# From Color-Avoiding to Color-Favored Percolation in Diluted Lattices

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Percolation and Directed Percolation

## 3. Self-organized Criticality and the Fragment Method

## 4. Color-Avoiding Percolation

#### Site and Bond Color-Avoiding Percolation

- Independent colors: nodes can have at the same time more than one vulnerability/color;

## 5. Applications

## 6. Multilayer Model

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Broadbent, S.R.; Hammersley, J.M. Percolation processes: I. Crystals and mazes. In Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambridge, UK, 1957; Volume 53, pp. 629–641. [Google Scholar] [CrossRef]
- Stauffer, D. Scaling theory of percolation clusters. Phys. Rep.
**1979**, 54, 1–74. [Google Scholar] [CrossRef] - Stauffer, D.; Aharony, A. Introduction to Percolation Theory; Taylor and Francis: Abingdon, UK, 1992. [Google Scholar]
- Bellini, E.; Bagnoli, F.; Massaro, E.; Rechtman, R. Percolation and Internet Science. Future Internet
**2019**, 11, 35. [Google Scholar] [CrossRef] [Green Version] - Albert, R.; Jeong, H.; Barabási, A.L. Error and attack tolerance of complex networks. Nature
**2000**, 406, 378–382. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Callaway, D.S.; Newman, M.E.; Strogatz, S.H.; Watts, D.J. Network robustness and fragility: Percolation on random graphs. Phys. Rev. Lett.
**2000**, 85, 5468. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Newman, M.E.; Watts, D.J. Scaling and percolation in the small-world network model. Phys. Rev. E
**1999**, 60, 7332. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cardy, J.L.; Grassberger, P. Epidemic models and percolation. J. Phys. Math. Gen.
**1985**, 18, L267. [Google Scholar] [CrossRef] - Kermack, W.O.; McKendrick, A.G. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond.
**1927**, 115, 700–721. [Google Scholar] [CrossRef] [Green Version] - Brauer, F.; Castillo-Chavez, C.; Castillo-Chavez, C. Mathematical Models in Population Biology and Epidemiology; Springer: Berlin/Heidelberg, Germany, 2012; Volume 2. [Google Scholar] [CrossRef]
- Murray, J.D. Mathematical Biology: I. An introduction; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2007; Volume 17. [Google Scholar]
- Kephart, J.; White, S. Directed-graph epidemiological models of computer viruses; IEEE computer society symposium on research in security and privacy. In Proceedings of the IEEE Computer Society Symposium on Research in Security and Privacy, Oakland, CA, USA, 20–22 May 1991; pp. 343–359. [Google Scholar] [CrossRef]
- Pastor-Satorras, R.; Vespignani, A. Epidemic Spreading in Scale-Free Networks. Phys. Rev. Lett.
**2001**, 86. [Google Scholar] [CrossRef] [Green Version] - Krause, S.M.; Danziger, M.M.; Zlatić, V. Hidden Connectivity in Networks with Vulnerable Classes of Nodes. Phys. Rev. X
**2016**, 6, 041022. [Google Scholar] [CrossRef] [Green Version] - Krause, S.M.; Danziger, M.M.; Zlatić, V. Color-avoiding percolation. Phys. Rev. E
**2017**, 96, 022313. [Google Scholar] [CrossRef] [Green Version] - Kadović, A.; Krause, S.M.; Caldarelli, G.; Zlatic, V. Bond and site color-avoiding percolation in scale-free networks. Phys. Rev. E
**2018**, 98, 062308. [Google Scholar] [CrossRef] [Green Version] - Giusfredi, M.; Bagnoli, F. A Self-organized Criticality Method for the Study of Color-Avoiding Percolation. In Proceedings of the Internet Science: 6th International Conference (INSCI 2019), Perpignan, France, 2–5 December 2019; p. 217. [Google Scholar] [CrossRef]
- Bagnoli, F.; Palmerini, P.; Rechtman, R. Algorithmic mapping from criticality to self-organized criticality. Phys. Rev. E
**1997**, 55, 3970–3976. [Google Scholar] [CrossRef] - Kinzel, W. Directed Percolation. In Percolation Structures and Processes; Deutscher, G., Zallen, R., Adler, J., Eds.; Adam Hilger: Bristol, UK, 1983; Volume 5. [Google Scholar]
- Kinzel, W. Phase transitions of cellular automata. Z. für Phys. B Condens. Matter
**1985**, 58, 229–244. [Google Scholar] [CrossRef] - Domany, E.; Kinzel, W. Equivalence of Cellular Automata to Ising Models and Directed Percolation. Phys. Rev. Lett.
**1984**, 53, 311–314. [Google Scholar] [CrossRef] - Hinrichsen, H. Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys.
**2000**, 49, 815–958. [Google Scholar] [CrossRef] - Martins, M.L.; Verona de Resende, H.F.; Tsallis, C.; de Magalhes, A.C.N. Evidence for a new phase in the Domany-Kinzel cellular automaton. Phys. Rev. Lett.
**1991**, 66, 2045–2047. [Google Scholar] [CrossRef] [Green Version] - Bagnoli, F.; Rechtman, R. Phase Transitions of Cellular Automata. In Probabilistic Cellular Automata: Theory, Applications and Future Perspectives; Louis, P.Y., Nardi, F.R., Eds.; Springer International Publishing: Cham, Switzerland, 2018; pp. 215–236. [Google Scholar] [CrossRef] [Green Version]
- Bagnoli, F.; Lio, P.; Sguanci, L. Risk perception in epidemic modeling. Phys. Rev. E
**2007**, 76, 061904. [Google Scholar] [CrossRef] [Green Version] - Bagnoli, F.; Bellini, E.; Massaro, E. Risk Perception and Epidemics in Complex Computer Networks. In Proceedings of the 2018 IEEE Workshop on Complexity in Engineering (COMPENG), Florence, Italy, 10–12 October 2018; pp. 1–5. [Google Scholar]
- Bak, P.; Tang, C.; Wiesenfeld, K. Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett.
**1987**, 59, 381–384. [Google Scholar] [CrossRef] - Bak, P.; Sneppen, K. Punctuated equilibrium and criticality in a simple model of evolution. Phys. Rev. Lett.
**1993**, 71, 4083–4086. [Google Scholar] [CrossRef] - Wilkinson, D.; Willemsen, J.F. Invasion percolation: A new form of percolation theory. J. Phys. Math. Gen.
**1983**, 16, 3365–3376. [Google Scholar] [CrossRef] - Grassberger, P.; Zhang, Y.C. Self-organized formulation of standard percolation phenomena. Phys. Stat. Mech. Its Appl.
**1996**, 224, 169–179. [Google Scholar] [CrossRef] - Aharony, A.; Harris, A.B. Absence of self-averaging and universal fluctuations in random systems near critical points. Phys. Rev. Lett.
**1996**, 77, 3700. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**The phase diagram of the DK model, $\alpha $ marks the critical line separating the active and inactive (or absorbing) phase, $\gamma $ marks the “chaotic” phase near the corner $p=1,q=0$. The simplest mean-field approximation gives ${p}_{c}=1/2$ independently of q.

**Figure 2.**Phase planes of the bond CAP for two exclusive colors (denoted color 1 and 2). assigned to sites with probability $p(1)$ and $p(2)=1-p(1)$ and different dilutions (different connectivities).

**Figure 3.**Phase planes of the bond CAP for three exclusive colors, in networks with average connectivity $k=5$, at different values of the dilution parameter $\varphi $.

**Figure 4.**Phase planes of the bond CAP for three exclusive colors, with one trusted color with fixed probability $p(3)=0.2$, varying the average connectivity k of the network.

**Figure 5.**Phase space of directed percolation with interacting colors, case (a). The vertical axes is p and the horizontal one is q. For moderate values of $\epsilon $ one simply has a shift of the critical line, but for $\epsilon \ge 0.75$ a region near the corner $p=q=0$ becomes active.

**Figure 6.**Phase space of directed percolation with interacting colors, case (b). The vertical axes is p and the horizontal one is q. In this case, (disruptive interference among layers) the active region tends to occupy the whole line $p=0$.

© 2020 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**

Giusfredi, M.; Bagnoli, F.
From Color-Avoiding to Color-Favored Percolation in Diluted Lattices. *Future Internet* **2020**, *12*, 139.
https://doi.org/10.3390/fi12080139

**AMA Style**

Giusfredi M, Bagnoli F.
From Color-Avoiding to Color-Favored Percolation in Diluted Lattices. *Future Internet*. 2020; 12(8):139.
https://doi.org/10.3390/fi12080139

**Chicago/Turabian Style**

Giusfredi, Michele, and Franco Bagnoli.
2020. "From Color-Avoiding to Color-Favored Percolation in Diluted Lattices" *Future Internet* 12, no. 8: 139.
https://doi.org/10.3390/fi12080139