SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Xiaobin Rui
China University of Mining and Technology
Xuzhou, Jiangsu, China
ruixiaobin@cumt.edu.cn
Zhixiao Wang*
zhxwang@cumt.edu.cn
Jiayu Zhao
zhaojy@cumt.edu.cn
Lichao Sun
Bethlehem, PA, USA
lis221@lehigh.edu
Wei Chen*
Beijing, China,
weic@microsoft.com
Given a graph  $G$ , a community structure  $\mathcal{C}$ , and a budget  $k$ , the fair influence maximization problem aims to select a seed set  $S(|S| \leq k)$  that maximizes the influence spread while narrowing the influence gap between different communities. While various fairness notions exist, the welfare fairness notion, which balances fairness level and influence spread, has shown promising effectiveness. However, the lack of efficient algorithms for optimizing the welfare fairness objective function restricts its application to small-scale networks with only a few hundred nodes. In this paper, we adopt the objective function of welfare fairness to maximize the exponentially weighted summation over the influenced fraction of all communities. We first introduce an unbiased estimator for the fractional power of the arithmetic mean. Then, by adapting the reverse influence sampling (RIS) approach, we convert the optimization problem to a weighted maximum coverage problem. We also analyze the number of reverse reachable sets needed to approximate the fair influence at a high probability. Further, we present an efficient algorithm that guarantees  $1 - 1 / e - \varepsilon$  approximation.
Influence maximization (IM) is a well-studied problem in the field of social network analysis. Given a graph  $G$  and a positive integer  $k$ , the problem asks to find a node set  $S(|S| \leq k)$  which can spread certain information to trigger the largest expected number of remaining nodes. There have been various IM variants, such as adaptive influence maximization [1], multi-round influence maximization [2], competitive influence maximization [3], and time-critical influence maximization [4]. Influence maximization and these variants have important applications in viral marketing, rumor control, health interventions, etc [5].
Considering the situation that when disseminating public health interventions, for example, suicide/HIV prevention [6] or community preparedness against natural disasters, we can select individuals who act as peer-leaders to spread such information to maximize the outreach following influence
maximization. However, it may lead to discriminatory solutions as individuals from racial minorities or LGBTQ communities may be disproportionately excluded from the benefits of the intervention [7]. Therefore, derived from such significant social scenarios, fairness in influence maximization has become a focus of attention for many researchers [8, 9, 7, 10].
Generally, fair influence maximization aims to improve the influenced fraction inside some communities where the coverage may get unfairly low. Currently, a universally accepted definition of fair influence maximization remains elusive, and recent work has incorporated fairness into influence maximization by proposing various notions of fairness, such as maximin fairness [9], diversity constraints [8], demographic parity [11], and welfare fairness [7]. Among these notions, welfare fairness show several attractive features. Its objective function is the weighted sum (with community size as the weight) of the fractional power ( $\alpha$  fraction) of the expected proportion of activated nodes within every community. The fractional exponent  $\alpha$  is the inequality aversion parameter, allowing one to balance between fairness and influence spread, with  $\alpha$  tending to 1 for influence spread and  $\alpha$  tending to 0 for fairness. The objective function enjoys both monotonicity and submodularity, enabling a greedy approach with  $1 - 1 / e - \varepsilon$  approximation.
However, even though Rahmattalabi et al. [9] has given a full analysis of welfare fairness, there is currently no efficient algorithm to optimize its objective function with provable guarantee, which restricts their applications to small-scale networks with only a few hundred nodes. In this paper, we propose an efficient algorithm for maximizing the welfare fairness based on reverse influence sampling (RIS) [12, 13, 14]. The main challenges in adapting the RIS approach to the welfare fairness objective include: (a) how to carry out the unbiased estimation of the fractional power of the expected proportion of activated nodes in each community, since simply obtaining an unbiased estimate of the expected proportion and then taking its fractional power is not an unbiased estimator; (b) how to integrate the unbiased estimation designed into the RIS framework. We address both challenges and propose a new scalable fair influence maximization with theoretical guarantees.
- We propose an unbiased estimator for the fractional power of the arithmetic mean by leveraging Taylor expansion techniques. The estimator enables us to accurately estimate the fair influence under welfare fairness.
- Based on the above unbiased estimator, we adapt the RIS approach to approximate the fair influence with Reverse Reachable (RR) sets and propose the FIMM algorithm that works efficiently while guaranteeing the  $(1 - 1 / e - \varepsilon)$ -approximation solution. Our theoretical analysis needs to address the concentration of the unbiased estimator of the fractional power and thus is much more involved than the standard RIS analysis.
- We carry out detailed experimental analysis on five real social networks to study the trade-off between fairness and total influence spread. We test different fairness parameters, influence probabilities, seed budget, and community structures to confirm the performance of our proposed algorithms.
Related Work Influence maximization (IM) is first studied as an algorithmic problem by Domingos and Richardson [15, 16]. Kempe et al. [17] mathematically formulate IM as a discrete optimization problem and prove it is NP-hard. They also provide a greedy algorithm with  $1 - 1 / e$  approximation based on the submodularity and monotonicity of the problem. Hence, many works have been proposed to improve the efficiency and scalability of influence maximization algorithms [18, 19, 20, 21, 22, 12, 13, 14]. Among these methods, the most recent and the state of the art is the reverse influence sampling (RIS) approach [12, 13, 14, 23], where the IMM algorithm [14] is one of the representative algorithms. The idea of RIS approaches is to generate an adequate number of reverse reachable sets (a.k.a. RR sets), and then the influence spread can be approximated at a high probability based on these RR sets. Therefore, the greedy approach can be easily applied by iteratively selecting the node that could bring the maximal marginal gain in terms of influence spread as a seed node.
Recent work has incorporated fairness directly into the influence maximization framework by relying on Rawlsian theory [9], game theoretic principles [8], and equity-based notion [24]. Based on the Rawlsian theory, maximin fairness [9, 8] aims to maximize the influence fraction of the worst-off community. Inspired by the game theoretic notion of core, diversity constraints [8] require that every community obtains an influenced fraction higher than when it receives resources proportional to its size and allocates them internally. Equity-based notion [24] strives for equal influenced fraction
across all communities. However, these notions can hardly balance fairness and total influence and usually lead to a high influence reduction. Especially, strict equity [24] is rather hard to achieve in influence maximization. To address these shortcomings, Rahmattalabi et al. [7] propose the welfare fairness that can control the trade-off between fairness and total influence by an inequality aversion parameter. Based on the cardinal welfare theory [25], the objective function of welfare fairness is to maximize the weighted summation over the exponential influenced fraction of all communities. Fish et al. [26] also follow welfare functions and propose  $\phi$ -mean fairness, where the objective function becomes MMF when  $\phi = -\infty$ . However, they do not address the challenge of unbiased estimation of the fractional power. In addition, none of the above studies address the scalability of the algorithms. Thus, to the best of our knowledge, we are the first to study scalability in the context of fair influence maximization.
2 Model and Problem Definition
Information Diffusion Model In this paper, we adopt the well-studied Independent Cascade (IC) model as the basic information diffusion model. Under IC model, a social network is modeled as a directed influence graph  $G = (V,E,p)$ , where  $V$  is the set of vertices (nodes) and  $E \subseteq V \times V$  is the set of directed edges that connect pairs of nodes. For an edge  $(v_i,v_j) \in E$ ,  $p(v_i,v_j)$  indicates the probability that  $v_i$  influences  $v_j$ . The diffusion of information or influence proceeds in discrete time steps. At time  $t = 0$ , the seed set  $S$  is selected to be active, denoted as  $A_0$ . At each time  $t \geq 1$ , all nodes in  $A_{t-1}$  try to influence their inactive neighbors following influence probability  $p$ . The set of activated nodes at step  $t$  is denoted as  $A_t$ . The diffusion process ends when there is no more node activated in a time step. An important metric in influence maximization is the influence spread, denoted as  $\sigma(S)$ , which is defined as the expected number of active nodes when the propagation from the given seed set  $S$  ends. For the IC model,  $\sigma(S) = \mathbb{E}[|A_0 \cup A_1 \cup A_2 \cup \ldots|]$ . We use  $ap(v,S)$  to represent the probability that node  $v$  is activated given the seed set  $S$ . Then we have  $\sigma(S) = \sum_{v \in V} ap(v,S)$ .
Live-edge Graph Given the influence probability  $p$ , we can construct the live-edge graph  $L = (V, E(L))$ , where each edge  $(v_i, v_j)$  is selected independently to be a live edge with the probability  $p(v_i, v_j)$ . The influence diffusion in the IC model is equivalent to the deterministic propagation via bread-first traversal in a random live-edge graph  $L$ . Let  $\Gamma(G, S)$  denote the set of nodes in graph  $G$  that can be reached from the node set  $S$ . By the above live-edge graph model, we have  $\sigma(S) = \mathbb{E}_L[|\Gamma(L, S)|] = \sum_L Pr[L|G] \cdot |\Gamma(L, S)|$ , where the expectation is taken over the distribution of live-edge graphs, and  $Pr[L|G]$  is the probability of sampling a live-edge graph  $L$  in graph  $G$ .
Approximation Solution A set function  $f: V \to \mathbb{R}$  is called submodular if for all  $S \subseteq T \subseteq V$  and  $u \in V \setminus T$ ,  $f(S \cup \{u\}) - f(S) \geq f(T \cup \{u\}) - f(T)$ . Intuitively, submodularity characterizes the diminishing return property often occurring in economics and operation research. Moreover, a set function  $f$  is called monotone if for all  $S \subseteq T \subseteq V$ ,  $f(S) \leq f(T)$ . It is shown in [17] that influence spread  $\sigma(\cdot)$  for the independent cascade model is a monotone submodular function. A non-negative monotone submodular function allows a greedy solution to its maximization problem with  $1 - 1/e$  approximation [27], which provides the technical foundation for most influence maximization tasks.
Fair Influence Maximization For a given graph  $G$  with  $n_G$  nodes, the classic influence maximization problem is to choose a seed set  $S$  consisting of at most  $k$  seeds to maximize the influence spread  $\sigma(S, G)$ . Assuming each node belongs to one of the disjoint communities  $c \in \mathcal{C} := \{1, 2, \dots, C\}$ , such that  $V_1 \cup V_2 \cup \dots \cup V_C = V$  where  $V_c$  ( $n_c = |V_c|$ ) denotes the set of nodes that belongs to community  $c$ . Generally, fair influence maximization (FIM) aims to narrow the influence gap between different communities while maintaining the total influence spread as much as possible. In this paper, we adopt the fair notion proposed by Rahmattalabi et al. [7], where the welfare function is used to aggregate the cardinal utilities of different communities. The goal is to select at most  $k$  seed nodes, such that the objective function  $F_{\alpha}(S)$  (also referred to as fair influence in this paper) is maximized, where  $F_{\alpha}(S) = \sum_{c \in \mathcal{C}} n_c u_c(S)^{\alpha}$ ,  $0 < \alpha < 1$ . The utility  $u_c(S)$  denotes the expected proportion of influenced nodes in the community  $c$  with the seed set  $S$ . Exponent  $\alpha$  is the inequality aversion parameter that controls the trade-off between fairness and total influence, with  $\alpha$  tending to 1 for influence spread and  $\alpha$  tending to 0 for fairness. We thus define the fair influence maximization problem in this paper as follows:
Definition 1. The Fair Influence Maximization (FIM) under the independent cascade model is the optimization task where the input includes the directed influence graph  $G = (V, E, p)$ , the nonoverlapping community structure  $\mathcal{C}$ , and the budget  $k$ . The goal is to find a seed set  $S^*$  to maximize the fair influence, i.e.,  $S^* = \operatorname{argmax}_{S:|S| \leq k} F_{\alpha}(S)$ .
According to [7], the fair influence  $F_{\alpha}(S)$  is both monotone and submodular, which provides the theoretical basis for our efficient algorithm design, to be presented in the next section.
The monotonicity and submodularity of the fair influence objective function enable a greedy approach for the maximization task. However, as commonly reported in influence maximization studies, naively implementing a greedy approach directly on the objective function will suffer a long running time. The reason is that accurate function evaluation requires a large number of Monte-Carlo simulations. In this section, we aim to significantly speed up the greedy approach by adapting the reverse influence sampling (RIS) [12, 13, 14], which provides both theoretical guarantee and high efficiency. We propose the FIMM algorithm that is efficient when the number of communities is small, which is hopefully a common situation such as gender and ethnicity.
For the convenience of reading, we list most important symbols featured in this paper in Table 1.
Table 1: Important symbols appeared in this paper.
Symbol	Explanation
G=(V,E,p)	A network;
V	Node set of the network;
E	Edge set of the network;
nG	The number of nodes in G, i.e. nG=|V|;
p(vi,vj)	The probability that vi influence vj;
C={c1,c2,···}	Community structure;
C	The number of communities in C;
Vc	The node set in community c;
nc	The number of nodes in community c, i.e. nc=|Vc|;
S	A seed set;
S*	The optimal seed set for fair influence maximization;
ap(v,S)	The expected probability that v is activated by S;
σ(S)	Influence spread of S, i.e. σ(S)=∑v∈V ap(v,S);
uc	The utility of c (expected fraction of influenced nodes in c);
Fα(S)	The fair influence of S;
R	A set of RR sets;
Rc	The set of RR sets rooted in community c;
ˆFα(S,R)	The unbiased estimator for fair influence of S based on R;
θ	The total number of RR sets;
θc	The number of RR sets rooted in community c;
α	The aversion parameter regarding fairness;
Q	The approximation parameter for Taylor expansion;
ε	The accuracy parameter;
l	The confidence parameter.
3.1 Unbiased Fair Influence
To estimate the influence spread, we may generate a number of live-edge graphs  $\mathcal{L} = \{L_1,L_2,\dots ,L_t\}$  as samples. Then, for a given seed set  $S$ ,  $\hat{\sigma} (\mathcal{L},S) = \frac{1}{t}\sum_{i = 1}^{t}|\Gamma (L_i,S)|$  is an unbiased estimator of  $\sigma (S)$ . However, situations are completely different for fair influence. For each community  $c$ , its fair influence is actually  $n_c u_c^\alpha$ . If we still generate a number of live-edge graphs and estimate  $\pmb{u}_c$  by  $\hat{\pmb{u}}_c(\mathcal{L},S) = \frac{1}{t}\sum_{i = 1}^{t}|\Gamma (L_i,S)\cap V_c| / |V_c|$ , then  $\hat{\pmb{u}}_c(\mathcal{L},S)$  is an unbiased estimator for  $\pmb{u}_c$ , but  $\hat{\pmb{u}}_c(\mathcal{L},S)^\alpha$  is actually a biased estimator of  $\pmb{u}_c^\alpha$  for  $0 < \alpha < 1$ . In fact, the value of  $\pmb{u}_c^\alpha$  is generally higher than the true value, which is revealed by Jensen's Inequality.
Fact 1. (Jensen's Inequality) If  $X$  is a random variable and  $\phi$  is a concave function, then
$$
\mathbb {E} [ \phi (X) ] \leq \phi (\mathbb {E} [ X ]).
$$
Therefore, our first challenge in dealing with the welfare fairness objective is to provide an unbiased estimator for the fractional power value of  $\pmb{u}_c^\alpha$ . We meet this challenge by incorporating Taylor expansion as in Lemma 1.
Lemma 1. For a given seed set  $S$  and an inequality aversion parameter  $\alpha$ , the fair influence
$$
F _ {\alpha} (S) = \sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \big (1 - \boldsymbol {u} _ {c} (S) \big) ^ {n}\right), \eta (n, \alpha) = \left\{ \begin{array}{l l} 1, & n = 1, \\ \frac {(1 - \alpha) (2 - \alpha) \ldots (n - 1 - \alpha)}{n !}, & n \geq 2. \end{array} \right.
$$
Proof. By Taylor expansion of binomial series, we have
$$
(1 + x) ^ {\alpha} = 1 + \sum_ {n = 1} ^ {\infty} \binom {\alpha} {n} x ^ {n}, \left( \begin{array}{c} \alpha \\ n \end{array} \right) = \frac {\alpha (\alpha - 1) \ldots (\alpha - n + 1)}{n !}.
$$
By definition of fair influence in Definition 1, we have
$$
\begin{array}{l} F _ {\alpha} (S) = \sum_ {c \in \mathcal {C}} n _ {c} \left(1 + \left(\boldsymbol {u} _ {c} (S) - 1\right)\right) ^ {\alpha} = \sum_ {c \in \mathcal {C}} n _ {c} \left(1 + \sum_ {n = 1} ^ {\infty} \binom {\alpha} {n} \left(\boldsymbol {u} _ {c} (S) - 1\right) ^ {n}\right) \\ = \sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \boldsymbol {u} _ {c} (S)\right) ^ {n}\right) \tag {1} \\ \end{array}
$$
$$
\eta (n, \alpha) = \left\{ \begin{array}{l l} 1, & n = 1, \\ \frac {(1 - \alpha) (2 - \alpha) \ldots (n - 1 - \alpha)}{n !}, & n \geq 2. \end{array} \right.
$$
Thus concludes the proof.
Lemma 1 demonstrates that the calculation of fair influence with fractional powers can be transformed into the summation of integral powers. Further, we can get an unbiased estimator for integral powers of arithmetic mean as given in Lemma 2.
Lemma 2. [28] Suppose that a simple random sample of size  $m$  is to be drawn, with replacement, in order to estimate  $\mu^n$ . An unbiased estimator for  $\mu^n$  ( $n \leq m$ ) is
$$
\hat {\mu} ^ {n} = \frac {(m - n) !}{m !} \left\{\sum x _ {i _ {1}} x _ {i _ {2}} \dots x _ {i _ {n}} \right\} \left(i _ {1} \neq i _ {2} \neq \dots \neq i _ {n}\right) \tag {2}
$$
where the summation extends over all permutations of all sets of  $n$  observations in a sample subject only to the restriction noted.
3.2 Unbiased Fair Influence with RR sets
Many efficient influence maximization algorithms such as IMM [14] are based on the RIS approach, which generates a suitable number of reverse-reachable (RR) sets for influence estimation.
RR set An RR set  $RR(v)$  (rooted at node  $v \in V$ ) can be generated by reversely simulating the influence diffusion process starting from the root  $v$ , and then adding all nodes reached by reversed simulation into this RR set. In this way,  $RR(v)$  is equivalent to collecting all nodes that can reach  $v$  in the random live-edge graph  $L$ , denoted by  $\Gamma'(L, v)$ . Intuitively, each node  $u \in RR(v)$  if selected as a seed would activate  $v$  in this random diffusion instance. We say that  $S$  covers a RR set  $RR(v)$  if  $S \cap RR(v) \neq \emptyset$ . The expected activated probability  $ap(v, S)$  is thus equivalent to the probability that  $S$  covers a randomly generated  $v$ -rooted RR set. In the following, we use  $RR(v)$  to represent a randomly generated RR set when  $v$  is not specified, i.e.,  $RR(v) = \Gamma_{L \sim \mathcal{U}(P_L)}'(L, v)$  where  $P_L$  is the space of all live-edge graphs and  $\mathcal{U}(\cdot)$  denotes the uniform distribution.
Let  $X_{c}$  be the random event that indicates whether a randomly selected node in community  $c$  would be influenced in a diffusion instance by the given seed set  $S$ . As mentioned above, an RR set maps to a random diffusion instance. Assuming we generate  $\mathcal{R}$  consisting of  $\theta$  RR sets in total and each community  $c$  gets  $\theta_{c}$  RR sets. Let  $\mathcal{R}_{c}$  be the set of RR sets that are rooted in the community  $c$ , then  $|\mathcal{R}_c| = \theta_c$ . Let  $X_{c}^{i} (i \in [\theta_{c}])$  be a random variable for each RR set  $R_{i} \in \mathcal{R}_{c}$ , such that  $X_{c}^{i} = 1$  if  $\mathcal{R}_{c}^{i} \cap S \neq \emptyset$ , and  $X_{c}^{i} = 0$  otherwise. Then, we have  $\mathbb{E}[X_{c}] = \boldsymbol{u}_{c}$  and  $\mathbb{E}[\overline{X_c}] = 1 - \boldsymbol{u}_c$ .
Based on Lemma 1 and Lemma 2, we can get the unbiased estimator of  $\mathbb{E}[X_c]^{\alpha}$  through RR sets as
$$
\begin{array}{l} \mathbb {E} \left[ X _ {c} \right] ^ {\alpha} = 1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(1 - \mathbb {E} \left[ X _ {c} \right]\right) ^ {n} \\ = 1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \frac {\left(\theta_ {c} - n\right) !}{\theta_ {c} !} \left\{\sum \overline {{X _ {c} ^ {i _ {1}}}} \cdot \overline {{X _ {c} ^ {i _ {2}}}} \dots \overline {{X _ {c} ^ {i _ {n}}} \right\}, \tag {3} \\ \end{array}
$$
Further, we can get the unbiased estimator of the fair influence  $F_{\alpha}(S)$  as
$$
\begin{array}{l} \hat {F} _ {\alpha} (S, \mathcal {R}) = \sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \frac {\left(\theta_ {c} - n\right) !}{\theta_ {c} !} \left\{\sum \overline {{X _ {c} ^ {i _ {1}}}} \cdot \overline {{X _ {c} ^ {i _ {2}}}} \dots \overline {{X _ {c} ^ {i _ {n}}}} \right\}\right) \\ = \sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\pi_ {c} - i}{\theta_ {c} - i}\right) \tag {4} \\ \end{array}
$$
where  $\pi_c = \theta_c - \sum_{i\in [\theta_c]}X_c^i$  , and
In the following, we consider Eq. 4 as our objective function to deal with the fair IM problem.
3.3 FIMM
For a given seed set  $S$ , let  $\varphi[c]$  denote the number of all  $u$ -rooted  $(u \in V_c)$  RR sets covered by  $S$ , and  $\kappa[v][c]$  denote the number of all  $u$ -rooted  $(u \in V_c)$  RR sets that covered by  $v$  ( $v \in V \setminus S$ ) but not by  $S$ , then the marginal fair influence gain of  $v$  is
$$
\begin{array}{l} \hat {F} _ {\alpha} (v | S) = \sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\theta_ {c} - \kappa [ v ] [ c ] - \varphi [ c ] - i}{\theta_ {c} - i}\right) \\ - \sum_ {c \in \mathcal {C}} n _ {c} \left(1 - \alpha \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \prod_ {i = 0} ^ {n - 1} \frac {\theta_ {c} - \varphi [ c ] - i}{\theta_ {c} - i}\right) \\ = \sum_ {c \in \mathcal {C}} \alpha n _ {c} \sum_ {n = 1} ^ {\infty} \eta (n, \alpha) \left(\prod_ {i = 0} ^ {n - 1} \frac {\theta_ {c} - \varphi [ c ] - i}{\theta_ {c} - i} - \prod_ {i = 0} ^ {n - 1} \frac {\theta_ {c} - \kappa [ v ] [ c ] - \varphi [ c ] - i}{\theta_ {c} - i}\right) \tag {5} \\ \end{array}
$$
Therefore, when generating RR sets, we have to count  $\kappa[v][c]$  which indicates the community-wise coverage for  $v$  and record  $\eta[v]$  which indicates the linked-list from  $v$  to all its covered RR sets, as shown in Algorithm 1. As shown in lines 6~9, when generating a random  $v$ -rooted RR set  $RR(v)$ , we count all nodes  $u \in RR(v)$  and raise all  $\kappa[u][c(v)]$  by 1, where  $c(v)$  indicates  $v$ 's community label. It should be noted that modifying  $\kappa[u][c(v)]$  can be accomplished simultaneously when generating  $RR(v)$  by the reverse influence sampling.