# Threshold Cascade Dynamics in Coevolving Networks

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Results

#### 3.1. On a Static Network

#### 3.2. Segregation of Adopting Nodes via Link Rewiring

#### 3.3. Phase Diagram for Global Cascades

#### 3.4. Non-Monotonicity in the Size of the Global Cascade

#### 3.5. Structure of Rewired Networks

#### 3.6. Mean-Field Approximations

## 4. Summary and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Granovetter, M. Threshold models of collective behavior. Am. J. Soc.
**1978**, 83, 1420–1443. [Google Scholar] [CrossRef] [Green Version] - Watts, D.J. A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. USA
**2002**, 99, 5766. [Google Scholar] [CrossRef] [Green Version] - Centola, D.; Macy, M. Complex Contagions and the Weakness of Long Ties. SSRN Electron. J.
**2007**, 113, 702. [Google Scholar] [CrossRef] [Green Version] - Centola, D. How Behavior Spreads: The Science of Complex Contagions How Behavior Spreads: The Science of Complex Contagions; Princeton University Press: Princeton, NJ, USA, 2018. [Google Scholar]
- Battiston, F.; Amico, E.; Barrat, A.; Bianconi, G.; de Arruda, G.F.; Franceschiello, B.; Iacopini, I.; Kéfi, S.; Latora, V.; Moreno, Y.; et al. The physics of higher-order interactions in complex systems. Nat. Phys.
**2021**, 17, 1093–1098. [Google Scholar] [CrossRef] - Centola, D. The Spread of Behavior in an Online Social Network Experiment. Science
**2010**, 329, 5996. [Google Scholar] [CrossRef] - Levine, J.M.; Bascompte, J.; Adler, P.B.; Allesina, S. Beyond pairwise mechanisms of species coexistence in complex communities. Nature
**2017**, 546, 56–64. [Google Scholar] [CrossRef] [Green Version] - Hébert-Dufresne, L.; Scarpino, S.V.; Young, J.-G. Macroscopic patterns of interacting contagions are indistinguishable from social reinforcement. Nat. Phys.
**2020**, 16, 426–431. [Google Scholar] [CrossRef] - Schelling, T. Hockey helmets, concealed weapons, and daylight saving: A study of binary choices with externalities. Confl. Resolut.
**1973**, 17, 381–428. [Google Scholar] [CrossRef] - Gleeson, J.P.; Cahalane, D.J. Seed size strongly affects cascades on random networks. Phys. Rev. E
**2007**, 75, 056103. [Google Scholar] [CrossRef] [Green Version] - Karsai, M.; Iñiguez, G.; Kaski, K.; Kertész, J. Complex contagion process in spreading of online innovation. J. R. Soc. Interface
**2014**, 11, 20140694. [Google Scholar] [CrossRef] - Nematzadeh, A.; Ferrara, E.; Flammini, A.; Ahn, Y.-Y. Optimal Network Modularity for Information Diffusion. Phys. Rev. Lett.
**2014**, 113, 088701. [Google Scholar] [CrossRef] [Green Version] - Auer, S.; Heitzig, J.; Kornek, U.; Schöll, E.; Kurths, J. The Dynamics of Coalition Formation on Complex Networks. Sci. Rep.
**2015**, 5, 13386. [Google Scholar] [CrossRef] [Green Version] - Kook, J.; Choi, J.; Min, B. Double transitions and hysteresis in heterogeneous contagion processes. Phys. Rev. E
**2021**, 104, 044306. [Google Scholar] [CrossRef] - Pastor-Satorras, R.; Castellano, C.; Van Mieghem, P.; Vespignani, A. Epidemic processes in complex networks. Rev. Mod. Phys.
**2015**, 87, 925–979. [Google Scholar] [CrossRef] [Green Version] - Friedman, N.; Ito, S.; Brinkman, B.A.; Shimono, M.; DeVille, R.L.; Dahmen, K.A.; Beggs, J.M.; Butler, T.C. Universal Critical Dynamics in High Resolution Neuronal Avalanche Data. Phys. Rev. Lett.
**2012**, 108, 208102. [Google Scholar] [CrossRef] [Green Version] - Lee, K.-M.; Yang, J.-S.; Kim, G.; Lee, J.; Goh, K.-I.; Kim, I.-M. Impact of the Topology of Global Macroeconomic Network on the Spreading of Economic Crises. PLoS ONE
**2011**, 6, e18443. [Google Scholar] [CrossRef] - Motter, A.E.; Lai, Y.-C. Cascade-based attacks on complex networks. Phys. Rev. E
**2002**, 66, 065102. [Google Scholar] [CrossRef] [Green Version] - Hackett, A.; Melnik, S.; Gleeson, J. Cascades on a class of clustered random networks. Phys. Rev. E
**2011**, 83, 056107. [Google Scholar] [CrossRef] [Green Version] - Lee, K.-M.; Brummitt, C.D.; Goh, K.-I. Threshold cascades with response heterogeneity in multiplex networks. Phys. Rev. E
**2014**, 90, 062816. [Google Scholar] [CrossRef] [Green Version] - Min, B.; Miguel, M.S. Competition and dual users in complex contagion processes. Sci. Rep.
**2018**, 8, 14580. [Google Scholar] [CrossRef] [Green Version] - Abella, D.; Miguel, M.S.; Ramasco, J.J. Aging in binary-state models: The Threshold model for complex contagion. Phys. Rev. E
**2023**, 107, 024101. [Google Scholar] [CrossRef] - Lee, K.-M.; Lee, S.; Min, B.; Goh, K.-I. Threshold cascade dynamics on signed random networks. Chaos Solitons Fractals
**2023**, 168, 113118. [Google Scholar] [CrossRef] - Diaz-Diaz, F.; Miguel, M.S.; Meloni, S. Echo chambers and information transmission biases in homophilic and heterophilic networks. Sci. Rep.
**2022**, 12, 9350. [Google Scholar] [CrossRef] - Czaplicka, A.; Toral, R.; Miguel, M.S. Competition of simple and complex adoption on interdependent networks. Phys. Rev. E
**2016**, 94, 062301. [Google Scholar] [CrossRef] [Green Version] - Min, B.; Miguel, M.S. Competing contagion processes: Complex contagion triggered by simple contagion. Sci. Rep.
**2018**, 8, 10422. [Google Scholar] [CrossRef] [Green Version] - Holme, P.; Saramaki, J. Temporal Networks; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Gross, T.; Sayama, H. Adaptive Networks: Theory, Models and Applications; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Gross, T.; Blasius, B. Adaptive coevolutionary networks: A review. J. R. Soc. Interface
**2007**, 5, 259–271. [Google Scholar] [CrossRef] [Green Version] - Funk, S.; Gilad, E.; Watkins, C.; Jansen, V.A.A. The spread of awareness and its impact on epidemic outbreaks. Proc. Natl. Acad. Sci. USA
**2009**, 106, 6872–6877. [Google Scholar] [CrossRef] [Green Version] - Holme, P.; Newman, M.E.J. Nonequilibrium phase transition in the coevolution of networks and opinions. Phys. Rev. E
**2006**, 74, 056108. [Google Scholar] [CrossRef] [Green Version] - Vazquez, F.; Eguíluz, V.M.; Miguel, M.S. Generic Absorbing Transition in Coevolution Dynamics. Phys. Rev. Lett.
**2008**, 100, 108702. [Google Scholar] [CrossRef] [Green Version] - Yi, S.D.; Baek, S.K.; Zhu, C.-P.; Kim, B.J. Phase transition in a coevolving network of conformist and constrarian voters. Phys. Rev. E
**2013**, 87, 012806. [Google Scholar] [CrossRef] [Green Version] - Diakonova, M.; Miguel, M.S.; Eguíluz, V.M. Absorbing and shattered fragmentation transitions in multilayer coevolution. Phys. Rev. E
**2014**, 89, 062818. [Google Scholar] [CrossRef] [Green Version] - Diakonova, M.; Eguíluz, V.M.; Miguel, M.S. Noise in coevolving networks. Phys. Rev. E
**2015**, 92, 032803. [Google Scholar] [CrossRef] [Green Version] - Min, B.; Miguel, M.S. Fragmentation transitions in a coevolving nonlinear voter model. Sci. Rep.
**2017**, 7, 12864. [Google Scholar] [CrossRef] [Green Version] - Raducha, T.; Min, B.; Miguel, M.S. Coevolving nonlinear voter model with triadic closure. EPL
**2018**, 124, 30001. [Google Scholar] [CrossRef] [Green Version] - Min, B.; Miguel, M.S. Multilayer coevolution dynamics of the nonlinear voter model. New J. Phys.
**2019**, 21, 035004. [Google Scholar] [CrossRef] - Raducha, T.; Wilinski, M.; Gubiec, T.; Stanley, H.E. Statistical mechanics of a coevolving spin system. Phys. Rev. E
**2018**, 98, 030301. [Google Scholar] [CrossRef] [Green Version] - Mandrà, S.; Fortunato, S.; Castellano, C. Coevolution of Glauber-like Ising dynamics and topology. Phys. Rev. E
**2009**, 80, 056105. [Google Scholar] [CrossRef] [PubMed] - Fu, F.; Wang, L. Coevolutionary dynamics of opinions and networks: From diversity to uniformity. Phys. Rev. E
**2008**, 78, 016104. [Google Scholar] [CrossRef] [Green Version] - Su, J.; Liu, B.; Li, Q.; Ma, H. Coevolution of Opinions and Directed Adaptive Networks in a Social Group. J. Artif. Soc. Soc. Simul.
**2014**, 17, 4. [Google Scholar] [CrossRef] - Gross, T.; D’lima, C.J.D.; Blasius, B. Epidemic Dynamics on an Adaptive Network. Phys. Rev. Lett.
**2006**, 96, 208701. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Marceau, V.; Noël, P.-A.; Hébert-Dufresne, L.; Allard, A.; Dubé, L.J. Adaptive networks: Coevolution of disease and topology. Phys. Rev. E
**2010**, 82, 036116. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shaw, L.B.; Schwartz, I.B. Fluctuating epidemics on adaptive networks. Phys. Rev. E
**2008**, 77, 066101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Vazquez, F.; Gonzalez-Avella, J.C.; Eguiluz, V.M.; San Miguel, M. Time scale competition leading to fragmentation and recombination transitions in the co-evolution of network and states. Phys. Rev. E
**2007**, 76, 046120. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Centola, D.; González-Avella, J.C.; Eguíluz, V.M.; Miguel, M.S. Homophily, Cultural Drift, and the Co-Evolution of Cultural Groups. J. Confl. Resolut.
**2007**, 51, 905–929. [Google Scholar] [CrossRef] [Green Version] - Casado, M.A.G.; Sánchez, A.; Miguel, M.S. Network coevolution drives segregation and enhances Pareto optimal equilibrium selection in coordination games. Sci. Rep.
**2023**, 13, 2866. [Google Scholar] [CrossRef] - Coelho, F.C.; Codeco, C.T. Dynamic modeling of vaccinating behavior as a function of individual beliefs. PLoS Comput. Biol.
**2009**, 5, e1000425. [Google Scholar] [CrossRef] - Granell, C.; Gómez, S.; Arenas, A. Dynamical Interplay between Awareness and Epidemic Spreading in Multiplex Networks. Phys. Rev. Lett.
**2013**, 111, 128701. [Google Scholar] [CrossRef] [Green Version] - Pires, M.A.; Oestereich, A.L.; Crokidakis, N.; Queirós, S.M.D. Antivax movement and epidemic spreading in the era of social networks: Nonmonotonic effects, bistability, and network segregation. Phys. Rev. E
**2021**, 104, 034302. [Google Scholar] [CrossRef] - Fang, F.; Ma, J.; Li, Y. The coevolution of the spread of a disease and competing opinions in multiplex networks. Chaos Solitons Fractals
**2023**, 170, 113376. [Google Scholar] [CrossRef] - Lambiotte, R.; González-Avella, J.C. On co-evolution and the importance of initial conditions. Physics A
**2010**, 390, 392–397. [Google Scholar] [CrossRef] - Karimi, F.; Holme, P. Threshold model of cascades in empirical temporal networks. Physics A
**2013**, 392, 3476–3483. [Google Scholar] [CrossRef] [Green Version] - Rosenthal, S.B.; Twomey, C.R.; Hartnett, A.T.; Wu, H.S.; Couzin, I.D. Revealing the hidden networks of interaction in mobile animal groups allows prediction of complex behavioral contagion. Proc. Natl. Acad. Sci. USA
**2015**, 112, 4690–4695. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Karsai, M.; Iñiguez, G.; Kikas, R.; Kaski, K.; Kertész, J. Local cascades induced global contagion: How heterogeneous thresholds, exogenous effects, and unconcerned behaviour govern online adoption spreading. Sci. Rep.
**2016**, 6, 27178. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mønsted, B.; Sapieżyński, P.; Ferrara, E.; Lehmann, S. Evidence of complex contagion of information in social media: An experiment using Twitter bots. PLoS ONE
**2017**, 12, e0184148. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Unicomb, S.; Iñiguez, G.; Karsai, M. Threshold driven contagion on weighted networks. Sci. Rep.
**2018**, 8, 1–10. [Google Scholar] [CrossRef] [Green Version] - Guibeault, D.; Centola, D. Topological measures for identifying and predicting the spread of complex contagions. Nat. Commun.
**2021**, 12, 4430. [Google Scholar] [CrossRef] - Aral, S.; Nicolaides, C. Exercise contagion in a global social network. Nat. Commun.
**2017**, 8, 14753. [Google Scholar] [CrossRef] [Green Version] - McPherson, M.; Smith-Lovin, L.; Cook, J.M. Birds of a Feather: Homophily in Social Networks. Annu. Rev. Sociol.
**2001**, 27, 415–444. [Google Scholar] [CrossRef] [Green Version] - Lee, E.; Karimi, F.; Wagner, C.; Jo, H.-H.; Strohmaier, M.; Galesic, M. Homophily and minority-group size explain perception biases in social networks. Nat. Hum. Behav.
**2019**, 3, 1078–1087. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**An example of the evolution rules of the coevolutionary dynamics of a threshold cascade model. A connected pair of an adopting (filled circles) and a non-adopting node (open circles) is removed with a probability p, and the non-adopting node establishes a connection to a new node that is not adopting, chosen randomly from the entire network. In addition, a non-adopting node becomes adopting if the fraction of adopting neighbors is larger than the threshold $\theta $. Once a node becomes adopting, the node is then permanently in this state.

**Figure 2.**(

**a**) The final fraction R of adopting nodes and the size S of the largest non-adopting cluster as a function of the network plasticity p. (

**b**) The number ${n}_{c}$ of clusters to network size N as a function of p. The dynamics starts with $\theta =0.18$ in ER networks with $N={10}^{5}$, $z=3$, and an initial fraction of seeds of ${R}_{0}=2\times {10}^{-4}$. The average values are obtained by ${10}^{4}$ independent runs with different network realizations for each run. Examples of network structures with $N=200$ at the steady state with (

**c**) $p=0.2$ and (

**d**) $p=0.8$. Red and blue nodes represent adopting and non-adopting states, respectively.

**Figure 3.**The final fraction R of adopting nodes in ER networks with $N={10}^{5}$ as a function of the average degree z and threshold $\theta $, with various rewiring probabilities, i.e., (

**a**) $p=0.2$, (

**b**) $p=0.4$, and (

**c**) $p=0.6$, in a steady state. Th dashed lines represent the transition points between the global cascade and no cascade phases in static networks, that is $p=0$, obtained from Equation (3). The solid lines represent the transition points with network plasticity p by using mean-field approximations. The numerical results are obtained by ${10}^{3}$ independent runs with different network realizations for each run.

**Figure 4.**(

**a**) The final size R of cascades as a function of network plasticity p and average degree z of the ER networks with threshold $\theta =0.18$ and seed fraction ${R}_{0}=2\times {10}^{-4}$. The dashed lines represent analytical predictions obtained by Equation (9). (

**b**) The size R as a function of the average degree z in ER networks for $p=0,0.2,0.4$, and $0.6$ with $\theta =0.18$. The lines represent analytical predictions based on Equations (4) and (5). (

**c**) Inset shows the size R with respect to the probability of link rewiring p for $z=2$ and $2.5$ and $\theta =0.1$. The numerical results were obtained with ${10}^{3}$ independent runs with different network realizations for each run.

**Figure 5.**Degree distribution $P\left(k\right)$ of the coevolving threshold model at the steady state in (

**a**) linear and (

**b**) log scales for $p=0,0.1,0.2,0.3$ (global cascade region) and (

**c**) linear scale for $p=0.4,0.5,0.6,0.7$ (no cascades region). The results were obtained from ER networks with $z=4$ and $N={10}^{5}$ with ${10}^{4}$ independent runs. The solid lines in (

**a**,

**c**) represent the Poisson distribution with $z=4$.

**Figure 6.**The critical values ${p}_{c}$ of network plasticity estimated by the mean-field approximation (

**a**) with respect to z with fixed $\theta =0.1,0.15,0.2$ and (

**b**) with respect to $\theta $ with fixed $z=2,4,6,8$.

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**

Min, B.; San Miguel, M.
Threshold Cascade Dynamics in Coevolving Networks. *Entropy* **2023**, *25*, 929.
https://doi.org/10.3390/e25060929

**AMA Style**

Min B, San Miguel M.
Threshold Cascade Dynamics in Coevolving Networks. *Entropy*. 2023; 25(6):929.
https://doi.org/10.3390/e25060929

**Chicago/Turabian Style**

Min, Byungjoon, and Maxi San Miguel.
2023. "Threshold Cascade Dynamics in Coevolving Networks" *Entropy* 25, no. 6: 929.
https://doi.org/10.3390/e25060929