We study a variant of the widely popular, fast and often used family of community detection procedures referred to as label propagation algorithms. These mechanisms also exhibit many parallels with models of opinion exchange dynamics and consensus mechanisms in distributed computing. Initially, given a network, each vertex starts with a random label in the interval $[0,1]$. Then, in each round of the algorithm, 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. We investigate the performance of this algorithm on the binomial random graph $\mathcal{G}(n,p)$. We show that for $np \ge n^{5/8+\varepsilon}$, the algorithm terminates with a single label a.a.s. (which was previously known only for $np\ge n^{3/4+\varepsilon}$). 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.

翻译：我们研究了一种广泛流行、快速且常用的社区检测算法变体，即标签传播算法系列。这些机制与分布式计算中的意见交换动态模型和共识机制也具有诸多相似之处。初始时，给定一个网络，每个顶点被随机赋予区间 $[0,1]$ 内的一个标签。随后，在算法的每一轮中，每个顶点将其标签切换为其邻域（包括自身标签）中的多数标签。在第一轮中，平局倾向于选择较小的标签，而在后续每一轮中，平局则随机均匀打破。我们研究了该算法在二项随机图 $\mathcal{G}(n,p)$ 上的性能。我们证明，当 $np \ge n^{5/8+\varepsilon}$ 时，算法几乎必然收敛于一个单一标签（此前仅已知 $np\ge n^{3/4+\varepsilon}$ 的情形）。此外，我们证明：若 $np\gg n^{2/3}$，该标签几乎必然是最小值；而若 $n^{5/8+\varepsilon}\le np\ll n^{2/3}$，则存活的标签几乎必然不是最小值。