图似然的最小值
The minimum of the graph likelihood
浏览论文内容
中文总结 AI 辅助
该研究证明平衡完全二部图在给定阶数完全二部图中最小化均匀顺序附着过程的图似然，发现n=15时存在反例，确定过程香农熵主项并给出顶点删除递推式。
中文摘要 AI 辅助
n个顶点的有限简单无向图G的似然，是均匀顺序附着过程输出与G同构的图的概率，该过程每一步将新顶点连接到已存在顶点中均匀随机大小的均匀随机子集。Dervovic、Mocherla和Severini猜想，似然由平衡完全二部图最小化。我们证明，给定阶数的完全二部图中，平衡图唯一最小化似然。精确计算显示，其在6到14阶的所有图中也最小化似然，首个反例出现在n=15时。5-环的大小为3的独立集的吹胀图，等价于15个顶点、连接集为{1,4,6}的循环图，其似然为K_{7,8}的0.20128…倍，且仍为无三角形图。我们证明该失败并非偶然：平衡完全二部图的似然为2^{-(1/2-1/(8ln2)+o(1))n²}，而n阶所有图的最小似然为2^{-(1/2+o(1))n²}，故猜想的最小器超出最小值的因子随n²指数增长。我们还确定了该过程的香农熵主项为n²/(4ln2)比特，表明猜想的最小器实际上比该过程的典型输出更可能出现。证明基于顶点删除递推式，其计算似然的时间复杂度为O(n2ⁿ)，且可闭合于任意固定基图的吹胀图。
英文摘要
The likelihood of a finite simple undirected graph $G$ on $n$ vertices is the probability that the uniform sequential attachment process, which at each step joins a new vertex to a uniformly random subset of uniformly random size of the vertices already present, outputs a graph isomorphic to $G$. Dervovic, Mocherla and Severini conjectured that the likelihood is minimised by the balanced complete bipartite graph. We prove that, among complete bipartite graphs of a given order, the balanced one uniquely minimises the likelihood. Exact computation shows that it also minimises over all graphs for every order from $6$ through $14$, and that the first counterexample occurs at $n=15$. The blow-up of the five cycle by independent sets of size three, equivalently the circulant on fifteen vertices with connection set $\{1,4,6\}$, has likelihood $0.20128\ldots$ times that of $K_{7,8}$, and it is again triangle-free. We show that the failure is not sporadic by proving that the likelihood of the balanced complete bipartite graph is $2^{-(1/2-1/(8\ln 2)+o(1))n^2}$, whereas the minimum over all graphs of order $n$ is $2^{-(1/2+o(1))n^2}$, so the conjectured minimiser exceeds the minimum by a factor exponential in $n^2$. We also determine the Shannon entropy of the process to leading order, namely $n^2/(4\ln 2)$ bits, which shows that the conjectured minimiser is in fact more likely than a typical output of the process. The proofs rest on a vertex deletion recurrence which evaluates the likelihood in time $O(n\,2^n)$ and which closes on the blow-ups of any fixed base graph.
