Author

Casper van Elteren

Background of models

Under construction 🚧.

Ising model

Named after Ernst Ising, the Ising model is a mathematical model of ferromagnetism. It consists of a set of binary distributed variables that follows the Gibbs distribution. Each variable updates through nearest neighbor interactions where for ferromagnetic systems the lowest energy state is achieved by aligning the spin state with its neighbors.

One of the major properties of the Ising model is the phase transition in two or more dimensions, i.e. as a function of noise (temperature) the model goes from an ordered phase, to an unordered phase.

It has bee shown that the Ising model falls within the same universality class as (directed) diffusion, and it has been extended it many different fields including studying neural behavior, voter dynamics and so on.

Potts model

The q-Potts model generalizes the Ising model. In the Potts model the spins are not binary but can take on arbitrary spin directions i.e.

\[\theta = \frac{q_i 2 \pi}{q}\]

\(q_i \in n = \{0, \dots, q - 1\}\). In the limit \(q \rightarrow \infty\) this model reduces to the XY model. One particular interesting extensions is the cellular Potts model used to model static and kinetic phenomena in biological morphogenesis.

Bornholdt model

Is an extensions of the traditional Ising model used for modeling financial systems. It adds a global cost term such that each variable “feels” the effect of the entire system. This causes “shocks” in the system similar to an abrupt financial crisis. Each variable can be given a strategy that either aligns, or skews it states according to this general magnetization. This causes a dynamic between traders (variables) that either want to be apart of the minority or the majority state.

AB voter model

todo

SIRS

Susceptible Infected Recovered (Susceptible) or SIRS model inspired by Youssef & Scolio (2011) The article describes an individual approach to SIR modeling which canonically uses a mean-field approximation. In mean-field approximations nodes are assumed to have ’homogeneous mixing’, i.e. a node is able to receive information from the entire network. The individual approach emphasizes the importance of local connectivity motifs in spreading dynamics of any process.

Cycledelic

A model for studying species dynamics. It is based on Reichenbach et al. 2007.The model was designed to understand the co-existance of interacting species in a spatially extended ecosystem. Each vertex point represents the locus of three species. The color (red, green, blue) are proportional to the density of the three species at each pixel (vertex point).

The model produces a wide range of different patterns based on three input parameters

  • Diffusion (\(D\)): mobility of species.

  • Predation (\(P\)): competition between the tree different species.

  • Competition (\(C\)): Competition among different specifies.

Each vertex in the system \(\sigma_i\) \(\in\) \(\sigma :=\{ \sigma_0, \dots, \sigma_n\}\) contains a vector with the density of the three “species”, i.e. rock (\(r\)), paper (\(g\)), or scissor (\(b\)). The concentration of each specie at vertex \(i\) is updated according to

\[\begin{split}\frac{d \sigma_i}{dt} = \scriptstyle \begin{cases} \frac{dr_i}{dt}& = ((\underbrace{P (g_i - b_i) + r_i}_{\textrm{predation}} - \underbrace{C (g_i + b_i) - r_i^2}_{\textrm{Competition}})r_i - \underbrace{D(\sum_{<i,j>} r_j r_i)}_{\textrm{mobility}}) \delta t \\\\\\ \frac{dg_i}{dt}& = ((P (b_i - r_i) + g_i - C (r_i + b_i) - g_i^2)g_i - D(\sum_{<i,j>} g_j g_i)) \delta t \\\\\\ \frac{db_i}{dt}& = ((P (r_i - g_i) + b_i - C (r_i + g_i) - b_i^2)b_i - D(\sum_{<i,j>} b_j b_i)) \delta t, \end{cases}\end{split}\]

where \(<i,j>\) indicates the nearest neighbors of variable \(i\).