A simple model of knowledge percolation

TL;DR

This work develops a simple bipartite model of knowledge percolation where each contribution spans $N$ items drawn from a universe of $L$ items and forms or fuses knowledge clusters when overlaps exceed a threshold $\Omega$, as quantified by $\omega_{ij}$. The authors derive a tractable description of corpus growth, yielding the analytic result $\overline{S}=L(1-e^{-Nt/L})$ (in the regime $L\gg N$) and show that $\Omega$ has limited influence on the total knowledge size while larger $N$ accelerates growth. They also analyze cluster dynamics, identifying three phases of cluster formation, growth, and merging, with the new-cluster rate $a_n(t)$ decaying roughly as $\exp(-g(N)t/L)$ where $g(N)\approx 3N-4$, and demonstrate a data-collapse scaling with $Q=(L/F)^\Omega$ and $X=N/(1+\Omega/4)$. Overall, the study reveals nontrivial time-dependent behavior in both knowledge accumulation and clustering, offering a conceptual and semi-quantitative framework for understanding percolation-like knowledge organization in collaborative or citation-style systems.

Abstract

We investigate how knowledge percolates and clusters in a given knowledge space. We introduce a simple model of knowledge organization in which each contribution spans a certain number of items. If this contribution overlaps with others above a certain threshold, they form a cluster. A contribution can also merge clusters together. We study the growth of global knowledge and the cluster dynamics, both showing a nontrivial behavior.

Figures (11)

• Figure 1: Schematic representation of the system in the case of a knowledge space with $L=26$ items, $N=3$ and $\Omega=2$ divided into two clusters $C_1=\{k_1, k_2\}$ of size $c_1=2$ and $C_2=\{k_3\}$ of size $c_2=1$.
• Figure 2: Knowledge size $S$ vs. time $t$ for $L=2000$, $N=8$, $\Omega=3$ (crosses) confronted with equation \ref{['eq:overlap']} (continuous line) with the same values of $N$.
• Figure 3: Knowledge $S$ vs. time $t$ with $L=500$ and $N=31$ for several values of $\Omega$.
• Figure 4: Knowledge $S$ vs. time $t$ with $L=6000$ and $\Omega=1$ for several values of $N$.
• Figure 5: Number of clusters $A$ vs. time $t$ for $L=600$, $N=2$ and $\Omega=1$.