{"id":30256,"date":"2021-11-24T19:07:13","date_gmt":"2021-11-24T11:07:13","guid":{"rendered":"https:\/\/swarma.org\/?p=30256"},"modified":"2021-11-24T19:07:13","modified_gmt":"2021-11-24T11:07:13","slug":"%e9%99%88%e5%85%b3%e8%8d%a3%ef%bc%9a%e6%8e%a2%e8%ae%a8%e5%a4%8d%e6%9d%82%e7%bd%91%e7%bb%9c%e7%9a%84%e9%ab%98%e9%98%b6%e6%8b%93%e6%89%91%e5%8f%8a%e5%85%b6%e5%ba%94%e7%94%a8","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=30256","title":{"rendered":"\u9648\u5173\u8363\uff1a\u63a2\u8ba8\u590d\u6742\u7f51\u7edc\u7684\u9ad8\u9636\u62d3\u6251\u53ca\u5176\u5e94\u7528"},"content":{"rendered":"
\n
\n
\n
\n

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

\n
\n
\n
\n
\n
\n

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

\u5bfc\u8bed<\/strong><\/p>\n<\/section>\n<\/section>\n

\n

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

\u7f51\u7edc\u79d1\u5b66\u6709\u52a9\u4e8e\u6211\u4eec\u66f4\u597d\u5730\u7406\u89e3\u5f53\u4eca\u9ad8\u5ea6\u4e92\u8054\u4e16\u754c\u7684\u6f14\u53d8\uff0c\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u5bf9\u7f51\u7edc\u7cfb\u7edf\u7684\u52a8\u529b\u5b66\u7814\u7a76\u4ea7\u751f\u4e86\u91cd\u5927\u5f71\u54cd\uff0c\u6253\u7834\u4e86\u539f\u59cb\u5355\u4e00\u7c7b\u578b\u6210\u5bf9\u4ea4\u4e92\u7684\u7f51\u7edc\u62d3\u6251\u5206\u6790\u7684\u5c40\u9650\u6027\uff0c\u4e3a\u590d\u6742\u7f51\u7edc\u7684\u4ea4\u4e92\u5206\u6790\u63d0\u4f9b\u4e86\u65b0\u7684\u601d\u8def\u3002\u672c\u6587\u6574\u7406\u4e86\u9648\u5173\u8363\u6559\u6388\u57282021\u5e74\u590d\u6742\u7f51\u7edc\u5b66\u672f\u4f1a\u8bae\u4e2d\u7684\u62a5\u544a\uff0c\u4ecb\u7ecd\u4e86\u590d\u6742\u7f51\u7edc\u9ad8\u9636\u62d3\u6251\u7684\u76f8\u5173\u6982\u5ff5\uff0c\u5982\u4f55\u5229\u7528\u9ad8\u9636\u62d3\u6251\u89e3\u51b3\u7f51\u7edc\u540c\u6b65\u95ee\u9898\uff0c\u5e76\u4e14\u6982\u8ff0\u4e86\u9ad8\u9636\u62d3\u6251\u5728\u8111\u79d1\u5b66\u3001\u751f\u6001\u5b66\u3001\u793e\u4f1a\u79d1\u5b66\u7b49\u591a\u9886\u57df\u7684\u5e94\u7528\u3002<\/span><\/strong><\/p>\n

 <\/strong><\/section>\n
\u96c6\u667a\u5b66\u56ed\u7279\u9080\u9648\u5173\u8363\u3001\u6a0a\u745b\u3001\u5468\u8fdb\u3001\u674e\u7fd4\u3001\u5f20\u6c5f\u3001\u95eb\u5c0f\u52c7\u3001\u5218\u5b97\u534e\u3001\u77f3\u5ddd\u3001\u865e\u6587\u6b66\u3001\u8d75\u6d77\u5174\u3001\u53f2\u5b9a\u534e\u7b49\u7f51\u7edc\u79d1\u5b66\u4e13\u5bb6\u4f5c\u4e3a\u5bfc\u5e08\uff0c\u81ea10\u670816\u65e5\u8d77\u5f00\u5c55\u201c\u7f51\u7edc\u79d1\u5b66\u7b2c\u4e09\u671f<\/a>\u201d\u7cfb\u5217\u5728\u7ebf\u8bfe\u7a0b\uff0c\u4ecb\u7ecd\u590d\u6742\u7f51\u7edc\u7684\u5efa\u6a21\u4e0e\u5e94\u7528\u4ee5\u53ca\u76f8\u5173\u8bfe\u9898\u3002\u6b22\u8fce\u5e0c\u671b\u8fdb\u5165\u7f51\u7edc\u79d1\u5b66\u9886\u57df\u3001\u63d0\u9ad8\u7f51\u7edc\u5206\u6790\u80fd\u529b\u3001\u4e0e\u4e00\u7ebf\u4e13\u5bb6\u63a2\u8ba8\u95ee\u9898\u7684\u670b\u53cb\u62a5\u540d\u53c2\u52a0\uff01<\/strong><\/section>\n


<\/p>\n

\u7814\u7a76\u9886\u57df\uff1a<\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong>\u7f51\u7edc\u79d1\u5b66\uff0c\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\uff0c\u7f51\u7edc\u540c\u6b65\u95ee\u9898\uff0c\u8111\u79d1\u5b66\uff0c\u751f\u6001\u5b66\uff0c\u793e\u4f1a\u79d1\u5b66<\/span><\/strong><\/span><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

\n
\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n
\n
<\/strong><\/section>\n<\/section>\n
\n

\u9648\u5173\u8363<\/span> | \u8bb2\u8005<\/strong><\/span><\/p>\n

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

\u6881\u91d1<\/span> | \u5ba1\u6821<\/strong><\/span><\/strong><\/span>
<\/p>\n

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


<\/section>\n
\u76ee\u5f55\uff1a
<\/span><\/strong><\/span><\/section>\n
\u4e00\u3001\u590d\u6742\u7f51\u7edc\u9ad8\u9636\u62d3\u6251\u6982\u8ff0<\/span><\/strong><\/section>\n
\n
\u4e8c\u3001\u9ad8\u9636\u62d3\u6251\u89e3\u51b3\u7f51\u7edc\u540c\u6b65\u95ee\u9898<\/span><\/strong><\/section>\n<\/section>\n
\n
\u4e09\u3001\u9ad8\u9636\u62d3\u6251\u7684\u591a\u9886\u57df\u5e94\u7528\uff1a\u8111\u79d1\u5b66\u3001\u751f\u6001\u5b66\u3001\u793e\u4f1a\u79d1\u5b66<\/span><\/strong><\/section>\n<\/section>\n
\n
\u56db\u3001\u603b\u7ed3\u5c55\u671b<\/span><\/strong><\/section>\n<\/section>\n

<\/section>\n

\n
\n
\n
\n
\n

<\/section>\n

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

\u4e00\u3001\u590d\u6742\u7f51\u7edc\u9ad8\u9636\u62d3\u6251\u6982\u8ff0<\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

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


<\/p>\n

\u5728\u8fc7\u5f80\u7684\u7814\u7a76\u4e2d\uff0c\u4eba\u4eec\u5e38\u5e38\u7528\u7f51\u7edc\u5c06\u8bb8\u591a\u81ea\u7136\u3001\u79d1\u5b66\u548c\u6280\u672f\u7b49\u7cfb\u7edf\u7684\u590d\u6742\u76f8\u4e92\u4f5c\u7528\u5f62\u8c61\u5730\u63cf\u8ff0\u51fa\u6765\u3002\u7136\u800c\uff0c\u4ece\u4eba\u7c7b\u4ea4\u6d41\u5230\u5316\u5b66\u53cd\u5e94\u548c\u751f\u6001\u7cfb\u7edf\uff0c\u76f8\u4e92\u4f5c\u7528\u7684\u5355\u4f4d\u5f80\u5f80\u4ee5\u4e09\u4e2a\u6216\u4e09\u4e2a\u4ee5\u4e0a\u7684\u8282\u70b9\u7ec4\u6210\uff0c\u4e0d\u80fd\u7b80\u5355\u5730\u7528\u4e24\u4e2a\u4e2a\u4f53\u7684\u5173\u7cfb\u6765\u63cf\u8ff0\u3002\u8fd1\u6765\uff0c\u771f\u5b9e\u590d\u6742\u7cfb\u7edf\u7684\u9ad8\u9636\u7ed3\u6784\u9010\u6e10\u88ab\u4eba\u4eec\u6240\u5173\u6ce8\u3002\u8d8a\u6765\u8d8a\u591a\u7684\u8bc1\u636e\u8868\u660e\uff0c\u5bf9\u590d\u6742\u7cfb\u7edf\u7684\u9ad8\u9636\u62d3\u6251\u7ed3\u6784\u7814\u7a76\uff0c\u53ef\u4ee5\u589e\u5f3a\u6211\u4eec\u9488\u5bf9\u590d\u6742\u7cfb\u7edf\u7684\u5efa\u6a21\u80fd\u529b\uff0c\u66f4\u597d\u5730\u7406\u89e3\u3001\u5206\u6790\u548c\u9884\u6d4b\u5176\u52a8\u6001\u884c\u4e3a<\/strong>\u3002\u5728\u8fd9\u91cc\u6211\u4eec\u5c1d\u8bd5\u7528\u590d\u6742\u7f51\u7edc\u7684\u9ad8\u9636\u62d3\u6251\u7ed3\u6784\uff0c\u660e\u786e\u800c\u53c8\u81ea\u7136\u5730\u63cf\u8ff0\u7fa4\u4f53\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u3002<\/span><\/section>\n
\n
\n
\n

<\/section>\n
\n

<\/section>\n
\n
\u56fe\u8bba\u57fa\u672c\u6982\u5ff5<\/strong><\/span><\/section>\n

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

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

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

\u5ea6\uff08Degree\uff09\uff1a\u8282\u70b9\u7684\u8fde\u8fb9\u6570\uff1b<\/span><\/p>\n

\u8ddd\u79bb\uff08Distance\uff09\uff1a\u4e24\u8282\u70b9\u4e4b\u95f4\u6700\u77ed\u8def\u5f84\u8fde\u8fb9\u6570\u7684\u603b\u6570\uff1b<\/span><\/p>\n

\u8def\u548c\uff08Path-sum\uff09\uff1a\u8282\u70b9\u5230\u5176\u5b83\u8282\u70b9\u8ddd\u79bb\u7684\u603b\u548c\uff1b<\/span><\/p>\n

\u5708\uff08Cycle\uff09\uff1a\u95ed\u8def\u5f84\uff1b<\/span><\/p>\n

\u5468\u957f\uff08Girth\uff09\uff1a\u7ecf\u8fc7\u67d0\u8282\u70b9\u7684\u6700\u77ed\u5708\u957f\uff1b<\/span><\/p>\n

\u56e2\uff08Clique\uff09\uff1a\u5168\u8fde\u63a5\u7684\u5b50\u56fe\uff1b\u56e2\u4e5f\u79f0\u4e3a\u5355\u7eaf\u5f62\u6216\u5355\u5f62\uff08Simplex\uff09\uff0c\u5728\u63a5\u4e0b\u6765\u7684\u5206\u6790\u8fc7\u7a0b\u4e2d\uff0c\u4ee5\u5355\u7eaf\u5f62\u4e3a\u8fb9\u754c\u7684\u590d\u5408\u56fe\u5f62\u79f0\u4e3a\u5355\u7eaf\u590d\u5f62\u3002<\/span><\/p>\n

\u6d1e\uff08Cavity\uff09\uff1a\u4e0d\u662f\u56e2\u7684\u6700\u5c0f\u65e0\u5173\u5708\u3002<\/span><\/p>\n


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

\uff08\u672c\u6587\u53ea\u8ba8\u8bba\u65e0\u5411\u7f51\u7edc\uff09<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe1 \u7f51\u7edc\u7ed3\u6784\u793a\u610f\u56fe<\/span><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

<\/section>\n

<\/section>\n
\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u7f51\u7edc\u7ed3\u6784\uff0c\u5f97\u5230\u5b83\u7684\u8282\u70b9\u5ea6\u77e9\u9635 D \u548c\u8868\u793a\u8282\u70b9-\u8282\u70b9\u5173\u7cfb\u7684<\/span>\u90bb\u63a5\u77e9\u9635<\/strong> A\uff0c\u4ee5\u53ca\u62c9\u666e\u62c9\u65af\u77e9\u9635<\/strong> L\uff0c\u5b83\u4eec\u4e4b\u95f4\u7684\u5173\u7cfb\uff1aL=D-A\u3002\u4e5f\u53ef\u4ee5\u5f97\u5230\u8868\u793a\u8282\u70b9-\u8fde\u8fb9\u5173\u7cfb\u7684\u5173\u8054\u77e9\u9635<\/strong> M\u3002\u90a3\u4e48\u9488\u5bf9\u9ad8\u9636\u60c5\u51b5\uff0c\u6211\u4eec\u53ef\u4ee5\u7814\u7a76\u8fde\u8fb9-\u4e09\u89d2\u5f62\u3001\u4e09\u89d2\u5f62-\u4e09\u89d2\u5f62\u3001\u4e09\u89d2\u5f62-\u56db\u9762\u4f53\u4ee5\u53ca\u56db\u9762\u4f53-\u56db\u9762\u4f53\u7b49\u9ad8\u9636\u56e2\u4e4b\u95f4\u7684\u5173\u7cfb\u6765\u6784\u5efa\u5173\u8054\u77e9\u9635\uff0c\u4ece\u800c\u8fdb\u884c\u590d\u6742\u7f51\u7edc\u7684\u5206\u6790\u3002<\/span><\/section>\n
 <\/span><\/section>\n

\u5177\u4f53\u8868\u793a\u65b9\u6cd5\u4e3a\uff1a\u4ee5\u201c\u56e2\u201d\u4e3a\u57fa\u672c\u5355\u5143\uff0c\u5efa\u7acb\u4e00\u7cfb\u5217\u4e8c\u5143\u57df\u4e0a\u7684\u5411\u91cf\u7a7a\u95f4\u3002\u5176\u4e2d\uff1a<\/span><\/p>\n

\u00b7 \u4ee5\u8282\u70b9\u4e3a\u57fa\u6240\u5f97\u5411\u91cf\u7a7a\u95f4 C0<\/sub>\uff0c\u5176\u7ef4\u6570\u662f\u7f51\u7edc\u4e2d\u7684\u8282\u70b9\u6570\u76ee<\/span><\/p>\n

\u00b7 \u4ee5\u8fde\u8fb9\u4e3a\u57fa\u6240\u5f97\u5411\u91cf\u7a7a\u95f4 C1<\/sub>\uff0c\u5176\u7ef4\u6570\u662f\u7f51\u7edc\u4e2d\u7684\u8fde\u7ebf\u6570\u76ee<\/span><\/p>\n

\u00b7 \u4ee5\u4e09\u89d2\u5f62\u4e3a\u57fa\u7684\u5411\u91cf\u7a7a\u95f4 C2<\/sub>\uff0c\u5176\u7ef4\u6570\u662f\u7f51\u7edc\u4e2d\u4e09\u89d2\u5f62\u6570\u76ee<\/span><\/p>\n

\u00b7 \u4ee5\u56db\u9762\u4f53\u4e3a\u57fa\u7684\u5411\u91cf\u7a7a\u95f4 C3<\/sub>\uff0c\u5982\u6b64\u7c7b\u63a8<\/span><\/section>\n
 <\/span><\/section>\n
\u5728\u6b64\u57fa\u7840\u4e0a\u5f15\u8fdb\u8fd0\u7b97\u89c4\u5219\uff1a\u4e8c\u5143\u57df<\/strong>\u662f\u6574\u6570\u6a21 2 \u7684\u57df\uff0c\u53ea\u6709\u5143\u7d20 0 \u548c 1\uff0c\u6ee1\u8db3 0+0=0, 0+1=1, 1+0=1, 1+1=0\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5411\u91cf\u7a7a\u95f4\u4e2d\u7684\u52a0\u6cd5\uff0c\u4e0e\u6570\u7684\u8ba1\u7b97\u4e0d\u540c\uff0c\u9700\u8981\u7528\u96c6\u5408\u8bba\u7684\u89c2\u70b9\u6765\u7406\u89e3\u3002\u4e24\u5411\u91cf c\u3001d \u7684\u52a0\u6cd5\uff0c\u5176\u548c\u662f\u4e24\u4e2a\u5b50\u96c6\u7684\u5bf9\u79f0\u5dee\uff0c\u5373 c+d=(c\u222ad)-(c\u2229d)\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u6839\u636e\u4e0a\u8ff0\u65b9\u6cd5\uff0c\u6784\u5efa\u4e86\u7ef4\u6570\u4e0d\u540c\u7684\u4e24\u4e2a\u7a7a\u95f4\u4e4b\u540e\uff0c\u4fbf\u53ef\u4ee5\u5f15\u5165\u8fb9\u754c\u7b97\u5b50<\/strong> <\/sub>\uff0c\u5efa\u7acb\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u8054\u7cfb\u3002\u8fd9\u91cc\u7684\u8fb9\u754c\u7b97\u5b50\u53ef\u4ee5\u8ba4\u4e3a\u662f\u4e00\u4e2a\u6295\u5f71\u7b97\u5b50\uff0c\u5c06\u9ad8\u9636\u5411\u91cf\u7a7a\u95f4\u4e2d\u4e00\u4e2a\u57fa\u672c\u5355\u4f4d\uff0c\u6295\u5f71\u5230\u4e0b\u4e00\u9636\u7684\u5411\u91cf\u7a7a\u95f4\u4e2d\uff0c\u523b\u753b\u5b83\u4eec\u4e4b\u95f4\u7684\u8fb9\u754c\u3002 <\/span><\/section>\n
 <\/span><\/section>\n
\u8fd9\u6837\uff0c\u53ef\u4ee5\u7528\u8fb9\u754c\u7b97\u5b50\u6765\u5b9a\u4e49\u5708\uff1a\u5982\u679c L \u5728\u8fb9\u754c\u7b97\u5b50\u4f5c\u7528\u540e\u7b49\u4e8e 0\uff0c\u5373<\/sub>\uff0c\u90a3\u4e48 L \u5c31\u662f\u4e00\u4e2a\u5708 \u3002\u8fdb\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u6e05\u6670\u5730\u7406\u89e3\u6d1e\u7684\u6982\u5ff5\u3002\u6d1e\u9996\u5148\u662f\u4e00\u4e2a\u5708\uff0c\u7136\u540e\u662f\u5728\u4e8c\u5143\u57df\u8fd0\u7b97\u610f\u4e49\u4e0b\u4e0e\u5176\u4ed6\u5708\u4e3a\u7ebf\u6027\u65e0\u5173\u7684\u4e00\u4e2a\u6700\u5c0f\u7684\u5708\uff0c\u6700\u540e\u5b83\u4e0d\u662f\u4e00\u4e2a\u56e2\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u56fe2. \u7f51\u7edc\u4e2d\u5404\u9636\u5708\u3001\u56e2\u3001\u6d1e\u3001\u8fb9\u754c\u793a\u610f\u56fe<\/span><\/section>\n

<\/span><\/section>\n
\u57fa\u4e8e\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8fb9\u754c\u7b97\u5b50\u628a\u4e32\u8054\u8d77\u6765\u7684\u5411\u91cf\u7a7a\u95f4\u7528\u8fb9\u754c\u77e9\u9635\u6765\u8868\u793a\uff0c\u5e76\u5728\u4e8c\u5143\u57df\u4e0a\u8fdb\u884c\u521d\u7b49\u884c\u5217\u53d8\u6362\uff0c\u8ba1\u7b97\u51fa\u77e9\u9635\u7684\u79e9\u3002\u6839\u636e\u7f51\u7edc\u7684\u7ef4\u6570\u548c\u79e9\uff0c\u6211\u4eec\u53ef\u4ee5\u5f15\u5165\u7f51\u7edc\u4e0d\u53d8\u91cf\uff1a<\/span><\/section>\n
 <\/span><\/section>\n

  \u00b7 \u793a\u6027\u6570<\/strong>\uff1a\u7f51\u7edc\u5404\u9636\u56e2\u7684\u6570\u76ee\u5206\u522b\u8bb0\u4e3a m0<\/sub><\/span>\uff08\u8282\u70b9\u6570\uff09<\/span>\uff0cm1<\/sub><\/span>\uff08\u8fde\u7ebf\u6570\uff09<\/span>\uff0cm2<\/sub><\/span>\uff08\u4e09\u89d2\u5f62\u6570\uff09<\/span>\uff0cm3<\/sub><\/span>\uff08\u56db\u9762\u4f53\u6570\uff09<\/span>\u7b49\u7b49\uff0c\u5219\u793a\u6027\u6570\u516c\u5f0f\u4e3a\uff1a<\/sub><\/span><\/p>\n

  \u00b7 \u8d1d\u8482\u6570<\/strong><\/span>\uff08Betti\uff09<\/span>\uff1a\u7f51\u7edc\u5404\u8fb9\u754c\u77e9\u9635\u7684\u79e9\u5206\u522b\u8bb0\u4e3a r0<\/sub>=0<\/span>\uff08\u7ea6\u5b9a\uff09<\/span>\uff0cr1<\/sub><\/span>\uff08\u70b9\u7ebf\u77e9\u9635\uff09<\/span>\uff0cr2<\/sub><\/span>\uff08\u7ebf\u9762\u77e9\u9635\uff09<\/span>\uff0cr3<\/sub><\/span>\uff08\u9762\u4f53\u77e9\u9635\uff09<\/span>\uff0c\u7b49\u7b49\u3002\u5219\u8d1d\u8482\u6570\u516c\u5f0f\u4e3a\uff1a<\/sub><\/span><\/p>\n

\u5176\u4e2d\u03b20<\/sub>\u4e3a\u8fde\u901a\u5b50\u5757\u6570\u76ee\uff0c\u03b21<\/sub>\u4e3a1\u6d1e\u7684\u6570\u76ee\uff0c\u03b22<\/sub>\u4e3a2\u6d1e\u7684\u6570\u76ee\uff0c\u7b49\u7b49\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u8fd9\u6837\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u6839\u636e\u4e0a\u9762\u6240\u9610\u8ff0\u7684\u6982\u5ff5\uff0c\u901a\u8fc7\u9ad8\u9636\u62d3\u8865\u7ed3\u6784\u6765\u5206\u6790\u590d\u6742\u7f51\u7edc\u3002<\/span><\/section>\n
 <\/span><\/section>\n
 <\/span><\/section>\n

\n
\n
\n
\n
\n

<\/section>\n

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

\u4e8c\u3001\u9ad8\u9636\u62d3\u6251\u89e3\u51b3\u7f51\u7edc\u540c\u6b65\u95ee\u9898<\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

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


<\/p>\n

\u4eba\u4eec\u5df2\u7ecf\u6ce8\u610f\u5230\uff0c\u5927\u91cf\u73b0\u5b9e\u4e16\u754c\u4e2d\u7684\u590d\u6742\u7f51\u7edc\u6216\u5176\u5b50\u7f51\u7edc\u5177\u6709\u8bf8\u591a\u4f18\u826f\u7684\u52a8\u529b\u5b66\u6027\u8d28\uff0c\u6bd4\u5982\u52a8\u6001\u540c\u6b65\u6027\u5f3a\u3001\u53ef\u63a7\u6027\u597d\u3001\u6297\u653b\u51fb\u80fd\u529b\u5f3a\u3001\u4fe1\u606f\u4f20\u8f93\u901f\u5ea6\u5feb\u3001\u534f\u4f5c\u81ea\u7136\u6d8c\u73b0\u7b49\u3002\u7f51\u7edc\u540c\u6b65\u95ee\u9898\u5df2\u7136\u6210\u4e3a\u7814\u7a76\u7684\u70ed\u70b9\u3002<\/span><\/section>\n
 <\/span><\/section>\n

\n
\u00b7 <\/span><\/strong>\u5168\u9f50\u7f51\u7edc\u662f\u540c\u6b65\u6700\u4f18\u7f51\u7edc<\/strong><\/span><\/section>\n<\/h2>\n

<\/section>\n
\u5168\u9f50\u7f51\u7edc\u662f\u6240\u6709\u8282\u70b9\u5177\u6709\u76f8\u540c\u7684\u5ea6<\/span>\uff08degree\uff09<\/span>\uff0c\u8def\u548c<\/span>\uff08path-sum\uff09<\/span>\uff0c\u5468\u957f<\/span>\uff08girth\uff09<\/span>\u7684\u7f51\u7edc\uff0c\u5177\u6709\u5bf9\u79f0\u548c\u5747\u5300\u7684\u7279\u70b9\u3002<\/span><\/section>\n
 <\/span><\/section>\n
<\/section>\n
\u56fe3 \u5168\u9f50\u7f51\u7edc\u793a\u4f8b<\/span><\/section>\n
 <\/span><\/section>\n
\u5c06\u5168\u9f50\u7f51\u7edc\u548c\u5176\u4ed6\u7f51\u7edc\u7ed3\u6784\u8fdb\u884c\u5bf9\u6bd4\uff0c\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u4e00\u822c\u89c4\u5f8b\uff1a\u5168\u9f50\u7f51\u7edc\u7684\u540c\u6b65\u80fd\u529b\u4f18\u4e8e\u968f\u673a\u7f51\u7edc\uff0c\u518d\u4f18\u4e8e\u5c0f\u4e16\u754c\u7f51\u7edc\uff0c\u8f83\u5dee\u7684\u4e3a\u89c4\u5219\u7f51\u7edc\u3002<\/span><\/section>\n
 <\/span><\/section>\n
<\/section>\n
\u56fe4 \u4e0d\u540c\u7f51\u7edc\u7684\u540c\u6b65\u80fd\u529b\u6bd4\u8f83<\/span><\/section>\n
\n
D.H. Shi, G. Chen, W.W.K. Thong, X.Y. Yan, Searching for optimal network topology with best possible synchronizability, IEEE Circ. Syst. Mag. 13(1): 66-75, 2013<\/span><\/section>\n
https:\/\/ieeexplore.ieee.org\/stamp\/stamp.jsp?tp=&arnumber=6468141<\/span><\/section>\n<\/blockquote>\n
 <\/span><\/section>\n
\u901a\u5e38\u5728\u7f51\u7edc\u540c\u6b65\u80fd\u529b\u7684\u6bd4\u8f83\u4e2d\uff0c\u8c31\u9699\u548c\u7279\u5f81\u6bd4\u8d8a\u5927\uff0c\u663e\u793a\u7f51\u7edc\u540c\u6b65\u80fd\u529b\u8d8a\u597d\u3002\u4f46\u662f\u56e0\u4e3a\u7279\u5f81\u6bd4\u7684\u540c\u6b65\u57df\u53d6\u503c\u533a\u95f4\u6709\u53ef\u80fd\u662f\u591a\u4e2a\u533a\u95f4\u7684\u5e76\u96c6\uff0c\u7528\u7279\u5f81\u6bd4\u4f5c\u4e3a\u7f51\u7edc\u80fd\u529b\u8bc4\u5224\u6807\u51c6\u65f6\uff0c\u5076\u5c14\u4f1a\u51fa\u73b0\u4e0d\u51c6\u786e\u6027\u3002\u7814\u7a76\u53d1\u73b0\u91c7\u7528\u793a\u6027\u6570\u53ef\u4ee5\u5341\u5206\u6e05\u6670\u51c6\u786e\u5730\u8868\u793a\u4e0d\u540c\u7f51\u7edc\u7684\u540c\u6b65\u80fd\u529b\u3002\u4e8e\u662f\u5f97\u5230\u4ee5\u4e0b\u7ed3\u8bba\uff1a\u7f51\u7edc\u89c4\u6a21\u76f8\u540c\u6761\u4ef6\u4e0b\uff0c\u5f71\u54cd\u7f51\u7edc\u540c\u6b65\u80fd\u529b\u7684\u5173\u952e\u56e0\u7d20\u662f\u7f51\u7edc\u4e0d\u53d8\u91cf\u4e4b\u4e00\u7684\u793a\u6027\u6570\uff0c\u793a\u6027\u6570\u8d8a\u5c0f\u7f51\u7edc\u8d8a\u5bb9\u6613\u540c\u6b65\u3002<\/strong><\/span><\/section>\n
\n
D. H. Shi, L. Y. Lyu, G. R. Chen, Totally homogeneous networks, National Science Review (2019)<\/span><\/section>\n
https:\/\/arxiv.org\/ftp\/arxiv\/papers\/1903\/1903.11289.pdf<\/span><\/section>\n<\/blockquote>\n

\n

<\/section>\n
\u00b7 \u590d\u6742\u7f51\u7edc\u4e2d\u7684\u9ad8\u9636\u7ec4\u7ec7\u7ed3\u6784<\/span><\/strong><\/section>\n<\/h2>\n

<\/section>\n
\u8bb8\u591a\u7f51\u7edc\u5c55\u793a\u4e86\u4e30\u5bcc\u7684\u3001\u53ef\u4ee5\u5728\u5355\u4e2a\u8282\u70b9\u7ea7\u522b\u6355\u83b7\u7684\u4f4e\u9636\u94fe\u63a5\u6a21\u5f0f\u548c\u8fb9\u754c\u4fe1\u606f\u3002\u4f46\u662f\u590d\u6742\u7f51\u7edc\u7684\u9ad8\u9636\u7ec4\u7ec7\u7ed3\u6784\u548c\u7279\u6027\u5f88\u5927\u7a0b\u5ea6\u4e0a\u4ecd\u7136\u672a\u77e5\u3002\u4e0b\u9762\u8fd9\u7bc7\u6587\u7ae0\u5efa\u7acb\u4e86\u4e00\u4e2a\u901a\u7528\u7684\u57fa\u4e8e\u9ad8\u9636\u8fde\u63a5\u6a21\u5f0f\u7684\u805a\u7c7b\u7f51\u7edc\u6a21\u578b\uff0c\u63ed\u793a\u4e86\u8bb8\u591a\u7f51\u7edc\u4e2d\u5b58\u5728\u7684\u9ad8\u9636\u7ec4\u7ec7<\/span>\uff08\u4f8b\u5982\u795e\u7ecf\u5143\u7f51\u7edc\u4fe1\u606f\u4f20\u64ad\u548c\u4ea4\u901a\u67a2\u7ebd\u7ed3\u6784\u7f51\u7edc\uff09<\/span>\uff0c\u53d1\u73b0\u7f51\u7edc\u6a21\u578b\u901a\u8fc7\u57fa\u4e8e\u9ad8\u9636\u805a\u7c7b\u8fde\u63a5\u6a21\u5f0f\u8868\u73b0\u51fa\u4e30\u5bcc\u7684\u9ad8\u9636\u7ec4\u7ec7\u7ed3\u6784\uff0c\u5e76\u4e14\u5bf9\u4e8e\u9ad8\u8fbe\u6570\u5341\u4ebf\u6761\u8fb9\u7684\u5927\u89c4\u6a21\u7f51\u7edc\u7684\u96c6\u7fa4\u6700\u4f18\u6027\u63d0\u4f9b\u6570\u5b66\u6839\u636e\u3002<\/span><\/section>\n
 <\/span><\/section>\n
<\/section>\n
\u56fe5 \u9ad8\u9636\u7f51\u7edc\u7ed3\u6784\u548c\u9ad8\u9636\u7f51\u7edc\u805a\u7c7b\u6846\u67b6<\/span><\/section>\n
\n
A.R. Benson, D.F. Gleich, J. Leskovec, Higher-order organization of complex networks, Science, 353: 163-166, 2016<\/span><\/section>\n
https:\/\/www.science.org\/doi\/epdf\/10.1126\/science.aad9029<\/span><\/section>\n<\/blockquote>\n
 <\/span><\/section>\n
\u6211\u4eec\u7279\u522b\u63a8\u8350\u4e00\u7bc7\u7efc\u8ff0\u6587\u7ae0\u201cNetworks beyond pairwise interactions: Structure and dynamics\u201d\uff0c\u53ef\u4ee5\u4ece\u8fd9\u7bc7\u6587\u7ae0\u5f00\u59cb\u4e86\u89e3\u9ad8\u9636\u62d3\u6251\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u8fd9\u7bc7\u7efc\u8ff0\u5168\u9762\u4ecb\u7ecd\u4e86\u65b0\u5174\u7684\u8d85\u8d8a\u8282\u70b9\u5bf9\u76f8\u4e92\u4f5c\u7528\u7684\u7f51\u7edc\u9886\u57df\uff0c\u8ba8\u8bba\u4e86\u5982\u4f55\u8868\u793a\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\uff0c \u5e76\u4ecb\u7ecd\u63cf\u8ff0\u9ad8\u9636\u7cfb\u7edf\u7684\u6846\u67b6<\/span>\uff08\u4f8b\u5982\u968f\u673a\u548c\u589e\u957f\u578b\u4e8c\u5206\u56fe\u3001\u8d85\u56fe\u548c\u5355\u7eaf\u590d\u5f62\uff09<\/span>\uff0c\u4ecb\u7ecd\u4e86\u9ad8\u9636\u52a8\u529b\u7cfb\u7edf\u548c\u62d3\u6251\u52a8\u529b\u5b66\uff0c\u7814\u7a76\u4e86\u9ad8\u9636\u4e92\u52a8\u4e0e\u7fa4\u96c6\u884c\u4e3a\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u7279\u522b\u7740\u91cd\u63cf\u8ff0\u4e86\u52a8\u6001\u8fc7\u7a0b\u7684\u65b0\u73b0\u8c61\uff0c\u5982\u6269\u6563\u3001\u540c\u6b65\u3001\u4f20\u64ad\u3001\u793e\u4f1a\u52a8\u6001\u548c\u535a\u5f08\uff0c\u4ee5\u53ca\u5f53\u5b83\u4eec\u6269\u5c55\u5230\u8d85\u51fa\u70b9\u5bf9\u76f8\u4e92\u4f5c\u7528\u65f6\u7684\u60c5\u5f62\uff0c\u5e76\u4ecb\u7ecd\u5b83\u4eec\u7684\u76f8\u5173\u5e94\u7528\u3002<\/span><\/section>\n
 <\/span><\/section>\n
<\/section>\n
\u56fe6 \u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u8868\u73b0\u5f62\u5f0f<\/span><\/section>\n
\n
F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lucas, A. Patania, J.-G. Young, G. Petri, Networks beyond pairwise interactions: Structure and dynamics, Physics Reports, 847: 1-92, 2020<\/span><\/section>\n
https:\/\/arxiv.org\/abs\/2006.01764v1<\/span><\/section>\n<\/blockquote>\n

\n

<\/section>\n
\u00b7 \u8026\u5408\u9ad8\u9636\u5355\u7eaf\u590d\u5f62\u4e0a\u62d3\u6251\u4fe1\u53f7\u7684\u540c\u6b65<\/span><\/strong><\/section>\n<\/h2>\n

<\/section>\n
\u4ece\u5927\u8111\u5230\u793e\u4ea4\u7f51\u7edc\uff0c\u5355\u7eaf\u590d\u5f62\u80fd\u5f88\u597d\u5730\u63cf\u8ff0\u590d\u6742\u7684\u5e95\u5c42\u7f51\u7edc\u62d3\u6251\u548c\u51e0\u4f55\u3002\u4ee3\u6570\u62d3\u6251\u662f\u4e00\u4e2a\u7814\u7a76\u5355\u7eaf\u590d\u5f62\u9ad8\u9636\u52a8\u529b\u5b66\u7684\u57fa\u672c\u5de5\u5177\u3002\u8003\u8651\u62d3\u6251\u4fe1\u53f7\uff0c\u5373\u5b9a\u4e49\u5728\u4e0d\u540c\u7684\u5355\u7eaf\u5f62\u4e0a\u7684\u52a8\u6001\u4fe1\u53f7\uff0c\u53d1\u73b0\u5728\u8282\u70b9\u548c\u8fde\u8fb9\u4e4b\u95f4\u8fb9\u754c\u8026\u5408\u4e0a\u5b9a\u4e49\u7684\u4fe1\u53f7\u53ef\u4ee5\u5bfc\u81f4\u7206\u70b8\u6027\u7684\u62d3\u6251\u540c\u6b65\u3001\u5b9e\u73b0\u76f8\u4f4d\u8f6c\u6362\u548c\u540c\u6b65\u3002\u53d1\u5c55\u4e86\u4e00\u4e2a\u5168\u9762\u7684\u7406\u8bba\u548c\u65b9\u6cd5\uff0c\u53ef\u5728\u5168\u8fde\u63a5\u7f51\u7edc\u548c\u968f\u673a\u7f51\u7edc\u4e0a\u9884\u6d4b\u8fd9\u79cd\u8f6c\u53d8\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\n
R. Ghorbanchian, J.G. Restrepo, J.J. Torres, G. Bianconi, Higher-order simplicial synchronization of coupled topological signals, Comm. Phys., 4: Paper 120, 2021<\/span><\/section>\n
https:\/\/arxiv.org\/abs\/2011.00897v2<\/span><\/section>\n<\/blockquote>\n

\n

<\/section>\n
\u00b7 \u9ad8\u7ef4\u5355\u7eaf\u590d\u5f62\u4e0a Kuramoto \u632f\u5b50\u7684\u76f8\u4f4d\u540c\u6b65<\/span><\/strong><\/section>\n

<\/span><\/section>\n<\/h2>\n
<\/section>\n
\u56fe7 Kuramoto \u6a21\u578b\u7684\u63a8\u5e7f<\/span><\/section>\n
 <\/span><\/section>\n
\u6b64\u6587\u4e2d\u7684\u7406\u8bba\u4e0d\u4ec5\u5b8c\u6574\u5730\u63cf\u8ff0\u548c\u63ed\u793a\u4e86\u8fc4\u4eca\u4e3a\u6b62\u901a\u8fc7\u6210\u5bf9\u76f8\u4e92\u4f5c\u7528\u6240\u89c2\u5bdf\u5230\u7684\u5404\u79cd\u73b0\u8c61\u7684\u5185\u5728\u539f\u56e0\u548c\u673a\u5236\uff0c\u800c\u4e14\u8fd8\u4e3a\u5355\u7eaf\u5f62\u7ed3\u6784\u4e2d\u4e00\u7cfb\u5217\u4e30\u5bcc\u800c\u65b0\u9896\u7684\u884c\u4e3a\u63d0\u4f9b\u4e86\u9884\u6d4b\u3002\u8fd9\u4e9b\u884c\u4e3a\u5305\u62ec\uff1a\u5728\u6240\u6709\u7ef4\u5ea6\u7684\u8026\u5408\u5f3a\u5ea6\u4e3a\u6b63\u503c\u65f6\u7684\u4e0d\u8fde\u7eed\u89e3\u7684\u540c\u6b65\u8fc7\u6e21\uff1b\u5728\u6240\u6709\u5947\u6570\u7ef4\u5ea6\u7684\u96f6\u8026\u5408\u5904\u7684\u8dc3\u8fc1\uff1b\u5728\u90e8\u5206\u540c\u6b65\u72b6\u6001\u51fa\u73b0\u65f6\uff0c\u751a\u81f3\u5bf9\u8026\u5408\u5f3a\u5ea6\u4e3a\u8d1f\u503c\uff0c\u8fd9\u79cd\u7279\u5f81\u5728\u6210\u5bf9\u76f8\u4e92\u4f5c\u7528\u4e2d\u662f\u4e0d\u53ef\u80fd\u7684\u3002\u6b64\u5916\uff0c\u8be5\u6587\u7684\u7406\u8bba\u89e3\u91ca\u4e86\u6d8c\u73b0\u884c\u4e3a\u7684\u51e0\u4e2a\u65b9\u9762: \u7cfb\u7edf\u4e0d\u80fd\u4ece\u65e0\u5e8f\u4e2d\u8fbe\u5230\u5b8c\u5168\u540c\u6b65\uff0c\u5e76\u4e14\u62e5\u6709\u6781\u7aef\u591a\u7a33\u5b9a\u6027\uff0c\u56e0\u4e3a\u6e10\u8fd1\u7a33\u5b9a\u72b6\u6001\u603b\u662f\u4f9d\u8d56\u4e8e\u521d\u59cb\u6761\u4ef6\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u7ed3\u8bba\uff1a\u5947\u6570\u7ef4\u548c\u5076\u6570\u7ef4\u7684\u632f\u5b50\u7f51\u7edc\u540c\u6b65\u7279\u6027\u975e\u5e38\u4e0d\u4e00\u6837\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\n
X. Dai, K. Kovalenko, M. Molodyk, Z. Wang, X. Li, D. Musatov, A.M. Raigorodskii, A. Bittner, G.D. Cooper, G. Bianconi, S. Boccaletti, D-dimensional oscillators in simplicial structures: Odd and even dimensions display different synchronization scenarios. Chaos, Solitons Fractals, 146: 110888, 2021<\/span><\/section>\n
https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0960077921002411<\/span><\/section>\n<\/blockquote>\n

\n

<\/section>\n
\u00b7 \u5e7f\u4e49\u7f51\u7edc\u540c\u6b65\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u65b0\u5f62\u6001<\/span><\/strong><\/section>\n<\/h2>\n

<\/span><\/section>\n
\u8fd9\u7bc7\u6587\u7ae0\u5efa\u7acb\u4e86\u4e00\u4e2a\u7b80\u5316\u96c6\u7fa4\u540c\u6b65\u6a21\u5f0f\u7684\u7a33\u5b9a\u6027\u5206\u6790\u7edf\u4e00\u6846\u67b6\uff0c\u53ef\u5e94\u7528\u4e8e\u5e7f\u4e49\u7f51\u7edc\uff0c\u5305\u62ec\u8d85\u56fe\u3001\u591a\u5c42\u7f51\u7edc\u548c\u65f6\u6548\u7f51\u7edc\u3002\u57fa\u4e8e\u5206\u5757\u5bf9\u89d2\u5316\u7f16\u7801\u540c\u6b65\u6a21\u5f0f\u548c\u7f51\u7edc\u62d3\u6251\u77e9\u9635\uff0c\u63ed\u793a\u4e86\u4e00\u79cd\u6709\u8da3\u7684\u3001\u4ec5\u5728\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u65f6\u51fa\u73b0\u7684\u590d\u6742\u5d4c\u5408\u72b6\u6001\u7c7b\u578b \u3002\u5efa\u7acb\u7684\u7edf\u4e00\u6846\u67b6\u53ef\u4ee5\u63a8\u5e7f\u5230\u5176\u4ed6\u52a8\u529b\u5b66\u8fc7\u7a0b\uff0c\u5e76\u4e14\u53ef\u4ee5\u5e2e\u52a9\u5728\u5177\u6709\u5e7f\u4e49\u76f8\u4e92\u4f5c\u7528\u7684\u590d\u6742\u7cfb\u7edf\u4e2d\u53d1\u73b0\u65b0\u7684\u7a81\u73b0\u73b0\u8c61\u3002<\/span><\/section>\n

<\/span><\/section>\n
\n
Y. Zhang, V. Latora, A.E. Motter, Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactions, Comm. Phys., 4: Article 195, 2021<\/span><\/section>\n
https:\/\/arxiv.org\/pdf\/2010.00613v2.pdf<\/span><\/section>\n<\/blockquote>\n

\n

<\/section>\n
\u00b7 \u9ad8\u9636\u7f51\u7edc\u540c\u6b65\u7684\u591a\u9636\u62c9\u666e\u62c9\u65af\u7b97\u5b50\u65b9\u6cd5<\/span><\/strong><\/section>\n<\/h2>\n

<\/span><\/section>\n
\u8bb8\u591a\u7269\u7406\u7cfb\u7edf\u4e2d\u5b58\u5728\u591a\u4e2a\u7fa4\u4f53\u4ea4\u4e92\u7684\u540c\u6b65\u73b0\u8c61\uff0c \u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u53ef\u4ee5\u660e\u663e\u5730\u5f71\u54cd\u7f51\u7edc\u7684\u65b0\u52a8\u6001\u3002\u8be5\u6587\u5bf9\u590d\u6742\u62d3\u6251<\/span>\uff08\u5982\u8d85\u56fe\uff09<\/span>\u5f15\u5165\u4e86\u591a\u9636\u62c9\u666e\u62c9\u65af\u7b97\u5b50\uff0c\u5176\u9891\u8c31\u51b3\u5b9a\u4e86\u540c\u6b65\u7684\u7a33\u5b9a\u6027\u3002\u5bf9\u63cf\u8ff0\u7315\u7334\u5927\u8111\u8fde\u63a5\u771f\u5b9e\u6570\u636e\u7684\u5206\u6790\uff0c\u53d1\u73b0\u5927\u8111\u8fde\u63a5\u4e2d\u7fa4\u7ec4\u7684\u91cd\u8981\u6027\uff0c\u5b83\u4eec\u63cf\u8ff0\u4e86\u7cfb\u7edf\u4e2d\u7684\u76f8\u4e92\u4f5c\u7528\u7684\u590d\u6742\u6027\u3002\u591a\u9636\u62c9\u666e\u62c9\u65af\u7b97\u5b50\u6709\u52a9\u4e8e\u83b7\u5f97\u4efb\u610f\u9ad8\u9636\u7f51\u7edc\u4e2d\u540c\u6b65\u7a33\u5b9a\u6027\u7684\u5b8c\u6574\u89e3\u6790\u8868\u5f81\uff0c\u5bf9\u8d85\u8d8a\u70b9\u5bf9\u76f8\u4e92\u4f5c\u7528\u7684\u52a8\u6001\u8fc7\u7a0b\u63d0\u4f9b\u4e00\u79cd\u4e00\u822c\u7684\u5904\u7406\u65b9\u6cd5\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe8 \u5355\u7eaf\u590d\u5f62\u4e0a\u5177\u6709\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u632f\u5b50\u7fa4<\/span><\/section>\n

<\/span><\/section>\n
\n
M. Lucas, G. Cencetti, F. Battiston, Multiorder Laplacian for synchronization in higher-order networks, Phys. Rev. Research, 2: 033410, 2020<\/span><\/section>\n
https:\/\/arxiv.org\/pdf\/2003.09734v3.pdf<\/span><\/section>\n<\/blockquote>\n

\n

<\/section>\n
\u00b7 \u5355\u7eaf\u590d\u5f62\u7f51\u7edc\u540c\u6b65\u7684\u7a33\u5b9a\u6027<\/span><\/strong><\/section>\n<\/h2>\n

<\/span><\/section>\n
\u4f20\u7edf\u8026\u5408\u52a8\u529b\u7cfb\u7edf\u7684\u7f51\u7edc\u6a21\u578b\u4e2d\uff0c\u4ee5\u8282\u70b9\u5bf9\u7684\u8fde\u63a5\u6765\u63cf\u8ff0\u76f8\u4e92\u4f5c\u7528\u3002\u6700\u8fd1\u7684\u7814\u7a76\u8868\u660e\uff0c\u9ad8\u9636\u591a\u4f53\u76f8\u4e92\u4f5c\u7528\u5b58\u5728\u4e8e\u793e\u4f1a\u7fa4\u4f53\u3001\u751f\u6001\u7cfb\u7edf\u548c\u4eba\u7c7b\u5927\u8111\u4e2d\u3002\u8be5\u6587\u5efa\u7acb\u4e86\u4e00\u4e2a\u8026\u5408\u52a8\u529b\u7cfb\u7edf\u7684\u4e00\u822c\u6846\u67b6\uff0c\u53ef\u4ee5\u63cf\u8ff0\u4efb\u610f\u987a\u5e8f\u7684\u76f8\u4e92\u4f5c\u7528\u3002\u8bc1\u660e\u4e86\u5b8c\u5168\u540c\u6b65\u4f5c\u4e3a\u4e0d\u53d8\u89e3\u7684\u5b58\u5728\u6027\uff0c\u5e76\u7ed9\u51fa\u5176\u7a33\u5b9a\u72b6\u6001\u7684\u5fc5\u8981\u6761\u4ef6\u3002<\/span><\/section>\n
\n
L.V. Gambuzza, F. Di Patti, L. Gallo, S.S. Lepri, M. Romance, R. Criado, M. Frasca, V. Latora, S. Boccaletti, Stability of synchronization in simplicial complexes, Nature Comm., 12: 1255, 2021<\/span><\/section>\n
https:\/\/www.nature.com\/articles\/s41467-021-21486-9.pdf<\/span><\/section>\n<\/blockquote>\n

\n
 <\/span><\/section>\n<\/h1>\n
 <\/span><\/section>\n

\n
\n
\n
\n
\n

<\/section>\n

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

\u4e09\u3001\u9ad8\u9636\u62d3\u6251\u7684\u591a\u9886\u57df\u5e94\u7528\uff1a<\/strong><\/strong><\/p>\n

\u8111\u79d1\u5b66\u3001\u751f\u6001\u5b66\u3001\u793e\u4f1a\u79d1\u5b66 <\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

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


<\/p>\n

\n
1\u3001\u8111\u79d1\u5b66<\/span><\/strong><\/span><\/section>\n<\/h2>\n

\n

<\/span><\/strong><\/section>\n
\u00b7 \u5927\u8111\u529f\u80fd\u7f51\u7edc\u7684\u5b9e\u8bc1\u7814\u7a76\u2014\u2014\u56e2\u548c\u6d1e\u7684\u91cd\u8981\u6027<\/span><\/strong><\/section>\n<\/h2>\n

<\/section>\n
\u8fd9\u7bc7\u6587\u7ae0\u5c06\u5927\u8111\u533a\u57df\u53ca\u5176\u8fde\u63a5\u7f16\u7801\u4e3a\u4e00\u4e2a\u7531\u8282\u70b9\u548c\u8fde\u8fb9\u7ec4\u6210\u7684\u7f51\u7edc\uff0c\u6355\u83b7\u4e86\u8bb8\u591a\u53ef\u80fd\u7684\u8def\u5f84\uff0c\u4fe1\u606f\u53ef\u4ee5\u901a\u8fc7\u8fd9\u4e9b\u8def\u5f84\u4f20\u8f93\u5e76\u6267\u884c\u590d\u6742\u7684\u884c\u4e3a\u3002\u5c3d\u7ba1\u4eba\u7c7b\u5927\u8111\u7684\u795e\u7ecf\u7f51\u7edc\u770b\u4e0a\u53bb\u9887\u4e3a\u7a00\u758f\uff0c\u5b83\u91cc\u9762\u7684\u56e2\u548c\u6d1e\u7ed3\u6784\u5374\u662f\u975e\u5e38\u7a20\u5bc6\u7684\u3002\u7814\u7a76\u53d1\u73b0\u5927\u8111\u76ae\u5c42\u56e2\u548c\u6d1e\u7ed3\u6784\u5728\u89c6\u89d2\u548c\u8ba4\u77e5\u529f\u80fd\u91cc\u8d77\u7740\u91cd\u8981\u4f5c\u7528\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u4ece\u4f20\u7edf\u56fe\u8bba\u7684\u89c2\u70b9\u6765\u770b\u8111\u7f51\u7edc\uff0c\u53ea\u80fd\u770b\u5230\u8282\u70b9\u5bf9\u4e4b\u95f4\u7684\u8fde\u63a5\uff0c\u800c\u8981\u4ece\u66f4\u6df1\u523b\u7684\u5c42\u6b21\u548c\u9ad8\u5ea6\u6765\u7406\u89e3\u5927\u8111\u7ed3\u6784\uff0c\u8fd8\u5f97\u4ece\u9ad8\u9636\u4ee3\u6570\u62d3\u6251\u7684\u89d2\u5ea6\u6765\u8003\u91cf\u3002\u5b9e\u8bc1\u7814\u7a76\u53d1\u73b0\uff0c\u5468\u957f\u8f83\u957f\u7684\u5708\u7ed3\u6784\u5728\u63a7\u5236\u5927\u8111\u76ae\u5c42\u6d3b\u52a8\u4e2d\u4e3e\u8db3\u8f7b\u91cd\uff0c\u800c\u4e14\u53d1\u73b0\u6d1e\u6240\u626e\u6f14\u7684\u89d2\u8272\u5c24\u4e3a\u91cd\u8981\uff0c\u5b83\u4e0e\u5927\u8111\u4fe1\u606f\u4f20\u64ad\u7684\u6a21\u5f0f\u5bc6\u5207\u76f8\u5173\u3002\u8fd9\u4e9b\u7ed3\u679c\u8bc1\u660e\u4e86\u4ee3\u6570\u62d3\u6251\u7684\u6280\u672f\u4e3a\u7ed3\u6784\u8fde\u63a5\u7ec4\u5408\u63d0\u4f9b\u4e86\u4e00\u4e2a\u5168\u65b0\u7684\u89c6\u89d2\uff0c\u7a81\u51fa\u4e86\u73af\u72b6\u8def\u5f84\u4f5c\u4e3a\u4eba\u7c7b\u5927\u8111\u7ed3\u6784\u7684\u5173\u952e\u7279\u5f81\uff0c\u6298\u5c04\u51fa\u5728\u8111\u79d1\u5b66\u7814\u7a76\u4e2d\u4ec5\u4ec5\u91c7\u7528\u7ecf\u5178\u56fe\u8bba\u7684\u7f3a\u9677\u548c\u8fdb\u4e00\u6b65\u91c7\u7528\u9ad8\u9636\u62d3\u6251\u56fe\u8bba\u7684\u5fc5\u8981\u3002<\/span><\/section>\n
 <\/span><\/section>\n
<\/section>\n
\u56fe9 \u5c0f\u56e2\u4f53\u662f\u5927\u8111\u7ed3\u6784\u7f51\u7edc\u4e2d\u7684\u5c40\u90e8\u793e\u533a\u7684\u7279\u5f81<\/span><\/section>\n
\n
A.E. Sizemore, C. Giusti, A. Kahn, J.M. Vettel, R.F. Betzel, D.S. Bassett, Cliques and cavities in the human connectome. J. Comput. Neurosci. 44(1): 115-145, 2018<\/span><\/section>\n
https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10827-017-0672-6.pdf<\/span><\/section>\n<\/blockquote>\n
 <\/span><\/section>\n
\u5bf9\u795e\u7ecf\u7f51\u7edc\u7ed3\u6784\u4e0e\u5176\u529f\u80fd\u6d8c\u73b0\u4e4b\u95f4\u8054\u7cfb\u8ba4\u77e5\u7684\u7f3a\u5931\u963b\u788d\u4e86\u6211\u4eec\u5bf9\u5927\u8111\u5982\u4f55\u5904\u7406\u4fe1\u606f\u7684\u7406\u89e3\u3002\u8fd9\u91cc\u8003\u8651\u4e86\u73b0\u6709\u901a\u8fc7\u7a81\u89e6\u4f20\u9012\u4fe1\u606f\u7684\u56e0\u7d20\uff0c\u6784\u5efa\u4e86\u53cd\u6620\u4fe1\u606f\u6d41\u65b9\u5411\u7684\u7f51\u7edc\u56fe\uff0c\u5e76\u4f7f\u7528\u4ee3\u6570\u62d3\u6251\u6765\u5206\u6790\u8fd9\u4e9b\u6709\u5411\u56fe\u3002\u6b64\u65b9\u6cd5\u5e94\u7528\u4e8e\u65b0\u76ae\u8d28\u4e2d\u7684\u5c40\u90e8\u795e\u7ecf\u5143\u7f51\u7edc\uff0c\u63ed\u793a\u4e86\u4e00\u4e2a\u975e\u5e38\u590d\u6742\u4e14\u4ee5\u524d\u7a81\u89e6\u8fde\u63a5\u770b\u4e0d\u89c1\u7684\u62d3\u6251\u7ed3\u6784\u3002\u5176\u4e2d\u7a81\u89e6\u7f51\u7edc\u5305\u542b\u5927\u91cf\u675f\u7f1a\u5728\u7a7a\u8154\u4e2d\u7684\u795e\u7ecf\u5143\uff0c\u8fd9\u4e9b\u7a7a\u8154\u5f15\u5bfc\u7740\u76f8\u5173\u6d3b\u52a8\u7684\u51fa\u73b0\u3002\u5728\u5bf9\u523a\u6fc0\u7684\u53cd\u5e94\u4e2d\uff0c\u76f8\u5173\u7684\u6d3b\u52a8\u5c06\u7a81\u89e6\u8fde\u63a5\u7684\u795e\u7ecf\u5143\u7ed1\u5b9a\u5230\u529f\u80fd\u6027\u7684\u5c0f\u56e2\u4f53\u548c\u7a7a\u6d1e\u4e2d\uff0c\u8fd9\u4e9b\u5c0f\u56e2\u4f53\u4ee5\u4e00\u79cd\u56fa\u5b9a\u7684\u987a\u5e8f\u8fdb\u5316\u5230\u6781\u7aef\u590d\u6742\u6027\u3002\u8be5\u6587\u8ba4\u4e3a\u5927\u8111\u901a\u8fc7\u5f62\u6210\u8d8a\u6765\u8d8a\u590d\u6742\u7684\u529f\u80fd\u56e2\u548c\u8154\u6765\u5904\u7406\u523a\u6fc0\u529f\u80fd\u3002<\/span><\/section>\n
\n
\u675f\u7f1a\u5728\u7a7a\u8154\u4e2d\u7684\u795e\u7ecf\u5143<\/span><\/section>\n
M.W. Reimann, M. Nolte, M. Scolamiero, K. Turner, R. Perin, G. Chindemi, P. D\u0142otko, R. Levi, K. Hess, H. Markram, Cliques of neurons bound into cavities provide a missing link between structure and function, Frontiers in Comput. Neurosci., 11: paper 48, 2017<\/span><\/section>\n
https:\/\/www.frontiersin.org\/articles\/10.3389\/fncom.2017.00048\/full<\/span><\/section>\n<\/blockquote>\n
 <\/span><\/section>\n

\n
2\u3001\u751f\u6001\u5b66<\/span><\/strong><\/span><\/section>\n
 <\/span><\/strong><\/section>\n<\/h2>\n

\n
\u00b7 \u7269\u79cd\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u51b3\u5b9a\u751f\u6001\u7cfb\u7edf\u7684\u591a\u6837\u6027<\/span><\/strong><\/section>\n<\/h2>\n

<\/section>\n
\u7269\u79cd\u4e4b\u95f4\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u591a\u6837\u6027\u548c\u7cfb\u7edf\u7684\u7a33\u5b9a\u6027\u4e00\u822c\u6765\u8bf4\u662f\u5bf9\u7acb\u7684\u3002\u867d\u7136\u968f\u7740\u7269\u79cd\u6570\u91cf\u7684\u589e\u52a0\uff0c\u7fa4\u843d\u4e2a\u4f53\u4e24\u4e24\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u53d8\u5f97\u66f4\u52a0\u654f\u611f\uff0c\u4f46\u5bf9\u4e09\u4e2a\u4e2a\u4f53\u76f8\u4e92\u4f5c\u7528\u7684\u654f\u611f\u6027\u5374\u4fdd\u6301\u4e0d\u53d8\u3002\u5b9e\u9645\u4e0a\uff0c\u5176\u654f\u611f\u6027\u5230\u56db\u4e2a\u4e2a\u4f53\u65f6\u7684\u76f8\u4e92\u4f5c\u7528\u53cd\u800c\u4f1a\u51cf\u5c11\u3002\u56e0\u6b64\uff0c\u5b83\u4eec\u7684\u7fa4\u4f53\u7ec4\u5408\u4e2d\u5b58\u5728\u6570\u91cf\u6709\u4e0b\u9650\u548c\u4e0a\u9650\u7684\u7269\u79cd\u3002\u8fd9\u4e9b\u53d1\u73b0\u8868\u660e\u7269\u79cd\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u51b3\u5b9a\u4e86\u81ea\u7136\u751f\u6001\u7cfb\u7edf\u7684\u591a\u6837\u6027\u3002<\/span><\/section>\n
\n
E. Bairey, E.D. Kelsic, R. Kishony, High-order species interactions shape ecosystem diversity, Nature Comm., 7: Article 12285, 2016<\/span><\/section>\n
https:\/\/www.nature.com\/articles\/ncomms12285.pdf<\/span><\/section>\n<\/blockquote>\n

\n

<\/section>\n
\u00b7 \u751f\u6001\u7cfb\u7edf\u4e2d\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u548c\u7fa4\u843d\u7a33\u5b9a\u6027<\/span><\/strong><\/section>\n<\/h2>\n

<\/span><\/section>\n
\u751f\u6001\u5b66\u5bb6\u957f\u671f\u4ee5\u6765\u4e00\u76f4\u5728\u5bfb\u627e\u5bf9\u81ea\u7136\u754c\u4e2d\u89c2\u5bdf\u5230\u7684\u751f\u7269\u591a\u6837\u6027\u7684\u5408\u7406\u89e3\u91ca\u3002\u76ee\u524d\u7b80\u5355\u7684\u76f8\u4e92\u4f5c\u7528\u7ade\u4e89\u8005\u6a21\u578b\u4e0d\u80fd\u4fdd\u8bc1\u975e\u5e38\u5927\u7684\u751f\u6001\u7fa4\u843d\u7684\u6301\u4e45\u7a33\u5b9a\u6027\uff0c\u73b0\u6709\u7684\u4e2d\u6027\u6a21\u578b\u4e5f\u89e3\u91ca\u4e0d\u4e86\u7269\u79cd\u4e0d\u76f8\u4e92\u4f5c\u7528\u65f6\u4ea7\u751f\u7684\u591a\u6837\u6027\u3002\u6700\u8fd1\u53d1\u73b0\uff0c\u7531\u4e8e\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u7a33\u5b9a\u6027\uff0c\u975e\u5e38\u591a\u6837\u5316\u7684\u793e\u56e2\u53ef\u4ee5\u6301\u7eed\u5b58\u5728\u3002\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u5bf9\u5c01\u95ed\u751f\u6001\u7fa4\u843d\u6a21\u578b\u6709\u5f3a\u5927\u5f71\u54cd\uff0c\u80fd\u4e0d\u65ad\u5f15\u8fdb\u65b0\u7269\u79cd\u5230\u793e\u56e2\u4e2d\uff0c\u5e76\u4e14\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u8fd8\u53ef\u4ee5\u5e2e\u52a9\u6a21\u578b\u7684\u53c2\u6570\u5316\u5e76\u8fdb\u884c\u9a8c\u8bc1\u3002<\/span><\/section>\n
\n
J. Grilli, G. Barab\u00e1s, M.J. Michalska-Smith, S. Allesina, Higher-order interactions stabilize dynamics in competitive network models, Nature, 548: 210-213, 2017<\/span><\/section>\n
https:\/\/www.nature.com\/articles\/nature23273.pdf<\/span><\/section>\n<\/blockquote>\n

\n

<\/section>\n
\u00b7 \u79c0\u4e3d\u9690\u6746\u7ebf\u866b\u7684\u9ad8\u9636\u795e\u7ecf\u7f51\u7edc\u7ed3\u6784\u4e0e\u529f\u80fd<\/span><\/strong><\/section>\n<\/h2>\n

<\/section>\n
\u4e86\u89e3\u52a8\u7269\u795e\u7ecf\u7cfb\u7edf\u7684\u8fde\u901a\u6027\u5bf9\u4e8e\u7406\u89e3\u5176\u529f\u80fd\u81f3\u5173\u91cd\u8981\u3002\u8fd9\u91cc\u5efa\u7acb\u4e86\u5305\u542b\u4ece\u611f\u89c9\u8f93\u5165\u5230\u672b\u7aef\u5668\u5b98\u8f93\u51fa\u7684\u6240\u6709\u8fde\u8fb9\u7684\u5b9a\u91cf\u90bb\u63a5\u77e9\u9635\u3002\u53d1\u73b0\u4e0d\u540c\u6027\u522b\u7ebf\u866b<\/span>\uff08C. elegans\uff09<\/span>\u7684\u795e\u7ecf\u7cfb\u7edf\u5728\u591a\u4e2a\u5c42\u9762\u4e0a\u662f\u4e0d\u540c\u7684\uff0c\u8fde\u63a5\u5f3a\u5ea6\u4e5f\u4e0d\u76f8\u540c\u3002\u9ad8\u9636\u5b9a\u91cf\u90bb\u63a5\u77e9\u9635\u5305\u62ec\u6240\u6709\u7684\u8fde\u63a5\u6570\u636e\uff0c\u662f\u7406\u89e3\u590d\u6742\u7cfb\u7edf\u7684\u9002\u5e94\u6027\u884c\u4e3a\u5982\u4f55\u4ea7\u751f\u7684\u57fa\u7840\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe10 \u6210\u5e74\u79c0\u4e3d\u9690\u6746\u7ebf\u866b\u795e\u7ecf\u7cfb\u7edf\u94fe\u63a5\u56fe<\/span><\/section>\n
\n
S.J. Cook, et al., Whole-animal connectomes of both Caenorhabditis elegans sexes, Nature, 571: 63-71, 2019<\/span><\/section>\n
https:\/\/www.nature.com\/articles\/s41586-019-1352-7.pdf<\/span><\/section>\n<\/blockquote>\n
 <\/span><\/section>\n

\n
3\u3001\u793e\u4f1a\u79d1\u5b66<\/span><\/strong><\/span><\/section>\n

<\/span><\/strong><\/section>\n<\/h2>\n

\n
\u00b7 \u793e\u4f1a\u7f51\u7edc\u4e2d\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u53d1\u5c55\u52a8\u529b\u5b66<\/span><\/strong><\/section>\n

<\/span><\/section>\n<\/h2>\n
\u4eba\u4eec\u5728\u7f51\u7edc\u4e2d\u751f\u6d3b\u548c\u5408\u4f5c\uff0c\u4f46\u521d\u7b49\u56fe\u8bba\u8fde\u63a5\u53ea\u5141\u8bb8\u4e24\u8282\u70b9\u95f4\u7684\u4e92\u52a8\uff0c\u8be5\u6846\u67b6\u53ea\u9002\u7528\u4e8e\u4e8c\u5143\u535a\u5f08\u800c\u4e0d\u9002\u7528\u4e8e\u7fa4\u7ec4\u535a\u5f08\u3002\u8fdb\u5316\u52a8\u529b\u5b66\u5728\u793e\u4f1a\u7cfb\u7edf\u4e2d\u901a\u8fc7\u516c\u5171\u7269\u54c1\u535a\u5f08<\/span>\uff08Public Good Game\uff09<\/span>\u5177\u6709\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u3002\u5747\u5300\u8d85\u56fe\u4e0a\u7684\u535a\u5f08\u5bf9\u5e94\u4e8e\u5145\u5206\u6df7\u5408\u6781\u9650\u4e0b\u7684\u590d\u5236\u52a8\u529b\u5b66\uff0c\u662f\u7f51\u7edc\u7fa4\u4f53\u5408\u4f5c\u7684\u7406\u8bba\u57fa\u7840\u3002\u8be5\u6587\u4e2d\u53d1\u73b0\u4e86\u67a2\u7ebd\u7fa4\u4f53\u7684\u5b58\u5728\u4ee5\u53ca\u5b83\u4eec\u4e0e\u4e0d\u540c\u89c4\u6a21\u7fa4\u4f53\u7684\u76f8\u4e92\u4f5c\u7528\u90fd\u4f1a\u5f71\u54cd\u5408\u4f5c\u7684\u53d1\u5c55\u3002\u901a\u8fc7\u5b9e\u9645\u79d1\u5b66\u548c\u6280\u672f\u9886\u57df\u7684\u6570\u636e\uff0c\u5206\u6790\u4e86\u7fa4\u4f53\u5408\u4f5c\u5982\u4f55\u4f9d\u8d56\u4e8e\u7fa4\u4f53\u89c4\u6a21\uff0c\u53ef\u4ee5\u63d0\u4f9b\u4e00\u79cd\u4fc3\u8fdb\u793e\u4f1a\u56e2\u4f53\u5408\u4f5c\u7684\u6709\u6548\u65b9\u6cd5\u3002<\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n
\u56fe11 \u9ad8\u9636\u76f8\u5bf9\u4e8e\u6210\u5bf9\u76f8\u4e92\u4f5c\u7528<\/span><\/section>\n
\n
U. Alvarez-Rodriguez, F. Battiston, G.F. de Arruda, Y. Moreno, M. Perc, V. Latora, Evolutionary dynamics of higher-order interactions in social networks, Nature Human Behaviour, 5: 586-595, 2021<\/span><\/section>\n
https:\/\/www.nature.com\/articles\/s41562-020-01024-1<\/span><\/section>\n<\/blockquote>\n
 <\/span><\/section>\n

\n
4\u3001\u5176\u4ed6\u65b0\u5e94\u7528\u573a\u666f<\/span><\/strong><\/span><\/section>\n<\/h2>\n

\n

<\/section>\n
\u00b7 \u8ba1\u7b97\u62d3\u6251\u5b66\u5f00\u59cb\u8fdb\u5165\u590d\u6742\u7f51\u7edc\u7814\u7a76\u9886\u57df\u2014\u2014\u5bfb\u627e\u9ad8\u9636\u62d3\u6251\u7279\u5f81<\/span><\/strong><\/section>\n<\/h2>\n

<\/section>\n
\u4ee3\u6570\u62d3\u6251\u7814\u7a76\u7684\u6301\u4e45\u540c\u8c03\u7279\u6027\u88ab\u7528\u6765\u8ba1\u7b97\u4e0d\u540c\u7a7a\u95f4\u7684\u5206\u8fa8\u7387\uff0c \u5728\u591a\u91cd\u7a7a\u95f4\u5c3a\u5ea6\u4e0a\u68c0\u6d4b\u5230\u66f4\u591a\u6301\u4e45\u7684\u62d3\u6251\u7279\u5f81\uff0c \u56e0\u800c\u66f4\u80fd\u4ee3\u8868\u5e95\u5c42\u7a7a\u95f4\u7814\u7a76\u5bf9\u8c61\u7684\u672c\u8d28\u3002\u5bf9\u65f6\u95f4\u5e8f\u5217\u6570\u636e\u6784\u6210\u7684\u529f\u80fd\u7f51\u7edc\u7684\u7814\u7a76\uff0c \u63ed\u793a\u4e86\u529f\u80fd\u7f51\u7edc\u4e2d\u7684\u56e2\u548c\u6d1e\u7684\u7ed3\u6784\u6bd4\u666e\u901a\u7684\u70b9\u5bf9\u8fde\u63a5\u7ed3\u6784\u5177\u6709\u66f4\u9ad8\u9636\u7684\u4e00\u4e9b\u5185\u5728\u7279\u6027\uff0c \u80fd\u591f\u63d0\u4f9b\u4e00\u4e9b\u66f4\u4e3a\u6709\u7528\u7684\u4fe1\u606f\u3002\u5bf9Kuramoto \u632f\u5b50\u540c\u6b65\u548c\u5927\u8111\u6838\u78c1\u5171\u632f\u6570\u636e\u7684\u7814\u7a76\uff0c \u53d1\u73b0\u6301\u4e45\u540c\u8c03\u7279\u6027\u53ef\u4ee5\u6e05\u6670\u5730\u533a\u5206\u5927\u8111\u5728\u5b66\u4e60\u8fc7\u7a0b\u4e2d\u7684\u4e00\u4e9b\u540c\u6b65\u89c4\u5f8b\u548c\u73b0\u8c61\uff0c \u63ed\u793a\u4e00\u4e9b\u5e38\u89c4\u4fe1\u53f7\u91c7\u6837\u548c\u566a\u58f0\u5206\u6790\u770b\u4e0d\u6e05\u695a\u7684\u7279\u6027<\/span>\uff08\u4f8b\u5982\uff0c \u5e38\u89c4\u7684\u57fa\u4e8e\u70b9\u5bf9\u8fde\u63a5\u7684\u7f51\u7edc\u5206\u6790\uff0c \u4e0d\u80fd\u63d0\u4f9b\u793e\u56e2\u4e4b\u95f4\u5f3a\u540c\u6b65\u4e0e\u5f31\u540c\u6b65\u7684\u4e00\u4e9b\u76f8\u5f02\u7279\u5f81\u4ee5\u53ca\u4eba\u5728\u5b66\u4e60\u8fc7\u7a0b\u4e2d\u8111\u529f\u80fd\u7f51\u7edc\u7684\u4e00\u4e9b\u91cd\u8981\u53d8\u5316\uff09<\/span>\u3002\u6301\u4e45\u540c\u8c03\u7279\u6027\u53ef\u7528\u4e8e\u62d3\u6251\u6570\u636e\u5206\u6790\uff0c \u4ece\u52a0\u6743\u3001 \u566a\u58f0\u6c61\u67d3\u3001 \u975e\u5300\u91c7\u6837\u7684\u590d\u6742\u6570\u636e\u4e2d\u63ed\u793a\u5c40\u90e8\u3001 \u4e2d\u5c3a\u5ea6\u53ca\u5168\u5c40\u6570\u636e\u591a\u79cd\u7279\u5f81<\/span>\uff08\u5728\u8111\u7535\u56fe\u6570\u636e\u5206\u6790\u4e2d\u5df2\u5f97\u5230\u9a8c\u8bc1\uff09<\/span> \u3002<\/span><\/section>\n
\n
B.J. Stolz, A.H. Heather, M.A. Porter, Persistent homology of time-dependent functional networks constructed from coupled time series. Chaos 27: 047410, 2017<\/span><\/section>\n
https:\/\/arxiv.org\/pdf\/1605.00562.pdf<\/span><\/section>\n
E. Ibanez-Marcelo, L. Campioni, A. Phinyomark, G. Petri, E.L. Santarcangelo, Topology highlights mesoscopic functional equivalence between imagery and perception. NeuroImage 200: 437-449, 2019<\/span><\/section>\n
https:\/\/www.researchgate.net\/publication\/344952631_Topology_highlights_mesoscopic_functional_equivalence_between_imagery_and_perception<\/span><\/section>\n<\/blockquote>\n

\n

<\/section>\n
\u00b7 \u8bc6\u522b\u8d85\u56fe\u4e2d\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u4fe1\u606f<\/span><\/strong><\/section>\n<\/h2>\n

<\/span><\/section>\n
\u7f51\u7edc\u4e92\u52a8\u4e0d\u4ec5\u9650\u4e8e\u4e8c\u5143\u7ec4\uff0c\u8fd8\u7ecf\u5e38\u6d89\u53ca\u4e09\u4e2a\u6216\u66f4\u591a\u3002\u8fd9\u4e9b\u6570\u636e\u53ef\u4ee5\u7528\u8d85\u56fe\u5f88\u597d\u5730\u63cf\u8ff0\uff0c\u5176\u4e2d\u8d85\u94fe\u63cf\u8ff0\u4e86\u8282\u70b9\u7ec4\u4e4b\u95f4\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\uff0c\u53ef\u4ee5\u901a\u8fc7\u8bc6\u522b\u8d85\u94fe\u6765\u53d1\u73b0\u76f8\u5173\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u3002\u6570\u636e\u5206\u6790\u8868\u660e\uff0c\u66f4\u591a\u7684\u8d85\u94fe\u53ef\u4ee5\u6bd4\u7528\u4e8c\u5143\u7ec4\u65b9\u6cd5\u83b7\u5f97\u66f4\u5927\u91cf\u7684\u4fe1\u606f\u3002<\/span><\/section>\n
\n
F. Musciotto, F. Battiston, R.N. Mantegna, Detecting informative higher-order interactions in statistically validated hypergraphs, Comm. Phys., 4: Paper 218, 2021<\/span><\/section>\n
https:\/\/www.nature.com\/articles\/s42005-021-00710-4.pdf<\/span><\/section>\n<\/blockquote>\n

\n

<\/span><\/section>\n
\u00b7 \u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u6539\u5584\u7f51\u7edc\u7fa4\u96c6\u884c\u4e3a\u7684\u4f18\u5316<\/span><\/strong><\/section>\n<\/h2>\n

<\/span><\/section>\n
\u7fa4\u96c6\u884c\u4e3a\u5728\u751f\u7406\u3001\u751f\u7269\u548c\u795e\u7ecf\u7cfb\u7edf\u529f\u80fd\u4e2d\u8d77\u7740\u5173\u952e\u4f5c\u7528\u3002\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u6307\u5728\u590d\u6742\u7f51\u7edc\u4e2d\u6709\u591a\u4e2a\u72ec\u7acb\u5355\u5143\u76f8\u4e92\u4f5c\u7528\u7684\u7ed3\u6784\u3002\u5b9e\u8bc1\u7814\u7a76\u63ed\u793a\u4e86\uff0c\u5b83\u4eec\u666e\u904d\u5b58\u5728\u4e8e\u5404\u79cd\u590d\u6742\u7cfb\u7edf\u4e4b\u4e2d\u3002\u7814\u7a76\u7f51\u7edc\u4e2d\u7fa4\u96c6\u884c\u4e3a\u7684\u4f18\u5316\uff0c\u53ef\u884c\u65b9\u6cd5\u5305\u62ec\u590d\u5408\u62c9\u666e\u62c9\u65af\u77e9\u9635\u548c\u591a\u9636\u76f8\u4e92\u4f5c\u7528\u5982\u4e8c\u5143\u548c\u4e09\u5143\u7684\u8026\u5408\u3002\u901a\u8fc7\u52a0\u5bbd\u590d\u5408\u62c9\u666e\u62c9\u65af\u77e9\u9635\u7684\u7279\u5f81\u503c\u8c31\uff0c\u53ef\u4ee5\u6539\u8fdb\u6700\u4f18\u7fa4\u96c6\u884c\u4e3a\u5e76\u62d3\u5bbd\u53ef\u80fd\u7684\u884c\u4e3a\u8303\u56f4\u3002\u5728\u7ea6\u675f\u4f18\u5316\u573a\u666f\u4e2d\uff0c\u4e00\u4e2a\u975e\u5e73\u51e1\u7684\u3001\u7406\u60f3\u7684\u6210\u5bf9\u548c\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u5e73\u8861\u5bfc\u81f4\u6700\u5f3a\u7684\u7fa4\u96c6\u884c\u4e3a\u3002\u4f18\u5316\u7cfb\u7edf\u4e0d\u540c\u7c7b\u578b\u7684\u5e73\u8861\u6216\u6269\u5927\u5176\u52a8\u6001\u884c\u4e3a\u8303\u56f4\uff0c\u53ef\u671b\u5e94\u7528\u4e8e\u81ea\u6211\u8c03\u8282\u7cfb\u7edf<\/span>\uff08\u5982\u5927\u8111\uff09<\/span>\u3002<\/span><\/section>\n
\n
P.S. Skardal, D. Taylor, L.A rola-Fernandez, A. Arenas, Higher-order interactions improve optimal collective dynamics on networks, arXiv 2021<\/span><\/section>\n<\/blockquote>\n

\n

<\/span><\/section>\n
\u00b7 \u9ad8\u9636\u56e2\u7f51\u7edc\u4e0e\u6e17\u6d41\u8fc7\u7a0b<\/span><\/strong><\/section>\n<\/h2>\n

<\/section>\n

\u8fd9\u7bc7\u6587\u7ae0\u7814\u7a76\u4e86\u4ee5\u56e2\u4e3a\u57fa\u7840\u7684\u7f51\u7edc\u5c40\u90e8\u7ed3\u6784\u5bf9\u7f51\u7edc\u6e17\u6d41\u8fc7\u7a0b\u7684\u5f71\u54cd\u3002\u6bd4\u8f83\u4e86\u5e26\u5c40\u90e8\u793e\u56e2\u7ed3\u6784\u7684\u968f\u673a\u56fe\u6a21\u578b\u548c\u5e26\u5c40\u90e8\u6811\u72b6\u7ed3\u6784\u7684\u968f\u673a\u56fe\u6a21\u578b\u3002\u53d1\u73b0\u4e86\u7f51\u7edc\u7684\u5e73\u5747\u5ea6\u6570\u8d8a\u5927\uff0c\u5176\u5c40\u90e8\u6811\u72b6\u7ed3\u6784\u5bf9\u6e17\u6d41\u8fc7\u7a0b\u7684\u5f71\u54cd\u5c31\u8d8a\u5c0f\u3002\u5728\u9ad8\u9636\u56e2\u7f51\u7edc\u4e0a\uff0c\u4e0e\u6811\u72b6\u7f51\u7edc\u6bd4\u8f83\uff0c\u6e17\u6d41\u7684\u4e0d\u540c\u884c\u4e3a\u51e0\u4e4e\u53ef\u4ee5\u5b8c\u5168\u5f52\u56e0\u4e8e\u5ea6\u5e8f\u5217\u800c\u4e0d\u662f\u793e\u56e2\u7ed3\u6784\u7684\u5dee\u5f02\u3002<\/span><\/p>\n

\u4f8b\u5b50\uff1aKuramoto \u632f\u5b50\u7f51\u7edc<\/span><\/section>\n
\n
C. Stegehuis, T. Peron, Network processes on clique-networks with high average degree: the limited effect of higher-order structure, arXiv 2021<\/span><\/section>\n
https:\/\/www.researchgate.net\/publication\/351279057_Network_processes_on_clique-networks_with_high_average_degree_the_limited_effect_of_higher-order_structure<\/span><\/section>\n<\/blockquote>\n

\n

<\/span><\/section>\n
\u00b7 \u590d\u6742\u7f51\u7edc\u4e2d\u7684\u6d1e\u4e0e\u9c81\u68d2\u6027<\/span><\/strong><\/section>\n<\/h2>\n

<\/section>\n
\u8282\u70b9\u7684\u9ad8\u9636\u805a\u5408\u7ed3\u6784\u53ef\u7528\u5355\u7eaf\u590d\u5f62\u6765\u5efa\u6a21\u63cf\u8ff0\u3002\u8282\u70b9\u7684\u79fb\u9664\u4f1a\u5f71\u54cd\u62d3\u6251\u4e0d\u53d8\u91cf\u7684\u503c<\/span>\uff08\u5982\u91cf\u5316\u7684\u9ad8\u9636\u6d1e\u7684\u6570\u91cf\u548c\u8d1d\u8482\u6570\uff09<\/span>\u3002\u6e17\u6d41\u7406\u8bba\u548c\u65b9\u6cd5\u53ef\u4ee5\u7528\u6765\u63cf\u8ff0\u67d0\u4e9b\u8282\u70b9\u7684\u79fb\u9664\u5bfc\u81f4\u65b0\u7684\u6d1e\u51fa\u73b0\u3002\u590d\u6742\u7f51\u7edc\u9c81\u68d2\u6027\u548c\u540c\u8c03\u6027\u4e4b\u95f4\u7684\u5173\u7cfb\u53ef\u4ee5\u901a\u8fc7\u5bf9\u7f51\u7edc\u7279\u5f81\u548c\u62d3\u6251\u4e0d\u53d8\u91cf\u7684\u8054\u7cfb\u52a0\u4ee5\u7814\u7a76\u3002\u901a\u8fc7\u6a21\u62df\u5bf9\u7f51\u7edc\u7684\u968f\u673a\u6545\u969c\u548c\u6076\u610f\u653b\u51fb\uff0c\u53d1\u73b0\u9c81\u68d2\u6027\u7684\u7f51\u7edc\u7279\u5f81\u53d8\u5316\u662f\u4e0d\u540c\u653b\u51fb\u7b56\u7565\u4e0b\u5404\u4e2a\u96c6\u7fa4\u4e2d\u6d1e\u7684\u6570\u91cf\u5206\u5e03\u51fa\u73b0\u4e86\u5dee\u5f02\u800c\u5e26\u6765\u7684\u540e\u679c\u3002\u62d3\u6251\u4e0d\u53d8\u91cf\u7684\u7814\u7a76\u6709\u52a9\u4e8e\u8fdb\u4e00\u6b65\u7406\u89e3\u590d\u6742\u7f51\u7edc\u4e0a\u53d1\u751f\u7684\u52a8\u6001\u53d8\u5316\u3002<\/span><\/section>\n
\n
A.D. Zhou, S. Maleti\u0107, Y. Zhao, Robustness and percolation of holes in complex networks, Physica A, 502: 459-468, 2018<\/span><\/section>\n
https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0378437118302188<\/span><\/section>\n<\/blockquote>\n

\n

<\/section>\n
\u00b7 \u79d1\u5b66\u4e0e\u6280\u672f\u77e5\u8bc6\u7f51\u7edc\u4e2d\u7684\u9ad8\u9636\u62d3\u6251\u7ed3\u6784<\/span><\/strong><\/section>\n<\/h2>\n

<\/span><\/section>\n
\u79d1\u5b66\u6280\u672f\u7684\u53d1\u5c55\u662f\u4e00\u4e2a\u91cd\u7ec4\u8fc7\u7a0b\uff0c\u5176\u4e2d\u65b0\u7684\u53d1\u73b0\u548c\u53d1\u660e\u662f\u6839\u636e\u5148\u9a8c\u77e5\u8bc6\u6784\u5efa\u7684\u3002\u4f7f\u7528\u4ee3\u6570\u62d3\u6251\u5de5\u5177\u53ef\u4ee5\u63cf\u8ff0\u8de8\u5c3a\u5ea6\u79d1\u5b66\u6280\u672f\u77e5\u8bc6\u7f51\u7edc\u7684\u9ad8\u9636\u7ed3\u6784\u3002\u89c2\u5bdf\u5230\u5f88\u591a\u79d1\u5b66\u548c\u6280\u672f\u9886\u57df\u77e5\u8bc6\u7684\u9ad8\u9636\u7ed3\u6784\u7684\u589e\u957f\uff0c\u800c\u8fd9\u79cd\u589e\u957f\u662f\u4f7f\u7528\u4f20\u7edf\u7684\u7f51\u7edc\u6d4b\u91cf\u65e0\u6cd5\u89c2\u5bdf\u5230\u7684\u3002\u8bc1\u660e\u4e86\u9ad8\u9636\u7ed3\u6784\u4e0e\u4f4e\u9636\u7ed3\u6784\u76f8\u543b\u5408\uff0c\u5e76\u4e14\u5728\u67d0\u79cd\u7a0b\u5ea6\u4e0a\u9ad8\u9636\u7ed3\u6784\u7684\u589e\u52a0\u4e0e\u66f4\u597d\u7684\u7ed3\u679c\u76f8\u5173<\/span>\uff08\u5982\u8bba\u6587\u548c\u4e13\u5229\u7684\u65b0\u9896\u6027\u548c\u5f71\u54cd\u529b\uff09<\/span>\u3002\u5728\u79d1\u5b66\u6280\u672f\u672c\u8d28\u4e0b\u4ea7\u751f\u7684\u9ad8\u9636\u5236\u5ea6\u8868\u73b0\u51fa\u66f4\u5927\u7684\u8bed\u8a00\u62bd\u8c61\u6027\u548c\u66f4\u5927\u7684\u77e5\u8bc6\u6d41\u3002<\/span><\/section>\n
\n
T. Gebhart, R.J. Funk, The emergence of higher-order structure in scientific and technological knowledge networks, arXiv 2020<\/span><\/section>\n
https:\/\/arxiv.org\/pdf\/2009.13620v1.pdf<\/span><\/section>\n<\/blockquote>\n

\n

<\/section>\n
\u00b7 \u6700\u4f18\u9ad8\u9636\u7f51\u7edc\u6a21\u578b\u53ca\u5176\u6570\u636e\u5206\u6790\u5e94\u7528<\/span><\/strong><\/section>\n<\/h2>\n

<\/span><\/section>\n
\u4e30\u5bcc\u7684\u6570\u636e\u8868\u660e\uff0c\u57fa\u4e8e\u8282\u70b9\u5bf9\u76f8\u4e92\u4f5c\u7528\u7684\u6a21\u578b\u65e0\u6cd5\u6355\u83b7\u7f51\u7edc\u8282\u70b9\u4e4b\u95f4\u7684\u590d\u6742\u4f9d\u8d56\u5173\u7cfb\u3002\u9ad8\u9636\u7f51\u7edc\u6a21\u578b\u8d85\u8d8a\u4e86\u8fd9\u4e9b\u9650\u5236\uff0c\u4e3a\u7406\u89e3\u590d\u6742\u7cfb\u7edf\u63d0\u4f9b\u4e86\u65b0\u89c6\u89d2\u3002\u7f51\u7edc\u63cf\u8ff0\u8282\u70b9\u5bf9\u7684\u76f8\u4e92\u4f5c\u7528\uff0c\u53ef\u4ee5\u89e3\u91ca\u7cfb\u7edf\u62d3\u6251\u7684\u57fa\u672c\u7279\u5f81\uff0c\u5e76\u4e3a\u7edf\u8ba1\u5206\u6790\u548c\u63a8\u7406\u63d0\u4f9b\u7406\u8bba\u57fa\u7840\u3002\u8d8a\u6765\u8d8a\u660e\u663e\u7684\u793e\u4f1a\u3001\u6280\u672f\u548c\u751f\u7269\u7cfb\u7edf\u6570\u636e\u8868\u660e\uff0c\u83b7\u53d6\u590d\u6742\u8def\u5f84\u548c\u76f8\u4e92\u4f5c\u7528\u7684\u4fe1\u606f\u9700\u8981\u5148\u8fdb\u7684\u5efa\u6a21\u6280\u672f\u3002\u6700\u4f18\u9ad8\u9636\u7f51\u7edc\u6a21\u578b\u5305\u542b\u6709\u591a\u5c42\u3001\u7ec4\u5408\u548c\u975e\u9a6c\u5c14\u53ef\u592b\u8fc7\u7a0b\uff0c\u4e30\u5bcc\u4e86\u7f51\u7edc\u79d1\u5b66\u7684\u7814\u7a76\u65b9\u5f0f\u3002<\/span><\/section>\n
\n
R. Lambiotte, M. Rosvall, I. Scholtes, From networks to optimal higher-order models of complex systems, Nature Physics, 15: 313\u2013320, 2019<\/span><\/section>\n
https:\/\/www.nature.com\/articles\/s41567-019-0459-y.pdf<\/span><\/section>\n<\/blockquote>\n

\n

<\/span><\/section>\n
\u00b7 \u590d\u6742\u7cfb\u7edf\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u7269\u7406\u5b66\u539f\u7406<\/span><\/strong><\/section>\n<\/h2>\n

<\/span><\/section>\n
\u590d\u6742\u7f51\u7edc\u5df2\u6210\u4e3a\u76f8\u4e92\u4f5c\u7528\u7cfb\u7edf\u52a8\u529b\u5b66\u5efa\u6a21\u7684\u4e3b\u8981\u8303\u5f0f\u3002\u7136\u800c\uff0c\u7f51\u7edc\u672c\u8d28\u4e0a\u4ec5\u9650\u4e8e\u63cf\u8ff0\u8282\u70b9\u5bf9\u7684\u76f8\u4e92\u4f5c\u7528\uff0c\u800c\u73b0\u5b9e\u4e16\u754c\u7684\u590d\u6742\u7cfb\u7edf\u901a\u5e38\u4ee5\u9ad8\u9636\u4e3a\u7279\u5f81\uff0c\u6d89\u53ca\u4e09\u4e2a\u6216\u66f4\u591a\u4e2a\u5355\u5143\u7ec4\u7684\u76f8\u4e92\u4f5c\u7528\u3002\u9ad8\u9636\u7ed3\u6784<\/span>\uff08\u5982\u8d85\u56fe\u548c\u5355\u7eaf\u590d\u5f62\uff09<\/span>\u662f\u63cf\u7ed8\u8bb8\u591a\u793e\u4f1a\u3001\u751f\u7269\u548c\u4eba\u9020\u7cfb\u7edf\u7b49\u771f\u5b9e\u7ec4\u7ec7\u66f4\u4e3a\u6709\u6548\u7684\u5de5\u5177\uff0c\u63d0\u4f9b\u4e86\u7531\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u5f15\u8d77\u7684\u7fa4\u4f53\u884c\u4e3a\u7684\u6700\u65b0\u8bc1\u636e\u3002\u8be5\u6587\u8fd8\u6982\u8ff0\u4e86\u7269\u7406\u5b66\u7684\u4e09\u4e2a\u5173\u952e\u6311\u6218\u6027\u9ad8\u9636\u7cfb\u7edf\u3002<\/span><\/section>\n
\n
F. Battiston et al., The physics of higher-order interactions in complex systems, Nature Physics, 17: 1093\u20131098, 2021<\/span><\/section>\n
https:\/\/www.nature.com\/articles\/s41567-021-01371-4.pdf<\/span><\/section>\n<\/blockquote>\n

\n

<\/span><\/section>\n
\u00b7 \u8ba1\u7b97\u590d\u6742\u7f51\u7edc\u4e2d\u7684\u9ad8\u9636\u56e2\u548c\u6d1e<\/span><\/strong><\/section>\n<\/h2>\n

<\/span><\/section>\n
\u5bfb\u627e\u6240\u6709\u56e2<\/span>\uff08\u6216\u6700\u5927\u56e2\uff09<\/span>\u662fNP-\u5b8c\u5168\u95ee\u9898\uff0c\u66f4\u4e0d\u5fc5\u8bf4\u9ad8\u9636\u6d1e\u3002\u672c\u6587<\/span>\uff08\u300a\u8ba1\u7b97\u590d\u6742\u7f51\u7edc\u4e2d\u7684\u9ad8\u9636\u56e2\u548c\u6d1e\u300b\uff09<\/span>\u901a\u8fc7 k-\u6838\u5206\u89e3\u6765\u51b3\u5b9a\u7f51\u7edc\u7684\u8ba1\u7b97\u53ef\u884c\u6027\u3002\u8bbe\u8ba1\u4e86\u5bfb\u627e\u5404\u9636\u56e2\u7684\u53ef\u884c\u7b97\u6cd5\uff0c\u53ef\u540c\u65f6\u83b7\u5f97 Euler \u793a\u6027\u6570\u548c Betti \u6570\u3002\u5bf9\u7ebf\u866b<\/span>\uff08C. elegans\uff09<\/span>\u771f\u5b9e\u5927\u6570\u636e\u627e\u5230\u6240\u6709\u5404\u9636\u56e2\u548c\u90e8\u5206\u9ad8\u9636\u6d1e\u3002<\/span><\/section>\n
\n
D.H. Shi*, Z.F. Chen, X. Sun, Q.H. Chen, C. Ma, Y. Lou, G.R. Chen*, Computing cliques and cavities in networks, Comm. Phys., accepted, 2021<\/span><\/section>\n
https:\/\/arxiv.org\/vc\/arxiv\/papers\/2101\/2101.00536v1.pdf<\/span><\/section>\n<\/blockquote>\n

\n

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

<\/span><\/section>\n<\/h1>\n

\n
\n
\n
\n
\n

<\/section>\n

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

\u56db\u3001\u603b\u7ed3\u5c55\u671b<\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

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


<\/p>\n

\u672c\u6b21\u62a5\u544a\u8be6\u7ec6\u4ecb\u7ecd\u4e86\u590d\u6742\u7f51\u7edc\u9ad8\u9636\u62d3\u6251\u7684\u76f8\u5173\u6982\u5ff5\uff0c\u5e76\u901a\u8fc7\u5927\u91cf\u6587\u732e\u4e3e\u4f8b\uff0c\u5728\u7f51\u7edc\u540c\u6b65\u95ee\u9898\u3001\u8111\u79d1\u5b66\u7f51\u7edc\u3001\u751f\u6001\u7cfb\u7edf\u7b49\u5206\u6790\u4ee5\u53ca\u7f51\u7edc\u52a8\u529b\u5b66\u7814\u7a76\u4e2d\uff0c\u5c55\u793a\u4e86\u9ad8\u9636\u62d3\u6251\u7684\u5e94\u7528\u3002\u7ecf\u9a8c\u8868\u660e\uff0c\u5f53\u8003\u8651\u5230\u66f4\u73b0\u5b9e\u7684\u76f8\u4e92\u4f5c\u7528\u6a21\u5f0f\u65f6\uff0c\u5229\u7528\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u5206\u6790\u53ef\u80fd\u4f1a\u5f15\u51fa\u65b0\u7684\u7814\u7a76\u95ee\u9898\u548c\u5bf9\u8c61\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u5bf9\u5177\u6709\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u7f51\u7edc\u7cfb\u7edf\u7814\u7a76\u4ecd\u5904\u4e8e\u521d\u7ea7\u9636\u6bb5\uff0c\u8fd8\u6709\u8bb8\u591a\u5f00\u653e\u7684\u548c\u672a\u63a2\u7d22\u7684\u65b9\u5411\u3002\u6211\u4eec\u5217\u51fa\u5176\u4e2d\u7684\u4e00\u4e9b\u4f9b\u8bfb\u8005\u63a2\u8ba8\uff1a<\/span><\/section>\n

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

\n
\u00b7 \u9ad8\u9636\u7ed3\u6784\u7684\u5ea6\u91cf<\/span><\/strong><\/section>\n

<\/span><\/strong><\/section>\n<\/h2>\n
\u5728\u590d\u6742\u7f51\u7edc\u4e2d\uff0c\u65f6\u6548\u3001\u6f14\u5316\u548c\u591a\u5c42\u6b21\u7684\u7f51\u7edc\u5ea6\u91cf\u65b9\u6cd5\uff0c\u6709\u5f88\u5927\u7684\u63a2\u7d22\u7a7a\u95f4\u3002\u4f8b\u5982\uff0c\u867d\u7136\u5bf9\u632f\u8361\u5668\u7f51\u7edc\u6a21\u578b\u4e2d\u8282\u70b9\u4e4b\u95f4\u7684\u540c\u6b65\u5df2\u6709\u4e00\u4e2a\u6e05\u6670\u7684\u7406\u89e3\uff0c\u4f46\u8981\u4ee5\u4e24\u8282\u70b9\u4ea4\u4e92\u7684\u89c2\u70b9\u548c\u65b9\u6cd5\u6765\u7406\u89e3\u5b83\u4eec\u4ee5\u56e2\u4e3a\u5355\u4f4d\u7684\u5185\u5916\u5f3a\u5f31\u540c\u6b65\u7684\u72b6\u6001\u662f\u56f0\u96be\u7684\u3002\u6b63\u786e\u5b9a\u4e49\u80fd\u591f\u6355\u83b7\u8fd9\u4e9b\u72b6\u6001\u7684\u5ea6\u91cf\u5c06\u662f\u63cf\u8ff0\u5206\u6790\u5176\u4f5c\u7528\u53ca\u5f71\u54cd\u7684\u4e00\u4e2a\u5fc5\u8981\u6b65\u9aa4\u3002<\/span><\/section>\n

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

\n
\u00b7 \u9ad8\u9636\u7ed3\u6784\u7684\u751f\u6210\u6a21\u578b<\/span><\/strong><\/section>\n

<\/span><\/strong><\/section>\n<\/h2>\n
\u80fd\u591f\u523b\u753b\u9ad8\u9636\u7ed3\u6784\u7684\u5404\u79cd\u7279\u5f81\u7684\u6a21\u578b\u662f\u81f3\u5173\u91cd\u8981\u7684\u3002\u5f00\u53d1\u65b0\u7684\u5ea6\u91cf\u548c\u751f\u6210\u6a21\u578b\u5bf9\u4e8e\u786e\u5b9a\u6211\u4eec\u89c2\u5bdf\u5230\u7684\u6a21\u5f0f\u80cc\u540e\u7684\u57fa\u672c\u673a\u5236\u662f\u975e\u5e38\u91cd\u8981\u7684\u3002\u76ee\u524d\u5355\u7eaf\u590d\u5f62\u7684\u968f\u673a\u6a21\u578b\u5f88\u5c11\u3002\u4e00\u4e9b\u73b0\u6709\u7684\u5355\u7eaf\u590d\u5f62\u6a21\u578b\u518d\u73b0\u4e86\u5c40\u90e8\u8fde\u63a5\u6a21\u5f0f\uff0c\u4f46\u6ca1\u6709\u4e00\u4e2a\u80fd\u591f\u518d\u73b0\u66f4\u7cbe\u7ec6\u7684\u62d3\u6251\u7ed3\u6784\u3002\u8003\u8651\u5230\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u65f6\u95f4\u548c\u591a\u91cd\u7ed3\u6784\u7684\u7f51\u7edc\u62d3\u6251\u6a21\u578b\u8fd8\u6709\u5f85\u7814\u7a76\u3002<\/span><\/section>\n

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

\n
\u00b7 \u6570\u636e\u63a8\u65ad<\/span><\/strong><\/section>\n

<\/span><\/strong><\/section>\n<\/h2>\n
\u5728\u8bb8\u591a\u590d\u6742\u7cfb\u7edf\u4e2d\uff0c\u4ea4\u4e92\u4f5c\u7528\u5c1a\u672a\u786e\u5b9a\uff0c\u9700\u8981\u4ece\u6570\u636e\u4e2d\u63a8\u65ad\u51fa\u6765\u3002\u5982\u679c\u53ef\u80fd\uff0c\u6211\u4eec\u9700\u8981\u4ec0\u4e48\u7c7b\u578b\u7684\u6570\u636e\u6765\u533a\u5206\u4f4e\u9636\u548c\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u5462\uff1f\u4e8b\u5b9e\u4e0a\uff0c\u5bf9\u4e8e\u6570\u636e\u5df2\u7ecf\u4ee5\u96c6\u5408\u7684\u5f62\u5f0f\u51fa\u73b0\u7684\u7cfb\u7edf\uff0c\u63d0\u53d6\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u56e0\u7d20\u5e76\u6d4b\u91cf\u5176\u5f3a\u5ea6\u662f\u5f88\u7b80\u5355\u7684\u3002\u4f46\u662f\u5728\u5b9e\u9645\u60c5\u51b5\u4e0b\uff0c\u7531\u4e8e\u5404\u79cd\u539f\u56e0\uff0c\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u5f88\u5c11\u88ab\u6570\u636e\u76f4\u63a5\u53cd\u6620\u51fa\u6765\u3002\u5173\u4e8e\u5177\u6709\u6216\u4e0d\u5177\u6709\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u52a8\u529b\u5b66\u6a21\u578b\u7684\u6570\u636e\u7f3a\u5931\u4f7f\u5f97\u4eba\u4eec\u65e0\u6cd5\u4e3a\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u5b58\u5728\u3001\u6027\u8d28\u548c\u5f3a\u5ea6\u5f15\u51fa\u9002\u5f53\u7684\u5206\u6790\u65b9\u6848\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u76ee\u524d\u7684\u56f0\u96be\u4e0d\u5355\u662f\u65b9\u6cd5\u8bba\u95ee\u9898\uff0c\u8fd8\u9700\u8981\u5c06\u9ad8\u9636\u62d3\u6251\u7406\u8bba\u5728\u66f4\u5e7f\u6cdb\u7684\u4e3b\u9898\u4e2d\u627e\u5230\u5177\u4f53\u7684\u5e94\u7528\u3002\u6211\u4eec\u53ef\u4ee5\u8bbe\u60f3\u5728\u751f\u7269\u5b66\u3001\u751f\u6001\u5b66\u3001\u4eba\u7c7b\u884c\u4e3a\u52a8\u529b\u5b66\u3001\u795e\u7ecf\u79d1\u5b66\u548c\u8ba1\u7b97\u793e\u4f1a\u79d1\u5b66\u7b49\u65b9\u5411\uff0c\u5229\u7528\u9ad8\u9636\u62d3\u6251\u4e3a\u8fdb\u4e00\u6b65\u7684\u6df1\u5165\u7814\u7a76\u63d0\u4f9b\u65b0\u9014\u5f84\u3002<\/span><\/section>\n
 <\/span><\/section>\n

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

\u53c2\u8003\u6587\u732e<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n

\n
\n
\n
\n

1\u3001\u7f51\u7edc\u540c\u6b65\u95ee\u9898<\/span><\/p>\n


<\/p>\n

\u5168\u9f50\u7f51\u7edc\u662f\u540c\u6b65\u6700\u4f18\u7f51\u7edc<\/span><\/p>\n

D.H. Shi, G. Chen, W.W.K. Thong, X.Y. Yan, Searching for optimal network topology with best possible synchronizability, IEEE Circ. Syst. Mag. 13(1): 66-75, 2013<\/span><\/p>\n

https:\/\/ieeexplore.ieee.org\/stamp\/stamp.jsp?tp=&arnumber=6468141<\/span><\/p>\n

 <\/span><\/p>\n

D. H. Shi, L. Y. Lyu, G. R. Chen, Totally homogeneous networks, National Science Review (2019)<\/span><\/p>\n

https:\/\/arxiv.org\/ftp\/arxiv\/papers\/1903\/1903.11289.pdf<\/span><\/p>\n

 <\/span><\/p>\n

\u590d\u6742\u7f51\u7edc\u4e2d\u7684\u9ad8\u9636\u7ec4\u7ec7\u7ed3\u6784<\/span><\/p>\n

A.R. Benson, D.F. Gleich, J. Leskovec, Higher-order organization of complex networks, Science, 353: 163-166, 2016<\/span><\/p>\n

https:\/\/www.science.org\/doi\/epdf\/10.1126\/science.aad9029<\/span><\/p>\n

 <\/span><\/p>\n

\u8d85\u51fa\u8282\u70b9\u5bf9\u95f4\u76f8\u4e92\u4f5c\u7528\u7684\u7f51\u7edc\uff1a\u7ed3\u6784\u4e0e\u52a8\u529b\u5b66\uff08\u7efc\u8ff0\uff09<\/span><\/p>\n

F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lucas, A. Patania, J.-G. Young, G. Petri, Networks beyond pairwise interactions: Structure and dynamics, Physics Reports, 847: 1-92, 2020<\/span><\/p>\n

https:\/\/arxiv.org\/abs\/2006.01764v1<\/span><\/p>\n

 <\/span><\/p>\n

\u8026\u5408\u9ad8\u9636\u5355\u7eaf\u590d\u5f62\u4e0a\u62d3\u6251\u4fe1\u53f7\u7684\u540c\u6b65<\/span><\/p>\n

R. Ghorbanchian, J.G. Restrepo, J.J. Torres, G. Bianconi, Higher-order simplicial synchronization of coupled topological signals, Comm. Phys., 4: Paper 120, 2021<\/span><\/p>\n

https:\/\/arxiv.org\/abs\/2011.00897v2<\/span><\/p>\n

 <\/span><\/p>\n

\u9ad8\u7ef4\u5355\u7eaf\u590d\u5f62\u4e0a Kuramoto \u632f\u5b50\u7684\u76f8\u4f4d\u540c\u6b65<\/span><\/p>\n

X. Dai, K. Kovalenko, M. Molodyk, Z. Wang, X. Li, D. Musatov, A.M. Raigorodskii, A. Bittner, G.D. Cooper, G. Bianconi, S. Boccaletti, D-dimensional oscillators in simplicial structures: Odd and even dimensions display different synchronization scenarios. Chaos, Solitons Fractals, 146: 110888, 2021<\/span><\/p>\n

https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0960077921002411<\/span><\/p>\n

 <\/span><\/p>\n

\u5e7f\u4e49\u7f51\u7edc\u540c\u6b65\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u65b0\u5f62\u6001<\/span><\/p>\n

Y. Zhang, V. Latora, A.E. Motter, Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactions, Comm. Phys., 4: Article 195, 2021<\/span><\/p>\n

https:\/\/arxiv.org\/pdf\/2010.00613v2.pdf<\/span><\/p>\n

 <\/span><\/p>\n

\u9ad8\u9636\u7f51\u7edc\u540c\u6b65\u7684\u591a\u9636\u62c9\u666e\u62c9\u65af\u7b97\u5b50\u65b9\u6cd5<\/span><\/p>\n

M. Lucas, G. Cencetti, F. Battiston, Multiorder Laplacian for synchronization in higher-order networks, Phys. Rev. Research, 2: 033410, 2020<\/span><\/p>\n

https:\/\/arxiv.org\/pdf\/2003.09734v3.pdf<\/span><\/p>\n

 <\/span><\/p>\n

\u5355\u7eaf\u590d\u5f62\u7f51\u7edc\u540c\u6b65\u7684\u7a33\u5b9a\u6027<\/span><\/p>\n

L.V. Gambuzza, F. Di Patti, L. Gallo, S.S. Lepri, M. Romance, R. Criado, M. Frasca, V. Latora, S. Boccaletti, Stability of synchronization in simplicial complexes, Nature Comm., 12: 1255, 2021<\/span><\/p>\n

https:\/\/www.nature.com\/articles\/s41467-021-21486-9.pdf<\/span><\/p>\n

 <\/span><\/p>\n

2\u3001\u8111\u79d1\u5b66\u3001\u751f\u6001\u5b66\u3001\u793e\u4f1a\u79d1\u5b66<\/span><\/p>\n

 <\/span><\/p>\n

\u5927\u8111\u529f\u80fd\u7f51\u7edc\u7684\u5b9e\u8bc1\u7814\u7a76\uff1a\u56e2\u548c\u6d1e\u7684\u91cd\u8981\u6027<\/span><\/p>\n

A.E. Sizemore, C. Giusti, A. Kahn, J.M. Vettel, R.F. Betzel, D.S. Bassett, Cliques and cavities in the human connectome. J. Comput. Neurosci. 44(1): 115-145, 2018<\/span><\/p>\n

https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10827-017-0672-6.pdf<\/span><\/p>\n

 <\/span><\/p>\n

\u675f\u7f1a\u5728\u7a7a\u8154\u4e2d\u7684\u795e\u7ecf\u5143\u56e2\u63d0\u4f9b\u4e86\u7ed3\u6784\u548c\u529f\u80fd\u4e4b\u95f4\u7f3a\u5931\u7684\u8054\u7cfb<\/span><\/p>\n

M.W. Reimann, M. Nolte, M. Scolamiero, K. Turner, R. Perin, G. Chindemi, P. D\u0142otko, R. Levi, K. Hess, H. Markram, Cliques of neurons bound into cavities provide a missing link between structure and function, Frontiers in Comput. Neurosci., 11: paper 48, 2017<\/span><\/p>\n

https:\/\/www.frontiersin.org\/articles\/10.3389\/fncom.2017.00048\/full<\/span><\/p>\n

 <\/span><\/p>\n

\u7269\u79cd\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u51b3\u5b9a\u751f\u6001\u7cfb\u7edf\u7684\u591a\u6837\u6027<\/span><\/p>\n

E. Bairey, E.D. Kelsic, R. Kishony, High-order species interactions shape ecosystem diversity, Nature Comm., 7: Article 12285, 2016<\/span><\/p>\n

https:\/\/www.nature.com\/articles\/ncomms12285.pdf<\/span><\/p>\n

 <\/span><\/p>\n

\u751f\u6001\u7cfb\u7edf\u4e2d\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u548c\u7fa4\u843d\u7a33\u5b9a\u6027<\/span><\/p>\n

J. Grilli, G. Barab\u00e1s, M.J. Michalska-Smith, S. Allesina, Higher-order interactions stabilize dynamics in competitive network models, Nature, 548: 210-213, 2017<\/span><\/p>\n

https:\/\/www.nature.com\/articles\/nature23273.pdf<\/span><\/p>\n

 <\/span><\/p>\n

\u79c0\u4e3d\u9690\u6746\u7ebf\u866b\u7684\u9ad8\u9636\u795e\u7ecf\u7f51\u7edc\u7ed3\u6784\u4e0e\u529f\u80fd<\/span><\/p>\n

S.J. Cook, et al., Whole-animal connectomes of both Caenorhabditis elegans sexes, Nature, 571: 63-71, 2019<\/span><\/p>\n

https:\/\/www.nature.com\/articles\/s41586-019-1352-7.pdf<\/span><\/p>\n

 <\/span><\/p>\n

\u793e\u4f1a\u7f51\u7edc\u4e2d\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u53d1\u5c55\u52a8\u529b\u5b66<\/span><\/p>\n

U. Alvarez-Rodriguez, F. Battiston, G.F. de Arruda, Y. Moreno, M. Perc, V. Latora, Evolutionary dynamics of higher-order interactions in social networks, Nature Human Behaviour, 5: 586-595, 2021<\/span><\/p>\n

https:\/\/www.nature.com\/articles\/s41562-020-01024-1<\/span><\/p>\n

 <\/span><\/p>\n

3\u3001\u65b0\u5e94\u7528\u573a\u666f\uff1a\u5176\u4ed6<\/span><\/p>\n

 <\/span><\/p>\n

\u8ba1\u7b97\u62d3\u6251\u5b66\u5f00\u59cb\u8fdb\u5165\u590d\u6742\u7f51\u7edc\u7814\u7a76\u9886\u57df\uff08\u5bfb\u627e\u9ad8\u9636\u62d3\u6251\u7279\u5f81\uff09<\/span><\/p>\n

B.J. Stolz, A.H. Heather, M.A. Porter, Persistent homology of time-dependent functional networks constructed from coupled time series. Chaos 27: 047410, 2017<\/span><\/p>\n

https:\/\/arxiv.org\/pdf\/1605.00562.pdf<\/span><\/p>\n

 <\/span><\/p>\n

E. Ibanez-Marcelo, L. Campioni, A. Phinyomark, G. Petri, E.L. Santarcangelo, Topology highlights mesoscopic functional equivalence between imagery and perception. NeuroImage 200: 437-449, 2019<\/span><\/p>\n

https:\/\/www.researchgate.net\/publication\/344952631_Topology_highlights_mesoscopic_functional_equivalence_between_imagery_and_perception<\/span><\/p>\n

 <\/span><\/p>\n

\u8bc6\u522b\u8d85\u56fe\u4e2d\u7684\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u4fe1\u606f<\/span><\/p>\n

F. Musciotto, F. Battiston, R.N. Mantegna, Detecting informative higher-order interactions in statistically validated hypergraphs, Comm. Phys., 4: Paper 218, 2021<\/span><\/p>\n

https:\/\/www.nature.com\/articles\/s42005-021-00710-4.pdf<\/span><\/p>\n

 <\/span><\/p>\n

\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u6539\u5584\u7f51\u7edc\u7fa4\u96c6\u884c\u4e3a\u7684\u4f18\u5316<\/span><\/p>\n

P.S. Skardal, D. Taylor, L.A rola-Fernandez, A. Arenas, Higher-order interactions improve optimal collective dynamics on networks, arXiv 2021<\/span><\/p>\n

https:\/\/arxiv.org\/abs\/2108.08190<\/span><\/p>\n

 <\/span><\/p>\n

\u9ad8\u9636\u56e2\u7f51\u7edc\u4e0e\u6e17\u6d41\u8fc7\u7a0b<\/span><\/p>\n

C. Stegehuis, T. Peron, Network processes on clique-networks with high average degree: the limited effect of higher-order structure, arXiv 2021<\/span><\/p>\n

https:\/\/www.researchgate.net\/publication\/351279057_Network_processes_on_clique-networks_with_high_average_degree_the_limited_effect_of_higher-order_structure<\/span><\/p>\n

 <\/span><\/p>\n

\u590d\u6742\u7f51\u7edc\u4e2d\u7684\u6d1e\u4e0e\u9c81\u68d2\u6027<\/span><\/p>\n

A.D. Zhou, S. Maleti\u0107, Y. Zhao, Robustness and percolation of holes in complex networks, Physica A, 502: 459-468, 2018<\/span><\/p>\n

https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0378437118302188<\/span><\/p>\n

 <\/span><\/p>\n

\u79d1\u5b66\u4e0e\u6280\u672f\u77e5\u8bc6\u7f51\u7edc\u4e2d\u7684\u9ad8\u9636\u62d3\u6251\u7ed3\u6784<\/span><\/p>\n

T. Gebhart, R.J. Funk, The emergence of higher-order structure in scientific and technological knowledge networks, arXiv 2020<\/span><\/p>\n

https:\/\/arxiv.org\/pdf\/2009.13620v1.pdf<\/span><\/p>\n

 <\/span><\/p>\n

\u6700\u4f18\u9ad8\u9636\u7f51\u7edc\u6a21\u578b\u53ca\u5176\u6570\u636e\u5206\u6790\u5e94\u7528<\/span><\/p>\n

R. Lambiotte, M. Rosvall, I. Scholtes, From networks to optimal higher-order models of complex systems, Nature Physics, 15: 313\u2013320, 2019<\/span><\/p>\n

https:\/\/www.nature.com\/articles\/s41567-019-0459-y.pdf<\/span><\/p>\n

 <\/span><\/p>\n

\u590d\u6742\u7cfb\u7edf\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u7269\u7406\u5b66\u539f\u7406<\/span><\/p>\n

F. Battiston et al., The physics of higher-order interactions in complex systems, Nature Physics, 17: 1093\u20131098, 2021<\/span><\/p>\n

https:\/\/www.nature.com\/articles\/s41567-021-01371-4.pdf<\/span><\/p>\n

 <\/span><\/p>\n

\u8ba1\u7b97\u590d\u6742\u7f51\u7edc\u4e2d\u7684\u9ad8\u9636\u56e2\u548c\u6d1e<\/span><\/p>\n

D.H. Shi*, Z.F. Chen, X. Sun, Q.H. Chen, C. Ma, Y. Lou, G.R. Chen*, Computing cliques and cavities in networks, Comm. Phys., accepted, 2021<\/span><\/p>\n

https:\/\/arxiv.org\/vc\/arxiv\/papers\/2101\/2101.00536v1.pdf<\/span><\/p>\n


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

\n
\n

\uff08\u53c2\u8003\u6587\u732e\u53ef\u4e0a\u4e0b\u6ed1\u52a8<\/strong>\u67e5\u770b\uff09<\/p>\n<\/section>\n<\/section>\n


<\/p>\n


<\/p>\n

\n
\n
\n
\n
\n

\u7f51\u7edc\u79d1\u5b66\u96c6\u667a\u8bfe\u5802\u7b2c\u4e09\u671f\u62a5\u540d\u4e2d<\/strong><\/span><\/p>\n

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


<\/section>\n<\/h2>\n
\u4ece\u73b0\u5b9e\u793e\u4f1a\u7684\u5173\u7cfb\u7f51\u5230\u865a\u62df\u7684\u4e92\u8054\u7f51\uff0c\u4ece\u7ebf\u4e0b\u5230\u7ebf\u4e0a\uff0c\u6211\u4eec\u7684\u751f\u6d3b\u59cb\u7ec8\u6ca1\u6709\u8131\u79bb\u590d\u6742\u7f51\u7edc\u3002\u771f\u5b9e\u7684\u590d\u6742\u7f51\u7edc\u4ece\u5176\u8bde\u751f\u5f00\u59cb\u5c31\u4e0d\u65ad\u5730\u6f14\u5316\u7740\u3002\u7f51\u7edc\u8282\u70b9\u4e0d\u65ad\u5730\u589e\u52a0\uff0c\u8282\u70b9\u4e4b\u95f4\u7684\u8fde\u63a5\u4e0d\u65ad\u5730\u589e\u957f\u3002\u7136\u800c\uff0c\u590d\u6742\u7f51\u7edc\u7684\u5f62\u6210\u673a\u5236\u662f\u4ec0\u4e48\uff1f\u5177\u6709\u4ec0\u4e48\u6837\u7684\u6f14\u5316\u89c4\u5f8b\uff1f\u5b83\u4eec\u7684\u6f14\u5316\u673a\u5236\u5bf9\u7f51\u7edc\u7684\u529f\u80fd\u548c\u52a8\u529b\u5b66\u884c\u4e3a\u6709\u4ec0\u4e48\u5f71\u54cd\uff1f\u4e3a\u4e86\u56de\u7b54\u8fd9\u4e9b\u95ee\u9898\uff0c\u79d1\u5b66\u5bb6\u4eec\u5bf9\u590d\u6742\u7f51\u7edc\u7684\u63a2\u7d22\u4ece\u672a\u505c\u6b62\u3002<\/span><\/section>\n

<\/section>\n
\u7f51\u7edc\u79d1\u5b66\u662f\u4e00\u4e2a\u84ec\u52c3\u53d1\u5c55\u7684\u5d2d\u65b0\u4ea4\u53c9\u5b66\u79d1\uff0c\u53ef\u4ee5\u770b\u505a\u590d\u6742\u7cfb\u7edf\u7684\u9aa8\u67b6\uff0c\u6838\u5fc3\u662f\u7814\u7a76\u5404\u79cd\u5927\u578b\u590d\u6742\u7f51\u7edc\u4e4b\u95f4\u7684\u5171\u6027\u548c\u5904\u7406\u5b83\u4eec\u7684\u666e\u9002\u65b9\u6cd5\uff0c\u5176\u7814\u7a76\u5bf9\u4e8e\u53d1\u5c55\u590d\u6742\u7cfb\u7edf\u7684\u57fa\u672c\u7406\u8bba\u53ca\u6784\u5efa\u4ea7\u751f\u4e86\u6781\u5927\u7684\u63a8\u52a8\u4f5c\u7528\u3002<\/span><\/section>\n

<\/section>\n
\u7f51\u7edc\u79d1\u5b66\u7684\u7b2c\u4e09\u4e2a\u5341\u5e74\uff0c\u5df2\u7ecf\u8fc7\u53bb\u4e86\u51e0\u5e74\u3002\u4ece\u56fd\u5185\u5916\u7f51\u7edc\u79d1\u5b66\u7814\u7a76\u7684\u53d1\u5c55\u8d8b\u52bf\u6765\u770b\uff0c\u5404\u79cd\u5404\u6837\u7684\u66f4\u590d\u6742\u7684\u7f51\u7edc\u6a21\u578b\u548c\u7ed3\u6784\u4ee5\u53ca\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u52a8\u529b\u5b66\u5f15\u8d77\u4e86\u4eba\u4eec\u7684\u6781\u5927\u5174\u8da3\u3002\u4e3a\u4e86\u56de\u5e94\u8fd9\u79cd\u8feb\u5207\u9700\u6c42\uff0c \u6211\u4eec\u7f51\u7edc\u79d1\u5b66\u7b2c\u4e09\u671f\u8bfe\u7a0b\u5c06\u56f4\u7ed5\u590d\u6742\u7f51\u7edc\u7684\u6570\u5b66\u5efa\u6a21\u4e0e\u5e94\u7528\u8fdb\u884c\u591a\u89d2\u5ea6\u7684\u4ecb\u7ecd\u3002<\/span><\/section>\n

<\/section>\n
\u96c6\u667a\u5b66\u56ed\u7279\u9080\u9648\u5173\u8363\u3001\u6a0a\u745b\u3001\u5468\u8fdb\u3001\u674e\u7fd4\u3001\u5f20\u6c5f\u3001\u95eb\u5c0f\u52c7\u3001\u5218\u5b97\u534e\u3001\u77f3\u5ddd\u3001\u865e\u6587\u6b66\u3001\u8d75\u6d77\u5174\u3001\u53f2\u5b9a\u534e\u52a0\u5165\u6253\u9020\u7b2c\u4e09\u671f\u8bfe\u7a0b<\/a>\uff0c\u6b22\u8fce\u4f60\u7684\u52a0\u5165\u3002<\/span><\/section>\n

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

<\/section>\n
\u8be6\u60c5\u8bf7\u70b9\u51fb\uff1a<\/span><\/section>\n
\u4ece\u6570\u5b66\u5efa\u6a21\u5230\u591a\u5b66\u79d1\u5e94\u7528\u2014\u2014\u7f51\u7edc\u79d1\u5b66\u00b7\u96c6\u667a\u8bfe\u5802\u5168\u65b0\u5347\u7ea7<\/a><\/span><\/strong><\/span><\/section>\n

<\/section>\n

<\/section>\n
\n

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