# Human Networks and Toxic Relationships

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Rewiev

## 3. Basic Model of a Toxic Relationship

- (i)
- ${x}_{1}\left(t\right)$ is the love, which consists of both passion and intimacy individual 1 has for the partner 2 at time t.
- (ii)
- $\alpha $ refers to the forgetting coefficient and measures how quickly the state of love will decrease, exponentially, in the absence of the partner.
- (iii)
- $\beta $ refers to the response of each partner to the love they receive from the other partner.
- (iv)
- $\overline{{x}_{2}}$ is the love the partner 2 puts in the relationship. We assume it is an exogenous variable of the system and it remains constant over time, so that $0\le \overline{{x}_{2}}<{x}_{1}\left(t\right)$, $\forall t\ge 0$.
- (v)
- ${A}_{1}\left(t\right)$ is the appeal factor, It is the partner 1’s perception of partner 2 at a time t. It is a subjective variable and depends on the perception of the partner at time t. It represents the measure of the partner’s power of attracting or arousing interest. This is affected by partner 2 possessing specific characteristics (physical, intellectual, financial etc.) that partner 1 is especially sensitive to.
- (vi)
- ${\gamma}_{1}$ refers to the response coefficients of the appeal of the other partner.
- (vii)
- k measures the sensitivity of an individual with respect to the excessive unreciprocated love given to the partner.
- (viii)
- $\widehat{{x}_{2}}$ is the minimum level of desired love by individual 1.
- (ix)
- ${M}_{2}$ measures the source of partner 2’s appeal, it represents the measure of wealth, beauty, etc. For simplicity we assume ${M}_{2}$ to be constant over time.

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

**Proposition**

**2.**

**Proof.**

**Definition**

**2.**

**Remark**

**2.**

## 4. Healing

**Proposition**

**3.**

**Proof.**

**Remark**

**3.**

**Proposition**

**4.**

**Proof.**

**Remark**

**4.**

**Proposition**

**5.**

**Proof.**

**Remark**

**5.**

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Rinaldi, S. Love dynamics. Appl. Math. Comput.
**1998**, 95, 181–192. [Google Scholar] - Glass, L. Toxic People. Your Total Image Publsihing. 1995. Available online: https://www.drlillianglass.com/wp-content/uploads/2015/06/Toxic-People_ebook.pdf (accessed on 8 September 2021).
- Browning, M.; Chiappori, P.A.; Weiss, Y. Economics of the Family; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Grossbard, S. The Economics of Marriage; The International Library of Critical Writings in Economics Series; San Diego State University: San Diego, CA, USA, 2015. [Google Scholar]
- Becker, G.S. A theory of marriage: Part I. J. Political Econ.
**1973**, 81, 813–846. [Google Scholar] [CrossRef] - Becker, G.S. A Theory of Marriage in Economics of the Family: Marriage, Children, and Human Capital; University of Chicago Press: Chicago, IL, USA, 1974; pp. 299–351. [Google Scholar]
- Brines, J.; Joyner, K. The ties that bind: Principles of cohesion in cohabitation and marriage. Am. Sociol. Rev.
**1999**, 64, 333–355. [Google Scholar] [CrossRef] - Johnson, M.D.; Horne, R.M.; Hardy, N.R.; Anderson, J.R. Temporality of couple conflict and relationship perceptions. J. Fam. Psychol.
**2018**, 32, 445–455. [Google Scholar] [CrossRef] [PubMed] - Motz, A. Toxic Couples: The Psychology of Domestic Violence; Routledge: Oxfordshire, UK, 2014. [Google Scholar]
- Houston, M.J. The Psychology of Abusive/Predatory Relationships; iUniverse: Bloomington, IN, USA, 2012. [Google Scholar]
- Sprott, C. Dynamics of Love and Happiness; Stenberg, R.J., Barnas, M.L., Eds.; Chaos and Complex Systems Seminar in Madison: Madison, WI, USA, 2001. [Google Scholar]
- Wauer, J.; Schwarzer, D.; Cai, G.Q.; Lin, Y.K. Dynamical models of love with time-varying fluctuations. Appl. Math. Comput.
**2007**, 188, 1535–1548. [Google Scholar] [CrossRef] [Green Version] - Regan, P.C. The Mating Game: A Primer on Love, Sex, and Marriage; Sage Publications: London, UK, 2008. [Google Scholar]
- Thibaut, J.W.; Kelley, H.H. The Social Psychology of Groups; Routledge: London, UK, 2007. [Google Scholar]
- Rusbult, C.E. A longitudinal test of the investment model: The development (and deterioration) of satisfaction and commitment in heterosexual involvements. J. Personal. Soc. Psychol.
**1983**, 45, 101–117. [Google Scholar] [CrossRef] - Satsangi, A.; Sinha, K. Dynamics of Love and Happiness: A Mathematical Analysis. Mod. Educ. Comput. Sci.
**2012**, 5, 31–37. [Google Scholar] [CrossRef] [Green Version]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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**

Solferino, N.; Tessitore, M.E.
Human Networks and Toxic Relationships. *Mathematics* **2021**, *9*, 2258.
https://doi.org/10.3390/math9182258

**AMA Style**

Solferino N, Tessitore ME.
Human Networks and Toxic Relationships. *Mathematics*. 2021; 9(18):2258.
https://doi.org/10.3390/math9182258

**Chicago/Turabian Style**

Solferino, Nazaria, and Maria Elisabetta Tessitore.
2021. "Human Networks and Toxic Relationships" *Mathematics* 9, no. 18: 2258.
https://doi.org/10.3390/math9182258