# Temporal, Structural, and Functional Heterogeneities Extend Criticality and Antifragility in Random Boolean Networks

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Random Boolean Networks

- Each gene is regulated by exactly K other genes.
- The K genes that regulate each node are chosen randomly using a uniform probability.
- All genes are updated synchronously, i.e., at the same timescale [47].
- Each gene is expressed (i.e., its Boolean value is 1) with probability p and is unexpressed with probability $1-p$.

#### 2.2. Criticality

#### 2.3. Antifragility

## 3. Results and Discussion

#### 3.1. Homogeneity vs. Heterogeneity

#### 3.2. Varying Functional and Temporal Parameters

#### 3.3. Antifragility

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RBN | Random Boolean network |

## Appendix A. Code

#### Appendix A.1. mi_ rbn.py

#### Appendix A.2. complexity.py

#### Appendix A.3. antifragility.py

## References

- Stanley, H.E. Introduction to Phase Transitions and Critical Phenomena; Oxford University Press: Oxford, UK, 1987. [Google Scholar]
- Haimovici, A.; Tagliazucchi, E.; Balenzuela, P.; Chialvo, D.R. Brain organization into resting state networks emerges at criticality on a model of the human connectome. Phys. Rev. Lett.
**2013**, 110, 178101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Setzler, M.; Marghetis, T.; Kim, M. Creative leaps in musical ecosystems: Early warning signals of critical transitions in professional jazz. In CogSci 2018. Changing/Minds. 40th Annual Cognitive Science Society Meeting, Madison WI, USA, 25–28 July 2018; Kalish, C., Rau, M., Zhu, J., Rogers, T., Eds.; pp. 2467–2472. Available online: https://cogsci.mindmodeling.org/2018/ (accessed on 1 September 2022).
- Christensen, K.; Moloney, N.R. Complexity and Criticality; World Scientific: Singapore, 2005. [Google Scholar]
- Mora, T.; Bialek, W. Are Biological Systems Poised at Criticality? J. Stat. Phys.
**2011**, 144, 268–302. [Google Scholar] [CrossRef] [Green Version] - Roli, A.; Villani, M.; Filisetti, A.; Serra, R. Dynamical Criticality: Overview and Open Questions. J. Syst. Sci. Complex.
**2018**, 31, 647–663. [Google Scholar] [CrossRef] [Green Version] - Taleb, N.N. Antifragile: Things That Gain From Disorder; Random House: London, UK, 2012. [Google Scholar]
- Aven, T. The concept of antifragility and its implications for the practice of risk analysis. Risk Anal.
**2015**, 35, 476–483. [Google Scholar] [CrossRef] [PubMed] - Danchin, A.; Binder, P.M.; Noria, S. Antifragility and tinkering in biology (and in business) flexibility provides an efficient epigenetic way to manage risk. Genes
**2011**, 2, 998–1016. [Google Scholar] [CrossRef] [Green Version] - Abid, A.; Khemakhem, M.T.; Marzouk, S.; Jemaa, M.B.; Monteil, T.; Drira, K. Toward antifragile cloud computing infrastructures. Procedia Comput. Sci.
**2014**, 32, 850–855. [Google Scholar] [CrossRef] [Green Version] - Jones, K.H. Engineering antifragile systems: A change in design philosophy. Procedia Comput. Sci.
**2014**, 32, 870–875. [Google Scholar] [CrossRef] [Green Version] - Kauffman, S.A. Metabolic Stability and Epigenesis in Randomly Constructed Genetic Nets. J. Theor. Biol.
**1969**, 22, 437–467. [Google Scholar] [CrossRef] - Aldana-González, M.; Coppersmith, S.; Kadanoff, L.P. Boolean Dynamics with Random Couplings. In Perspectives and Problems in Nonlinear Science. A Celebratory Volume in Honor of Lawrence Sirovich; Applied Mathematical Sciences Series; Kaplan, E., Marsden, J.E., Sreenivasan, K.R., Eds.; Springer: Berlin, Germany, 2003. [Google Scholar] [CrossRef] [Green Version]
- Gershenson, C. Introduction to Random Boolean Networks. In Workshop and Tutorial Proceedings, Ninth International Conference on the Simulation and Synthesis of Living Systems (ALife IX), Boston, MA, USA, 12–15 September 2004; Bedau, M., Husbands, P., Hutton, T., Kumar, S., Suzuki, H., Eds.; ISAL: Boston, MA, USA, 2004; pp. 160–173. [Google Scholar] [CrossRef]
- Luque, B.; Solé, R.V. Phase Transitions in Random Networks: Simple Analytic Determination of Critical Points. Phys. Rev. E
**1997**, 55, 257–260. [Google Scholar] [CrossRef] [Green Version] - Gershenson, C. Guiding the Self-organization of Random Boolean Networks. Theory Biosci.
**2012**, 131, 181–191. [Google Scholar] [CrossRef] [Green Version] - von Neumann, J. The Theory of Self-Reproducing Automata; Burks, A.W., Ed.; University of Illinois Press: Champaign, IL, USA, 1966. [Google Scholar]
- Wolfram, S. Statistical mechanics of cellular automata. Rev. Mod. Phys.
**1983**, 55, 601–644. [Google Scholar] [CrossRef] - Wuensche, A.; Lesser, M. The Global Dynamics of Cellular Automata; An Atlas of Basin of Attraction Fields of One-Dimensional Cellular Automata; Santa Fe Institute Studies in the Sciences of Complexity, Addison-Wesley: Reading, MA, USA, 1992. [Google Scholar]
- Griffiths, R.B. Nonanalytic Behavior Above the Critical Point in a Random Ising Ferromagnet. Phys. Rev. Lett.
**1969**, 23, 17–19. [Google Scholar] [CrossRef] - Bray, A. Nature of the Griffiths phase. Phys. Rev. Lett.
**1987**, 59, 586. [Google Scholar] [CrossRef] [PubMed] - Albert, R.; Barabási, A.L. Statistical mechanics of complex networks. Rev. Mod. Phys.
**2002**, 74, 47–97. [Google Scholar] [CrossRef] [Green Version] - Newman, M.E.J. The structure and function of complex networks. SIAM Rev.
**2003**, 45, 167–256. [Google Scholar] [CrossRef] [Green Version] - Newman, M.; Barabási, A.L.; Watts, D.J. (Eds.) The Structure and Dynamics of Networks; Princeton Studies in Complexity, Princeton University Press: Princeton, NJ, USA, 2006. [Google Scholar]
- Munoz, M.A.; Juhász, R.; Castellano, C.; Ódor, G. Griffiths phases on complex networks. Phys. Rev. Lett.
**2010**, 105, 128701. [Google Scholar] [CrossRef] [Green Version] - Barabási, A.L. Network Science; Cambridge University Press: Cambridge, UK, 2016. [Google Scholar]
- Gershenson, C.; Helbing, D. When slower is faster. Complexity
**2015**, 21, 9–15. [Google Scholar] [CrossRef] [Green Version] - Carreón, G.; Gershenson, C.; Pineda, L.A. Improving public transportation systems with self-organization: A headway-based model and regulation of passenger alighting and boarding. PLoS ONE
**2017**, 12, e0190100. [Google Scholar] [CrossRef] - Santos, F.C.; Pacheco, J.M.; Lenaerts, T. Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc. Natl. Acad. Sci. USA
**2006**, 103, 3490–3494. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhou, B.; Lu, X.; Holme, P. Universal evolution patterns of degree assortativity in social networks. Soc. Netw.
**2020**, 63, 47–55. [Google Scholar] [CrossRef] - Oosawa, C.; Savageau, M.A. Effects of Alternative Connectivity on Behavior of Randomly Constructed Boolean Networks. Phys. D
**2002**, 170, 143–161. [Google Scholar] [CrossRef] - Aldana, M. Boolean Dynamics of Networks with Scale-Free Topology. Phys. D
**2003**, 185, 45–66. [Google Scholar] [CrossRef] - Gershenson, C.; Kauffman, S.A.; Shmulevich, I. The Role of Redundancy in the Robustness of Random Boolean Networks. In Artificial Life X, Proceedings of the Tenth International Conference on the Simulation and Synthesis of Living Systems; Rocha, L.M., Yaeger, L.S., Bedau, M.A., Floreano, D., Goldstone, R.L., Vespignani, A., Eds.; MIT Press: Bloomington, IN, USA, 2006; pp. 35–42. [Google Scholar] [CrossRef]
- Sánchez-Puig, F.; Zapata, O.; Pineda, O.K.; Iñiguez, G.; Gershenson, C. Heterogeneity Extends Criticality. arXiv
**2022**, arXiv:2208.06439. [Google Scholar] - Schlosser, G.; Wagner, G.P. Modularity in Development and Evolution; The University of Chicago Press: Chicago, IL, USA, 2004. [Google Scholar]
- Callebaut, W.; Rasskin-Gutman, D. Modularity: Understanding the Development and Evolution of Natural Complex Systems; Vienna Series in Theoretical Biology; The MIT Press: Cambridge, MA, USA, 2005. [Google Scholar]
- Poblanno-Balp, R.; Gershenson, C. Modular Random Boolean Networks. In Artificial Life XII Proceedings of the Twelfth International Conference on the Synthesis and Simulation of Living Systems, Odense, Denmark, 19–23 August 2010; Fellermann, H., Dörr, M., Hanczyc, M.M., Laursen, L.L., Maurer, S., Merkle, D., Monnard, P.A., Sty, K., Rasmussen, S., Eds.; MIT Press: Odense, Denmark, 2010; pp. 303–304. [Google Scholar] [CrossRef]
- Smith, C.; Pechuan, X.; Puzio, R.S.; Biro, D.; Bergman, A. Potential unsatisfiability of cyclic constraints on stochastic biological networks biases selection towards hierarchical architectures. J. R. Soc. Interface
**2015**, 12, 20150179. [Google Scholar] [CrossRef] [PubMed] - Ódor, G.; Hartmann, B. Heterogeneity effects in power grid network models. Phys. Rev. E
**2018**, 98, 022305. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Vazquez, F.; Bonachela, J.A.; López, C.; Munoz, M.A. Temporal griffiths phases. Phys. Rev. Lett.
**2011**, 106, 235702. [Google Scholar] [CrossRef] [Green Version] - Iñiguez, G.; Pineda, C.; Gershenson, C.; Barabási, A.L. Universal dynamics of ranking. Nat. Commun.
**2022**, 13, 1646. [Google Scholar] [CrossRef] - Wang, S.J.; Zhou, C. Hierarchical modular structure enhances the robustness of self-organized criticality in neural networks. New J. Phys.
**2012**, 14, 023005. [Google Scholar] [CrossRef] [Green Version] - Moretti, P.; Muñoz, M.A. Griffiths phases and the stretching of criticality in brain networks. Nat. Commun.
**2013**, 4, 1–10. [Google Scholar] [CrossRef] [Green Version] - Ratnayake, P.; Weragoda, S.; Wansapura, J.; Kasthurirathna, D.; Piraveenan, M. Quantifying the Robustness of Complex Networks with Heterogeneous Nodes. Mathematics
**2021**, 9, 2769. [Google Scholar] [CrossRef] - Sormunen, S.; Gross, T.; Saramäki, J. Critical drift in a neuro-inspired adaptive network. arXiv
**2022**, arXiv:2206.10315v1. [Google Scholar] - Drossel, B. Random Boolean networks. In Reviews of Nonlinear Dynamics and Complexity; Schuster, H.G., Ed.; Wiley-VCH Verlag GmbH & Co: Weinheim, Germany, 2008; pp. 69–110. [Google Scholar] [CrossRef]
- Gershenson, C. Updating Schemes in Random Boolean Networks: Do They Really Matter? In Artificial Life IX Proceedings of the Ninth International Conference on the Simulation and Synthesis of Living Systems, Boston, MA, USA, 12–15 September 2004; Pollack, J., Bedau, M., Husbands, P., Ikegami, T., Watson, R.A., Eds.; MIT Press: Cambridge, MA, USA, 2004; pp. 238–243. [Google Scholar]
- Derrida, B.; Pomeau, Y. Random Networks of Automata: A Simple Annealed Approximation. Europhys. Lett.
**1986**, 1, 45–49. [Google Scholar] [CrossRef] - Gershenson, C. Classification of Random Boolean Networks. In Artificial Life VIII: Proceedings of the Eight International Conference on Artificial Life, Sydney, Australia, 9–13 December 2002; Standish, R.K., Bedau, M.A., Abbass, H.A., Eds.; MIT Press: Cambridge, MA, USA, 2002; pp. 1–8. [Google Scholar] [CrossRef]
- Lloyd, S. Measures of Complexity: A Non-Exhaustive List; Technical Report; Department of Mechanical Engineering, Massachusetts Institute of Technology: Cambridge, MA, USA, 2001. [Google Scholar]
- Edmonds, B. Syntactic Measures of Complexity. Ph.D. Thesis, University of Manchester, Manchester, UK, 1999. [Google Scholar]
- Prokopenko, M.; Boschetti, F.; Ryan, A.J. An Information-Theoretic Primer On Complexity, Self-Organisation And Emergence. Complexity
**2009**, 15, 11–28. [Google Scholar] [CrossRef] - De Domenico, M.; Camargo, C.; Gershenson, C.; Goldsmith, D.; Jeschonnek, S.; Kay, L.; Nichele, S.; Nicolás, J.; Schmickl, T.; Stella, M.; et al. Complexity Explained: A Grassroot Collaborative Initiative to Create a Set of Essential Concepts of Complex Systems. Available online: https://complexityexplained.github.io (accessed on 1 September 2022). [CrossRef]
- Lopez-Ruiz, R.; Mancini, H.L.; Calbet, X. A statistical measure of complexity. Phys. Lett. A
**1995**, 209, 321–326. [Google Scholar] [CrossRef] [Green Version] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423, 623–656. [Google Scholar] [CrossRef] - Fernández, N.; Maldonado, C.; Gershenson, C. Information Measures of Complexity, Emergence, Self-organization, Homeostasis, and Autopoiesis. In Guided Self-Organization: Inception; Emergence, Complexity and Computation; Prokopenko, M., Ed.; Springer: Berlin/Heidelberg, Germany, 2014; Volume 9, pp. 19–51. [Google Scholar] [CrossRef] [Green Version]
- Santamaría-Bonfil, G.; Gershenson, C.; Fernández, N. A Package for Measuring Emergence, Self-organization, and Complexity Based on Shannon Entropy. Front. Robot. AI
**2017**, 4, 10. [Google Scholar] [CrossRef] - Pineda, O.K.; Kim, H.; Gershenson, C. A Novel Antifragility Measure Based on Satisfaction and Its Application to Random and Biological Boolean Networks. Complexity
**2019**, 2019, 10. [Google Scholar] [CrossRef] [Green Version] - Monton, B. God, fine-tuning, and the problem of old evidence. Br. J. Philos. Sci.
**2006**, 57, 405–424. [Google Scholar] [CrossRef] [Green Version] - Solé, R.V.; Luque, B.; Kauffman, S.A. Phase Transitions in Random Networks with Multiple States; Technical Report 00-02-011; Santa Fe Institute: Santa Fe, NM, USA, 2000. [Google Scholar]
- Kosmann-Schwarzbach, Y. The Noether Theorems. In The Noether Theorems; Springer: New York, NY, USA; Dordrecht/Heidelberg, Germany; London, UK, 2011; pp. 55–64. [Google Scholar] [CrossRef]
- Luque, B.; Solé, R.V. Lyapunov exponents in random Boolean networks. Phys. A Stat. Mech. Its Appl.
**2000**, 284, 33–45. [Google Scholar] [CrossRef] [Green Version] - Roy, M.; Pascual, M.; Franc, A. Broad scaling region in a spatial ecological system. Complexity
**2003**, 8, 19–27. [Google Scholar] [CrossRef] [Green Version] - Zapata, O.; Gershenson, C. Random Fuzzy Networks. In Artificial Life 14: Proceedings of the Fourteenth International Conference on the Synthesis and Simulation of Living Systems, New York, NY, USA, 30 July–2 August 2014; Sayama, H., Rieffel, J., Risi, S., Doursat, R., Lipson, H., Eds.; MIT Press: Cambridge, MA, USA, 2014; pp. 427–428. [Google Scholar] [CrossRef]
- Zapata, O.; Kim, H.; Gershenson, C. k On two information-theoretic measures of random fuzzy networks. In Artificial Life Conference Proceedings; MIT Press: Cambridge, MA, USA, 2020; Volume 32, pp. 623–625. [Google Scholar] [CrossRef]
- Hoel, E.P.; Albantakis, L.; Tononi, G. Quantifying causal emergence shows that macro can beat micro. Proc. Natl. Acad. Sci. USA
**2013**, 110, 19790–19795. [Google Scholar] [CrossRef] [PubMed] - Pascual, M.; Roy, M.; Laneri, K. Simple models for complex systems: Exploiting the relationship between local and global densities. Theor. Ecol.
**2011**, 4, 211–222. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gross, T.; Sayama, H. (Eds.) Adaptive Networks: Theory, Models and Applications; Understanding Complex Systems; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar] [CrossRef]
- Khajehabdollahi, S.; Prosi, J.; Giannakakis, E.; Martius, G.; Levina, A. When to Be Critical? Performance and Evolvability in Different Regimes of Neural Ising Agents. Artif. Life
**2022**, 28, 458–478. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Illustration of a random Boolean network with $N=9$ nodes and $K=2$ inputs per node (self-connections are allowed). The node rules are commonly represented by lookup tables, which associate a 1-bit output (the node’s future state) to each ${2}^{K}$ possible K-bit input configuration. The out-column is commonly called the “rule” of the node.

**Figure 2.**Phase diagram described by Derrida and Pomeau [48]. The solid curve represents the critical connectivity ${K}_{c}={\left[2p(1-p)\right]}^{-1}$. When $K<{K}_{c}$, the network exhibits ordered dynamics (small shaded area below ${K}_{c}$), while for $K>{K}_{c}$, the network exhibits chaotic dynamics (large shaded area above ${K}_{c}$).

**Figure 3.**Example of three regimes of CRBN using 50 nodes ($N=50$) with 200 steps each (time flows downwards). The resulting complexities C (see Equation (3)) are: for $K=1$, $C=0.0513$, for $K=2$, $C=0.8651$, and for $K=5$, $C=0.3079$.

**Figure 4.**Average complexity C of RBNs as the average connectivity K is increased for the eight cases. The labels describe the combinations used; He stands for heterogeneity, Ho for homogeneity, T for temporal, S for structural, and F for functional. The dissimilarities of these curves for different numbers of nodes in the network are shown. With $N=100$ nodes (

**a**), it is observed that, approximately from $K=6$, the triple heterogeneity (3He) dominates the other cases. For $N=75$ (

**b**), the order of the curves is preserved, but the value of K needed to demarcate the dominance of 3He is higher, reaching a value close to $K=7$. With $N=50$ nodes (

**c**), the curves look much flatter; despite the increase of noise in the curves, the order of the curves is still preserved, but now, an approximate value of $K=8$ is needed to visualize that 3He outperforms the other cases. At $N=25$ (

**d**), too much noise accumulates in the curves; however, by following the trend of the 3He curve, it can be stated that the order will still be preserved, but now, a much larger K is required to display it. For all curves, a step of $\mathsf{\Delta}K=0.2$ was used, giving a total of 45 K-connectivities. For each of these connectivity values, 1000 networks were generated, and their complexities were averaged. To obtain these results, 2000 time steps were used.

**Figure 5.**State transitions by iteration for functional homogeneity. Time flows down (the initial state is the top row). In the upper part of the subfigures, we indicate the distributions used for each case. “Poi” stands for Poisson, “Exp” for exponential, “Out” for out-degree, and “Non” for none. At the bottom of each subfigure, the average complexity (per iteration) for each case is shown. The iterations for HoSHoTHoF (

**a**) resemble white noise; this coincides with the fact that the complexity obtained is very low. Adding only temporal heterogeneity (

**b**) gives a little more structure, but the complexity is still very low. Adding only structural heterogeneity (

**c**) does not seem to gain as much structure in the iterations, but still, the complexity is higher than in HoSHeTHoF. Adding both temporal and structural heterogeneity (

**d**) gives more structure to the dynamics, and the complexity increases more. To generate these images, we considered a network with 100 nodes, 100 steps each (time flows downwards), and a connectivity of $K=10$.

**Figure 6.**State transitions by iteration for functional heterogeneity. In the upper part of the subfigures, we indicate the distributions used for each case. “Poi” stands for Poisson, “Exp” for exponential, “Out” for out-degree, “Tri” for triangular, and “Non” for none. At the bottom of each subfigure, the average complexity (per iteration) for each case is shown. When only functional heterogeneity is added (

**a**), greater complexity is observed with respect to HoSHeTHoF and HeSHoTHoF. Adding functional and structural heterogeneity (

**b**) results in similar complexity to the HeSHeTHoF case, but having functional and temporal heterogeneity (

**c**) keeps the complexity at an intermediate value. This is consistent with what we saw previously: HeS increases the complexity more than HeT. Finally, HeSHeTHeF (

**d**) is the case that gives the major structure to the iterations. In fact, complexity is close to maximal for this case. To generate these images, we considered a network with 100 nodes, 100 steps each (time flows downwards), and a connectivity of $K=10$.

**Figure 7.**Average complexity of RBNs as the average connectivity K is increased for variations in the functional parameters of the triple heterogeneity (3He) case. (

**a**) shows the changes obtained by testing seven different functional distributions. Point-like indicates the extreme case when the standard deviation is zero; this is equivalent to the homogeneous functional case. When the distribution is Gaussian, convergence is observed after a certain value for the standard deviation. This behavior is not generalizable, because, with the uniform distribution, something similar is obtained, and its standard deviation is smaller. By varying only the mean of the functional distribution (

**b**), we observe symmetric behavior, so that varying the mean only matters how far we are from $\mu =0.5$. (

**c**,

**d**) show what happens when the domain of the functional distribution is modified. In both cases, a similar order in the curves is observed, and also, as we make the interval shorter, the curves stick together. For all curves, a step of $\mathsf{\Delta}K=0.2$ was used, and in every step, a total of 1000 iterations were averaged.

**Figure 8.**Average complexity of RBNs as the average connectivity K is increased for two temporal strategies using two functional distributions. For the triangular distribution, out-degree achieves a higher complexity for relatively large K. However, it is clear that, for most values of K, the Ceil strategy is dominant. For the uniform distribution, Ceil is always superior to out-degree. For very large values of K, a tie is observed, but out-degree never seems to completely outperform Ceil. For all curves, a step of $\mathsf{\Delta}K=0.2$ was used, and in every step, a total of 1000 iterations were averaged.

**Figure 9.**Average antifragility of ordered, critical, and chaotic RBNs depending on X and O for the 3Ho and 3He cases. In the homogeneous case (

**a**,

**b**), only the ordered and critical networks achieve antifragility intervals, the former reaching much larger values. Adding triple heterogeneity (

**c**,

**d**) shows a reversal in the order of the previously chaotic networks, and in this case, two of the three curves for $K>2$ reach antifragility intervals. Although, in the heterogeneous case, the number of curves attaining antifragility increases and the length of the intervals with antifragility is extended, in the homogeneous case, much larger antifragility values are obtained. To generate the fragility curves against O, a value of $X=40$ was used. To generate the fragility curves against X, a value of $O=1$ was used. For all curves, steps of $\mathsf{\Delta}X=1$, $\mathsf{\Delta}O=1$ were used, and in every step, a total of 1000 iterations were averaged. To obtain these results, 200 time steps were used.

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**MDPI and ACS Style**

López-Díaz, A.J.; Sánchez-Puig, F.; Gershenson, C.
Temporal, Structural, and Functional Heterogeneities Extend Criticality and Antifragility in Random Boolean Networks. *Entropy* **2023**, *25*, 254.
https://doi.org/10.3390/e25020254

**AMA Style**

López-Díaz AJ, Sánchez-Puig F, Gershenson C.
Temporal, Structural, and Functional Heterogeneities Extend Criticality and Antifragility in Random Boolean Networks. *Entropy*. 2023; 25(2):254.
https://doi.org/10.3390/e25020254

**Chicago/Turabian Style**

López-Díaz, Amahury Jafet, Fernanda Sánchez-Puig, and Carlos Gershenson.
2023. "Temporal, Structural, and Functional Heterogeneities Extend Criticality and Antifragility in Random Boolean Networks" *Entropy* 25, no. 2: 254.
https://doi.org/10.3390/e25020254