1 Introduction
The prosperous development of online social networks (OSNs) has derived a number of famous social platforms, such as Facebook, LinkedIn, Twitter and WeChat, which have become the main means of communication. There are more than 1.52 billion users active daily on Facebook and 321 million users active monthly on Twitter. These social platforms have been taken as an effective advertising channel by many companies to promote their products through ”wordofmouth” effect. It motivates the researches about viral marketing. The OSNs can be represented as an undirected graph, where nodes are users and edges denote the friendships between two users. Viral marketing, proposed by Domingos and Richardson [2] [3], aims to maximize the followup users by giving rewards, coupons, or discounts to a subset of users who are the most influential. Then, Kempe et al. [4] formulated Influence Maximization (IM) problem as a combinatorial problem: selects a subset of users as the seed set under the cardinality budget, such that the expected number of users who are influenced by this seed set can be maximized. They proposed two information diffusion models that were accepted by most researchers in subsequent researches: Independent Cascade model (ICmodel) and Linear Threshold model (LTmodel), and proved IM is NPhard and its objective function is monotone submodular under these two models, thus, a good approximation can be obtained by natural greedy algorithm [5]. After this milestone work, a series of variant problems based on IM model and adapted to different application scenarios were emerged. For example, Revenue Maximization (RM), also called Profit Maximization, is a representative among them. RM problem is commonly studied in the literature [6] [7] [8] [9] [10] [11]. More different factors, such as product price, discount, cost [12] [13] [14] [15] and their impact on propagation, need to be considered in maximizing revenue.
From the perspective of companies, their objecitves are to maximize the exptected total revenue by selecting a subset of users from the whole network. However, most existing researches about RM problem were based on nonadaptive seeding strategy, where we select all seed nodes in one batch but do not consider the influence diffusion process. Thus, nonadaptive seeding is not the best choice to solve RM problem. Compared to that, adaptive one can response with a better seeding strategy because of making decision according to the realtime feedback from the users. Not only can it take advantage of the limited budget more wisely, but also adapt to the dynamic features of a social network. For example, the network topology is changing at any time because users join or leave, and friendships construct or terminate. In addition, most previous researches were based on the simple information diffusion model, such as ICmodel or LTmodel, where they considered the number of followup users that accept our information cascade. It applies user as a measurement unit to quantify the revenue, where each user corresponds to a fixed value of revenue. Sometimes, this is not enough and we need to apply activity as a measurement unit to quantify the revenue. Suppose some user in facebook is invited by a company to initiate an activity, after accepting and initiating this activity, the nerghbor users of this initiator would be infected. The initiator’s friends, or friends of friends, may be participate in this activity. Thus, Guo et al. [1] propose Collaborate Game model to characterize this scene, where the total revenue gained by the company is correlated to the number of successfully initiated activity. The number of participants is different among different activities. For a single activity, the revenue we can obtain from this activity is related to the number of its participants, but not simple linear relationship. In general, the closer the distance is from the initiator, the more business benefit this participant will provide.
Based on the Collaborate Game model, we propose the Revenue Maximization under the Size Budget (RMSB) problem. RMSB aims to maximize the total revenue by inviting a small number of users to initiate an activity in an adaptive strategy. We adopt adaptive seeding strategy because: before sending an invitation to a potential initiator, we do not know whether she will accept it and how many users around her will follow and join together. By adaptive strategy, we can observe the actual state of users and edges after sending an invitation. Besides, for a company, they should not only consider the largest revenue, but also consider the promotion effect of this product. According to that, we propose the Revenue Maximization under the Community Budget (RMCB) problem, where the number of invited users in each community of the targeted network is constrained. In this way, we can make sure there exists several activities happened in every area of the entire network, thus, the balance of influence distribution is guaranteed. Relied on the adaptive greedy policy, we design RMSBSolver and RMCBSolver algorithms to solve RMSB and RMCB respectively. The objective function of RMSB and RMCB problem is adaptive monotone and not adaptive submodular, but in some special cases, it is adaptive submodular. In this paper, the community budget of RMCB can be represented by use of partition matroid, and we can obtain a constant approximation ratio for RMCB problem under the special submodular cases. For the RMSB problem, it is an open question left by [1] how to get a theoretical bound for general nonsubmodular cases. In this paper, we attempt to solve it. Smith et al. [16] proposed the concept of adaptive primal curvature and adaptive total primal curvature. We give a bound for adaptive total primal curvature and then, with the help of that, we can obtain a datadependent approximation ratio for RMSB problem. Our contributions are summarized as follows:

Based on Collaborate Game model, we propose RMSB and RMCB problem, which are adaptive monotone but not adaptive submodular.

We design RMSBSolver and RMCBSolver algorithms to RMSB and RMCB problem, generalize RMCB to partition matroid, and obtain a approximation under the special submodular cases.

To RMSB problem, we prove the solution returned by RMCBSolver satisfies a datadependent approximation under the general nonsubmodular cases.

Our proposed algorithms are evaluted on realworld datasets, and it shows that our algorithms are effective and better than other baseline algorithms.
Organiztion: In Section 2, we survey the related work in RM and adaptive submodular optimization. We then present RMSB and RMCB problem in Section 3, discuss the algorthms in Section 4 and introduce the comprehensive theoretical analysis in Section 5. Finally, we conduct experiments and conclude in Section 6 and Section 7.
2 Related Work
Domingos and Richardson [2] [3] were the first to study viral marketing and the value of customers in social networks. Kempe et al. [4] studied IM as a discrete optimization problem and generalized ICmodel and LTmodel to triggering model, who provided us with a greedy algorithm and a constant approximation ratio. RM is an important variant problem of IM, and its existing researches focused on in the nonadaptive setting. [7] [8] studied the problem that selects quality seed users such that maximizing revenue with the help of influence diffusion. Lu et al. [7]
extended the LTmodel to include prices and valuation, who used a heuristic unbedgeted greedy framework to solve this problem. Tang et al.
[8] applied deterministic and randomized double greedy algorithms [17] to solve the RM problem. If the objective function is nonnegative and submodular, they obtained a  and approximation ratio respectively. Zhang et al. [9] investigated RM problem with multiple adoptions, which aimed at maximizing the overall profit across all products. Liu et al. [18] considered RM with coupons in his new model, independent cascade model with coupons and valuations (ICCV), and solved it based on local search algorithm [19]. Recently, Tong et al. [20] designed randomized algorithms, called simulationbased and realizationbased RM, to address the RM with coupons and achieved aapproximation with high probability. Guo et al.
[21] proposed a composed influence model with complementary products and gave a solution by use of sandwich approximation framework [22].In the adaptive setting, Golovin et al. [23] were the first to study the adaptive submodular optimization problem. Similar to the monotonicity and submodularity of set function, they extended these two concepts to adaptive version, adaptive monotonicity and adaptive submodularity, and proved that the solution returned by adaptive greedy policy is a approximation if the objective function is adaptive monotone and adaptive submodular. Applied it to social networks, Tong et al. [24] provided a systematic study on the adaptive influence maximization problem with different (partial or full) feedback model, especially for the algorithmic analysis of the scenarios when it is not adaptive submodular. Smith et al. [16] introduced two important concepts, adaptive primal curvature and adaptive total primal curvature, to obtain a valid approximation ratio for adaptive greedy policy when the objective function is adaptive monotone but not adaptive submodular. Further, when the objective function is not adaptive monotone, but adaptive submodular, Gotoves et al. [25] extended random greedy algorithm to adaptive policy and obtained a approximation. Other researches on the application of adaptive strategy can refer to [26] [27] [28] [29].
3 Problem and Preliminaries
In this section, we talk about some preliminary knowledges and concepts to the rest of paper, which maily includes model review and problem definition.
3.1 Model Recapitulation
The problem in this paper is an extension based on Collaborate Game model [1], thus, we need to review it briefly here. If you want to know more details about Collaborate Game model, please read [1]. The Collaborate Game model is defined in the targeted network, which is an undirected graph where is the set of users and is the set of edges. If the edge exists, it means user and are friend, and there is a probability representing their intimacy. For each user , we denote as the set of users who are the neighbor (friend) of .
Definition 1 (Collaborate Game [1]).
When a game company invites a user to play their game, she accepts this invitation to launch this game with probability . If she accepts it, we call her as ”initiator”, and in round , the initiator will invite each of her friend to participate in this game with success probability . We call user as 0hop participant and user who accepts the invitation from as 1hop participant. In round , each (i1)hop participant will invite each of her friend to participate in this game with success probability . Then, the process terminates until finishing round .
Then, we have a ”Tendency Assumption”: assuming that a user is an ihop participant to one initiator and a jhop participant to another initiator , where , at this time, we consider user will choose to be an ihop participant to initiator . There are two additional parameters associated to our model:

Acceptance vector
, where each element is an acceptance probability for user . It quantifies the likelihood for user to accept to be an initiator when she receives the invitation from the company. In this paper, we assumeis uniformly distributed in interval
. 
Revenue vector , where each element is the revenue the company can gain from an ihop participant. In this paper, we assume according to the ”Benefit Diminishing” assumption [1].
Remark 1.
The Collaborate Game model is based on such a scenario: A game company wants to promote their new multiplayer game over the targeted network by inviting some users to play this game. Once they accept it, they will attract their friends recursively to participate in this game. Then, game company can obtain revenue from it. Since we assume that the initiator’s influence range is at most khop from her, this model is called khop Collaborate Game model as well. Thus, the ”Benefit Diminishing” assumption means those participants whose are far from the initiator will provide less benefit to the company.
From the perspective of companies, they are not sure whether she will accept it before sending an invitation to a potential initiator, and do not know how many users around her will follow and join together. Therefore, they need to adopt an adaptive strategy. Before determining who should be the next potential initiator, they need to make an observation to the change of states of both users and networks from the last invitation. Given targeted network , for each user , the state of can be denoted by , where means user accepts to be an initiator because of the company’s invitation and means user rejects to be an initiator. means user is unknown, who did not receive an invitation from the company. The states of all users are at the beginning. Similarly, for each edge , the state of can be denoted by , where indicates user (resp. ) did not accept the invitation from user (resp. ) and indicates user (resp. ) is willing to play the game with user (resp. ). Once determined, it cannot be changed. indicates there is no invitation that happens between user and user . The states of all edges are at the beginning.
After defining the states of user and edge, we have a function be all possible states, which is called as realization. Thus, we say that is the state of user and is the state of edge under the realization . We denote by a random realization and by
the probability distribution over all realizations. Besides, each realization should be consistent. Here, each user can only be one of states in
, and each edge can only be one of states in identically. After each pick, our observations so far can be represented as a partial realization , which is a function from observed objects to their states. Then, is the domain of , that is, the observed objects in . A partial realization is consistent with a realization if they are equal everywhere in the domain of , denoted by . If and are both consistent with some , and , we say is a subrealization of , denoted by . Besides, we denote the observed users in the domain of by , and the observed edges in the domain of by . [1]3.2 Problem Definition
At each pick, the game company attempts to send an invitation to a user according to current partial realization , and observe the ’s state . If , we need to update , in other words, update the states of edges whose distance are less than khop from user . In this process, user attracts her nearby users to play together. If , we do nothing and go to the next pick. An example is shown as Fig. 1. Let be such an adaptive policy, and be the set of users who are invited by the game company according to policy under the realization . The total revenue gained according to policy under the realization can be defined as follows: [1]
(1) 
where and is the set that contains all ihop participants to the company according to policy under the realization , thus, we have
(2)  
(3) 
Finally, we can evaluate the performance of a policy by its expected revenue, and we have
(4) 
where the expectation is taken with respect to . Then, under the khop Collaborate Game model, the Revenue Maximization under the Size Budget (RMSB) [1] is formulated as follows:
Problem 1.
Given a targeted network , an acceptance vector , a size budget and a revenue vector , we aim to find a policy such that maximizing the expected revenue under the khop Collaborate Game model, , subject to for all realization .
However, a size budget is not enough because not only we want to maximize the revenue, but also need to consider the influence distribution. Generally, there exists a community structure given a social network, which is a partition of this network. Given a targeted network , we assume it exists a unique community structure as associated with , where is a partition of . In other words, it means that and for any . Denote community budget by vector , . It means that the number of users that invited to be initiators by the company in each community cannot be larger than budget . Then, under the khop Collaborate Game model, the Revenue Maximization under the Community Budget (RMCB) is formulated as follows:
Problem 2.
Given a targeted network , a community structure associated with , an acceptance vector , a community budget and a revenue vector , we aim to find a policy such that maximizing the expected revenue under the khop Collaborate Game model, , subject to for any and all realization .
Community budget in RMCB problem can be generalized to the matroid constraint, and we introduce some besic concepts about matroid here. A matroid
is an order pair
, where is the ground set, node set in this paper, and is the collection of independent sets, which satisfies: (1) For all , if then ; (2) For all with , we have such that . Given a matroid , the matroid constraint means that a feasible solution . The bases of matroid, denoted by , are those satisfy and such that . All bases of a matroid have the same size. There is a kind of special matroid, Partition Matroid, that is related to our problem: The ground set is partitioned into disjoint sets , community structure in this paper, where . Let , community budget in this paper, be an integer with for any . is an indepndent set, , if and only if for any . The matroid whose independet sets are defined in this form is called partition matroid. Therefore, we want to find such that for all realization , and(5) 
where is a collection of independent sets defined by partition matroid we talk about before.
4 Algorithm
In this section, we propose our algorithms to solve RMSB and RMCB problem. Adaptive greedy policy, proposed by Golovin et al. [23], is an effective method to solve such problems. It can be divided into two steps at each iteration:

Send an invitation to the user who has the largest expected increment of revenue based on the partial realization .

Observe the state of this user, if she accepts this invitation, update the states of edges whose distance within khop from her; Otherwise, back to (1) to invite the next user.
The Conditional Exepected Marginal Benefit under the adaptive surrounding can be shown as follows:
Definition 2 (Conditional Expected Marginal Benefit).
Given a partial realization and an user , the conditional expected marginal benefit of conditioned on observed is
(6) 
In our RMSB and RMCB problem, is the expected revenue increment of user by sending an invitation to initiate user . It conditions on the previous observation to users and edges, partial realization , and the expectation is taken over all realization that are consistant with .
Based on adaptive greedy policy [23], RMSBSolver and RMCBSolver is proposed, shown as Algorithm 1 and Algorihtm 2. In algorithm 1, we invite a user with the largest expected revenue increment at each iteration until the number of invitation is larger than size budget . However, in Algorithm 2, it is more complicated. At each iteration, we need to check whether the candidate set is empty. This candidate set is determined by the collection of independent set , shown as Equation (5), given community budget . If this candidate set is empty, it means that the number of invitations at each community reaches its budget constraint, thus, stop inviting at that time.
At each iteration, we need to compute the value of given current partial realization, which can be done by MonteCarlo simulation: running this diffusion process many times and then take the average of them. In order to achieve satisfactory accuracy, the number of simulations cannot be too small, however, it increases the running time greatly. To improve its scabability, a simplified computing method was proposed in Algorithm 2, Section 4.2 of [1]. Its core idea is computing the value of by use of a technique similar to breathfirst searching instead of Monto carlo simulation. It fixes all potential ihop participants for in advance, ignores some cases where the probability of occurrence is low and computes the probability of potential participants to join this game from up to down. The time complexity to compute each is reduced to . The effectiveness of this method was validated by experiments on realworld dataset, and it reduces the running time significantly under the premise of ensuring accuracy. Besides, similar to [1], we do not need to compute for each user repeatedly at each iteration. If, at iteration , the selected user at the last iteration rejects the invitation, we do not need to update any . Otherwise, we only need to update for those users whose distences from is less than or equal to at iteration . It improves the efficieny of Algorithm 1 and Algorithm 2 further.
5 Theoretical Analysis
In this section, we discuss some related conclusions about RMSB and RMCB problem, and then extend them to generalized nonsubmodular cases.
5.1 Related Conclusions
Before starting our discussion, we introduce two important concepts, defined in [23], as follows:
Definition 3 (Adaptive Monotone).
A function is adaptive monotone with respect to distribution if the conditional expected marginal benefit of any user is nonnegative, i.e., for all with and all , we have
(7) 
Definition 4 (Adaptive Submodular).
A function is adaptive submodular with respect to distribution if the conditional expected marginal benefit of any user does not increase as more states of users and edges are observed, i.e., for all and such that is a subrealization of () and all , we have
(8) 
Our RMSB and RMCB problem are two instances of adaptive optimization problem, thus, determined theoretical bounds can be obtained for both instances by adaptive greedy policy if our objective function, Equation (4), is adapative monotone and adaptive submodular. Based on [1], we have seveal conclusions as follows:
Theorem 1.
The RMSB and RMCB problem is NPhard, and there are no solutions with polynomial time unless NPP.
Proof.
The RMSB is NPhard [1], because it can be reduced to Maximum Coverage problem when setting revenue vector , acceptance vector and for each . Then, when there is only one community in targeted network
, and community budget is a scalar, at this moment, RMCB can be reduced to RMSB problem. Thus, RMCB problem is NPhard as well. ∎
Lemma 1.
The objective function of the RMSB and RMCB problem is adaptive monotone.
Lemma 2.
The objective function of the RMSB and RMCB problem is not adaptive submodular.
Lemma 3 ([1]).
If the RMSB and RMCB problem conform one of following two special cases:


For all ,
The objective function is adaptive submodular.
Those properties suitable to RMSB problem can be applied to RMCB problem, because their objective function is identical, but constraints are different. Therefore, for RMSB problem, we have
Theorem 2.
The adaptive greedy policy , given by Algorithm 1, for RMSB problem is a approximate solution when or for each . Hence, we have
(9) 
where is optimal policy s.t. for all .
Proof.
From Lemma 1, the objective function is adaptive monotone, and from Lemma 3, is adaptive submodular when or for each . The adaptive greedy policy is a approximation under the size constraint according to the conclusion of [23]. ∎
In [30], Golovin et al. analyzed the theoretical performance of adaptive submodular optimization under pindependence system. The main conclusions are summarized as follows:
Lemma 4 ([30]).
Given an adaptive monotone and submodular function and a pindependence system , the adaptive approximate greedy policy under the constraint yields and approximation. Hence, we have
(10) 
where is optimal policy s.t. for all .
Therefore, for RMCB problem, we have
Theorem 3.
The adaptive greedy policy , given by Algorithm 2, for RMCB problem is a approximate solution when or for each . Hence, we have
(11) 
where is optimal policy s.t. for any and all .
Proof.
From Lemma 1, the objective function is adaptive monotone, and from Lemma 3, is adaptive submodular when or for each
. As we said before, the constraint of community budget can be classified to partition matroid. The independent set
in Lemma 4 can be defined by Equation (5). It is 1independence system, thus, we have . And the adaptive greedy policy is exact, which means that . According to Lemma 4, Equation (10), we have finally. ∎5.2 NonSubmodularity
In addition to the special cases we discuss in the last subsection, the objective function of RMSB problem is not adaptive submodular, thus, the approximation ratio in Theorem 2 cannot be held in general cases. In order to deal with the general cases that is not adaptive submodular, we get help from the concept of adaptive primal curvature, proposed by [16], to obtain the approximation bound for adaptive greedy policy without adaptive submodularity. Wang et al. [31] were the first to propose the concept of elemental curvature, which is the maximum ratio of the marginal gain of an element at any subset and for any element . Extended to adaptive case, adaptive primal curvature [16] was formulated, which is the ratio of the marginal gain of element under partial realization and , where is possible state for any element . Thus, we have
Definition 5 (Adaptive Primal Curvature [16]).
Given a adaptive momotone function , the adaptive primal curvature is
(12) 
where is the state set of and is conditional expected marginal benefit, defined as Definition 2.
From Equation (12), the measures the changing of the conditional expected marginal benefit because of added to the solution previously. Under the background of RMSB and RMCB problem, the meaning of is a little different. Here, and represent the users that are invited to be initiators. Let us look at the term: , can be or . When , means we only add state to . But when , the meaning of will be more complicated, where we not only add the state to , but also all state change of edges since becomes an initiator should be added to . Thus, the expectation of primal curvature is with respect to the acceptance probability of and , and edge probabilities. In the nonadaptive case, is determined, the elemental curvature is the maximum primal curvature. To measure the total change form partial realization to , total primal curvature [16] is defined as follows:
Definition 6 (Adaptive Total Primal Curvature [16]).
Let and let represents the set of possible state sequences changing from partial realization to . Then the adaptive total primal curvature is
where and
which is a simplified notation of the adaptive primal curvature corresponded to a single state .
In the following, we can find a constant upper bound of the adaptive total primal curvature for the RMSB problem and a relationship between any policy with budget and adaptive greedy policy with any budget. Then, according to that, the approximation ratio of general RMSB problem can be found later.
Lemma 5.
The adaptive total primal curvature for any , and is upper bounded by , in the general cases for RMSB problem, where
(13) 
Proof.
The adaptive total primal curvature can be denoted by . Let us assume sequence , we have
From above, the product for any sequence from partial realization to can be reduced to trivially. Then, we have
(14) 
where is the marginal gain assuming that user accepts to be the initiator. For , the revenue from user is at most and from each of user in is at most . Considering an extreme example, there exists an edge from user to each of user and the edge probability is equal to for each edge. Thus, the total revenue because of selecting as initiator could be at most . For , we need to know the current participation role of user , maybe user is a xhop participant, before selecting as initiator. The revenue from user is equal to . Based on Benefit diminishing assumption, , the revenue from user is at least . Considering an extreme example, all neighbors of user have been 1hop participants before selecting as initiator, there is no revenue for other users. Thus, the total revenue because of selecting as initiator could be at most . Combined with the above analysis, the Lemma is held. ∎
Let us denote by any policy that inviting exact users, and by the adaptive greedy policy that inviting users. Besides, we define as the user who is invited to be an initiator in adaptive greedy policy. Then, we can bound the adaptive greedy policy that inviting users with any policy that inviting exact users.
Lemma 6.
The difference of expected revenue bewtween any policy and adaptive greedy policy in the general cases for RMSB problem can be bounded by
(15) 
where we assume and is defined as .
Proof.
We know because is adaptive monotone [23]. From it, we have
The difference of expected revenue between and can be bounded by running after running . It means that we need to send invitations again. Supposing that is the partial realization generated by , the initiators selected by is on partial realization such that . Then, we have if and , because from Lemma 5. Therefore,
where the first inequality is from seeding invitations again, and other equality from Definition 2 and Equation (4). ∎
So far, we can obtain the main theorem in this paper that gets the approximation of RMSB problem in the general cases that is not adaptive submodular.
Theorem 4.
The approximation performance of Adaptive greedy policy , given by Algorithm 1, in the general cases for RMSB problem satisfies the following:
(16) 
when we hypothesize .
Proof.
According to the Lemma 6, can be bounded by , we have
(17)  
(18) 
where and . We note that . Multiply both side by of Inequality (17) and sum from to . The left hand side of Inequality (17) can be reduced to
Similarly, the right hand side of Inequality (17) can be written as follows:
(19) 
Then, (19) can be reduced to
Rearrange and separate the term , then combine its coefficients, we have
Here, if we assume
(20) 
Then, we have
Therefore, combining with above, we have
(21) 
Then, we can obtain the conclusion, Inequality (16), when from Inequality (21). ∎
5.3 Further Estimation
From the Lemma 5 in the last subsection, it gives us an upper bound of for any , and
. However, this estimation is too rough to get a valid approximation ratio. A natural question is whether we can obtain a tight upper bound. For
, there exists a more precise solution which depends on the structure of network dataset. Given a user , all the potential following participants can be found by breathfirst searching, here we assume the edge probability in network is equal to . Then, we can determine the hop of these followers from this initiator . The maximum revenue from user and her following participants can be known. Finally, we choose the largest one among all users as the upper bound of . Given targeted network and revenue vector , the calculation steps are as follows:
For each user , we find all potential followers by breathfirst searching, defined as , where and , , is the set of potential jhop participants to user . It means that the length of the shortest path from a user in to user is equal to .

For each user , we compute the maximum possible revenue from and her followers, which can be expressed as , where
(22) 
Finally, we select the largest as the upper bound of .
Thus, from above process, we can get a estimated vector , which can be computed directly according to the graph dataset. Then, we have
Lemma 7.
The adaptive total primal curvature for any , and is upper bounded by , in the general cases for RMSB problem, where
(23) 
The value of is datadependent. When is smaller and the degree distribution is more uniform, the upper bound is more tight. It is in line with our intuition.
6 Experiment
In this section, we conduct several experiments to validate the correctness and efficiency of our proposed algorithms on real datasets. It evaluates Algorithm 1 and 2 with some common used baseline algorithms.
6.1 Dataset Description and Statistics
Our experiments are relied on the datasets from networkrepository.com [32], an online network repository. There are three datasets used in our experiments: (1) Dataset1: A coauthorship network, where each edge is a coauthorship among scientists to publish papers in the area of network theory. (2) Dataset2: A Wiki network, which is a whovotesonwhom network collected from Wikipedia. (3) A Collaboration network extracted from Arxiv General Relativity. The statistics information of the three datasets is represented in table I.
Dataset  n  m  Type  Average degree 

Dataset1  0.4K  1.01K  undirected  4 
Dataset2  1.0K  3.15K  undirected  6 
Dataset3  5.2K  14.5K  undirected  5 
6.2 Experimental Settings
As mentioned earlier, our proposed algorithms are based on the following parameters: Hop number (at most k hop participants follow an initiator), Acceptance vector , Revenue vector , budget and edge probability. There are two experiments we perform to test Algorithm 1 and 2 used to solve RMSB and RMCB problem, called submodualr performance and nonsubmodular performance. Shown as Lemma 3, the objective function of RMSB and RMCB is adaptive submodular when or for all . Thus, for submodular performance, we set (1) edge probability for all and ; (2) edge probability for all and . And in Algorithm 1 and 2, we estimate the Conditional Expected Marginal Benefit for each by the simplifed method of Section 4.2 in [1]. For nonsubmodular performance, we set (1) edge probability for all and ; (2) edge probability for all and . We conduct these two experiments on the three datasets shown as above, and analyze their experimental results.
For submodular performance and nonsubmodular performance, we observe the outcomes of our Algorithm 1, Algorithm 2 and some common used heuristic algorithms, then we compare the results of their performance. It aims to evaluate the effectiveness of the adaptive greedy policy on adaptive (non)submodular cases under the size and community budget. The Revenue vector can be set as: (1) when ; (2) when ; (3) when . It means that the revenue of 0hop participant is 8 units, 1hop is 6 units, 2hop is 4 units and 3hop is 2 units. The baseline algorithms are shown as follows:

MaxDegree: Invite the user with maximum degree at each step within budget .

Random: Invite a user randomly from at each step within budget .

MaxProb: Invite the user with maximum acceptance probability at each step within budget .
We use python programming to test each algorithm. The simulation is run on a Windows machine with a 3.40GHz, 4 core Intel CPU and 16GB RAM.
6.3 Experimental Results
In our experiments, the whole datasets are considered as the targeted networks. These adaptive algorithms, including our proposed algorithms and other baseline algorithms, are run 50 times on each network, and we take the average of them as the final results. To size budget, it is easy to understand that the number of invitations is limited as . However, for RMCB problem, we need to know how to define this community budget . Here, we adopt such a strategy: Suppose the total number of invitations in budget is predefined, the budget for each community can be represented as follows: Given and budget vector , we have
(24) 
It is possible that is not equal to the total invitations of . We need to make some adjustments such that is equal to the total invitation of : add one or mimus one to each community’s budget from large community to small community until satisfying equality. Let us look at a specific example:
Example 1.
For submodular performance, the experimental results achieved by different algorithms on these three datasets are shown as Fig. 2 and Fig. 3. The left columns of Fig. 2 and Fig. 3 are under the size budget, and we can see that the expected total revenues returned by RMSBSolver is larger than other policies. The right columns of Fig. 2 and Fig. 3 are under the community budget, and the performance of RMCBSolver is better than other policies as well. Especially on Dataset3, the gap between RMSBSolver and other baseline algorithms is very significant, which may be related to the size of the dataset and the network structure. The performance of baseline algorithm, such as MaxDegree, is on and off, good for Dataset2 and bad for Dataset3. Thus, we cannot predict whether it is good or bad in advance. An interesting discovery is that, intuitively, the total revenue of RMCB should be smaller than that of RMSB, because the constraint of RMCB is more strict and the approximation performance of RMSBSolver is better than that of RMCBSolver. But, from the results, their expected total revenues are very close regardless of adaptive greedy strategy or other heuristic policies, even sometimes the revenues obtained under the community budget are better. For example, on Dataset3, the performance of MaxDegree algorithm under the community budget is apparently better than that under the size budget. Therefore, considering the distribution of influence and the average revenue comprehensively, community budget is a more sensible choice to us.
For nonsubmodular performance, the experimental results achieved by different algorithms on these three datasets are shown as Fig. 4. We can see that the expected total revenues returned by RMSBSolver is larger than other policies. Same as before, on Dataset3, the gap between RMSBSolver and other baseline algorithms is very significant. Even though the objective function of RMSB is not adaptive submodular, from the actual performance, this effect does not seem obvious. In other words, the objective function of RMSB is close to adaptive submodular. According to Lemma 7, the bound of these three datasets are shown as follows:
Dataset1  Dataset2  Dataset3  

106  310  247  
199  1060  820 
Unfortunately, this results is frustrating. Even if the bound given by Lemma 7 is much smaller than that given by Lemma 5, it is still too large to get a satisfactory approximation ratio. Thus, we need to look for a more compact estimation method of bound further so as to get a practical and meaningful approximation ratio.
7 Conclusion
In this paper, we propose RMSB problem based on Collaborate Game model. In order to consider the distribution of influence and total revenue, we extend RMSB to RMCB problem by use of community budget. We proposed RMSBSolver and RMCBSolver to address both of them according to the adaptive greedy policy. The objective function of RMSB and RMCB is adaptive monotone and not adaptive submodular, but adaptive submodular in some special cases. Then, we reduce the community budget of RMCB to partition matroid, which can be solved within approximation under the special submodular cases. For RMSB problem under the general nonsubmodular cases, we give a datadependent approximation through bounding the adaptive total primal curvature by . The good performance of our algorithms is verified by our experiments on three real network datasets. The performance under the community budget is not worse than that under the size budget, thus, community budget is a better choice. However, the bound is not satisfactory, in future, we need to improve it further such that getting a smaller one.
Acknowledgments
This work is partly supported by National Science Foundation under grant 1747818.
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