Scalable CostAware MultiWay Influence Maximization
Abstract
I Introduction
Viral marketing is different from other marketing strategies since it leverages the influence power in intimate relationship, e.g., close friends, family members, couples. Due to the development and popularity of social networking services, such as Facebook, Twitter, and Pinterest, the new notion of “social media marketing” has appeared in recent years and presents new opportunities for enabling largescale and prevalent viral marketing online. To boost the growth of their sales, business is embracing social media in a big way. According to USA Today, the sales of software to run corporate social networks will grow 61% a year and be a billion business by 2016^{1}^{1}1http://usatoday30.usatoday.com/money/economy/story/20120514/socialmediaeconomycompanies/55029088/1. On the other hand, general advertisement channels such as TV, and newspaper are not dead yet. While there has been a significant drop since the rise of the Internet age, 116.3 million Americans have a television according to a Nielsen 2014 report.^{2}^{2}2http://www.nielsen.com/us/en/insights/news/2014/nielsenestimates1163milliontvhomesintheus.html Despite of the prevalence of social media marketing, TV is still an important traditional marketing method companies should consider advertising on. Advertising efforts on TV or radio has the benefit of reaching a mass audience. A recent line of research also focuses on how to model and quantify the influence from external (e.g., TV, Online News) and internal (i.e., friends, followees) exposures together [8, 4].
Consider the following scenario as a motivating example. A telecom firm attempts to market the new plan through multiple ways, e.g., TV commercial, social media, cold calls. The company has limited budgets on each advertising way such that it can only broadcast the TV commercials several times or select some initial users in the online network to adopt it (by giving them discount or free phone accessories). The ideal case of the company is that the TV audiences love the advertisement and adopt the new plan or selected seed users love the new plan. Afterward, the initial users start canvassing their friends for the new plan on the social network, and their friends will influence their friends’ friends and so on, and thus through the viral marketing a large population in the social network would adopt the new plan.
The above marketing strategy can be regarded as a combination of traditional media marketing and social media marketing. The problems are how many the budgets should be allocated to each advertising way and whom to select as the initial users so that they eventually influence the largest number of people in the network. Moreover, since the goal is to maximize the revenue, it is desirable to construct a precise cost model. Take the telecom case as an example, when users became the member of the company, the fees may be reduced due to the discounts on all calls made within the intranetwork.
With this objective in mind, we formulate a new fundamental optimization problem, named CostAware MultiWay Influence maXimization (CAMAIX). The problem is given a social graph , where each node represents a candidate person and is associated with an activation probability vector of traditional media, and each edge has a social influence probability to indicate the influence power between the two persons. Given the userspecified budget upper bound for each advertising way and the precise cost model, the goal of CAMAIX is to automatically allocate budgets and select seed users which maximizes the total revenue.
There are three major challenges of CAMAIX: i) The spread maximization problem in the Independent Cascade (IC) model [6] suffers from the expensive computation problem since the difficulty of the influence spread given a seed set is hard. Also, instead of finding an exact algorithm, MonteCarlo simulations of the influence cascade model run a large number of times in order to obtain a correct estimation of the spread. ii) The number of seed nodes is nonfixed, which is different from traditional influence maximization problem and complicates the problem. Let denote the number of nodes in . The enumeration approach for selecting seed nodes needs to evaluate candidate groups, whereas the enumeration approach for selecting nonfixed seed nodes is .^{3}^{3}3It is worth noting that the cost of activating seed nodes is considered so that the total revenue will be reduced when wasting budgets on users who are really not willing to use. iii) The budget allocation problem needs to deal with the interplay between different advertising ways and be extended to adopt more complicated real settings.
Aiming to efficiently solve the multiway influence maximization with more sophisticated real settings, we systematically explore various heuristics, including Social Influence Pruning (SIP) and Adaptive Budget Allocation (ABA), to design our algorithm Intermediate Seeds Selection with Budget Allocation (ISSBA). The idea of SIP is to incrementally construct the best seed set by maintaining a number of intermediate subsets. Therefore, by iteratively expanding the best intermediate seed sets from the subsets obtained in previous iterations, SIP finds seed sets with good quality efficiently. Also, we prove the performance bound of the proposed algorithm is . On the other hand, given the upper bound of the budget for each advertising way, ABA efficiently calculates the optimal budget for each advertising way via dynamic programming.
The contribution of this paper is listed as follows.

We formulate a new optimization problem, namely CAMAIX, to consider the multiway influence maximization with a sophisticated cost model, which is hard. To the best of the our knowledge, there is no real system or existing work in the literature that efficiently addresses the issue of multiway influence maximization based on real settings.

We design Algorithm ISSBA to find the solution to CAMAIX with an approximation ratio. Experimental results demonstrate that the solution returned by ISSBA it perform the baseline algorithms in both solution quality and execution time on the largescale datasets.
Ii Preliminary
Iia Problem Definition
Let denote the seed set. A user has probability to be activated as a seed, i.e., , if the advertisement is sent to through a multiple advertising ways as follows.
(1) 
The activation probability of broadcasting advertisements , such as TV and billboard, is the product of the probability that the advertisement broadcasts to user and the probability of user being activated by broadcasting advertisement. If a user is activated, the propagation starts from to his neighbors with probability . Moreover, let denote the cost for each advertising way on node , where is the advertise way set. Notice that for broadcasting advertisements is the total advertisement expense divided by the number of seed nodes .
Given a directed social network , where node and each edge are associated with an initial fee and an influence probability that user activates respectively, an advertising way set , and a budget and a cost for each advertising way on node , this paper studies a new optimization problem called CostAware MultiWay Influence Maximization (CAMWIM) for finding the optimal budget and the seed set of vertices to maximize the revenue , i.e.,
where and represent the probability that the user will be activated with the seed set and the discount of activating related to its neighbor , respectively. The discount is suitable for many different scenarios, such as telecom (intranetwork free) and direct sale (agent commission), and is set as for nodiscount cases.
IiB Related Work
Influence maximization is to find a set of influential nodes, which are targeted as initial active nodes, to maximize the spread. The problem has been connected to the Independent Cascading (IC) model and the Linear Threshold (LT) Model models in [6]. D. Kempe et al. [6] show that the influence maximization problem is NPHard and propose a greedy algorithm for both IC and LT models, with the guarantee of the solution quality. However, the greedy algorithm needs Monte Carlo simulations to estimate the expected spread, which is time consuming. J. Leskovec et al. [7] proposes CELF to further speed up Monte Carlo simulations. Nevertheless, for large scale social networks, CELF is still not efficient enough. Several heuristic methods are proposed, such as degree discount [3], PMIA [2], and IRIE [5], to find initial active nodes very efficiently.
Iii CostAware Influence Maximization
To tackle CAMAIX, a basic approach is to enumerate all possible seeds and combinations of budgets, and retrieve the one with largest revenue. However, the enumerative approach is not scalable since there are combinations for seed selection. To address the challenges, we propose a framework called Intermediate Seeds Selection with Budget Allocation (ISSBA) including Social Influence Pruning (SIP) and Adaptive Budget Allocation (ABA). SIP iteratively expands the best intermediate seed sets from the subset obtained in previous iterations. Moreover, we leverage the merit of MIA model to efficiently approximate the computation of MonteCarlo simulation. Finally, ABA exploits dynamicprogramming for allocating the budgets adaptively.
Iiia Social Influence Pruning with Quality Guarantee
Here, we describe our proposed SIP in detail. SIP first constructs all oneitem subsets , where each contains exactly one item . The corresponding join group for each are computed and the best subpacks with the largest join groups are reserved ^{4}^{4}4The calculation process to obtain each is described in LABEL:wp later. For ease of understanding, we rename the best subpacks as , , ….. IPO then generates 2item subpacks by adding every possible items into the best separately. For example, IPO expands into subpacks by adding each item into . Note that during the generation of subpacks, multiple subpacks that contain the same items may be generated. IPO discards those additional duplicate subpacks. Similarly, IPO computes the join group for each 2item subpacks and reserves the ones with the largest join groups for generating 3item subpacks ^{5}^{5}5For ease of understanding, the best subpacks are renamed as , , ….. The process runs iteratively until the item subpacks are generated and the best item subpack is returned. The pseudo code of IPO is showed as Algorithm LABEL:alg:poa.
IiiB Approximate Influence Maximization
The spread maximization problem in the Independent Cascade (IC) model [6] suffers from the expensive computation problem since the difficulty of the influence spread given a seed set is hard. To efficiently address this issue, an approximate IC model, called MIA, has been proposed [2, 1]. The social influence from a person to another person is effectively approximated by their maximum influence path (MIP), where the social influence on the path (,) is the maximum weight among all the possible paths from to . MIA creates a maximum influence inarborescence, i.e., a directed tree, MIIA(,) including the union of every MIP to with the probability of social influence at least from a set of leaf nodes. The MIA model has been widely adopted to describe the phenomenon of social influence in the literature with the following definition on activation probability, which is basically the same as the acceptance probability if broadcasts friending invitations to all nodes in .
Definition 1
The activation probability of a node v in is =
Note that is the joint probability that is activated and successfully influences , and can never influence if it is not activated. Therefore, the activation probability of a node can be derived according to the activation probability of all its inneighbors, i.e., the child nodes in the tree. Since is the set the leaf nodes, the activation probabilities of all nodes in can be efficiently derived in a bottomup manner from toward .
IiiC Computation Reduction of Budget Allocation via Dynamic Programming
Iv Experiment
Iva Experiment Setting
call detail records collected by a telecom operator
As [3] [4], we study telecommunications social networks extracted from a large amount of Call Detail Records (CDRs).
V Conclusion
The conclusion goes here. this is more of the conclusion
The authors would like to thank… more thanks here
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