# Systemic States of Spreading Activation in Describing Associative Knowledge Networks II: Generalisations with Fractional Graph Laplacians and q-Adjacency Kernels

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Associative Knowledge Network to Be Explored

`Newton`–

`Gravitation`) between persons, ideas, inventions, and events during the three centuries. In finding the items to match in pairs, students used Wikipedia as a source most often [18]. The set of data in the form of connected (dyadic) pairs provided by 25 students was then agglomerated to obtain an agglomerated AKN consisting of all different connections to be found in data sets from 25 individual students. The agglomerated AKN turned out to have a rich community structure, organized around temporal periods, and, structurally, it was found to be a heavy-tailed network [17,18].

#### 2.2. Diffusion Models and Generalised Systemic States

#### 2.3. Activity Centrality Based on Systemic States

## 3. Results

#### 3.1. An Example: An Agglomerated Associative Knowledge Network

`Boyle`and

`Bernoulli`(italicized in Table 2) are exceptions and not found among the top ranking 33 items in all models. Consequently, the top rankings are not model dependent.

#### 3.2. Changes in Rankings

`Newton`with d = 64 to rank 40 with d = 16 of item

`Scientific method`. The next band of items from 147 up to 240 has a more pronounced change in rankings, occurring in clear bands. With lower rankings and lower degrees, changes become more common, and nearly all items eventually change rankings. The largest changes of rankings in the band from 241 up to 597 are about ±150, meaning quite large differences in rankings for different models. In all cases, however, the degree of nodes constrains the changes in rankings, and, generally (although not exactly), changes in rankings occur within bands of constant degrees and only a few border crossings between different bands occur.

## 4. Discussion

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The agglomerated AKN representing students’ associative knowledge of the history of science. Only the most central part of the network is shown for better visibility. The size of the nodes is proportional to the degree d (i.e., number of links attached) of the node. The abbreviations are explained in Table 2.

**Figure 2.**Scatter plots of return and escape probabilities ${\pi}_{R}$ and ${\pi}_{E}$ (scaled in range $[0,1]$ with increasing values of $\tau $ scaled to maximum ${\tau}_{\mathrm{max}}$, with $\tau /{\tau}_{\mathrm{max}}$ =0.1 (

**a**), 0.3 (

**b**), 0.6 (

**c**) and 1.0 (

**d**). The cases shown are for a baseline model with $\alpha =1$ and $\gamma =1$. Stabilization of spreading activation is obtained when correlation between escape and return probabilities becomes high and results collapse onto a single regular curve (Figure 2d).

**Figure 3.**Kendall rank correlation coefficients between vectors $\overline{K}=({K}_{1},{K}_{2},\dots ,{K}_{N})$ and $\overline{K}\prime =({K}_{1}^{\prime},{K}_{2}^{\prime},\cdots ,{K}_{N}^{\prime})$, where $K\prime $ refers to values for each instant of $\tau $ with parameters $(\alpha ,\gamma )$ and K to values at the final stationary state at ${\tau}_{\mathrm{max}}$ of reference state with $(\alpha ,\gamma )=(1,1)$. Results are shown for $\alpha =$ 1.0 (

**a**), 0.75 (

**b**), 0.50 (

**c**) and 0.20 (

**d**). Different parameter combinations are indicated in figure legends. In all cases, p-values for correlations are below ${10}^{-6}$.

**Figure 4.**Fingerprint maps of changes in rankings (based on activity centrality K) of items 1–330 for different models corresponding parameters: (

**a**) ($\alpha =1,\gamma =1$); (

**b**) ($\alpha =0.5,\gamma =1$); (

**c**) ($\alpha =1,\gamma =1.75$); and (

**d**) ($\alpha =0.2,\gamma =1.75$). The decrease in rankings is in blue and the increase in red, while white regions correspond to changes in rankings, which are low in comparison to reference state (final state of model A). The bar legend explains the colour coding.

**Figure 5.**Fingerprint maps of changes in rankings (based on activity centrality K) of all items 1–597 for a range of values $\alpha $=1.0, 0.75, 0.50, 0.25, 0.20 and 0.05 and $\gamma $ = 1.0 (

**a**); 1.25 (

**b**); 1.50 (

**c**); 1.75 (

**d**) and 2.00 (

**e**). The decrease in rankings is in blue and the increase in red, while regions in light colours correspond to changes in rankings, which are low in comparison to the reference state (final state of model A). The bar legends at right explain the colour coding. Changes in rankings are normalized to a maximum of 150.

**Table 1.**A schematic representation of the generalized model. In a network, where nodes are connected through adjacency matrix $\mathbf{A}$, direct jumps are restricted to sites connected by links ${\left[\mathbf{A}\right]}_{ij}\ne 0$ (solid lines). Long jumps occur between distant nodes, when the fractional graph Laplacian (F.g. Laplacian) has a nonzero element between nodes that are not directly connected (${\left[\mathbf{A}\right]}_{ij}=0$), but connected through intermediate steps (two examples shown by a dashed line). Then, the systemic state also has non-zero elements between the nodes and allowing spreading activation between the nodes. The function ${e}_{q}\left(x\right)$ is the generalised q-exponential.

Scheme of Long Jumps | Description | Symbol |
---|---|---|

Adjacency matrix | $\mathbf{A}$ | |

Elements of $\mathbf{A}$ | ${\left[\mathbf{A}\right]}_{ij}={a}_{ij}$ | |

Degree matrix | $\mathbf{D}$ | |

Graph Laplacian $\mathbf{L}$ | $\mathbf{L}=\mathbf{D}-\mathbf{A}$ | |

F. g. Laplacian ${\mathbf{\mathcal{L}}}^{\left(\alpha \right)}$ | ${\mathbf{D}}^{-\frac{\alpha}{2}}{\mathbf{L}}^{\alpha}{\mathbf{D}}^{-\frac{\alpha}{2}}$ | |

Elements of ${\mathbf{\mathcal{L}}}^{\left(\alpha \right)}$ | ${\left[{\mathbf{\mathcal{L}}}^{\left(\alpha \right)}\right]}_{ij}$ | |

Systemic state ${\mathit{\rho}}_{\alpha \gamma}$ | ${Z}_{\alpha \gamma}^{-1}\phantom{\rule{0.277778em}{0ex}}{\mathrm{e}}_{\gamma}^{-\tau {\mathbf{\mathcal{L}}}^{\left(\alpha \right)}}$ | |

Elements of ${\mathit{\rho}}_{\alpha \gamma}$ | ${\left[{\mathit{\rho}}_{\alpha \gamma}\right]}_{ij}$ |

**Table 2.**Rankings of the highest ranking 33 items and their degree d. Ranking is based activity centrality K in the case of normal diffusive spreading corresponding to parameters $\alpha =1$ and $\gamma =1$. Acronyms (Ac) refer to Figure 1. The items that have the same rankings in all models with different parameters $\alpha $ and $\gamma $ are in boldface.

Item | d | Ac | Item | d | Ac | Item | d | Ac |
---|---|---|---|---|---|---|---|---|

1. Newton | 64 | Nw | 12. Bernoulli | 23 | Be | 23. Optics | 20 | op |

2. Galilei | 44 | Ga | 13. Faraday | 22 | Fa | 24. Bacon | 20 | Ba |

3. Huygens | 41 | Hu | 14. Electrodynamics | 22 | EM | 25. Scientific revol. | 20 | sR |

4.Hooke | 36 | Hk | 15. Industrial revol. | 22 | IR | 26. Brahe | 19 | Br |

5. Leibniz | 32 | Lb | 16. Kepler | 22 | Kp | 27. Planet motion | 19 | PM |

6. Descartes | 31 | De | 17. Gravitation law | 22 | gl | 28. Euler | 19 | Eu |

7. Gravitation | 30 | Gg | 18. Steam engine | 22 | st | 29. Electric current | 18 | EC |

8. Enlightenment | 29 | EN | 19. Royal Society | 22 | RS | 30. Electricity | 18 | ED |

9. Mechanics | 27 | Me | 20. French revol. | 21 | FR | 31. Thermodynamics | 18 | TD |

10. Empiricism | 23 | em | 21. French Academy | 21 | FA | 32. Reformation | 17 | RF |

11. Boyle | 23 | Bo | 22. Heliocentricity | 20 | H | 33. Locke | 17 | Lo |

**Table 3.**Summary of 13 highest ranking items based on activity centrality K calculated for reference case with parameters $\alpha =1$ and $\gamma =1$. The 13 items shown are common in sets 25 of the highest ranking in a given band I–V of items and with all parameters in $\alpha \in [0,1]$ and $\gamma \in [1,2]$. The total number of shared items out of the 25 highest ranking items are given in parenthesis in form (

**shared**/all). Numbering of items is based on the reference case. The items that have the same rankings in all models with different parameters $\alpha $ and $\gamma $ are in boldface. Abbreviations refer to Figure 1.

Key Items in Item Bands I–V | ||||
---|---|---|---|---|

I 33–46 (13/13) | II 47–60 (13/13) | III 61–100 (20/25) | IV 101–140 (18/25) | V 141–200 (16/25) |

Locke (Lo) | Thermometer | Beeckman | Finnish War | Thought experim. |

Halley (Ha) | Volta | Wave thr. light | Electric potential | Chatelet |

Watt (Wa) | Earth magn field | Atmsph. pressure | Kirchhoff | Medical science |

Cath. church (CC) | Stevin | Telescope | Hydrost. pressure | Gauss |

Pressure (pr) | Ludvig XIV | Fire of London | Gay-Lussac law | Mozart |

Laplace (LP) | Refraction law | Oersted | Spinoza | Electric charge |

Lagrange (Lg) | Theory of light | d’Alambert | Speed of light | Electric motor |

Scientific method | Pendulum | Liberalism | Magnetic field | Carnot’s engine |

Napoleon | Mathematics | Pascal | Pendulum clock | Locomotive |

Gilbert | Newton’s laws | Differential calc. | Maupertuis | Year 1848 |

Kant (Ka) | Gravity | Hobbes | Cavendish | Swedish empire |

30 Y. War (TW) | Copernicus | Rationalism | Spectrum of light | Carnot’s process |

Magnetism | Wren | Marx | Elizabeth I | Galvani |

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**MDPI and ACS Style**

Koponen, I.T.
Systemic States of Spreading Activation in Describing Associative Knowledge Networks II: Generalisations with Fractional Graph Laplacians and q-Adjacency Kernels. *Systems* **2021**, *9*, 22.
https://doi.org/10.3390/systems9020022

**AMA Style**

Koponen IT.
Systemic States of Spreading Activation in Describing Associative Knowledge Networks II: Generalisations with Fractional Graph Laplacians and q-Adjacency Kernels. *Systems*. 2021; 9(2):22.
https://doi.org/10.3390/systems9020022

**Chicago/Turabian Style**

Koponen, Ismo T.
2021. "Systemic States of Spreading Activation in Describing Associative Knowledge Networks II: Generalisations with Fractional Graph Laplacians and q-Adjacency Kernels" *Systems* 9, no. 2: 22.
https://doi.org/10.3390/systems9020022