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Diffusion of tacit knowledge in a company: a mathematical model based on diffusion on graphs

T0 review · 3 major / 0 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Questionnaire scores become a directed knowledge graph whose Jordan blocks tell managers which employees can be scheduled together so tacit knowledge equalizes.

desk verdict Clean questionnaire-to-Jordan scheduling heuristic for tacit-knowledge sessions; the heat-diffusion equalizing claim is only sketched and unproven. read the letter →

arxiv 2607.10919 v1 pith:GI33YBCO submitted 2026-07-12 econ.TH

Rados{l}aw A. Kycia , Agnieszka Niemczynowicz , Andrzej Buszko This is my paper · ORCID
classification econ.TH
keywords tacitknowledgemanagementgraphmodeldiffusionofdynamicsonJordandecompositiontransferschedulingeigenvectorcentrality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that tacit knowledge can be treated like heat: self-reported excess or shortage of expertise, multiplied by willingness to share, yields directed gradients between employees. Those gradients define a Tacit Knowledge Transfer Graph. The adjacency matrix of the graph, once put into Jordan form, decomposes into invariant subspaces; the employees appearing in each subspace form a natural learning group that should be scheduled in the same time window. The authors prove that following these groups equalizes knowledge deficiencies inside each group, and they show that the same construction also ranks employees by demand (via centrality or PageRank) and works for school timetable planning. A sympathetic reader cares because the method turns ordinary survey data into concrete, optimizable schedules and risk maps without requiring full NP-hard timetable solvers.

What carries the argument

The Tacit Knowledge Transfer Graph (TKTG) and its adjacency matrix of knowledge gradients (Definition 1); Algorithm 1 extracts the employee components of each Jordan block and traces the action of the matrix to obtain closed learning groups.

What would settle it

Collect the a and b questionnaires in a real work team, form the TKTG, schedule the Jordan-derived groups, re-survey knowledge levels after a fixed period, and test whether measured deficiencies inside those groups have equalized relative to a control set of randomly scheduled sessions.

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Extended reading notes

Core claim

A Tacit Knowledge Transfer Graph whose edge weights are the knowledge gradients ∇aij built from questionnaire excess/shortage and propensity scores, when analyzed by Jordan decomposition of its adjacency matrix, produces invariant employee groups that can be scheduled together for knowledge-transfer sessions and that equalize knowledge deficiencies.

Load-bearing premise

That the product of self-reported excess or shortage and propensity to share, after one of four simple sign adjustments, really behaves like a heat-style gradient that drives equalizing diffusion of knowledge.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 0 minor

Summary. The paper proposes a Tacit Knowledge Transfer Graph (TKTG) whose directed edge weights are knowledge gradients ∇aij constructed from questionnaire measures of excess/shortage (aij) and propensity to share (bij). Four sign-function variants of the gradient (Definition 1) and a Minimal Case are introduced. The adjacency matrix M of the TKTG is analyzed by Jordan decomposition; the resulting (generalized) eigenvectors identify invariant employee groups that Algorithm 1 schedules for knowledge-transfer sessions in the same period. Eigenvector centrality and PageRank are offered as scores of knowledge demand. Appendix C sketches a Knowledge Diffusion Equation by analogy with Fourier’s law, inserts an ad-hoc non-negativity correction because knowledge is non-conservative, and states Theorem C.1 that the scheduling procedure equalizes knowledge deficiencies. Small analytic examples, a Monte-Carlo study of eigenspace size versus sparsity, and a school-timetabling illustration are supplied. Empirical validation is explicitly deferred.

Significance. If the construction is sound, the paper supplies a concrete, questionnaire-driven linear-algebra pipeline that converts self-reported knowledge imbalances into schedulable learning groups and demand rankings—something the knowledge-management literature largely lacks. The Jordan-block heuristic is a potentially useful preprocessing step for the NP-hard University Course Timetable Problem, and the Monte-Carlo observation that sparse demand matrices yield smaller average groups is a falsifiable organizational prediction. The linear-algebra apparatus itself (adjacency matrix, Jordan form, centrality, PageRank) is correctly stated and immediately implementable. The main limitation is that the equalizing claim (Theorem C.1) that justifies the scheduling heuristic remains an informal analogy rather than a proved discrete-diffusion result.

major comments (3)
  1. Appendix C / Theorem C.1: the equalizing property that underwrites Algorithm 1 is only sketched. After writing the continuity equation the authors replace the right-hand side by its positive part because knowledge is non-conservative, then assert that the resulting operator on the finite directed graph smooths deficiencies. No discrete Green identity, maximum principle, or spectral argument is given showing that the patched operator actually drives the deficiency vector toward uniformity. Either supply a rigorous discrete-calculus proof for the directed graph, or restate Theorem C.1 as a conjecture and present Algorithm 1 purely as a scheduling heuristic whose equalization properties remain to be verified.
  2. Section 4.2, Definition 1: the four sign-function modifications of the gradient are introduced as policy choices, yet no criterion is offered for selecting among them, nor is any sensitivity analysis performed. Because the subsequent Jordan groups and centrality scores depend on which case is chosen, the manuscript should either (i) recommend a default case with empirical or theoretical justification or (ii) demonstrate that the qualitative scheduling recommendations are robust across the four cases on the same questionnaire data.
  3. Section 5 examples and Algorithm 1: the claim that Jordan-invariant subspaces yield “optimal” same-day learning groups is illustrated only on tiny or randomly generated matrices; no comparison with any standard UCTP heuristic or with a simple greedy matching baseline is provided. Without such a benchmark it is impossible to assess whether the extra computational cost of Jordan form (O(n^{3})) improves scheduling quality. A minimal computational experiment on a realistic-sized instance would strengthen the central practical claim.

Circularity Check

0 steps flagged · score 1.0 of 10

Definitional graph construction from questionnaires yields Jordan groups by linear algebra; no fitted-as-prediction or self-referential forcing of the equalization claim.

full rationale

The paper builds the TKTG adjacency matrix M directly from questionnaire-derived aij and bij via the four explicit gradient formulas of Definition 1 (or the Minimal Case), then extracts invariant employee sets from the Jordan form of M by Algorithm 1. These steps are ordinary linear-algebra consequences of the chosen matrix; they do not re-label a fit as a prediction, invoke a uniqueness theorem from the authors’ prior work, or smuggle an ansatz via self-citation. Theorem C.1’s equalization claim is only an informal sketch that imports the smoothing property of the heat equation after an ad-hoc positive-part correction for non-conservativeness; that sketch is weak but not circular—the claimed dynamics are not already assumed in the definition of the gradient. No load-bearing self-citations appear; the single author-overlapping reference is used only for a peripheral remark on Generation Z. The model is therefore self-contained and definitional in the ordinary modeling sense, not circular under the listed patterns. Score 1 reflects only the mild definitional character of any pure construction paper.

Assumptions & free parameters 4 free parameters · 4 assumptions · 3 invented entities

The central claim rests on a modeling analogy (knowledge behaves like heat) plus several free modeling choices for how questionnaire answers become edge weights. No external empirical constants are fitted; the free parameters are the survey-derived aij, bij and the four discrete cases of the gradient. Invented entities are the TKTG itself and the latent continuum quantities introduced only in the appendix.

free parameters (4)
  • aij(1), aij(2)
    Self-reported excess/shortage scores normalized to [-1,1]; free because they are obtained from Likert items whose wording and rescaling are chosen by the modeler.
  • bij(1), bij(2)
    Self-reported propensity-to-share scores normalized to [0,1]; free survey parameters.
  • gradient case (I–IV or Minimal)
    Four mutually exclusive sign-function modifications of the basic product a·b; the choice is a free modeling decision that changes which edges appear.
  • αij, βij, κij, τij
    Latent continuum coefficients introduced in Appendix C to complete the heat-equation analogy; never measured and left for future work.
assumptions (4)
  • ad hoc to paper Knowledge imbalance between two employees can be represented by a scalar gradient that drives directed transfer exactly as a temperature gradient drives heat flux (Fourier’s law).
    Stated as the motivating analogy in §4.2 and used to define ∇aij; no independent derivation is offered.
  • ad hoc to paper Knowledge is a non-conservative ‘liquid’ whose reservoir never decreases during transfer; the continuity equation is therefore replaced by its positive part.
    Introduced in Appendix C to avoid the unphysical implication that a teacher loses knowledge; required for Theorem C.1.
  • domain assumption Jordan blocks of the (generally non-symmetric) adjacency matrix identify closed sets of employees that should be scheduled together.
    Standard linear-algebra fact about invariant subspaces, applied without further justification to the knowledge-transfer interpretation.
  • domain assumption Questionnaire answers on a Likert scale, after linear rescaling, faithfully quantify both knowledge excess/shortage and interpersonal propensity.
    Assumed throughout §4.1; social-desirability and Dunning–Kruger biases are noted but not corrected for in the model.
invented entities (3)
  • Tacit Knowledge Transfer Graph (TKTG)
    purpose: Directed weighted graph whose edges encode knowledge gradients and whose Jordan structure yields learning cohorts.
    Central new construct of the paper; defined from the a/b questionnaire matrix.
  • knowledge flux q⃗ij and knowledge reservoir Qij
    purpose: Latent continuum quantities that let the authors write a discrete heat equation on the graph.
    Introduced only in Appendix C; never observed or measured.
  • knowledge conductivity κij, absorptive capacity αij, quality βij, diffusion coefficient τij
    purpose: Material coefficients that complete the formal analogy with Fourier’s law.
    Postulated in Appendix C; left for future empirical determination.

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Pith. "Pith review of Diffusion of tacit knowledge in a company: a mathematical model based on diffusion on graphs." pith.science (2026). https://pith.science/paper/GI33YBCO

@misc{pith2026260710919,
  author       = {Pith},
  title        = {Pith review of: Diffusion of tacit knowledge in a company: a mathematical model based on diffusion on graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GI33YBCO}},
  note         = {Machine review of arXiv:2607.10919}
}
read the original abstract

This article is devoted to the process of diffusing tacit knowledge. This intangible asset has proven crucial for achieving a competitive advantage among market-oriented companies. A novel model of tacit knowledge diffusion is presented, employing the concept of heat diffusion from physics. Furthermore, graph theory and the dynamics it defines are utilized. We defined a Tacit Knowledge Transfer Graph that encodes data from questionnaires. It enables us to identify employees' learning needs, allowing for the planning of classes within the same period, such as a day, to schedule them optimally. Moreover, the model can be used to identify employees with high knowledge demands. The application is not limited to companies; a simple example of planning classes in a school/university is provided. The presented model can help with optimal scheduling, enhance operational efficiency, and improve talent management within the company. It can also identify risks associated with critical sources of knowledge and help improve organizational culture and knowledge management policy.

Figures

Figures reproduced from arXiv: 2607.10919 by the authors.

Figure 2
Figure 2. The knowledge transfer relation between employees i and j. The construction is a part of a directed graph called a Tacit Knowledge Transfer Graph (TKTG) that is presented in [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. An example of the tacit knowledge transfer graph. It represents a situation of two domains (d1 and d2) with the employees {p1, p2} in d1, and {p3, p4} in d2. When gradient vanished the arrows are omitted for clarity. TKTG in [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Knowledge transfer graph. Diagonalizing the adjacency matrix M one gets the following Jordan form J= [ 0 0 0 1 0 0 0 1 0] , with the following (cyclic) eigenvectors 29 [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figures from the paper (5 more)
Figure 3
Figure 3. Figure 3: The groups consists of a single employee, which is reflected in the zero eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p030_3.png]
Figure 5
Figure 5. Figure 5: Randomly generated adjacency matrix [PITH_FULL_IMAGE:figures/full_fig_p033_5.png]
Figure 6
Figure 6. Figure 6: Adjacency matrix – graphical representation. As a result of Jordan decomposition we obtain 10 eigenvalues defining one-dimensional eigenspaces. Analyzing dynamics of adjacency matrix one obtains the same 9 result not 33 [PITH_FULL_IMAGE:figures/full_fig_p033_6.png]
Figure 7
Figure 7. Figure 7: Monte Carlo algorithm for calculating average groups depending on the percentage number of zeros [figure generated using Gemini AI based on the code in Wolfram Mathematica/Wolfram Language]. We selected adjacency matrix dimensions n=10, 20, 30, 40. The results are pres…
Figure 8
Figure 8. Figure 8: The averaged size of set of unique base vectors for each eigenspace (OY axis) with respect to the zeroProbablity (OX axis). Each point represents 100 Monte Carlo runs. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_8.png]

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