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DCBM

Model description

The Directed Binary Configuration Model (DBCM) is a maximum-entropy null model for undirected networks. It is based on the idea of fixing the in- and outdegree sequence of the network, i.e., the number of incoming and outgoing edges incident to each node, and then randomly rewiring the edges while preserving the degree sequence. The model assumes that the edges are unweighted and that the network is simple, i.e., it has no self-loops or multiple edges between the same pair of nodes [1,2].

We define the parameter vector as $\theta = [\alpha ; \beta]$, where $\alpha$ and $\beta$ denote the parameters associated with the out- and indegree respectively.

DescriptionFormula
Constraints$\forall i: \begin{cases} k_{i, out}(A^{*}) = \sum_{j=1}^{N} a^{*}_{ij} \\ k_{i, in}(A^{*}) = \sum_{j=1}^{N} a^{*}_{ji} \end{cases}$
Hamiltonian$H(A, \Theta) = H(A, \alpha, \beta) = \sum_{i=1}^{N} \alpha_i k_{i,out}(A) + \beta_i k_{i,in}(A)$
Factorized graph probability$P(A | \Theta) = \prod_{i=1}^{N}\prod_{j=1, j \ne i}^{N} p_{ij}^{a_{ij}} (1 - p_{ij})^{1-a_{ij}}$
$\langle a_{ij} \rangle$$p_{ij} = \frac{e^{-\alpha_i - \beta_j}}{1+e^{-\alpha_i - \beta_j}}$
Log-likelihood$\mathcal{L}(\Theta) = -\sum_{i=1}^{N} \left[ \alpha_i k_{i,out}(A^{*}) + \beta_i k_{i,in}(A^{*}) \right] - \sum_{i=1}^{N} \sum_{j=1, j\ne i}^{N} \ln \left( 1+e^{-\alpha_i - \beta_j} \right)$
$\langle a_{ij}^{2} \rangle$$\langle a_{ij} \rangle$
$\langle a_{ij}a_{ts} \rangle$$\langle a_{ij} \rangle \langle a_{ts} \rangle$
$\sigma^{*}(X)$$\sqrt{\sum_{i,j} \left( \sigma^{*}[a_{ij}] \frac{\partial X}{\partial a_{ij}} \right)^{2}_{A = \langle A^{*} \rangle} + \dots }$
$\sigma^{*}[a_{ij}]$$\frac{\sqrt{e^{-\alpha_i - \beta_j}}}{1+e^{-\alpha_i - \beta_j}}$

Creation

using Graphs
using MaxEntropyGraphs

# define the network
G = SimpleDiGraph(rhesus_macaques())

# instantiate a UBCM model
model = DBCM(G)

Obtaining the parameters

# solve using default settings
solve_model!(model)

Sampling the ensemble

# generate 10 random instance of the ensemble
rand(model, 10)

Model comparison

# compute the AIC  
AIC(model)

Counting network motifs

The count of a specific network motif can be computed by using M{motif_number}. The motif numbers match the patterns shown on the image below. So for example if you want to compute the number of reciprocated triangles, you would use M13(model).

source

# Compute the number of occurences of M13
M13(model)

References