Modification of Social Dominance in Autocratic Networks

Abstract

The previous two chapters focused on establishing the properties of dynamics of the individuals’ social powers, including convergence results. This chapter returns to the original DeGroot–Friedkin model with constant relative interaction topology, and considers a network modification problem. In particular, the star topology is studied, because this special type of topology leads to a single individual (the centre node of the star) accumulating all of the social power. Different strategies, via the insertion of new individuals and/or new interpersonal relationships, are proposed to modify the star topology so that the centre individual no longer has the largest social power. Necessary and sufficient conditions on the strength of the new interpersonal relationships are obtained for each strategy. Interpretations are provided in the context of social networks, which leads to several insightful conclusions.

7.6 Appendix: Proofs

The proofs extensively use Theorem 7.1, Statement (ii), which states that $x_i^*> x_j^* \Leftrightarrow \gamma _i > \gamma _j$ and $x_i^* = x_j^* \Leftrightarrow \gamma _i = \gamma _j$.

1.1 7.6.1 Proofs for Theorem 7.2 and Corollary 7.1

The expression $\varvec{\gamma }^\top = \varvec{\gamma }^\top \varvec{C}$, where $\varvec{C}$ is given in Eq. (7.3), yields

$$\begin{aligned} \gamma _1&= \sum _{i = 2}^{n-2} \gamma _i + (1-\beta )\gamma _{n-1} \end{aligned}$$

(7.4a)

$$\begin{aligned} \gamma _i&= c_{1i}\gamma _1,\quad \forall \, i \ne 1,n-1,n \end{aligned}$$

(7.4b)

$$\begin{aligned} \gamma _{n-1}&= c_{1,n-1}\gamma _1 + \gamma _n \end{aligned}$$

(7.4c)

$$\begin{aligned} \gamma _n&= \beta \gamma _{n-1}. \end{aligned}$$

(7.4d)

Statement (i) is obtained from Eq. (7.4b), where it is concluded that $\gamma _i < \gamma _1$ because $c_{1i} < 1$ for all $i \ne 1, n-1, n$, and from Eq. (7.4d), which allows one to conclude that $\gamma _n < \gamma _{n-1}$ for all $\beta \in (0,1)$. For Statement (ii), sub stituting $\gamma _n$ from Eqs. (7.4d) into (7.4c) yields $\gamma _{n-1} = c_{1,n-1}\gamma _1 + \beta \gamma _{n-1}$. This is rearranged to obtain $\gamma _1 = \gamma _{n-1}(1-\beta )/c_{1,n-1}$. Recalling that $0 < c_{1,n-1}$ and $0< \beta < 1$, it follows that $\gamma _1 < \gamma _{n-1}$ if and only if $\beta > 1 - c_{1,n-1}$. Similarly, one can obtain that $\gamma _1 > \gamma _n$ if and only if $\beta < 1/(1+c_{1,n-1})$, which proves Statement (iii). Corollary 7.1 is a generalisation of Statement (ii) obtained by observing that $\text {argmin}_j (1-c_{1,j}) = \text {argmax}_j c_{1,j}$. $\square$

1.2 7.6.2 Proofs for Theorem 7.3 and Corollary 7.2

For Topology Variation 2, the relative interaction matrix $\varvec{C}$ is given by

$$\begin{aligned} \varvec{C}(\beta ) = \begin{bmatrix} 0&c_{12}&c_{13}&\ldots&c_{1,n-1}&0&0 \\ 1&0&0&\ldots&0&0&0 \\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots \\ 1-(\beta _1+\beta _2)&0&0&\ldots&0&\beta _1&\beta _2 \\ 0&0&0&\ldots&1&0&0 \\ 0&0&0&\ldots&1&0&0 \end{bmatrix}. \end{aligned}$$

(7.5)

From $\varvec{\gamma }^\top \varvec{C} = \varvec{\gamma }^\top$, it follows that

$$\begin{aligned} \gamma _1&= \sum _{i = 2}^{n-3} \gamma _i + (1-\beta _1-\beta _2)\gamma _{n-2} \end{aligned}$$

(7.6a)

$$\begin{aligned} \gamma _i&= c_{1i}\gamma _1,\quad \forall \, i \ne 1,n-2, n-1,n \end{aligned}$$

(7.6b)

$$\begin{aligned} \gamma _{n-2}&= c_{1,n-2}\gamma _1 + \gamma _{n-1} + \gamma _n \end{aligned}$$

(7.6c)

$$\begin{aligned} \gamma _{n-1}&= \beta _1 \gamma _{n-2} \end{aligned}$$

(7.6d)

$$\begin{aligned} \gamma _n&= \beta _2 \gamma _{n-2}. \end{aligned}$$

(7.6e)

Statement (i) is obtained from Eqs. (7.6a) and (7.6d) and (7.6e), using the same arguments as in the proof for Theorem 7.2. Regarding Statement (ii), substitute Eqs. (7.6d) and (7.6e) into (7.6c) and rearrange to obtain $\gamma _{n-2} = c_{1,n-2}\gamma _1/(1-\beta _1 - \beta _2)$. The statement is then straightforwardly obtained. For Statement (iii), in regards to $\gamma _n$, substitute $\gamma _{n-2} = c_{1,n-2}\gamma _1/(1-\beta _1 - \beta _2)$ into the right hand side of Eq. (7.6e) to obtain $\gamma _n = \beta _2 c_{1,n-2} \gamma _1/ (1 - \beta _1 - \beta _2)$. One can verify that $\beta _2 > (1- \beta _1)/(1+c_{1,n-2})$ implies $\beta _2 c_{1,n-2}/ (1 - \beta _1 - \beta _2) > 1$, which in turn implies $\gamma _n > \gamma _1$. The inequality that ensures $\gamma _{n-1} > \gamma _1$ can be similarly found. Observe that $1- \beta _1 < 1$, $1-\beta _2 < 1$ and $1 < 1+c_{1,n-2}$. There must also hold $\beta _1 + \beta _2 < 1$. This implies that for any value $c_{1,n-2}$, there always exist $\beta _1, \beta _2$ which ensures $\gamma _{n-1} > \gamma _1$ and $\gamma _n > \gamma _1$. From Eqs. (7.6d) and (7.6e), one has that $\gamma _{n-1}/\gamma _{n} = \beta _1/\beta _2$. Statement (iv) follows immediately. Corollary 7.2 is a generalisation of Statement (ii) by observing that $\text {argmin}_j (1-c_{1,j}) = \text {argmax}_j c_{1,j}$. $\square$

1.3 7.6.3 Proof for Theorem 7.4

The equation $\varvec{\gamma }^\top \varvec{C}= \varvec{\gamma }^\top$, with $\varvec{C}$ given by

$$\begin{aligned} \varvec{C}(\beta _1,\beta _2)= \begin{bmatrix} 0&c_{1,2}&\cdots&c_{1,n-3}&c_{1,n-2}&0&0 \\ 1&0&\cdots&0&0&0&0 \\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots \\ 1-\beta _1&0&\cdots&0&0&\beta _1&0 \\ 1-\beta _2&0&\cdots&0&0&0&\beta _2 \\ 0&0&\cdots&1&0&0&0 \\ 0&0&\cdots&0&1&0&0 \end{bmatrix}, \end{aligned}$$

(7.7)

yields $\gamma _1 = (1-\beta _1)\gamma _{n-3} + (1-\beta _2)\gamma _{n-2} + \sum _{i=2}^{n-4} \gamma _i$, along with the equalities:

$$\begin{aligned} \gamma _i&= c_i\gamma _1, \;\;\; i\in \{2,\ldots ,n-4\} \end{aligned}$$

(7.8a)

$$\begin{aligned} \gamma _{n-3}&= c_{1,n-3}\gamma _{1} +\gamma _{n-1} \end{aligned}$$

(7.8b)

$$\begin{aligned} \gamma _{n-2}&= c_{1,n-2}\gamma _{1} +\gamma _{n} \end{aligned}$$

(7.8c)

$$\begin{aligned} \gamma _{n-1}&= \beta _1\gamma _{n-3} \end{aligned}$$

(7.8d)

$$\begin{aligned} \gamma _{n}&= \beta _2\gamma _{n-2} \end{aligned}$$

(7.8e)

From (7.8a), since $c_{1,i} \in (0,1)$ for all $i\in \{2,\ldots ,n-4\}$, it follows that $\gamma _i<\gamma _0$ for all $i\in \{2,\ldots ,n-4\}$. From (7.8d) and (7.8e), since $\beta _1,\beta _2\in (0,1)$, it follows that $\gamma _{n-1}<\gamma _{n-3}$ and $\gamma _{n}<\gamma _{n-2}$. Thus, Statement (i) is true.

From (7.8b) and (7.8d), one has that $\frac{ \gamma _{n-3} }{\gamma _1} = \frac{c_{1,n-3}}{1-\beta _1}$, which implies that $\gamma _1>\gamma _{n-3}$ if and only if $\beta _1 < 1-c_{1,n-3}$. Similarly, from (7.8c) and (7.8e), one has that $\gamma _1 >\gamma _{n-2}$ if and only if $\beta _2 < 1-c_{1,n-2}$. One then concludes that for $i\in \{1,2\}$, if $\beta _i > 1 - c_{1,n-4+i}$, then $x_{n-4+i}^*> x_1^*$. Therefore, Statement (ii) is true.

From (7.8b) and (7.8d), one obtains that $\frac{\gamma _{n-1}}{\gamma _1} = \frac{\beta _1 c_{1,n-3} }{1-\beta _1}$. It follows that $\gamma _{n-1}>\gamma _1$ if and only if $\beta _1>1/(1+c_{1,n-3})$. Similarly, from (7.8c) and (7.8e), one concludes that $\gamma _{n}>\gamma _1$ if and only if $\beta _2>1/(1+c_{1,n-2})$. Thus, Statement (iii) is true.

Since $\frac{\gamma _{n-3}}{\gamma _1} = \frac{c_{1,n-3}}{1-\beta _1}$ and $\frac{\gamma _{n-2}}{\gamma _1} = \frac{c_{1,n-2}}{1-\beta _2}$, it follows that $\frac{\gamma _{n-3}(1-\beta _1)}{c_{1,n-3}} = \frac{\gamma _{n-2}(1-\beta _2)}{c_{1,n-2}}$, which implies that $\frac{\gamma _{n-3}}{\gamma _{n-2}} = \frac{c_{1,n-3}(1-\beta _2)}{c_{1,n-2}(1-\beta _1)}$. Then, $\gamma _{n-3} > \gamma _{n-2}$ if and only if $\frac{1-\beta _2}{1-\beta _1}>\frac{c_{1,n-2}}{c_{1,n-3}}$. Therefore, Statement (iv) is true. $\square$

1.4 7.6.4 Proof for Theorem 7.5

For Topology Variation 4, the relative interaction matrix $\varvec{C}$ is expressed as

$$\begin{aligned} \varvec{C}(\beta _1, \beta _2) = \begin{bmatrix} 0&c_{12}&c_{13}&\ldots&c_{1,n-1}&c_{1,n} \\ 1&0&0&\ldots&0&0 \\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots \\ 1-\beta _1&0&0&\ldots&0&\beta _1 \\ 1- \beta _2&0&0&\ldots&\beta _2&0 \end{bmatrix}, \end{aligned}$$

(7.9)

where $\beta _1 = c_{n-1,n}$ and $\beta _2 = c_{n,n-1}$. The expression $\varvec{\gamma }^\top \varvec{C} = \varvec{\gamma }^\top$ yields

$$\begin{aligned} \gamma _1&= \sum _{i =2 }^{n-2} \gamma _i + (1-\beta _1)\gamma _{n-1} + (1-\beta _2) \gamma _n \end{aligned}$$

(7.10a)

$$\begin{aligned} \gamma _i&= c_{1i}\gamma _1,\quad \forall \, i \ne 1, n-1,n \end{aligned}$$

(7.10b)

$$\begin{aligned} \gamma _{n-1}&= c_{1,n-1}\gamma _1 + \beta _2\gamma _n \end{aligned}$$

(7.10c)

$$\begin{aligned} \gamma _{n}&= c_{1,n}\gamma _1 + \beta _1 \gamma _{n-1}. \end{aligned}$$

(7.10d)

Again, Statement (i) is obtained from Eq. (7.10b). Substitute Eqs. (7.10c) into (7.10d) and rearrange for $\gamma _n$ to obtain

$$\begin{aligned} \gamma _n = \left( \frac{c_{1,n} + \beta _1 c_{1,n-1} }{ 1-\beta _1\beta _2 } \right) \gamma _1, \end{aligned}$$

(7.11)

and it follows that $\gamma _n > \gamma _1$ is implied by

$$\begin{aligned} c_{1,n}+ \beta _1 c_{1, n-1}&> 1 - \beta _1 \beta _2 \end{aligned}$$

(7.12)

$$\begin{aligned} \beta _1&> \frac{1-c_{1,n}}{c_{1,n-1} + \beta _2 } \end{aligned}$$

(7.13)

$$\begin{aligned} \beta _2&> \frac{1 - c_{1,n} - c_{1,n-1}\beta _1 }{\beta _1 }. \end{aligned}$$

(7.14)

Consider Eq. (7.13). Observe that $(1-c_{1,n})/(c_{1,n-1} + \beta _2) \ge 1 \Leftrightarrow 1 - c_{1,n} - c_{1,n-1} \ge \beta _2 \Leftrightarrow \sum _{i = 2}^{n-2} c_{1,i} \ge \beta _2$. Recalling that $\beta _1 \in (0,1)$, one concludes that $\gamma _n > \gamma _1$ is possible only if $\beta _2 > \sum _{i = 2}^{n-2} c_{1,i}$. Alternatively, one can consider Eq. (7.14) and similarly derive that $\gamma _n > \gamma _1$ if $\beta _2 > (1-c_{1,n} - \beta _1 c_{1,n-1})/\beta _1$ and $\beta _1 > (1-c_{1,n})/(1+c_{1,n-1})$. The inequality conditions for ensuring $\gamma _{n-1} > \gamma _1$ are derived in a similar manner. $\square$

1.5 7.6.5 Proof for Theorem 7.6

The relative interaction matrix for Topology Variation 5 is given by

$$\begin{aligned} \varvec{C}(\beta _1, \beta _2) = \begin{bmatrix} 0&c_{12}&c_{13}&\ldots&\beta _1&0&\ldots&0 \\ 1&0&0&\ldots&0&0&\ldots&0 \\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots \\ 1&0&0&\ldots&0&0&\ldots&0 \\ \beta _2&0&0&\ldots&0&c_{n+1,n+2}&\ldots&c_{n+1,n} \\ 0&0&0&\ldots&1&0&\ldots&0 \\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots \\ 0&0&0&\ldots&1&0&\ldots&0 \end{bmatrix} \end{aligned}$$

(7.15)

And the expression $\varvec{\gamma }^\top \varvec{C} = \varvec{\gamma }^\top$ yields the following equalities

$$\begin{aligned} \gamma _1&= \sum _{ 1 < i \le n } \gamma _i + \beta _2\gamma _{n+1} \end{aligned}$$

(7.16a)

$$\begin{aligned} \gamma _i&= c_{1,i}\gamma _1,\quad \forall \, i \in \{2,\ldots , n\} \end{aligned}$$

(7.16b)

$$\begin{aligned} \gamma _{n+1}&= \sum _{ n + 1 < i \le n+m } \gamma _i + \beta _1\gamma _1 \end{aligned}$$

(7.16c)

$$\begin{aligned} \gamma _{i}&= c_{n+1,i} \gamma _{n+1},\quad \forall \, i \in \{n+2,\ldots , n+m\} \end{aligned}$$

(7.16d)

Statement (i) is obtained trivially from Eqs. (7.16b) and (7.16d). In regards to Statement (ii), first substitute Eqs. (7.16b) into (7.16a) to obtain $\gamma _1 = \beta _2 \gamma _{n+1} + \sum _{ 1 < i \le n } c_{1,i} \gamma _1$ which is rearranged to yield $\gamma _1 (1 - \sum _{ 1 < i \le n } c_{1,i}) = \beta _2 \gamma _{n+1}$, and which is equivalent to $\beta _1\gamma _1 = \beta _2 \gamma _{n+1}$ because $1 - \sum _{ 1 < i \le n } c_{1,i} = \beta _1$. Statement (iii) is obtained by substituting $\gamma _1 = \beta _2 \gamma _{n+1}/ \beta _1$ into Eq. (7.16d). $\square$