# Spreading of Information on a Network: A Quantum View

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Model and Its Dynamics

## 3. The Network Model

- the inertia parameters associated with Agent 1 are equal (a form of neutrality of the transmitter); Agent 2 (Agent 5, respectively) has the higher value of the inertia parameter for fake news (for good news, respectively); for the remaining agents, the inertia parameters associated with fake news are smaller than those associated with good news;
- the interaction parameters responsible for the conversion of good news into fake news and vice versa are all equal, except for Agents 2 and 5, where this interaction is absent;
- the coefficients related to the diffusion of good and fake news between the transmitter and the agents in the intermediate layer are all equal, except the coefficients involving Agents 2 and 5.

**Rule 1:**- $$\begin{array}{cc}& {\omega}_{\alpha}^{\left(f\right)}\to {\omega}_{\alpha}^{\left(f\right)}(1+{\kappa}_{\alpha}{\delta}_{\alpha}^{\left(f\right)}),\hfill \\ & {\omega}_{\alpha}^{\left(g\right)}\to {\omega}_{\alpha}^{\left(g\right)}(1+\frac{{\kappa}_{\alpha}}{2}{\delta}_{\alpha}^{\left(g\right)});\hfill \end{array}$$
**Rule 2:**- $$\begin{array}{cc}& {\omega}_{\alpha}^{\left(f\right)}\to {\omega}_{\alpha}^{\left(f\right)}(1+\frac{{\kappa}_{\alpha}}{2}{\delta}_{\alpha}^{\left(f\right)}),\hfill \\ & {\omega}_{\alpha}^{\left(g\right)}\to {\omega}_{\alpha}^{\left(g\right)}(1+{\kappa}_{\alpha}{\delta}_{\alpha}^{\left(g\right)});\hfill \end{array}$$
**Rule 3:**- $$\begin{array}{cc}& {\omega}_{\alpha}^{\left(f\right)}\to {\omega}_{\alpha}^{\left(f\right)}(1+{\kappa}_{\alpha}{\delta}_{\alpha}^{\left(f\right)}),\hfill \\ & {\omega}_{\alpha}^{\left(g\right)}\to {\omega}_{\alpha}^{\left(g\right)}(1+\frac{{\kappa}_{\alpha}}{2}{\left({\delta}_{\alpha}^{\left(g\right)}\right)}^{2});\hfill \end{array}$$
**Rule 4:**- $$\begin{array}{cc}& {\omega}_{\alpha}^{\left(f\right)}\to {\omega}_{\alpha}^{\left(f\right)}(1+{\kappa}_{\alpha}{\left({\delta}_{\alpha}^{\left(f\right)}\right)}^{2}),\hfill \\ & {\omega}_{\alpha}^{\left(g\right)}\to {\omega}_{\alpha}^{\left(g\right)}(1+\frac{{\kappa}_{\alpha}}{2}{\delta}_{\alpha}^{\left(g\right)});\hfill \end{array}$$
**Rule 5:**- $$\begin{array}{cc}& {\omega}_{\alpha}^{\left(f\right)}\to {\omega}_{\alpha}^{\left(f\right)}(1+\frac{{\kappa}_{\alpha}}{2}{\delta}_{\alpha}^{\left(f\right)}),\hfill \\ & {\omega}_{\alpha}^{\left(g\right)}\to {\omega}_{\alpha}^{\left(g\right)}(1+{\kappa}_{\alpha}{\left({\delta}_{\alpha}^{\left(g\right)}\right)}^{2});\hfill \end{array}$$
**Rule 6:**- $$\begin{array}{cc}& {\omega}_{\alpha}^{\left(f\right)}\to {\omega}_{\alpha}^{\left(f\right)}(1+\frac{{\kappa}_{\alpha}}{2}{\left({\delta}_{\alpha}^{\left(f\right)}\right)}^{2}),\hfill \\ & {\omega}_{\alpha}^{\left(g\right)}\to {\omega}_{\alpha}^{\left(g\right)}(1+{\kappa}_{\alpha}{\delta}_{\alpha}^{\left(g\right)}).\hfill \end{array}$$

#### Numerical Simulations

## 4. Dynamics of the System Using the GKSL Equations

#### 4.1. Experiment I

#### 4.2. Experiment II

#### 4.3. Experiment III

## 5. Conclusions

## Author Contributions

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Jin, F.; Dougherty, E.; Saraf, P.; Cao, Y.; Ramakrishnan, N. Epidemiological Modeling of News and Rumors on Twitter. In Proceedings of the SNAKDD ’13: Proceedings of the 7th Workshop on Social Network Mining and Analysis, Chicago, IL, USA, 11 August 2013; Volume 8, pp. 1–8. [Google Scholar]
- Lerman, K. Social Information Processing in News Aggregation. IEEE Internet Comput.
**2007**, 11, 16–28. [Google Scholar] [CrossRef] - Abdullah, S.; Wu, X. An Epidemic Model for News Spreading on Twitter. In Proceedings of the IEEE 23rd International Conference on Tools with Artificial Intelligence, Boca Raton, FL, USA, 7–9 November 2001; pp. 163–169. [Google Scholar]
- Doerr, B.; Fouz, M.; Friedrich, T. Why rumors spread so quickly in social networks. Commun. ACM
**2012**, 55, 70–75. [Google Scholar] [CrossRef] - Bagarello, F.; Gargano, F.; Oliveri, F. Spreading of competing information on a network. Entropy
**2020**, 22, 1169. [Google Scholar] [CrossRef] [PubMed] - Bagarello, F.; Gargano, G.; Oliveri, F. Quantum Tools for Macroscopic Systems; Synthesis Lectures on Mathematics & Statistics; Springer: Cham, Switzerland, 2023. [Google Scholar]
- Robinson, T.R. The Quantum Nature of Things: How Counting Leads to the Quantum; CRC Press: Boca Raton, FL, USA, 2023. [Google Scholar]
- Bagarello, F.; Salvo, R.D.; Gargano, F.; Oliveri, F. (H,ρ)–induced dynamics and large time behaviors. Phys. A
**2018**, 505, 355–373. [Google Scholar] [CrossRef] - Manzano, D. A Short Introduction to the Lindblad Master Equation. AIP Adv.
**2020**, 10, 025106. [Google Scholar] [CrossRef] - Bagarello, F. Quantum Dynamics for Classical Systems: With Applications of the Number Operator; Wiley: New York, NY, USA, 2012. [Google Scholar]
- Bagarello, F. Quantum Concepts in the Social, Ecological and Biological Sciences; Cambridge University Press: Cambridge, UK, 2019. [Google Scholar]
- Di Salvo, R.; Gorgone, M.; Oliveri, F. Generalized Hamiltonian for a two-mode fermionic model and asymptotic equilibria. Phys. A
**2020**, 540, 12032. [Google Scholar] [CrossRef] - Bagarello, F.; Gargano, F. Dynamics for a quantum parliament. Stud. Appl. Math.
**2023**, 150, 1182–1200. [Google Scholar] [CrossRef] - Asano, M.; Ohya, M.; Tanaka, Y.; Basieva, I.; Khrennikov, A. Quantum-like model of brain’s functioning: Decision making from decoherence. J. Theor. Biol.
**2011**, 281, 56–64. [Google Scholar] [CrossRef] [PubMed] - Asano, M.; Basieva, I.; Ohya, A.K.; Tanaka, Y.; Yamato, I. A model of epigenetic evolution based on theory of open quantum systems. Syst. Synth. Biol.
**2013**, 7, 161–173. [Google Scholar] [CrossRef] [PubMed] - Nava, A.; Giuliano, D.; Papa, A.; Rossi, M. Traffic models and traffic-jam transition in quantum (N+1)-level systems. SciPost Phys. Core
**2022**, 5, 22. [Google Scholar] [CrossRef] - Nava, A.; Giuliano, D.; Papa, A.; Rossi, M. Understanding traffic jams using lindblad superoperators. Int. J. Theor. Phys.
**2023**, 62, 2. [Google Scholar] [CrossRef] - Basieva, I.; Khrennikov, A. “What Is Life?”: Open Quantum Systems Approach. Open Syst. Inf. Dyn.
**2023**, 29, 2250016. [Google Scholar] [CrossRef]

**Figure 1.**Schematic representation of the network composed of three layers. The top and bottom layers consist of only one agent (Agent 1, $\mathcal{T}$, the transmitter, and Agent 6, $\mathcal{R}$, the receiver, respectively). The middle layer is composed of four agents interacting with the top and bottom layers. Links between the various agents are also shown.

**Figure 2.**Time evolution of the mean values of fake and good news for Agent 1 (

**top left**), Agent 6 (

**top right**), agents of the middle layer (

**bottom left**), and all agents (

**bottom right**).

**Figure 3.**Time evolution of the mean values of fake and good news for Agent 1 (

**top left**), Agent 6 (

**top right**), agents of the middle layer (

**bottom left**), and all agents (

**bottom right**); rule 1 with $\tau =1$.

**Figure 4.**Time evolution of the mean values of fake and good news for Agent 1 (

**top left**), Agent 6 (

**top right**), agents of the middle layer (

**bottom left**), and all agents (

**bottom right**); rule 2 with $\tau =1$.

**Figure 5.**Time evolution of the mean values of fake and good news for Agent 1 (

**top left**), Agent 6 (

**top right**), agents of the middle layer (

**bottom left**), and all agents (

**bottom right**); rule 3 with $\tau =1$.

**Figure 6.**Time evolution of the mean values of fake and good news for Agent 1 (

**top left**), Agent 6 (

**top right**), agents of the middle layer (

**bottom left**), and all agents (

**bottom right**); rule 4 with $\tau =1$.

**Figure 7.**Time evolution of the mean values of fake and good news for Agent 1 (

**top left**), Agent 6 (

**top right**), agents of the middle layer (

**bottom left**), and all agents (

**bottom right**); rule 5 with $\tau =1$.

**Figure 8.**Time evolution of the mean values of fake and good news for Agent 1 (

**top left**), Agent 6 (

**top right**), agents of the middle layer (

**bottom left**), and all agents (

**bottom right**); rule 6 with $\tau =1$.

**Figure 9.**Time evolutions of (

**a**) the mean values ${G}_{1}\left(t\right)={F}_{1}\left(t\right),{G}_{6}\left(t\right),{F}_{6}\left(t\right)$; (

**b**) the mean values ${G}_{3}\left(t\right),{F}_{3}\left(t\right),{G}_{4}\left(t\right),{F}_{4}\left(t\right)$. In (

**c**), the asymptotic value ${\overline{G}}_{6}$ of ${G}_{6}\left(t\right)$ versus the parameter ${p}_{3,3}^{\left(g\right)}$. Initial conditions ${G}_{1}\left(0\right)={F}_{1}\left(0\right)=0.5$ and other mean values are equal to zero. Model parameters: all ${p}_{\alpha ,\beta}^{(g,f)}=0.5$ in Equations (20)–(23), ${p}_{3,3}^{\left(g\right)}=2$, ${p}_{4,4}^{\left(f\right)}=0.05$, all ${\omega}_{\alpha}^{\left(g\right)}={\omega}_{\alpha}^{\left(f\right)}=1$ in Equation (19).

**Figure 10.**Time evolutions of the mean values ${G}_{1}\left(t\right),{F}_{1}\left(t\right),{G}_{6}\left(t\right),{F}_{6}\left(t\right)$ and ${F}_{3}\left(t\right)$. Model parameters: all ${p}_{\alpha ,\beta}^{(g,f)}=0.5$ in Equations (20)–(23), except ${p}_{1,3}^{\left(f\right)}=5$, ${p}_{3,3}^{\left(g\right)}=2$, ${p}_{4,4}^{\left(f\right)}=0.05$; all other parameters as in Experiment I. In the inset, the time evolution for small times of ${F}_{1}\left(t\right)$ and ${F}_{3}\left(t\right)$.

**Figure 11.**(

**a**) Time evolutions of the mean values ${G}_{1}\left(t\right),{F}_{1}\left(t\right),{G}_{6}\left(t\right),{F}_{6}\left(t\right)$ when considering the Lindblad operators ${\lambda}_{1}^{\left(g\right)}{g}_{1}^{\u2020}$. Model parameters: ${\lambda}_{1}^{\left(g\right)}=0.1$, ${p}_{3,3}^{\left(g\right)}=0$, ${p}_{4,4}^{\left(f\right)}=0.0$; all other parameters as in Experiment I. In the inset, the time evolution for small time (

**b**). Same as panel (

**a**) but with ${\lambda}_{1}^{\left(g\right)}=0.5$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bagarello, F.; Gargano, F.; Gorgone, M.; Oliveri, F.
Spreading of Information on a Network: A Quantum View. *Entropy* **2023**, *25*, 1438.
https://doi.org/10.3390/e25101438

**AMA Style**

Bagarello F, Gargano F, Gorgone M, Oliveri F.
Spreading of Information on a Network: A Quantum View. *Entropy*. 2023; 25(10):1438.
https://doi.org/10.3390/e25101438

**Chicago/Turabian Style**

Bagarello, Fabio, Francesco Gargano, Matteo Gorgone, and Francesco Oliveri.
2023. "Spreading of Information on a Network: A Quantum View" *Entropy* 25, no. 10: 1438.
https://doi.org/10.3390/e25101438