二项随机图上的标签传播
Label propagation on binomial random graphs
AI总结：
该研究分析标签传播算法在Erdős-Rényi随机图上的收敛性，通过多阶段边暴露方法突破先前阈值，证明在$np \ge n^{5/8+\varepsilon}$时算法以单一标签终止，并刻画存活标签的性质。
摘要：
我们研究了标签传播算法（LPA）在Erdős-Rényi随机图 $\mathcal{G}(n,p)$ 上的行为。初始时，给定一个网络，每个顶点在区间 $[0,1]$ 内随机分配一个标签。然后，在LPA的每一轮中，每个顶点将其标签切换为其邻域（包括其自身标签）中的多数标签。在第一轮中，平局被打破为较小的标签，而在随后的每一轮中，平局被均匀随机打破。算法在所有标签在连续两次迭代中保持不变时终止。LPA在实践中成功用于检测网络中的社区（对应于算法终止后具有相同标签的顶点集合）。也许令人惊讶的是，LPA在稠密随机图上的性能难以分析，目前仅在 $np\ge n^{3/4+\varepsilon}$ 时已知收敛到共识，此时LPA在三轮内收敛。通过定义一种替代的标签归属过程，该过程在三轮后收敛到标签传播算法，对边进行仔细的多阶段暴露使我们能够突破 $n^{3/4+\varepsilon}$ 的障碍，并证明当 $np \ge n^{5/8+\varepsilon}$ 时，算法以单个标签终止（a.a.s.）。此外，我们证明，如果 $np\gg n^{2/3}$，则该标签是最小的（a.a.s.），而如果 $n^{5/8+\varepsilon}\le np\ll n^{2/3}$，则存活的标签不是最小的（a.a.s.）。顺便，我们证明了二项随机变量的一个可能新的单调性引理，该引理可能具有独立的意义。
英文摘要：
We study the behavior of a label propagation algorithm (LPA) on the Erdős-Rényi random graph $\mathcal{G}(n,p)$. Initially, given a network, each vertex starts with a random label in the interval $[0,1]$. Then, in each round of LPA, every vertex switches its label to the majority label in its neighborhood (including its own label). At the first round, ties are broken towards smaller labels, while at each of the next rounds, ties are broken uniformly at random. The algorithm terminates once all labels stay the same in two consecutive iterations. LPA is successfully used in practice for detecting communities in networks (corresponding to vertex sets with the same label after termination of the algorithm). Perhaps surprisingly, LPA's performance on dense random graphs is hard to analyze, and so far convergence to consensus was known only when $np\ge n^{3/4+\varepsilon}$, where LPA converges in three rounds. By defining an alternative label attribution procedure which converges to the label propagation algorithm after three rounds, a careful multi-stage exposure of the edges allows us to break the $n^{3/4+\varepsilon}$ barrier and show that, when $np \ge n^{5/8+\varepsilon}$, a.a.s.\ the algorithm terminates with a single label. Moreover, we show that, if $np\gg n^{2/3}$, a.a.s.\ this label is the smallest one, whereas if $n^{5/8+\varepsilon}\le np\ll n^{2/3}$, the surviving label is a.a.s.\ not the smallest one. En passant, we show a presumably new monotonicity lemma for Binomial random variables that might be of independent interest.
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