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\u53f2\u8d35\u5143<\/strong>\uff0c\u7406\u5b66\u535a\u58eb\uff0c\u73b0\u4efb\u5317\u4eac\u5e08\u8303\u5927\u5b66\u6587\u7406\u5b66\u9662\u7cfb\u7edf\u79d1\u5b66\u7cfb\u7279\u8058\u526f\u7814\u7a76\u5458\u3001\u7855\u58eb\u751f\u5bfc\u5e08\u3002\u5728Physical Review Letters\u7b49\u520a\u7269\u53d1\u8868\u5b66\u672f\u6587\u7ae0\u5341\u4f59\u7bc7\u3002\u7814\u7a76\u65b9\u5411\uff1a\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u7406\u8bba\u4e0e\u5e94\u7528\uff0c\u590d\u6742\u7f51\u7edc\u3002<\/span><\/section>\n

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<\/section>\n
\u90b1\u4ef2\u666e<\/strong>\uff0c\u5317\u4eac\u5e08\u8303\u5927\u5b66\u7cfb\u7edf\u79d1\u5b66\u5b66\u9662\u7cfb\u7edf\u7406\u8bba\u4e13\u4e1a\u535a\u58eb\u751f\uff0c\u5bfc\u5e08\u4e3a\u6a0a\u4eac\u82b3\u6559\u6388\u3002\u672c\u79d1\u5c31\u8bfb\u4e8e\u56db\u5ddd\u5927\u5b66\u7269\u7406\u5b66\uff08\u8bd5\u9a8c\u73ed\uff09\u4e13\u4e1a\uff0c\u66fe\u5165\u90092018\u5e74\u5ea6\u57fa\u7840\u5b66\u79d1\u62d4\u5c16\u5b66\u751f\u57f9\u517b\u8bd5\u9a8c\u8ba1\u5212\uff08\u73e0\u5cf0\u8ba1\u5212\uff09\u3002\u4e3b\u8981\u4ece\u4e8b\u76f8\u53d8\u4e0e\u4e34\u754c\u73b0\u8c61\u3001\u5730\u7403\u7cfb\u7edf\u590d\u6742\u6027\u4e0e\u4e34\u754c\u6027\u5206\u6790\u3001\u590d\u6742\u7f51\u7edc\u4e0a\u7684\u540c\u6b65\u4e0e\u6d8c\u73b0\u52a8\u529b\u5b66\u7b49\u65b9\u9762\u7684\u7814\u7a76\u3002<\/span><\/section>\n

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\u674e\u660e<\/strong>\uff0c\u5408\u80a5\u5de5\u4e1a\u5927\u5b66\u7269\u7406\u7cfb\uff0c\u7814\u7a76\u5458\u3002\u672c\u79d1\u6bd5\u4e1a\u4e8e\u5b89\u5fbd\u5927\u5b66\uff0c\u535a\u58eb\u6bd5\u4e1a\u4e8e\u4e2d\u56fd\u79d1\u5b66\u6280\u672f\u5927\u5b66\u3002\u5173\u6ce8\u76f8\u53d8\u4e0e\u4e34\u754c\u73b0\u8c61\u3001\u590d\u6742\u7f51\u7edc\u3001\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u7b49\u7814\u7a76\u65b9\u5411\u3002<\/span><\/section>\n

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\u5f20\u7ae0<\/strong>\uff0c\u5317\u4eac\u5e08\u8303\u5927\u5b66\u7cfb\u7edf\u79d1\u5b66\u5b66\u9662\u535a\u58eb\u751f\u3002\u7814\u7a76\u5174\u8da3\u96c6\u4e2d\u4e8e\u590d\u6742\u7f51\u7edc\u4e0e\u6df1\u5ea6\u5b66\u4e60\u7684\u4ea4\u53c9\u9886\u57df\uff0c\u5177\u4f53\u5305\u62ec\u673a\u5668\u5b66\u4e60\uff0c\u590d\u6742\u7cfb\u7edf\u81ea\u52a8\u5efa\u6a21\u7b49\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u5f20\u6bc5\u8d85<\/strong>\uff0c\u535a\u58eb\uff0c\u540c\u6d4e\u5927\u5b66\u526f\u6559\u6388\uff0c\u535a\u5bfc\u3002\u4e3b\u8981\u7814\u7a76\u9886\u57df\u5305\u62ec\u57fa\u4e8e\u56fe\u7684\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u5b66\u4e60\u7b97\u6cd5\u3001\u94fe\u8def\u53ca\u5176\u6743\u91cd\u9884\u6d4b\u3001\u52a0\u6743\u7f51\u7edc\u5efa\u6a21\u4e0e\u968f\u673a\u6269\u6563\u3001\u793e\u4ea4\u7f51\u7edc\u4e0a\u7684\u4fe1\u606f\u6269\u6563\u3001\u7f51\u7edc\u535a\u5f08\u3001\u91d1\u878d\u6570\u636e\u5206\u6790\u3001\u57ce\u5e02\u9632\u707e\u3002\u7814\u7a76\u65b9\u5411\u5305\u62ec\u590d\u6742\u7f51\u7edc\u3001\u673a\u5668\u5b66\u4e60\u3002\u66fe\u4e00\u4f5c\u6216\u901a\u8baf\u5171\u53d1\u8868\u76f8\u5173\u6587\u7ae020\u4f59\u7bc7\uff0c\u4e13\u522913\u9879\uff0c\u8f6f\u84571\u9879\uff0c\u4e13\u84571\u672c\uff1b\u4e00\u7bc7\u4f1a\u8bae\u8bba\u6587\u83b7IEEE Outstanding Paper Award\uff1b\u4e00\u7bc7\u8bba\u6587\u5165\u7f16\u591a\u9662\u9662\u58eb\u9648\u5173\u8363\u6559\u6388\u300a\u590d\u6742\u7f51\u7edc\u57fa\u7840\u300b\u4e00\u4e66\uff1b\u4e3b\u6301\u56fd\u5bb6\u81ea\u7136\u79d1\u5b66\u57fa\u91d1\u9752\u5e74\u9879\u76ee1\u9879\uff0c\u4e0a\u6d77\u5e02\u81ea\u7136\u79d1\u5b66\u57fa\u91d1\u9762\u4e0a1\u9879\uff0c\u9aa8\u5e72\u53c2\u4e0e\u79d1\u6280\u90e8\u56fd\u5bb6\u91cd\u70b9\u7814\u53d1\u8ba1\u52121\u9879\uff0c\u56fd\u5bb6\u81ea\u7136\u79d1\u5b66\u57fa\u91d1\u91cd\u70b9\u9879\u76ee1\u9879\uff0c\u9762\u4e0a\u9879\u76ee\u591a\u9879\u3002<\/span><\/section>\n

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\u62a5\u540d\u53c2\u4e0e\u8bfb\u4e66\u4f1a<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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


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

\u8fd0\u884c\u6a21\u5f0f<\/strong><\/span><\/h3>\n
\u4ece2025\u5e743\u67087\u65e5\u5f00\u59cb<\/strong>\uff0c\u6bcf\u5468\u4e9419:30-21:30\uff0c\u6301\u7eed\u65f6\u95f4\u9884\u8ba1 8-10 \u5468\u5de6\u53f3\uff0c\u6309\u8bfb\u4e66\u4f1a\u6846\u67b6\u8bbe\u8ba1\uff0c\u6bcf\u5468\u8fdb\u884c\u7ebf\u4e0a\u4f1a\u8bae\uff0c\u4e0e\u4e3b\u8bb2\u4eba\u7b49\u793e\u533a\u6210\u5458\u5f53\u9762\u4ea4\u6d41\uff0c\u4f1a\u540e\u53ef\u4ee5\u83b7\u5f97\u89c6\u9891\u56de\u653e\u6301\u7eed\u5b66\u4e60\u3002<\/span><\/section>\n


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

\u62a5\u540d\u65b9\u5f0f<\/strong><\/span><\/h3>\n

\u7b2c\u4e00\u6b65\uff1a\u626b\u7801\u586b\u5199\u62a5\u540d\u4fe1\u606f\u3002<\/span><\/p>\n

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https:\/\/pattern.swarma.org\/study_group\/59?from=wechat<\/span><\/section>\n
\u626b\u7801\u62a5\u540d\uff08\u53ef\u5f00\u53d1\u7968\uff09<\/span><\/section>\n

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

\u7b2c\u4e8c\u6b65\uff1a\u586b\u5199\u4fe1\u606f\u540e\uff0c\u4ed8\u8d39\u62a5\u540d\u3002<\/span><\/p>\n

\u5982\u9700\u7528\u652f\u4ed8\u5b9d\u652f\u4ed8\uff0c\u8bf7\u5728PC\u7aef\u8fdb\u5165\u8bfb\u4e66\u4f1a\u9875\u9762\u62a5\u540d\u652f\u4ed8\u3002<\/em><\/span><\/p>\n

\u7b2c\u4e09\u6b65\uff1a\u6dfb\u52a0\u8fd0\u8425\u8d1f\u8d23\u4eba\u5fae\u4fe1\uff0c\u83b7\u53d6\u6240\u6709\u63a8\u8350\u8bba\u6587\u8d44\u6e90\u5305\uff0c\u62c9\u5165\u5bf9\u5e94\u4e3b\u9898\u7684\u8bfb\u4e66\u4f1a\u793e\u533a\uff08\u5fae\u4fe1\u7fa4\uff09\u3002<\/span><\/p>\n

PS\uff1a\u4e3a\u786e\u4fdd\u4e13\u4e1a\u6027\u548c\u8ba8\u8bba\u7684\u805a\u7126\uff0c\u672c\u8bfb\u4e66\u4f1a\u8c22\u7edd\u8131\u79bb\u8bfb\u4e66\u4f1a\u4e3b\u9898\u548c\u590d\u6742\u79d1\u5b66\u95ee\u9898\u672c\u8eab\u7684\u7a7a\u6cdb\u7684\u54f2\u5b66\u548c\u601d\u8fa8\u5f0f\u8ba8\u8bba\uff1b\u5982\u679c\u51fa\u73b0\u8ba8\u8bba\u5185\u5bb9\u4e0d\u7b26\u5408\u8981\u6c42\u3001\u7ecf\u63d0\u9192\u65e0\u6548\u8005\uff0c\u4f1a\u88ab\u79fb\u9664\u7fa4\u804a\u5e76\u5bf9\u672a\u53c2\u4e0e\u90e8\u5206\u9000\u8d39\u3002<\/strong><\/span><\/section>\n


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

\u52a0\u5165\u793e\u533a\u540e\u53ef\u4ee5\u83b7\u5f97\u7684\u8d44\u6e90<\/strong><\/span><\/h3>\n

\u5b8c\u6574\u6743\u9650\uff0c\u5305\u62ec\u7ebf\u4e0a\u95ee\u7b54\u3001\u5f55\u64ad\u56de\u770b\u3001\u8d44\u6599\u5171\u4eab\u3001\u793e\u7fa4\u4ea4\u6d41\u3001\u4fe1\u606f\u540c\u6b65\u3001\u5171\u521b\u4efb\u52a1\u83b7\u53d6\u79ef\u5206\u7b49<\/strong>\u3002<\/span><\/p>\n

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

\u53c2\u4e0e\u5171\u521b\u4efb\u52a1\u83b7\u53d6\u79ef\u5206\uff0c\u5171\u5efa\u5b66\u672f\u793e\u533a<\/span><\/strong><\/span><\/h3>\n

\u8bfb\u4e66\u4f1a\u91c7\u7528\u5171\u5b66\u5171\u7814\u673a\u5236\uff0c\u6210\u5458\u901a\u8fc7\u5185\u5bb9\u5171\u521b\u83b7\u79ef\u5206\uff08\u5b57\u5e55\u4fee\u6539\u3001\u8bfb\u4e66\u4f1a\u7b14\u8bb0\u3001\u8bba\u6587\u901f\u9012\u3001\u516c\u4f17\u53f7\u6587\u7ae0\u3001\u96c6\u667a\u767e\u79d1\u3001\u8bba\u6587\u89e3\u8bfb\u7b49\u5171\u521b\u4efb\u52a1<\/strong>\uff09\uff0c\u79ef\u5206\u7b26\u5408\u6761\u4ef6\u5373\u53ef\u9000\u8d39\u3002\u53d1\u8d77\u4eba\u548c\u4e3b\u8bb2\u4eba\u540c\u6837\u9075\u5faa\u6b64\u673a\u5236\uff0c\u65e0\u989d\u5916\u91d1\u94b1\u6fc0\u52b1\u3002<\/span><\/p>\n

<\/p>\n

PS\uff1a\u5177\u4f53\u53c2\u4e0e\u65b9\u5f0f\u53ef\u4ee5\u52a0\u5165\u8bfb\u4e66\u4f1a\u540e\u67e5\u770b\u5bf9\u5e94\u7684\u5171\u521b\u4efb\u52a1\u5217\u8868\uff0c\u9886\u53d6\u4efb\u52a1\uff0c\u4e0e\u8fd0\u8425\u8d1f\u8d23\u4eba\u6c9f\u901a\u8be6\u60c5\uff0c\u4e0a\u8ff0\u89c4\u5219\u7684\u6700\u7ec8\u89e3\u91ca\u6743\u5f52\u96c6\u667a\u4ff1\u4e50\u90e8\u6240\u6709\u3002<\/span><\/section>\n


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\u8bba\u6587\u6e05\u5355<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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


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\u9605\u8bfb\u6750\u6599\u8f83\u957f\uff0c\u4e3a\u4e86\u66f4\u597d\u7684\u9605\u8bfb\u4f53\u9a8c\uff0c\u5efa\u8bae\u60a8\u524d\u5f80\u96c6\u667a\u6591\u56fe\u6c89\u6d78\u5f0f\u9605\u8bfb\uff0c\u5e76\u53ef\u6536\u85cf\u611f\u5174\u8da3\u7684\u8bba\u6587\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
https:\/\/pattern.swarma.org\/article\/335?from=wechat<\/span><\/section>\n


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

\u6a21\u5757\u4e00\uff1a\u76f8\u53d8\u4e0e\u4e34\u754c<\/span><\/strong><\/span><\/h3>\n


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

1. \u590d\u6742\u7cfb\u7edf\u7684\u591a\u7a33\u6001\u4e0e\u4e34\u754c\u70b9<\/strong><\/span><\/strong><\/h3>\n

<\/strong><\/span><\/section>\n
\u4e3b\u9898\u53d1\u8d77\u4eba\uff1a<\/strong>\u53f2\u8d35\u5143<\/span><\/section>\n

<\/strong><\/span><\/section>\n
\u4e3b\u9898\u63a8\u8350\u8bed\uff1a<\/strong><\/span><\/section>\n
\u5728\u8bb8\u591a\u73b0\u5b9e\u7cfb\u7edf\u4e2d\uff0c\u7531\u4e8e\u7cfb\u7edf\u5404\u7ec4\u5206\u95f4\u5b58\u5728\u975e\u7ebf\u6027\u76f8\u4e92\u4f5c\u7528\u4e0e\u53cd\u9988\u5faa\u73af\uff0c\u7cfb\u7edf\u901a\u5e38\u5b58\u5728\u591a\u4e2a\u7a33\u6001<\/strong>\u3002\u4e00\u65e6\u8d8a\u8fc7\u7279\u5b9a\u7684\u4e34\u754c\u70b9\uff0c\u7cfb\u7edf\u4fbf\u4f1a\u4ece\u4e00\u4e2a\u7a33\u6001\u5207\u6362\u81f3\u53e6\u4e00\u4e2a\u7a33\u6001\uff0c\u8fd9\u53ef\u80fd\u7ed9\u81ea\u7136\u548c\u793e\u4f1a\u5e26\u6765\u4e0d\u53ef\u9006\u7684\u5f71\u54cd\u3002\u5f53\u7ed9\u5b9a\u52a8\u529b\u5b66\u65b9\u7a0b\u7684\u5f62\u5f0f\uff0c\u4ee5\u53ca\u63cf\u8ff0\u4e24\u4e24\u4e4b\u95f4\u76f8\u4e92\u4f5c\u7528\u5173\u7cfb\u7684\u590d\u6742\u7f51\u7edc\u65f6\uff0c\u6211\u4eec\u80fd\u591f\u4ece\u7406\u8bba\u5c42\u9762\u5bf9\u4e34\u754c\u70b9\u7684\u53d6\u503c\u6216\u8303\u56f4\u3001\u4e0d\u540c\u7a33\u6001\u4e0b\u7684\u8282\u70b9\u72b6\u6001\u7b49\u8fdb\u884c\u5b9a\u91cf\u5206\u6790\uff0c\u8fd8\u53ef\u4ee5\u8bc4\u4f30\u5f71\u54cd\u7cfb\u7edf\u72b6\u6001\u7684\u5173\u952e\u8282\u70b9\u548c\u53c2\u6570\u7b49\u8981\u7d20\u3002\u7136\u800c\u5728\u8bb8\u591a\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u65e0\u6cd5\u7cbe\u786e\u83b7\u77e5\u52a8\u529b\u5b66\u65b9\u7a0b\u53ca\u5404\u7ec4\u5206\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u5173\u7cfb\uff0c\u4ec5\u80fd\u76d1\u6d4b\u5230\u8282\u70b9\u72b6\u6001\u968f\u65f6\u95f4\u7684\u53d8\u5316\u3002\u4e3a\u6b64\uff0c\u7814\u7a76\u8005\u4eec\u5f00\u53d1\u51fa\u4e00\u7cfb\u5217\u65b9\u6cd5\uff0c\u7528\u4e8e\u4ece\u8282\u70b9\u72b6\u6001\u7684\u65f6\u95f4\u5e8f\u5217\u4e2d\u8bc6\u522b\u51fa\u7cfb\u7edf\u63a5\u8fd1\u4e34\u754c\u70b9\u7684\u65e9\u671f\u9884\u8b66\u4fe1\u53f7\u3002<\/strong>\u603b\u4e4b\uff0c\u591a\u7a33\u6001\u73b0\u8c61\u662f\u8bb8\u591a\u73b0\u5b9e\u7cfb\u7edf\u7684\u91cd\u8981\u7279\u5f81\uff0c\u800c\u4e34\u754c\u70b9\u662f\u8fd9\u4e9b\u7cfb\u7edf\u81f3\u5173\u91cd\u8981\u7684\u4fe1\u606f\u4e4b\u4e00\u3002<\/strong>\u7814\u7a76\u590d\u6742\u7cfb\u7edf\u591a\u7a33\u6001\u548c\u4e34\u754c\u70b9\uff0c\u6709\u52a9\u4e8e\u6211\u4eec\u6784\u5efa\u8d77\u91cf\u53d8\u5f15\u53d1\u8d28\u53d8\u7684\u7406\u8bba\u8ba4\u8bc6\uff0c\u5bf9\u9884\u9632\u7cfb\u7edf\u6027\u98ce\u9669\u3001\u5c06\u7cfb\u7edf\u8c03\u63a7\u81f3\u9884\u671f\u72b6\u6001\u7b49\u4e5f\u6709\u6307\u5bfc\u4f5c\u7528\u3002<\/span><\/section>\n

<\/strong><\/span><\/section>\n
\u4e3b\u9898\u91cd\u70b9\u5173\u6ce8\uff1a<\/strong><\/span><\/section>\n
1. \u81ea\u7136\u548c\u793e\u4f1a\u4e2d\u7684\u591a\u7a33\u6001\u73b0\u8c61<\/span><\/section>\n
2. \u591a\u7a33\u6001\u73b0\u8c61\u80cc\u540e\u7684\u6570\u5b66\u539f\u7406<\/span><\/section>\n
3. \u4e34\u754c\u70b9\u9644\u8fd1\u7cfb\u7edf\u72b6\u6001\u7684\u4e3b\u8981\u7279\u5f81<\/span><\/section>\n

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

\u91cd\u8981\u7efc\u8ff0\uff1a<\/strong><\/span><\/p>\n

[1] Scheffer, Marten, et al. “Anticipating critical transitions.” science 338.6105 (2012): 344-348.<\/span><\/p>\n

\u8fd9\u662f\u4e34\u754c\u8f6c\u53d8\u73b0\u8c61\u7684\u65e9\u671f\u7efc\u8ff0\uff0c\u7cfb\u7edf\u9610\u8ff0\u4e86\u8bc6\u522b\u5e76\u9884\u6d4b\u4e34\u754c\u8f6c\u53d8\u7684\u6838\u5fc3\u601d\u60f3\u548c\u65b9\u6cd5\u3002<\/span><\/p>\n

[2] Dakos, Vasilis, et al. “Ecosystem tipping points in an evolving world.” Nature ecology & evolution 3.3 (2019): 355-362.<\/span><\/p>\n

\u63a2\u8ba8\u751f\u6001\u7cfb\u7edf\u5728\u52a8\u6001\u73af\u5883\u4e2d\u53ef\u80fd\u906d\u9047\u7684\u4e34\u754c\u70b9\u53ca\u5176\u5e94\u5bf9\u7b56\u7565\uff0c\u4e3a\u7406\u89e3\u590d\u6742\u751f\u6001\u6f14\u5316\u63d0\u4f9b\u4e86\u91cd\u8981\u89c6\u89d2\u3002<\/span><\/p>\n

[3] Liu, Xueming, et al. “Network resilience.” Physics Reports 971 (2022): 1-108.<\/span><\/p>\n

\u5168\u9762\u5730\u56de\u987e\u4e86\u7f51\u7edc\u97e7\u6027\u9886\u57df\u4e2d\u591a\u7ef4\u52a8\u529b\u5b66\u65b9\u7a0b\u5efa\u6a21\u3001\u9ad8\u7ef4\u7cfb\u7edf\u97e7\u6027\u73b0\u8c61\u4e0e\u4e34\u754c\u8f6c\u53d8\u5206\u6790\u7684\u7814\u7a76\u8fdb\u5c55\u3002<\/span><\/section>\n

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

\u57fa\u4e8e\u7f51\u7edc\u7ed3\u6784\u548c\u52a8\u529b\u5b66\u65b9\u7a0b\u7684\u4e34\u754c\u70b9\u5206\u6790\uff1a<\/strong><\/span><\/p>\n

[4] Gao, Jianxi, Baruch Barzel, and Albert-L\u00e1szl\u00f3 Barab\u00e1si. “Universal resilience patterns in complex networks.” Nature 530.7590 (2016): 307-312.<\/span><\/p>\n

\u901a\u8fc7\u4e8c\u9636\u5e73\u5747\u573a\u7406\u8bba\uff08\u5c06\u90bb\u5c45\u7684\u90bb\u5c45\u89c6\u4e3a\u5e73\u5747\u7684\u8282\u70b9\uff09\u548c\u52a8\u529b\u5b66\u65b9\u7a0b\u51c6\u7ebf\u6027\u8fd1\u4f3c\uff0c\u9996\u6b21\u63d0\u51fa\u9ad8\u7ef4\u975e\u7ebf\u6027\u52a8\u529b\u7cfb\u7edf\u964d\u7ef4\u7684\u901a\u7528\u6846\u67b6\u3002<\/span><\/p>\n

[5] Morone, Flaviano, Gino Del Ferraro, and Hern\u00e1n A. Makse. “The k-core as a predictor of structural collapse in mutualistic ecosystems.” Nature physics 15.1 (2019): 95-102.<\/span><\/p>\n

\u901a\u8fc7\u9636\u68af\u51fd\u6570\u8fd1\u4f3c\uff0c\u63d0\u51fa\u4e92\u60e0\u4e92\u5229\u751f\u6001\u7cfb\u7edf\u5d29\u6e83\u7684\u4e34\u754c\u70b9\u53ef\u4ee5\u7528\u7f51\u7edc\u7684\u6700\u5927k-core\u6765\u89e3\u6790\u8868\u793a\u3002<\/span><\/p>\n

[6] Wu, Rui-Jie, et al. “Rigorous criteria for the collapse of nonlinear cooperative networks.” Physical Review Letters 130.9 (2023): 097401.<\/span><\/p>\n

\u8bc1\u660e\u4e86\u9ad8\u7ef4\u975e\u7ebf\u6027\u52a8\u529b\u7cfb\u7edf\u7684\u4e34\u754c\u70b9\u4e00\u5b9a\u843d\u5728\u7f51\u7edc\u7684\u6700\u5927\u7279\u5f81\u503c\u548c\u6700\u5927k-core\u4e4b\u95f4\u3002<\/span><\/section>\n

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

\u57fa\u4e8e\u8282\u70b9\u72b6\u6001\u65f6\u95f4\u5e8f\u5217\u7684\u4e34\u754c\u70b9\u9884\u6d4b\uff1a<\/strong><\/span><\/p>\n

[7] Scheffer, Marten, et al. “Early-warning signals for critical transitions.” Nature 461.7260 (2009): 53-59.<\/span><\/p>\n

\u8be6\u7ec6\u7efc\u8ff0\u4e86\u4ece\u65f6\u95f4\u5e8f\u5217\u7edf\u8ba1\u7279\u5f81\u4e2d\u8bc6\u522b\u4e34\u754c\u8f6c\u53d8\u5373\u5c06\u53d1\u751f\u65e9\u671f\u4fe1\u53f7\u7684\u7ecf\u5178\u65b9\u6cd5\u3002<\/span><\/p>\n

[8] Liu, Zijia, et al. “Early predictor for the onset of critical transitions in networked dynamical systems.” Physical Review X 14.3 (2024): 031009.<\/span><\/p>\n

\u4f7f\u7528\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\u65e9\u671f\u5b9a\u91cf\u9884\u6d4b\u590d\u6742\u7cfb\u7edf\u4e2d\u4e34\u754c\u70b9\uff0c\u8be5\u65b9\u6cd5\u5728\u5404\u79cd\u4e0d\u540c\u7cfb\u7edf\u4e0a\u8fdb\u884c\u9a8c\u8bc1\uff0c\u5e76\u5bf9\u566a\u58f0\u548c\u4e0d\u5b8c\u6574\u6570\u636e\u7b49\u5177\u6709\u9c81\u68d2\u6027\uff0c\u4e14\u6210\u529f\u9884\u6d4b\u4e86\u771f\u5b9e\u975e\u6d32\u690d\u88ab\u751f\u6001\u7cfb\u7edf\u4e2d\u68ee\u6797\u5411\u7a00\u6811\u8349\u539f\u7684\u4e34\u754c\u8f6c\u53d8\u3002<\/span><\/section>\n

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

\u8c03\u63a7\u7cfb\u7edf\u4f7f\u5176\u5904\u5728\u671f\u671b\u7684\u7a33\u6001\uff1a<\/strong><\/span><\/p>\n

[9] Sanhedrai, Hillel, et al. “Reviving a failed network through microscopic interventions.” Nature Physics 18.3 (2022): 338-349.<\/span><\/p>\n

\u901a\u8fc7\u5bf9\u5c40\u90e8\u8282\u70b9\u65bd\u52a0\u7ec6\u5fae\u5e72\u9884\u624b\u6bb5\u5b9e\u73b0\u7f51\u7edc\u590d\u82cf\uff0c\u5c55\u73b0\u4e86\u5728\u7cfb\u7edf\u5d29\u6e83\u540e\u91cd\u5efa\u7a33\u6001\u7684\u53ef\u884c\u8def\u5f84\u3002<\/span><\/section>\n

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2. \u7f51\u7edc<\/span>\u540c\u6b65<\/strong><\/strong><\/span><\/section>\n

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\u4e3b\u9898\u63a8\u8350\u8bed\uff1a<\/strong>\u590d\u6742\u7cfb\u7edf\u7531\u591a\u4e2a\u76f8\u4e92\u4f5c\u7528\u7684\u5355\u5143\u6784\u6210\u3002\u5b83\u4eec\u5728\u52a8\u6001\u6f14\u5316\u4e2d\u76f8\u4e92\u534f\u8c03\uff0c\u4ece\u800c\u6d8c\u73b0\u51fa\u96c6\u4f53\u884c\u4e3a\u6a21\u5f0f\u3002\u540c\u6b65\u662f<\/strong>\u975e\u5e73\u8861<\/strong>\u6d8c\u73b0\u5f62\u5f0f\u4e2d\u6700\u7b80\u5355\u7684\u4e00\u79cd<\/strong>\uff0c\u6d89\u53ca\u901a\u8fc7\u5fae\u89c2\u8026\u5408\u5b9e\u73b0\u7cfb\u7edf\u7684\u5b8f\u89c2\u72b6\u6001\u4e00\u81f4\u6027<\/strong>\u3002\u4ee5 Kuramoto \u6a21\u578b\u4e3a\u4ee3\u8868\u7684\u8026\u5408\u968f\u673a\u81ea\u7136\u9891\u7387\u632f\u5b50\u7cfb\u7edf\u662f\u540c\u6b65\u7814\u7a76\u9886\u57df\u4e2d\u7684\u8303\u5f0f\u6027\u6a21\u578b\u3002\u5728\u8fd9\u4e00\u6846\u67b6\u4e0b\uff0c\u632f\u5b50\u4eec\u901a\u8fc7\u76f8\u4e92\u4f5c\u7528\u514b\u670d\u81ea\u8eab\u81ea\u7136\u9891\u7387\u4e4b\u95f4\u7684\u5f02\u8d28\u6027\uff0c\u53d1\u751f\u540c\u6b65\u76f8\u53d8\uff0c\u4f7f\u5f97\u79e9\u5e8f<\/strong>\uff08order\uff09<\/strong>\u4ece\u65e0\u5e8f<\/strong>(disorder)<\/strong>\u4e2d\u6d8c\u73b0<\/strong>\u51fa\u6765\u3002Kuramoto\u6a21\u578b\u7684<\/strong>\u5404<\/strong>\u79cd\u53d8\u4f53\u53ef\u4ee5\u63cf\u8ff0\u81ea\u7136\u548c\u793e\u4f1a\u4e2d\u7684<\/strong>\u591a\u79cd\u591a\u6837\u7684<\/strong>\u540c\u6b65\u73b0\u8c61<\/strong>\uff0c\u5176\u4e2d\u7684\u82e5\u5e72\u5206\u652f\u4f9d\u7136\u5404\u81ea\u6210\u4e3a\u4e86\u76f8\u5bf9\u72ec\u7acb\u7684\u5b50\u65b9\u5411\u3002\u6b64\u5916\uff0c\u6df1\u523b\u800c\u65b0\u9896\u7684\u6982\u5ff5\u65b9\u6cd5\u4e5f\u5c42\u51fa\u4e0d\u7a77\uff0c\u5982\u81ea\u6d3d\u6027\u65b9\u7a0b\u3001\u4e3b\u7a33\u5b9a\u51fd\u6570\uff08Master Stability Function, MSF\uff09\u3001OA \u62df\u8bbe\u548c\u91cd\u6574\u5316\u7fa4\u7b49\u3002<\/span><\/p>\n

\u8fd1\u5e74\u6765\uff0c\u8fd9\u4e00\u9886\u57df\u4e0e\u591a\u4e2a\u5b66\u79d1\u4ea4\u53c9\uff0c\u4ea7\u751f\u4e86\u4e00\u4e9b\u7cfb\u5217\u65b0\u6a21\u578b\u3001\u65b0\u6982\u5ff5\u548c\u65b0\u65b9\u6cd5\u3002\u5b83\u4eec\u52a0\u6df1\u4e86\u6211\u4eec\u5bf9\u4e8e\u540c\u6b65\u73b0\u8c61\u548c\u7406\u8bba\u7684\u7406\u89e3\uff0c\u503c\u5f97\u66f4\u591a\u5173\u6ce8\u3002<\/span><\/section>\n

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

\u4e3b\u9898\u91cd\u70b9\u5173\u6ce8\uff1a<\/strong><\/span><\/p>\n

    \n
  1. \n

    \u540c\u6b65\u4e0e\u7edf\u8ba1\u7269\u7406\u7684\u4ea4\u53c9\u9886\u57df\u2014\u2014\u540c\u6b65\u76f8\u53d8\u7684\u4e34\u754c\u6027\u8d28\uff1b<\/span><\/p>\n<\/li>\n

  2. \n

    \u65b0\u7684\u5185\u7980\u81ea\u7531\u5ea6\u2014\u2014D\u7ef4Kuramoto\u6a21\u578b\u548c\u8026\u5408D\u7ef4\u632f\u5b50\u7cfb\u7edf\uff1b<\/span><\/p>\n<\/li>\n

  3. \n
    \u540c\u6b65\u4e0e\u7fa4\u4f53\u8fd0\u52a8\u7684\u4ea4\u53c9\u9886\u57df\u2014\u2014 \u96c6\u7fa4\u632f\u5b50\u6a21\u578b\uff1b<\/span><\/section>\n<\/li>\n<\/ol>\n

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

    \u540c\u6b65\u76f8\u53d8\u7684\u4e34\u754c\u6027\u8d28\uff1a\u4e34\u754c\u7ef4\u6570\u548c\u666e\u9002\u7c7b<\/strong><\/span><\/p>\n

    [1] Daido, H. Lower Critical Dimension for Populations of Oscillators with Randomly Distributed Frequencies: A Renormalization-Group Analysis. Phys. Rev. Lett.<\/em> 61, 231\u2013234 (1988).<\/span><\/p>\n

    \u901a\u8fc7\u91cd\u6574\u5316\u7fa4\u5206\u6790\u63ed\u793a\u4e86\u968f\u673a\u5206\u5e03\u81ea\u7136\u9891\u7387\u632f\u5b50\u7cfb\u7edf\u7684\u4e0b\u4e34\u754c\u7ef4\u6570\uff0c\u5960\u5b9a\u4e86\u7406\u89e3\u540c\u6b65\u76f8\u53d8\u4e34\u754c\u884c\u4e3a\u7684\u57fa\u77f3\u3002<\/span><\/p>\n

    [2] Acebr\u00f3n, J. A., Bonilla, L. L., P\u00e9rez Vicente, C. J., Ritort, F. & Spigler, R. The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Mod. Phys.<\/em> 77, 137\u2013185 (2005).<\/span><\/p>\n

    \u7cfb\u7edf\u56de\u987e\u4e86Kuramoto\u6a21\u578b\u7684\u57fa\u672c\u7406\u8bba\u4e0e\u62d3\u5c55\u5e94\u7528\uff0c\u662f\u7814\u7a76\u540c\u6b65\u73b0\u8c61\u7684\u7ecf\u5178\u7efc\u8ff0\u4e0e\u5165\u95e8\u6307\u5357\u3002<\/span><\/p>\n

    [3] Hong, H., Chat\u00e9, H., Park, H. & Tang, L.-H. Entrainment Transition in Populations of Random Frequency Oscillators. Phys. Rev. Lett.<\/em> 99, 184101 (2007).<\/span><\/p>\n

    \u91cd\u70b9\u8ba8\u8bba\u4e86\u968f\u673a\u9891\u7387\u632f\u5b50\u7cfb\u7edf\u4e2d\u7684\u534f\u9891\u8f6c\u53d8\uff08entrainment transition\uff09<\/em>\uff0c\u89e3\u6790\u6c42\u89e3\u4e86\u4e0a\u4e34\u754c\u7ef4\u6570\uff0c\u53d1\u73b0\u5b58\u5728\u4e24\u79cd\u4e0d\u540c\u7684\u534f\u9891\u8f6c\u53d8\u6a21\u5f0f\u3002<\/span><\/p>\n

    [4] Hong, H., Chat\u00e9, H., Tang, L.-H. & Park, H. Finite-size scaling, dynamic fluctuations, and hyperscaling relation in the Kuramoto model. Phys. Rev. E<\/em> 92, 022122 (2015).<\/span><\/p>\n

    \u901a\u8fc7\u6709\u9650\u5c3a\u5bf8\u6807\u5ea6\u5206\u6790\u6c42\u89e3\u52a8\u529b\u5b66\u6da8\u843d\u7684\u4e34\u754c\u6307\u6570\u3002<\/span><\/p>\n

    [5] Daido, H. Susceptibility of large populations of coupled oscillators. Phys. Rev. E<\/em> 91, 012925 (2015).<\/span><\/p>\n

    \u7814\u7a76\u4e86\u5927\u89c4\u6a21\u8026\u5408\u632f\u5b50\u7cfb\u7edf\u5728\u63a5\u8fd1\u4e34\u754c\u70b9\u65f6\u7684\u54cd\u5e94\u7279\u6027\uff0c\u6c42\u89e3\u4e86\u654f\u611f\u6027(susceptibility)\u548c\u5916\u573a\u5bf9\u5e94\u7684\u4e34\u754c\u6307\u6570\u3002<\/span><\/section>\n

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

    D \u7ef4 Kuramoto \u6a21\u578b\u548c\u8026\u5408 D \u7ef4\u632f\u5b50\u7cfb\u7edf<\/strong><\/span><\/p>\n

    [6] Chandra, S., Girvan, M. & Ott, E. Continuous versus Discontinuous Transitions in the $D$-Dimensional Generalized Kuramoto Model: Odd D is Different. Phys. Rev. X<\/em> 9, 011002 (2019).<\/span><\/p>\n

    \u7cfb\u7edf\u6027\u5730\u7814\u7a76\u4e86D-\u7ef4\u5e7f\u4e49Kuramoto\u6a21\u578b\u5728\u5947\u6570D-\u7ef4\u4e0e\u5076\u6570D-\u7ef4\u4e0b\u7684\u540c\u6b65\u8f6c\u53d8\u65b9\u5f0f\uff0c\u63ed\u793a\u4e86\u5e8f\u53c2\u91cf\u7ef4\u6570\u5947\u5076\u6027\u5bf9\u4e8e\u7cfb\u7edf\u540c\u6b65\u884c\u4e3a\u7684\u663e\u8457\u5dee\u5f02\u3002<\/span><\/p>\n

    [7] Dai, X. et al.<\/em> Discontinuous Transitions and Rhythmic States in the D-Dimensional Kuramoto Model Induced by a Positive Feedback with the Global Order Parameter. Phys. Rev. Lett.<\/em> 125, 194101 (2020).<\/span><\/p>\n

    \u5728\u5168\u5c40\u5e8f\u53c2\u91cf\u4e0a\u5f15\u5165\u6b63\u53cd\u9988\uff0c\u53d1\u73b0\u4e86D-\u7ef4\u5e7f\u4e49Kuramoto\u6a21\u578b\u4e2d\u7684\u65b0\u7684\u8282\u5f8b\u72b6\u6001\uff0c\u533a\u522b\u4e8e\u4f20\u7edf\u7684\u5947\u7f8e\u62c9\u6001\u3002<\/span><\/p>\n

    [8] Kovalenko, K. et al.<\/em> Contrarians Synchronize beyond the Limit of Pairwise Interactions. Phys. Rev. Lett.<\/em> 127, 258301 (2021).<\/span><\/p>\n

    \u63a2\u8ba8\u4e86\u201c\u5531\u53cd\u8c03\u8005\u201d\uff08contrarians\uff09\u5728\u591a\u4f53\u76f8\u4e92\u4f5c\u7528\u60c5\u51b5\u4e0b\u7684\u540c\u6b65\u73b0\u8c61\uff0c\u7a81\u7834\u4e86\u4f20\u7edf\u4e8c\u4f53\u76f8\u4e92\u4f5c\u7528\u7684\u9650\u5236\u3002<\/span><\/p>\n

    [9] Dai, X. et al.<\/em> D-dimensional oscillators in simplicial structures: Odd and even dimensions display different synchronization scenarios. Chaos, Solitons & Fractals<\/em> 146, 110888 (2021).<\/span><\/p>\n

    \u9762\u5411\u5355\u7eaf\u5f62\u7ed3\u6784\u4e2d\u7684\u9ad8\u7ef4\u632f\u5b50\uff0c\u63ed\u793a\u4e86\u5947\u6570\u548c\u5076\u6570\u7ef4\u7cfb\u7edf\u5728\u540c\u6b65\u52a8\u529b\u5b66\u4e0a\u5c55\u73b0\u7684\u4e0d\u540c\u573a\u666f\u4e0e\u673a\u5236\u3002<\/span><\/p>\n

    [10] Zou, W., He, S., Senthilkumar, D. V. & Kurths, J. Solvable Dynamics of Coupled High-Dimensional Generalized Limit-Cycle Oscillators. Phys. Rev. Lett.<\/em> 130<\/strong>, 107202 (2023).<\/span><\/p>\n

    \u63d0\u51fa\u4e86\u4e00\u7c7b\u53ef\u89e3\u6790\u7684\u9ad8\u7ef4\u8026\u5408\u6781\u9650\u73af\u632f\u5b50\u6a21\u578b\uff0c\u662f\u65af\u56fe\u52a0\u817e-\u6717\u9053\u632f\u5b50\u7684\u9ad8\u7ef4\u63a8\u5e7f\uff0c\u4e3a\u7814\u7a76\u9ad8\u7ef4\u540c\u6b65\u4e0e\u632f\u8361\u7279\u6027\u63d0\u4f9b\u4e86\u65b0\u5de5\u5177\u3002<\/span><\/section>\n

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

    \u96c6\u7fa4\u632f\u5b50\u6a21\u578b<\/strong><\/span><\/p>\n

    [11] O\u2019Keeffe, K. P., Hong, H. & Strogatz, S. H. Oscillators that sync and swarm. Nat Commun<\/em> 8, 1504 (2017).<\/span><\/p>\n

    \u63d0\u51fa\u96c6\u7fa4\u632f\u5b50\u6a21\u578b\u201cSwarmalator\u201d\u6982\u5ff5\uff0c\u5c06\u7a7a\u95f4\u96c6\u7fa4\u4e0e\u76f8\u4f4d\u540c\u6b65\u7ed3\u5408\uff0c\u5f00\u8f9f\u4e86\u65f6\u7a7a\u534f\u540c\u81ea\u7ec4\u7ec7\u7814\u7a76\u7684\u65b0\u9886\u57df\u3002<\/span><\/p>\n

    [12] Yoon, S., O\u2019Keeffe, K. P., Mendes, J. F. F. & Goltsev, A. V. Sync and Swarm: Solvable Model of Nonidentical Swarmalators. Phys. Rev. Lett.<\/em> 129, 208002 (2022).<\/span><\/p>\n

    \u9488\u5bf9\u975e\u540c\u8d28\u6027Swarmalator\u7cfb\u7edf\u5efa\u7acb\u53ef\u89e3\u6790\u6a21\u578b\uff0c\u6df1\u5165\u63ed\u793a\u7fa4\u4f53\u52a8\u529b\u5b66\u4e0e\u4e2a\u4f53\u5f02\u8d28\u6027\u4e4b\u95f4\u7684\u76f8\u4e92\u5f71\u54cd\u3002<\/span><\/p>\n

    [13] Anwar, M. S., Sar, G. K., Perc, M. & Ghosh, D. Collective dynamics of swarmalators with higher-order interactions. Commun Phys<\/em> 7<\/strong>, 1\u201311 (2024).<\/span><\/p>\n

    \u5c06\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u5f15\u5165Swarmalator\u6a21\u578b\uff0c\u63ed\u793a\u590d\u6742\u62d3\u6251\u4e0b\u96c6\u7fa4\u632f\u5b50\u66f4\u52a0\u4e30\u5bcc\u591a\u6837\u7684\u540c\u6b65\u6a21\u5f0f\u4e0e\u52a8\u529b\u5b66\u884c\u4e3a\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    3. \u7f51\u7edc<\/span>\u903e\u6e17<\/strong><\/strong><\/span><\/strong><\/section>\n

    <\/strong><\/strong><\/span><\/section>\n
    \u4e3b\u9898\u53d1\u8d77\u4eba\uff1a<\/strong>\u674e\u660e<\/span><\/section>\n

    <\/strong><\/span><\/section>\n
    \u4e3b\u9898\u63a8\u8350\u8bed\uff1a<\/strong>\u903e\u6e17\uff08percolation\uff0c\u4e5f\u5e38\u8bd1\u4f5c\u6e17\u6d41\uff09\u76f8\u53d8\u662f\u7edf\u8ba1\u7269\u7406\u548c\u76f8\u53d8\u7406\u8bba\u4e2d\u7684\u7ecf\u5178\u95ee\u9898\uff0c\u63cf\u8ff0\u4e86\u7cfb\u7edf\u4e2d\u968f\u673a\u8fde\u63a5\u5f62\u6210\u5927\u89c4\u6a21\u8fde\u901a\u7ed3\u6784\u7684\u4e34\u754c\u884c\u4e3a<\/strong>\u3002\u8fd9\u4e00\u8fc7\u7a0b\u5e7f\u6cdb\u5b58\u5728\u4e8e\u81ea\u7136\u754c\u548c\u6280\u672f\u7f51\u7edc\u4e2d\uff0c\u5982\u6750\u6599\u5bfc\u7535\u6027\u3001\u6d41\u4f53\u6e17\u900f\u3001\u75be\u75c5\u4f20\u64ad\u3001\u4e43\u81f3\u590d\u6742\u7f51\u7edc\u7684\u5f62\u6210\u4e0e\u5206\u89e3\u3002\u901a\u8fc7\u5206\u6790\u903e\u6e17\u6a21\u578b\u7684\u4e34\u754c\u6027\u8d28\uff0c\u6211\u4eec\u4e0d\u4ec5\u80fd\u63ed\u793a\u4ece\u5fae\u89c2\u968f\u673a\u6027\u5230\u5b8f\u89c2\u6709\u5e8f\u7684\u8f6c\u53d8\u673a\u5236\uff0c\u8fd8\u80fd\u63a2\u7d22\u76f8\u5173\u7684\u666e\u9002\u6027\u548c\u4e34\u754c\u73b0\u8c61\u3002\u672c\u90e8\u5206\u5c06\u56f4\u7ed5\u903e\u6e17\u76f8\u53d8\u7684\u6838\u5fc3\u7406\u8bba\u3001\u6700\u65b0\u8fdb\u5c55\u4ee5\u53ca\u5176\u5728\u590d\u6742\u7cfb\u7edf\u4e2d\u7684\u5e94\u7528\u5c55\u5f00\u3002<\/span><\/section>\n

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

    \u4e3b\u9898\u91cd\u70b9\u5173\u6ce8\uff1a<\/strong><\/span><\/p>\n

      \n
    1. \n

      \u7206\u70b8\u903e\u6e17\u4e34\u754c\u884c\u4e3a\u7684\u5173\u952e\u7279\u5f81\uff1f<\/span><\/p>\n<\/li>\n

    2. \n

      \u4e0d\u540c\u7c7b\u578b\u7684\u7ea7\u8054\u8fc7\u7a0b\u5bf9\u903e\u6e17\u76f8\u53d8\u7684\u5f71\u54cd\u6709\u4f55\u5f02\u540c\uff1f<\/span><\/p>\n<\/li>\n

    3. \n

      \u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u5f71\u54cd\u80fd\u5426\u7b49\u6548\u4e3a\u82e5\u5e72\u7b80\u5355\u673a\u5236\u7684\u53e0\u52a0\uff1f<\/span><\/p>\n<\/li>\n<\/ol>\n

      \u903e\u6e17\u6700\u521d\u5728\u89c4\u5219\u683c\u5b50\u4e0a\u7814\u7a76\u8fde\u901a\u6027\u4e0e\u4e34\u754c\u76f8\u53d8\uff0c\u4f46\u5982\u4eca\u5df2\u6269\u5c55\u5230\u590d\u6742\u7f51\u7edc\u53ca\u591a\u9886\u57df\u5e94\u7528\u3002\u4e0b\u9762\u674e\u660e\u8001\u5e08\u7cbe\u9009\u4ece2001\u52302024\u5173\u4e8e\u903e\u6e17\u7684\u6587\u732e\uff0c\u5171\u8ba146\u7bc7\uff0c\u7cfb\u7edf\u5448\u73b0\u51fa\u9886\u57df\u7684\u8109\u7edc\u3002<\/span><\/section>\n

      <\/strong><\/span><\/section>\n
      \u6a21\u62df\u7b97\u6cd5<\/strong>\uff0c\u9664\u4e86\u666e\u9002\u7684\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\uff08BFS\uff09\uff0c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff08DFS\uff09\uff0c\u5728\u7f51\u7edc\u7cfb\u7edf\u4e2d\uff0c\u4e00\u822c\u8fd8\u5e38\u7528\u5e76\u67e5\u96c6\uff08disjoint set\uff09\u7684\u65b9\u6cd5\u52a8\u6001\u8bb0\u5f55\u903e\u6e17\u96c6\u56e2\u7684\u4fe1\u606f\uff0c\u5177\u4f53\u89c1Newman-Ziff \u7b97\u6cd5[Physical Review E<\/em>, 64(1):016706,2001]<\/span><\/section>\n

      <\/span><\/section>\n
      \u8ba1\u7b97\u65b9\u6cd5<\/strong>\uff0c\u751f\u6210\u51fd\u6570\u65b9\u6cd5\uff08Phys. Rev. E 64 (2001) 026118\uff09\uff0c\u6d88\u606f\u4f20\u9012\u65b9\u6cd5\uff08Phys. Rev. Lett. 113 (2014) 208702\uff0cPhys. Rev. E 90 (2014) 052824\uff0cPhys. Rev. Lett. 113 (2014) 208701\uff0cPhys. Rev. E 91 (2015) 010801\uff0cPhys. Rev. E 95 (2017) 042322\uff09<\/span><\/section>\n

      <\/span><\/section>\n
      \u5176\u884d\u751f\u6a21\u578b<\/strong>\u5305\u62ec\uff1a<\/span><\/section>\n
        \n
      • \n

        \u7206\u70b8\u903e\u6e17<\/strong>\uff08Explosive Percolation\uff09\uff0c\u901a\u8fc7\u5f15\u5165\u96c6\u56e2\u751f\u957f\u7684\u5f02\u8d28\u673a\u5236\uff0c\u4ea7\u751f\u4e86\u4e30\u5bcc\u7684\u4e34\u754c\u73b0\u8c61[Science 323, 1453 (2009)]\u3002\u6700\u521d\u88ab\u8ba4\u4e3a\u662f\u4e0d\u8fde\u7eed\u76f8\u53d8[Phys. Rev. Lett. 103, 255701 (2009); Phys. Rev. Lett. 103, 045701 (2009); Phys. Rev. Lett. 103, 135702 (2009); Phys. Rev. Lett. 103, 168701 (2009); Phys. Rev. E 82, 051105 (2010); Phys. Rev. E 81, 036110 (2010); Phys. Rev. Lett. 104, 195702 (2010); Phys. Rev. Lett. 107, 275703 (2011).]\uff0c\u540e\u88ab\u8bc1\u660e\u4e3a\u8fde\u7eed\u76f8\u53d8[Science 333, 322 (2011); Phys. Rev. Lett. 105, 255701 (2010); Phys. Rev. E 84, 020101(R) (2011)]\uff0c\u8fdb\u800c\u5927\u91cf\u6570\u503c\u7ed3\u679c\u663e\u793a\u5176\u6709\u5f02\u5e38\u7684\u6709\u9650\u5c3a\u5ea6\u6807\u5ea6\u884c\u4e3a[Phys. Rev. Lett. 106, 225701 (2011);  Nat. Phys. 11, 531 (2015)]\uff0c\u800c\u8fd1\u671f\u88ab\u8bc1\u5b9e\u5728\u57fa\u4e8e\u4e8b\u4ef6\u7684\u7cfb\u7efc\u4e0b\uff0c\u4ecd\u670d\u4ece\u6807\u51c6\u7684\u6709\u9650\u5c3a\u5ea6\u6807\u5ea6\u7406\u8bba[Phys. Rev. Lett. 130, 147101 (2023); Phys. Rev. Research. 6, 033319 (2024)];<\/span><\/p>\n<\/li>\n

      • \n

        k\u6838\u903e\u6e17<\/strong>\uff08k-core Percolation\uff09\uff0c\u5173\u6ce8\u66f4\u7a20\u5bc6\u6216\u7d27\u81f4\u7684\u8fde\u901a\u96c6\u56e2\u7684\u6d8c\u73b0\u3002\u4e34\u754c\u73b0\u8c61\u7684\u8ba8\u8bba\uff08\u6df7\u5408\u76f8\u53d8\uff09[Phys. Rev. Lett. 96 (2006) 040601; Phys. Rev. E 94 (2016) 062307; Phys. Rev. E 78 (2008) 022101; Phys. Rev. Lett. 122 (2019) 108301; New J. Phys. 26, 013006 (2024); Nat. Comm. 15 (2024) 5850]; \u8fde\u7eed\u4e0e\u4e0d\u8fde\u7eed\u76f8\u53d8\u7684\u4ea4\u8de8\u884c\u4e3a[Phys. Rev. Lett. 107 (2011) 175703; Phys. Rev. E 83 (2011) 051134; Phys. Rev. E 87 (2013) 022134]\uff1bk\u6838\u5728\u590d\u6742\u7f51\u7edc\u4e0a\u5e94\u7528\u7684\u7efc\u8ff0[Phys. Rep. 832 (2019) 1-32]<\/span><\/p>\n<\/li>\n

      • \n

        \u76f8\u4f9d\u7f51\u7edc\u903e\u6e17<\/strong>\uff08Percolation on interdependent Networks\uff09\uff0c\u5f3a\u8c03\u591a\u5c42\u6216\u4e92\u8054\u7f51\u7edc\u4e4b\u95f4\u7684\u8026\u5408\u5931\u6548\u5bf9\u6574\u4f53\u8fde\u901a\u6027\u7684\u5f71\u54cd\u3002\u6a21\u578b\u7684\u63d0\u51fa[Nature 464 (2010) 1025; Phys. Rev. Lett. 105 (2010) 048701]\uff1b\u66f4\u7b80\u5355\u7684\u89e3\u6790\u65b9\u6cd5 [Europhys. Lett. 97 (2012) 16006]\uff1b\u6df7\u5408\u76f8\u53d8\u7684\u8ba8\u8bba[Phys. Rev. Lett. 109 (2012) 248701; Phys. Rev. Lett. 129, 268301,2022]<\/span><\/p>\n<\/li>\n

      • \n
        \u8d85\u7f51\u7edc\u903e\u6e17<\/strong>\uff08Percolation on hypergraphs\uff09\uff0c\u8d85\u7f51\u7edc\u903e\u6e17\u5173\u6ce8\u5177\u6709\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u7cfb\u7edf\u3002[Phys. Rev. E 104, (2021) 034306, Phys. Rev. E 109, (2024)014306] \u7ed9\u51fa\u4e86\u5355\u4e2a\u7f51\u7edc\u548c\u591a\u4e2a\u7f51\u7edc\u7684\u903e\u6e17\u7406\u8bba\u3002\u7136\u800c\uff0c\u8d85\u56fe\u4e2d\u7684\u8d85\u8fb9\u5f80\u5f80\u5305\u542b\u591a\u4e2a\u8282\u70b9\uff0c\u4e0d\u540c\u4e8e\u7b80\u5355\u7f51\u7edc\u4e2d\u4e00\u6761\u8fb9\u4e2d\u4e00\u4e2a\u8282\u70b9\u7684\u5220\u9664\u5fc5\u7136\u4f1a\u5bfc\u81f4\u8fd9\u6761\u8fb9\u88ab\u5220\u9664\uff0c[Chaos, Solitons & Fractals,173(2023), 113645] \u63d0\u51fa\u4e86 (k,q)-core\u5206\u89e3\u7684\u65b9\u6cd5\u6765\u627e\u5230\u7531\u81f3\u5c11\u8fde\u63a5k\u6761\u57fa\u6570\u5927\u4e8e\u7b49\u4e8em\u7684\u8d85\u8fb9\u7684\u8282\u70b9\u6240\u7ec4\u6210\u7684\u5de8\u5206\u652f\uff0c[Nature Communications,14 (2023), 6223] \u9a8c\u8bc1\u4e86(k,q)-core\u5728\u8d85\u56fe\u4e2d\u7684\u9ad8\u4f20\u64ad\u80fd\u529b\u548c\u5c40\u57df\u6027\u3002\u6b64\u5916\uff0c\u5728\u6e17\u6d41\u8fc7\u7a0b\u4e2d\uff0c\u8282\u70b9\u8fd8\u53ef\u80fd\u5bf9\u8d85\u8fb9\u5177\u6709\u4e0d\u540c\u7684\u4f5c\u7528\u673a\u5236\uff0c\u4e00\u79cd\u4f5c\u7528\u673a\u5236\u662f\u4e00\u4e2a\u6216\u90e8\u5206\u8282\u70b9 [Chaos, Solitons & Fractals, 173, (2023), 113746] \u7684\u5c31\u4f1a\u5bfc\u81f4\u8d85\u8fb9\u4e2d\u7684\u5269\u4f59\u8282\u70b9\u7684\u5220\u9664,\u800c\u53e6\u4e00\u79cd\u673a\u5236\u662f\u4e00\u4e2a\u6216\u90e8\u5206\u8282\u70b9\u7684\u5220\u9664\u4f1a\u5bfc\u81f4\u8d85\u8fb9\u7684\u89e3\u4f53\uff0c\u800c\u5176\u4f59\u8282\u70b9\u5e76\u4e0d\u4f1a\u88ab\u5220\u9664 [Chaos, Solitons & Fractals, 177, (2023), 114246; Chaos 34, (2024)043148]\u3002\u53e6\u5916\uff0c[Phys. Rev. Lett. 132, 087401 (2024)]\u7814\u7a76\u4e86\u8d85\u56fe\u4e2d\u7531\u9ad8\u9636\u8fde\u63a5\u6240\u5f62\u6210\u7684\u9ad8\u9636\u5206\u652f(higher-order components) \u5bf9\u8d85\u56fe\u4e0a\u9ad8\u9636\u4f20\u67d3\u52a8\u529b\u5b66\u7684\u7206\u53d1\u6a21\u5f0f\u7684\u5f71\u54cd\u3002(\u6b64\u6761\u7531\u676d\u5dde\u5e08\u8303\u5927\u5b66\u5218\u6da6\u7136\u6559\u6388\u6574\u7406)<\/span><\/section>\n<\/li>\n<\/ul>\n

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

        \u4e66\u7c4d\u6216\u7efc\u8ff0\uff1a<\/strong><\/span><\/p>\n

        [1] Stauffer D, Aharony A. Introduction to percolation theory[M]. Taylor & Francis, 2018.<\/span><\/p>\n

        \u504f\u5411\u57fa\u7840\u4e0e\u7ecf\u5178\uff0c\u662f\u7cfb\u7edf\u3001\u5165\u95e8\u7684\u4e66\u7c4d\uff0c\u4ece\u57fa\u672c\u6a21\u578b\u5230\u4e34\u754c\u7406\u8bba\u90fd\u6709\u6e05\u6670\u7684\u8bb2\u8ff0\u3002<\/span><\/p>\n

        [2] Li M, Liu R R, L\u00fc L, et al. Percolation on complex networks: Theory and application[J]. Physics Reports, 2021, 907: 1-68.<\/span><\/p>\n

        \u9762\u5411\u590d\u6742\u7f51\u7edc\u9886\u57df\uff0c\u5c06\u903e\u6e17\u7406\u8bba\u4e0e\u7f51\u7edc\u79d1\u5b66\u6df1\u5ea6\u7ed3\u5408\uff0c\u6db5\u76d6\u4e86\u7406\u8bba\u6a21\u578b\u3001\u4eff\u771f\u5206\u6790\u53ca\u5e94\u7528\u6848\u4f8b\uff0c\u53ef\u5e2e\u52a9\u7406\u89e3\u7f51\u7edc\u9c81\u68d2\u6027\u3001\u4f20\u64ad\u52a8\u529b\u5b66\u7b49\u95ee\u9898\u3002<\/span><\/section>\n

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

        \u903e\u6e17\u76f8\u53d8\u4e0e\u673a\u5668\u5b66\u4e60\uff1a<\/strong><\/span><\/p>\n

        \u8bf4\u660e\uff1a\u4ee5\u4e0b\u7814\u7a76\u4ee5\u7ecf\u5178\u7cfb\u7edf\u4e3a\u7814\u7a76\u5bf9\u8c61\uff0c\u800c\u975e\u7f51\u7edc\u7cfb\u7edf\uff0c\u4f9b\u53c2\u8003\u3002<\/span><\/p>\n

        [3] Zhang W, Liu J, Wei T C. Machine learning of phase transitions in the percolation and XY models[J]. Physical Review E, 2019, 99(3): 032142.<\/span><\/p>\n

        \u91c7\u7528\u673a\u5668\u5b66\u4e60\u8bc6\u522b\u903e\u6e17\u4e0eXY\u6a21\u578b\u7684\u4e34\u754c\u70b9\uff0c\u5c55\u793a\u4eba\u5de5\u667a\u80fd\u5728\u590d\u6742\u76f8\u53d8\u95ee\u9898\u4e0a\u7684\u53ef\u884c\u6027\u4e0e\u9ad8\u7cbe\u5ea6\u6f5c\u80fd\uff0c\u9002\u5408\u63a2\u7d22\u591a\u6a21\u578b\u901a\u7528\u6027\u7814\u7a76\u3002<\/span><\/p>\n

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        \u4ece\u5168\u5c40\u4fe1\u606f\u51fa\u53d1\uff0c\u7528\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\u63ed\u793a\u903e\u6e17\u76f8\u53d8\u7279\u5f81\uff0c\u4e3a\u5927\u89c4\u6a21\u7f51\u7edc\u6216\u590d\u6742\u7cfb\u7edf\u4e2d\u5173\u952e\u7ed3\u6784\u68c0\u6d4b\u63d0\u4f9b\u5168\u5c40\u6027\u89c6\u89d2\u3002<\/span><\/p>\n

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        \u5c06\u6df1\u5ea6\u5b66\u4e60\u878d\u5165\u76f8\u53d8\u3001\u903e\u6e17\u4e0e\u6750\u6599\u65ad\u88c2\u7814\u7a76\uff0c\u63a2\u7d22\u591a\u5c3a\u5ea6\u590d\u6742\u6027\u4e0b\u7684\u9884\u6d4b\u4e0e\u6a21\u62df\uff0c\u5bf9\u6750\u6599\u5de5\u7a0b\u4e0e\u5730\u8d28\u5206\u6790\u5177\u6709\u91cd\u8981\u53c2\u8003\u4ef7\u503c\u3002<\/span><\/p>\n

        [6] Shen J, Liu F, Chen S, et al. Transfer learning of phase transitions in percolation and directed percolation[J]. Physical Review E, 2022, 105(6): 064139.<\/span><\/p>\n

        \u5c06\u6df1\u5ea6\u5b66\u4e60\u878d\u5165\u76f8\u53d8\u3001\u903e\u6e17\u4e0e\u6750\u6599\u65ad\u88c2\u7814\u7a76\uff0c\u63a2\u7d22\u591a\u5c3a\u5ea6\u590d\u6742\u6027\u4e0b\u7684\u9884\u6d4b\u4e0e\u6a21\u62df\uff0c\u5bf9\u6750\u6599\u5de5\u7a0b\u4e0e\u5730\u8d28\u5206\u6790\u5177\u6709\u91cd\u8981\u53c2\u8003\u4ef7\u503c\u3002<\/span><\/p>\n

        [7] Shen J, Li W, Deng S, et al. Supervised and unsupervised learning of directed percolation[J]. Physical Review E, 2021, 103(5): 052140.<\/span><\/p>\n

        \u7ed3\u5408\u6709\u76d1\u7763\u4e0e\u65e0\u76d1\u7763\u5b66\u4e60\u624b\u6bb5\uff0c\u6df1\u6316\u5b9a\u5411\u903e\u6e17\u7684\u4e34\u754c\u884c\u4e3a\uff0c\u4e3a\u57fa\u4e8e\u6570\u636e\u9a71\u52a8\u7684\u975e\u5e73\u8861\u76f8\u53d8\u63a2\u7d22\u63d0\u4f9b\u4e86\u65b0\u601d\u8def\u4e0e\u5b9e\u7528\u65b9\u6cd5\u3002<\/span><\/p>\n

        [8] Zhang J, Zhang B, Xu J, et al. Machine learning for percolation utilizing auxiliary Ising variables [J]. Physical Review E, 2022, 105(2): 024144.<\/span><\/p>\n

        [9] Hu G, Sun Y, Liu T, et al. Universality class of machine learning for critical phenomena [J]. Science China Physics, Mechanics & Astronomy, 2023, 66(12): 120511.<\/span><\/section>\n


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

        \u6a21\u5757\u4e8c\uff1a\u6269\u6563\u4e0e\u968f\u673a\u6e38\u8d70<\/span><\/strong><\/span><\/h3>\n

        <\/strong><\/span><\/section>\n
        \u4e3b\u9898\u53d1\u8d77\u4eba\uff1a<\/strong>\u5f20\u7ae0<\/span><\/section>\n

        <\/strong><\/span><\/section>\n
        \u4e3b\u9898\u63a8\u8350\u8bed\uff1a<\/strong><\/span><\/section>\n
        \u590d\u6742\u7f51\u7edc\u7ed3\u6784<\/strong>\u4e0e\u6269\u6563\/\u968f\u673a\u884c\u8d70\u7b49\u5178\u578b\u52a8\u529b\u5b66\u8fc7\u7a0b<\/strong>\u4e4b\u95f4\u7684\u5b58\u5728\u591a\u5c42\u4ea4\u4e92\u3002\u65e0\u8bba\u662f\u4fe1\u606f\u6d41\u52a8\u3001\u5408\u4f5c\u8fdb\u5316\uff0c\u8fd8\u662f\u7cfb\u7edf\u7a33\u5b9a\u6027\u4e0e\u529f\u80fd\u6027\u56e2\u7c07\u7684\u5f62\u6210\uff0c\u7f51\u7edc\u7684\u62d3\u6251\u7279\u5f81\uff08\u5c42\u6b21\u6a21\u5757\u5316\u3001\u5f02\u8d28\u5ea6\u3001\u52a0\u6743\u65b9\u5f0f\u3001\u9690\u85cf\u51e0\u4f55\u7b49\uff09\u90fd\u8d77\u5230\u81f3\u5173\u91cd\u8981\u7684\u4f5c\u7528\u3002\u4e0e\u6b64\u540c\u65f6\uff0c\u52a8\u529b\u5b66\u8fc7\u7a0b\u672c\u8eab\u7684\u7c7b\u578b\uff08\u7ebf\u6027\/\u975e\u7ebf\u6027\u3001\u5c40\u90e8\/\u5168\u5c40\u8026\u5408\u3001\u6f14\u5316\u535a\u5f08\u7b49\uff09\u4e5f\u4f1a\u663e\u8457\u6539\u53d8\u7f51\u7edc\u529f\u80fd\u8868\u73b0\uff0c\u4ece\u800c\u5f62\u6210\u4e30\u5bcc\u591a\u6837\u7684\u590d\u6742\u6d8c\u73b0\u73b0\u8c61\u3002<\/span><\/section>\n

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

        \u4e3b\u9898\u91cd\u70b9\u5173\u6ce8\uff1a<\/strong><\/span><\/p>\n

          \n
        1. \n

          \u591a\u5c42\u5f02\u8d28\u7f51\u7edc\u4e2d\u7684\u6269\u6563\u4e0e\u7a33\u5b9a\u6027\u8026\u5408\u673a\u5236<\/span><\/p>\n<\/li>\n

        2. \n

          \u4ece\u5c40\u90e8\u4fe1\u606f\u5230\u5168\u5c40\u529f\u80fd\uff1a\u9690\u85cf\u5ea6\u91cf\u7a7a\u95f4\u4e0e\u4fe1\u606f\u4f20\u9012\u6548\u7387<\/span><\/p>\n<\/li>\n

        3. \n

          \u52a8\u529b\u5b66\u4e0e\u8c31\u7279\u6027\uff1a\u4ece\u7ed3\u6784\u71b5\u5230\u529f\u80fd\u96c6\u7fa4<\/span><\/p>\n<\/li>\n<\/ol>\n

          [1] Snijders, T. A. The statistical evaluation of social network dynamics. Sociological methodology 31,361\u2013395 (2001). https:\/\/www.jstor.org\/stable\/3097281<\/span><\/p>\n

          \u5728\u793e\u4f1a\u7f51\u7edc\u5206\u6790\u4e2d\uff0c\u4e3a\u7406\u89e3\u7f51\u7edc\u968f\u65f6\u95f4\u7684\u52a8\u6001\u53d8\u5316\u63d0\u4f9b\u4e86\u53ef\u4f30\u8ba1\u3001\u53ef\u6a21\u62df\u7684\u7edf\u8ba1\u6846\u67b6\uff1b\u80fd\u591f\u540c\u65f6\u8003\u8651\u7f51\u7edc\u6548\u5e94\uff08\u5982\u4e92\u60e0\u3001\u4f20\u9012\u6027\u7b49\uff09\u548c\u5916\u90e8\u534f\u53d8\u91cf\uff08\u4e2a\u4f53\u6216\u4e8c\u5143\u6027\u7279\u5f81\uff09\u3002<\/span><\/p>\n

          [2] Tyson, J. J., Chen, K. & Novak, B. Network dynamics and cell physiology. Nature reviews Molecular cell biology 2, 908\u2013916 (2001). https:\/\/pubmed.ncbi.nlm.nih.gov\/11733770\/<\/span><\/p>\n

          \u5f3a\u8c03\u201c\u52a8\u529b\u5b66\u7cfb\u7edf\u201d\u7684\u5efa\u6a21\u65b9\u6cd5\u5bf9\u7ec6\u80de\u7f51\u7edc\u7684\u7406\u89e3\u5177\u6709\u91cd\u8981\u610f\u4e49\uff1b\u4e3a\u751f\u7269\u9886\u57df\u4e2d\u5c06\u7f51\u7edc\u7ed3\u6784\u4e0e\u52a8\u529b\u5b66\u8fc7\u7a0b\u76f8\u7ed3\u5408\u7684\u7814\u7a76\u5f00\u8f9f\u4e86\u9053\u8def\u3002<\/span><\/p>\n

          [3] Guimer`a, R., D\u00b4\u0131az-Guilera, A., Vega-Redondo, F., Cabrales, A. & Arenas, A. Optimal network topologies for local search with congestion. Physical Review Letters 89 (2002). https:\/\/doi.org\/10.1103\/physrevlett.89.248701.<\/span><\/p>\n

          \u63ed\u793a\u5728\u5e76\u884c\u641c\u7d22\u8d1f\u8f7d\u8f83\u4f4e\u65f6\uff0c\u661f\u72b6\u7f51\u7edc\u6700\u4f18\uff1b\u4f46\u5f53\u641c\u7d22\u91cf\u8f83\u5927\u65f6\uff0c\u5747\u5300-\u5404\u5411\u540c\u6027\u7f51\u7edc\u62d3\u6251\u5219\u66f4\u5177\u4f18\u52bf\u3002\u5bf9\u4e92\u8054\u7f51\u3001\u4ea4\u901a\u4ee5\u53ca\u793e\u4f1a\u7f51\u7edc\u4e2d\u7684\u641c\u7d22\u4e0e\u62e5\u585e\u95ee\u9898\u6709\u542f\u793a\u610f\u4e49\u3002<\/span><\/p>\n

          [4] Chavez, M., Hwang, D.-U., Amann, A., Hentschel, H. G. E. & Boccaletti, S. Synchronization is enhanced in weighted complex networks. Physical Review Letters 94 (2005). https:\/\/doi.org\/10.1103\/physrevlett.94.218701.<\/span><\/p>\n

          \u5c55\u793a\u4e86\u7f51\u7edc\u7ed3\u6784\uff08\u5c24\u5176\u662f\u52a0\u6743\u65b9\u5f0f\u7684\u7cbe\u5de7\u8bbe\u8ba1\uff09\u5bf9\u540c\u6b65\u8fc7\u7a0b\u7684\u4fc3\u8fdb\u6548\u5e94\uff0c\u4e3a\u7814\u7a76\u591a\u632f\u5b50\u6216\u591a\u4ee3\u7406\u7684\u540c\u6b65\u73b0\u8c61\u63d0\u4f9b\u4e86\u65b0\u7684\u601d\u8def\u3002<\/span><\/p>\n

          [5] Arenas, A., D\u00b4\u0131az-Guilera, A. & P\u00b4erez-Vicente, C. J. Synchronization reveals topological scales in complex networks. Physical Review Letters 96 (2006).  https:\/\/doi.org\/10.1103\/physrevlett.96.114102.<\/span><\/p>\n

          \u5728\u7406\u8bba\u4e0e\u6570\u503c\u5b9e\u9a8c\u5c42\u9762\uff0c\u5c06\u540c\u6b65\u8fc7\u7a0b\u4e0e\u8c31\u56fe\u5206\u6790\uff08\u6bd4\u5982\u62c9\u666e\u62c9\u65af\u77e9\u9635\u8c31\uff09\u7d27\u5bc6\u8054\u7cfb\u8d77\u6765\uff0c\u63ed\u793a\u6a21\u5757\u5316\u6216\u591a\u5c42\u6b21\u7ed3\u6784\u5728\u7f51\u7edc\u52a8\u529b\u5b66\u6f14\u5316\u4e2d\u7684\u91cd\u8981\u6027\u3002<\/span><\/p>\n

          [6] Bogu\u02dcn\u00b4a, M., Krioukov, D. & Claffy, K. C. Navigability of complex networks. Nature Physics 5, 74\u201380(2008).  https:\/\/doi.org\/10.1038\/nphys1130.<\/span><\/p>\n

          \u8bc1\u660e\u4e86\u81ea\u7136\u754c\u548c\u4eba\u9020\u7cfb\u7edf\u4e2d\u7684\u8bb8\u591a\u7f51\u7edc\u4e4b\u6240\u4ee5\u80fd\u591f\u5728\u4ec5\u4f7f\u7528\u5c40\u90e8\u4fe1\u606f\u7684\u6761\u4ef6\u4e0b\u5b9e\u73b0\u9ad8\u6548\u8def\u7531\uff0c\u662f\u56e0\u4e3a\u5b83\u4eec\u5177\u6709\u5185\u5728\u7684\u51e0\u4f55\u7279\u5f81\u3002\u4e3a\u7406\u89e3\u968f\u673a\u884c\u8d70\u3001\u8d2a\u5fc3\u8def\u7531\u4e0e\u5ea6\u91cf\u51e0\u4f55\u5728\u7f51\u7edc\u4e2d\u7684\u7edf\u4e00\u673a\u5236\u5960\u5b9a\u4e86\u57fa\u7840\u3002<\/span><\/p>\n

          [7] G\u00b4omez-Garde\u02dcnes, J., Campillo, M., Flor\u00b4\u0131a, L. M. & Moreno, Y. Dynamical organization of cooperation in complex topologies. Physical Review Letters 98 (2007).  https:\/\/doi.org\/10.1103\/physrevlett.98.108103.<\/span><\/p>\n

          \u5b9a\u91cf\u5c55\u793a\u4e86\u7f51\u7edc\u8282\u70b9\u5ea6\u5206\u5e03\u5bf9\u5408\u4f5c\u5f62\u6210\u548c\u6f14\u5316\u7684\u5f71\u54cd\uff1a\u5728\u65e0\u6807\u5ea6\u7f51\u7edc\u4e2d\uff0c\u9ad8\u5ea6\u8282\u70b9\uff08hub\uff09\u503e\u5411\u4e8e\u6210\u4e3a\u5408\u4f5c\u8005\u5e76\u5f62\u6210\u5355\u4e00\u5927\u578b\u5408\u4f5c\u96c6\u7fa4\u3002<\/span><\/p>\n

          [8] Barzel, B. & Barab\u00b4asi, A.-L. Universality in network dynamics. Nature physics 9, 673\u2013681 (2013). https:\/\/www.nature.com\/articles\/nphys2741<\/span><\/p>\n

          \u63d0\u4f9b\u4e86\u4e00\u4e2a\u6db5\u76d6\u591a\u79cd\u8fde\u7eed\u52a8\u529b\u5b66\u8fc7\u7a0b\u7684\u7edf\u4e00\u6846\u67b6\uff1b\u901a\u8fc7\u6570\u503c\u9a8c\u8bc1\u53ca\u5bf9\u751f\u7269\u3001\u793e\u4f1a\u7cfb\u7edf\u7684\u5b9e\u6d4b\u6570\u636e\u5206\u6790\uff0c\u663e\u793a\u51fa\u8be5\u65b9\u6cd5\u53ef\u8bc6\u522b\u4e0d\u540c\u7f51\u7edc\u52a8\u529b\u5b66\u7684\u666e\u9002\u6027\u7279\u5f81\u3002<\/span><\/p>\n

          [9] Domenico, M. D. & Biamonte, J. Spectral entropies as information-theoretic tools for complex network comparison. Physical Review X 6 (2016).  https:\/\/doi.org\/10.1103\/physrevx.6.041062.<\/span><\/p>\n

          \u4ece\u7f51\u7edc\u7684\u62c9\u666e\u62c9\u65af\u8c31\u6216\u76f8\u5e94\u7b97\u5b50\u8c31\u51fa\u53d1\uff0c\u901a\u8fc7\u4fe1\u606f\u71b5\u6765\u8861\u91cf\u7f51\u7edc\u7684\u7ed3\u6784\u5dee\u5f02\uff1b\u53ef\u5e94\u7528\u4e8e\u7f51\u7edc\u6a21\u578b\u53c2\u6570\u4f30\u8ba1\u3001\u6a21\u578b\u9009\u62e9\u4ee5\u53ca\u591a\u5c42\u7f51\u7edc\u7684\u805a\u7c7b\u5206\u6790\u7b49\u3002<\/span><\/p>\n

          [10] Harush, U. & Barzel, B. Dynamic patterns of information flow in complex networks. Nature Communications 8 (2017).  https:\/\/doi.org\/10.1038\/s41467-017-01916-3.<\/span><\/p>\n

          \u53d1\u73b0\u5bf9\u4e8e\u4e00\u7c7b\u5e38\u89c1\u7684\u975e\u7ebf\u6027\u8026\u5408\u7cfb\u7edf\uff0c\u4fe1\u606f\u6d41\u5e76\u975e\u603b\u662f\u7ecf\u8fc7\u4e2d\u5fc3\u8282\u70b9\uff08hub\uff09\uff0c\u751a\u81f3\u4f1a\u51fa\u73b0\u201c\u4fe1\u606f\u7ed5\u5f00\u4e2d\u5fc3\u201d\u800c\u901a\u8fc7\u5916\u56f4\u8def\u5f84\u6d41\u52a8\u7684\u60c5\u5f62\uff1b\u5bf9\u4e8e\u7406\u89e3\u7f51\u7edc\u4e2d\u7684\u771f\u5b9e\u4fe1\u606f\u4f20\u64ad\u81f3\u5173\u91cd\u8981\u3002<\/span><\/p>\n

          [11] Domenico, M. D. Diffusion geometry unravels the emergence of functional clusters in collective phenomena. Physical Review Letters 118 (2017).  https:\/\/doi.org\/10.1103\/physrevlett.118.168301.<\/span><\/p>\n

          \u63d0\u4f9b\u4e86\u4e00\u4e2a\u5c06\u529f\u80fd\u6027\u5206\u533a\uff08functional clusters\uff09\u4e0e\u201c\u6269\u6563\u51e0\u4f55\u201d\uff08diffusion geometry\uff09\u76f8\u7ed3\u5408\u7684\u7edf\u4e00\u89c6\u89d2\uff0c\u6709\u52a9\u4e8e\u7406\u89e3\u8bb8\u591a\u751f\u7269\u3001\u793e\u4f1a\u3001\u5de5\u7a0b\u7f51\u7edc\u4e2d\u7684\u591a\u5c3a\u5ea6\u529f\u80fd\u5212\u5206\u3002<\/span><\/p>\n

          [12] Meena, C. et al. Emergent stability in complex network dynamics. Nature Physics 19, 1033\u20131042(2023). https:\/\/doi.org\/10.1038\/s41567-023-02020-8.<\/span><\/p>\n

          \u8bf4\u660e\u5927\u89c4\u6a21\u3001\u5f02\u8d28\u5ea6\u9ad8\u7684\u7f51\u7edc\u5728\u4e00\u5b9a\u6761\u4ef6\u4e0b\u4f1a\u81ea\u52a8\u83b7\u5f97\u7a33\u5b9a\u6027\uff08\u4e0e\u53c2\u6570\u548c\u5916\u90e8\u6270\u52a8\u65e0\u5173\uff09\uff0c\u800c\u53e6\u4e00\u4e9b\u5219\u6c38\u8fdc\u65e0\u6cd5\u7a33\u5b9a\u6216\u5bf9\u6270\u52a8\u6781\u4e3a\u654f\u611f\u3002\u4e3a\u7406\u89e3\u81ea\u7136\u4e0e\u4eba\u9020\u7cfb\u7edf\u7684\u7a33\u6001\u8c03\u63a7\u4e0e\u7a33\u5b9a\u6027\u8d77\u6e90\u63d0\u4f9b\u4e86\u7406\u8bba\u6846\u67b6\u3002<\/span><\/section>\n


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

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

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          <\/section>\n
          \u4e3b\u9898\u63a8\u8350\u8bed\uff1a<\/strong><\/span><\/section>\n
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          <\/strong><\/span><\/section>\n

          \u4e3b\u9898\u91cd\u70b9\u5173\u6ce8\uff1a<\/strong><\/span><\/p>\n

            \n
          1. \n

            \u7f51\u7edc\u535a\u5f08\u5efa\u6a21\u7684\u590d\u6742\u6027\u4e0e\u52a8\u6001\u6027\uff0c\u4ece\u590d\u6742\u7684\u591a\u65b9\u4ea4\u4e92\u3001\u52a8\u6001\u7f51\u7edc\u73af\u5883\u5230\u9ad8\u7ef4\u7b56\u7565\u7a7a\u95f4\uff1b<\/span><\/p>\n<\/li>\n

          2. \n

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            \u6784\u5efa\u4e86\u7f51\u7edc\u535a\u5f08\u6846\u67b6\uff0c\u9996\u6b21\u63d0\u51fa\u7f51\u7edc\u7ed3\u6784\u4f1a\u5bf9\u4e2a\u4f53\u95f4\u5408\u4f5c\u4ea7\u751f\u663e\u8457\u7684\u5f71\u54cd\u3002<\/span><\/p>\n

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            [3] D. G. Rand, S. Arbesman, and N. A. Christakis, \u201cDynamic social networks promote cooperation in experiments with humans,\u201d Proc. Nat. Acad. Sci. USA<\/em>, vol. 108, pp. 19193\u201319198, 2011.<\/span><\/p>\n

            \u8bbe\u8ba1\u4e86\u5b9e\u8bc1\u5b9e\u9a8c\uff0c\u8bc1\u660e\u4e86\u7f51\u7edc\u7684\u9ad8\u52a8\u6001\u6027\u53ef\u4ee5\u663e\u8457\u63d0\u9ad8\u4eba\u7c7b\u7fa4\u4f53\u5408\u4f5c\u6c34\u5e73\u3002<\/span><\/p>\n

            [4] W. Press and F. Dyson, \u201cIterated prisoner\u2019s dilemma contains strategies that dominate any evolutionary opponent,\u201d Proc. Nat. Acad. Sci. USA, vol. 109, pp. 10409\u201310413, 2012.<\/span><\/p>\n

            2010\u5e74\u4ee5\u540e\uff0c\u91cd\u590d\u535a\u5f08\u7406\u8bba\u7814\u7a76\u7684\u4ee3\u8868\u6027\u6210\u679c\uff0c\u8bc1\u660e\u4e86\u57fa\u4e8e\u7279\u5b9a\u7b56\u7565\u535a\u5f08\u4e00\u65b9\u53ef\u4ee5\u9650\u5236\u53e6\u4e00\u65b9\u7684\u6536\u76ca\u3002<\/span><\/p>\n

            [5] Y. Zhang, G. Chen, J. Guan, Z. Zhang, S. Zhou, \u201cUnfavorable individuals in social gaming networks,\u201d Scientific Reports<\/em>, vol. 5, p. 17481, 2015.<\/span><\/p>\n

            \u6784\u5efa\u4e86\u5206\u6cbb\u535a\u5f08\u6846\u67b6\uff0c\u5206\u6790\u4e2a\u4f53\u7684\u5ea6\u5bf9\u5176\u535a\u5f08\u6536\u76ca\u6ce2\u52a8\u8303\u56f4\u7684\u5f71\u54cd\u3002<\/span><\/p>\n

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            \u63ed\u793a\u521d\u59cb\u62d3\u6251\u7ed3\u6784\u662f\u5426\u5177\u6709\u8f83\u9ad8\u7684\u7fa4\u805a\u6027\u5bf9\u4e8e\u52a8\u6001\u7f51\u7edc\u4e2d\u7684\u5408\u4f5c\u6c34\u5e73\u6ca1\u6709\u5f71\u54cd\uff0c\u4f46\u5bf9\u9759\u6001\u7f51\u7edc\u5408\u4f5c\u6c34\u5e73\u7684\u5f71\u54cd\u663e\u8457\u3002\u6b64\u5916\uff0c\u58f0\u8a89\u4f1a\u5f71\u54cd\u5408\u4f5c\u4f19\u4f34\u7684\u9009\u62e9\u4f46\u4e0d\u5f71\u54cd\u5408\u4f5c\u6c34\u5e73\u3002\u5206\u6cbb\u6216\u7edf\u4e00\u51b3\u7b56\u5bf9\u4e8e\u52a8\u6001\u7f51\u7edc\u4e2d\u5408\u4f5c\u6c34\u5e73\u7684\u5f71\u54cd\u4e0d\u5927\uff0c\u4f46\u5bf9\u9759\u6001\u7f51\u7edc\u5408\u4f5c\u6c34\u5e73\u7684\u5f71\u54cd\u663e\u8457\u3002<\/span><\/p>\n

            [8] M. Perc, J. J. Jordan, D. G. Rand, Z. Wang, S. Boccaletti, A. Szolnoki, \u201cStatistical physics of human cooperation,\u201d Physics Reports<\/em>, vol. 687, pp. 1-51, 2017.<\/span><\/p>\n

            \u4ece\u65b9\u6cd5\u8bba\u7684\u5c42\u9762\u4ecb\u7ecd\u7814\u7a76\u4eba\u7c7b\u5408\u4f5c\u7684\u5404\u79cd\u7edf\u8ba1\u7269\u7406\u65b9\u6cd5\u3002<\/span><\/p>\n

            [9] Y. Zhang, G. Wen, G. Chen, J. Wang, M. Xiong, J. Guan, and S. Zhou, \u201cGaming temporal networks,\u201d IEEE Transactions on Circuits and Systems II: Express Briefs<\/em>, vol. 66, no. 4, pp. 672\u2013676, 2018.<\/span><\/p>\n

            \u5728\u5206\u6cbb\u535a\u5f08\u7684\u6846\u67b6\u4e0b\u63ed\u793a\u4eba\u7c7b\u534f\u4f5c\u9635\u53d1\u6027\u6e90\u4e8e\u7406\u6027\u4e92\u52a8\uff0c\u4ece\u65f6\u95f4\u7684\u89d2\u5ea6\u91cd\u65b0\u8ba4\u8bc6\u5206\u6cbb\u535a\u5f08\u7684\u52a8\u529b\u5b66\u7279\u5f81\u3002<\/span><\/p>\n

            [10] Y. Zhang, J. Wang, G. Wen, J. Guan, S. Zhou, G. Chen, K. Chatterjee, and M. Perc, \u201cLimitation of Time Promotes Cooperation in Structured Collaboration Systems,\u201d IEEE Transactions on Network Science and Engineering<\/em>, vol. 12, no. 1, pp. 4-12, 2025.<\/span><\/p>\n

            \u6784\u5efa\u4e86\u65f6\u5e8f\u535a\u5f08\u6846\u67b6\uff0c\u63ed\u793a\u4e86\u6709\u9650\u65f6\u95f4\u8d44\u6e90\u662f\u63d0\u5347\u7ed3\u6784\u5316\u534f\u4f5c\u7cfb\u7edf\u4e2d\u4eba\u7c7b\u5408\u4f5c\u6c34\u5e73\u7684\u91cd\u8981\u56e0\u7d20\u3002<\/span><\/section>\n

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

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

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

            1. \u63a2\u7d22\u7f51\u7edc\u52a8\u529b\u5b66\u7cfb\u5217\u8bfe\u7a0b<\/a><\/span><\/p>\n

            2. \u590d\u6742\u7f51\u7edc\u4e2d\u7684\u52a8\u529b\u5b66\u73b0\u8c61\uff1a\u57fa\u7840\u4e0e\u5e94\u7528<\/a><\/span><\/p>\n

            3. \u642d\u5efa\u56fe\u4e0e\u7f51\u7edc\u4e4b\u95f4\u7684\u6865\u6881\uff1a\u7f51\u7edc\u79d1\u5b66\u7684\u4e0b\u4e00\u4e2a\u7a81\u7834\u5728\u54ea\u91cc\uff1f<\/a><\/span><\/p>\n

            4. Nature \u5386\u5e74\u7f51\u7edc\u79d1\u5b66\u6587\u96c6\uff1a\u7f51\u7edc\u5c42\u6b21\u548c\u7ed3\u6784<\/a><\/span><\/p>\n

            5. Nature \u5386\u5e74\u7f51\u7edc\u79d1\u5b66\u6587\u96c6\uff1a\u52a8\u529b\u5b66\u4e0e\u4f20\u64ad<\/a><\/span><\/p>\n

            6. Barab\u00e1si\u7b97\u6cd5+Physics Reports\u7cbe\u9009\uff0c\u7f51\u7edc\u79d1\u5b66\u7efc\u8ff010\u5e74Top10 | \u5999\u7b97\u590d\u6742<\/a><\/span><\/p>\n


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

            <\/p>\n


            <\/p>\n


            <\/p>\n

            \u70b9\u51fb\u201c\u9605\u8bfb\u539f\u6587\u201d\uff0c\u62a5\u540d\u8bfb\u4e66\u4f1a<\/span><\/strong><\/span><\/strong><\/span><\/strong><\/p>\n

            <\/mp-style-type><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

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