

\u56fe3 \u732b\u8111\u76ae\u5c42\u7f51\u7edc<\/span><\/span><\/p>\n

<\/section>\n

\u56fe4 \u732b\u8111\u76ae\u5c42\u7f51\u7edc\u7684H3-\u6838(\u7565\u56fe)<\/span><\/span><\/p>\n


<\/span><\/strong><\/h3>\n

\n
\n
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\n

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

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

\u603b\u7ed3\u4e0e\u5c55\u671b<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

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

<\/span><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


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

\u540c\u8c03<\/span><\/span>H(k)<\/span><\/span><\/em>-\u6838\u5206\u89e3\u65b9\u6cd5\u4e3a\u7f51\u7edc\u9ad8\u9636\u62d3\u6251\u7279\u5f81\u5206\u6790\u63d0\u4f9b\u4e86\u65b0\u5de5\u5177\u3002\u8be5\u65b9\u6cd5\u4e0d\u4ec5\u80fd\u6709\u6548\u7b80\u5316\u9ad8\u9636\u6d1e\u7684\u641c\u7d22\uff0c\u8fd8\u53ef\u62d3\u5c55\u81f3\u7f51\u7edc\u6392\u5e8f\u3001\u6e17\u6d41\u76f8\u53d8\u7b49\u52a8\u529b\u5b66\u7814\u7a76\u3002\u5b9e\u4f8b\u8868\u660e\uff0c<\/span><\/span>H(k)<\/span><\/span><\/em>-\u6838\u6392\u5e8f\u80fd\u8bc6\u522b\u5173\u952e\u8282\u70b9\uff0c\u5176\u6e17\u6d41\u7279\u6027\u53ef\u5448\u73b0<\/span><\/span>k<\/span><\/span><\/em>\u22652\u65f6\u7684\u6df7\u5408\u76f8\u53d8\u3002\u8fdb\u4e00\u6b65\u7814\u7a76\u53ef\u63a2\u7d22\u8be5\u65b9\u6cd5\u5728\u4f20\u64ad\u6e90\u8bc6\u522b\u3001\u7f51\u7edc\u63a7\u5236\u7b49\u9886\u57df\u7684\u5e94\u7528\uff0c\u5e76\u63a8\u5e7f\u81f3\u4e0a\u540c\u8c03\u3001\u8def\u5f84\u540c\u8c03\u3001\u952f\u9f7f\u540c\u8c03\u3001\u8d85\u7f51\u540c\u8c03\u7b49\u8303\u7574\u3002\u67cf\u62c9\u56fe\u591a\u9762\u4f53\u7f51\u7edc\u7684\u62d3\u6251\u7279\u6027\u63ed\u793a\u4e86\u5bf9\u79f0\u6027\u4e0e\u9ad8\u9636\u6d1e\u7684\u6df1\u523b\u8054\u7cfb\uff0c\u4e3a\u590d\u6742\u7f51\u7edc\u62d3\u6251\u7814\u7a76\u63d0\u4f9b\u4e86\u65b0\u89c6\u89d2\u3002<\/span><\/span><\/span><\/p>\n


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


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\u8bba\u6587\u4f5c\u8005<\/span><\/span><\/strong><\/p>\n

\u53f2\u5b9a\u534e\u6559\u6388<\/span><\/span><\/strong>\uff0c\u66fe\u662f\u4e0a\u6d77\u5927\u5b66\u8fd0\u7b79\u5b66\u4e0e\u63a7\u5236\u8bba\u535a\u58eb\u70b9\u7684\u5b66\u79d1\u5e26\u5934\u4eba\uff0c\u5728\u53ef\u9760\u6027\u3001\u6392\u961f\u8bba\u3001\u968f\u673a\u8fc7\u7a0b\u3001\u590d\u6742\u7f51\u7edc\u7b49\u76f8\u5173\u9886\u57df\u53d6\u5f97\u8bb8\u591a\u521b\u65b0\u6027\u6210\u679c\uff0c\u662f\u590d\u6742\u7f51\u7edc\u9886\u57df\u7684\u8457\u540d\u5b66\u8005\uff0c\u53d1\u8868\u5b66\u672f\u8bba\u6587\u767e\u4f59\u7bc7\uff0c\u8457\u8bd1\u4f5c\u591a\u90e8\uff0c\u5e76\u83b7\u4e0a\u6d77\u5e02\u81ea\u7136\u79d1\u5b66\u4e8c\u7b49\u5956\u30021992\u5e74\u8d77\u8363\u83b7\u56fd\u52a1\u9662\u9881\u53d1\u7684\u653f\u5e9c\u7279\u6b8a\u6d25\u8d34\u3002<\/span><\/span>\u9648\u5173\u8363\u6559\u6388<\/span><\/span><\/strong>\uff0c\u662f\u4e00\u4f4d\u5bf9\u6df7\u6c8c\u7406\u8bba\u505a\u51fa\u8d21\u732e\u7684\u4e2d\u56fd\u6570\u5b66\u5bb6\uff0c\u7814\u7a76\u9886\u57df\u4e3b\u8981\u96c6\u4e2d\u5728\u975e\u7ebf\u6027\u7cfb\u7edf\u7684\u52a8\u529b\u5b66\u5206\u6790\u53ca\u63a7\u5236\u3001\u7f51\u7edc\u79d1\u5b66\u4e0e\u5de5\u7a0b\u3002\u81ea2000\u5e74\u4ee5\u6765\uff0c\u4ed6\u4e00\u76f4\u62c5\u4efb\u9999\u6e2f\u57ce\u5e02\u5927\u5b66\u6df7\u6c8c\u4e0e\u590d\u6742\u7f51\u7edc\u4e2d\u5fc3\u7684\u4e3b\u5e2d\u6559\u6388\u548c\u521b\u59cb\u4e3b\u4efb\u3002\u9648\u4e8e2014\u5e74\u5f53\u9009\u4e3a\u6b27\u6d32\u79d1\u5b66\u9662\u9662\u58eb\uff0c\u4e8e2015\u5e74\u5f53\u9009\u4e3a\u4e16\u754c\u79d1\u5b66\u9662\u9662\u58eb\uff0c\u4e8e1997\u5e74\u5f53\u9009\u4e3aIEEE\u9662\u58eb\u3002\u4ed6\u76ee\u524d\u662f\u300a\u56fd\u9645\u5206\u53c9\u4e0e\u6df7\u6c8c\u6742\u5fd7\u300b\u7684\u4e3b\u7f16\u3002<\/span><\/span><\/span><\/p>\n

\u8d75\u9633<\/span><\/span><\/strong>\uff0c2021\u5e74\u4e8e\u6e56\u5357\u79d1\u6280\u5927\u5b66\u83b7\u63a7\u5236\u79d1\u5b66\u4e0e\u5de5\u7a0b\u7855\u58eb\u5b66\u4f4d\uff0c\u73b0\u4e3a\u590d\u65e6\u5927\u5b66\u7535\u8def\u4e0e\u7cfb\u7edf\u4e13\u4e1a\u535a\u58eb\u7814\u7a76\u751f\uff0c\u4e3b\u8981\u7814\u7a76\u65b9\u5411\u4e3a\u9ad8\u9636\u7f51\u7edc\u4e0a\u7684\u6d41\u884c\u75c5\u4f20\u64ad\u52a8\u529b\u5b66\u3002<\/span><\/span><\/span><\/p>\n


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\u53c2\u8003\u6587\u732e\uff1a<\/span><\/span><\/p>\n

1. Wang X, Chen G. Synchronization in scale-free dynamical networks: Robustness and fragility. IEEE Trans Circuits Syst I 2002; 49: 54\u201362.<\/span><\/span><\/span><\/p>\n

2. Shi D, L\u00fc L, Chen G. Totally homogeneous networks. Natl Sci Rev. 2019; 6(5):962\u2013969.<\/span><\/span><\/p>\n

3. Shi D, Chen Z, Ma C, Chen G. Computing persistent homology by spanning trees and critical simplices.<\/span><\/span><\/p>\n

4. Research A Science Partner Journal 2023; 6: 0230.<\/span><\/span><\/p>\n

5. Shi D, Chen Z, Sun X, Chen Q, Ma C, Lou Y, Chen G. Computing cliques and cavities in networks.<\/span><\/span><\/p>\n

6. Commun Phys. 2021; 4:249.<\/span><\/span><\/p>\n

7. Battiston F, Petri G. Higher-order systems. Complexity. Berlin (Germany): Springer; 2022.<\/span><\/span><\/span><\/p>\n


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\u9ad8\u9636\u7f51\u7edc\u793e\u533a<\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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\u968f\u7740\u5bf9\u73b0\u5b9e\u4e16\u754c\u63a2\u7d22\u7684\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<\/span><\/span><\/section>\n

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\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\u5728\u96c6\u667a\u4ff1\u4e50\u90e8\u8054\u5408\u53d1\u8d77\u4e86\u3010\u9ad8\u9636\u7f51\u7edc\u8bfb\u4e66\u4f1a<\/a><\/span><\/span>\u3011\u3002\u8bfb\u4e66\u4f1a\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\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\u8bfb\u4e66\u4f1a\u7b2c\u4e00\u5b63\u5df2\u7ecf\u5706\u6ee1\u7ed3\u675f\uff0c\u7b2c\u4e8c\u5b63\u6b63\u5728\u7b79\u5907\u4e2d\u3002\u73b0\u5728\u62a5\u540d\u52a0\u5165\u53ef\u4ee5\u89e3\u9501\u7b2c\u4e00\u5b63\u5168\u90e8\u5f55\u64ad\u89c6\u9891\u5e76\u52a0\u5165\u793e\u7fa4\u4ea4\u6d41\u3002<\/span><\/span><\/section>\n


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\"\u56fe\u7247\"<\/span><\/a><\/section>\n
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\u8be6\u60c5\u8bf7\u89c1\uff1a<\/span><\/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<\/a><\/span><\/strong>
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\u7f51\u7edc\u79d1\u5b66\u96c6\u667a\u8bfe\u5802\u7b2c\u4e09\u671f\uff1a<\/span><\/strong><\/span><\/p>\n

\u4ece\u6570\u5b66\u5efa\u6a21\u5230\u591a\u5b66\u79d1\u5e94\u7528<\/span><\/strong><\/span><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


<\/span><\/section>\n<\/h2>\n
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<\/span><\/section>\n
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\"\u56fe\u7247\"<\/span><\/span><\/span><\/section>\n

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\u63a8\u8350\u9605\u8bfb<\/span><\/span><\/strong><\/strong><\/strong><\/section>\n
<\/strong><\/section>\n
1. \u8d85\u56fe\u4e0a\u7684\u81ea\u4e3e\u6e17\u6d41\uff1a\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u5982\u4f55\u5f71\u54cd\u7f51\u7edc\u6297\u6bc1\u6027\uff1f  Chaos\u901f\u9012<\/span><\/a><\/span><\/strong>
<\/span><\/strong><\/span><\/section>\n
2. Nat. Rev. Phys.\u901f\u9012\uff1a\u9ad8\u9636\u7f51\u7edc\u4e0a\u7684\u4f20\u67d3\u52a8\u529b\u5b66<\/span><\/a><\/span><\/strong>
<\/span><\/strong><\/span><\/section>\n
3. \u62d3\u6251\u6df1\u5ea6\u5b66\u4e60\u6355\u6349\u6570\u636e\u7684\u9ad8\u9636\u5173\u7cfb\uff1a\u4fe1\u606f\u4f20\u9012\u62d3\u6251\u795e\u7ecf\u7f51\u7edc\u6982\u8ff0<\/span><\/a><\/span><\/strong><\/section>\n
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<\/strong>5. <\/span><\/span><\/strong>AI\u65f6\u4ee3\u7684\u5b66\u4e60\uff1a\u5171\u63a2\u4eba\u7c7b\u5b66\u4e60\u7684\u590d\u6742\u6027<\/span><\/a><\/span><\/strong><\/section>\n

6. <\/span><\/span><\/strong>\u63a2\u7d22\u8005\u8ba1\u5212 | \u96c6\u667a\u4ff1\u4e50\u90e82025\u5185\u5bb9\u56e2\u961f\u62db\u52df\uff08\u5168\u804c&\u517c\u804c\uff09<\/span><\/a><\/span><\/strong><\/span><\/p>\n

7.<\/span><\/span><\/strong><\/strong><\/span> <\/span><\/span><\/strong><\/strong><\/span>\u7cfb\u7edf\u79d1\u5b66\u524d\u6cbf\u5341\u8bb2\uff1a\u7406\u89e3\u81ea\u7136\u3001\u751f\u547d\u4e0e\u793e\u4f1a\u7684\u590d\u6742\u6027<\/span><\/a><\/span><\/strong><\/strong><\/span><\/p>\n


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\u70b9\u51fb\u201c\u9605\u8bfb\u539f\u6587\u201d\uff0c\u62a5\u540d\u8bfb\u4e66\u4f1a<\/span><\/span><\/section>\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

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