Influence Maximization with Priority in Online Social Networks

Abstract

The Influence Maximization ($\mathsf{IM}$) problem, which finds a set of k nodes (called seedset) in a social network to initiate the influence spread so that the number of influenced nodes after propagation process is maximized, is an important problem in information propagation and social network analysis. However, previous studies ignored the constraint of priority that led to inefficient seed collections. In some real situations, companies or organizations often prioritize influencing potential users during their influence diffusion campaigns. With a new approach to these existing works, we propose a new problem called Influence Maximization with Priority ($\mathsf{IMP}$) which finds out a set seed of k nodes in a social network to be able to influence the largest number of nodes subject to the influence spread to a specific set of nodes U (called priority set) at least a given threshold T in this paper. We show that the problem is NP-hard under well-known $\mathsf{IC}$ model. To find the solution, we propose two efficient algorithms, called Integrated Greedy ($\mathsf{IG}$) and Integrated Greedy Sampling ($\mathsf{IGS}$) with provable theoretical guarantees. $\mathsf{IG}$ provides a $\left(1-{(1-\frac{1}{k})}^t\right)$-approximation solution with t is an outcome of algorithm and $t\ge 1$. The worst-case approximation ratio is obtained when $t=1$ and it is equal to $1/k$. In addition, $\mathsf{IGS}$ is an efficient randomized approximation algorithm based on sampling method that provides a $\left(1-{(1-\frac{1}{k})}^t-ϵ\right)$-approximation solution with probability at least $1-\delta$ with $ϵ>0,\delta \in (0,1)$ as input parameters of the problem. We conduct extensive experiments on various real networks to compare our $\mathsf{IGS}$ algorithm to the state-of-the-art algorithms in $\mathsf{IM}$ problem. The results indicate that our algorithm provides better solutions interns of influence on the priority sets when approximately give twice to ten times higher than threshold T while running time, memory usage and the influence spread also give considerable results compared to the others.

1. Introduction

Presently, Online Social Networks (OSNs) have become an important platform in communication as well as e-commerce. Companies and businesses have leveraged a rapid spread of information thanks to the “word of mouth” effect among friends in social networks as a powerful tool for viral marketing. For instance, companies can provide some ones with free samples over an OSN so that much more people may know about their products and they have more chances to sell them. Influence Maximization ($\mathsf{IM}$) problem [1], a key problem in viral marketing, has been extensively studied for this decade due to its tremendous value in business, viral marketing and influence propagation. Basically, $\mathsf{IM}$ aims to find some nodes (called seedset) in a social network to inject opinion, innovation or influence that can effect the largest the number of nodes. Kempe et al. [1] first studied $\mathsf{IM}$ as an optimization problem combined with two well-known models, Independent Cascade ($\mathsf{IC}$) and Linear Threshold ($\mathsf{LT}$). Since $\mathsf{IM}$ is NP-hard, they designed a native greedy algorithm that returned an $(1-1/e)$-approximation solution. The research shows that $\mathsf{IM}$ is not only a potential commercial role in viral marketing [2,3] but also a foundation of various applications in many fields such as epidemics control in social network [4,5,6,7,8], social network monitoring [9,10], recommendation system [11], etc. Hence, $\mathsf{IM}$ has been extensively studied recently [2,4,12,13,14,15,16,17,18,19].

Although $\mathsf{IM}$ has a lot of great applications in viral marketing, previous studies ignored considering the impact on priority users who could play an important role for effectiveness of viral marketing campaigns. In fact, companies often prioritize specific potential customers, who are financially competent or suitable for their products. For examples, if a company produces baby diapers, they tend to introduce the product to married women aged 20 to 45. Supposing that they have some data about user accounts on a social network, hence they launch a promotion with suitable amount of gifts to married female users via this social network. If we only care about the number of influenced individuals, as in the case of $\mathsf{IM}$, we will not evaluate the impact to the potential users and lead to wrong selection of a seed set. Figure 1 shows an example. This network contains 8 nodes and 9 edges, the priority set is $\{b,d\}$ and the weight of each edge (or influence probability) is assigned to 1. Considering the case when the budget $k=1$ (number of seed nodes), the optimal solution of $\mathsf{IM}$ is $\left\{f\right\}$ influences to 6 nodes including $\{f,d,g,c,e,h\}$ except b. Hence, $\mathsf{IM}$ cannot take effect to all priority nodes.The solution must be $\left\{a\right\}$ that has the total influence is only 5.

Figure 1. A toy example shows the difference between the influence maximization and our proposed problem.

Motivated by such interesting scenarios, in this paper we investigate the Influence Maximization with Priority ($\mathsf{IMP}$) problem, which takes into account the priority constraint for influence process. Given a social network $G=(V,E)$, a priority set $U\subset V$, a budget k and a priority threshold $T,(T\le k)$, the goal is to find the seed set S sized at k so that it influences to U at least T and the influence of the cascade is maximized. In fact, $\mathsf{IMP}$ is more suitable than $\mathsf{IM}$. Besides, it generalizes $\mathsf{IM}$ problem. Nevertheless this problem faces with complicated challenges caused by the constraint of priority. To address this problem, we propose two approximation algorithms, Integrated Greedy ($\mathsf{IG}$) and Integrated Greedy-based Sampling ($\mathsf{IGS}$), with provable theoretical guarantees. $\mathsf{IG}$ meets the theoretical guarantee based on a modification of the natural greedy algorithm while $\mathsf{IGS}$ is an efficient randomized approximation algorithm based on sampling method [13,14,15,20]. This algorithm combines two novel techniques. Firstly, we propose Targeted Reverse Reachable (TRR) concept by modifying the Reverse Reachable Sampling (RR) technique [13,14,15,20] to estimate influence from a seed set to a given priority set. Secondly, we develop a new strategy to select a set of seeds in accordance with the priority constraint and set the number of samples to give a theoretical guarantees. Because $\mathsf{IMP}$ is a separate case of $\mathsf{IM}$, we have built extensive experiments on various real networks to compare our $\mathsf{IGS}$ algorithm to the state-of-the-art algorithms for $\mathsf{IM}$ problem such as $\mathsf{DSSA}$ [15], $\mathsf{BCT}$ [2], $\mathsf{OPIM}$ about the influence on a given priority set, running time and memory used while the influence spread approximations are ensures as in $\mathsf{IM}$.

Our contributions are summarized as follows:

• We propose the Influence Maximization with Priority ($\mathsf{IMP}$) problem that considers priority constraint in Influence Maximization ($\mathsf{IM}$) problem. It means we expand the $\mathsf{IM}$ by adding a constraint to influence on a given set of users. $\mathsf{IMP}$ aims to find the seed set S with size k so that total influence of priority users is at least a given threshold $T,(k\ge T)$ and still maintain the influence of cascade maximized.
• We propose two approximation algorithms, $\mathsf{IG}$ and $\mathsf{IGS}$, for the $\mathsf{IMP}$ problem. $\mathsf{IG}$ algorithm provides an approximation ratio of $\left(1-{(1-\frac{1}{k})}^t\right)$, where $t\ge k-T$ is an output of the algorithm. In addition, $\mathsf{IGS}$ is a randomized approximation algorithm providing an approximation ratio of $\left(1-{(1-\frac{1}{k})}^t-ϵ\right)$ with probability at least $1-\delta$, where $ϵ>0,\delta \in (0,1)$ are input parameters and t is an output of algorithm.
• We conduct extensive experiments on various real networks such as netHEPT, netPHY, Email-Enron, DBLP, and Twitter ReTweet. The results indicate that our algorithm, $\mathsf{IGS}$, often outperforms state-of-the-art $\mathsf{IM}$ algorithms in terms of influence, running time and memory used. In particular, $\mathsf{IGS}$ provides the solution which ensures that the influence on the priority set is approximately from twice to 10 times greater than its threshold T while still maintains influence spread approximations as in $\mathsf{IM}$ algorithms. Further, we also demonstrate that $\mathsf{IGS}$ is faster and uses lower memory than the others in a lot of cases. On the whole, although $\mathsf{IGS}$ has to care about how influences to a target given users, $\mathsf{IGS}$ still gives considerable fast runtime, low memory used and high maximized influence on all nodes such as state-of-the-art algorithms such as DSSA, BCT, OPIM-C. It proves that $\mathsf{IGS}$ has been very well designed.

Related work. Kempe et al. [1] first studied the Influence Maximization ($\mathsf{IM}$) problem inspired by exploiting the influence among users in social networks for viral marketing [21]. They formulated $\mathsf{IM}$ as a discrete optimization problem under two classical information diffusion models, Independent Cascade ($\mathsf{IC}$) and Linear Threshold ($\mathsf{LT}$). They proved that $\mathsf{IM}$ could be approximated within a ratio of $1-1/e+ϵ$ for any $ϵ\in (0,1)$ and proposed a greedy algorithm that provided an approximation ratio of $1-1/e-ϵ$ for $ϵ>0$. Later, Chen et al. [12,16] continued to study $\mathsf{IM}$ and proved that to calculate exactly the influence spread of a seeding set was #P-Hard. Hence although many heuristics algorithms have been proposed to solve this problem in large networks, they still have failed to retain the approximation ratio of $1-1/e-ϵ$ and have provided a low quality solutions such as the cost-effective lazy-forward heuristic (CELF) proposed by Leskovec et al. [22] which is based on improving greedy algorithm to get 700 times faster than the greedy algorithm with Mote-Carlo simulation; a fast heuristics algorithm called PMIA proposed by Chen et al. [12] which constructs a directed acyclic graph to estimate the influence under $\mathsf{IC}$ model or the algorithm proposed by the authors in [16] which uses a local directed acyclic graphs (LDAG) to calculate the local influence of nodes under $\mathsf{LT}$ model. To keep the $1-1/e-ϵ$ ratio, research on the approximation approach continues to be explored. Borgs et al. [13] first presented an $(1-1/e-ϵ)$-approximation algorithm with probability at least $1-\delta$ in $O(k{l}^2(m+n){log}^{2}n/{ϵ}^3)$ time complexity by introducing Reverse Influence Sampling (RIS) model. This model has formed the foundation for further algorithm development. [14,15,20,23].

From then on, many works expanded $\mathsf{IM}$ in contexts of viral marketing. Nguyen et al. [24] investigated the Budged Influence Maximization (BIM) problem which considered the cost of selecting a node and proposed a $(1-1/\sqrt{e}-ϵ)$ approximation algorithm. The authors in [2] studied the a generalization of $\mathsf{IM}$ and BIM problems, called Cost-aware Targeted Viral Marketing (CTVM). In this work, each node u had an arbitrary cost $c\left(u\right)$ and a benefit $b\left(u\right)$ and the goal of CTVM was to select a seed set within a given budget so that the total benefit was maximized. We believe that this is the closest problem to our work. In CTVM problem, we can set parameters that maximize the influence on a given target set of users but cannot simultaneously maximize the influence of the others as in our problem. Later, several works improve the approximation as well as the scalability of CTVM algorithms [25,26].

Moreover, there are also many variants of $\mathsf{IM}$ problem that were studied. Some works studied the constraints of $\mathsf{IM}$ such as [17,18,27], in which edges were associated with a topic influence weight. These problems aimed to find a set of k users that maximized influenced users according to a topic query. However, the proposed algorithms did not provide any theoretical guarantee. Li et al. [28] proposed the Location-aware Influence Maximization (LIM) problem with the goal was to select the k-seed set so that the number of influenced nodes in the given query region was maximized. [29] investigated the Distance-aware Influence Maximization (DAIM) problem which considered the role of distance between users and the promoted location in seed selection. They extended a RIS process model and provided an unbiased estimator for the DAIM problem.

Besides, some works investigated the problem of Competitive Influence Maximization (CIM), which considered the context of $\mathsf{IM}$ under the competition of many rivals. Bharathi et al. [30] first formulated the CIM problem under a new competitive propagation model which was an extension of $\mathsf{IC}$ model. Chen et al. [12] investigated CIM under the combating with negative opinions based on an assumption that negative information was often more attractive than official information. Some authors considered the problem under many different cases in viral marketing, such as proposing a distance-aware problem [31], expanding the $\mathsf{LT}$ model to reflect competition [13,32,33,34], proposing a heuristic algorithm [35], etc.

Recently, some authors studied the selection of seed nodes in a social network to influence groups of users or communities instead of individuals [36,37,38,39]. They argue that in real-world scenarios, creating impact on groups is more beneficial than the individuals in a network. Tsang et al. [36] investigated the Fairness Group Maximization problem with two fairness criteria including maximin fairness and diversity. While the maximin fairness aimed to maximize the minimum influence nodes of any per their population, the criterion of diversity was an alternate fairness concept by extending the notion of individual rationality to group rationality. They proposed an approximation algorithm based on multi-submodular objective function processing techniques. More recent, the authors in [37] proposed exact algorithms for fairness group influence with multiple criteria based on mix integer linear programming formulation on a specific set of sample graphs under $\mathsf{IC}$ model. In [38], the authors characterize the intricate relationship between diversity and efficiency, which sometimes may be at odds but may also reinforce each other. Nguyen et al. [39] considered the Influence Maximization problem at the Community level problem, which found seed set of k nodes that influenced to largest number of communities. They showed that the objective function was neither sub-modular nor super-modular and proposed some approximation algorithms with provable guarantees. Different to our studied problem in this paper, these studies did not address the priority set in influence maximization context. Hence the proposed algorithms cannot be applied to the $\mathsf{IMP}$ problem.

Organization. The rest of the paper is organized as follows: Section 2 presents information diffusion model and problem definitions. Section 3 and Section 4 present our proposed Integrated Greedy and Integrated Greedy-based Sampling algorithms for $\mathsf{IMP}$ problem with the theoretical analysis. Experimental results are shown in Section 5. In Section 6 we discuss the future work and conclude this paper.

2. Model and Problem Definition

In this section, we introduce about network model and the well-known Independent Cascade ($\mathsf{IC}$) diffusion information model [1]. Under $\mathsf{IC}$ model, we formally define the Influence Maximization with Priority ($\mathsf{IMP}$) problem.

2.1. Graph Notation and Independent Cascade Model

Let $G=(V,E)$ be a directed graph representing a social network with a node set V and a directed edge set E, $\left|V\right|=n$ and $\left|E\right|=m$. Let $N_{in}\left(v\right)$ and $N_{out}\left(v\right)$ be two sets of in-neighbors and out-neighbor of a node v, respectively. The notations of S and $S^{*}$ represent to a seed set that is a solution and an optimal solution of $\mathsf{IMP}$, respectively. We also note $\mathsf{OPT}=\sigma \left(S^{*}\right)$ is the influence of an optimal solution.

In Independent Cascade ($\mathsf{IC}$) model, each edge $e=(u,v)\in E$ has an influence probability $p(u,v)\in (0,1)$ that represents the information transmission from u to v. Each node $v\in V$ has two possible states, active and inactive. Given a seed set $S\subseteq V$, the diffusion process from S happens in discrete steps $t=0,1,\dots$, as follow:

• At step $t=0$, all nodes in S is activated.
• At step $t\ge 1$, for an activated node u in previous steps, it has a single chance to activate each inactive neighbour v with the successful probability $p(u,v)$. An activated node remains $active$ till the end of the diffusion process.
• The propagation process ends when no more node is activated.

Kempe et al. [1] show that $\mathsf{IC}$ model is equivalent to live-edge model and estimating the quantity of influence nodes can be done as follows. We first generate a sample graph g from original graph G by selecting each edge $e=(u,v)\in E$, independently, with probability $p(u,v)$, and no select edge $(u,v)$ with probability $1-p(u,v)$. The probability that a realization g can be generated from G (denoted as $g\sim G$) is

$$\begin{array}{c}\hfill Pr[g\sim G]=\prod _{e\in E\left(g\right)}p(u,v)\prod _{e\in E\backslash E\left(g\right)}(1-p(u,v))\end{array}$$

(1)

In this equation, $E\left(g\right)$ is the set edge of g. The number of sample graphs is $2^{\left|E\right|}$. The influence spread of a seed set S in G is calculated as follows:

$$\begin{array}{c}\hfill \sigma \left(S\right)=\sum _{g\sim G}Pr[g\sim G]\left|R\right(g,S\left)\right|\end{array}$$

(2)

where $R(g,S)$ denotes the set of reachable nodes from S in g. For a set of priority nodes U, the influence spread of S to U is calculated as follows:

$$\begin{array}{c}\hfill {\sigma }_U\left(S\right)=\sum _{g\sim G}Pr[g\sim G]\left|R\right(g,S\to U\left)\right|\end{array}$$

(3)

where $R_g(S\to P)$ denotes the set of nodes in U that can reach from S in g. Kempe et al. [1] also show that, $\sigma (·)$ is a monotone and sub-modular function, i.e, for any $A\subset V$, and $v\notin V\backslash B$, we have:

$$\begin{array}{c}\hfill \sigma (A+\{v\left\}\right)\ge \sigma \left(A\right)\end{array}$$

(4)

and for any $A\subseteq B\subset V$, and $v\notin V\backslash B$, we have:

$$\begin{array}{c}\hfill \sigma (A+\{v\left\}\right)-\sigma \left(A\right)\ge \sigma (B+\{v\left\}\right)-\sigma \left(B\right)\end{array}$$

(5)

We also easy to see that ${\sigma }_U(·)$ is a monotone and sub-modular function.

2.2. Problem Definition

We investigate Influence Maximization with Priority ($\mathsf{IMP}$) defined as follows:

Definition 1

($\mathsf{IMP}$ problem). Given a graph $G=(V,E)$ under $\mathsf{IC}$ model, a positive integer k (budget), the priority set $U\subset V$, and the threshold T with $T\le k,T\le \left|U\right|$. $\mathsf{IMP}$ problem asks to find the seed set $S\subset V$, with $\left|S\right|\le k$ and ${\sigma }_U\left(S\right)\ge T$ so that influence spread, $\sigma \left(S\right)$, is maximized, i.e, find S that is the solution to the following optimization problem:

$$\begin{array}{cc}\hfill \mathit{maximize}:& \sigma \left(S\right)\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \mathit{subject}\mathit{to}:& \left|S\right|\le k\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & {\sigma }_U\left(S\right)\ge T\hfill \end{array}$$

(8)

$\mathsf{IMP}$ becomes $\mathsf{IM}$ problem when $U=\varnothing$. Therefore, $\mathsf{IM}$ is a special case of $\mathsf{IMP}$ and $\mathsf{IMP}$ is also NP-hard. In addition, the calculation of the influence function from the seed set is proven to be #P-hard [12]. Thus finding the solution to the problem within the time allowed is very challenging.

3. Integrated Greedy Algorithm

In this section, we first propose Integrated Greedy ($\mathsf{IG}$) Algorithm which is well-known to resolve monotone and sub-modular problems that ensures an lower-bounded of optimization solution. The details of algorithm is described in Algorithm 1.

Algorithm 1: Integrated Greedy ($\mathsf{IG}$) algorithm

Assume $S_1$ is the solution of the problem that finds the minimum seed nodes such that the influence on the priority set is greater than threshold T, and $S_2$ is a solution of $\mathsf{IM}$ problem. The main idea of this algorithm is to modify the native greedy algorithm [1] by combining two above solutions.

The algorithm is divided into two main phases. In the first phase, it tries to find a solution $S_1$ by a greedy strategy (line 2–4). In each iterator, the algorithm chooses a node u with largest influence incremental to set U into $S_1$ (line 3–4) until the ${\sigma }_U\left(S_1\right)\ge T$. Since $T<k$, $|S_1|\le T<k$. Denote $t=k-T$ as the remaining budget (line 6). The algorithm next finds the candidate solution $S_2$ for $\mathsf{IM}$ with the remaining budget t by using a greedy method in the second phase (line 6–10). In each iterator i, it selects a node u with largest influence incremental (line 7). If u already belongs to $S_1$, the algorithm increases t by 1 (line 8–9). This phase ends when the remaining budget t is exhausted (line 6). Finally, the algorithm returns the solution S which unites $S_1$ and $S_2$. It is easy to confirm that $\left|S\right|=k$, and $t>T-k\ge 1$ since $k>T$. Theorem 1 shows the approximation guarantee of $\mathsf{IG}$ algorithm.

Theorem 1.

$\mathsf{IG}$ algorithm returns $(S,t)$, where S is a feasible solution and $t\ge 1$, satisfies:

$$\sigma \left(S\right)\ge \left(1-{\left(1-\frac{1}{k}\right)}^t\right)\sigma \left(S^{*}\right)$$

The worst-case approximation ratio is obtained when t = 1 and it is equal to 1/k.

Proof. 

Denote $S_{\mathsf{IM}}^{*}=\{s_1,s_2,\dots ,s_k\}$ is an optimal solution of $\mathsf{IM}$ problem for input data of Algorithm 1 (the graph G and budget k). Obviously, we have $\sigma \left(S_{\mathsf{IM}}^{*}\right)\ge \sigma \left(S^{*}\right)$. After ending the second phase, assume that $S_2=\{u_2^1,u_2^2,\dots ,u_2^t\}$, $S_2^i=\{u_2^1,u_2^2,\dots ,u_2^i\}$, and $S_2^0=\varnothing$. In the second phase, the algorithm repeatedly selects a node u of which incremental influence gain is largest and due to the function $\sigma (·)$ is monotone and sub-modular [1], so we have:

$$\begin{array}{cc}\hfill \sigma \left(S_{\mathsf{IM}}^{*}\right)-\sigma \left(S_2^i\right)& \le \sigma (S_{\mathsf{IM}}^{*}\cup S_2^i)-\sigma \left(S_2^i\right)\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \le \sum _{j=1}^k\left(\sigma (S_2^i\cup \{s_1,s_2,\dots ,s_j\})-\sigma (S_2^i\cup \{s_1,s_2,\dots ,s_{j-1}\})\right)\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \le \sum _{j=1}^k\left(\sigma (S_2^i\cup \left\{s_j\right\})-\sigma \left(S_2^i\right)\right)(\mathrm{Dueto} \sigma \mathrm{is} \mathrm{a} \mathrm{sub-modular} \mathrm{function})\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & \le k·\underset{s\in S_{\mathsf{IM}}^{*}}{max}\left(\sigma (S_2^i\cup \left\{s\right\})-\sigma \left(S_2^i\right)\right)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \le k·\left(\sigma \left(S_2^{i+1}\right)-\sigma \left(S_2^i\right)\right)\hfill \end{array}$$

(13)

Therefore, for any $i=0,\dots ,t-1$, we have

$$\begin{array}{c}\hfill \sigma \left(S_2^{i+1}\right)-\sigma \left(S_2^i\right)\ge \frac{1}{k}(\sigma \left(S_{\mathsf{IM}}^{*}\right)-\sigma \left(S_2^i\right))\end{array}$$

(14)

Minus two inequality terms to $\sigma \left(S_{\mathsf{IM}}^{*}\right)$, we have:

$$\begin{array}{c}\hfill \sigma \left(S_2^{i+1}\right)-\sigma \left(S_2^i\right)-\sigma \left(S_{\mathsf{IM}}^{*}\right)\ge \frac{1}{k}\sigma \left(S_{\mathsf{IM}}^{*}\right)-\sigma \left(S_{\mathsf{IM}}^{*}\right)-\frac{1}{k}\sigma \left(S_2^i\right)\end{array}$$

(15)

Rearrange the terms of the above inequality, we have

$$\begin{array}{cc}\hfill \sigma \left(S_2^{i+1}\right)-\sigma \left(S_{\mathsf{IM}}^{*}\right)& \ge \left(1-\frac{1}{k}\right)(\sigma \left(S_2^i\right)-\sigma \left(S_{\mathsf{IM}}^{*}\right))\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & \ge {\left(1-\frac{1}{k}\right)}^t(\sigma \left(S_2^0\right)-\sigma \left(S_{\mathsf{IM}}^{*}\right))\hfill \end{array}$$

(17)

Together with the fact that $S_2^0=\varnothing$ and $\sigma (\varnothing )=0$, the above inequality implies

$$\begin{array}{c}\hfill \sigma \left(S_2\right)=\sigma \left(S_2^t\right)\ge \left(1-{\left(1-\frac{1}{k}\right)}^t\right)\sigma \left(S_{\mathsf{IM}}^{*}\right)\end{array}$$

(18)

Since ${\sigma }_U\left(S_1\right)\ge T$ and $S=S_1\cup S_2$, S is feasible solution of $\mathsf{IMP}$, and

$$\begin{array}{cc}\hfill \sigma \left(S\right)\ge \sigma \left(S_2^t\right)& \ge \left(1-{\left(1-\frac{1}{k}\right)}^t\right)\sigma \left(S_{\mathsf{IM}}^{*}\right)\ge \left(1-{\left(1-\frac{1}{k}\right)}^t\right)\sigma \left(S^{*}\right)\hfill \end{array}$$

(19)

which proves the theorem! ☐

Although Algorithm 1 can provide an approximation guarantee, but it cannot work with real-social networks because the calculation of the influence function $\sigma \left(S\right)$ is $\#P$-hard under $\mathsf{IC}$ model [12]. To overcome this challenge, we propose a randomize algorithm with provable approximation guarantee based on combining $\mathsf{IG}$ with a sampling technique.

4. Sampling Algorithm with Provable Guarantees

In this section, we present an efficient algorithm for $\mathsf{IMP}$ problem called Integrated Greedy Sampling ($\mathsf{IGS}$) algorithm that can provide an guarantee theoretical. In addition, we show that our algorithm can also be applied to large networks in experiments.

4.1. Estimator of Influence Functions

Firstly, we recap the concept of Reachable Reverse (RR) set [40] to estimate influence function $\sigma (·)$. Base on that, we propose the concept of Targeted Reachable Reverse (TRR) set to estimate influence function ${\sigma }_U\left(S\right)$. Then we propose $\mathsf{IGS}$ algorithm and provide theoretical analysis based on statistical evidence.

Definition 2 

(Reachable Reverse (RR) set [40]). Given a graph $G=(V,E)$ under $\mathsf{IC}$ model. A random RR set $R_j$ is generated from G by:

1. 

Picking a source node u with probability $\frac{1}{n}$.

2. 

Generating a sample graph g from G, and returning $R_j$ as nodes which can be reached from u in g.

For a random RR set $R_j$, define a random variable $X_g\left(S\right)=min\{1,|R_g\cap S\left|\right\}$. Borgs et al. [40] show that RR samples can be used to estimate the influence function by applying the following Lemma.

Lemma 1.

For any set of nodes $S\subseteq V$, we have $\sigma \left(S\right)=n·\mathbb{E}\left[X_g\left(S\right)\right]$.

Given a set of RR set $\mathcal{R}$, and a set node S, we can approximate the value of $\sigma \left(S\right)$ by $\widehat{\sigma }\left(S\right)$ defined as follow:

$$\begin{array}{c}\hfill \widehat{\sigma }\left(S\right)=\frac{n}{\left|\mathcal{R}\right|}\sum _{R_g\in \mathcal{R}}{X}_g\left(S\right)\end{array}$$

(20)

Generating RR sets can be accomplished by using $\mathsf{IM}$ algorithms in [13,14,15,20,23]. The common algorithm for generating RR set $R_j$ is described in Algorithm 2. This algorithm first selects a source node u with a probability $\frac{1}{n}$ to add into $R_j$. The algorithm uses a queue Q to store the visited nodes. Initially, u is also added to Q. The algorithm next retrieves each node v in Q and picks an incoming node x with probability $p(x,v)$ (line 6). If successful, it adds x in to Q and $R_j$. This process takes place until the set Q is empty.

Algorithm 2: Generating RR sample under $\mathsf{IC}$ model

We now introduce the definition of Targeted Reachable Reverse (TRR) Set on the basis of modifying RR concept.

Definition 3 

(Targeted Reachable Reverse (TRR) Set). Given a graph $G=(V,E)$ under $\mathsf{IC}$ model. A random TRR set $R_j^U$ is generated from G by:

1. 

Picking a source node $u\in U$ with probability $\frac{1}{\left|U\right|}$.

2. 

Generating a sample graph g from G, and returning $R_g^U$ as nodes which can be reached from u in g.

We define a random variable $Y_g\left(A\right)=min\{1,|R_g^U\cap S\left|\right\}$. Similar to Lemma 1, Lemma 2 shows that we can use the value of $Y_g\left(S\right)$ to estimate function ${\sigma }_U\left(S\right)$.

Lemma 2.

For any set of nodes $S\subseteq V$, we have ${\sigma }_U\left(S\right)=\left|U\right|·\mathbb{E}\left[Y_g\left(S\right)\right]$

Proof. 

Denote $R_g^U\left(u\right)$ is a TRR sample with a source node u for the sample graph g, we have:

$$\begin{array}{cc}\hfill {\sigma }_U\left(S\right)& =\sum _{g\sim G}\left|R(g,S\to U)\right|\hfill \\ \hfill & =\sum _{u\in U}\sum _{g\sim G}Pr[g\sim G][\exists v\in S:u\mathrm{is}\mathrm{reached}\mathrm{from}v]\hfill \\ \hfill & =\sum _{u\in U}\sum _{g\sim G}Pr[g\sim G][\exists v\in S:v\in R_g^U\left(u\right)]\hfill \\ \hfill & =\left|U\right|\sum _{u\in V}\frac{1}{\left|U\right|}\sum _{g\sim G}Pr[g\sim G]Y_g\left(S\right)\hfill \\ \hfill & =\left|U\right|\sum _{u\in V}\sum _{g\sim G}Pr\left[u\mathrm{is}\mathrm{source}\mathrm{node}\right]Pr[g\sim G]Y_g\left(S\right)\hfill \\ \hfill & =\left|U\right|·\mathbb{E}\left[Y_g\left(S\right)\right]\hfill \end{array}$$

The transition from the second equality to the third equality comes from the definition of $R_g^U\left(u\right)$ and from the third to the fourth then to the fifth is caused by the distribution of choosing a node u as a source node. ☐

Given a set of TRR samples $\mathcal{R}$ and a set node S, we define and an approximation value of ${\sigma }_U\left(S\right)$ as follow:

$$\begin{array}{c}\hfill {\widehat{\sigma }}_U\left(S\right)=\frac{\left|U\right|}{\left|\mathcal{R}\right|}\sum _{R_g^U\in \mathcal{R}}{Y}_g\left(S\right)\end{array}$$

(21)

From Lemma 2, we can give a good approximation of ${\sigma }_U(·)$ when the number of TRR samples is large enough. We can re-use Algorithm 2 to generate a TRR set $R_j^U$ by a modification. We replace line 1 in the algorithm by picking source node $u\in U$ with probability $\frac{1}{\left|U\right|}$ and leave the rest as is.

4.2. Algorithm Description and Theoretical Analysis

Algorithm description. The algorithm is detailed in Algorithm 3. It generates the set of $N_U$ TRR sets ${\mathcal{R}}_1$, and set two candidate solutions $S_1$, $S_2$ empty at first. Then the body of the algorithm divides into two phases. In phase 1, it finds a candidate solution $S_1$ with minimum-size so that $\widehat{\sigma }\left(S\right)\ge (1+\alpha )T$ by using a greedy strategy with potential function $\widehat{\sigma }$ over ${\mathcal{R}}_1$. In each iterator, it selects a node u with maximal incremental value of the potential function (line 4) until $\widehat{\sigma }\left(S\right)\ge (1+\alpha )T$. The candidate solution $S_1$ obtained by this phase satisfies the priority constraint, ${\sigma }_U\left(S_1\right)\ge T$ with probability at least $1-\delta$ (Lemma 4).

The phase 2 selects a candidate solution $S_2$ with the remaining budget ($t=k-|S_1|$) so that the influence spread $\sigma (·)$ is maximized. In this phase, it first sets the parameters ${ϵ}_1,t_{max}$, $N_{max}$ and generates $N_1$ set of RR samples ${\mathcal{R}}_2$. The main of this phase operates in several iterators (line 12–27) until meeting the stopping condition (line 22). In each iterator, it finds a candidate solution $S_2$ by a greedy strategy. It picks a node u with maximal incremental of approximation influence $\widehat{\sigma }(·)$ over ${\mathcal{R}}_2$ (line 12) until t nodes are selected. Similar to $\mathsf{IG}$ algorithm, if u already belongs to $S_1$, the algorithm increases t by 1. After that, the algorithm checks the quality of candidate solution $S_2$ (line 17). It calculates $F_l(S_2,{\mathcal{R}}_2,\delta )$- a lower-bounded of $\sigma \left(S_2\right)$, and $F_u(S_2,{\mathcal{R}}_2,\delta )$-an upper-bounded of an optimal solution respect to $\mathsf{IMP}$ problem. These functions ensure the statistical criterion, which are claimed in the Lemmas 5 and 6. If solution $S_2$ meets the approximation condition (line 19), the algorithm returns $S_2$. If not, it moves to the next iterator and stops when the number of TRR samples is at least $N_{max}$ (line 21).

Algorithm 3: Integrated Greedy -based Sampling ($\mathsf{IGS}$) algorithm

Theoretical analysis. Fortunately, the sequence of random variables $X_g\left(S\right)$ and $Y_g\left(S\right)$ constructed from the RR and TRR samples can be shown to form a martingale. For any random variable $X_g\left(S\right)\in [0,1]$, let a random variable $M_i={\sum }_{j=1}^i(X_g^i\left(S\right)-\mu ),\forall i\ge 1$, where $\mu =\mathbb{E}\left[X_g\right]$. For a sequence of random variables $M_1,M_2,\dots$ we have $\mathbb{E}\left[M_i\right|M_1,\dots ,M_{j-1}]=\mathbb{E}\left[M_{i-1}\right]+\mathbb{E}[X_g^i\left(S\right)-\mu ]=\mathbb{E}\left[M_{i-1}\right]$. Hence, $M_1,M_2,\dots$ be a form of martingale [41]. Similarly, $Y_g$ is also a form of martingale. Therefore, the following concentration inequality [41] applies:

Lemma 3.

If $M_1,M_2,\dots$ be a form of martingale, $|M_1|\le a$, $|M_j-M_{j-1}|\le a$ for $j\in [1,i]$, and

$$\begin{array}{c}\hfill \mathsf{Var}\left[M_1\right]+\sum _{j=2}^i\mathsf{Var}\left[M_j\right|M_1,M_2,\dots ,M_{j-1}]=b\end{array}$$

(22)

where $\mathsf{Var}[·]$ denotes the variance of a random variable. Then, for any λ, we have:

$$\begin{array}{c}\hfill Pr[M_i-\mathbb{E}\left[M_i\right]\ge \lambda ]\le exp\left(-\frac{{\lambda }^2}{\frac{2}{3}a\lambda +2b}\right)\end{array}$$

(23)

Apply this Lemma with $|M_1|=|X_g^1\left(S\right)|\le 1$, $|M_j-M_{j-1}|=|X_g^j\left(S\right)-X_g^{j-1}\left(S\right)|\le 1$, $\mathsf{Var}\left[M_1\right]=\mathsf{Var}[X_g^1\left(S\right)-\mu ]=\mathsf{Var}\left[X_g\left(S\right)\right]$, $\mathsf{Var}\left[M_j\right|M_1,M_2,\dots ,M_{j-1}]=\mathsf{Var}[X_g^j\left(S\right)-\mu ]=\mathsf{Var}\left[X_g\left(S\right)\right]$, and $\mathsf{Var}\left[X_g\left(S\right)\right]\le \mu (1-\mu )\le \mu$, we have:

$$\begin{array}{cc}\hfill Pr\left[\sum _{i=1}^{\left|\mathcal{R}\right|}{X}_g^i\left(S\right)-\left|\mathcal{R}\right|·\mu \ge \lambda \right]& \le exp\left(-\frac{{\lambda }^2}{\frac{2}{3}\lambda +2\mu \left|\mathcal{R}\right|}\right)\hfill \end{array}$$

(24)

Similarly, $-M_1,\dots ,-M_i,\dots$ also form a Martingale, so apply Lemma 3, we have:

$$\begin{array}{cc}\hfill Pr\left[\sum _{i=1}^{\left|\mathcal{R}\right|}{X}_g^i\left(S\right)-\left|\mathcal{R}\right|·\mu \le -\lambda \right]& \le exp\left(-\frac{{\lambda }^2}{2\mu \left|\mathcal{R}\right|}\right)\hfill \end{array}$$

(25)

Let $\lambda =ϵ\mu \left|\mathcal{R}\right|$ and put it in two above inequalities, we have:

$$\begin{array}{c}\hfill Pr[\sum _{i=1}^{\left|\mathcal{R}\right|}{X}_g^i-\left|\mathcal{R}\right|·\mu \ge ϵ\left|\mathcal{R}\right|\mu ]\le exp\left(-\frac{{ϵ}^2\left|\mathcal{R}\right|\mu }{2+\frac{2}{3}ϵ}\right)\end{array}$$

(26)

$$\begin{array}{c}\hfill Pr[\sum _{i=1}^{\left|\mathcal{R}\right|}{X}_g^i-\left|\mathcal{R}\right|·\mu \le -ϵ\left|\mathcal{R}\right|\mu ]\le exp\left(-\frac{{ϵ}^2\left|\mathcal{R}\right|\mu }{2}\right)\end{array}$$

(27)

The following Lemma shows the lower-bound of the influence of candidate solution $S_1$.

Lemma 4.

The candidate solution $S_1$ obtained by phase 1 of Algorithm 3 satisfies $Pr[{\sigma }_U\left(S_1\right)\ge T]\ge 1-\delta$.

Proof. 

Denote ${\mu }_Y=\mathbb{E}\left[Y_g\right]=\frac{{\sigma }_U\left(S_1\right)}{\left|U\right|}$, and ${\widehat{\mu }}_Y=\frac{1}{N_U}{\sum }_{i=1}^{N_U}{Y}_g^i=\frac{{\widehat{\sigma }}_U\left(S_1\right)}{\left|U\right|}\ge \frac{(T+\alpha T)}{\left|U\right|}$. Apply (27) for set ${\mathcal{R}}_1$, we have:

$$\begin{array}{cc}\hfill Pr[{\widehat{\mu }}_Y\le (1-\alpha ){\mu }_Y]& =Pr\left(\sum Y_g^i-N_U{\mu }_Y\le -\alpha N_U{\mu }_Y\right)\hfill \\ \hfill & \le exp\left(\frac{-{ϵ}^2{N}_U{\mu }_Y}{2}\right)\hfill \\ \hfill & \le exp\left(\frac{-{ϵ}^2{\widehat{\mu }}_Y{N}_U}{2(1-\alpha )}\right)\hfill \\ \hfill & \le exp\left(\frac{-{ϵ}^2(T+\alpha T)}{2(1-\alpha )\left|U\right|}{N}_U\right)\hfill \\ \hfill & \le exp\left(\frac{-(2+\frac{2}{3}\alpha )ln(\left(\genfrac{}{}{0pt}{}{\left|U\right|}{\lfloor \left|U\right|/2\rfloor }\right)/\delta )}{2(1-\alpha )}\right)\hfill \\ \hfill & \le exp\left(-ln(\left(\genfrac{}{}{0pt}{}{\left|U\right|}{\lfloor \left|U\right|/2\rfloor }\right)/\delta )\right)\le \frac{\delta }{\lfloor \left|U\right|/2\rfloor }\hfill \end{array}$$

We assume that the event ${\widehat{\mu }}_Y\le (1-\alpha ){\mu }_Y$ happens, apply (26) for set ${\mathcal{R}}_1$, we have:

$$\begin{array}{cc}\hfill Pr[{\sigma }_U\left(S_1\right)\le T]& \le Pr\left({\sigma }_U\left(S_1\right)\le \frac{{\widehat{\sigma }}_U\left(S_1\right)}{1+\alpha }\right)\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & =Pr\left({\widehat{\sigma }}_U\left(S_1\right)\ge (1+\alpha ){\sigma }_U\left(S_1\right)\right)\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & =Pr\left(\frac{\left|U\right|}{N_U}\sum _{i=1}^{N_U}{Y}_g^i-\left|U\right|{\mu }_Y\ge \left|U\right|\alpha {\mu }_Y\right)\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & =Pr\left(\sum _{i=1}^{N_U}{Y}_g^i-N_U{\mu }_Y\ge N_U\alpha {\mu }_Y\right)\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \le exp\left(-\frac{{\alpha }^2{\mu }_Y}{2+\frac{2}{3}\alpha }{N}_U\right)\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & \le exp\left(-\frac{{\alpha }^2{\widehat{\mu }}_Y}{(2+\frac{2}{3}\alpha )(1-\alpha )}{N}_U\right)\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & \le exp\left(-\frac{{\alpha }^2{\widehat{\sigma }}_U\left(S_1\right)}{(2+\frac{2}{3}\alpha )(1-\alpha )(T+\alpha T)}{N}_U\right)\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & \le exp\left(-ln(\left(\genfrac{}{}{0pt}{}{\left|U\right|}{\lfloor \left|U\right|/2\rfloor }\right)/\delta )\right)\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & \le \frac{\delta }{\left(\genfrac{}{}{0pt}{}{\left|U\right|}{\lfloor \left|U\right|/2\rfloor }\right)}\hfill \end{array}$$

(36)

Assume that $|S_1|=k_1$, there are at most $\left(\genfrac{}{}{0pt}{}{n}{k_1}\right)$ possibilities for the candidate solutions $S_1$. Therefore,

$$\begin{array}{c}\hfill Pr[\exists S_1:{\sigma }_U\left(S_1\right)\le T]\le \left(\genfrac{}{}{0pt}{}{n}{k_1}\right)\frac{\delta }{\left(\genfrac{}{}{0pt}{}{\left|U\right|}{\lfloor \left|U\right|/2\rfloor }\right)}\le \delta \end{array}$$

(37)

 ☐

Lemma 5

(Lower-bound). For any $\delta \in (0,1)$, a set of RR samples $\mathcal{R}$, let $c=ln\left(\frac{1}{\delta }\right)$, and

$$\begin{array}{cc}\hfill & F_l(\mathcal{R},S,\delta )=min\left\{\widehat{\sigma }\left(S\right)-\frac{nc}{3\left|\mathcal{R}\right|},\widehat{\sigma }\left(S\right)+\frac{n}{\left|\mathcal{R}\right|}\left(\frac{2c}{3}-\sqrt{\frac{4c^2}{9}+2\left|\mathcal{R}\right|c\frac{\widehat{\sigma }\left(S\right)}{n}}\right)\right\}\hfill \end{array}$$

(38)

We have $Pr[\sigma \left(S\right)\ge F_l(\mathcal{R},\delta )]\ge 1-\delta$.

Proof. 

Denote $\mu =\mathbb{E}\left[X_g\left(S\right)\right]=\frac{\sigma \left(S\right)}{n}$ and $\widehat{\mu }=\frac{1}{n}{\sum }_{R_g\in \mathcal{R}}{X}_g\left(S\right)=\frac{\widehat{\sigma }\left(S\right)}{n}$. Apply (24) with $\lambda =\frac{c}{3}+\sqrt{\frac{c^2}{9}+2c\mu \left|\mathcal{R}\right|}$, we have:

$$\begin{array}{cc}\hfill Pr\left[\sum _{j=1}^T{X}_g\left(S\right)-\left|\mathcal{R}\right|·\mu \ge \lambda \right]& \le \delta \hfill \end{array}$$

(39)

Therefore, the following event happens with probability at least $1-\delta$

$$\begin{array}{c}\hfill \sum _{j=1}^T{Z}_j\left(S\right)-\left|\mathcal{R}\right|·\mu \le \lambda \iff \left|\mathcal{R}\right|\widehat{\mu }-\left|\mathcal{R}\right|\mu -\frac{c}{3}\le \sqrt{\frac{c^2}{9}+2c\mu \left|\mathcal{R}\right|}\end{array}$$

(40)

We consider two following cases:

Case 1: 

If $\left|\mathcal{R}\right|\widehat{\mu }-\left|\mathcal{R}\right|\mu -\frac{c}{3}\le 0$, then $\mu \ge \widehat{\mu }-\frac{c}{3\left|\mathcal{R}\right|}$.

Case 2: 

If $\left|\mathcal{R}\right|\widehat{\mu }-\left|\mathcal{R}\right|\mu -\frac{c}{3}>0$, (40) becomes:

$$\begin{array}{cc}\hfill & {\left(\left|\mathcal{R}\right|\widehat{\mu }-\left|\mathcal{R}\right|\mu -\frac{c}{3}\right)}^2\le \frac{c^2}{9}+2c\mu \left|\mathcal{R}\right|\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \iff & {(\widehat{\mu }-\mu )}^2\left|\mathcal{R}\right|+\frac{4c}{3}(\widehat{\mu }-\mu )-2c\widehat{\mu }\le 0\hfill \end{array}$$

(42)

Solve the above inequality for $\mu$, we obtain:

$$\begin{array}{c}\hfill \mu \ge \widehat{\mu }+\frac{1}{T}\left(\frac{2c}{3}-\sqrt{\frac{4c^2}{9}+2\left|\mathcal{R}\right|c\widehat{\mu }}\right)\end{array}$$

(43)

Combine two above cases and replace $\mu =\frac{\sigma \left(S\right)}{n},\widehat{\mu }=\frac{\widehat{\sigma }\left(S\right)}{n}$, we obtain the proof. ☐

Lemma 6

(Upper-bound). For any $\delta \in (0,1)$, in an iterator t of Algorithm 3, denote ${\mathcal{R}}_2^t$ is a set of RR samples with $N_t=\left|{\mathcal{R}}_t\right|$, $S_2^t$ is a candidate solution of phase 2, and

$$\begin{array}{c}\hfill F_u({\mathcal{R}}_2^t,S_2^t,\delta )=\frac{\widehat{\sigma }\left(S_2^t\right)}{1-{\left(1-\frac{1}{k}\right)}^t}+\frac{n}{N_t}\left(\sqrt{c^2+2N_{t}c\frac{\widehat{\sigma }\left(S_2^t\right)}{\left(1-{\left(1-\frac{1}{k}\right)}^t\right)n}}-c\right)\end{array}$$

We have $Pr[\mathsf{OPT}\le F_u({\mathcal{R}}_2^t,S_2^t,\delta )]\ge 1-\delta$.

Proof. 

Let $\lambda =\sqrt{2c\mu N_t}$, apply inequality (25), we have:

$$\begin{array}{cc}\hfill Pr\left[\sum _{i=1}^{N_t}{X}_g^i\left(S\right)-\left|\mathcal{R}\right|·\mu \ge -\lambda \right]& \le exp\left(-\frac{{\lambda }^2}{2\mu \left|\mathcal{R}\right|}\right)\le \delta \hfill \end{array}$$

(44)

Therefore, the following event happens with the probability at least $1-\delta$:

$$\begin{array}{c}\hfill N_t\widehat{\mu }-N_t\mu \le -\sqrt{2c\mu N_t}\iff -N_t(\widehat{\mu }-\mu )\ge \sqrt{2c\mu N_t}\end{array}$$

(45)

Solve the above quadratic inequality for $\mu$, we obtain upper-bound for $\mu$ is,

$$\begin{array}{cc}\hfill \mu & \le max\left\{\widehat{\mu },\widehat{\mu }+\frac{1}{N_t}\left(\sqrt{c^2+2N_{t}c\widehat{\mu }}-c\right)\right\}\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & =\widehat{\mu }+\frac{1}{N_t}\left(\sqrt{c^2+2N_{t}c\widehat{\mu }}-c\right)\hfill \end{array}$$

(47)

Denote $S^0=arg{max}_{S,\left|S\right|\le k}\widehat{\sigma }\left(S\right)$, where $\widehat{\sigma }$ is calculated over ${\mathcal{R}}_t^2$. Since the phase of Algorithm 3 selects a candidate solution $S_2^t$ by a greedy strategy. Similar to Theorem 1, we have:

$$\begin{array}{c}\hfill \widehat{\sigma }\left(S_2^t\right)\ge {\left(1-(1-\frac{1}{k})\right)}^t\widehat{\sigma }\left(S^0\right)\ge {\left(1-(1-\frac{1}{k})\right)}^t\widehat{\sigma }\left(S^{*}\right)\end{array}$$

(48)

Replace $\mu =\frac{\sigma \left(S_2^t\right)}{n},\widehat{\mu }=\frac{\widehat{\sigma }\left(S_2^t\right)}{n}$ into (47) and combine it with (48), we have:

$$\begin{array}{cc}\hfill \mathsf{OPT}=\sigma \left(S^{*}\right)& \le \widehat{\sigma }\left(S^{*}\right)+\frac{n}{N_t}\left(\sqrt{c^2+2N_{t}c\frac{\widehat{\sigma }\left(S^{*}\right)}{n}}-c\right)\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & \le \frac{\widehat{\sigma }\left(S_2^t\right)}{{\left(1-(1-\frac{1}{k})\right)}^t}+\frac{n}{N_t}\left(\sqrt{c^2+2N_{t}c\frac{\widehat{\sigma }\left(S_2^t\right)}{n{\left(1-(1-\frac{1}{k})\right)}^t}}-c\right)\hfill \end{array}$$

(50)

which completes the proof. ☐

Based on above theoretical analysis, the following Theorem Approximation guarantee of $\mathsf{IGS}$ algorithm.

Theorem 2.

The Algorithm 3 provides a solution S and an integer t, satisfies:

• $Pr[{\sigma }_U\left(S\right)\ge T]\ge 1-\delta$
• $Pr[\sigma \left(S\right)\ge \left(1-{(1-\frac{1}{k})}^t\right)\mathsf{OPT}]\ge 1-\delta$

Proof. 

Since $S=S_1\cup S_2$ and Lemma 4, we have:

$$\begin{array}{c}\hfill Pr[{\sigma }_U\left(S\right)\ge T]\ge Pr[{\sigma }_U\left(S_1\right)\ge T]\ge 1-\delta \end{array}$$

We consider two following cases:

Case 1: 

If the algorithm stops with the condition $|{\mathcal{R}}_2^t|\ge N_{max}$, apply (26) with set $S^{*}$ and ${\mathcal{R}}_2$, we have:

$$\begin{array}{cc}\hfill Pr[\widehat{\sigma }\left(S^{*}\right)\le (1-{ϵ}_1)\sigma \left(S^{*}\right)]& \le exp\left\{\frac{-{ϵ}_1^2{N}_{max}\mathsf{OPT}}{2n}\right\}\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill & \le exp\left\{\frac{-{ϵ}_1^2{N}_{max}k}{2n}\right\}\left(\mathrm{Due} \mathrm{to} \mathsf{OPT}\ge k\right)\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill & \le exp\left\{\frac{-{ϵ}_1^2{N}_{max}{t}_0}{2n}\right\}\left(\mathrm{Due} \mathrm{to} k\ge t_0\right)\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill & \le {\delta }_2/\left(\genfrac{}{}{0pt}{}{n}{t_{max}}\right)\le {\delta }_2\hfill \end{array}$$

(54)

From (27), we have:

$$\begin{array}{cc}\hfill Pr[\sigma \left(S_2^t\right)\le \frac{\widehat{\sigma }\left(S_2^t\right)}{(1+{ϵ}_1)}]& \le exp\left\{\frac{-{ϵ}_1^2{N}_{max}\sigma \left(S_2^t\right)}{(2+\frac{2}{3}{ϵ}_1)n}\right\}\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill & \le exp\left\{\frac{-{ϵ}_1^2{N}_{max}t}{(2+\frac{2}{3}{ϵ}_1)n}\right\}\left(\mathrm{Due} \mathrm{to} \sigma \left(S_2^t\right)\ge t\right)\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill & \le {\delta }_2/\left(\genfrac{}{}{0pt}{}{n}{t_{max}}\right)\le {\delta }_2\hfill \end{array}$$

(57)

Apply an union probability that the events (54) and (57) happen with the probability at most ${\delta }_1+{\delta }_1=\delta /3$. Assume that they do not happen, we have:

$$\begin{array}{cc}\hfill \sigma \left(S_2^t\right)& \ge \frac{\widehat{\sigma }\left(S_2^t\right)}{1+{ϵ}_1}\ge \frac{{\left(1-(1-\frac{1}{k})\right)}^t\widehat{\sigma }\left(S^0\right)}{1+{ϵ}_1}\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill & \ge \frac{{\left(1-(1-\frac{1}{k})\right)}^t\widehat{\sigma }\left(S^{*}\right)}{1+{ϵ}_1}\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill & \ge \frac{1-{ϵ}_1}{1+{ϵ}_1}{\left(1-(1-\frac{1}{k})\right)}^t\sigma \left(S^{*}\right)\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill & =\left({\left(1-(1-\frac{1}{k})\right)}^t-\frac{2{ϵ}_1}{1+{ϵ}_1}{\left(1-(1-\frac{1}{k})\right)}^t\right)\sigma \left(S^{*}\right)\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill & \ge \left({\left(1-(1-\frac{1}{k})\right)}^t-\frac{2{ϵ}_1}{1+{ϵ}_1}(1-\frac{1}{e})\right)\sigma \left(S^{*}\right)\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill & \ge \left({\left(1-(1-\frac{1}{k})\right)}^t-ϵ\right)\sigma \left(S^{*}\right)\hfill \end{array}$$

(63)

Hence, in this case the algorithm satisfies approximation guarantee with probability at least $1-\frac{\delta }{3}$.

Case 2: 

If the algorithm stops at any iterator $i,i=1,2,\dots ,i_{max}$. At this iterator, the condition in line 19 is satisfied, apply Lemma 5 and Lemma 6, the following thing happens with the probability at least $1-2i_{max}{\delta }_2=1-2\delta /3$:

$$\begin{array}{c}\hfill \frac{\sigma \left(S_2^t\right)}{\mathsf{OPT}}\ge \frac{F_l(S_2^t,{\mathcal{R}}_2,{\delta }_2)}{F_u(S_2^t,{\mathcal{R}}_2,{\delta }_2)}\ge {\left(1-(1-\frac{1}{k})\right)}^t-ϵ\end{array}$$

(64)

Combine two above cases, the algorithm meets the approximation ratio condition with the probability at least $1-\delta /3-2\delta /3=1-\delta$. ☐

5. Experiments

In this section, we implement and compare our algorithm $\mathsf{IGS}$ to other influence maximization methods about the influence in general, the influence on priority nodes, running time and memory usage. The dataset includes several network databases with thousands or even millions nodes and edges (Table 1).

Table 1. Dataset’s statistics.

Database	#Nodes	#Edges	Types	Avg. Degree
netHEPT [15]	15 K	59 K	directed	4.1
ENRON [15]	37 K	184 K	directed	5
netPHY [15]	37 K	181 K	directed	13.4
DBLP [15]	655 K	2 M	directed	6.1
TWITTER RETWEET [42]	1 M	2 M	directed	4

5.1. Experimental Settings

All the implementations are on Linux machine with configurations are 2× Intel(R) Xeon(R) CPU E5-2697 v4 @ 2.30GHz and 4 × 16 GB DIMM ECC DDR4 @ 2400MHz.

Algorithm comparisons. Since $\mathsf{IMP}$ is an expansion of $\mathsf{IM}$, we compare $\mathsf{IGS}$ algorithm with several state-of-the-art $\mathsf{IM}$ algorithms including: $\mathsf{DSSA}$ [15], $\mathsf{BCT}$ [2], $\mathsf{OPIM}$-C [23]. In addition, we use the basic algorithm, Max degree ($\mathsf{Degree}$), which is the common baseline for information diffusion problems. In $\mathsf{IMP}$, there are two factors that impact the solution in practice: the budget (k) of selecting seed node and the priority set of nodes (U). As a result, these two factors also affect the algorithms. From the above observation, we conduct experiments under two settings: varies k and fixed T; varies T and fixed k.

The dataset. For experimental purpose, we choose 5 types of databases from various resources: NetHept, NetPhy, DBLP are citation networks, Email-Enron is communication network [15] and Twitter Retweet is online social networks [42]. The brief of these ones are described on Table 1. These databases are experimented because they are popular in information diffusion problems, especially used in the state-of-the-art algorithms what we are comparing.

Parameter Settings. Graphs are formatted as each edge $e=(u,v)\in E$ has the weight $w(u,v)$ formulated as $w(u,v)={\displaystyle \frac{1}{d_{in}\left(v\right)}}$ where $d_{in}\left(v\right)$ is the in-degree of node v [14,15,20].

For the first case, k is assigned with 150, 160, 170, 180, 190 and 200, respectively, while T is fixed at 100. In addition, set U is generated with 200 nodes. With the second case, the value of k is fixed at 500. U set includes about 1000 nodes. We change the value of T increasing from 100 to 500. In all experiments, we keep $ϵ=0.1$, $\delta =1/n$ according setting for $\mathsf{IM}$ algorithms [14,15,20] and $\alpha =0.01$.

5.2. Experimental Results

We install $\mathsf{IGS}$ to compare with state-of-the-art algorithms such as $\mathsf{BCT}$, $\mathsf{DSSA}$, $\mathsf{OPIM}-C$ and $\mathsf{Degree}$ then calculate the spread of influence on all nodes and to U, the priority set, $U\subset V$. Results are shown in following tables and figures.

The Influence. The Figure 2 and the Table 2 indicate $\mathsf{IGS}$ outperforms the others when influencing to priority nodes by a given threshold T.

Figure 2. Comparisons of Influence Spreading with $k=100\to 500$, T = 100 and U size = 200

Table 2. Comparisons about $\sigma \left(S\right)$ and ${\sigma }_U\left(S\right)$ between $\mathsf{IGS}$ and the others with k = 500, U size = 1 K and $T=100\to 500$.

	Dataset
	T		NetHept	Enron	netPHY	DBLP	RETWEET
$\mathsf{IGS}$	100	$\sigma \left(S\right)$	5666.16	14,267.40	1865.92	54,033.50	17,307.70
		${\sigma }_U\left(S\right)$	1482.04	1075.77	1192.84	1271.62	511.08
	200	$\sigma \left(S\right)$	5581.34	14,162.20	1805.26	53,553.90	18,581.50
		${\sigma }_U\left(S\right)$	1478.93	1079.74	1175.32	1267.52	491.35
	300	$\sigma \left(S\right)$	5645.40	14,284.80	1773.33	53,240.50	19,459.10
		${\sigma }_U\left(S\right)$	1476.08	1074.30	1153.32	1264.79	492.39
	400	$\sigma \left(S\right)$	5640.21	14,196.50	1688.53	52,918.80	18,832.20
		$\sigma \left(S\right)$	1468.48	1075.68	1125.69	1260.31	490.46
	500	$\sigma \left(S\right)$	5039.45	14,245.50	1593.66	52,130.90	228,801.00
		${\sigma }_U\left(S\right)$	1238.54	1079.28	1104.20	1252.70	994.40
DSSA		$\sigma \left(S\right)$	4098.63	9960.35	3230.27	58,197.7	38,253.7
		${\sigma }_U\left(S\right)$	1093.7	857.608	174.479	474.635	168.087
BCT		$\sigma \left(S\right)$	11,088.10	19,901.70	6675.95	117,197.00	77,316.90
		${\sigma }_U\left(S\right)$	1280.54	1701.60	386.49	474.635	159.77
OPIM-C		$\sigma \left(S\right)$	3779.09	19,326.3	6262.5	112,334	72,026.1
		${\sigma }_U\left(S\right)$	600.93	894.18	194.04	459.801	173.41
Degree		$\sigma \left(S\right)$	3824.44	19,349.10	6345.86	114,249	73,936
		${\sigma }_U\left(S\right)$	292.82	779.84	164	260.94	22.77

The above figure gives information about the influence values in case k changes from 150 to 200, U includes 200 nodes and the threshold T is 100. The terms “infU”, “inf” mean the influences to set U (${\sigma }_U\left(S\right)$ ) and to all nodes ($\sigma \left(S\right)$), respectively. These algorithms output differently on various databases. Looking at red bars, we can see $\mathsf{IGS}$ approximately affects the set U twice the value of the threshold T on most databases except Re-Tweet but still higher than T. Conversely, the influence on U of the remaining sharply fluctuate according to the databases. While $\mathsf{DSSA}$ and $\mathsf{BCT}$ influence on U over T with netHEPT and ENRON, they work quite low with the others. $\mathsf{OPIM}-C$ and $\mathsf{Degree}$ often affect U much lower than T. Besides, the $\sigma \left(S\right)$ of $\mathsf{BCT}$ is highest on netHEPT whereas the one of $\mathsf{IGS}$ keeps at top in all other cases. In general, the values of $\sigma \left(S\right)$ of $\mathsf{DSSA}$, $\mathsf{OPIM}-C$ and $\mathsf{Degree}$ have similarities with each others.

Besides, Table 2 describes the experiment while T comes from 100 to 500, k = 500 and enlarge U up to 1K nodes. This setting is to check the case when U is large and when the threshold T is incremental. Certainly, the condition that $k\ge T$ has to be maintained so we fixed k = 500. Looking at bold values, we can see although U and S both become large and T increments gradually, the influence on U of $\mathsf{IGS}$ is always significantly higher than T, even up to more than ten times. $\mathsf{DSSA}$, $\mathsf{BCT}$ and $\mathsf{OPIM}-C$ also give the outputs over threshold T in many cases, they still have values lower than T = 500 on netPHY, DBLP and RETWEET however. The ${\sigma }_U\left(S\right)$ of $\mathsf{Degree}$ is lowest, especially, is only 22.77 on Re-Tweet.

From Figure 2 and Table 2, we can see ${\sigma }_U\left(S\right)$ of $\mathsf{IGS}$ is significantly higher than T and produces better results than the state-of-the-art algorithms. This is because $\mathsf{IGS}$ always prioritizes affecting U until over the threshold T then affects other nodes as well even with large values of k, U size and T. The other algorithms show that they are not always possible to influence U to exceed the desired threshold. On the whole, the state-of-the-art $\mathsf{IM}$ algorithms cannot influence the given priority set as well as $\mathsf{IGS}$ can.

Running time.Figure 3 compares running time of these algorithms. They indicate time of $\mathsf{IGS}$ gives lowest values on netHEPT, ENRON and netPHY databases. Nevertheless, $\mathsf{IGS}$ stays at top 3 on DBLP while it costs highest running time on the remaining of the dataset to find 150 and 160 seed sets but return to top 3 at the other values of budget k. $\mathsf{IGS}$ only takes about 0.1 s to find out the seed set in most cases except RETWEET. Besides, the figures also give information about the other algorithms. First, $\mathsf{BCT}$ runs significantly slow on netHEPT than the others. This method often stays at top 3 or top 4 on ENRON, DBLP and RETWEET. Second, running time of $\mathsf{DSSA}$ and $\mathsf{IGS}$ look similiar, while that of $\mathsf{OPIM}$-C and $\mathsf{Degree}$ is usually higher than the above two algorithms. As the whole, $\mathsf{IGS}$’s running time gives the most stable results and usually runs around the 0.1-s mark.

Figure 3. Comparisons about Runtime (s) with k varies from 150 to 200 between $\mathsf{IGS}$ and the others.

The time of $\mathsf{IGS}$ is fast and stable because of parallel programming and this algorithm costs most of time to find out $S1$ while the loop to calculate $S2$ usually stops at 1–2 rounds. The TRR sampling technique also helps to quickly identify which seeds will affect to the priority U.

Memory Usage. The Table 3 illustrates the memory consumption of $\mathsf{IGS}$ and state-of-the-art methods including $\mathsf{DSSA}$, $\mathsf{BCT}$, $\mathsf{OPIM}-C$ and $\mathsf{Degree}$. The smallest numbers are highlighted in bold while the largest ones are in red. The output shows that $\mathsf{IGS}$ outperforms the others, especially on small databases with tens of thousands of nodes and from tens to hundreds of thousands of edges such as netHEPT, ENRON, and netPHY. $\mathsf{IGS}$ also consumes sharply less memory than $\mathsf{OPIM}-C$ and $\mathsf{Degree}$ when testing with larger databases such as DBLP and RETWEET. When $\mathsf{IGS}$ spends only more than 130 MB and more than 200 MB, $\mathsf{OPIM}-C$ and $\mathsf{Degree}$ spend about four times higher with DBLP and RETWEET, respectively. Besides, $\mathsf{DSSA}$ also results less expensive memory usage in all cases. $\mathsf{BCT}$ is less stable than $\mathsf{IGS}$ and $\mathsf{DSSA}$ because it works as $\mathsf{DSSA}$ does on ENRON, netPHY, DBLPB and RETWEET but suddenly costs the most memory in NetHEPT.

Table 3. Memory usage (MB) comparisons between $\mathsf{IGS}$ and the others.

Dataset	Algorithm	Budget k
		150	160	170	180	190	200
NetHEPT	IGS	9.90	9.90	9.90	9.89	9.89	9.95
	DSSA	22.84	22.84	22.84	22.84	22.84	22.84
	BCT	1023.79	1017.52	1021.60	1012.21	1020.18	1020.74
	OPIM-C	47.76	47.91	48.03	48.11	48.30	48.46
	Degree	49.14	49.18	49.48	49.68	49.86	50.13
ENRON	IGS	16.82	16.79	16.81	16.81	16.82	16.82
	DSSA	30.48	28.07	28.07	28.07	28.07	30.48
	BCT	30.35	30.35	30.39	30.39	30.39	30.39
	OPIM-C	27.16	27.20	42.00	27.22	27.25	27.30
	Degree	27.98	28.08	43.77	28.19	28.27	28.41
NetPHY	IGS	15.18	15.18	15.18	15.18	15.18	15.04
	DSSA	52.12	52.12	52.12	52.12	38.50	52.14
	BCT	34.82	34.82	34.82	34.82	34.82	34.80
	OPIM-C	87.88	88.39	88.92	89.31	90.26	90.51
	Degree	92.26	92.71	93.33	93.88	94.68	94.98
DBLP	IGS	138.66	138.66	138.66	138.66	138.66	138.66
	DSSA	152.90	152.87	152.87	152.91	152.91	152.83
	BCT	162.88	162.87	162.87	162.88	162.88	162.89
	OPIM-C	475.05	373.72	373.78	373.95	477.18	477.51
	Degree	500.87	395.00	394.26	395.35	504.52	505.26
RETWEET	IGS	214.67	214.67	214.67	214.67	214.67	214.67
	DSSA	253.14	253.14	253.14	253.14	253.14	253.14
	BCT	282.50	282.50	282.50	282.47	282.50	282.48
	OPIM-C	877.31	874.20	722.91	876.99	886.78	877.80
	Degree	918.53	916.23	756.93	920.00	930.33	921.95

TRR sampling technique focuses on finding the seeds that influence the priority U first then Algorithm 3 explores another seeds to push on the seed set. Hence the algorithm 3 saves memory to run loop more than the others because of must not check whether a seed node influences to U set or not. Moreover, the condition of $\frac{F_l(S_2,{\mathcal{R}}_2,\delta )}{F_u(S_2,{\mathcal{R}}_2,\delta )}\ge 1-{(1-\frac{1}{k})}^t-ϵ$ helps $S_2$ generated soon without waiting for the stop condition of the repeat.

Finally, our algorithm, $\mathsf{IGS}$, was designed very well to get a balance between the target to influence on the given priority set and the influence that has to propagate to the largest number of nodes. Hence, running time, memory used and the influence of $\mathsf{IGS}$ give significantly high results and even more steadily rather than the others in general.

6. Conclusions

In this paper, we investigate the $\mathsf{IMP}$ problem, which is a variant of the $\mathsf{IM}$ problem with priority constraint that arises in a realistic scenario in which companies or organizations often prioritize influencing potential users during their viral marketing campaigns. The goal of the $\mathsf{IMP}$ problem is to select a seed set with k nodes can influence of a given priority set U greater than a threshold T which adjusts the influence of the seed set to the priority set. Although the objective function (influence spread function) is still a monotone and sub-modular function, but when considering the priority constraint the state-of-the-art $\mathsf{IM}$ algorithms cannot be applied.

To address this challenge, we propose two algorithms with provable theoretical guarantees, called $\mathsf{IG}$ and $\mathsf{IGS}$. We show that $\mathsf{IG}$ provides a $\left(1-{(1-\frac{1}{k})}^t\right)$-approximation solution; $\mathsf{IGS}$ is an efficient randomized approximation algorithm based on sampling method that returns a $\left(1-{(1-\frac{1}{k})}^t-ϵ\right)$-approximation solution with probability at least $1-\delta$ with $ϵ>0,\delta \in (0,1)$ as input parameters of the problem. Experiments on real world social networks show our algorithm outperforms state-of-the-art $\mathsf{IM}$ algorithms including $\mathsf{DSSA}$ [15], $\mathsf{BCT}$ [2] and $\mathsf{OPIM}$ [23] in terms of influences, running time, and memory used.

In the future, we are going to improve our algorithm to expand it with large networks to billions scale with acceptable time. In addition, the problem with multiple priority user sets and thresholds is going to be considered.