# Glassy States of Aging Social Networks

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

”Yesterdays’ friend (enemy) rarely become tomorrows’ enemy (friend).”

## 2. The Evolving Network

## 3. Results and Discussion

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Heider, F. Attitudes and cognitive organization. J. Psychol.
**1946**, 21, 107–112. [Google Scholar] [CrossRef] [PubMed] - Wang, Z.; Szolnoki, A.; Perc, M. Self-organization towards optimally interdependent networks by means of coevolution. New J. Phys.
**2014**, 16, 033041. [Google Scholar] [CrossRef] - Perca, M.; Szolnokib, A. Coevolutionary games—A mini review. BioSystems
**2010**, 99, 109–125. [Google Scholar] [CrossRef] [PubMed] - Kirman, A.; Markose, S.; Giansante, S.; Pin, P. Marginal contribution, reciprocity and equity in segregated groups: Bounded rationality and self-organization in social networks. J. Econ. Dyn. Control
**2007**, 31, 2085–2107. [Google Scholar] [CrossRef] - Ramasco, J.J.; Dorogovtsev, S.N.; Pastor-Satorras, R. Self-organization of collaboration networks. Phys. Rev. E
**2004**, 70, 036106. [Google Scholar] [CrossRef] [PubMed] - Becker, H.S. Notes on the concept of commitment. Am. J. Sociol.
**1960**, 66, 32–40. [Google Scholar] [CrossRef] - Stanley, S.M.; Markman, H.J. Assessing commitment in personal relationships. J. Marriage Fam.
**1992**, 54, 595–608. [Google Scholar] [CrossRef] - Clements, R.; Swensen, C.H. Commitment to one’s spouse as a predictor of marital quality among older couples. Curr. Psychol.
**2000**, 19, 110–119. [Google Scholar] [CrossRef] - Granovetter, M. The strength of weak ties. Am. J. Sociol.
**1973**, 78, 1360–1380. [Google Scholar] [CrossRef] - Stehlé, J.; Barrat, A.; Bianconi, G. Dynamical and bursty interactions in social networks. Phys. Rev. E
**2010**, 81, 035101–035104. [Google Scholar] [CrossRef] [PubMed] - Zhao, K.; Stehlé, J.; Bianconi, G.; Barrat, A. Social network dynamics of face-to-face interactions. Phys. Rev. E
**2011**, 83, 056109–056127. [Google Scholar] [CrossRef] [PubMed] - Karsai, M.; Kaski, K.; Kertész, J. Correlated Dynamics in Egocentric Communication Networks. PLoS ONE
**2012**, 7, e40612. [Google Scholar] [CrossRef] [PubMed] - Rybski, D.; Buldyrev, S.; Havlin, S.; Liljeros, F.; Makse, H. Communication Activity in a Social Network: Relation Between Long-Term Correlations and Inter-Event Clustering; Scientific Reports; Nature Publishing Group: London, UK, 2012; Volume 2, p. 560. [Google Scholar]
- Shirazi, A.H.; Namaki, A.; Roohi, A.A.; Jafari, G.R. Transparency effect in emergence of monopolies in social networks. J. Artif. Soc. Soc. Simul.
**2013**, 6, 1–10. [Google Scholar] [CrossRef] - Karsai, M.; Kaski, K.; Barabási, A.; Kertész, J. Universal Features of Correlated Bursty Behavior; Scientific Reports; Nature Publishing Group: London, UK, 2012; Volume 2, p. 397. [Google Scholar]
- Vestergaard, C.; Génois, M.; Barrat, A. How memory generates heterogeneous dynamics in temporal networks. Phys. Rev. Lett.
**2014**, 90, 042805. [Google Scholar] [CrossRef] [PubMed] - Karsai, M.; Perra, N.; Vespignani, A. Time Varying Networks and the Weakness of Strong Ties; Scientific Reports; Nature Publishing Group: London, UK, 2014; Volume 4, pp. 4001–4007. [Google Scholar]
- Saeedian, M.; Khaliqi, M.; Azimi-Tafreshi, N.; Jafari, G.R.; Ausloos, M. Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model. Phys. Rev. E
**2017**, 95, 022409. [Google Scholar] [CrossRef] [PubMed] - Dezso, Z.; Almaas, E.; Lukacs, A.; Racz, B.; Szakadat, I.; Barabási, A. Dynamics of information access on the web. Phys. Rev. E
**2008**, 73, 066132–066137. [Google Scholar] [CrossRef] [PubMed] - Ebadi, E.; Saeedian, M.; Ausloos, M.; Jafari, G.R. Effect of memory in non-Markovian Boolean networks illustrated with a case study: A cell cycling process. EPL
**2016**, 116, 30004. [Google Scholar] [CrossRef] - Lipowski, A.; Gontarek, K.; Ausloos, M. Statistical mechanics approach to a reinforcement learning model with memory. Physica A
**2009**, 388, 1849–1856. [Google Scholar] [CrossRef] - Szolnoki, A.; Perc, M.; Szabó, G.; Stark, H.-U. Impact of aging on the evolution of cooperation in the spatial prisoner’s dilemma game. Phys. Rev. E
**2009**, 80, 021901. [Google Scholar] [CrossRef] [PubMed] - Aguiar, F.; Parravano, A. Tolerating the Intolerant: Homophily, Intolerance, and Segregation in Social Balanced Networks. J. Confl. Resolut.
**2013**, 59, 29–50. [Google Scholar] [CrossRef] - Van de Rijt, A. The Micro-Macro Link for the Theory of Structural Balance. J. Math. Sociol.
**2011**, 35, 94–113. [Google Scholar] [CrossRef] - Summers, T.H.; Shames, I. Active influence in dynamical models of structural balance in social networks. Europhys. Lett.
**2013**, 103, 18001. [Google Scholar] [CrossRef] - Hassanibesheli, F.; Hedayatifar, L.; Gawroński, P.; Stojkow, M.; Żuchowska-Skiba, D.; Kulakowski, K. Gain and loss of esteem, direct reciprocity and Heider balance. Physica A
**2017**, 468, 334–339. [Google Scholar] [CrossRef] - Moore, M. An international application of Heider’s balance theory. Eur. J. Soc. Psychol.
**1978**, 8, 401–405. [Google Scholar] [CrossRef] - Esmailian, P.; Abtahi, S.E.; Jalili, M. Mesoscopic analysis of online social networks: The role of negative ties. Phys. Rev. E
**2014**, 90, 042817. [Google Scholar] [CrossRef] [PubMed] - Szell, M.; Lambiotte, R.; Thurner, S. Multirelational organization of large-scale social networks in an online world. Proc. Natl. Acad. Sci. USA
**2010**, 107, 13636–13641. [Google Scholar] [CrossRef] [PubMed] - Zheng, X.; Zeng, D.; Wang, F.Y. Social balance in signed networks. Inf. Syst. Front.
**2014**, 17, 1–19. [Google Scholar] [CrossRef] - Kunegis, J.; Lommatzsch, A.; Bauckhage, C. The slashdot zoo: Mining a social network with negative edges. In Proceedings of the 18th International Conference on World Wide Web—WWW 2009, Madrid, Spain, 20–24 April 2009; Association for Computing Machinery: New York, NY, USA, 2009; pp. 741–750. [Google Scholar]
- Doreian, P. Evolution of Human Signed Networks. Metodol. Zv.
**2004**, 1, 277–293. [Google Scholar] - Guha, R.V.; Kumar, R.; Raghavan, P.; Tomkins, A. Propagation of trust and distrust. In Proceedings of the 13th International Conference on World Wide Web, New York, NY, USA, 17–20 May 2004. [Google Scholar]
- Marvel, S.A.; Kleinberg, J.; Strogatz, S.H. The energy landscape of social balance. Phys. Rev. Lett.
**2009**, 103, 198701. [Google Scholar] [CrossRef] [PubMed] - Antal, T.; Krapivsky, P.; Redner, S. Social balance on networks: The dynamics of friendship and enmity. Physica D
**2006**, 224, 130–136. [Google Scholar] [CrossRef] - Hedayatifar, L.; Hassanibesheli, F.; Shirazi, A.H.; Vasheghani Farahani, S.; Jafari, G.R. Pseudo paths toward minimum energy states in network dynamics. Physica A
**2017**. [Google Scholar] [CrossRef] - Safdari, H.; Chechkin, A.V.; Jafari, G.R.; Metzler, R. Aging Scaled Brownian Motion. Phys. Rev. E
**2015**, 91, 042107. [Google Scholar] [CrossRef] [PubMed] - Gallos, L.K.; Rybski, D.; Liljeros, F.; Havlin, S.; Makse, H.A. How People Interact in Evolving Online Affiliation Networks. Phys. Rev. X
**2012**, 2, 031014. [Google Scholar] [CrossRef] - Livina, V.; Havlin, S.; Bunde, A. Memory in the Occurrence of Earthquakes. Phys. Rev. Lett.
**2005**, 95, 208501. [Google Scholar] [CrossRef] [PubMed] - Kemuriyama, T.; Ohta, H.; Sato, Y.; Maruyama, S.; Tandai-Hiruma, M. A power-law distribution of inter-spike intervals in renal sympathetic nerve activity in salt-sensitive hypertension-induced chronic heart failure. Biosystems
**2010**, 101, 144–147. [Google Scholar] [CrossRef] [PubMed] - Siwy, Z.; Ausloos, M.; Ivanova, K. Correlation studies of open and closed states fluctuations in an ion channel: Analysis of ion current through a large-conductance locust potassium channel. Phys. Rev. E
**2002**, 65, 031907. [Google Scholar] [CrossRef] [PubMed] - Kułakowski, K. Some recent attempts to simulate the Heider balance problem. Comput. Sci. Eng.
**2007**, 9, 80–85. [Google Scholar] [CrossRef] - Kułakowski, K.; Gawronski, P.; Gronek, P. The Heider balance-a continuous approach. Int. J. Mod. Phys. C
**2005**, 16, 707–716. [Google Scholar] [CrossRef] - Marvel, S.A.; Kleinberg, J.; Kleinberg, R.D.; Strogatz, S.H. Continuous-time model of structural balance. Proc. Natl. Acad. Sci. USA
**2011**, 108, 1771–1776. [Google Scholar] [CrossRef] [PubMed] - Altafini, C. Dynamics of Opinion Forming in Structurally Balanced Social Networks. PLoS ONE
**2012**, 7, e38135. [Google Scholar] [CrossRef] [PubMed] - Ausloos, M.; Petroni, F. Threshold Model for Triggered Avalanches on Networks; Stock Markets, F., Prattico, P.F., D’Amico, G., Eds.; Nova Scotia: New York, NY, USA, 2013; pp. 83–101. [Google Scholar]
- Sousa, A.O.; Yu-Song, T.; Ausloos, M. Propaganda spreading or running away from frustration effects in Sznajd model. Eur. Phys. J. B
**2008**, 66, 115–124. [Google Scholar] [CrossRef] - Newcomb, T.M.; Turner, R.H.; Converse, P.E. Social Psychology: The Study of Human Interaction; Holt, Rinehart and Winston: New York, NY, USA, 1965. [Google Scholar]
- Cartwright, D.; Harary, F. Structure balance: A generalization of Heider’s theory. Psychol. Rev.
**1956**, 63, 277–293. [Google Scholar] [CrossRef] [PubMed] - Saeedian, M.; Azimi-Tafreshi, N.; Jafari, G.R.; Kertesz, J. Epidemic spreading on evolving signed networks. Phys. Rev. E
**2017**, 95, 022314. [Google Scholar] [CrossRef] [PubMed] - Barrat, A.; Barthelemy, M.; Pastor-Satorras, R.; Vespignani, A. The architecture of complex weighted networks. Proc. Natl. Acad. Sci. USA
**2004**, 101, 3747–3752. [Google Scholar] [CrossRef] [PubMed] - Horvath, S. Weighted Network Analysis. Applications in Genomics and Systems Biology; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Gligor, M.; Ausloos, M. Clusters in weighted macroeconomic networks: The EU case. Introducing the overlapping index of GDP/capita fluctuation correlations. Eur. Phys. J. B
**2008**, 63, 533–539. [Google Scholar] [CrossRef] - Goychuk, I. Viscoelastic subdiffusion: From anomalous to normal. Phys. Rev. E
**2009**, 80, 046125. [Google Scholar] [CrossRef] [PubMed] - Jeon, J.H.; Metzler, R. Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries. Phys. Rev. E
**2010**, 81, 021103. [Google Scholar] [CrossRef] [PubMed] - West, B.J.; Turalska, M.; Grigolini, P. Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics. New J. Phys.
**2015**, 17, 045009. [Google Scholar] [CrossRef] - Safdari, H.; Kamali, M.Z.; Shirazi, A.H.; Khaliqi, M.; Jafari, G.R.; Ausloos, M. Fractional Dynamics of Network Growth Constrained by Aging Node Interactions. PLoS ONE
**2016**, 11, e0154983. [Google Scholar] [CrossRef] [PubMed] - Caputo, M. Linear Models of Dissipation whose Q is almost Frequency Independent-II. Geophys. J. R. Astron. Soc.
**1967**, 13, 529–539. [Google Scholar] [CrossRef] - Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; North-Holland Mathematics Studies; Elsevier Science: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Garrappa, R. On linear stability of predictor-corrector algorithms for fractional differential equations. Int. J. Comput. Math.
**2010**, 87, 2281–2290. [Google Scholar] [CrossRef] - Diethelm, K.; Freed, A.D. The FracPECE subroutine for the numerical solution of differential equations of fractional order. In Forschung und Wissenschaftliches Rechnen; Heinzel, S., Plesser, T., Eds.; Gessellschaft für Wissenschaftliche Datenverarbeitung: Göttingen, Germany, 1998; pp. 57–71. [Google Scholar]
- Lubich, C. A stability analysis of convolution quadratures for Abel-Volterra integral equations. IMA J. Numer. Anal.
**1986**, 6, 87–101. [Google Scholar] [CrossRef] - Traag, V.; Van Dooren, P.; De Leenheer, P. Dynamical Models Explaining Social Balance and Evolution of Cooperation. PLoS ONE
**2013**, 8, e60063. [Google Scholar] [CrossRef] [PubMed] - Krawczyk, M.J.; Castillo-Mussot, M.; Hernández-Ramírez, E.; Naumis, G.G.; Kułakowski, K. Heider balance, asymmetric ties, and gender segregation. Physica A
**2015**, 439, 66–74. [Google Scholar] [CrossRef]

**Figure 1.**In terms of social networks dynamics, links can carry various information such as type, age, and strength of relations. This figure illustrates a network with two types of relations (for instance, friendship and animosity) which are denoted by solid and dashed lines, respectively. Here colors display the gradient of age from young (red) to old (blue); the weight (strength) of links are represented by the line thickness. Increasing the age or weight of links can lead to decreasing the tendency towards modifying relationships.

**Figure 2.**The maximum time interval during which a network resists against any change (long-lived states). The left panel depicts this time interval for networks with 21 nodes in various paths (10,000 realizations). The right panel illustrates that the maximum time intervals follow a Poisson distribution.

**Figure 3.**Energy of final states (Glassy, Balance and Jammed states) versus mean of positive and negative links. We show these final states for networks with 45 nodes at different $\alpha $ (strength of memory) values. With increasing $\alpha $ values, the system becomes more flexible against changes and the probability of reaching balance and jammed states increases. The right bottom panel shows a sudden "phase transition" in the percentage (density) of glassy states to balanced states about $\alpha \sim 0.6$.

© 2017 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hassanibesheli, F.; Hedayatifar, L.; Safdari, H.; Ausloos, M.; Jafari, G.R. Glassy States of Aging Social Networks. *Entropy* **2017**, *19*, 246.
https://doi.org/10.3390/e19060246

**AMA Style**

Hassanibesheli F, Hedayatifar L, Safdari H, Ausloos M, Jafari GR. Glassy States of Aging Social Networks. *Entropy*. 2017; 19(6):246.
https://doi.org/10.3390/e19060246

**Chicago/Turabian Style**

Hassanibesheli, Foroogh, Leila Hedayatifar, Hadise Safdari, Marcel Ausloos, and G. Reza Jafari. 2017. "Glassy States of Aging Social Networks" *Entropy* 19, no. 6: 246.
https://doi.org/10.3390/e19060246