SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} P r \left\{1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} <   \left(1 - \varepsilon_ {1}\right) \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n}\right) \right\} \\ = P r \left\{1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} <   1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n} - \varepsilon_ {1} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n}\right) \right\} \\ = \Pr \left\{\alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} > \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n} + \varepsilon_ {1} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n}\right) \right\} \\ = P r \left\{\sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} > \left(1 + \varepsilon_ {1} ^ {\prime}\right) \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n} \right\} \\ \leq P r \left\{\sum_ {n = 1} ^ {\pi_ {c}} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} > \left(1 + \varepsilon_ {1} ^ {\prime}\right) \sum_ {n = 1} ^ {\pi_ {c}} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n} \right\} \tag {7} \\ \end{array}
$$
$$
\begin{array}{l} \leq P r \left\{\sum_ {n = 1} ^ {\pi_ {c}} \eta (n, \alpha) \left(\frac {\pi_ {c}}{\theta}\right) ^ {n} > \left(1 + \varepsilon_ {1} ^ {\prime}\right) \sum_ {n = 1} ^ {\pi_ {c}} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n} \right\} \\ \leq 1 - P r \left\{\left(\frac {\pi_ {c}}{\theta}\right) ^ {\pi_ {c}} <   \left(1 + \varepsilon_ {1} ^ {\prime}\right) \left(1 - \boldsymbol {u} _ {c} (S ^ {*})\right) ^ {\pi_ {c}} \right\} \\ = P r \left\{\left(\frac {\pi_ {c}}{\theta}\right) ^ {\pi_ {c}} \geq \left(1 + \varepsilon_ {1} ^ {\prime}\right) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {\pi_ {c}} \right\} \\ \left(\text {L e t} 1 + \varepsilon_ {0} = \sqrt [ \pi c ]{1 + \varepsilon_ {1} ^ {\prime}}\right) \\ = P r \left\{\frac {\pi_ {c}}{\theta} \geq \left(1 + \varepsilon_ {0}\right) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) \right\} \\ = \Pr \left\{\pi_ {c} - \theta_ {c} \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) \geq \varepsilon_ {0} \theta_ {c} \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) \right\} \\ \leq e x p \left(- \frac {\varepsilon_ {0} ^ {2}}{3} \theta_ {c} \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right)\right) \tag {8} \\ \end{array}
$$
Since  $0 \leq \frac{\epsilon}{2x} \leq \sqrt[x]{1 + \epsilon} - 1 \leq \frac{\epsilon}{x}$  for  $0 \leq \epsilon \leq 1$  and  $x \geq 1$ , it holds  $\frac{2x}{\epsilon} \geq \frac{1}{\sqrt[x]{1 + \epsilon} - 1}$ . Let  $\theta_c \geq \frac{12\pi_c^2 \ln(C / \delta_1)}{\varepsilon_1^2(1 - u_c(S^*))} \geq \frac{3 \ln(C / \delta_1)}{(\pi_c \sqrt{1 + \varepsilon_1'} - 1)^2(1 - u_c(S^*))} = \frac{3 \ln(C / \delta_1)}{\varepsilon_0^2(1 - u_c(S^*))}$ , then
$$
\begin{array}{l} E q. 8 = \exp \left(- \frac {\varepsilon_ {0} ^ {2}}{3} \theta_ {c} (1 - \boldsymbol {u} _ {c} (S ^ {*}))\right) \\ \leq e x p \left(- \frac {\varepsilon_ {0} ^ {2}}{3} \frac {3 \ln (C / \delta_ {1})}{\varepsilon_ {0} ^ {2} \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right)} \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right)\right) \\ = \delta_ {1} / C \tag {9} \\ \end{array}
$$
$$
E q. (6) \leq 1 - \prod_ {c \in \mathcal {C}} \left(1 - \delta_ {1} / C\right) \leq \delta_ {1} \tag {10}
$$
To limit Eq.(7) to the first  $Q$ $(Q\geq 2)$  terms, Eq.(7) becomes
$$
\begin{array}{l} P r \left\{\sum_ {n = 1} ^ {Q} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} > \left(1 + \varepsilon_ {1} ^ {\prime}\right) \sum_ {n = 1} ^ {\pi_ {c}} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n} \right\} \\ \leq P r \left\{\sum_ {n = 1} ^ {Q} \eta (n, \alpha) \left(\frac {\pi_ {c}}{\theta}\right) ^ {n} > \left(1 + \varepsilon_ {1} ^ {\prime}\right) \sum_ {n = 1} ^ {\pi_ {c}} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n} \right\} \\ \leq 1 - \Pr \left\{\left(\frac {\pi_ {c}}{\theta}\right) ^ {Q} <   \left(1 + \varepsilon_ {1} ^ {\prime}\right) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {Q} \right\} \\ \leq P r \left\{\left(\frac {\pi_ {c}}{\theta}\right) ^ {Q} \geq \left(1 + \varepsilon_ {1} ^ {\prime}\right) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {Q} \right\} \\ \left(\text {L e t} 1 + \varepsilon_ {0} = \sqrt [ q ]{1 + \varepsilon_ {1} ^ {\prime}}\right) \\ = \Pr \left\{\pi_ {c} - \theta_ {c} \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) \geq \varepsilon_ {0} \theta_ {c} \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) \right\} \\ \leq \exp \left(- \frac {\varepsilon_ {0} ^ {2}}{3} \theta_ {c} \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right)\right) (11) \\ \left(\operatorname {L e t} \theta_ {c} \geq \frac {1 2 Q ^ {2} \ln (C / \delta_ {1})}{\varepsilon_ {1} ^ {2} \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right)} \geq \frac {3 \ln (C / \delta_ {1})}{\left(\sqrt [ 4 ]{1 + \varepsilon_ {1} ^ {\prime}} - 1\right) ^ {2} \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right)} = \frac {3 \ln (C / \delta_ {1})}{\varepsilon_ {0} ^ {2} \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right)}\right) \\ \leq \delta_ {1} / C (12) \\ \end{array}
$$
$$
E q. (6) \leq \delta_ {1} \tag {13}
$$
It indicates  $Pr\left\{\hat{F}_{\alpha}(S^{*}, \mathcal{R}) < (1 - \varepsilon_{1}) \cdot OPT_{F}\right\} \geq 1 - \delta_{1}$ , thus concludes the proof.
Lemma 4. Let  $\delta_2 \in (0,1)$ ,  $\varepsilon_2 = \left(\frac{e}{e - 1}\right)\varepsilon - \varepsilon_1$ , and  $\theta_2 = \frac{8Q^2\ln(C\binom{n_G}{k} / \delta_2)}{\varepsilon_2^2(1 - b_0)}$  where  $Q$  is the approximation parameter;  $b_0 = \max(\mathbf{u}_c(S^\#))$ ,  $\forall c \in \mathcal{C}$  where  $S^\#$  could be an arbitrary fair solution. For each bad  $S$  (which indicates  $F_\alpha(S) < (1 - 1/e - \varepsilon) \cdot OPT_F$ ,  $\hat{F}_\alpha(S, \mathcal{R}) \geq (1 - 1/e)(1 - \varepsilon_1) \cdot OPT_F$  holds at most  $\delta_2 / \binom{n_G}{k}$  probability if  $\theta \geq C\theta_2$ .
Proof. Let  $X_{c}^{i}$  be the random variable for each  $R_{i} \in \mathcal{R}$  ( $R_{i}$  rooted in  $c$ ), such that  $X_{c}^{i} = 1$  if  $S \cap R_{c}(i) \neq \emptyset$ , and  $X_{c}^{i} = 0$  otherwise. Let  $\pi_{c} = \theta_{c} - \sum_{i \in [\theta_{c}]} X_{c}^{i}$ .
$$
\begin{array}{l} P r \left\{\hat {F} _ {\alpha} (S, \mathcal {R}) \geq \left(1 - \frac {1}{e}\right) \left(1 - \varepsilon_ {1}\right) \cdot O P T _ {F} \right\} \\ \leq P r \left\{\hat {F} _ {\alpha} (S, \mathcal {R}) \geq (1 + \varepsilon_ {2}) F _ {\alpha} (S) \right\} \\ = \Pr \left\{\sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i}\right) \geq (1 + \varepsilon_ {2}) \sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n}\right) \right\} \\ \leq 1 - \prod_ {c \in \mathcal {C}} \left(1 - P r \left\{1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} \geq \left(1 + \varepsilon_ {2}\right) \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n}\right) \right\} \right. \tag {14} \\ \end{array}
$$
For each community  $c$ , let  $\varepsilon_2' = \frac{1 - \alpha \sum_{n=1}^{\infty} \eta(n, \alpha) (1 - \mathbf{u}_c(S))^n}{\alpha \sum_{n=1}^{\infty} \eta(n, \alpha) (1 - \mathbf{u}_c(S))^n} \cdot \varepsilon_2$ , thus  $\varepsilon_2' \geq \varepsilon_2$  when  $\alpha \leq 1/2$ , and
$$
\begin{array}{l} P r \left\{1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} \geq \left(1 + \varepsilon_ {2}\right) \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n}\right) \right\} \\ = P r \left\{1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} \geq 1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n} + \varepsilon_ {2} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n}\right) \right\} \\ = \Pr \left\{\alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} \leq \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n} - \varepsilon_ {2} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n}\right) \right\} \\ = \Pr \left\{\sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} \leq \left(1 - \varepsilon_ {2} ^ {\prime}\right) \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n} \right\} \tag {15} \\ \end{array}
$$
To limit Eq. 15 to the first  $Q$  ( $Q \geq 2$ ) terms, let  $y = \frac{(1 - \pmb{u}_c(S))^Q + 1}{(Q + 1)\pmb{u}_c(S)}$ ,  $x = \frac{y\theta_c^2}{\theta_c - \pi_c + y\theta_c}$ , Eq. 15 becomes
$$
\begin{array}{l} P r \left\{\sum_ {n = 1} ^ {Q} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} \leq \left(1 - \varepsilon_ {2} ^ {\prime}\right) \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n} \right\} \\ = P r \left\{\frac {\pi_ {c}}{\theta_ {c}} - (1 - \varepsilon_ {2} ^ {\prime}) \sum_ {n = Q + 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n} + \sum_ {n = 2} ^ {Q} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} \leq \left(1 - \varepsilon_ {2} ^ {\prime}\right) \sum_ {n = 1} ^ {Q} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n} \right\} \\ \leq P r \left\{\frac {\pi_ {c}}{\theta_ {c}} - y + \sum_ {n = 2} ^ {Q} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} \leq \left(1 - \varepsilon_ {2} ^ {\prime}\right) \sum_ {n = 1} ^ {Q} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n} \right\} \\ \leq P r \left\{\frac {\pi_ {c} - x}{\theta_ {c} - x} + \sum_ {n = 2} ^ {Q} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} \leq \left(1 - \varepsilon_ {2} ^ {\prime}\right) \sum_ {n = 1} ^ {Q} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n} \right\} \\ \leq P r \left\{\sum_ {n = 1} ^ {Q} \eta (n, \alpha) \left(\frac {\pi_ {c} - Q + 1}{\theta_ {c} - Q + 1}\right) ^ {n} \leq \left(1 - \varepsilon_ {2} ^ {\prime}\right) \sum_ {n = 1} ^ {Q} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n} \right\} (\text {w h e n} x \leq Q - 1) \\ \leq 1 - \Pr \left\{\left(\frac {\pi_ {c} - Q + 1}{\theta_ {c} - Q + 1}\right) ^ {Q} > \left(1 - \varepsilon_ {2} ^ {\prime}\right) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {Q} \right\} \\ \end{array}
$$
$$
\begin{array}{l} = P r \left\{\left(\frac {\pi_ {c} - Q + 1}{\theta_ {c} - Q + 1}\right) ^ {Q} \leq \left(1 - \varepsilon_ {2} ^ {\prime}\right) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {Q} \right\} \\ \left(\text {L e t} 1 - \varepsilon_ {0} = \sqrt [ q ]{1 - \varepsilon_ {2} ^ {\prime}}\right) \\ = \Pr \left\{\frac {\pi_ {c} - Q + 1}{\theta_ {c} - Q + 1} \leq \left(1 - \varepsilon_ {0}\right) \left(1 - \boldsymbol {u} _ {c} (S)\right) \right\} \tag {16} \\ \end{array}
$$
Let  $\varepsilon_0^{\prime} = \varepsilon_0 + \frac{\theta_c}{\pi_c}\frac{\pi_c - Q + 1}{\theta_c - Q + 1} -1$ $\varepsilon_0\geq \varepsilon_0^{\prime}\geq 1 - \frac{\theta_c}{\pi_c}\frac{\pi_c - Q}{\theta_c - Q}$  Eq. 16 becomes
$$
\begin{array}{l} P r \left\{\frac {\pi_ {c} - Q + 1}{\theta_ {c} - Q + 1} \leq \left(1 - \varepsilon_ {0}\right) \left(1 - \boldsymbol {u} _ {c} (S)\right) \right\} \\ = P r \left\{\frac {\pi_ {c}}{\theta_ {c}} \leq \left(1 - \frac {\pi_ {c}}{\theta_ {c}} \frac {\theta_ {c} - Q + 1}{\pi_ {c} - Q + 1} \varepsilon_ {0} ^ {\prime}\right) \left(1 - \boldsymbol {u} _ {c} (S)\right) \right\} \\ \leq e x p \left(- \frac {\left(\frac {\pi_ {c}}{\theta_ {c}} \frac {\theta_ {c} - Q + 1}{\pi_ {c} - Q + 1} \varepsilon_ {0} ^ {\prime}\right) ^ {2}}{2} \theta_ {c} (1 - \boldsymbol {u} _ {c} (S))\right) \\ \leq e x p \left(- \frac {\varepsilon_ {0} ^ {\prime 2}}{2} \theta_ {c} (1 - \boldsymbol {u} _ {c} (S))\right) \\ \end{array}
$$
$$
\begin{array}{l} \left(\operatorname {L e t} \theta_ {c} \geq \frac {8 Q ^ {2} \ln \left(C \binom {n _ {G}} {k} / \delta_ {2}\right)}{\varepsilon_ {2} ^ {2} (1 - \boldsymbol {u} _ {c} (S))} \geq \frac {8 \ln \left(C \binom {n _ {G}} {k} / \delta_ {2}\right)}{\left(1 - \sqrt [ q ]{1 - \varepsilon_ {2} ^ {\prime}}\right) ^ {2} (1 - \boldsymbol {u} _ {c} (S))} \geq \frac {2 \ln \left(C \binom {n _ {G}} {k} / \delta_ {2}\right)}{\varepsilon_ {0} ^ {\prime 2} (1 - \boldsymbol {u} _ {c} (S))}\right) \\ \leq \delta_ {2} / C \binom {n _ {G}} {k} \tag {17} \\ \end{array}
$$
$$
E q. (1 4) \leq 1 - \prod_ {c \in \mathcal {C}} \left(1 - \delta_ {2} / C \binom {n _ {G}} {k}\right) \leq \delta_ {2} / \binom {n _ {G}} {k} \tag {18}
$$
It indicates  $Pr\left\{\hat{F}_{\alpha}(S,\mathcal{R}) \geq (1 - 1 / e)(1 - \varepsilon_1) \cdot OPT_F\right\} \leq \delta_2 / \binom{n_G}{k}$ , thus concludes the proof.
B.1 Running Time
The number of RR sets is mainly determined by both the size of the network and the number of communities. In the following, we exhibit the runtime of our algorithm with the scale of networks. Note that our algorithm is currently implemented in Matlab 2022a, thus it costs more time to generate RR sets (generating RR sets in  $\mathrm{C + + }$  could be at least 100 times faster). RRsets refers to the time (seconds) used to generate RR sets, IMM and FIMM denote the time used to select seeds based on the generated RR sets for IMM and FIMM, respectively.
Table 2: Running time (seconds).
Network	nG	C	RRsets	IMM	FIMM
Email	1,005	42	14.281	0.011	0.020
Amazon	12,698	509	82.204	0.028	0.283
Youtube	30,696	1,157	162.533	0.052	3.592
DBLP	72,875	1,352	735.755	0.063	18.259
Note that the proposed algorithm is more suitable and could be much more efficient under scenarios where the number of communities is rather limited (e.g., a constant).
B.2 Comparison with Equality-based methods
Our proposed algorithm is based on the notion of welfare fairness, which is in favor of fair result-aware seeding. In this subsection, we aim to explore the difference between such results and community-aware seeding, which is based on the notion of equality. The Equality asks to divide the budget  $k$  proportionally to the cluster sizes, i.e.,  $|S \cap V_{c}| \approx k \cdot n_{c} / n_{G}$ . We adopt two strategies to select seeds under Equality: picking nodes with the highest degree per community and running IMM inside each community.
The dataset tested is the Email network (since the number of communities of other datasets is more than 50 and  $k$  is set to 50). The experiment settings are the same as the setting for Email in our Section 4 ( $\alpha = 0.5$  and  $p$  ranges from 0.001 to 0.01 with the step of 0.001). The Table 3 below exhibits the results, where  $S_{\text{ours}}$ ,  $S_{\text{C-HD}}$  and  $S_{\text{C-IMM}}$  refer to seeds selected by our method, seeds selected by community-aware highest degree, and community-aware IMM, respectively.  $PoF$  and  $EoF$  are calculated w.r.t IMM in the classic influence maximization problem.
Table 3: Comparison with equality-based methods
p	PoF	EoF
Sours	SC-HD	SC-IMM	Sours	SC-HD	SC-IMM
0.001	21.77%	31.99%	21.56%	51.91%	45.08%	45.77%
0.002	16.92%	31.50%	20.56%	42.68%	37.31%	38.91%
0.003	12.11%	30.73%	20.76%	37.44%	31.76%	34.23%
0.004	10.08%	29.63%	19.11%	28.10%	20.97%	25.94%
0.005	9.22%	28.85%	17.63%	26.23%	16.49%	23.63%
0.006	6.31%	28.25%	17.96%	22.54%	3.82%	18.06%
0.007	5.48%	27.06%	16.21%	19.25%	N/A	13.57%
0.008	4.49%	26.03%	15.77%	17.11%	N/A	7.99%
0.009	3.70%	24.65%	14.88%	13.89%	N/A	N/A
0.01	2.57%	24.07%	15.64%	12.37%	N/A	N/A
As can be seen from the results, both  $PoF$  (the lower the better) and  $RoF$  of  $S_{\text{ours}}$  (the higher the better) are always better than that of both  $S_{\text{C-HD}}$  and  $S_{\text{C-IMM}}$ . In other words,  $S_{\text{C-HD}}$  pays more price of fairness yet achieves a lower degree of fairness. Compared with  $S_{\text{C-HD}}$ ,  $S_{\text{C-IMM}}$  yields a better performance. Moreover, as  $p$  increases, the fair influence of both  $S_{\text{C-HD}}$  and  $S_{\text{C-IMM}}$  is even lower (leading to N/A of  $EoF$ ) than that of IMM, which does not even contribute to fairness at all. The reason is that community-aware seeding highlights fairness in the process of seed allocation but not selection, while welfare fairness (also, MMF and DC) highlights fairness in the spreading results. The former could have an explicit fair distribution in seeding, but may still lead to unfair results.