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机器学习遇见复杂网络：通过双曲插值的方法.html
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机器学习遇见复杂网络：通过双曲插值的方法.html
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<div id='write' class = 'is-mac show-fences-line-number'><h1><a name='header-n0' class='md-header-anchor '></a>机器学习遇见复杂网络：通过双曲空间的协同插值的方法</h1><p> </p><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n0"><a class="md-toc-inner" style="" href="#header-n0">机器学习遇见复杂网络：通过双曲空间的协同插值的方法</a></span><span class="md-toc-item md-toc-h2" data-ref="n61"><a class="md-toc-inner" style="" href="#header-n61">points</a></span><span class="md-toc-item md-toc-h2" data-ref="n23"><a class="md-toc-inner" style="" href="#header-n23">摘要翻译</a></span><span class="md-toc-item md-toc-h2" data-ref="n35"><a class="md-toc-inner" style="" href="#header-n35">logics</a></span><span class="md-toc-item md-toc-h2" data-ref="n212"><a class="md-toc-inner" style="" href="#header-n212">algorithms</a></span><span class="md-toc-item md-toc-h2" data-ref="n313"><a class="md-toc-inner" style="" href="#header-n313">Results</a></span><span class="md-toc-item md-toc-h3" data-ref="n357"><a class="md-toc-inner" style="" href="#header-n357">coalescent embedding protocol</a></span><span class="md-toc-item md-toc-h3" data-ref="n665"><a class="md-toc-inner" style="" href="#header-n665">在PSO网络上的表现</a></span><span class="md-toc-item md-toc-h3" data-ref="n381"><a class="md-toc-inner" style="" href="#header-n381">Greedy routing performance（未看）</a></span><span class="md-toc-item md-toc-h3" data-ref="n392"><a class="md-toc-inner" style="" href="#header-n392">真实网络的社区识别</a></span><span class="md-toc-item md-toc-h3" data-ref="n403"><a class="md-toc-inner" style="" href="#header-n403">大于2维……</a></span><span class="md-toc-item md-toc-h2" data-ref="n825"><a class="md-toc-inner" style="" href="#header-n825">Method</a></span><span class="md-toc-item md-toc-h3" data-ref="n828"><a class="md-toc-inner" style="" href="#header-n828">2D融合插值算法</a></span><span class="md-toc-item md-toc-h3" data-ref="n833"><a class="md-toc-inner" style="" href="#header-n833">3D融合插值算法</a></span><span class="md-toc-item md-toc-h3" data-ref="n838"><a class="md-toc-inner" style="" href="#header-n838">基于流形的插值算法</a></span><span class="md-toc-item md-toc-h3" data-ref="n843"><a class="md-toc-inner" style="" href="#header-n843">最小曲度插值算法</a></span><span class="md-toc-item md-toc-h3" data-ref="n860"><a class="md-toc-inner" style="" href="#header-n860">HyperMap/HyperMap-CN</a></span><span class="md-toc-item md-toc-h3" data-ref="n865"><a class="md-toc-inner" style="" href="#header-n865">LPCS</a></span><span class="md-toc-item md-toc-h2" data-ref="n876"><a class="md-toc-inner" style="" href="#header-n876">Matlab代码/软硬件</a></span></p></div><ul><li>from nature communications</li></ul><h2><a name='header-n61' class='md-header-anchor '></a>points</h2><ul><li><p>利用pop-sim模型生成的图作为原始数据。</p></li><li><p>思路：</p><ol><li>将无权拓扑网络（文中用的是PSO网络，popularity similarity optimization）根据<em>距离、连接关系、中心性等</em>要素转换成有权重网络</li><li>然后进行非线性降维，得到角度参数。</li><li>再调整角参数（目的：在插值空间上满足所希望的分布）</li><li>根据度得到参数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-564-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.693ex" height="1.41ex" viewBox="0 -504.6 729 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E848-MJMATHI-72" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E848-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E848-MJMATHI-72" x="0" y="0"></use><use xlink:href="#E848-MJMAIN-2E" x="451" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-564">r.</script> 得到coalescent embedding的结果。</li></ol></li><li><p>机器学习中，分类算法与回归问题应该是矛盾的，一个属于supervised一个属于unsupervised. 那么是否提取层级结构的时候，我们应该先用（基于树的）分类算法来提取层级结构？</p><ul><li>本文与Facebook团队的结果不同点？</li></ul></li></ul><h2><a name='header-n23' class='md-header-anchor '></a>摘要翻译</h2><p>物理学家最近发现，真实的复杂网络是双曲几何模型的离散采样：到圆心的距离代表中心性（centrality）；夹角代表两个结点之间的拓扑相似性(topological proximity). 建立双曲圆环模型的目的是建立一个全局空间来表示和分析真实网络。但是，由真实网络推断角度参数一直是一个很难的问题。本文提出，利用无监督学习的方法，可以找出大数据内部的similarity结构，称为angular coalescence. 基于这个结果，作者提出了一类算法，来快速而准确地给出插值结果。这个计算的解法可以应用于多种复杂系统。</p><p> </p><h2><a name='header-n35' class='md-header-anchor '></a>logics</h2><ul><li><p>复杂网络的全局性质是近20年的研究重点。</p></li><li><p>很多复杂网络背后的度量是双曲的。这可以用于链路预测/交通拥堵/时变控制系统的的分析。</p></li><li><p>流行性-相似性优化(PSO)模型很重要，近期，有一些这个框架的变体。</p></li><li><p>无监督非线性降维的<strong>流形学习</strong>算法是拓扑机器学习算法中很重要的一部分。这种算法可以学到在多维特征空间背后的特征流形上的非线性的similarities，同时也可以被可视化成一个二维的图像。三个步骤：</p><ol><li>特征流形的重建：建立一个高维空间中点的最近邻网络/图。（如果网络的形式已经是一个无权重的邻接矩阵，我们不用执行第一步。）</li><li>估计这个新图中，每两个结点的拓扑相似性。并把这些相似性数据存在一个距离矩阵中(kernal). </li><li>利用矩阵分解的手段实现降维（通常是2维）。</li></ol></li></ul><p>这个算法很快，复杂度为<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-563-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.4ex" height="2.811ex" viewBox="0 -906.7 3186.1 1210.2" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E847-MJMATHI-4F" d="M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 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104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E847-MJMATHI-4F" x="0" y="0"></use><use xlink:href="#E847-MJMAIN-28" x="763" y="0"></use><g transform="translate(1152,0)"><use xlink:href="#E847-MJMATHI-4E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E847-MJMAIN-32" x="1291" y="513"></use></g><use xlink:href="#E847-MJMAIN-29" x="2519" y="0"></use><use xlink:href="#E847-MJMAIN-2E" x="2908" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-563">O(N^2).</script> </p><p> </p><h2><a name='header-n212' class='md-header-anchor '></a>algorithms</h2><ol><li><p>Popularity versus Similarity Optimization(PSO)</p></li><li><p>Isomap(ISO,等距映射)</p><ul><li>这是流形技术的基础，将邻接矩阵映射到原平面，同时保持结点间的距离不变，还要使得图中心化。<strong>？</strong> </li></ul></li><li><p>circular organization：降维到一个圆环上</p></li><li><p>根据下面的公式，给出参数<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-564-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.693ex" height="1.41ex" viewBox="0 -504.6 729 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E848-MJMATHI-72" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E848-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E848-MJMATHI-72" x="0" y="0"></use><use xlink:href="#E848-MJMAIN-2E" x="451" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-564">r.</script> </p></li><li><p>Noncentered minimum curvilinearity embedding (ncMCE, 非中心最小弯曲度插值)，纵坐标的差代表相似程度。第一维没有意义。可以换成Newtonian Model。</p></li></ol><h2><a name='header-n313' class='md-header-anchor '></a>Results</h2><h3><a name='header-n357' class='md-header-anchor '></a>coalescent embedding protocol</h3><p> </p><p>作者自称选取了表现最好的非线性拓扑降维方法。模型特点：无监督，而且是无参的。</p><p> </p><ul><li><p>三种基于流形的降维方法：</p><ul><li>Isomap(ISO)：从PSO无权邻接矩阵 到 <span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-565-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.394ex" height="1.41ex" viewBox="0 -504.6 600 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E849-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E849-MJMATHI-6E" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-565">n</script> -分类。</li><li>noncentered Isomap(ncISO)：</li><li>Laplacian eigenmaps(LE)</li></ul></li><li><p>两种基于最小曲率的降维方法（将信息压缩到<strong>一个</strong>维度，方式是<a href='https://en.wikipedia.org/wiki/Minimum_spanning_tree'>最小生成树</a>算法(minimum spinning tree，MST)）：这种方式的<strong>优点</strong>：最小生成树提供了一个<u><strong>层次结构</strong></u>。而基于流形的方法完全不能做这种事（流形使得模型连续化）。</p><ul><li>minimum curvilinear embedding(MCE)</li><li>noncentered minimum curvilinear embedding(ncMCE)：线性化</li></ul><p> </p></li></ul><p> </p><p><img src='/Users/GeorgeXiu/Desktop/屏幕快照 2018-08-17 下午6.38.07.png' alt='mappings' referrerPolicy='no-referrer' /></p><p> </p><ul><li><p>结果1: coalescent embedding</p><p>作者使用不同的方法对双曲网络进行插值。结论是一直的：结点会在插值空间积累成图案（我猜作者的意思是<u>连续性</u>），成为angular coalescence（按角度融合）. 作者提出一个词：coalescent embedding 以表示一类可以形成angular coalescence的二维网络插值算法。</p><p>由于基于流形的算法和（尤其是）基于层次结构的算法都可以提取出 angular coalescence ，作者认为这是一种 <em>二级现象</em>（作者原文用的是epiphenomenon，附带现象，与前后文不相符合。Page4para2）。</p></li><li><p><img src='/Users/GeorgeXiu/Desktop/屏幕快照 2018-08-17 下午7.12.55.png' alt='流程' referrerPolicy='no-referrer' /></p></li></ul><p> </p><ul><li><p>流程</p><ul><li><p>pre-weighting：无权图表示方便，但是实用性很差，看不出结构。</p><p><span> </span>预先加入权：可以大概地表示出连接的结点之间的距离。</p><p><span> </span>作者给出两种方法：</p><ol><li>repulsion-attraction rule：权重只与局部信息有关系。<strong>度很大的结点之间的距离应该比较大，因为它们代表了不同的hub。</strong> 根据网络的navigability理论，每个hub都倾向于控制自己的一片区域，而hub之间离得比较远。这就是repulsion<em>厌恶</em>。而共同邻居比较多的结点应该比较近这就是attraction<em>吸引</em>。补充材料图1显示出这个处理能显著提高效果。</li><li>EBC边介数中心性：这是一种全局性质。介数高的边适合用来连接距离远的边，边缘附近的边适合连接局部的边。</li></ol><p>作者不确定保持角参数是最好的策略，因为它们的“互利角距离”会受短距离的噪声影响。作者提出了一种策略来对圆周上的结点进行re-organize，使它们等距离分布（这回层次结构肯定是没有了）。</p><p>作者用上图中的流程（包括几种变体）来对生成网络与真实网络进行了变换。并用这种方法与Hypermap进行了对比。</p></li></ul></li></ul><p> </p><p> </p><h3><a name='header-n665' class='md-header-anchor '></a>在PSO网络上的表现</h3><p><img src='/Users/GeorgeXiu/Desktop/屏幕快照 2018-08-17 下午7.54.41.png' alt='' referrerPolicy='no-referrer' /></p><p> </p><ul><li><p>指标：</p><ul><li>HD-correlation（上图）：每一对结点之间距离(Hyperbolic distance)的Pearson相关系数。</li><li>C-Score（下图）：点对在映射前后的angular relationship不变的比例。即保夹角性强弱。C-Score=0代表完全不对。如果蓝色点大概是一个直线，那么说明效果很好。</li></ul><p><img src='/Users/GeorgeXiu/Desktop/屏幕快照 2018-08-17 下午8.08.19.png' alt='' referrerPolicy='no-referrer' /></p></li></ul><p>如果没有RA技术（突出hub位置），所有模型的表现就都会显著变差。</p><p>温度比较高的时候，随机性比较大，clustering比较弱。所以相关系数都会降低。可见映射算法对低温模型（clustering比较强的模型）更好用一些。</p><p>这些算法都比HyperMap快。这些算法可以处理超过10000个结点的网络，一分钟之内就可以算完；30000个结点的网络几分钟就可以算完。而HyperMap对1000个结点的网络就需要超过3个小时。</p><p>相关性（HD-correlation）对于大网络依然保持得很好。</p><h3><a name='header-n381' class='md-header-anchor '></a>Greedy routing performance（未看）</h3><p>navigability。<a href='http://www.nature.com/ncomms/2015/150703/ncomms8651/full/ncomms8651.html'>博弈论那篇</a>的topic。另外有一篇<a href='http://www.nature.com/nphys/journal/v5/n1/full/nphys1130.html'>navigability of complex networks</a>. </p><p>还没看。</p><h3><a name='header-n392' class='md-header-anchor '></a>真实网络的社区识别</h3><p>主要用的是基于树的方法。</p><p><img src='/Users/GeorgeXiu/Desktop/屏幕快照 2018-08-17 下午8.25.24.png' alt='' referrerPolicy='no-referrer' /></p><h3><a name='header-n403' class='md-header-anchor '></a>大于2维……</h3><p>主要是基于流形学习的方法</p><p><img src='/Users/GeorgeXiu/Desktop/屏幕快照 2018-08-17 下午8.26.09.png' alt='' referrerPolicy='no-referrer' /></p><p>作者做了几个可视化。</p><p> </p><p> </p><h2><a name='header-n825' class='md-header-anchor '></a>Method</h2><h3><a name='header-n828' class='md-header-anchor '></a>2D融合插值算法</h3><ol><li><p>赋予权重</p><ol><li><p>厌恶-喜好规则（local）</p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block" id="mathjax-n928" cid="n928" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-559-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="86.381ex" height="12.616ex" viewBox="0 -2967.3 37191.8 5432" role="img" focusable="false" style="vertical-align: -5.724ex;"><defs><path stroke-width="0" id="E827-MJMAIN-28" d="M94 250Q94 319 104 381T127 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y="0"></use></g></g><use transform="scale(0.707)" xlink:href="#E827-MJMAIN-3D" x="1707" y="0"></use><use transform="scale(0.707)" xlink:href="#E827-MJMAIN-31" x="2485" y="0"></use><use transform="scale(0.707)" xlink:href="#E827-MJMAIN-7D" x="2985" y="0"></use></g></g></g></g></g></g></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-559">x_{ij}^{RA1}=\frac{d_i+d_j+d_id_j}{1+CN_{ij}}\cdot \mathbb 1_{\{x_{ij}=1\}} \\
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2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E860-MJMATHI-58" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-576">X</script>中。<em>某种意义上相当于距离矩阵。</em></p></li><li><p>非线性降维</p><ol><li>Laplacian特征映射</li><li>Isomap</li><li>无中心Isomap</li><li>最小曲度降维</li><li>无中心最小曲度降维</li></ol></li><li><p>角参数</p><ol><li>圆周调整</li><li>等距离化调整</li></ol></li><li><p>半径参数</p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block" id="mathjax-n1072" cid="n1072" mdtype="math_block"><span class="MathJax_Preview"></span><span class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-561-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="89.883ex" height="5.612ex" viewBox="0 -1459.5 38699.6 2416.4" role="img" focusable="false" style="vertical-align: -2.012ex; margin-bottom: -0.211ex;"><defs><path stroke-width="0" id="E840-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E840-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 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318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E840-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E840-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 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d="M22 710V750H159V-250H22V-210H119V710H22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(37421,0)"><g id="mjx-eqn-3" transform="translate(0,23)"><use xlink:href="#E840-MJMAIN-28"></use><use xlink:href="#E840-MJMAIN-33" x="389" y="0"></use><use xlink:href="#E840-MJMAIN-29" x="889" y="0"></use></g></g><g transform="translate(13533,0)"><g transform="translate(-15,0)"><g transform="translate(0,23)"><use xlink:href="#E840-MJMATHI-72" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E840-MJMATHI-69" x="637" y="-213"></use><use xlink:href="#E840-MJMAIN-3D" x="1072" y="0"></use><g transform="translate(1850,0)"><g transform="translate(397,0)"><rect stroke="none" width="620" height="60" x="0" y="220"></rect><use xlink:href="#E840-MJMAIN-32" x="60" y="676"></use><use xlink:href="#E840-MJMATHI-3B6" x="74" y="-686"></use></g></g><use xlink:href="#E840-MJMAIN-5B" x="2988" y="0"></use><use xlink:href="#E840-MJMATHI-3B2" x="3266" y="0"></use><g transform="translate(4006,0)"><use xlink:href="#E840-MJMAIN-6C"></use><use xlink:href="#E840-MJMAIN-6E" x="278" y="0"></use></g><use xlink:href="#E840-MJMATHI-69" x="5006" y="0"></use><use xlink:href="#E840-MJMAIN-2B" x="5574" y="0"></use><use xlink:href="#E840-MJMAIN-28" x="6574" y="0"></use><use xlink:href="#E840-MJMAIN-31" x="6963" y="0"></use><use xlink:href="#E840-MJMAIN-2212" x="7685" y="0"></use><use xlink:href="#E840-MJMATHI-3B2" x="8685" y="0"></use><use xlink:href="#E840-MJMAIN-29" x="9258" y="0"></use><g transform="translate(9814,0)"><use xlink:href="#E840-MJMAIN-6C"></use><use xlink:href="#E840-MJMAIN-6E" x="278" y="0"></use></g><use xlink:href="#E840-MJMATHI-4E" x="10815" y="0"></use><use xlink:href="#E840-MJMAIN-5D" x="11703" y="0"></use></g></g></g></g></svg></span></span><script type="math/tex; mode=display" id="MathJax-Element-561">r_i = \frac{2}{\zeta}[\beta\ln i +(1-\beta)\ln N]</script></div><p>其中<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-577-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.062ex" height="1.877ex" viewBox="0 -755.9 888 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E861-MJMATHI-4E" d="M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></path></defs><g 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682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E862-MJMATHI-3B6" x="0" y="0"></use><use xlink:href="#E862-MJMAIN-3D" x="748" y="0"></use><use xlink:href="#E862-MJMAIN-2212" x="1804" y="0"></use><use xlink:href="#E862-MJMATHI-4B" x="2582" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-578">\zeta=-K</script>，<span class="MathJax_Preview"></span><span class="MathJax_SVG" id="MathJax-Element-579-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.065ex" height="1.877ex" viewBox="0 -755.9 889 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E863-MJMATHI-4B" d="M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 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-188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E865-MJMATHI-3B3" x="0" y="0"></use></g></svg></span><script type="math/tex" id="MathJax-Element-581">\gamma</script>为度分布的幂律指数。</p></li></ol><h3><a name='header-n833' class='md-header-anchor '></a>3D融合插值算法</h3><h3><a name='header-n838' class='md-header-anchor '></a>基于流形的插值算法</h3><h3><a name='header-n843' class='md-header-anchor '></a>最小曲度插值算法</h3><h3><a name='header-n860' class='md-header-anchor '></a>HyperMap/HyperMap-CN</h3><h3><a name='header-n865' class='md-header-anchor '></a>LPCS</h3><p>……写了好多，每个算法的思路都给出来了。还没看完。</p><h2><a name='header-n876' class='md-header-anchor '></a>Matlab代码/软硬件</h2><p><a href='https://github.com/biomedical-cybernetics/coalescent_embedding' target='_blank' class='url'>https://github.com/biomedical-cybernetics/coalescent_embedding</a></p><p>Win8 Pro 工作站</p><p>Unless stated otherwise MATLAB code was used for all the methods and simulations, which were carried out on a workstation under Windows 8.1 Pro with 512 GB of RAM and 2 Intel(R) Xenon(R) CPU E5-2687W v3 processors with 3.10 GHz. </p></div>
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