We start our analysis with the unobserved block connectivity probability matrix \({\textbf{B}}\) for SBM and then illustrate how to migrate the proposed methods for real applications when we have the observed adjacency matrix \({\textbf{A}}\).
Consider the K-block SBM parametrized by the block connectivity probability matrix \({\textbf{B}} \in (0, 1)^{K \times K}\) and the vector of block assignment probabilities \(\varvec{\pi } \in (0, 1)^K\) with \(K > 2\). Given initial sampling parameter \(p_0 \in (0, 1)\), initial sampling is uniformly at random, i.e.,
$$\begin{aligned} {\textbf{B}}_0 = p_0 {\textbf{B}}. \end{aligned}$$
(13)
This initial sampling simulates the case when one only obersves a partial graph with a small portion of the edges instead of the entire graph with all existing edges.
Theorem 2
For K-block SBMs, given two block connectivity probability matrices \({\textbf{B}}, p{\textbf{B}} \in (0, 1)^{K \times K}\) with \(p \in (0, 1)\) and a vector of block assignment probabilities \(\varvec{\pi } \in (0, 1)^K\), we have \({\textbf{B}} \succ p {\textbf{B}}\).
The proof of Theorem 2 can be found in Appendix. As an illustration, consider a 4-block SBM parametrized by block connectivity probability matrix \({\textbf{B}}\) as
$$\begin{aligned} {\textbf{B}} = \begin{bmatrix} 0.04 &{} 0.08 &{} 0.10 &{} 0.18 \\ 0.08 &{} 0.16 &{} 0.20 &{} 0.36 \\ 0.10 &{} 0.20 &{} 0.25 &{} 0.45 \\ 0.18 &{} 0.36 &{} 0.45 &{} 0.81 \end{bmatrix}. \end{aligned}$$
(14)
Figure 1 shows Chernoff information \(\rho\) as in Eq. (10) corresponding to \({\textbf{B}}\) as in Eq. (14) and \(p {\textbf{B}}\) for \(p \in (0, 1)\). In addition, Fig. 1a assumes \(\varvec{\pi } = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4})\) and Fig. 1b assumes \(\varvec{\pi } = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8})\). As suggested by Theorem 2, for any \(p \in (0, 1)\) we have \(\rho _{B} > \rho _{pB}\) and thus \({\textbf{B}} \succ p {\textbf{B}}\).
Now given dynamic network sampling parameter \(p_1 \in (0, 1-p_0)\), the baseline sampling scheme can proceed uniformly at random again, i.e.,
$$\begin{aligned} {\textbf{B}}_1 = {\textbf{B}}_0 + p_1 {\textbf{B}} = (p_0 + p_1) {\textbf{B}}. \end{aligned}$$
(15)
This dynamic network sampling simulates the situation when one is given limited resources to sample some extra edges after observing the partial graph with only a small portion of the edges. Since we only have limited budget to sample another small portion of edges, one would benefit from identifying vertex pairs that have much influence on the community structure. In other words, the baseline sampling scheme just randomly choosing vertex pairs without using the information from the initial observed graphs and our goal is to design an alternative scheme to optimize this dynamic network sampling procedure so that one could have a better block recovery even with limited resources to only observe a partial graph with a small portion of the edges.
Corollary 1
For K-block SBMs, given block connectivity probability matrix \({\textbf{B}} \in (0, 1)^{K \times K}\) and a vector of block assignment probabilities \(\varvec{\pi } \in (0, 1)^K\). We have \({\textbf{B}} \succ {\textbf{B}}_1 \succ {\textbf{B}}_0\) where \({\textbf{B}}_0\) is defined as in Eq. (13) with \(p_0 \in (0, 1)\) and \({\textbf{B}}_1\) is defined as in Eq. (15) with \(p_1 \in (0, 1-p_0)\).
The proof of Corollary 1 can be found in Appendix. This corollay implies that we can have a better block recovery from \({\textbf{B}}_1\) than \({\textbf{B}}_0\).
Assumption 1
The Chernoff-active blocks after initial sampling is unique, i.e., there exists an unique pair \(\left( k_0^*, \ell _0^* \right) \in \{(k, \ell ) \; | \; 1 \le k < \ell \le K \}\) such that
$$\begin{aligned} \left( k_0^*, \ell _0^* \right) = \arg \min _{k \ne l} C_{k ,\ell }({\textbf{B}}_0, \varvec{\pi }), \end{aligned}$$
(16)
where \({\textbf{B}}_0\) is defined as in Eq. (13) and \(\varvec{\pi }\) is the vector of block assignment probabilities.
To improve this baseline sampling scheme, we concentrate on the Chernoff-active blocks \(\left( k_0^*, \ell _0^* \right)\) after initial sampling assuming Assumption 1 holds. Instead of sampling from the entire block connectivity probability matrix \({\textbf{B}}\) like the baseline sampling scheme as in Eq. (15), we only sample the entries associated with the Chernoff-active blocks. As a competitor to \({\textbf{B}}_1\), our Chernoff-optimal dynamic network sampling scheme is then given by
$$\begin{aligned} \widetilde{{\textbf{B}}}_1 = {\textbf{B}}_0 + \frac{p_1}{\left( \pi _{k_0^*} + \pi _{\ell _0^*}\right) ^2 } {\textbf{B}} \circ {\textbf{1}}_{k_0^*, \ell _0^*}, \end{aligned}$$
(17)
where \(\circ\) denotes Hadamard product, \(\pi _{k_0^*}\) and \(\pi _{\ell _0^*}\) denote the block assignment probabilities for block \(k_0^*\) and \(\ell _0^*\) respectively, and \({\textbf{1}}_*\) is the \(K \times K\) binary matrix with 0’s everywhere except for 1’s associated with the Chernoff-active blocks \(\left( k_0^*, \ell _0^* \right)\), i.e., for any \(i, j \in \{1, \cdots , K \}\)
$$\begin{aligned} {\textbf{1}}_{k_0^*, \ell _0^*}[i, j] = {\left\{ \begin{array}{ll} 1 &{} \text {if} \;\; (i, j) \in \left\{ \left( k_0^*, k_0^* \right) , \; \left( k_0^*, \ell _0^* \right) , \; \left( \ell _0^*, k_0^* \right) , \; \left( \ell _0^*, \ell _0^* \right) \right\} \\ 0 &{} \text {otherwise} \end{array}\right. } . \end{aligned}$$
(18)
Note that the multiplier \(\frac{1}{\left( \pi _{k_0^*} + \pi _{\ell _0^*}\right) ^2}\) on \(p_1 {\textbf{B}} \circ {\textbf{1}}_*\) assures that we sample the same number of potential edges with \(\widetilde{{\textbf{B}}}_1\) as we do with \({\textbf{B}}_1\) in the baseline sampling scheme. In addition, to avoid over-sampling with respect to \({\textbf{B}}\), i.e., to ensure \(\widetilde{{\textbf{B}}}_1[i, j] \le {\textbf{B}}[i, j]\) for any \(i, j \in \{1, \cdots , K \}\), we require
$$\begin{aligned} p_1 \le p_1^{\text {max}} = \left( 1 - p_0 \right) \left( \pi _{k_0^*} + \pi _{\ell _0^*}\right) ^2. \end{aligned}$$
(19)
Assumption 2
For K-block SBMs, given a block connectivity probability matrix \({\textbf{B}} \in (0, 1)^{K \times K}\) and a vector of block assignment probabilities \(\varvec{\pi } \in (0, 1)^K\). Let \(p_1^* \in (0, p_1^{\text {max}}]\) be the smallest positive \(p_1 \le p_1^{\text {max}}\) such that
$$\begin{aligned} \arg \min _{k \ne l} C_{k ,\ell }(\widetilde{{\textbf{B}}}_1, \varvec{\pi }) \end{aligned}$$
(20)
is not unique where \(p_1^{\text {max}}\) is defined as in Eq. (19) and \(\widetilde{{\textbf{B}}}_1\) is defined as in Eq. (17). If the arg min is always unique, let \(p_1^* = p_1^{\text {max}}\).
For any \(p_1 \in (0, p_1^*)\), we can have a better block recovery from \(\widetilde{{\textbf{B}}}_1\) than \({\textbf{B}}_1\), i.e., our Chernoff-optimal dynamic network sampling sheme is better than the baseline sampling scheme in terms of block recovery.
As an illustaration, consider the 4-block SBM with initial sampling parameter \(p_0 = 0.01\) and block connectivity probability matrix \({\textbf{B}}\) as in Eq. (14). Figure 2 shows the Chernoff information \(\rho\) as in Eq. (10) corresponding to \({\textbf{B}}\) as in Eq. (14), \({\textbf{B}}_0\) as in Eq. (13), \({\textbf{B}}_1\) as in Eq. (15), and \(\widetilde{{\textbf{B}}}_1\) as in Eq. (17) with dynamic network sampling parameter \(p_1 \in (0, p_1^*)\) where \(p_1^*\) is defined as in Assumption 2. In addition, Figure 2a assumes \(\varvec{\pi } = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4})\) and Fig. 2b assumes \(\varvec{\pi } = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8})\). Note that for any \(p_1 \in (0, p_1^*)\) we have \(\rho _{B}> \rho _{{\widetilde{B}}_1}> \rho _{B_1} > \rho _{B_0}\) and thus \({\textbf{B}} \succ \widetilde{{\textbf{B}}}_1 \succ {\textbf{B}}_1 \succ {\textbf{B}}_0\). That is, in terms of Chernoff information, when given same amount of resources, the proposed Chernoff-optimal dynamic network sampling scheme can yield better block recovery results. In other words, to reach the same level of performance, in terms of Chernoff information, the proposed Chernoff-optimal dynamic network sampling scheme needs less resources.
As described earlier, it may be the case that \(p_1^* < p_1^{\text {max}}\) at which point Chernoff-active blocks change to \((k_1^*, \ell _1^*)\). This potential non-uniquess of the Chernoff argmin is a consequence of our dynamic network sampling scheme. In the case of \(p_1 > p_1^*\), our Chernoff-optimal dynamic network sampling scheme is adopted as
$$\begin{aligned} \widetilde{{\textbf{B}}}_1^* = {\textbf{B}}_0 + \left( p_1 - p_1^* \right) {\textbf{B}} + \frac{p_1^*}{\left( \pi _{k_0^*} + \pi _{\ell _0^*}\right) ^2 } {\textbf{B}} \circ {\textbf{1}}_{k_0^*, \ell _0^*}, \end{aligned}$$
(21)
Similarly, the multiplier \(\frac{1}{\left( \pi _{k_0^*} + \pi _{\ell _0^*}\right) ^2}\) on \(p_1^* {\textbf{B}} \circ {\textbf{1}}_{k_0^*, \ell _0^*}\) assures that we sample the same number of potential edges with \(\widetilde{{\textbf{B}}}_1^*\) as we do with \({\textbf{B}}_1\) in the baseline sampling scheme. In addition, to avoid over-sampling with respect to \({\textbf{B}}\), i.e., \(\widetilde{{\textbf{B}}}_1^*[i, j] \le {\textbf{B}}[i, j]\) for any \(i, j \in \{1, \cdots , K \}\), we require
$$\begin{aligned} p_1 \le p_{11}^{\text {max}} = 1 - p_0 - \frac{p_1^*}{\left( \pi _{k_0^*} + \pi _{\ell _0^*}\right) ^2 } + p_1^*. \end{aligned}$$
(22)
For any \(p_1 \in [p_1^*, p_{11}^{\text {max}}]\), we can have a better block recovery from \(\widetilde{{\textbf{B}}}_1^*\) than \({\textbf{B}}_1\), i.e., our Chernoff-optimal dynamic network sampling sheme is again better than the baseline sampling scheme in terms of block recovery.
As an illustration, consider a 4-block SBM with initial sampling parameter \(p_0 = 0.01\) and block connectivity probability matrix \({\textbf{B}}\) as in Eq. (14). Figure 3 shows the Chernoff information \(\rho\) as in Eq. (10) corresponding to \({\textbf{B}}\) as in Eq. (14), \({\textbf{B}}_0\) as in Eq. (13), \({\textbf{B}}_1\) as in Eq. (15), and \(\widetilde{{\textbf{B}}}_1^*\) as in Eq. (21) with dynamic network sampling parameter \(p_1 \in [p_1^*, p_{11}^{\text {max}}]\) where \(p_1^*\) is defined as in Assumption 2 and \(p_{11}^{\text {max}}\) is defined as in Eq. (22). In addition, Fig. 3a assumes \(\varvec{\pi } = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4})\) and Fig. 3b assumes \(\varvec{\pi } = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8})\). Note that for any \(p_1 \in [p_1^*, p_{11}^{\text {max}}]\) we have \(\rho _{B}> \rho _{{\widetilde{B}}_1^*}> \rho _{B_1} > \rho _{B_0}\) and thus \({\textbf{B}} \succ \widetilde{{\textbf{B}}}_1^* \succ {\textbf{B}}_1 \succ {\textbf{B}}_0\). That is, the adopted Chernoff-optimal dynamic network sampling scheme can still yield better block recovery results, in terms of Chernoff information, given the same amout of resources.
Now we illustrate how the proposed Chernoff-optimal dynamic network sampling sheme can be migrated for real applications. We summarize the uniform dynamic sampling scheme (baseline) as Algorithm 1 and our Chernoff-optimal dynamic network sampling scheme as Algorithm 2. Recall given potential edge set E and initial sampling parameter \(p_0 \in (0, 1)\), we have the initial edge set \(E_0 \subset E\) with \(|E_0 |= p_0 |E |\). The goal is to dynamically sample new edges from the potential edge set so that we can have a better block recovery given limited resources.