In this sub-section, we analyze the sparsity reduction process by promoting some of the periphery agents to core agents in each component. For example, Figure 1 illustrates the process of increasing the number of core agents from 2s = 2, 2s = 4, and then 2s = 6, while keeping component size 2p = 8 fixed. As two previous periphery agents are promoted to core status in each component, sparsity of $W_i$ decreases due to the formation of new links between the newly designated core agents and their corresponding remaining periphery sets.

In general, the formula for network sparsity is given by:

$\text{Sparsity}=\frac{N\left(N-1\right)-\left[4s\left(p-s\right)+2s\left(2s-1\right)\right]B}{N\left(N-1\right)}=1-\frac{\left(2p-1\right)s}{p\left(N-1\right)}$ (8)

We have $\frac{\partial \text{Sparsity}}{\partial s}=-\frac{2p-1}{p\left(N-1\right)}<0$, as p > 1. This confirms that sparsity is a decreasing function of the number of core agents s, implying that an increase in the number of core agents leads to a denser network. Consequently, s can serve as an inverse measure of sparsity.

Proposition 1. W is a block-diagonal matrix, where each block $W_i$ represents an incomplete core-periphery component of size 2p. Each $W_i$ consists of two interconnected sets of core agents, each connecting to its corresponding set of periphery agents. The structure of $W_i$ is as follows:

$W_i={\left(\begin{array}{cc}{1}_0{}_{{}_{2s\times 2s}}& R_0{}_{{}_{2s\times 2\left(p-s\right)}}\\ {R^{\prime }}_{0_{2\left(p-s\right)\times 2s}}& 0_{2\left(p-s\right)\times 2\left(p-s\right)}\end{array}\right)}_{2p\times 2p}$ (9)

where

$R_0{}_{{}_{2s\times 2\left(p-s\right)}}={\left(\begin{array}{cc}{1}_{s\times \left(p-s\right)}& 0_{s\times \left(p-s\right)}\\ 0_{s\times \left(p-s\right)}& 1_{s\times \left(p-s\right)}\end{array}\right)}_{2s\times 2\left(p-s\right)}$.

If $S\stackrel{\text{def}}{=}{\left(I-\rho W\right)}^{-1}$, then $S$ is also a block-diagonal matrix with submatrix $S_i$ of the following form:

$\begin{array}{c}{S}_i={\left(\begin{array}{cc}{A}_{2s\times 2s}& B_{2s\times 2\left(p-s\right)}\\ {B^{\prime }}_{2\left(p-s\right)\times 2s}& D_{2\left(p-s\right)\times 2\left(p-s\right)}\end{array}\right)}_{p\times p}\\ =\left(\begin{array}{cc}\begin{array}{cc}{M}_{s\times s}& F_{s\times s}\\ F_{s\times s}& M_{s\times s}\end{array}& \begin{array}{cc}{J}_{s\times \left(p-s\right)}& H_{s\times \left(p-s\right)}\\ H_{s\times \left(p-s\right)}& J_{s\times \left(p-s\right)}\end{array}\\ \begin{array}{cc}{J^{\prime }}_{\left(p-s\right)\times s}& {H^{\prime }}_{\left(p-s\right)\times s}\\ {H^{\prime }}_{\left(p-s\right)\times s}& {J^{\prime }}_{\left(p-s\right)\times s}\end{array}& \begin{array}{cc}{K}_{\left(p-s\right)\times \left(p-s\right)}& G_{\left(p-s\right)\times \left(p-s\right)}\\ G_{\left(p-s\right)\times \left(p-s\right)}& K_{\left(p-s\right)\times \left(p-s\right)}\end{array}\end{array}\right)\end{array}$ (10)

where

$A_{2s\times 2s}=\left(\begin{array}{cc}{M}_{s\times s}& F_{s\times s}\\ F_{s\times s}& M_{s\times s}\end{array}\right)$

$B_{2s\times 2\left(p-s\right)}=\left(\begin{array}{cc}{J}_{s\times \left(p-s\right)}& H_{s\times \left(p-s\right)}\\ H_{s\times \left(p-s\right)}& J_{s\times \left(p-s\right)}\end{array}\right)$

$D_{2\left(p-s\right)\times 2\left(p-s\right)}=\left(\begin{array}{cc}{K}_{\left(p-s\right)\times \left(p-s\right)}& G_{\left(p-s\right)\times \left(p-s\right)}\\ G_{\left(p-s\right)\times \left(p-s\right)}& K_{\left(p-s\right)\times \left(p-s\right)}\end{array}\right)$

$M_{s\times s}=a{I}_{s\times s}+c{1}_0{}_{{}_{s\times s}}$

$F_{s\times s}=f{1}_{s\times s}$

$K_{\left(p-s\right)\times \left(p-s\right)}=b{I}_{\left(p-s\right)\times \left(p-s\right)}+d{1}_0{}_{{}_{\left(p-s\right)\times \left(p-s\right)}}$

$G_{\left(p-s\right)\times \left(p-s\right)}=g{1}_{\left(p-s\right)\times \left(p-s\right)}$

$J_{s\times \left(p-s\right)}=e{1}_{s\times \left(p-s\right)}$

$H_{s\times \left(p-s\right)}=h{1}_{s\times \left(p-s\right)}$

$I$ is an identity matrix and $1_0$ consists of all ones except that the diagonal terms are zeros.

Detailed formulae of a to h are in the Appendix A.1.

From Proposition 1, we can calculate network impacts based on definition 2 in Section 2 and prove the following proposition in the Appendix A.1.

Proposition 2. Given the size p of incomplete core-periphery components, if the number of its core agents s increases that leads to a decrease in network sparsity, then the average total impact of the network increases.

Since average direct and indirect impacts, and their derivatives with respect to s are complicated, we rely on simulations to verify their signs. In the simulation analysis, we also verify the total impact to confirm Proposition 2.

We conducted two batches of simulations to examine the signs of these network impacts. In the first batch of simulation, we utilized the closed-form solutions for average direct, indirect, and total impacts from Proposition 2, as derived in Appendix A.1. By calculating their derivatives with respect to the number of core agents 𝑠—an inverse metric of network sparsity—we identified positive derivatives across all measures (see Panel A, B, and C of Table 1 ). This finding is consistent with our conclusion: as the number of core agents 𝑠 increases, leading to reduced sparsity, all measures of network impact enlarge.

Panel A. Derivatives of direct impact with respect to s (the number of core agents).

Panel B. Derivatives of indirect impact with respect to s (the number of core agents).

Panel C. Derivatives of total impact with respect to s (the number of core agents).

Panel D. Relationship between sparsity and network impacts.

In the second batch of simulations, we calculated sparsity and network impacts, presenting the results in Panel D of Table 1 and Figure 2 . It shows a negative relationship between sparsity and network impacts. Additionally, we observe from Panel D of Table 1 and Figure 2 that direct impacts contribute substantially to total impacts, while the reduction in sparsity was relatively small. This is because our simulated networks are highly sparse by our experiment design, indicating indirect impacts to be relatively low.

However, we find that the magnitude of the derivatives for indirect impacts significantly outweighs that of total impacts. In fact, the derivatives of total impacts in Panel C closely mirror those of indirect impacts presented in Panel B of Table 1 . This indicates that indirect impacts are more sensitive to changes in sparsity than direct impacts. This finding is akin to those reported in Jie and Ma [6] who examined financial networks with complete core-periphery components. Taking together, it accentuates the importance of investigating both direct and indirect impacts to get a full picture of the network total impacts. We have arrived at the following conclusion.

Conclusion 1. Given the size p of incomplete core-periphery components, if the number of its core agents s increases that leads to a reduction of network sparsity, then both the average direct and indirect impacts of the network increase.

This sub-section explores the second strategy to reduce network sparsity: merging multiple small incomplete core-periphery components into a larger incomplete core-periphery component. For example, Figure 3 illustrates the merging process from two core agents (2s = 2) to four core agents (2s = 4). Before the merger, the network consists of B incomplete core-periphery components. Each component has a size of 6 with 2 core agents: the first core agent is connected to two periphery agents, while the second core agent is linked to the other two periphery agents. After the merger, every two components are merged into a single larger incomplete core-periphery component, increasing the number of core agents to four (2s = 4).

In this new structure, two core agents are connected to the four periphery agents, while the other two core agents are linked to the remaining four periphery agents. The four core agents are also interconnected. The sparsity of the new network decreases as new links are formed among core agents and between core and periphery agents. The number of components B is halved, while the component size p doubles.

Suppose before merging, the adjacency matrix contains $2s_0$ core agents per component, with each component size being $2p_0$. After merging $\mu$ components into a larger component, the number of core agents becomes $2s=2\mu s_0$ and the component size becomes $2p=2\mu p_0$. That is, $p=s{p}_0/s_0=\lambda s$ where $\lambda =p_0/s_0>1$ is fixed. In this circumstance, p has a linear relationship with s.

In general, the network sparsity becomes

$\text{Sparsity}=\frac{N\left(N-1\right)-\left[4s\left(p-s\right)+2s\left(2s-1\right)\right]B}{N\left(N-1\right)}=1-\frac{2\lambda s-1}{\lambda \left(N-1\right)}$ (11)

The formula shows that sparsity decreases as s increases: $\frac{\partial \text{Sparsity}}{\partial s}=-\frac{2\lambda }{\lambda \left(N-1\right)}<0$ as $\lambda >1$. Therefore, the number of core agents s can serve as an inverse measure of sparsity in the network.

Proposition 3. W is a block-diagonal matrix, where each block $W_i$ represents an incomplete core-periphery component of size 2p. Each $W_i$ consists of two interconnected sets of core agents, each connecting to a separate set of periphery agents. The structure of $W_i$ is as follows:

$W_i={\left(\begin{array}{cc}{1}_0{}_{{}_{2s\times 2s}}& R_0{}_{{}_{2s\times 2\left(p-s\right)}}\\ {R^{\prime }}_{0_{2\left(p-s\right)\times 2s}}& 0_{2\left(p-s\right)\times 2\left(p-s\right)}\end{array}\right)}_{2p\times 2p}$ (12)

where

$R_0{}_{{}_{2s\times 2\left(p-s\right)}}={\left(\begin{array}{cc}{1}_{s\times \left(p-s\right)}& 0_{s\times \left(p-s\right)}\\ 0_{s\times \left(p-s\right)}& 1_{s\times \left(p-s\right)}\end{array}\right)}_{2s\times 2\left(p-s\right)}$.

If $S\stackrel{\text{def}}{=}{\left(I-\rho W\right)}^{-1}$ then $S$ is also a block-diagonal matrix with submatrix $S_i$ of the following form:

$\begin{array}{c}{S}_i={\left(\begin{array}{cc}{A}_{2s\times 2s}& B_{2s\times 2\left(p-s\right)}\\ {B^{\prime }}_{2\left(p-s\right)\times 2s}& D_{2\left(p-s\right)\times 2\left(p-s\right)}\end{array}\right)}_{p\times p}\\ =\left(\begin{array}{cc}\begin{array}{cc}{M}_{s\times s}& F_{s\times s}\\ F_{s\times s}& M_{s\times s}\end{array}& \begin{array}{cc}{J}_{s\times \left(p-s\right)}& H_{s\times \left(p-s\right)}\\ H_{s\times \left(p-s\right)}& J_{s\times \left(p-s\right)}\end{array}\\ \begin{array}{cc}{J^{\prime }}_{\left(p-s\right)\times s}& {H^{\prime }}_{\left(p-s\right)\times s}\\ {H^{\prime }}_{\left(p-s\right)\times s}& {J^{\prime }}_{\left(p-s\right)\times s}\end{array}& \begin{array}{cc}{K}_{\left(p-s\right)\times \left(p-s\right)}& G_{\left(p-s\right)\times \left(p-s\right)}\\ G_{\left(p-s\right)\times \left(p-s\right)}& K_{\left(p-s\right)\times \left(p-s\right)}\end{array}\end{array}\right)\end{array}$ (13)

where

$A_{2s\times 2s}=\left(\begin{array}{cc}{M}_{s\times s}& F_{s\times s}\\ F_{s\times s}& M_{s\times s}\end{array}\right)$

$B_{2s\times 2\left(p-s\right)}=\left(\begin{array}{cc}{J}_{s\times \left(p-s\right)}& H_{s\times \left(p-s\right)}\\ H_{s\times \left(p-s\right)}& J_{s\times \left(p-s\right)}\end{array}\right)$

$D_{2\left(p-s\right)\times 2\left(p-s\right)}=\left(\begin{array}{cc}{K}_{\left(p-s\right)\times \left(p-s\right)}& G_{\left(p-s\right)\times \left(p-s\right)}\\ G_{\left(p-s\right)\times \left(p-s\right)}& K_{\left(p-s\right)\times \left(p-s\right)}\end{array}\right)$

$M_{s\times s}=a{I}_{s\times s}+c{1}_0{}_{{}_{s\times s}}$

$F_{s\times s}=f{1}_{s\times s}$

$K_{\left(p-s\right)\times \left(p-s\right)}=b{I}_{\left(p-s\right)\times \left(p-s\right)}+d{1}_0{}_{{}_{\left(p-s\right)\times \left(p-s\right)}}$

$G_{\left(p-s\right)\times \left(p-s\right)}=g{1}_{\left(p-s\right)\times \left(p-s\right)}$

$J_{s\times \left(p-s\right)}=e{1}_{s\times \left(p-s\right)}$

$H_{s\times \left(p-s\right)}=h{1}_{s\times \left(p-s\right)}$

$I$ is an identity matrix and $1_0$ consists of all ones except that the diagonal terms are zeros.

Detailed formulae of a to h can be found in the Appendix A.2.

From Proposition 3, we can calculate network impacts based on definition 2 in Section 2. We can prove the following proposition in the Appendix A.2.

Proposition 4. Given the network size N, if a merger of incomplete core-periphery components that leads to a decrease in network sparsity, then the average total impact of the network increases.

Since both derivatives of direct and indirect impact with respect to s are complicated, we employ a simulation method to determine their signs. We also verify the total impact to support Proposition 4. The simulation results presented in Table 2 indicate that if the number of core agents s increases due to the merger of incomplete core-periphery components, then network sparsity decreases and all impact measures increase. Figure 4 illustrates the inverse relationship between sparsity and network impacts, further validating Proposition 4 and leading to the following conclusion.

Conclusion 2. Given the network size N, if a merger of incomplete core-periphery components that leads to a reduction of network sparsity, then both average direct and indirect impacts of the network increase.

Proof for Proposition 1:

The adjacency matrix $W_i$ of each core-periphery component $i$ is as follows:

$W_i=\left(\begin{array}{cc}CC& CP\\ PC& PP\end{array}\right)=\left(\begin{array}{cc}{1}_0& R\\ R^{\prime }& 0\end{array}\right)$ (1’)

In this case, $CC=1_0$ is a $2s\times 2s$ matrix with all ones except the diagonal terms. $PP$ is a $\left(2p-2s\right)\times \left(2p-2s\right)$ zero matrix. $CP$ is a $2s\times 2\left(p-s\right)$ matrix with all ones. $PC$ is the transpose matrix of $CP$. Thus, we have

$I-\rho W_i=\left(\begin{array}{cc}I-\rho 1_0& -\rho R\\ -\rho R^{\prime }& I\end{array}\right)$ (2’)

$S_i\stackrel{\text{def}}{=}{\left(I-\rho W_i\right)}^{-1}={\left(\begin{array}{cc}I-\rho 1_0& -\rho R\\ -\rho R^{\prime }& I\end{array}\right)}^{-1}=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$ (3’)

where

$A={\left(I-\rho 1_0-{\rho }^{2}R{R}^{\prime }\right)}^{-1}=\left(\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}a& c\\ c& a\end{array}& \begin{array}{cc}\cdots & c\\ \cdots & c\end{array}\\ \begin{array}{cc}⋮& ⋮\\ c& c\end{array}& \begin{array}{cc}& ⋮\\ \cdots & a\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}f& f\\ f& f\end{array}& \begin{array}{cc}\cdots & f\\ \cdots & f\end{array}\\ \begin{array}{cc}⋮& ⋮\\ f& f\end{array}& \begin{array}{cc}& ⋮\\ \cdots & f\end{array}\end{array}\\ \begin{array}{cc}\begin{array}{cc}f& f\\ f& f\end{array}& \begin{array}{cc}\cdots & f\\ \cdots & f\end{array}\\ \begin{array}{cc}⋮& ⋮\\ f& f\end{array}& \begin{array}{cc}& ⋮\\ \cdots & f\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}a& c\\ c& a\end{array}& \begin{array}{cc}\cdots & c\\ \cdots & c\end{array}\\ \begin{array}{cc}⋮& ⋮\\ c& c\end{array}& \begin{array}{cc}& ⋮\\ \cdots & a\end{array}\end{array}\end{array}\right)$,

$a=\frac{1}{2\left(1+\rho \right)}\left\{2+\frac{{\rho }^2\left(-p+s\right)}{-1+\rho \left[-1+\rho \left(p-s\right)s\right]}+\frac{\rho \left(-2-p\rho +\rho s\right)}{-1+\rho \left[-1+s\left(2+p\rho -\rho s\right)\right]}\right\}$

$c=\frac{\rho \left(1+\rho +p\rho +p{\rho }^2-\rho s-{\rho }^{2}s-2p{\rho }^{2}s-p^2{\rho }^{3}s+2{\rho }^2{s}^2+2p{\rho }^3{s}^2-{\rho }^3{s}^3\right)}{\left(1+\rho \right)\left(1+\rho -p{\rho }^{2}s+{\rho }^2{s}^2\right)\left(1+\rho -2\rho s-p{\rho }^{2}s+{\rho }^2{s}^2\right)}$

$f=\frac{\rho }{\left(1+\rho -p{\rho }^{2}s+{\rho }^2{s}^2\right)\left(1+\rho -2\rho s-p{\rho }^{2}s+{\rho }^2{s}^2\right)}$

$\begin{array}{c}B={\left(I-\rho 1_0\right)}^{-1}\rho R{\left[I-{\rho }^2{R}^{\prime }{\left(I-\rho 1_0\right)}^{-1}R\right]}^{-1}\\ =\left(\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}e& e\\ e& e\end{array}& \begin{array}{cc}\cdots & e\\ \cdots & e\end{array}\\ \begin{array}{cc}⋮& ⋮\\ e& e\end{array}& \begin{array}{cc}& ⋮\\ \cdots & e\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}h& h\\ h& h\end{array}& \begin{array}{cc}\cdots & h\\ \cdots & h\end{array}\\ \begin{array}{cc}⋮& ⋮\\ h& h\end{array}& \begin{array}{cc}& ⋮\\ \cdots & h\end{array}\end{array}\\ \begin{array}{cc}\begin{array}{cc}h& h\\ h& h\end{array}& \begin{array}{cc}\cdots & h\\ \cdots & h\end{array}\\ \begin{array}{cc}⋮& ⋮\\ h& h\end{array}& \begin{array}{cc}& ⋮\\ \cdots & h\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}e& e\\ e& e\end{array}& \begin{array}{cc}\cdots & e\\ \cdots & e\end{array}\\ \begin{array}{cc}⋮& ⋮\\ e& e\end{array}& \begin{array}{cc}& ⋮\\ \cdots & e\end{array}\end{array}\end{array}\right)\end{array}$

$e=\frac{\rho \left(1+\rho -\rho s-p{\rho }^{2}s+{\rho }^2{s}^2\right)}{\left(1+\rho -p{\rho }^{2}s+{\rho }^2{s}^2\right)\left(1+\rho -2\rho s-p{\rho }^{2}s+{\rho }^2{s}^2\right)}$

$h=\frac{{\rho }^{2}s}{\left(1+\rho -p{\rho }^{2}s+{\rho }^2{s}^2\right)\left(1+\rho -2\rho s-p{\rho }^{2}s+{\rho }^2{s}^2\right)}$

$C=\rho R^{\prime }{\left(I-\rho 1_0-{\rho }^{2}R{R}^{\prime }\right)}^{-1}=B^{\prime }$

$D={\left[I-{\rho }^2{R}^{\prime }{\left(I-\rho 1_0\right)}^{-1}R\right]}^{-1}=\left(\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}b& d\\ d& b\end{array}& \begin{array}{cc}\cdots & d\\ \cdots & d\end{array}\\ \begin{array}{cc}⋮& ⋮\\ d& d\end{array}& \begin{array}{cc}& ⋮\\ \cdots & b\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}g& g\\ g& g\end{array}& \begin{array}{cc}\cdots & g\\ \cdots & g\end{array}\\ \begin{array}{cc}⋮& ⋮\\ g& g\end{array}& \begin{array}{cc}& ⋮\\ \cdots & g\end{array}\end{array}\\ \begin{array}{cc}\begin{array}{cc}g& g\\ g& g\end{array}& \begin{array}{cc}\cdots & g\\ \cdots & g\end{array}\\ \begin{array}{cc}⋮& ⋮\\ g& g\end{array}& \begin{array}{cc}& ⋮\\ \cdots & g\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}b& d\\ d& b\end{array}& \begin{array}{cc}\cdots & d\\ \cdots & d\end{array}\\ \begin{array}{cc}⋮& ⋮\\ d& d\end{array}& \begin{array}{cc}& ⋮\\ \cdots & b\end{array}\end{array}\end{array}\right)$

$b=1+\frac{1}{2}{\rho }^{2}s\left(\frac{1}{1+\rho +{\rho }^{2}s\left(-p+s\right)}+\frac{1}{1+\rho +\rho s\left(-2-p\rho +\rho s\right)}\right)$

$d=\frac{{\rho }^{2}s\left(1+\rho -\rho s-p{\rho }^{2}s+{\rho }^2{s}^2\right)}{\left(1+\rho -p{\rho }^{2}s+{\rho }^2{s}^2\right)\left(1+\rho -2\rho s-p{\rho }^{2}s+{\rho }^2{s}^2\right)}$

$g=\frac{{\rho }^3{s}^2}{\left(1+\rho -p{\rho }^{2}s+{\rho }^2{s}^2\right)\left(1+\rho -2\rho s-p{\rho }^{2}s+{\rho }^2{s}^2\right)}$

QED

Proof for Proposition 2:

Based on definition 2 in Section 2, we can calculate the average direct/indirect/total impacts with normalized ${\beta }_k=1$ as follows. However, as will be shown, we can only provide a theoretical conclusion regarding the average total impact of the network. The formulae of average direct and indirect impacts are too complicated to obtain a theoretical result. We conduct numerical simulations and present the results in Table 1 .

First, we have average total impact with normalized ${\beta }_k=1$:

$\text{Average total impact}=\frac{p+p\rho -\rho s}{p\left(-1-\rho +2\rho s+p{\rho }^{2}s-{\rho }^2{s}^2\right)}$ (4’)

We take the derivative of total impact with respect to $s$:

$\begin{array}{l}\frac{\partial \text{Average Total Impact}}{\partial s}\\ =\frac{\rho \left(-1+2p-\rho +2p\rho +p^2\rho +p^2{\rho }^2-2p\rho s-2p{\rho }^{2}s+{\rho }^2{s}^2\right)}{p{\left(-1-\rho +2\rho s+p{\rho }^{2}s-{\rho }^2{s}^2\right)}^2}\\ =\frac{\rho \left(2p-1+\left(2p-1\right)\rho +p\rho \left(p-2s\right)+{\rho }^2{\left(p-s\right)}^2\right)}{p{\left(-1-\rho +2\rho s+p{\rho }^{2}s-{\rho }^2{s}^2\right)}^2}\end{array}$ (5’)

Since $2p-1+p\rho \left(p-2s\right)>2p-1+p\rho \left(s-2s\right)=2p-1-p\rho s$

$=p\left(2-\rho s\right)-1\ge p-1\ge 0$, as $0<\rho <\frac{1}{N-1}<\frac{1}{s}$

(see Jie and Ma [6] , Corollary 1 of Appendix A.1).

Thus $\frac{\partial \text{Average Total Impact}}{\partial s}>0$. (6’)

Next, we have the following two complicated formulae for average direct and indirect impacts with normalized ${\beta }_k=1$. Their numerical simulations results are presented in Table 1 .

$\begin{array}{l}\text{Average direct impact}=\left(\frac{1}{N}\underset{i=1}{\overset{N}{{\displaystyle \sum }}}{s}_{ii}\right)\beta =\frac{1}{p}\left[sa+\left(p-s\right)b\right]\\ =\frac{1}{2p}\{-2+2p-\frac{2+2\rho s}{1+\rho }+\frac{2+\rho }{1+\rho +{\rho }^{2}s\left(-p+s\right)}\\ +\frac{2+\rho -2\rho s}{1+\rho +\rho s\left(-2-p\rho +\rho s\right)}\}\end{array}$ (7')

$\begin{array}{l}\text{Average indirect impact}=\left(\frac{1}{N}\underset{i=1}{\overset{N}{{\displaystyle \sum }}}\underset{j\ne i}{\overset{}{{\displaystyle \sum }}}{s}_{ij}\right)\\ =\frac{1}{p}\left[\left(p-s\right)\left(p-s-1\right)d+2s\left(p-s\right)\left(e+h\right)+{\left(p-s\right)}^{2}g+s^{2}f+s\left(s-1\right)c\right]\\ =\frac{1}{2p}\{2-2p+\frac{2+2\rho s}{1+\rho }-\frac{2+\rho }{1+\rho +{\rho }^{2}s\left(-p+s\right)}\\ -\frac{2+\rho -2p\left(1+\rho \right)}{1+\rho +\rho s\left(-2-p\rho +\rho s\right)}\}\end{array}$ (8')

QED

Proof for Proposition 3:

The adjacency matrix $W_i$ of each core-periphery component $i$ is as follows:

$W_i=\left(\begin{array}{cc}CC& CP\\ PC& PP\end{array}\right)=\left(\begin{array}{cc}{1}_0& R\\ R^{\prime }& 0\end{array}\right)$ (9’)

In this case, $CC=1_0$ is a $2s\times 2s$ matrix with all ones except the diagonal terms. $PP$ is a $\left(2p-2s\right)\times \left(2p-2s\right)$ zero matrix. $CP$ is a $2s\times 2\left(p-s\right)$ matrix with all ones. $PC$ is the transpose matrix of $CP$. Suppose the number of cores in each component in the initial adjacency matrix is $2s_0$ and the size of component in the initial adjacency matrix is $2p_0$, the number of cores in each component is $2s=2\mu s_0$ after merging $\mu$ small components into one big component with the size of $2p=2\mu p_0$. That is, $p=\frac{p_0}{s_0}s=\lambda s$ where $\lambda =\frac{p_0}{s_0}>1$ is fixed. In this circumstance, $p$ has a linear relationship with $s$. Thus, we have

$I-\rho W_i=\left(\begin{array}{cc}I-\rho 1_0& -\rho R\\ -\rho R^{\prime }& I\end{array}\right)$ (10’)

$S_i\stackrel{\text{def}}{=}{\left(I-\rho W_i\right)}^{-1}={\left(\begin{array}{cc}I-\rho 1_0& -\rho R\\ -\rho R^{\prime }& I\end{array}\right)}^{-1}=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$ (11’)

where

$A={\left(I-\rho 1_0-{\rho }^{2}R{R}^{\text{'}}\right)}^{-1}=\left(\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}a& c\\ c& a\end{array}& \begin{array}{cc}\cdots & c\\ \cdots & c\end{array}\\ \begin{array}{cc}⋮& ⋮\\ c& c\end{array}& \begin{array}{cc}& ⋮\\ \cdots & a\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}f& f\\ f& f\end{array}& \begin{array}{cc}\cdots & f\\ \cdots & f\end{array}\\ \begin{array}{cc}⋮& ⋮\\ f& f\end{array}& \begin{array}{cc}& ⋮\\ \cdots & f\end{array}\end{array}\\ \begin{array}{cc}\begin{array}{cc}f& f\\ f& f\end{array}& \begin{array}{cc}\cdots & f\\ \cdots & f\end{array}\\ \begin{array}{cc}⋮& ⋮\\ f& f\end{array}& \begin{array}{cc}& ⋮\\ \cdots & f\end{array}\end{array}& \begin{array}{cc}\begin{array}{cc}a& c\\ c& a\end{array}& \begin{array}{cc}\cdots & c\\ \cdots & c\end{array}\\ \begin{array}{cc}⋮& ⋮\\ c& c\end{array}& \begin{array}{cc}& ⋮\\ \cdots & a\end{array}\end{array}\end{array}\right)$

$a=\frac{1}{2\left(1+\rho \right)}\left\{2+\frac{{\rho }^2\left(1-\lambda \right)s}{-1+\rho \left[-1+\rho \left(\lambda -1\right)s^2\right]}+\frac{\rho \left(-2+\left(1-\lambda \right)\rho s\right)}{-1+\rho \left[-1+s\left(2+\left(\lambda -1\right)\rho s\right)\right]}\right\}$