# The Theoretical and Statistical Ising Model: A Practical Guide in R

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Theoretical Ising Model

#### 2.1. A Conceptual Introduction

#### 2.2. A Model of Alignment

## 3. Simulating Ising Dynamics

#### 3.1. Network Structures

#### 3.2. Equilibrium Configurations

**Box 2.**How to sample states from an Ising system [33].

#### 3.3. Dynamics of $\beta $: A Pitchfork Bifurcation

**not**been quantitatively identified for social or psychological phenomena.

#### 3.4. Dynamics of $\alpha $: Hysteresis

#### 3.5. Summarising the Dynamics as a Cusp

#### 3.6. Variable Encoding

**not**"want" to align at 0, although it does want to align at 1. This is useful when we model a binary system that only "wants" to align in one direction. Present versus absent symptoms have been modelled this way [18]. In contrast, if the system "wants" to align in either of the directions, then the (−1, 1) encoding provides a better description of the system. Positive versus negative political attitudes have been modelled with this encoding [8,17]. The crucial choice of whether to use the $(0,1)$ versus $(-1,1)$ configurations depends on whether one theoretically expects negative states to increase alignment; for political attitudes, the hypothesis that negative feelings of a political candidate increase the negative cognitions of that candidate is plausible; meanwhile, for depression, the hypothesis that the absence of one problems causes the absence of another may be less plausible.

#### 3.7. Theoretical Applications

## 4. The Statistical Ising Model

#### 4.1. A Decade of Statistical Ising Models

#### 4.2. eLASSO Estimation

**Box 3.**How to perform eLASSO estimation with IsingFit.

#### 4.3. Bayesian Estimation

**Box 4.**How to perform Bayesian Ising estimation using rbinnet. We show this information in three edge-uncertainty graphs in Figure 9.

**Figure 9.**Result of rbinnet used to construct edge uncertainty plots. The edges of coloured graphs represent the inclusion Bayes factor $BF=\frac{P({\omega}_{ij}\ne 0)}{P({\omega}_{ij}=0)}$. The red graph indicates the evidence of absence of an edge ($BF<0.1$). The blue graph provides the evidence of inclusion of an edge ($BF>10$). The grey graph shows the absence of evidence ($0.1<BF<10$).

**Figure 10.**Robustness analysis based on rbinnet. The robustness of posterior inclusion probabilities is studied for various combinations of prior inclusion probabilities and precisions. We see that four edges have consistently high posterior inclusion probabilities (robust)—which are also the strongest in the assumed structure. Two edges have their probabilities “smeared out”, indicating non-robustness. Finally, nine edges have robustly low posterior inclusion probabilities.

**Box 6.**Bayesian Ising estimation using BGGM.

#### 4.4. Maximum Likelihood Estimation

#### 4.5. Summary of Recommendation

**Box 7.**Pruned maximum likelihood estimation using psychometrics. The comparison results are shown in Table 3.

**Table 3.**Output of model comparison using psychometrics. We estimated a saturated model (DF = 0) and a pruned model (DF = 39) from data generated in Text Box 2. A comparison of the two models’ fit indices (AIC and BIC) shows that the pruned model is preferred as it has both lower AIC and BIC.

Model | DF | BIC | AIC |
---|---|---|---|

Model 1: saturated | 0 | 4965 | 4862 |

Model 2: Pruned | 11 | 4909 | 4858 |

## 5. The Practical Gap between Statistical and Theoretical Ising Use

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MFA | Mean-field approximation |

MLE | Maximum likelihood estimation |

## Appendix A

**Figure A2.**The bifurcation plot resulting from plotting equilibrium points of the MFA while varying the alignment weight $\beta $.

## References

- Brush, S.G. History of the Lenz-Ising model. Rev. Mod. Phys.
**1967**, 39, 883. [Google Scholar] [CrossRef] - Ernest, I. Ising: Beitrag zur theorie des ferromagnetismus. Z. Für Phys.
**1926**, 31, 253–258. [Google Scholar] - Onsager, L. Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition. Phys. Rev.
**1944**, 65, 117–149. [Google Scholar] [CrossRef] - Bhattacharjee, S.M.; Khare, A. Fifty years of the exact solution of the two-dimensional Ising model by Onsager. Curr. Sci.
**1995**, 69, 816–821. [Google Scholar] - Sole, R.V. Phase Transitions; Primers in Complex Systems; OCLC: Ocn757257299; Princeton University Press: Princeton, NJ, USA, 2011. [Google Scholar]
- Galam, S. Sociophysics: A review of Galam models. Int. J. Mod. Phys. C
**2008**, 19, 409–440. [Google Scholar] [CrossRef] - Kruis, J.; Maris, G. Three representations of the Ising model. Sci. Rep.
**2016**, 6, 1–11. [Google Scholar] [CrossRef] [PubMed][Green Version] - van der Maas, H.L.J.; Dalege, J.; Waldorp, L. The polarization within and across individuals: The hierarchical Ising opinion model. J. Complex Netw.
**2020**, 8, cnaa010. [Google Scholar] [CrossRef] - Duke, T.A.J.; Bray, D. Heightened sensitivity of a lattice of membrane receptors. Proc. Natl. Acad. Sci. USA
**1999**, 96, 10104–10108. [Google Scholar] [CrossRef][Green Version] - Bornholdt, S.; Wagner, F. Stability of money: Phase transitions in an Ising economy. Phys. A Stat. Mech. Appl.
**2002**, 316, 453–468. [Google Scholar] [CrossRef][Green Version] - Stauffer, D. Social applications of two-dimensional Ising models. Am. J. Phys.
**2008**, 76, 470–473. [Google Scholar] [CrossRef][Green Version] - Roudi, Y. Statistical physics of pairwise probability models. Front. Comput. Neurosci.
**2009**, 3, 22. [Google Scholar] [CrossRef][Green Version] - Martyushev, L.M.; Seleznev, V.D. Maximum entropy production principle in physics, chemistry and biology. Phys. Rep.
**2006**, 426, 1–45. [Google Scholar] [CrossRef] - Jaynes, E.T. On the rationale of maximum-entropy methods. Proc. IEEE
**1982**, 70, 939–952. [Google Scholar] [CrossRef] - Borsboom, D.; Cramer, A.O. Network analysis: An integrative approach to the structure of psychopathology. Annu. Rev. Clin. Psychol.
**2013**, 9, 91–121. [Google Scholar] [CrossRef][Green Version] - Robinaugh, D.J.; Hoekstra, R.H.A.; Toner, E.R.; Borsboom, D. The network approach to psychopathology: A review of the literature 2008–2018 and an agenda for future research. Psychol. Med.
**2020**, 50, 353–366. [Google Scholar] [CrossRef] [PubMed] - Brandt, M.J.; Sleegers, W.W. Evaluating belief system networks as a theory of political belief system dynamics. Personal. Soc. Psychol. Rev.
**2021**, 25, 159–185. [Google Scholar] [CrossRef] - Cramer, A.O.J.; van Borkulo, C.D.; Giltay, E.J.; van der Maas, H.L.J.; Kendler, K.S.; Scheffer, M.; Borsboom, D. Major Depression as a Complex Dynamic System. PLoS ONE
**2016**, 11, e0167490. [Google Scholar] [CrossRef] - Dalege, J.; Borsboom, D.; Harreveld, F.V.; Lunansky, G.; Maas, H.L.J.v.d. The Attitudinal Entropy (AE) Framework: Clarifications, Extensions, and Future Directions. Psychol. Inq.
**2018**, 29, 218–228. [Google Scholar] [CrossRef] - Epskamp, S.; Maris, G.K.J.; Waldorp, L.J.; Borsboom, D. The Wiley Handbook of Psychometric Testing: A Multidisciplinary Reference on Survey, Scale and Test Development; Wiley: Hoboken, NJ, USA, 2016; pp. 953–986. [Google Scholar]
- Fried, E.I.; Bockting, C.; Arjadi, R.; Borsboom, D.; Amshoff, M.; Cramer, A.O.; Epskamp, S.; Tuerlinckx, F.; Carr, D.; Stroebe, M. From loss to loneliness: The relationship between bereavement and depressive symptoms. J. Abnorm. Psychol.
**2015**, 124, 256. [Google Scholar] [CrossRef] [PubMed][Green Version] - Boschloo, L.; Borkulo, C.D.v.; Borsboom, D.; Schoevers, R.A. A Prospective Study on How Symptoms in a Network Predict the Onset of Depression. Psychother. Psychosom.
**2016**, 85, 183–184. [Google Scholar] [CrossRef] [PubMed] - Epskamp, S.; Borsboom, D.; Fried, E.I. Estimating psychological networks and their accuracy: A tutorial paper. Behav. Res. Methods
**2018**, 50, 195–212. [Google Scholar] [CrossRef][Green Version] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2020. [Google Scholar]
- Newman, M.; Watts, D. Renormalization group analysis of the small-world network model. Phys. Lett. A
**1999**, 263, 341–346. [Google Scholar] [CrossRef][Green Version] - Broido, A.D.; Clauset, A. Scale-free networks are rare. Nat. Commun.
**2019**, 10, 1–10. [Google Scholar] [CrossRef] [PubMed] - Watts, D.J.; Strogatz, S.H. Collective dynamics of ‘small-world’ networks. Nature
**1998**, 393, 440–442. [Google Scholar] [CrossRef] [PubMed] - Dalege, J.; Borsboom, D.; Van Harreveld, F.; Van den Berg, H.; Conner, M.; Van der Maas, H.L. Toward a formalized account of attitudes: The Causal Attitude Network (CAN) model. Psychol. Rev.
**2016**, 123, 2. [Google Scholar] [CrossRef] - Borsboom, D.; Cramer, A.O.; Schmittmann, V.D.; Epskamp, S.; Waldorp, L.J. The small world of psychopathology. PLoS ONE
**2011**, 6, e27407. [Google Scholar] [CrossRef][Green Version] - Csardi, G.; Nepusz, T. The igraph software package for complex network research. InterJournal
**2006**, 1695, 1–9. [Google Scholar] - Strogatz, S.H. Nonlinear Dynamics and Chaos; Taylor & Francis Inc.: Abingdon, UK, 2014. [Google Scholar]
- Guastello, S.J.; Koopmans, M.; Pincus, D. Chaos and Complexity in Psychology; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Epskamp, S. parSim: Parallel Simulation Studies. Available online: cran.r-project.org/web/packages/parSim/parSim.pdf (accessed on 29 September 2021).
- Dalege, J.; van der Maas, H.L.J. Accurate by Being Noisy: A Formal Network Model of Implicit Measures of Attitudes. Soc. Cogn.
**2020**, 38, s26–s41. [Google Scholar] [CrossRef] - Scheffer, M.; Carpenter, S.; Foley, J.A.; Folke, C.; Walker, B. Catastrophic shifts in ecosystems. Nature
**2001**, 413, 591–596. [Google Scholar] [CrossRef] - Galam, S.; Gefen (Feigenblat), Y.; Shapir, Y. Sociophysics: A new approach of sociological collective behaviour. I. mean-behaviour description of a strike. J. Math. Sociol.
**1982**, 9, 1–13. [Google Scholar] [CrossRef] - Ditzinger, T.; Haken, H. Oscillations in the perception of ambiguous patterns a model based on synergetics. Biol. Cybern.
**1989**, 61, 279–287. [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] - Waldorp, L.; Kossakowski, J. Mean field dynamics of stochastic cellular automata for random and small-world graphs. J. Math. Psychol.
**2020**, 97, 102380. [Google Scholar] [CrossRef] - van der Maas, H.L.J.; Kolstein, R.; van der Pligt, J. Sudden Transitions in Attitudes. Sociol. Methods Res.
**2003**, 32, 125–152. [Google Scholar] [CrossRef] - Siegenfeld, A.F.; Bar-Yam, Y. An Introduction to Complex Systems Science and its Applications. arXiv
**2019**, arXiv:1912.05088. [Google Scholar] - Haslbeck, J.M.; Epskamp, S.; Marsman, M.; Waldorp, L.J. Interpreting the Ising model: The input matters. Multivar. Behav. Res.
**2020**, 56, 303–313. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kruis, J. Transformations of mixed spin-class Ising systems. arXiv
**2020**, arXiv:2006.13581. [Google Scholar] - Meadows, D.H.; Meadows, D.L.; Randers, J.; Behrens, W.W. The Limits to Growth; Yale University Press: New Haven, CT, USA, 1972. [Google Scholar]
- Oberauer, K.; Lewandowsky, S. Addressing the theory crisis in psychology. Psychon. Bull. Rev.
**2019**, 26, 1596–1618. [Google Scholar] [CrossRef] - Fried, E.I. Lack of theory building and testing impedes progress in the factor and network literature. PsyArXiv
**2020**. [Google Scholar] [CrossRef][Green Version] - Mischel, W. The Toothbrush Problem; Association for Psychological Science: Washington, DC, USA, 2008; Volume 21. [Google Scholar]
- Borsboom, D.; van der Maas, H.; Dalege, J.; Kievit, R.; Haig, B. Theory Construction Methodology: A practical framework for theory formation in psychology. PsyArXiv
**2020**. [Google Scholar] [CrossRef][Green Version] - Robinaugh, D.; Haslbeck, J.M.B.; Waldorp, L.; Kossakowski, J.J.; Fried, E.I.; Millner, A.; McNally, R.J.; van Nes, E.H.; Scheffer, M.; Kendler, K.S.; et al. Advancing the Network Theory of Mental Disorders: A Computational Model of Panic Disorder. PsyArXiv
**2019**. [Google Scholar] [CrossRef] - Muthukrishna, M.; Henrich, J. A problem in theory. Nat. Hum. Behav.
**2019**, 3, 221–229. [Google Scholar] [CrossRef] [PubMed][Green Version] - Zhou, W.X.; Sornette, D. Self-organizing Ising model of financial markets. Eur. Phys. J. B
**2007**, 55, 175–181. [Google Scholar] [CrossRef] - Hosseiny, A.; Bahrami, M.; Palestrini, A.; Gallegati, M. Metastable Features of Economic Networks and Responses to Exogenous Shocks. PLoS ONE
**2016**, 11, e0160363. [Google Scholar] [CrossRef] - Weber, M.; Buceta, J. The cellular Ising model: A framework for phase transitions in multicellular environments. J. R. Soc. Interface
**2016**, 13, 20151092. [Google Scholar] [CrossRef] - Matsuda, H. The Ising model for population biology. Prog. Theor. Phys.
**1981**, 66, 1078–1080. [Google Scholar] [CrossRef][Green Version] - Nareddy, V.R.; Machta, J.; Abbott, K.C.; Esmaeili, S.; Hastings, A. Dynamical Ising model of spatially coupled ecological oscillators. J. R. Soc. Interface
**2020**, 17, 20200571. [Google Scholar] [CrossRef] [PubMed] - Wang, Y.; Badea, T.; Nathans, J. Order from disorder: Self-organization in mammalian hair patterning. Proc. Natl. Acad. Sci. USA
**2006**, 103, 19800–19805. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bialek, W.; Cavagna, A.; Giardina, I.; Mora, T.; Silvestri, E.; Viale, M.; Walczak, A.M. Statistical mechanics for natural flocks of birds. Proc. Natl. Acad. Sci. USA
**2012**, 109, 4786–4791. [Google Scholar] [CrossRef][Green Version] - Maas, H.L.J.V.D.; Dolan, C.V.; Grasman, R.P.P.P.; Wicherts, J.M.; Huizenga, H.M.; Raijmakers, M.E.J. A dynamical model of general intelligence: The positive manifold of intelligence by mutualism. Psychol. Rev.
**2006**, 113, 842–861. [Google Scholar] [CrossRef] - Dalege, J.; Borsboom, D.; van Harreveld, F.; van der Maas, H.L. The Attitudinal Entropy (AE) Framework as a general theory of individual attitudes. Psychol. Inq.
**2018**, 29, 175–193. [Google Scholar] [CrossRef][Green Version] - Milkoreit, M.; Hodbod, J.; Baggio, J.; Benessaiah, K.; Calderón-Contreras, R.; Donges, J.F.; Mathias, J.D.; Rocha, J.C.; Schoon, M.; Werners, S.E. Defining tipping points for social-ecological systems scholarship—an interdisciplinary literature review. Environ. Res. Lett.
**2018**, 13, 033005. [Google Scholar] [CrossRef] - Scheffer, M.; Bascompte, J.; Brock, W.A.; Brovkin, V.; Carpenter, S.R.; Dakos, V.; Held, H.; van Nes, E.H.; Rietkerk, M.; Sugihara, G. Early-warning signals for critical transitions. Nature
**2009**, 461, 53–59. [Google Scholar] [CrossRef] [PubMed] - Lenton, T.M.; Rockström, J.; Gaffney, O.; Rahmstorf, S.; Richardson, K.; Steffen, W.; Schellnhuber, H.J. Climate tipping points—Too risky to bet against. Nature
**2019**, 575, 592–595. [Google Scholar] [CrossRef] - Lenton, T.M. Early warning of climate tipping points. Nat. Clim. Chang.
**2011**, 1, 201–209. [Google Scholar] [CrossRef] - Otto, I.M.; Donges, J.F.; Cremades, R.; Bhowmik, A.; Hewitt, R.J.; Lucht, W.; Rockström, J.; Allerberger, F.; McCaffrey, M.; Doe, S.S. Social tipping dynamics for stabilizing Earth’s climate by 2050. Proc. Natl. Acad. Sci. USA
**2020**, 117, 2354–2365. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bentley, R.A.; Maddison, E.J.; Ranner, P.H.; Bissell, J.; Caiado, C.; Bhatanacharoen, P.; Clark, T.; Botha, M.; Akinbami, F.; Hollow, M. Social tipping points and Earth systems dynamics. Front. Environ. Sci.
**2014**, 2, 35. [Google Scholar] [CrossRef][Green Version] - Galesic, M.; Olsson, H.; Dalege, J.; van der Does, T.; Stein, D.L. Integrating social and cognitive aspects of belief dynamics: Towards a unifying framework. J. R. Soc. Interface
**2021**, 18, 20200857. [Google Scholar] [CrossRef] - Page, S.E. The Model Thinker: What You Need to Know to Make Data Work for You, 1st ed.; OCLC: on1028523969; Basic Books: New York, NY, USA, 2018. [Google Scholar]
- Borsboom, D. A network theory of mental disorders. World Psychiatry
**2017**, 16, 5–13. [Google Scholar] [CrossRef][Green Version] - Van Borkulo, C.D.; Borsboom, D.; Epskamp, S.; Blanken, T.F.; Boschloo, L.; Schoevers, R.A.; Waldorp, L.J. A new method for constructing networks from binary data. Sci. Rep.
**2014**, 4, 1–10. [Google Scholar] [CrossRef][Green Version] - Fried, E.I. Theories and Models: What They Are, What They Are for, and What They Are About. Psychol. Inq.
**2020**, 31, 336–344. [Google Scholar] [CrossRef] - Kindermann, R.; Snell, J.L. Markov Random Fields and Their Applications; OCLC: 1030357447; American Mathematical Society: Providence, RI, USA, 2012. [Google Scholar]
- Cox, D.R.; Wermuth, N. Linear dependencies represented by chain graphs. Stat. Sci.
**1993**, 8, 204–218. [Google Scholar] [CrossRef] - Anderson, C.J.; Vermunt, J.K. Log-Multiplicative Association Models as Latent Variable Models for Nominal and/or Ordinal Data. Sociol. Methodol.
**2000**, 30, 81–121. [Google Scholar] [CrossRef][Green Version] - Wickens, T.D. Multiway Contingency Tables Analysis for the Social Sciences; Psychology Press: East Sussex, UK, 2014. [Google Scholar]
- Marsman, M.; Borsboom, D.; Kruis, J.; Epskamp, S.; van Bork, R.; Waldorp, L.J.; Maas, H.L.J.v.d.; Maris, G. An Introduction to Network Psychometrics: Relating Ising Network Models to Item Response Theory Models. Multivar. Behav. Res.
**2018**, 53, 15–35. [Google Scholar] [CrossRef] - Ravikumar, P.; Wainwright, M.J.; Lafferty, J.D. High-dimensional Ising model selection using ι1-regularized logistic regression. Ann. Stat.
**2010**, 38, 1287–1319. [Google Scholar] [CrossRef][Green Version] - Haslbeck, J.M.B.; Waldorp, L.J. mgm: Estimating Time-Varying Mixed Graphical Models in High-Dimensional Data. arXiv
**2020**, arXiv:1510.06871. [Google Scholar] [CrossRef] - Hosmer, D.W., Jr.; Lemeshow, S.; Sturdivant, R.X. Applied Logistic Regression; John Wiley & Sons: Hoboken, NJ, USA, 2013; Volume 398. [Google Scholar]
- Epskamp, S.; Kruis, J.; Marsman, M. Estimating psychopathological networks: Be careful what you wish for. PLoS ONE
**2017**, 12, e0179891. [Google Scholar] [CrossRef] - Meehl, P.E. Why Summaries of Research on Psychological Theories are Often Uninterpretable. Psychol. Rep.
**1990**, 66, 195–244. [Google Scholar] [CrossRef] - Williams, D.R.; Briganti, G.; Linkowski, P.; Mulder, J. On Accepting the Null Hypothesis of Conditional Independence in Partial Correlation Networks: A Bayesian Analysis. PsyArXiv 2021. Available online: psyarxiv.com/7uhx8 (accessed on 27 July 2021).
- Marsman, M.; Huth, K.; Waldorp, L.; Ntzoufras, I. Objective Bayesian Edge Screening and Structure Selection for Networks of Binary Variables. PsyArXiv
**2020**, 26. Available online: psyarxiv.com/dg8yx/ (accessed on 15 July 2021). - Huth, K.; Luigjes, J.; Goudriaan, A.; van Holst, R. Modeling Alcohol Use Disorder as a Set of Interconnected Symptoms-Assessing Differences between Clinical and Population Samples and Across External Factors. PsyArXiv 2021. Available online: psyarxiv.com/93t2f/ (accessed on 25 June 2021).
- Williams, D.R.; Mulder, J. Bayesian hypothesis testing for Gaussian graphical models: Conditional independence and order constraints. J. Math. Psychol.
**2020**, 99, 102441. [Google Scholar] [CrossRef] - Epskamp, S.; Isvoranu, A.M.; Cheung, M. Meta-analytic Gaussian Network Aggregation. PsyArXiv
**2020**. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**A simple network with five nodes and six edges. Its current configuration is $\mathit{x}=[-1,1,1,1,1]$.

**Figure 2.**On the left, a two-dimensional lattice structure. On the right, a small-world and a random network are created from a ring lattice by varying the rewiring probability of edges.

**Figure 3.**Result of sampling $\overline{\mathit{x}}$ from an Ising distribution, with $\alpha =0$ and varying $\beta $ between 0.00 and 0.06. The simulation assumes the small-world network structure of 40 variables from Text Box 1. Deviations from $\overline{\mathit{x}}$ become more likely for smaller networks and the pitchfork shape less pronounced.

**Figure 4.**Result of sampling $\overline{\mathit{x}}$ from an Ising distribution with $\beta =2$ (ordered phase) and varying $\alpha $ between −6 and 6. The simulation assumes the small-world network structure of 40 variables from Text Box 1. We see the outline of a hysteresis effect.

**Figure 5.**A series of ambiguous stimuli used by [37] to illustrate hysteresis. By looking across the illustrations in the figure, one will experience a transition in perception. This transition point depends on the direction we started from. Figures that are in between the transition points are ambiguous.

**Figure 6.**The cusp catastrophe model (

**C**) presents a unified picture of the alignment dynamics arising from the dynamical Ising model. It encompasses pitchfork bifurcation (

**B**) and hysteresis (

**A**).

**Figure 7.**With a pseudo-likelihood estimation, we first estimate the neighbourhood of each node. This is performed through one logistic regression per node (here, we consider a simple three-node example). The bottom panel shows how we combine the neighbourhoods into a single network model, $\omega $, through the AND rule: an edge is present if both ${\beta}_{ij}$ and ${\beta}_{ji}$ are non-zero. This step is necessary because each node is both the dependent and independent variables; hence, we have two $\beta $ estimates.

**Figure 11.**With $BGGM$, we compute three hypotheses per edge: ${H}_{1}:{\omega}_{ij}=0$ (grey), ${H}_{2}:{\omega}_{ij}>1$ (blue), and ${H}_{3}:{\omega}_{ij}<0$ (red). The absence of evidence then amounts to equal probabilities across the edge hypotheses.

**Table 1.**Software packages in R relevant in the simulation of Ising Dynamics. We relate functions to their packages using the ‘::’ notation from R. To access a function, we first install its package with install.package(“Example”) then load the package with library(“Example”).

Package::Function() | Description |
---|---|

IsingSampler::Isingsampler() | Flexible Ising state sampler |

bayess:isinghm() | Metropolis–Hastings Sampler |

igraph::make_lattice() | N dimensional lattice structures |

igraph::sample_small_world() | Watts–Strogatz model |

parSim::parSim() | Easy simulations and multi-core |

ggplot2::ggplot() | Visualisation |

set.seed() | Reproduces random numbers |

Package::Function() | Description | Pros | Cons | Encoding |
---|---|---|---|---|

IsingFit::IsingFit() | eLASSO estimation | Small–medium samples detecting present edges Applicable to >20 variables | Large samples Interpreting absent edges | (1, 0) |

psychometrics::Ising() | Full maximum likelihood | Large samples Extensive further analysis options | Small samples Max of 20 variables | Any |

mgm::mgm() | eLASSO | For mixtures of binary, continuous, and ordinal variables | Large samples Interpreting absent edges | (1, 0) |

rIsing::Ising() | eLASSO | Small–medium samples Detecting present edges Applicable to >20 variables | Large samples Interpreting absent edges | (1, 0) |

rbinnet::select_structure() | Bayesian estimation Slap and spike prior | Evidence of absent edges Model uncertainty Prior information use | Work in progress Prior information dependent | (1, 0) |

BGGM::explore(type = “binary”) | Bayesian estimation F-matrix prior | Evidence of absent edges Model uncertainty Prior information use | Prior information dependent | (1, 0) |

BDgraph::bdgraph() | Bayesian model selection G-wishart Prior | Model uncertainty Prior information use | Prior information dependent | (1, 0) |

IsingFit::LinTransform() | Transforms between (1, 0) and (1, −1) encodings | Works with unregularised models (psychometrics) | Any | |

NetworkComparisonTest::NCT() | Group comparison test | Works with eLASSO models |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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**

Finnemann, A.; Borsboom, D.; Epskamp, S.; van der Maas, H.L.J. The Theoretical and Statistical Ising Model: A Practical Guide in *R*. *Psych* **2021**, *3*, 593-617.
https://doi.org/10.3390/psych3040039

**AMA Style**

Finnemann A, Borsboom D, Epskamp S, van der Maas HLJ. The Theoretical and Statistical Ising Model: A Practical Guide in *R*. *Psych*. 2021; 3(4):593-617.
https://doi.org/10.3390/psych3040039

**Chicago/Turabian Style**

Finnemann, Adam, Denny Borsboom, Sacha Epskamp, and Han L. J. van der Maas. 2021. "The Theoretical and Statistical Ising Model: A Practical Guide in *R*" *Psych* 3, no. 4: 593-617.
https://doi.org/10.3390/psych3040039