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Triggered urn models for frequently asked questions (FAQ)

Irene Crimaldi, Andrea Ghiglietti, Leen Hatem, Hosam Mahmoud

TL;DR

The paper introduces a generalized triggered urn model for frequently asked questions (FAQ) where triggering events add new colors and non-trigger steps reinforce existing colors via a flexible update function $F$. It provides a comprehensive asymptotic analysis, including almost-sure growth regimes for the number of colors, stationary color-frequency distributions, and frequency-rank (Zipf-type) laws, with connections to central limit theorems and Poisson approximations. A detailed combinatorial treatment of the Simon FAQ urn yields exact distributions and recursions for key quantities, while dynamic variations with time-dependent trigger probabilities are explored. Real-data validation on Amazon Q&A datasets demonstrates that the model captures observed linear-logarithmic scaling and rank-frequency patterns, underpinning practical short-term predictions for FAQ systems. The results illuminate fundamental links between growth mechanisms, Zipf-type laws, and Heap's exponent in evolving novelty-driven urns with broad applicability to web services and information retrieval.

Abstract

We investigate a nonclassic urn model with triggers that increase the number of colors. The scheme has emerged as a model for web services that set up frequently asked questions (FAQ). We present a thorough asymptotic analysis of the FAQ urn scheme in generality that covers a large number of special cases, such as Simon urn. For instance, we consider time dependent triggering probabilities. We identify regularity conditions on these probabilities that classify the schemes into those where the number of colors in the urn remains almost surely finite or increases to infinity and conditions that tell us whether all the existing colors are observed infinitely often or not. We determine the rank curve, too. In view of the broad generality of the trigger probabilities, a spectrum of limit distributions appears, from central limit theorems to Poisson approximation, to power-laws, revealing connections to Heap's exponent and Zipf's law. A combinatorial approach to the Simon urn is presented to indicate the possibility of such exact analysis, which is important for short-term predictions. Extensive simulations on real datasets (from Amazon sales) as well as computer-generated data clearly indicate that the asymptotic and exact theory developed agrees with practice.

Triggered urn models for frequently asked questions (FAQ)

TL;DR

The paper introduces a generalized triggered urn model for frequently asked questions (FAQ) where triggering events add new colors and non-trigger steps reinforce existing colors via a flexible update function . It provides a comprehensive asymptotic analysis, including almost-sure growth regimes for the number of colors, stationary color-frequency distributions, and frequency-rank (Zipf-type) laws, with connections to central limit theorems and Poisson approximations. A detailed combinatorial treatment of the Simon FAQ urn yields exact distributions and recursions for key quantities, while dynamic variations with time-dependent trigger probabilities are explored. Real-data validation on Amazon Q&A datasets demonstrates that the model captures observed linear-logarithmic scaling and rank-frequency patterns, underpinning practical short-term predictions for FAQ systems. The results illuminate fundamental links between growth mechanisms, Zipf-type laws, and Heap's exponent in evolving novelty-driven urns with broad applicability to web services and information retrieval.

Abstract

We investigate a nonclassic urn model with triggers that increase the number of colors. The scheme has emerged as a model for web services that set up frequently asked questions (FAQ). We present a thorough asymptotic analysis of the FAQ urn scheme in generality that covers a large number of special cases, such as Simon urn. For instance, we consider time dependent triggering probabilities. We identify regularity conditions on these probabilities that classify the schemes into those where the number of colors in the urn remains almost surely finite or increases to infinity and conditions that tell us whether all the existing colors are observed infinitely often or not. We determine the rank curve, too. In view of the broad generality of the trigger probabilities, a spectrum of limit distributions appears, from central limit theorems to Poisson approximation, to power-laws, revealing connections to Heap's exponent and Zipf's law. A combinatorial approach to the Simon urn is presented to indicate the possibility of such exact analysis, which is important for short-term predictions. Extensive simulations on real datasets (from Amazon sales) as well as computer-generated data clearly indicate that the asymptotic and exact theory developed agrees with practice.
Paper Structure (19 sections, 8 theorems, 82 equations, 6 figures, 2 tables)

This paper contains 19 sections, 8 theorems, 82 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A database for inquiries
  3. The FAQ triggered urn model
  4. Rationale underlying the FAQ urn
  5. An illustrative example
  6. Analysis of the FAQ triggered urn scheme
  7. Asymptotic approximation of the distribution of $C_n$
  8. The stationary color-frequency distribution
  9. The frequency-rank curve
  10. Simon FAQ urn
  11. The number of colors
  12. Recursion of the average number of balls of a specific color
  13. Dynamic variations on Simon urn
  14. Applications to real data
  15. Data description and preprocessing
  16. ...and 4 more sections

Key Result

Theorem 3.1

(Asymptotic behavior of $C_n$) We have one the following two cases:

Figures (6)

  • Figure 1: Automotive dataset: overview of the results.
  • Figure 2: Cell Phones dataset: overview of the results.
  • Figure 3: Electronics dataset: overview of the results.
  • Figure 4: Example 3.1 with $F(x)=x$ and $p_n=0.3$: overview of the simulation results.
  • Figure 5: Example 3.1 with $F(x)=x+1$ and $p_n=0.3$: overview of the simulation results.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Remark 3.1
  • Theorem 3.4
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • ...and 7 more