Setup and models
We mostly follow the notation of Jin, Ke, and Luo (2019) with some minor deviations (e.g., we use \(\lambda\) to denote the expected average degree of the network). They consider the degree-corrected mixed membership model (DCMM) with the following expectation for the adjacency matrix \[\begin{align*} \mathop{\mathrm{\mathbb E}}[A_{ij} \mid \theta, \pi] = \theta_i \theta_j \,\pi_i^T P \pi_j, \quad i <j, \quad \pi_i \sim F \end{align*}\] where \(\pi_i\) is a random element of \([0,1]^K\). In the special case of the DCSBM, we have \(\pi_i = e_{z_i}\) where \(z_i\) is a random label in \([K]\) and \(e_k\) is the \(k\)th basis vector of \(\mathbb R^K\). This special case is what we consider here. For identifibaility, it is assumed that \(\|P \|_\infty = \max_{k\ell} |P_{k\ell}|= 1\). They also let \[h = \mathop{\mathrm{\mathbb E}}[\pi_i]\] denote the class prior.
For model 1, we have a DCSBM with \(K=1\) and node connection propensity parameter \(\theta_i\), that is, \[ \mathop{\mathrm{\mathbb E}}[A_{ij} \mid \theta] = \theta_i \theta_j, \quad i < j. \] Let us denote the degree of node \(i\) as \(\mathop{\mathrm{deg}}(i)\). Under model 1 is \[\mathop{\mathrm{\mathbb E}}[\mathop{\mathrm{deg}}(i)] = \theta_i \sum_{j \neq i} \theta_j = \theta_i (\|\theta\|_1 - \theta_i) .\]
For model 2, Jin, Ke, and Luo (2019) propose to use random propensity parameters \(\widetilde \theta= (\widetilde \theta_j)\), with \[ \widetilde \theta_i = d_{z_i} \theta_i \] where \(\{d_k, k \in [K]\}\) is to be determined. For model 2, we have \(\mathop{\mathrm{\mathbb E}}[A_{ij} \mid \widetilde \theta, z] = \widetilde \theta_i \widetilde \theta_j \,e_{z_i}^T P e_{z_j}\) for \(i < j\). The expected degree of node \(i\) under model 2 is \[ \mathop{\mathrm{\mathbb E}}[\mathop{\mathrm{deg}}(i)] = \sum_{j\ne i}\mathop{\mathrm{\mathbb E}}[A_{ij}] = \mathop{\mathrm{\mathbb E}}[\widetilde \theta_i e_{z_i}^T]\, \sum_{j\neq i} P \mathop{\mathrm{\mathbb E}}[\widetilde \theta_j e_{z_j}] \] using implicit independence assumptions. We have \[ \mathop{\mathrm{\mathbb E}}[\widetilde \theta_i e_{z_i}] = \mathop{\mathrm{\mathbb E}}[ d_{z_i} \theta_i e_{z_i}] = \theta_i \mathop{\mathrm{\mathbb E}}[d_{z_i} e_{z_i}] = \theta_i \sum_k h_k d_k e_k = \theta_i D h \] where \(D = \mathop{\mathrm{diag}}(d_k)\). Plugging-in we have, under model 2, \[ \mathop{\mathrm{\mathbb E}}[\mathop{\mathrm{deg}}(i)] = \theta_i \Big(\sum_{j \neq i} \theta_j \Big) h^T D P D h. \] Since \(h^T 1_K = 1_K\), if we choose \(D\) such that \(D P D h = 1_K\), then model 2 will have the same expected degree for each node as model 1.
Sinkhorn construction
Since \(P\) is symmetric and has nonnegative entries, under mild conditions on \(P\) and \(h\), the symmetric version of Sinkhorn theorem guarantees the existence of a diagonal matrix \(D = \mathop{\mathrm{diag}}(d_k)\) with positive entries such that \(D P D 1_K = h\) that is, \(D P D\) has row sums \(h\). See Idel (2016) for a review.
Let us use the elementwise notation \(h /
d\) and \(h d\) to denote the
vectors with entries \(h_k / d_k\) and
\(h_k d_k\). The Sinkhorn equation can
be written as \[
P d = h / d
\] or equivalently as the fixed point equation \(d = h / Pd\) which can be solved by
repeated application of the operator \(d
\mapsto h/Pd\) to an initial vector, say, \(1_K\). The iteration in fact oscillates
between the the two solutions \(d_1\)
and \(d_2\) of the nonsymmetric
Sinkhorn equation \(P d_2 = h/ d_1\)
(Knight 2008; Bradley
2010) and can be made to converge by averaging every two
steps. This is implemented in sinkhorn_knopp function in
the nett package.
To get the version we need, that is, \(D P D h = 1_K\), let us write Sinkhorn equation with \(d\) renamed to \(r\). Then, \[ P r = h / r \iff P ((r/h) h) = \frac{1_K}{r/h} \] This shows that defining \(d = r/h\), that is, \(d_k = r_k / h_k\) and letting \(D = \mathop{\mathrm{diag}}(d_k)\) we have the desired scaling \(D P D h = 1_K\). That is, we solve the symmetric Sinkhorn problem with row sum data \(h\), and rescale the output \(r\) to \(r/h\) to get the desired \(d\) for our problem.
Let us verify these ideas:
Ktru = 4 # number of true communities
oir = 0.15 # out-in-ratio
h = 1:Ktru
h = h / sum(h) # class prior or `pri` in our notation
P_diag = rep(1, Ktru)
P = oir + diag(P_diag - oir)
r = nett::sinkhorn_knopp(P, sums=h, sym=T)$r
d = r / h
diag(d) %*% P %*% diag(d) %*% h # verify Sinkhorn's scaling
#> [,1]
#> [1,] 1
#> [2,] 1
#> [3,] 1
#> [4,] 1Generating from the models
The expected average degree of model 1 is \(\alpha^2 := \frac1n(\|\theta\|_1^2 - \|\theta\|_2^2)\). We can rescale \(\theta\) to \(\sqrt{\lambda} \theta / \alpha\) to get a desired expected average degree \(\lambda\):
n = 2000 # number of nodes
lambda = 5 # expected average degree
# Generate from the degree-corrected Erdős–Rényi model
set.seed(1234)
theta <- EnvStats::rpareto(n, 2/3, 3)
alpha2 = (sum(theta)^2 - sum(theta^2))/n
theta = theta * sqrt(lambda/alpha2)
A = nett::sample_dcer(theta) # sample the adjacency matrix matrix
mean(Matrix::rowSums(A))
#> [1] 4.931
hist(Matrix::rowSums(A), 15, main = "Degree distribution", xlab = "degree")
z = sample(Ktru, n, replace=T, prob=h) # randomly smaple "true community labels"
tht = d[z] * theta
A = nett::sample_dcsbm(z, P, tht) # sample the adjacency matrix
hist(Matrix::rowSums(A), 15, main = "Degree distribution", xlab = "degree")
Verifying equality of expected degrees
nrep = 10
degs1 = degs2 = rep(0, n)
for (r in 1:nrep) {
degs1 = degs1 + Matrix::rowSums(nett::sample_dcer(theta))
z = sample(Ktru, n, replace=T, prob=h) # randomly smaple "true community labels"
degs2 = degs2 + Matrix::rowSums(nett::sample_dcsbm(z, P, d[z] * theta))
}
degs1 = degs1 / nrep
degs2 = degs2 / nrep
# cbind(degs1, degs2)
mean(abs(degs1 - degs2)/(degs1 + degs2)/2)
#> [1] 0.04304521
Comments
Theorem 3.2 of Jin, Ke, and Luo (2019) states that under the condition that \(\|\theta\|_\infty \lesssim 1\) and \(\min_k h_k \gtrsim 1\) if \[\|\theta\|\cdot |\mu_2(P)| \to 0, \] the two hypotheses are indistinguisiable (in the sense that their \(\chi^2\)-distance goes to zero). Here \(\mu_2(P)\) is the 2nd largest eigenvalue of \(P\) in magnitude. In our seteup, \(\mu_2(P) \asymp 1\). So the condition is essentially \(\|\theta\| \to 0\).
Note that the expected average degree of the null model is \[\lambda \sim \frac{\|\theta\|_1^2}{n}.\] We have \(\|\theta\|_2 \ge \frac{\|\theta\|_1}{\sqrt n}\), hence, in the wrost case \(\|\theta\|_2 = \frac{\|\theta\|_1}{\sqrt n} \asymp \sqrt{\lambda}\) and the condition is equivalent to \(\sqrt{\lambda} \to 0\), that is, the expected average degree \(\lambda\) vanishes asympototically. It is quite reasolable that most methods would fail when \(\lambda \to 0\).
In all other cases, the situation is better. In particular, the above experiment shows that even for a sufficiently small \(\lambda = 5\) and a fairly large \(n = 2000\), NAC tests can still be perfect for this problem.