SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Based on the RR sets generated by Algorithm 1, we present our FIMM algorithm (Algorithm 2) to select  $k$  seed nodes that maximize Eq. 4 through a greedy approach, i.e., iteratively selecting a node with the maximum alternative marginal fair influence gain as presented in Eq. 5. Apparently, it costs  $O(C)$  to calculate  $\hat{F}(v|S)$  for any  $v$  where  $C$  is the number of communities. When  $C$  is small (i.e., a constant), it would be efficient to compute  $\hat{F}(v|S)$  for all  $v \in V$  in  $O(Cn_G)$ . Besides, since
Algorithm 1: RR-Generate: Generate RR sets
Input: Graph  $G = (V,E,p)$  , community  $\mathcal{C}$  budget  $k$  number of RR sets for each community  $\theta_{c}$  Output: RR sets  $\mathcal{R}$  community-wise coverage  $\kappa$  linked-list  $\eta$  from nodes to covered RR sets 1 Initialize  $\kappa [v][c] = 0$  for all  $v\in V$ $c\in \mathcal{C}$  .   
2 Initialize  $\eta [v] = \emptyset$  for all  $v\in V$  .   
3  $\mathcal{R} = \varnothing$  .   
4 for  $c\in \mathcal{C}$  do   
5 for  $i = 1$  to  $\theta_c$  do   
6 Select a random node  $v$  in community  $c$  .   
7 Sample a random RR set  $R = RR(v)$  .   
8 for  $u\in R$  do   
9  $\begin{array}{r}\lfloor \kappa [u][c(v)] = \kappa [u][c(v)] + 1; \end{array}$    
10  $\mathcal{R} = \mathcal{R}\cup \{R\}$  .   
11  $\eta [v] = \eta [v]\cup \{R\}$
$\hat{F}_{\alpha}(S,\mathcal{R})$  is submodular and monotone, we can adopt a lazy-update strategy [18] that selects  $v$  with the maximal  $\hat{F}_{\alpha}(v|S)$  as a seed node if  $\hat{F}_{\alpha}(v|S)$  is still the maximum after updating. This lazy-update strategy (lines  $10\sim 12$ ) can cut down a great amount of redundant time cost that can be empirically up to 700 times faster than a simple greedy algorithm [18].
There are two vital counting arrays in Algorithm 2, i.e.,  $\varphi[c]$  and  $\kappa[v][c]$ .  $\varphi[c]$  records and updates the number of RR sets covered by  $S$  in community-wise. By lines  $20\sim 24$ ,  $\kappa[v][c]$  keeps updating and always indicates the extra coverage of  $v$  on all  $u$ -rooted  $(u \in V_c)$  RR sets besides  $S$ . It establishes a
Algorithm 2: FIMM: Fair Influence Maximization
Input: Graph  $G = (V,E,p)$  , community  $\mathcal{C}$  budget  $k$  , approximation parameter  $Q$  Output: Seed set  $S$    
1  $(\mathcal{R},\kappa ,\eta) = \mathbb{R}\mathcal{R}$  -Generate  $(G,\mathcal{C},k,\theta_c)$    
2 Initialize  $\varphi [c] = 0$  for all  $c\in \mathcal{C}$  ;//indicating the number of covered RR sets rooted in  $c$    
3 Initialize  $\gamma (v)$  according to Eq.(5) for all  $v\in V$  ;//indicating initial marginal gain   
4 Initialize covered  $[R] =$  false for all  $R\in \mathcal{R}$  ;//indicating whether  $R$  is covered   
5 Initialize updated[v]  $=$  true for all  $v\in V$  ;//indicating whether  $\kappa [v]$  is updated   
6  $S = \emptyset$  .   
7 for  $i = 1$  to k do   
8 while true do   
9  $v = \arg \max_{u\in V\backslash S}\gamma (u);$  if updated(v)  $= =$  false then Updating  $\gamma (v)$  according to Eq.(5); updated(v)  $=$  true; else   
14  $S = S\cup \{v\} ;$  for  $v\in V$  do updated(v)  $=$  false; break;   
for  $c\in \mathcal{C}$  do   
19  $\varphi [c] = \varphi [c] + \kappa [v][c];$    
20 for all  $R\in \eta [v]\wedge covered[R] ==$  false do covered[  $R] =$  true; r  $=$  root(R); for all  $u\in R\land u\neq v$  do  $\kappa [u][c(r)] = \kappa [u][c(r)] - 1;$
convenient way for updating  $\varphi(c)$  that only needs to increase  $\varphi(c)$  by  $\kappa[v][c]$  where  $v$  is the newly selected node for all  $c \in \mathcal{C}$ . If we denote the original community-wise coverage as  $\kappa'$ , which means  $(\sim, \kappa', \sim) = \mathsf{RR}$ -Generate  $(G, \mathcal{C}, k, \theta_c)$ , then it holds  $\kappa'[v][c] = \kappa[v][c] + \varphi[c]$  for all  $v \in V$  and  $c \in \mathcal{C}$  in Algorithm 2.
3.4 Number of RR sets
In this subsection, we discuss the number of RR sets needed to approximate the fair influence with high probability. Let  $OPT_F$  denote the optimal solution and  $S^*$  denote the corresponding optimal seed set for the fair influence maximization problem defined in this paper, i.e.,  $OPT_F = F_{\alpha}(S^{*}) = \sum_{c \in \mathcal{C}} n_c \pmb{u}_c(S^{*})^{\alpha} = \sum_{c \in \mathcal{C}} n_c \left(1 - \alpha \sum_{n=1}^{\infty} \eta(n, \alpha) (1 - \pmb{u}_c(S^{*}))^n \right)$ . Since this paper deals with the fair influence maximization problem, we thus assume that the maximal community utility  $\max_{c \in \mathcal{C}} \pmb{u}_c(S^{\#}) \geq \max_{c \in \mathcal{C}} \pmb{u}_c(S^{*})$  of an arbitrary seed set  $S^{\#}$  would not be too big.
Lemma 3. Let  $\delta_1 \in (0,1)$ ,  $\varepsilon_1 \in (0,1)$ , and  $\theta_1 = \frac{12Q^2\ln(C / \delta_1)}{\varepsilon_1^2(1 - b)}$  where  $Q$  is the approximation parameter;  $b = \max(\boldsymbol{u}_c(S^*)), \forall c \in \mathcal{C}$ , and  $S^* = \operatorname{argmax}_{S:|S| \leq k} F_\alpha(S)$  denotes the optimal solution for the FIM problem based on  $\mathcal{R}$ , then  $\hat{F}_\alpha(S^*,\mathcal{R}) \geq (1 - \varepsilon_1) \cdot OPT_F$  holds at least  $1 - \delta_1$  probability if  $\theta \geq C\theta_1$ .
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  $\theta \geq C\theta_2$ .
Please refer to Appendix for the detailed proof of Lemma 3 and Lemma 4.
Theorem 1. For every  $\varepsilon >0$ ,  $\ell >0$ ,  $0 < \alpha < 1$ , and  $Q\geq 2$ , by setting  $\delta_{1} = \delta_{2} = 1 / 2n_{G}^{\ell}$  and  $\theta \geq C\cdot \max (\theta_1,\theta_2)$ , the output  $S$  of FIMM satisfies  $F_{\alpha}(S)\geq (1 - 1 / e - \varepsilon)F_{\alpha}(S^{*})$ , where  $S^{*}$  denotes the optimal solution with probability at least  $1 - 1 / n_G^\ell$ .
Proof. Combining Lemma 3 and Lemma 4, we have  $\hat{F}_{\alpha}(S,\mathcal{R})\geq (1 - 1 / e - \varepsilon)\cdot OPT_{F}$  at least  $1 - \delta_1 - \delta_2$  probability based on the union bound. If we set  $\delta_1 = \delta_2 = 1 / 2n_G^\ell$ , then, following the standard analysis of IMM, our FIMM algorithm provides  $(1 - 1 / e - \varepsilon)$ -approximation with probability at least  $1 - 1 / n_G^\ell$ .
If we set  $\delta_1 = \delta_2 = \frac{1}{2n_G^\ell}$  and  $\varepsilon_1 = \varepsilon \cdot \frac{e}{e - 1} \cdot \frac{\sqrt{3}\tau_1}{\sqrt{3}\tau_1 + \sqrt{2}\tau_2}$  where  $\tau_1 = \sqrt{\ln C + \ell\ln n_G + \ln 2}$  and  $\tau_2^2 = \tau_1^2 + \ln \binom{n_G}{k}$ , then a possible setting of  $\theta$  could be  $\theta = (\frac{e - 1}{e})^2 \cdot \frac{4CQ^2(\sqrt{3}\tau_1 + \sqrt{2}\tau_2)^2}{\varepsilon^2(1 - b_0)}$ .
Email The Email dataset [29] is generated using email data from a large European research institution, where every node is a member of the research institution and an directed edge  $(v, u)$  indicates that  $v$  has sent  $u$  at least one email. It contains 1,005 nodes and 25,571 directed edges. Moreover, this dataset also contains "ground-truth" community memberships of nodes, where each member belongs to exactly one of 42 departments at the research institute.
Flixster The Flixster dataset [30] is a network of American social movie discovery services. To transform the dataset into a weighted graph, each user is represented by a node, and a directed edge from node  $u$  to  $v$  is formed if  $v$  rates one movie shortly after  $u$  does so on the same movie. It contains 29,357 nodes and 212,614 directed edges. It also provides the learned influence probability between each node pair, which can be incorporated into the IC model. Since it has no community information, we construct the biased community structure by categorizing individuals according to their susceptibility of being influenced to highlight the level of inequality and get 100 communities. Moreover, this dataset contains the learned influence probability between each node pair, which can be incorporated into the IC model.
Amazon The Amazon dataset [31] is collected based on Customers Who Bought This Item Also Bought feature of the Amazon website. If a product  $i$  is frequently co-purchased with product  $j$ , the graph contains an undirected edge between  $i$  to  $j$ . The dataset also provides the ground-truth community structure which indicates the product categories. The original network has 334,863 nodes and 925,872 undirected edges. After Pruning low-quality communities (whose size is no more than 10 nodes), the Amazon network tested in our experiments 12,698 nodes, 40,096 edges, and 509 communities.
Youtube The Youtube dataset [31] is a network of the video-sharing web site that includes social relationships. Users form friendship each other and users can create groups which other users can join. The friendship between users is regarded as undirected edges and the user-defined groups are considered as ground-truth communities. The original network has 1,134,890 nodes and 2,987,624 undirected edges. After screening high-quality communities, it remains 30,696 nodes, 198,867 edges, and 1,157 communities.
DBLP The DBLP dataset [31] is the co-authorship network where two authors are connected if they ever published a paper together. Publication venues, such as journals or conferences, defines an individual ground-truth community and authors who published to a certain journal or conference form a community. The original network has 717,080 nodes and 1,049,866 undirected edges. We also perform the network pruning and finally obtain 72,875 nodes, 268,346 edges, and 1,352 communities.
Let  $S_{I}$  denote the seed set returned by IMM [14],  $S_{F}$  denote the seed set returned by FIMM, the performance of  $S_{F}$  towards fairness can be evaluated via the Price of Fairness (PoF) and the Effect of Fairness (EoF) as
$$
P o F = \frac {\sigma (S _ {I}) - \sigma (S _ {F})}{\sigma (S _ {I}) - k}, E o F = \left(\frac {F _ {\alpha} (S _ {F}) - F _ {\alpha} (S _ {I})}{F _ {\alpha} (S _ {I}) - k}\right) ^ {\alpha},
$$
where  $|S_I| = |S_F| = k$ ,  $\sigma(\cdot)$  denotes the influence spread and  $F_{\alpha}(\cdot)$  denotes the fair influence.
Intuitively,  $PoF$  implies how much price it costs to access fairness and  $EoF$  implies to what extent it steps towards fairness.
We test IMM and our proposed FIMM algorithm in the experiment. In all tests, we run 10,000 Monte-Carlo simulations to evaluate both the influence spread and the fair influence under IC model. We also test influence probability  $p$ , inequality aversion parameter  $\alpha$  and the seed budget  $k$ .
4.3.1 Email & Flixster
For the Email network, we set  $\alpha = 0.5$ . Since the network is small, we apply the Uniformed IC model where the influence probability is the same across all edges. We test different probabilities that range
(a) Email
(b) Flixster
Figure 1: Results of testing influence probability  $p$  on Email and aversion parameter  $\alpha$  on Flixster.
from 0.001 to 0.01 with the step of 0.001. For the Fixel network, we test the inequality aversion parameter  $\alpha$  which ranges from 0.1 to 0.9 with the step of 0.1. We set  $k = 50$  for both networks and the results are shown in Figure 1.
As the influence probability  $p$  increases, both  $PoF$  and  $EoF$  show a downward trend. This may be attributed to the increased challenges faced by disadvantaged communities in being influenced when  $p$  is small. Similarly, both  $PoF$  and  $EoF$  also show a downward trend with the increase of the aversion parameter  $\alpha$ . The reason lies that communities experience greater promotions in fair influence when the aversion parameter is smaller, resulting in higher  $EoF$  and  $PoF$ . Moreover, there is hardly any fairness when  $\alpha \geq 0.7$  where the gap between  $u^{\alpha}$  and  $u$  is just too small.
4.3.2 Amazon, Youtube & DBLP
For Amazon, Youtube, and DBLP networks, we set  $\alpha = 0.5$  and  $p(v_{i},v_{j}) = 1 / d_{in}(v_{j})$  where  $d_{in}$  denotes the in-degree as the influence probability following the weighted IC model [17]. We test different seed budget  $k$  that ranges from 5 to 50 with the step of 5. Results are shown in Figure 2.
(b) Youtube
(c) DBLP
Figure 2: Results of testing seed budget  $k$  on Amazon, Youtube, and DBLP.
Generally, FIMM tends to produce a noticeably fairer output when  $k$  is small. It reflects the idea that enforcing fairness as a constraint becomes easier when there is an abundance of resources available. However, there are also some exceptions where smaller  $k$  leads to a lower EoF, e.g.,  $k = 20$  in Figure 2(b) and  $k = 5, 15$  in Figure 2(c). This may be attributed to the fact that the seed selection in FIMM follows a pattern of remedying the previously fair solutions in each round.
This paper focuses on the fair influence maximization problem with efficient algorithms. We first tackle the challenge of carrying out the unbiased estimation of the fractional power of the expected proportion of activated nodes in each community. Then, we deal with the challenge of integrating unbiased estimation into the RIS framework and propose an  $(1 - 1 / e - \varepsilon)$  approximation algorithm FIMM. We further give a theoretical analysis that addresses the concentration of the unbiased estimator of the fractional power. The experiments validate that our algorithm is both scalable and effective, which is consistent with our theoretical analysis.
There are several future directions from this research. One direction is to find some other unbiased estimators for the fair influence that would be easier to calculate through RIS. Another direction is to explore a more efficient seed selection strategy. The fairness bound is also a meaningful research direction.
**Limitation** The limitations of our work are mainly two points. The first limitation is that our algorithm can only be efficient when the number of communities is small (e.g., a constant). The second limitation is that our algorithm is based on the assumption that the minimal community utility of an arbitrary seed set would not be too big.
This work was supported by the Fundamental Research Funds for the Central Universities (No.2022QN1093).
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Fact 2. (Chernoff bound) Let  $X_{1}, X_{2}, \ldots, X_{R}$  be  $R$  independent random variables with  $X_{i}$  having range [0, 1], and there exists  $\mu \in [0,1]$  making  $\mathbb{E}[X_i] = \mu$  for any  $i \in [R]$ . Let  $Y = \sum_{i=1}^{R} X_{i}$ , for any  $\gamma > 0$ ,
$$
\operatorname * {P r} \{Y - t \mu \geq \gamma \cdot t \mu \} \leq \exp (- \frac {\gamma^ {2}}{2 + \frac {2}{3} \gamma} t \mu).
$$
For any  $0 < \gamma < 1$
$$
\operatorname * {P r} \{Y - t \mu \leq - \gamma \cdot t \mu \} \leq \exp (- \frac {\gamma^ {2}}{2} t \mu).
$$
Lemma 3. Let  $\delta_1 \in (0,1)$ ,  $\varepsilon_1 \in (0,1)$ , and  $\theta_1 = \frac{12Q^2\ln(C / \delta_1)}{\varepsilon_1^2(1 - b)}$  where  $Q$  is the approximation parameter;  $b = \max(\mathbf{u}_c(S^*))$ ,  $\forall c \in \mathcal{C}$ , and  $S^* = \operatorname{argmax}_{S:|S| \leq k} F_\alpha(S)$  denotes the optimal solution for the FIM problem based on  $\mathcal{R}$ , then  $\hat{F}_\alpha(S^*,\mathcal{R}) \geq (1 - \varepsilon_1) \cdot OPT_F$  holds at least  $1 - \delta_1$  probability if  $\theta \geq C\theta_1$ .
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  $\pi_{c} = \theta_{c} - \sum_{i \in [\theta_{c}]} X_{c}^{i}$ .
$$
\begin{array}{l} P r \left\{\hat {F} _ {\alpha} \left(S ^ {*}, \mathcal {R}\right) <   \left(1 - \varepsilon_ {1}\right) \cdot O P T _ {F} \right\} \\ = \Pr \left\{\sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \frac {\left(\theta_ {c} - n\right) !}{\theta_ {c} !} \left\{\sum \overline {{X _ {c} ^ {i _ {1}}}} \cdot \overline {{X _ {c} ^ {i _ {2}}}} \dots \overline {{X _ {c} ^ {i _ {n}}}} \right\}\right) <   \left(1 - \varepsilon_ {1}\right) \cdot F _ {\alpha} \left(S ^ {*}\right) \right\} \\ = P r \left\{\sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \frac {\left(\theta_ {c} - n\right) !}{\theta_ {c} !} \left\{\sum \overline {{X _ {c} ^ {i _ {1}}}} \cdot \overline {{X _ {c} ^ {i _ {2}}}} \dots \overline {{X _ {c} ^ {i _ {n}}} \right\}\right) <   \left(1 - \varepsilon_ {1}\right) \sum_ {c \in \mathcal {C}} n _ {c} \left(\boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {\alpha} \right\} \\ = \Pr \left\{\sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i}\right) <   (1 - \varepsilon_ {1}) \sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n}\right) \right\} \\ \leq 1 - \prod_ {c \in \mathcal {C}} \left(1 - P r \left\{1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i} <   \left(1 - \varepsilon_ {1}\right) \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} \left(S ^ {*}\right)\right) ^ {n}\right) \right\}\right) \tag {6} \\ \end{array}
$$
For each community  $c$ , let  $\varepsilon_1' = \frac{1 - \alpha \sum_{n=1}^{\infty} \eta(n, \alpha) (1 - \pmb{u}_c(S^*))^n}{\alpha \sum_{n=1}^{\infty} \eta(n, \alpha) (1 - \pmb{u}_c(S^*))^n} \cdot \varepsilon_1$ , thus  $\varepsilon_1' \geq \varepsilon_1$  when  $\alpha \leq 1/2$ , and