A rich line of research has studied the effectiveness of marketing strategies based on person-to-person recommendation within a social network — a process often termed viral marketing  and closely connected to the broader sociological literature on the diffusion of innovations in social networks . A key genre of theoretical question that emerged early in this literature is the problem of optimally “seeding” a product in a social network through the selection of a set of initial adopters [9, 15, 24]. In this class of questions, we consider a firm that has a product they would like to market to a group of agents on a social network; it is often the case that the firm cannot target all the participants in the network, and so they seek to target the most influential ones so as to maximize exposure and create a cascade of adoptions. Approaches to this question have generally been based on objective functions in which the goal is to maximize the number of people who are reached by the network cascade — or more generally, in which the objective function monotonically increases in the number of people reached.
The dangers of overexposure. Separately from this, lines of research in both marketing and in the dynamics of on-line information have provided diverse evidence that the benefits of a marketing campaign are not in fact purely increasing in the number of people reached. An influential example of such a finding is the Groupon effect, in which viral marketing via Groupon coupons leads to lower Yelp ratings. In , they note the negative effect Groupon has on average Yelp ratings and provide arguments for the underlying mechanism; one of their central hypotheses is that by using Groupon as a matchmaker, businesses may be attracting customers from a portion of the population that is less inclined to like the product. In another example, Kovcs and Sharkey  discuss a setting on Goodreads where books that win prestigious awards (or are short-listed for them) attract more readers following the announcement, which again leads to a drop in the average rating of the book on the platform. Aizen et al.  show a similar effect for on-line videos and other media; they receive a discontinuous drop in their ratings when a popular blog links to them, driving users to the item who may not be interested in it.
Research in marketing has shown that exposure to different groups and influence between such groups can help or hurt adoption [5, 11, 12]. For example, Hu and Van den Bulte  argue that agents adopt products to boost their status; and so as word-of-mouth effects for a product become stronger among middle-income individuals, there might be a negative impact on adoption among higher-income individuals. Similar behavior is observed in health campaigns. In , Wakefield et al. discuss the importance of segmenting populations and exposing groups to anti-smoking campaigns whose themes the group is most susceptible to in order to maximize the impact of future campaigns.
We think of these effects collectively as different forms of overexposure; while reaching many potential customers is not a concern in and of itself, the empirical research above suggests that there may exist particular subsets of the population — potentially large subsets — who will react negatively to the product. When a marketing cascade reaches members of this negatively inclined subset, the marketing campaign can suffer negative payoff that may offset the benefits it has received from other parts of the population. This negative payoff can come in the form of harm to the firm’s reputation, either through latent consumer impressions and effects on brand loyalty [7, 8] or through explicitly visible negative reviews on rating sites.
Despite the importance of these considerations in marketing, they have not been incorporated into models of influence-based marketing in social networks. What types of algorithmic issues arise when we seek to spread a word-of-mouth cascade through a network, but must simultaneously ensure that it reaches the “right” part of the audience — the potential customers who will like the product, rather than those who will react negatively to it?
The present work: A model of cascades with the risk of overexposure. In this paper, we propose a basic theoretical model for the problem of seeding a cascade when there are benefits from reaching positively inclined customers and costs from reaching negatively inclined customers.
There are many potential factors that play a role in the distinction between positively and negatively inclined customers, and for our model we focus on a stylized framework in which each product has a known parameter in the interval that serves as some measure for the breadth of its appeal. At this level of generality, this parameter could serve as a proxy for a number of things, including quality; or a one-dimensional combination of price and quality; or — in the case where the social network represents a population defined by a specific interest — compatibility with the core interests of network’s members.
Each node in the network is an agent who will evaluate the product when they first learn of it; agents differ in how critical they are of new products, with agents of low criticality tending to like a wider range of products and agents of high criticality tending to reject more products. Thus, each agent has a criticality parameter in the interval ; since we assume that the firm has a history of marketing products to this network over a period of time, it knows this parameter . When exposed to a product, an agent accepts the product if and advertises the product to their neighbors, leading to the potential for a cascade. However, if , then the agent rejects the product, which results in a negative payoff to the firm; the cascade stops at such agents , since they do not advertise it to their neighbors.
The firm’s goal is to advertise the product to a subset of the nodes in the network — the seed set — resulting in a potential cascade of further nodes who learn about the product, so as to maximize its overall payoff. This payoff includes a positive term for each agent who sees the product and has , and a negative term for each agent who sees the product and has ; agents who are never reached by the cascade never find out about the product, and the firm gets zero payoff from them.
Overview of Results. We obtain theoretical results for two main settings of this problem: the unbudgeted case, in which the firm can initially advertise the product to an arbitrary seed set of nodes, and the budgeted case, in which the firm can advertise the project to at most nodes, for a given parameter . We note that typically in influence maximization problems, the unbudgeted case is not interesting: if the payoff is monotonically increasing in the number of nodes who are exposed to the product, then the optimal unbudgeted strategy is simply to show the product to everyone. In a world with negative payoffs from overexposure, however, the unbudgeted optimzation problem becomes non-trivial: we must tradeoff the benefits of showing the product to customers who will like it against the negatives that arise when these customers in turn share it with others who do not.
For the unbudgeted problem, we give a polynomial-time algorithm for finding the optimal seed set. The algorithm uses network flow techniques on a graph derived from the underlying social network with the given set of parameters . In contrast, we provide an NP-hardness result for the budgeted problem.
We then provide a natural generalization of the model: rather than each agent exhibiting only two possible behaviors (rejecting the product, or accepting it and promoting it), we allow for a wider range of agent behaviors. In particular, we will assume each agent has three parameters which control whether the agent ignores the product, views but rejects the product, accepts the product but does not broadcast it to its neighbors, and accepts the product and advertises it to neighbors. We show how to extend our results to this more general case, obtaining a polynomial-time algorithm for the unbudgeted case and an NP-hardness result for the budgeted case.
Finally, we perform computational simulations of our algorithm for the unbudgeted case on sample network topologies derived from moderately-sized social networks. We find an interesting effect in which the performance of the optimal algorithm transitions between two behaviors as varies. For small the payoff grows slowly while a baseline that promotes the product to every agent with achieves negative payoff (reflecting the consequences of overexposure). Then, for large , the payoff grows quickly, approaching a simple upper bound consisting of all for which .
There is a product with a parameter , measuring the breadth of its appeal. is an unweighted, undirected graph with agents as its nodes. For each agent , the agent’s criticality parameter measures the minimum threshold for the agent demands before adoption. Thus, higher values of correspond to more critical agents. We assume that these values are fixed and known to the firm.
The firm chooses an initial set of agents to “seed” with the product. If an agent sees the product, it accepts it if and rejects it if . We say that an agent is accepting in the former case and rejecting in the latter case. Each accepting agent who is exposed to the product advertises it to their neighbors, who then, recursively, are also exposed to the product. We will assume throughout that the firm chooses a seed set consisting entirely of accepting nodes (noting, of course, that rejecting nodes might subsequently be exposed to the product after nodes in the seed set advertise it to their neighbors).
We write for the set of agents exposed to the product if the seed set is . Formally, is the set of all agents who have a path to some node in such that all of the internal nodes on the - path are accepting agents; this is the “chain of recommendations” by which the product reached . Among the nodes in , we define to be the set of agents who accept the product and to be the set of agents who reject the product.
The payoff function associated with seed set is:
We can, more generally, assume that there is a payoff of to accepting the product and a negative payoff of to rejecting the product, and we set the payoff function to be:
We will call this the generalized payoff function, and simply refer to Equation 1 as the payoff function. The overarching question then is:
Given a set of agents with criticality parameters () on a social network , and given a product of quality , what is the optimal seed set that the firm should target in order to maximize the payoff given by Equation (2)?
In contrast to much of the influence maximization literature, we assume that the agents’ likelihood of adoption, once exposed to this product, is not affected by which of their neighbors have accepted or rejected the product. This differs from, for instance, models in which each agent requires a certain fraction (or number) of its neighbors to have accepted the product before it does; or models where probabilistic contagion takes place across the edges. These all form interesting directions for further work; here, however, we focus on questions in which the intrinsic appeal of the product, via , determines adoption decisions, and the social network provides communication pathways for other agents to hear about the product.
Before proceeding to the main result, we develop some further terminology that will be helpful in reasoning about the seed sets.
Let be an accepting node, and let . Then we say that is the cluster of , denoted by ; we call the interior of and denote it by , and we call the boundary of and denote it by .
We denote the payoff corresponding to the seed set by . Note that,
Given an accepting node , and a node , we have .
If is in the interior of , then there exists a path in , where and such that each node along the path has . (That is, each node is exposed to and accepts the product as a result of ’s advertisement.) We would like to prove that if , then would be exposed to the product. Equivalently, we want to show there exists a path from to of nodes with ; but this is precisely the path . ∎
For an arbitrary seed set , the set may consist of multiple interior-disjoint clusters, which we label by , where . Note that each of these clusters might be associated with more than one agent in the seed set and that . (Likewise, and .)
Given a seed-set and corresponding clusters, a direct consequence of Lemma 3 is that adding more nodes already contained in these clusters to the seed-set does not change the payoff.
Given a set of accepting nodes such that , we have .
It therefore suffices to seed a single agent within a cluster. Given a cluster , we will simply pick an arbitrary node in the interior of the cluster to be the canonical node and use that to refer to the cluster even if is formed as a result of seeding another node .
3 Main Model
Given that all are known to the firm, a naive approach would suggest to seed all where . While this is guaranteed to give a nonnegative payoff, need not be optimal.
Consider the graph below, where nodes in blue accept the product and those in red reject the product.
Suppose . Then, a naive approach would set , since each of the resulting clusters to the blue nodes has negative payoff. However, setting has payoff .
This phenomenon is a result of the fact that the clusters might have boundaries that intersect non-trivially. Thus, there could be agents whose , but is in a sense “paid for” by seeding other agents; and hence we could have a net-positive payoff from including subject to seeding other agents whose cluster boundaries intersect with . Using this observation, we will give a polynomial-time algorithm for finding the optimal seed set under the generalized payoff function using a network flow argument. We first begin by constructing a flow network.
Given an instance defined by and , we let be the set of all distinct clusters in , with disjoint interiors. We form a flow network as follows: set corresponding to the canonical nodes of the clusters above and be the set of agents in the boundaries of all clusters. We add an edge from the source node to each node with capacity and label this value by , and an edge from each node to with capacity . We add an edge between and if and only if , and set these edges to have infinite capacity. We denote this corresponding flow network by .
For the above example, is:
In this example, the edges out of have capacities and edges into have capacities . Edges between blue and red notes have infinite capacity. Assuming , the min-cut has and all other nodes in .
Given a min-cut in , the optimal seed set in is .
The min-cut must have value at most since we can trivially obtain that by setting the cut to be . Given a node , which we recall corresponds to the canonical node of a cluster, if a node is exposed to the product as a result of seeding any node in the cluster, then . Otherwise, we would have edge included in the cut, which has infinite capacity contradicting the minimality of the cut . Therefore, the min-cut will include all nodes in the seed set as well as all nodes that are exposed to the product as a result of the corresponding seed set in .
Note that the edges across the cut are of two forms: or , where and . The first set of edges contribute (recall ) and the latter contributes . Therefore, the objective for finding a min-cut can be equally stated as minimizing,
over cuts . Note that
Therefore, we have:
Note the term is precisely what the payoff objective function is maximizing, giving a correspondence between the min-cut and optimal seed set. ∎
We therefore have the main result of this section:
There is a polynomial-time algorithm for computing the optimal seed set for Problem 1 when there are no budgets for the size of the seed set.
An interesting phenomenon is that the payoff is not monotone in even when considering optimal seed sets. Take the following example:
Suppose we are given the network below with the numbers specifying the for each corresponding node:
If , we cannot do better than the empty-set. If , the seed set that includes either of the two left-most nodes gives a payoff of , which is optimal. For the case where , the empty-set is again optimal.
This example gives a concrete way to think about overexposure phenomena such as the Groupon effect  discussed in the introduction. Viewed in the current terms, we could say that by using Groupon, one could increase the broad-appeal measure of the product (e.g., cheaper, signaling higher quality, etc), which therefore exposes the product to portions of the market that would have previously not been exposed to it, and this could lead to a worse payoff.
4 Generalized Model
We now consider the generalized model where there are three parameters corresponding to each agent , . An agent considers a product if , adopts a product if , and advertises it to their friends if . If an agent is exposed to a product but , then the payoff associated with the agent is 0. If , then the agent rejects the product, for a payoff of . As before, there is a payoff of if the agent accepts the product; however, the agent only advertises the product to its neighbors after accepting if . We therefore have four types of agents:
Type I: Agents for which ,
Type II: Agents for which ,
Type III: Agents for which , and
Type IV: Agents for which .
We denote the set of all agents of Type I by (and likewise for the other types). The basic model above is the special case where and for all . In this instance, we only have agents of Types II and IV.
Given any seed set , we note:
These follow from the observation that:
Agents of Type I are those that do not look at the product since is below their threshold , and thus do not affect the payoff function when added to any seed set.
Agents of Type III are those for which , and therefore they accept the product, but do not advertise it to their friends. Therefore, adding such an agent to the seed set increases the payoff by exactly per agent added.
In the simplest case, we have and , such that agents will be either of Type I or III, and the optimal seed set is precisely . That is, agents only view a product if they are going to accept it and they never advertise it to their neighbors, and there is no cascade triggered as a result of seeding agents.
When this is not the case, we will note that the results in the previous section can be adapted to this setting to find an optimal seed set efficiently. Given a network in this generalized setting, consider a corresponding network , which is the subgraph of consisting of agents of only Types II and IV. We can then apply the algorithm in the previous section to this subgraph to find an optimal seed set. We claim that the union of this with agents of Type III yields an optimal seed set in .
Given a product with a value and a network with agents of parameters and , there is a polynomial-time algorithm for finding an optimal seed set to maximize the payoff function.
Given such a graph , and a corresponding subgraph with an optimal seed set , we argue that is an optimal seed set in .
For the sake of contradiction, suppose is an optimal seed set in , such that . By Lemma 10, we can assume that does not include any agent of Type I and includes all agents of Type III. This assumption implies . But since , this contradicts the optimality of .
Returning to the implications for the Groupon effect, we note that a firm can efficiently maximize its payoff over the choice of both the seed set and . Suppose the firm knows the parameters for each of the agents . Given agents, these values divide up the unit interval into at most subintervals . It is easy to see that the payoff depends only on which subinterval is contained in, but does not vary within a subinterval. Earlier, we saw the payoff need not be monotone in ; but by trying values of in each of the subintervals, the firm can determine a value of and a seed set that maximize its payoff.
5 Budgeted Seeding is Hard
In this section, we show that the seeding problem, even for the initial model, is NP-hard if we consider the case where there is a budget for the size of the seed set and we want to find subject to the constraint that . In the traditional influence maximization literature, we leverage properties of the payoff function such as its submodularity or supermodularity to give algorithms that find optimal or near-optimal seed sets. The payoff function here, however, is neither submodular nor supermodular.
Take the network shown in figure below. Set :
Supermodularity states that given and ,
Set , , and let be node . Then, supermodularity would give:
Submodularity states that,
A counterexample to this is obtained by setting , and .
Here, we show an even stronger hardness result: it is NP-hard to decide if there is a (budgeted) set yielding positive payoff. Since it is NP-hard to tell whether the optimum in any instance is positive or negative, it is therefore also NP-hard to provide an approximation algorithm with any multiplicative guarantee — a sharp contrast with the multiplicative approximation guarantees available for budgeted problems in more traditional influence maximization settings.
The decision problem of whether there exists a seed set with and is NP-complete.
We will prove this using a reduction from the NP-complete Clique problem on -regular graphs: given a -regular graph and a number , the question is to determine whether there exists a -clique — a set of nodes that are all mutually adjacent. (We also require .)
We will reduce an instance of -clique on -regular graphs to an instance of the decision version of our budgeted seed set problem as follows. Given such an instance of Clique specified by a -regular graph and a number , we construct an instance of the budgeted seed set problem on a new graph obtained from as follows: we replace each , with two new edges , where is a new node introduced by subdividing . Let be the set of nodes originally in , and the set of nodes introduced by subdividing. In the seed set instance on , we define and such that for all and for all . We define the payoff coefficients by and for some .
We will show that has a -clique if and only if there is a seed set of size at most in with positive payoff.
First, suppose that is a set of nodes in that are all mutually adjacent, and consider the corresponding set of nodes in . As a seed set, has accepting nodes and rejecting neighbors, since is -regular but the nodes on the subdivided edges are double-counted. Thus the payoff from is , which is positive by our choice of .
For the converse, suppose has size and has positive payoff in . Since the seed set consists entirely of accepting neighbors (any others can be omitted without decreasing the payoff), , and hence so the neighbors of reject the product. If induces edges in , then the payoff from includes a negative term from each neighbor, with the nodes on the subdivided edges double-counted, so the payoff is . If , then since , the payoff is at most , which is negative by our choice of . If and , then the payoff is at most , which again is negative by our choice of . Thus it must be that and , so induces a -clique as required. ∎
6 Experimental Results
In this section, we present some computational results using datasets obtained from SNAP (Stanford Network Analysis Project). In particular, we consider an email network from a European research institution [18, 22] and a text message network from a social-networking platform at UC-Irvine .
The former is a directed network of emails sent between employees over an 803-day period, with nodes and directed edges. The latter is a directed network of text messages sent between students through an online social network at UC Irvine over a 193-day period, with nodes and directed edges. In both networks, we use the edge to indicate that sent at least one email or text to node over the time period considered.
For both of these networks, we present results corresponding to the general model. We consider 100 evenly-spaced values of in and compare the seed set obtained by our algorithm with some natural baselines. The parameters
for each agent are chosen as follows: we draw three numbers independently from an underlying distribution (we analyze both the uniform distribution on
and the Gaussian distribution with mean); we then sort these three numbers in non-decreasing order and set them to be and respectively.
For each we run 100 trials and present the average payoff. The average time to run one simulation is seconds for the text network and seconds for the email network. This includes the time to read the data and assemble the network; the average time spent only on computing the min-cut for the corresponding network is and seconds respectively.
For each of these figures, we give a natural upper-bound which is the number of agents such that . This includes agents of Type III and IV. We note that the seed set obtained by our algorithm often gives a payoff close to this upper bound. We compare this to two natural baselines: the first sets agents of Type III to be the seed set and the second sets agents of both Types III and IV to be the seed set. We show that the first baseline performs well for lower values of , where the second baseline underperforms significantly; and, the second picks up performance significantly for higher values of while the first baseline suffers. The seed set obtained by our algorithm, on the other hand, outperforms both baselines by a notable margin for moderate values of . This gap in performance corresponds to the overexposure effect in our models.
Each of the figures show that the seed set chosen by our algorithm outperforms these natural baselines. In Figure 2, we note for the optimal seed set obtained through our algorithm is close to picking only agents of Type III. Adding on agents of Type IV performs worse than both seeding just agents of Type III or the optimal seed set. This changes for values over . Here, the number of agents of Type III drops, and thus the payoff obtained by seeding agents of Type III drops with it. On the other hand, the seed set consisting of all agents of Types III and IV picks up performance, coming close to the optimal seed set for . This behavior appears in both networks.
7 Further Related Work
As noted in the introduction, our work — through its focus on selecting a seed set of nodes with which to start a cascade — follows the motivation underlying the line of theoretical work on influence maximization [9, 15, 24]. There has been some theoretical work showing the counter-intuitive outcome where increased effort results in a less successful spread. An example is , where they show that due to the separation of the infection and viral stage, there are cases where an increased effort can result in a lower rate of spread. A related line of work has made use of rich datasets on digital friend-to-friend recommendations on e-commerce sites to analyze the flow of product recommendations through an underlying social network . Further work has experimentally explored influence strategies, with individuals either immediately broadcasting their product adoption to their social network, or selecting individuals to recommend the product to .
The consequences of negative consumer reactions have been analyzed in a range of different domains. In the introduction we noted examples involving Groupon , book prizes , and on-line media collections . Although experimentally introduced negative ratings tend to be compensated for in later reviews, positive reviews can lead to herding effects . There has also been research seeking to quantify the economic impact of negative ratings, in contexts ranging from seller reputations in on-line auctions [4, 23] to on-line product reviews . This work has been consistent in ascribing non-trivial economic consequences to negative consumer impressions and their articulation through on-line ratings and reviews. Recent work has also considered the rate at which social-media content receives “likes” as a fraction of its total impressions, for quantifying a social media audience’s response to cascading content .
The literature on pricing goods with network effects is another domain that has developed models in which consumers are heterogeneous in their response to diffusing content. The underlying models are different from what we pursue here; a canonical structure in the literature on pricing with network effects is a set of consumers with different levels of willingness to pay for a product . This willingness to pay can change as the product becomes more popular; a line of work has thus considered how a product with network effects can be priced adaptively over time as it diffuses through the network [3, 10]. The variation in willingness to pay can be viewed as a type of “criticality,” with some consumers evaluating products more strictly and others less strictly. But a key contrast with our work is that highly critical individuals in these pricing models do not generally confer a negative payoff when they refuse to purchase an item.
Theoretical models of viral marketing in social networks have generally used the assumption that all exposures to a product are beneficial to the firm conducting the marketing. A separate line of empirical research in marketing, however, provides a more complex picture, in which different potential customers may have either positive or negative reactions to a product, and it can be a mistake to pursue a strategy that elicits too many negative reactions from potential customers.
In this work, we have proposed a new set of theoretical models for viral marketing, by taking into account these types of overexposure effects. Our models make it possible to consider the optimization trade-offs that arise from trying to reach a large set of positively inclined potential customers while reducing the number of negatively inclined potential customers who are reached in the process. Even in the case where the marketer has no budget on the number of people it can expose to the product, this tension between positive and negative reactions leads to a non-trivial optimization problem. We provide a polynomial-time algorithm for this problem, using techniques from network flow, and we prove hardness for the case in which a budget constraint is added to the problem formulation. Computational experiments show how our polynomial-time algorithm yields strong results on network data.
Our framework suggests many directions for future work. It would be interesting to integrate the role of negative payoffs in our model here with other technical components that are familiar from the literature on influence maximization, particularly the use of richer (and potentially probabilistic) functions governing the spread from one participant in the network to another. For example, when nodes have non-trivial thresholds for adoption — requiring both that they evaluate the product positively and also that they have heard about it from at least other people, for some — how significantly do the structures of optimal solutions change?
It will also be interesting to develop richer formalisms for the process by which positive and negative reactions arise when potential customers are exposed to good or bad products. With such extended formalisms we can more fully bring together considerations of overexposure and reputational costs into the literature on network-based marketing.
The first author was supported in part by a Google scholarship, a Facebook scholarship, and a Simons Investigator Award and the third author was supported in part by a Simons Investigator Award, an ARO MURI grant, a Google Research Grant, and a Facebook Faculty Research Grant.
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