networksns.statistical_models.dar_tgrg_directed

networksns.statistical_models.dar_tgrg_directed(time_series, tol=0.01, maxit=100.0)

Estimate, by an expectation-maximization algorithm, the parameters of the Discrete Auto Regressive Temporally Generalized Random Graph model (\(DAR\)-\(TGRG\)).

The \(DAR\)-\(TGRG\) model [1] can be interpreted as a mixture of \(DAR\) and \(TGRG\) models where the persistence pattern associated with the copying mechanism of the \(DAR\) model coexists with the node fitnesses evolving in time according to the \(TGRG\) model.

In the \(DAR\)-\(TGRG\) model for directed graphs, with temporal network described by a time series of adjacency matrices \(\{A_{ij}^t\}_{i,j=1,\ldots, n}^{t=1,\ldots,s}\), each node \(i\) is characterized by two latent variables, \(\theta_i^{t,in}\) and \(\theta_i^{t,out}\), both of them evolving in time by following a covariance stationary autoregressive process \(AR(1)\):

\(\phantom{aaaaaaaaaaaaaaaaaaa}\theta_i^{t, in} = \varphi_{0,i}^{in} + \varphi_{1,i}^{in} \theta_i^{t-1, in} + \epsilon_i^{t, in},\)

\(\phantom{aaaaaaaaaaaaaaaaaaa}\theta_i^{t, out} = \varphi_{0,i}^{out} + \varphi_{1,i}^{out} \theta_i^{t-1, out} + \epsilon_i^{t, out},\)

where \(\varphi_{0,i}^{in}, \varphi_{0,i}^{out}\in \mathbb{R}\), \(|\varphi_{1,i}^{in}|, |\varphi_{1,i}^{out}|<1\), and i.i.d. normal innovations \(\epsilon_i^{t, in}\sim \mathcal{N}(0, {\sigma_i^{in}}^{2})\) and \(\epsilon_i^{t, out}\sim \mathcal{N}(0, {\sigma_i^{out}}^2)\).

Then, the observation equation for the network snapshot at time \(t\) is given by \(N(N-1)\) independent Bernoulli trials whose conditional probability is:

\(\phantom{aaaa}\mathbb{P}(A^t| \Theta^t, A^{t-1}, \mathbf{\alpha}) = \prod_{i<j}\left( \alpha_{ij}\mathbb{I}_{A^t_{ij}A^{t-1}_{ij}} + (1-\alpha_{ij}) \frac{e^{A^t_{ij}(\theta_i^{t, out} + \theta_j^{t, in})}}{1 + e^{\theta_i^{t, out} + \theta_j^{t, in}}}\right)\),

where \(\Theta^t \equiv \{\theta_i^{t, in}, \theta_i^{t, out}\}_{i = 1, \dots, n}\) and \(\alpha \equiv \{\alpha_{ij}\}_{i,j = 1, \dots, n}\) with \(0<\alpha_{ij}<1\).

Parameters:

  • time_series (List object) – List of adjacency matrices [\(A_1, \dots, A_T\)] with binary entries and zero diagonal.

  • tol (float) – Relative error of the estimated parameters. Default: 1e-2.

  • maxit (integer) – Maximum number of iterations in the learning process. Default: 1e2.

Returns:

  • phi_0 (array) – Vector of length \(2n\) with the estimated values of \(\varphi_0\). The first \(n\) entries contain \(\varphi_0^{out}\), while the last \(n\) entries contain \(\varphi_0^{in}\).

  • phi_1 (array) – Vector of length \(2n\) with the estimated values of \(\varphi_1\). The first \(n\) entries contain \(\varphi_1^{out}\), while the last \(n\) entries contain \(\varphi_1^{in}\).

  • sigma (array) – Vector of length \(2n\) with the estimated values of \(\sigma\). The first \(n\) entries contain \(\sigma^{out}:\), while the last \(n\) entries contain \(\sigma^{in}\).

  • theta_naive (array_like) – Matrix of size \(2n \times T\), that in the entry \((i, t)\) has a naive estimation of the \(\theta\) parameters. The first \(2n\) rows contain the values of the \({\theta^{t}_i}^{out}\) while the last \(n\) rows contain the \({\theta^{t}_i}^{in}\).

  • theta (array_like) – Matrix of size \(2n \times T\), that in the entry \((i, t)\) has the estimation of the \(\theta\) parameters. The first \(2n\) rows contain the values of the \({\theta^{t}_i}^{out}\) while the last \(n\) rows contain the \({\theta^{t}_i}^{in}\).

  • alpha (array_like) – Matrix with the estimated values of \(\alpha_{ij}\).

Examples

>>>  from networksns import statistical_models as sm
>>>  import numpy as np

Define input parameters

>>>    n = 60
>>>    T = 150
>>>    phi0 = np.ones(n) * 0.2

Create temporal network

>>>    time_series = dar_tgrg_directed_simulation(n, T, phi_0_in=phi0, phi_0_out=phi0)

Estimate the \(DAR\)-\(TGRG\) model parameters

>>>    phi_0, phi_1, sigma, theta_naive, theta, alpha = sm.tgrg(time_series)

References