In Fagan-Lab/dynet: Network Dynamics

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(dynet)

Ising Glauber

Implementation of the Ising-Glauber model. Each node in the Ising-Glauber model has a binary state. The nodes in the graph switch their state with certain probability at every time step. But for inactive nodes, this probability is $\frac{1}{1+e^{\beta(k - 2m)/k}}$ where the $\beta$ parameter is a likelehood tuning of switching state, and $k$ is the degree of the node and $m$ is the number of neighbors that activate. The switch-state probability for active nodes is $1-\frac{1}{1+e^{\beta(k-2m)/k}}$ instead. The resulting list stores the initial $N\times N$ ground truth network, $N\times L$ time series synthetically generated

Params

• input_matrix The input (ground-truth) adjacency matrix of a graph with $N$ nodes. Must be valid square adjacency matrix.
• L Integer length of the desired synthetically generated time series.
• init Vector initial condition, which must be $0$ or $1$.
• beta Float Inverse temperature tuning the likelihood that a node switches its state. Default to $2$.

Returns

• List with the initial $N\times N$ ground truth network, $N\times L$ time series synthetically generated

Code Example

input <- matrix(
  cbind(
    c(1.0, 1.0, 2.0, 3.0),
    c(0.0, 0.0, 1.0, 0.0),
    c(0.0, 0.0, 0.0, 1.0),
    c(0.0, 0.0, 0.0, 0.0)
  ),
  nrow = 4
)
dynet::simulate_ising(input, L = 3, init = NULL, beta = 2)
dynet::simulate_ising(input, L = 3, init = c(1, 2, 3, 4), beta = 4)

Kuramoto

Implememation of Kuramoto oscillators model. Each node at each step adjusts its phase $\theta_i$ according to the equation $\theta_i=\omega_i + \frac{\lambda}{N}\sum_{j=1}^{N}\sin\left(\theta_j - \theta_i\right)$ where $\lambda$, is a coupling $strength$ parameter and the internal frequency of each node is $\omega_i$. The $freqs$ function parameter is an optional user-defined initialization frequencies. The $phases$ parameter enables each node's initial phase $\theta_{i0}$ to be randomly initialized. The resulting list stores the initial $N\times N$ ground truth network, $N\times L$ time series synthetically generated, and $N$ the internal frequencies.

Params

• input_matrix The input (ground-truth) adjacency matrix of a graph with $N$ nodes. Must be valid square adjacency matrix.
• L Integer length of the desired synthetically generated time series.
• dt Float size of timestep for numerical integration.
• strength Float coupling strength (prefactor for interaction terms).
• phases Vector of of initial phases.
• freqs Vector of internal frequencies.

Returns

• List with the initial $N\times N$ ground truth network, $N\times L$ time series synthetically generated, and $N$ the internal frequencies.

Code Example

input <- matrix(
  cbind(
    c(1,0,0),
    c(0,1,0),
    c(0,1,0)
  ),
  ncol = 3
)
dynet::simulate_kuramoto(input, 3, dt = 1, strength = 1, phases = c(1,2,3), freqs = c(1.0, 1.05,1.06))
dynet::simulate_kuramoto(input, 3)

Lotka-Volterra

Implementation of the Lotka-Volterra model. Species abundances in an ecosystem is described by the Lotka-Volterra model describes dynamics. Species $i$s abundance change per time is $\frac{dX_i}{d t}=r_iX_i(1-\frac{X_i}{K_i}+\sum_{j\neq i}W_{ij}\frac{X_j}{K_i})$ where $r_i$ and $K_i$ are the respective growth rate carrying capacity of the species $i$, and $W_{i,j}$ are relative interaction strengths of the species $j$ on $i$. The resulting list stores the initial $N\times N$ ground truth network, $N\times L$ time series synthetically generated, and the number of time steps.

Params

• input_matrix The input (ground-truth) adjacency matrix of a graph with $N$ nodes. Must be valid square adjacency matrix.
• L Integer length of the desired synthetically generated time series.
• init Initial condition vector. If not specified an initial condition is uniformly generated from 0 to the nodes' carrying capacity.
• gr Growth rate vector. If not specified, default to 1 for all nodes.
• cap Carrying capacity vector. If not specified, default to 1 for all nodes.
• inter $N\times N$ Matrix of interaction weights between nodes. If not specified, default to a zero-diagonal matrix whose ${i,j}$ entry is $(j - i)/(N-1)$
• dt Float or vector of sizes of time steps when simulating the continuous-time dynamics.
• stochastic Boolean determining whether to simulate the stochastic or deterministic dynamics.
• pertb Vector of perturbation magnitude of nodes' growth. If not specified, default to 0.01 for all nodes.

Returns

• List with the initial $N\times N$ ground truth network, $N\times L$ time series synthetically generated, and the number of time steps.

Code Example

input <- matrix(
  cbind(
    c(1,0,0),
    c(0,1,0),
    c(0,1,0)
  ),
  ncol = 3
)
dynet::simulate_lotka(input, L = 3, init = c(0.9662720, 0.7750888, 0.9066647), dt = c(5,3,1))
dynet::simulate_lotka(input, L = 10)

Sherrington-Kirkpatrick

Implementation of simulating the Sherrington-Kirkpatrick dynamics on an initial $N\times N$ ground truth. The resulting list stores the initial $N\times N$ ground truth network, $N\times L$ time series synthetically generated.

Params

• input_matrix The input (ground-truth) adjacency matrix of a graph with $N$ nodes. Must be valid square adjacency matrix.
• L Integer length of the desired synthetically generated time series.
• noise TRUE or FALSE value to generate noise

Returns

• List with the initial $N\times N$ ground truth network, $N\times L$ time series synthetically generated

Code Example

input <- matrix(
  cbind(
    c(1.0, 1.0, 2.0, 3.0),
    c(0.0, 0.0, 1.0, 0.0),
    c(0.0, 0.0, 0.0, 1.0),
    c(0.0, 0.0, 0.0, 0.0)
  ),
  nrow = 4
)
dynet::simulate_sherrington(input, 5, noise = TRUE)
dynet::simulate_sherrington(input, 10)

Single Unbaised Random Walker

Implementation of simulating the single random-walker dynamics on an initial $N\times N$ ground truth. This function generates an $N\times L$ time series generated synthetically $TS$ with $TS[j,t]==1$ if the walker is at node $j$ at time $t$, and $TS[j,t]==0$ otherwise. The resulting list stores the initial $N\times N$ ground truth network, $N\times L$ time series synthetically generated.

Params

• input_matrix The input (ground-truth) adjacency matrix of a graph with $N$ nodes. Must be valid square adjacency matrix.
• L Integer length of the desired synthetically generated time series.
• initial_node Starting node for walk

Returns

• An $N\times L$ matrix of synthetic time series data.

Code Example

input <- matrix(
  cbind(
    c(1, 1, 0, 0),
    c(1, 1, 0, 0),
    c(0, 0, 1, 1),
    c(0, 0, 1, 1)
  ),
  nrow = 4
)
dynet::single_unbiased_random_walker(input, 4)
dynet::single_unbiased_random_walker(input, 5, 2)

Voter

Implementation of simulating voter dynamics on a network. The nodes in the initial ground truth network are randomly assigned a state of either ${-1, 1}$. During every time all nodes are updated asynchronously by choosing new state uniformly from their neighbors. The resulting list stores the initial $N\times N$ ground truth network, $N\times L$ time series synthetically generated.

Params

• input_matrix The input (ground-truth) adjacency matrix of a graph with $N$ nodes. Must be valid square adjacency matrix.
• L Integer length of the desired synthetically generated time series.
• noise If noise is present, with this probability a node's state will be randomly redrawn from $(-1,1)$ independent of its neighbors' states. If omitted, set noise to $1/N$.

Returns

• List with the initial $N\times N$ ground truth network, $N\times L$ time series synthetically generated.

Code Example

input <- matrix(
  cbind(
    c(1.0, 1.0, 2.0, 3.0),
    c(0.0, 0.0, 1.0, 0.0),
    c(0.0, 0.0, 0.0, 1.0),
    c(0.0, 0.0, 0.0, 0.0)
  ),
  nrow = 4
)
dynet::voter(input, 5, .5)
dynet::voter(input, 10)

Fagan-Lab/dynet documentation built on March 3, 2021, 2:53 a.m.