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<p>There exist numerous conflicting heuristic models to explain Zipf’s law, which is an empirical inverse frequency power law that is ubiquitous in biological, social, and other natural phenomena characterized by complex system dynamics. Given its applicability in diverse systems, it should be possible to derive it as a model-free signature of self-organized criticality. This is what we do in this present paper by proposing a new law <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f\left( x \right) \propto x^{ - e/3}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mfenced close=")" open="("> <mi>x</mi> </mfenced> <mo>∝</mo> <msup> <mi>x</mi> <mrow> <mo>-</mo> <mi>e</mi> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, which is based on information-theoretic considerations, and it is nearly identical to Zipf’s law. The new law provides a basis for universal criticality as an indication of information optimality, and it also provides a specific mathematical function for Zipf’s law, which is normally stated only as an inverse correlation.</p>

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An information-theoretic optimal power law

  • 期刊论文

  • 发表日期：

    2026年4月1日

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  • Subhash Kak

摘要

There exist numerous conflicting heuristic models to explain Zipf’s law, which is an empirical inverse frequency power law that is ubiquitous in biological, social, and other natural phenomena characterized by complex system dynamics. Given its applicability in diverse systems, it should be possible to derive it as a model-free signature of self-organized criticality. This is what we do in this present paper by proposing a new law \(f\left( x \right) \propto x^{ - e/3}\) f x ∝ x - e / 3 , which is based on information-theoretic considerations, and it is nearly identical to Zipf’s law. The new law provides a basis for universal criticality as an indication of information optimality, and it also provides a specific mathematical function for Zipf’s law, which is normally stated only as an inverse correlation.