Zipf's law and criticality in multivariate data without fine-tuning.
Zipf's law and criticality in multivariate data without fine-tuning.
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DOI：
10.1103/physrevlett.113.068102
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发表时间：
2014-08-08
影响因子：
8.6
通讯作者：
Mehta P
中科院分区：
文献类型：
--
作者：
Schwab DJ;Nemenman I;Mehta P
生物系统中许多自由度的状态的联合概率分布，例如神经网络或抗体序列组合物中的发射模式，通常遵循齐普夫定律，其中在秩-频率图上观察到幂律。这种行为已经被证明意味着这些系统位于唯一的临界点附近，在该临界点处熵和能量的广泛部分完全相等。在这里，我们通过分析和数值模拟表明，如果有一个波动的未观察到的变量（或变量）影响系统，例如一个共同的输入刺激，导致单个神经元以随时间变化的速率发射，则Zipf概率分布会自然出现。在统计学和机器学习中，这些被称为潜在变量或混合模型。我们表明，齐普夫定律一般出现在大型系统，没有微调参数到一个点。我们的工作使人们深入了解齐普夫定律在广泛的系统中的普遍性。
The joint probability distribution of states of many degrees of freedom in biological systems, such as firing patterns in neural networks or antibody sequence compositions, often follows Zipf’s law, where a power law is observed on a rank-frequency plot. This behavior has been shown to imply that these systems reside near a unique critical point where the extensive parts of the entropy and energy are exactly equal. Here, we show analytically, and via numerical simulations, that Zipf-like probability distributions arise naturally if there is a fluctuating unobserved variable (or variables) that affects the system, such as a common input stimulus that causes individual neurons to fire at time-varying rates. In statistics and machine learning, these are called latent-variable or mixture models. We show that Zipf’s law arises generically for large systems, without fine-tuning parameters to a point. Our work gives insight into the ubiquity of Zipf’s law in a wide range of systems.