Mathematicians Build Long-Awaited Graph Sandwich
Kristina Armitage/ Quanta Magazine
克里斯蒂娜·阿米蒂奇/广达杂志
In 2004, two mathematicians hypothesized a powerful kind of sandwich.
2004 年，两位数学家假设了一种威力强大的三明治。
They were studying graphs, which are collections of points (called vertices) and lines (called edges).
他们正在研究图形，图形是点（称为顶点）和线（称为边）的集合。
Graphs might represent anything from social groups to the internet to neurons in the brain.
图表可以代表从社会群体到互联网再到大脑神经元的任何事物。
The mathematicians hoped to understand properties of one type of graph — a type that’s ubiquitous in mathematics and computer science but difficult to analyze — by sandwiching it, in a mathematically rigorous way, between two simpler graphs.
数学家希望通过以数学上严格的方式将一种图形夹在两个更简单的图形之间来理解一种图形的属性（这种图形在数学和计算机科学中普遍存在，但难以分析）。
If researchers could prove the existence of such a sandwich, they wouldn’t just be showing that the middle graph has one property of interest; they’d be showing that it has all sorts of important properties.
如果研究人员能够证明这种三明治的存在，他们将不仅仅证明中间的图具有一个令人感兴趣的属性；他们会证明它具有各种重要的特性。
In doing so, they’d also be demonstrating that two very different random processes that mathematicians like to study are connected in a deeper and more elegant way than they’d imagined.
在这样做的过程中，他们还将证明数学家喜欢研究的两个截然不同的随机过程以比他们想象的更深入、更优雅的方式联系在一起。
“The notion is so beautiful,” said Pu Gao , a mathematician at the University of Waterloo in Canada who has worked on the problem.
“这个想法太美妙了，”研究这个问题的加拿大滑铁卢大学数学家高普说。
“What attracts me most is actually the beauty of it.”
“最吸引我的其实是它的美丽。”
In the past two decades, mathematicians made progress on the “sandwich conjecture,” which says that so long as the graph you’re interested in is large enough, you can always create the needed sandwich.
在过去的二十年里，数学家在“三明治猜想”上取得了进展，即只要你感兴趣的图足够大，你总能创造出所需的三明治。
But no one could prove it in full.
但没有人能够完全证明这一点。
Then in 2025, three mathematicians found a way to push their field’s techniques to their limits, and completed the quest.
然后在 2025 年，三位数学家找到了一种将他们领域的技术推向极限的方法，并完成了任务。
In the late 1950s, the American mathematician Edgar Gilbert was studying telephone networks at Bell Labs.
20 世纪 50 年代末，美国数学家埃德加·吉尔伯特 (Edgar Gilbert) 在贝尔实验室研究电话网络。
To better understand those networks, he came up with a simple model of a “random” graph, in which vertices connect to other vertices at random.
为了更好地理解这些网络，他提出了一个简单的“随机”图模型，其中顶点随机连接到其他顶点。
(The mathematicians Paul Erdős and Alfréd Rényi independently came up with a similar model at around the same time.)
（数学家 Paul Erdős 和 Alfred Rényi 大约在同一时间独立提出了类似的模型。）
To make one of these graphs, start with a set of vertices.
要制作其中一个图，请从一组顶点开始。
Choose any pair of vertices in your set, then flip a (potentially biased) coin.
选择集合中的任意一对顶点，然后掷一枚（可能有偏差的）硬币。
If you get heads, draw an edge between them; otherwise, move on.
如果你得到了正面，就在它们之间画一条边；否则，继续。
Repeat this step for every pair of vertices in the graph.
对图中的每对顶点重复此步骤。
These graphs, known as random binomial graphs, turned out to provide a useful — if imperfect — way to represent networks.
这些图被称为随机二项式图，结果提供了一种有用的（尽管不完美）表示网络的方法。
They were relatively easy to analyze, and mathematicians proved many interesting things about them.
它们相对容易分析，数学家证明了它们的许多有趣的事情。
By the 1970s, for instance, they’d discovered under what conditions a random binomial graph will contain a Hamiltonian cycle, a path that visits each vertex exactly once.
例如，到了 20 世纪 70 年代，他们发现了在什么条件下随机二项式图将包含哈密顿循环，即访问每个顶点一次的路径。
But this isn’t the only type of random graph.
但这并不是随机图的唯一类型。
Mathematicians were also curious about random graphs in which all vertices have the same number of edges.
数学家也对所有顶点具有相同边数的随机图感到好奇。
These so-called regular graphs provide a better understanding of random structure than binomial graphs.
这些所谓的正则图比二项式图可以更好地理解随机结构。
And they’re often much more accurate at modeling real-world networks.
而且它们在对现实世界网络进行建模时通常更加准确。
But because their edges form more constrained, interdependent patterns, they’re also much harder to analyze.
但由于它们的边缘形成了更受约束、相互依赖的模式，因此它们也更难分析。
It took an additional 20 years of work after the question about Hamiltonian cycles was answered for binomial graphs before mathematicians could do the same for regular graphs.
在回答了二项式图的哈密顿循环问题之后，数学家们又花了 20 年的时间才能够对正则图做同样的事情。
But what if you can approximate random regular graphs with random binomial graphs?
但是，如果您可以用随机二项式图来近似随机正则图呢？
If that’s possible, then mathematicians can get many hard-to-prove properties of a regular graph from the matching binomial graph — for free.
如果可能的话，那么数学家就可以从匹配的二项式图中免费获得正则图的许多难以证明的属性。
In the early 2000s, Jeong Han Kim , then at Microsoft Research, and Van Ha Vu , then at the University of California, San Diego, showed how to do this by making a graph sandwich .
2000 年代初期，当时在微软研究院工作的 Jeong Han Kim 和当时在加州大学圣地亚哥分校工作的 Van Ha Vu 展示了如何通过制作图形三明治来做到这一点。
The idea, loosely stated, was to find a single recipe — a random process — to build a binomial graph and a regular graph at the same time.
笼统地说，这个想法是找到一个单一的方法——一个随机过程——来同时构建二项式图和正则图。
Not only does this recipe need to generate the right kinds of graphs, but those graphs must also fit together in just the right way.
这个秘诀不仅需要生成正确类型的图表，而且这些图表还必须以正确的方式组合在一起。
If you can do this, then when you prove results about the binomial graph, which is relatively easy to analyze, those results will also hold for the regular graph.
如果你能做到这一点，那么当你证明相对容易分析的二项式图的结果时，这些结果也适用于正则图。
In the sandwich analogy, it’s like proving things about one of the slices of bread and knowing that those results will also hold true for the cheese in the middle.
在三明治的类比中，这就像证明一片面包的情况，并知道这些结果也适用于中间的奶酪。
But how do those graphs need to fit together, exactly?
但这些图表到底需要如何组合在一起呢？
You have to come up with a recipe that layers the cheese on each slice of bread separately.
你必须想出一个食谱，将奶酪分别铺在每片面包上。
First, you need a recipe that gives you a regular graph that contains a binomial graph.
首先，您需要一个方法来为您提供包含二项式图的正则图。
That is, the binomial graph’s edges form a subset of the edges that make up the regular graph.
也就是说，二项式图的边形成了构成正则图的边的子集。
If that binomial graph has any property that is more likely to appear when you add edges to it, then your regular graph will also have that property.
如果该二项式图具有在向其添加边时更有可能出现的任何属性，那么您的常规图也将具有该属性。
This is the bottom half of Kim and Vu’s sandwich.
这是 Kim 和 Vu 三明治的下半部分。
Similarly, you need a recipe that gives you a regular graph that is contained within a binomial graph.
同样，您需要一个方法来为您提供包含在二项式图中的正则图。
If this bigger binomial graph has properties that are more likely to appear when you remove edges from it, then your regular graph must also have these properties.
如果这个更大的二项式图具有当您从中删除边时更有可能出现的属性，那么您的常规图也必须具有这些属性。
This is the top half of your sandwich.
这是三明治的上半部分。
Kim and Vu conjectured that so long as your regular graph has a reasonable number of edges, you can almost always build this sandwich .
Kim 和 Vu 推测，只要你的正则图具有合理数量的边，你几乎总能构建出这个三明治。
That’s no easy task, given that your recipe needs to create the binomial and regular graphs simultaneously, even though they usually get built using completely different random processes.
这不是一件容易的任务，因为你的配方需要同时创建二项式和正则图，即使它们通常是使用完全不同的随机过程构建的。
Over the years, mathematicians proved that the bottom half of Kim and Vu’s sandwich existed, and they proved the upper half in some settings.
多年来，数学家证明了 Kim 和 Vu 的三明治的下半部分存在，并且在某些情况下证明了上半部分。
“It was a sequence of ideas building upon one another,” said Michael Krivelevich , a mathematician at Tel Aviv University who has worked on the problem.
“这是一系列相互构建的想法，”特拉维夫大学数学家迈克尔·克里维列维奇（Michael Krivelevich）说，他一直致力于解决这个问题。
Each step “requires a very good technique.
每一步“都需要非常好的技术。
It requires ingenuity.”
这需要独创性。”
But the sandwich was not yet complete.
但三明治还没有完成。
The proof of the conjecture would require a way to closely connect the bread and cheese of any sandwich.
该猜想的证明需要一种将任何三明治的面包和奶酪紧密连接在一起的方法。
In particular, the layers would be built up in tandem, guaranteeing that they would always fit together.
特别是，这些层将串联构建，确保它们始终配合在一起。
In 2023, three mathematicians — Richard Montgomery of the University of Warwick; Natalie Behague , his postdoctoral researcher at the time; and Daniel Iľkovič , his doctoral student — started to think about ways to build a random regular graph and a random binomial graph edge by edge, ensuring that at each step the regular graph would contain the binomial one.
2023年，三位数学家——华威大学的理查德·蒙哥马利； Natalie Behague，他当时的博士后研究员；和他的博士生 Daniel Iľkovič 开始思考如何逐条构建随机正则图和随机二项式图，确保每一步正则图都包含二项式图。
It’s a bit like making your sandwich out of tiny bits of shredded cheese, placing them on the bread one by one, rather than slapping a whole slice on at once.
这有点像用一小块切碎的奶酪制作三明治，将它们一片一片地放在面包上，而不是一次拍上一整片。
Richard Montgomery helped craft a recipe for a mathematical sandwich that’s powerful but difficult to make.
理查德·蒙哥马利帮助制定了一种数学三明治的配方，该三明治功能强大但难以制作。
To follow their recipe (which, the mathematicians note, is heavily adapted from a 2019 result by Gao and two colleagues), start with two sets of vertices without edges.
为了遵循他们的秘诀（数学家指出，该秘诀很大程度上改编自高和两位同事 2019 年的结果），从两组没有边的顶点开始。
One set will ultimately become your binomial graph, the other your regular graph.
一组最终将成为您的二项式图，另一组将成为您的常规图。
Now build your binomial graph in the usual way.
现在以通常的方式构建二项式图。
That is, choose a pair of vertices and flip a weighted coin.
也就是说，选择一对顶点并掷一枚加权硬币。
If your coin lands on heads, add an edge to the binomial graph.
如果您的硬币正面朝上，请在二项式图中添加一条边。
Add one to the regular graph as well.
也将一个添加到常规图表中。
If the coin lands on tails, don’t add the edge in the binomial graph.
如果硬币反面朝上，则不要在二项式图中添加边。
But you may or may not need to add an edge to the regular graph.
但是您可能需要也可能不需要向常规图形添加边。
After all, a regular graph is defined by the property that every vertex has the same number of edges.
毕竟，正则图是由每个顶点具有相同数量的边的属性定义的。
You need to make sure that all the required edges are there.
您需要确保所有必需的边缘都在那里。
Natalie Behague called the conjecture she recently helped prove “almost too good to be true.”
娜塔莉·贝哈格称她最近帮助证明的猜想“几乎好得令人难以置信”。
Kyran O’Brien/Dublin City University
凯兰·奥布莱恩/都柏林城市大学
So when your coin lands on tails, ignore your binomial graph, but flip a second weighted coin to decide whether to add an edge to your regular graph.
因此，当您的硬币反面朝上时，请忽略您的二项式图表，而是翻转第二个加权硬币来决定是否在常规图表中添加边。
The weight of this second coin will change as you build up your graph.
当您构建图表时，第二枚硬币的重量将会发生变化。
Behague, Iľkovič, and Montgomery came up with a clever way to estimate the weight of the coin as you add more edges to your graphs so that you’re guaranteed to get a truly regular graph.
贝哈格、伊科维奇和蒙哥马利想出了一种巧妙的方法来估计硬币的重量，因为您可以在图表中添加更多边，从而保证获得真正规则的图表。
In addition, you also guarantee that your regular graph contains the binomial one, giving you the lower part of the sandwich.
此外，您还可以保证您的正则图包含二项式图，从而为您提供三明治的下部。
To build the upper part, the mathematicians then reversed their entire process.
为了建造上面的部分，数学家们颠倒了整个过程。
They began with two graphs that contained every possible edge.
他们从包含所有可能边的两张图开始。
They then removed edges one by one until they ended up with a regular graph and a binomial graph that contained it.
然后，他们一条一条地删除边，直到最终得到一个正则图和一个包含它的二项式图。
They had finished their sandwich.
他们已经吃完了三明治。
“The conjecture is in some way very natural.
“这个猜想在某种程度上是非常自然的。
It was kind of annoying not to have it proven yet,” Krivelevich said.
尚未得到证实有点令人恼火，”克里维列维奇说。
When he saw the trio’s new result, he was filled with “some kind of relief.”
当他看到三人组的新成果时，他心中充满了“某种解脱”。
With the sandwich conjecture resolved, mathematicians no longer have to prove every property of random regular graphs from scratch.
随着三明治猜想的解决，数学家不再需要从头开始证明随机正则图的每个属性。
They can now draw on the vast literature that’s been written about random binomial graphs and get all sorts of properties automatically.
他们现在可以利用有关随机二项式图的大量文献并自动获取各种属性。
That means they can rewrite scores of results about regular graphs in a single, streamlined proof.
这意味着他们可以在一个简化的证明中重写有关常规图的大量结果。
And new results are already starting to appear .
新的结果已经开始出现。
Moreover, the proof of this “meta-theorem,” as Gil Kalai of the Hebrew University of Jerusalem put it, offers a set of methods that “enriches our toolbox” and “sharpens our technical teeth.” Those methods might allow mathematicians to understand even more about the structure of networks than they originally set out to.
此外，正如耶路撒冷希伯来大学的吉尔·卡莱所说，这个“元定理”的证明提供了一套“丰富我们的工具箱”和“磨砺我们的技术牙齿”的方法。这些方法可能会让数学家比他们最初设想的更多地了解网络结构。
In the meantime, researchers hope to make even more complicated sandwiches, filled with alternating layers of binomial and regular graphs, or with other ingredients.
与此同时，研究人员希望制作出更复杂的三明治，其中填充有交替的二项式和正则图层，或其他成分。
In doing so, they’re continuing to explore the ways in which seemingly different random processes — one very constrained, the other not — are more similar than they look.
在此过程中，他们继续探索看似不同的随机过程（一个非常受约束，另一个不受约束）比看起来更相似的方式。
“That sort of deep connection between the two,” Behague said, “seems almost too good to be true.” And yet it is.
贝哈格说：“两者之间的这种深厚联系似乎好得令人难以置信。”但事实确实如此。
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