Forbidden interactions in the tripartite network have a probability of 0.
A tripartite network has the following structure, with $O$ being a zero matrix and $A_1$ and $A_2$ being the matrix of potential interactions between the three sets of species.
$$
\left(\begin{array}{cc}
O & A_1 & O \
O & O & A_2 \
O & O & O \
\end{array}\right)
$$
4 null models to explore:
Every potential interaction in $A_1$ and $A_2$ is equal to the average of the connectance of $A_1$ and $A_2$.
Every potential interaction in $A_1$ is equal to the connectance of $A_1$ (same for $A_2$).
Observed interactions have a probability of $1 - v$, where $v$ is a penalty of e.g. $0.1$ and unobserved potential interactions have a probability of $vL / (S_1 S_2 + S_2 S_3 - L)$ (redistributed probability)
Fixed false positive and false negative rates.
Local networks
The probability that an interaction is realized locally is given by the probability that the two species interact given that they can, times the probability that they can.
This probabiliity is fixed to arbitrary numbers lower than 1, which could be chosen to conserve the expected number of links.
This probability is obtained using the method of probability redistribution of metawebs.
Optional. This probability takes into account local observations of interactions. Using a Beta prior and a Bernoulli likelihood, this would give the following posterior (conjugate prior):
Metawebs
Forbidden interactions in the tripartite network have a probability of 0.
A tripartite network has the following structure, with $O$ being a zero matrix and $A_1$ and $A_2$ being the matrix of potential interactions between the three sets of species.
$$ \left(\begin{array}{cc} O & A_1 & O \ O & O & A_2 \ O & O & O \ \end{array}\right) $$
4 null models to explore:
Local networks
The probability that an interaction is realized locally is given by the probability that the two species interact given that they can, times the probability that they can.
$$P_N(i \rightarrow j) = PN(i \rightarrow j | M{i \rightarrow j}= 1) P_M(i \rightarrow j)$$
The probability that two species can interact $P_M(i \rightarrow j)$ is given by the metaweb.
The probability that the two species interact given that they can may be obtained in three different ways.
$$PN(i \rightarrow j | M{i \rightarrow j}= 1) = 1$$
This probabiliity is fixed to arbitrary numbers lower than 1, which could be chosen to conserve the expected number of links.
This probability is obtained using the method of probability redistribution of metawebs.
Optional. This probability takes into account local observations of interactions. Using a Beta prior and a Bernoulli likelihood, this would give the following posterior (conjugate prior):
$$PN(i \rightarrow j | M{i \rightarrow j}= 1) \propto Bernoulli(φ) \times Beta(\alpha, \beta)$$
$$PN(i \rightarrow j | M{i \rightarrow j}= 1) \propto Beta(\sum y_i + \alpha, \sum (1 - y_i) + \beta)$$
Using α = β = 2 seems reasonable.