# Systemic States of Spreading Activation in Describing Associative Knowledge Networks: From Key Items to Relative Entropy Based Comparisons

## Abstract

**:**

## 1. Introduction

## 2. Methods: Systemic States and Their Comparison

**A**and diagonal matrix

**D**for node degrees, and on basis of them, graph Laplacian

**L**describing diffusion-like spreading is introduced. Second, based on graph Laplacian

**L**, generalized systemic states ${\mathit{\rho}}_{q}$ are constructed. Third, and, finally, relative information theoretic entropy (generalized Jensen–Shannon–Tsallis divergence) is introduced to compare systemic states.

#### 2.1. Graph Laplacian and Diffusive Spreading

**L**, can now v described by discrete difference equation [20,21] (see also refs. [18,19])

#### 2.2. Systemic States and Activity

#### 2.3. Divergence for Comparisons

## 3. Results of Application: Associative Knowledge Network

#### 3.1. Networks and Key Items

#### 3.2. Activity of a Node in Spreading Activation

#### 3.3. Role of Sub-Networks: Divergences

## 4. Discussion

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Sub-networks (see legends) of complete associative knowledge network. The nodes corresponding to persons are shown in red, science in green and history in blue. The size of the node is in proportional to a number of links (degree) attached to it. In each of the six cases, the links forming the sub-networks are shown in bold black lines.

**Figure 2.**Activity ${\alpha}_{i}=\text{\u2212}\mathrm{log}{\left[{\mathit{\rho}}_{q}\right]}_{ii}$ of nodes i for systemic states ${\mathit{\rho}}_{q}$ for $q\to 1$ and $q=1.1,1.5,1.7$, and $1.9$ (from top to bottom) for 63 most important key items (nodes). With red tones are shown the high activity nodes, while blue indicates nodes of lower activity. The first 20 nodes fall in three categories, as follows: persons (1–6, 11, 12, 14); science (7,9,10, 13, 15, 16, 18); and, history (8, 17–20).

**Figure 3.**Divergences ${J}_{{q}^{*}}\left[\mathrm{X},\mathrm{Y}\right]$ where X and Y persons (P), science (S) or history (H). The results are shown for $q\text{*}=2-q$ for $q\to 1$ (

**a**), $q=1.10$ (

**b**), $q=1.30$ (

**c**), $q=1.50$ (

**d**), $q=1.70$ (

**e**), $q=1.80$ (

**f**), $q=1.90$ (

**g**), and $q=1.95$ (

**h**). Divergences are scaled to maximum value 1 of the largest divergence (P-S) and they are shown as a function of $\beta $ also scaled value of 1 corresponding to the maximum of the largest divergence (but, for $q<1.3$, corresponding to maximum of q = 1.3).

**Table 1.**Key quantities and their mathematical symbols used in definitions. A brief description is provided for each quantity and the range of variation of parameters are given.

Quantity and Symbol | Parameters | Description | |
---|---|---|---|

Adjacency matrix | A | - | Matrix of links (entries 0 or 1) between nodes |

Degree matrix | D | - | Matrix (diagonal) containing degrees of nodes |

Laplacian matrix | L | - | Matrix (diagonal) for diffusion (Laplacian) operator |

Systemic state | ${\mathit{\rho}}_{q}$ | $q\text{\u2208}]1,2]\text{,}\phantom{\rule{0.222222em}{0ex}}\beta >0$ | q-generalized systemic state of spreading activation |

Activity of node | ${\alpha}_{i}$ | Activity of a node in spreading activation | |

Tsallis-entropy | ${H}_{{q}^{*}}$ | ${q}^{*}=2-q\in [0,1[$ | Non-extensive entropy with index ${q}^{*}$ |

q-JST divergence | ${J}_{{q}^{*}}$ | ${q}^{*}=2-q\in [0,1[$ | q-generalized Jensen-Shannon-Tsallis divergence |

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**MDPI and ACS Style**

Koponen, I.T.
Systemic States of Spreading Activation in Describing Associative Knowledge Networks: From Key Items to Relative Entropy Based Comparisons. *Systems* **2021**, *9*, 1.
https://doi.org/10.3390/systems9010001

**AMA Style**

Koponen IT.
Systemic States of Spreading Activation in Describing Associative Knowledge Networks: From Key Items to Relative Entropy Based Comparisons. *Systems*. 2021; 9(1):1.
https://doi.org/10.3390/systems9010001

**Chicago/Turabian Style**

Koponen, Ismo T.
2021. "Systemic States of Spreading Activation in Describing Associative Knowledge Networks: From Key Items to Relative Entropy Based Comparisons" *Systems* 9, no. 1: 1.
https://doi.org/10.3390/systems9010001