# Generalized Ising Model on a Scale-Free Network: An Interplay of Power Laws

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## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Ising Model on an Annealed Network

#### 2.2. Ising Model with Random Spin Length on an Annealed Network

#### 2.3. Self-Averaging

## 3. Thermodynamic Functions

## 4. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Integral I μ (ε)

#### Integral I λ,μ (ε)

- $2<\lambda <3$:$$\begin{array}{c}\hfill {I}_{\lambda ,\mu}\left(\epsilon \right)-{i}_{\lambda ,\mu}=\left(\right)open="\{"\; close>\begin{array}{ccc}& {i}_{\lambda}\frac{{\epsilon}^{\lambda -\mu}}{\mu -\lambda}+O\left({\epsilon}^{\lambda -\mu +2}\right),\hfill & 2\mu 3\phantom{\rule{0.166667em}{0ex}},\hfill \\ & {i}_{\lambda}\frac{{\epsilon}^{\lambda -3}}{3-\lambda}+\frac{ln\epsilon}{3-\lambda}{\epsilon}^{3-\lambda}+O\left({\epsilon}^{\lambda -1}\right),\hfill & \mu =3\phantom{\rule{0.166667em}{0ex}},\hfill \\ & {i}_{\lambda}\frac{{\epsilon}^{\lambda -\mu}}{\mu -\lambda}-\frac{{\epsilon}^{6-\lambda -\mu}}{2(6-\lambda -\mu )(\mu -3)}+O\left({\epsilon}^{\lambda -\mu +2}\right),\hfill & 3\mu 5\phantom{\rule{0.166667em}{0ex}},\hfill \\ & {i}_{\lambda}\frac{{\epsilon}^{\lambda -5}}{5-\lambda}-\frac{{\epsilon}^{1-\lambda}}{4(1-\lambda )}+O\left({\epsilon}^{\lambda -3}\right),\hfill & \mu =5\phantom{\rule{0.166667em}{0ex}},\hfill \\ & {i}_{\lambda}\frac{{\epsilon}^{\lambda -\mu}}{\mu -\lambda}-\frac{{\epsilon}^{6-\lambda -\mu}}{2(6-\lambda -\mu )(\mu -3)}+O\left({\epsilon}^{\lambda -\mu +2}\right),\hfill & 5\mu 7\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}\end{array}$$
- $\lambda =3$:$$\begin{array}{c}\hfill {I}_{\lambda ,\mu}\left(\epsilon \right)-{i}_{\lambda ,\mu}=\left(\right)open="\{"\; close>\begin{array}{ccc}& \frac{{\epsilon}^{3-\mu}}{3-\mu}ln\epsilon -{\epsilon}^{3-\mu}[\frac{{i}_{3}}{3-\mu}+\frac{1}{2{(\mu -3)}^{2}}]+O({\epsilon}^{\mu -1},{\epsilon}^{7-\mu}),\hfill & 3\mu 5\phantom{\rule{0.166667em}{0ex}},\hfill \\ & {\epsilon}^{-2}[{i}_{3}/2-1/8]-{\epsilon}^{-2}ln\epsilon /2+O\left({\epsilon}^{2}\right),\hfill & \mu =5\phantom{\rule{0.166667em}{0ex}},\hfill \\ & \frac{{\epsilon}^{3-\mu}}{3-\mu}ln\epsilon -{\epsilon}^{3-\mu}[\frac{{i}_{3}}{3-\mu}+\frac{1}{2{(\mu -3)}^{2}}]+O\left({\epsilon}^{7-\mu}\right),\hfill & 5\mu 7\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}\end{array}$$
- $3<\lambda <5$:$$\begin{array}{c}\hfill {I}_{\lambda ,\mu}\left(\epsilon \right)-{i}_{\lambda ,\mu}=\left(\right)open="\{"\; close>\begin{array}{ccc}& {i}_{\lambda}\frac{{\epsilon}^{\lambda -\mu}}{\mu -\lambda}+\frac{{\epsilon}^{6-\lambda -\mu}}{2(\lambda -3)(\mu -3)}+O\left({\epsilon}^{\lambda -\mu +2}\right),\hfill & 3\mu 5\phantom{\rule{0.166667em}{0ex}},\hfill \\ & {i}_{\lambda}\frac{{\epsilon}^{\lambda -5}}{5-\lambda}+\frac{{\epsilon}^{1-\lambda}}{4(\lambda -3)}+O\left({\epsilon}^{\lambda -3}\right),\hfill & \mu =5\phantom{\rule{0.166667em}{0ex}},\hfill \\ & {i}_{\lambda}\frac{{\epsilon}^{\lambda -\mu}}{\mu -\lambda}+\frac{{\epsilon}^{6-\lambda -\mu}}{2(\lambda -3)(\mu -3)}-\hfill \\ & \frac{{\epsilon}^{10-\lambda -\mu}}{12(10-\lambda -\mu )(\mu -5)}+O\left({\epsilon}^{\lambda -\mu +2}\right),\hfill & 5\mu 7\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}\end{array}$$
- $\lambda =5$, $5<\mu <7$:$${I}_{\lambda ,\mu}\left(\epsilon \right)-{i}_{5,\mu}=\frac{{\epsilon}^{5-\mu}}{6(5-\mu )}ln\epsilon -{\epsilon}^{5-\mu}[\frac{{i}_{5}}{5-\mu}+\frac{1}{12{(\mu -5)}^{2}}]+\frac{{\epsilon}^{1-\mu}}{4(\mu -3)}+O\left({\epsilon}^{7-\mu}\right).$$
- $5<\lambda <7$, $5<\mu <7$:$${I}_{\lambda ,\mu}\left(\epsilon \right)-{i}_{\lambda ,\mu}={i}_{\lambda}\frac{{\epsilon}^{\mu -\lambda}}{\lambda -\mu}+\frac{{\epsilon}^{6-\lambda -\mu}}{2(\lambda -3)(\mu -3)}-\frac{{\epsilon}^{10-\lambda -\mu}}{12(\lambda -5)(\mu -5)}+O\left({\epsilon}^{\lambda -\mu +2}\right).$$

## References

- Brush, S.G. History of the Lenz-Ising Model. Rev. Mod. Phys.
**1967**, 39, 883–893. [Google Scholar] [CrossRef] - Martin, N. History of the Lenz-Ising Model 1920-1950: From Ferromagnetic to Cooperative Phenomena. Arch. Hist. Exact Sci.
**2005**, 59, 267–318. [Google Scholar] [CrossRef][Green Version] - Martin, N. History of the Lenz–Ising Model 1950–1965: From irrelevance to relevance. Arch. Hist. Exact Sci.
**2009**, 63, 243. [Google Scholar] [CrossRef] - Martin, N. History of the Lenz-Ising model 1965-1971: The role of a simple model in understanding critical phenomena. Arch. Hist. Exact Sci.
**2011**, 65, 625–658. [Google Scholar] [CrossRef] - Kobe, S. History of the Lenz-Ising Model. J. Stat. Phys.
**1997**, 88, 1572–9613. [Google Scholar] [CrossRef][Green Version] - Sornette, D. Physics and financial economics (1776–2014): Puzzles, Ising and agent-based models. Rep. Prog. Phys.
**2014**, 77, 062001. [Google Scholar] [CrossRef] - Available online: http://www.icmp.lviv.ua/ising/ (accessed on 5 September 2021).
- Holovatch, Y. Order, Disorder and Criticality. Advanced Problems of Phase Transition Theory; World Scientific: Singapore, 2004; Volume 1, p. 304. [Google Scholar] [CrossRef]
- Holovatch, Y. Order, Disorder and Criticality. Advanced Problems of Phase Transition Theory; World Scientific: Singapore, 2007; Volume 2, p. 308. [Google Scholar] [CrossRef]
- Holovatch, Y. Order, Disorder and Criticality. Advanced Problems of Phase Transition Theory; World Scientific: Singapore, 2012; Volume 3, p. 248. [Google Scholar] [CrossRef][Green Version]
- Holovatch, Y. Order, Disorder and Criticality. Advanced Problems of Phase Transition Theory; World Scientific: Singapore, 2015; Volume 4, p. 232. [Google Scholar] [CrossRef]
- Holovatch, Y. Order, Disorder and Criticality. Advanced Problems of Phase Transition Theory; World Scientific: Singapore, 2018; Volume 5, p. 412. [Google Scholar] [CrossRef]
- Holovatch, Y. Order, Disorder and Criticality. Advanced Problems of Phase Transition Theory; World Scientific: Singapore, 2020; Volume 6, p. 296. [Google Scholar] [CrossRef]
- Ising, E. Beitrag zur Theorie des Ferromagnetismus. Zeitschrift fur Physik
**1925**, 31, 253–258. [Google Scholar] [CrossRef] - Ising, E. Beitrag zur Theorie des Ferro- und Paramagnetismus. Ph.D. Dissertation, Hamburgischen Universität, Hamburg, Germany, Grete & Tiedel. 1924. Available online: http://www.icmp.lviv.ua/ising/books/isingshort.pdf (accessed on 5 September 2021).
- Jane (Johanna) Ehmer Ising. Walk on a Tightrope or Paradise Lasted a Year and a Half. Unpublished, 112p.. Available online: http://www.icmp.lviv.ua/ising/books/Jana_Ehmer_Ising.pdf (accessed on 5 September 2021).
- Ising, T.; Folk, R.; Kenna, R.; Berche, B.; Holovatch, Y. The Fate of Ernst Ising and the Fate of his Model. J. Phys. Stud.
**2017**, 21, 4001. [Google Scholar] [CrossRef] - Available online: http://www.lfour.org/CDFA-DFDK/index.html (accessed on 5 September 2021).
- Stauffer, D. Grand unification of exotic statistical physics. Phys. A Stat. Mech. Appl.
**2000**, 285, 121–126. [Google Scholar] [CrossRef] - Holovatch, Y.; Kenna, R.; Thurner, S. Complex systems: Physics beyond physics. Eur. J. Phys.
**2017**, 38, 023002. [Google Scholar] [CrossRef] - Potts, R. Some generalized order-disorder transformations. Math. Proc. Camb. Philos. Soc.
**1952**, 48, 106–109. [Google Scholar] [CrossRef] - Wu, F.Y. The Potts model. Rev. Mod. Phys.
**1982**, 54, 235–268. [Google Scholar] [CrossRef] - Stanley, H.E. Dependence of Critical Properties on Dimensionality of Spins. Phys. Rev. Lett.
**1968**, 20, 589–592. [Google Scholar] [CrossRef] - Stanley, H. Phase Transitions and Critical Phenomena; Clarendon Press: Oxford, UK, 1971. [Google Scholar]
- Dorogovtsev, S.; Goltsev, A.V.; Mendes, J. Critical phenomena in complex networks. Rev. Mod. Phys.
**2008**, 80, 1275–1335. [Google Scholar] [CrossRef][Green Version] - Krasnytska, M.; Berche, B.; Holovatch, Y.; Kenna, R. Ising model with variable spin/agent strengths. J. Phys. Complex.
**2020**, 1, 035008. [Google Scholar] [CrossRef] - Mattis, D. Solvable spin systems with random interactions. Phys. Lett. A
**1976**, 56, 421–422. [Google Scholar] [CrossRef] - Bianconi, G. Mean field solution of the Ising model on a Barabási–Albert network. Phys. Lett. A
**2002**, 303, 166–168. [Google Scholar] [CrossRef][Green Version] - Pastur, L.; Figotin, A. Exactly soluble model of a spin glass. Sov. J. Low Temp. Phys.
**1977**, 3, 378–383. [Google Scholar] - Pastur, L.; Figotin, A. On the theory of disordered spin systems. Theor. Math. Phys.
**1978**, 35, 403–414. [Google Scholar] [CrossRef] - Hopfield, J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA
**1982**, 79, 2554–2558. [Google Scholar] [CrossRef][Green Version] - Mezard, M.; Parisi, G.; Virasoro, M. Spin Glass Theory and Beyond. An Introduction to the Replica Method and Its Applications; World Scientific: Singapore, 1986; Volume 9, p. 476. [Google Scholar] [CrossRef]
- Dotsenko, V. An Introduction to the Theory of Spin Glasses and Neural Networks; World Scientific: Singapore, 1994. [Google Scholar]
- Folk, R.; Holovatch, Y.; Yavors’kii, T. Critical exponents of a three dimensional weakly diluted quenched Ising model. Physics-Uspiekhi
**2003**, 46, 169–191. [Google Scholar] [CrossRef] - Tadić, B.; Malarz, K.; Kułakowski, K. Magnetization Reversal in Spin Patterns with Complex Geometry. Phys. Rev. Lett.
**2005**, 94, 137204. [Google Scholar] [CrossRef][Green Version] - Tadić, B.; Gupte, N. Hidden geometry and dynamics of complex networks: Spin reversal in nanoassemblies with pairwise and triangle-based interactions. EPL (Europhys. Lett.)
**2020**, 132, 60008. [Google Scholar] [CrossRef] - Galam, S. Sociophysics: A Physicists Modeling of Psycho—Political Phenomena; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Hołyst, J.A. (Ed.) Cyberemotions. Collective Emotions in Cyberspace; Springer Series “Understanding Complex Systems”; Springer: Berlin/Heidelberg, Germany, 2017; p. 318. [Google Scholar] [CrossRef]
- Leone, M.; Vázquez, A.; Vespignani, A.; Zecchina, R. Ferromagnetic ordering in graphs with arbitrary degree distribution. Eur. Phys. J. B
**2002**, 28, 191–197. [Google Scholar] [CrossRef][Green Version] - Goltsev, A.V.; Dorogovtsev, S.N.; Mendes, J.F.F. Critical phenomena in networks. Phys. Rev. E
**2003**, 67, 026123. [Google Scholar] [CrossRef][Green Version] - Palchykov, V.; von Ferber, C.; Folk, R.; Holovatch, Y. Coupled order-parameter system on a scale-free network. Phys. Rev. E
**2009**, 80, 011108. [Google Scholar] [CrossRef][Green Version] - Lee, S.H.; Ha, M.; Jeong, H.; Noh, J.D.; Park, H. Critical Behavior of the Ising model in annealed scale-free networks. Phys. Rev. E
**2009**, 80, 051127. [Google Scholar] [CrossRef][Green Version] - Bianconi, G. Superconductor-insulator transition on annealed complex networks. Phys. Rev. E
**2012**, 85, 061113. [Google Scholar] [CrossRef][Green Version] - Krasnytska, M.; Berche, B.; Holovatch, Y.; Kenna, R. Violation of Lee-Yang circle theorem for Ising phase transitions on complex networks. Europhys. Lett.
**2015**, 111, 60009. [Google Scholar] [CrossRef] - Krasnytska, M.; Berche, B.; Holovatch, Y.; Kenna, R. Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks. J. Phys. A Math. Theor.
**2016**, 49, 135001. [Google Scholar] [CrossRef][Green Version] - Brout, R. Statistical Mechanical Theory of a Random Ferromagnetic System. Phys. Rev.
**1959**, 115, 824–835. [Google Scholar] [CrossRef] - Dorogovtsev, S.N.; Goltsev, A.V.; Mendes, J.F.F. Ising model on networks with an arbitrary distribution of connections. Phys. Rev. E
**2002**, 66, 016104. [Google Scholar] [CrossRef][Green Version] - Cohen, R.; Ben Avraham, D.; Havlin, S. Percolation critical exponents in scale-free networks. Phys. Rev. E
**2002**, 66, 036113. [Google Scholar] [CrossRef][Green Version] - Aiello, W.; Park, F.; Lu, L. Proceedings of the STOC ’00: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing; Portland, OR, USA, 21–23 May 2000, Association for Computing Machinery: New York, NY, USA, 2000. [Google Scholar]
- Krasnytska, M.; Berche, B.; Holovatch, Y. Phase transitions in the Potts model on complex networks. Condens. Matter Phys.
**2013**, 16, 23602. [Google Scholar] [CrossRef] - von Ferber, C.; Folk, R.; Holovatch, Y.; Kenna, R.; Palchykov, V. Entropic equation of state and scaling functions near the critical point in uncorrelated scale-free networks. Phys. Rev. E
**2011**, 83, 061114. [Google Scholar] [CrossRef][Green Version] - Berlin, T.H.; Kac, M. The Spherical Model of a Ferromagnet. Phys. Rev.
**1952**, 86, 821–835. [Google Scholar] [CrossRef] - Kac, M. On the Partition Function of a One-Dimensional Gas. Phys. Fluids
**1959**, 2. [Google Scholar] [CrossRef] - Baker, G.A. One-Dimensional Order-Disorder Model Which Approaches a Second-Order Phase Transition. Phys. Rev.
**1961**, 122, 1477–1484. [Google Scholar] [CrossRef] - Baker, G.A. Ising Model with a Long-Range Interaction in the Presence of Residual Short-Range Interactions. Phys. Rev.
**1963**, 130, 1406–1411. [Google Scholar] [CrossRef] - Kac, M.; Helfand, E. Study of Several Lattice Systems with Long-Range Forces. J. Math. Phys.
**1963**, 4, 1078. [Google Scholar] [CrossRef] - Kac, M.; Uhlenbeck, G.E.; Hemmer, P.C. On the van der Waals Theory of the Vapor-Liquid Equilibrium. I. Discussion of a One-Dimensional Model. J. Math. Phys.
**1963**, 4, 216. [Google Scholar] [CrossRef] - Kac, M.; Thompson, J. Critical Behavior of Several Lattice Models with Long-Range Interaction. J. Math. Phys.
**1969**, 10, 1373. [Google Scholar] [CrossRef] - Kenna, R. Universal Scaling Relations for Logarithmic-Correction Exponents. In Order, Disorder and Criticality; World Scientific: Singapore, 2012; Volume 3, pp. 1–46. [Google Scholar] [CrossRef][Green Version]
- Kenna, R.; Johnston, D.; Janke, W. Scaling Relations for Logarithmic Corrections. Phys. Rev. Lett.
**2006**, 96, 115701. [Google Scholar] [CrossRef][Green Version] - Kenna, R.; Johnston, D.A.; Janke, W. Self-Consistent Scaling Theory for Logarithmic-Correction Exponents. Phys. Rev. Lett.
**2006**, 97, 155702. [Google Scholar] [CrossRef][Green Version] - Kenna, R.; Johnston, D.A.; Janke, W. Publisher’s Note: Self-Consistent Scaling Theory for Logarithmic-Correction Exponents. Phys. Rev. Lett.
**2006**, 97, 169901E. [Google Scholar] [CrossRef] - Harris, A.B. Effect of random defects on the critical behaviour of Ising models. J. Phys. C Solid State Phys.
**1974**, 7, 1671–1692. [Google Scholar] [CrossRef] - Fisher, M.E. Renormalization of Critical Exponents by Hidden Variables. Phys. Rev.
**1968**, 176, 257–272. [Google Scholar] [CrossRef] - Kenna, R.; Hsu, H.P.; von Ferber, C. Fisher renormalization for logarithmic corrections. J. Stat. Mech. Theory Exp.
**2008**, 2008, L10002. [Google Scholar] [CrossRef][Green Version] - Kenna, R.; Berche, B. Scaling and Finite-Size Sclaing above the upper critical dimension. In Order, Disorder and Criticality; World Scientific: Singapore, 2015; Volume 4, pp. 1–54. [Google Scholar] [CrossRef]

**Figure 1.**Ising model with varying spin length (strength) as a model for a social phenomenon. Each individual is represented as a complex network node of a given degree ${k}_{i}$ (i.e., a number of persons connected to it via social links) and given strength ${\mathcal{S}}_{i}$. One may consider spreading of positive (spins up) and negative (spins down) emotions in a social network.

**Figure 2.**Phase diagram of the generalized Ising model with power-law distributed spin strength on a scale-free network (

**b**) is compared with those for the Ising model on a scale-free network (

**a**) and generalized Ising model with power-law distributed spin strength on a complete graph (

**c**). Asymptotics of the order parameter in different regions of $\mu $, $\lambda $ are shown explicitly. Corresponding asymptotics at marginal values of $\mu $, $\lambda $ (lines and points in the plot) are summarized in Table 1.

**Table 1.**Temperature behavior of the order parameter m at different values of $\mu $ and $\lambda $. The asymptotic is governed by the smaller parameter from the pair ($\mu ,\lambda $).

$2<\mathit{\lambda}<3$ | $\mathit{\lambda}=3$ | $3<\mathit{\lambda}<5$ | $\mathit{\lambda}=5$ | $\mathit{\lambda}>5$ | |
---|---|---|---|---|---|

$2<\mu <3$ | Equation (40) | ${T}^{\frac{1}{\mu -3}}$ | ${T}^{\frac{1}{\mu -3}}$ | ${T}^{\frac{1}{\mu -3}}$ | ${T}^{\frac{1}{\mu -3}}$ |

$\mu =3$ | ${T}^{\frac{1}{\lambda -3}}$ | ${e}^{-bT}$ | ${e}^{-bT}$ | ${e}^{-bT}$ | ${e}^{-bT}$ |

$3<\mu <5$ | ${T}^{\frac{1}{\lambda -3}}$ | ${e}^{-bT}$ | Equation (41) | ${\tau}^{\frac{1}{\mu -3}}$ | ${\tau}^{\frac{1}{\mu -3}}$ |

$\mu =5$ | ${T}^{\frac{1}{\lambda -3}}$ | ${e}^{-bT}$ | ${\tau}^{\frac{1}{\lambda -3}}$ | ${\tau}^{\frac{1}{2}}{|ln\tau |}^{-1}$ | ${\tau}^{\frac{1}{2}}{|ln\tau |}^{-\frac{1}{2}}$ |

$\mu >5$ | ${T}^{\frac{1}{\lambda -3}}$ | ${e}^{-bT}$ | ${\tau}^{\frac{1}{\lambda -3}}$ | ${\tau}^{\frac{1}{2}}{|ln\tau |}^{-\frac{1}{2}}$ | ${\tau}^{\frac{1}{2}}$ |

**Table 2.**Critical indices of the generalized model with power-law distributed spin strength on an annealed scale-free network in different regions of the phase diagram Figure 2b. Line 4: $3<(\lambda ,\mu )<5$, $\lambda =\mu $; region III: $3<\mu <5$, $\mu <\lambda $; region IV: $3<\lambda <5$, $\lambda <\mu $; region V: $\lambda ,\mu \ge 5$.

$\mathit{\alpha}$ | ${\mathit{\alpha}}_{\mathit{c}}$ | $\mathit{\gamma}$ | ${\mathit{\gamma}}_{\mathit{c}}$ | $\mathit{\beta}$ | $\mathit{\delta}$ | $\mathit{\omega}$ | ${\mathit{\omega}}_{\mathit{c}}$ | |
---|---|---|---|---|---|---|---|---|

Line 4 ($\mu =\lambda $) | $\frac{\lambda -5}{\lambda -3}$ | $\frac{\lambda -5}{\lambda -2}$ | 1 | $\frac{\lambda -3}{\lambda -2}$ | $\frac{1}{\lambda -3}$ | $\lambda -2$ | $\frac{\lambda -4}{\lambda -3}$ | $\frac{\lambda -4}{\lambda -2}$ |

Region III | $\frac{\lambda -5}{\lambda -3}$ | $\frac{\lambda -5}{\lambda -2}$ | 1 | $\frac{\lambda -3}{\lambda -2}$ | $\frac{1}{\lambda -3}$ | $\lambda -2$ | $\frac{\lambda -4}{\lambda -3}$ | $\frac{\lambda -4}{\lambda -2}$ |

Region IV | $\frac{\mu -5}{\mu -3}$ | $\frac{\mu -5}{\mu -2}$ | 1 | $\frac{\mu -3}{\mu -2}$ | $\frac{1}{\mu -3}$ | $\mu -2$ | $\frac{\mu -4}{\mu -3}$ | $\frac{\mu -4}{\mu -2}$ |

Region V, Lines 5–6, B | 0 | 0 | 1 | 2/3 | 1/2 | 3 | 1/2 | 1/3 |

**Table 3.**Logarithmic correction exponents of the generalized model with power-law distributed spin strength on an annealed scale-free network in different regions. Exponents for lines 5–6 coincide with those found previously [39,40,41]. Here, we find two new sets of exponents that govern logarithmic corrections along line 4 and in point B.

$\widehat{\mathit{\alpha}}$ | $\widehat{{\mathit{\alpha}}_{\mathit{c}}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{{\mathit{\gamma}}_{\mathit{c}}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\delta}}$ | $\widehat{\mathit{\omega}}$ | $\widehat{{\mathit{\omega}}_{\mathit{c}}}$ | |
---|---|---|---|---|---|---|---|---|

Line 4 ($\mu =\lambda $) | $-\frac{3}{\lambda -2}$ | $-\frac{3}{\lambda -2}$ | 0 | $-\frac{\lambda -3}{2(\lambda -2)}$ | $-\frac{1}{\lambda -3}$ | $-\frac{1}{\lambda -2}$ | $-\frac{\lambda -4}{\lambda -3}$ | $-2\frac{\lambda -4}{\lambda -2}$ |

Point B | $-2$ | $-2$ | 0 | $-2/3$ | $-1$ | $-2/3$ | $-1$ | $-4/3$ |

Lines 5–6 | $-1$ | $-1$ | 0 | $-1/3$ | $-1/2$ | $-1/3$ | $-1/2$ | $-2/3$ |

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Krasnytska, M.; Berche, B.; Holovatch, Y.; Kenna, R.
Generalized Ising Model on a Scale-Free Network: An Interplay of Power Laws. *Entropy* **2021**, *23*, 1175.
https://doi.org/10.3390/e23091175

**AMA Style**

Krasnytska M, Berche B, Holovatch Y, Kenna R.
Generalized Ising Model on a Scale-Free Network: An Interplay of Power Laws. *Entropy*. 2021; 23(9):1175.
https://doi.org/10.3390/e23091175

**Chicago/Turabian Style**

Krasnytska, Mariana, Bertrand Berche, Yurij Holovatch, and Ralph Kenna.
2021. "Generalized Ising Model on a Scale-Free Network: An Interplay of Power Laws" *Entropy* 23, no. 9: 1175.
https://doi.org/10.3390/e23091175