2. Literature and research problem

Social networks became important elements of marketing campaigns. Targeting individuals based on their position in the network and their capacity to influence others is an essential marketing tool and an active area of network science (Aral, Reference Aral2011; Watts and Dodds, Reference Watts and Dodds2007). However, marketing strategies often focus on communities and not on individuals. For example, health interventions target specific groups and by changing their behavior also aim to have a wider impact that can spread in the network (Gold et al., Reference Gold, Pedrana, Sacks-Davis, Hellard, Chang, Howard and Stoove2011). Similarly, a frequently used technique of brand management is the identification of communities of customers that engage with the brand and can also influence others (Gensler et al., Reference Gensler, Völckner, Liu-Thompkins and Wiertz2013).

Here we argue that the market segmentation and influence maximization techniques should be combined in marketing campaigns run on social networks. However, such an endeavor requires a better understanding of what groups of individuals should be selected for targeting such that social influence is maximized. To demonstrate this research niche, we shortly review grouping techniques and then introduce the research niche in group targeting.

3. Audience Selection vs Influence Maximization

Since the proposed problem of Audience Selection is closely related to Influence Maximization, it is worth to explore the differences between the two approaches. Let us consider a small example in which grouping is based on co-location of individuals. Figure 1 depicts a social network with additional geographic information. Colors signify different locations, say towns. Each town has three inhabitants (agents). Red agents are all connected to all the orange and blue agents. The top green agent is connected to all the orange agents and two blue ones, while the bottom green is connected to all the blue agents and two orange. Finally, the middle green agent is connected to all the pink agents.

Figure 1. Sketch of a social network with five geographical locations (red, orange, blue, green, and pink) each accommodating three agents.

In the Audience Selection problem, we have to find the most suitable locations for a series of marketing events. Agents corresponding to the same location can be reached simultaneously. Let us assume that the linear threshold diffusion model (a standard choice in the literature, see details in Section 5) adequately represents influence spreading in this network. In our example, one event is organized and every user is activated at the selected location. Which town should be chosen?

Formulating the question like this, we are asking which town provides the best audience. In contrast, the Influence Maximization problem tries to find the $k$ most pivotal agents in a diffusion.

Let us try to answer our questions from the perspective of Influence Maximization. Since each town accommodates three agents, let us fix $k=3$ . Kempe et al. (Reference Kempe, Kleinberg and Tardos2003) proposed a greedy algorithm to approximate the best set of $k$ agents. Since then, many clever heuristics have been invented to improve either the approximation or the running time of the greedy algorithm. However, for such a small example we do not need sophisticated techniques. We can try each combination of agent triplets in a diffusion simulation. It turns out that choosing any two of the red agents, and the middle green agent is the best: On average, they can reach (activate) 78.6% of the nodes.

Since the majority of the most influential agents are red, we are inclined to choose the red town. However, that would be a mistake. Green agents can activate on average 74.0% of all the nodes, while red agents only score 63.6%. Thus, the green town outperforms the red one by a hefty 10%.

Let us approach the question from another angle. Influence Maximization techniques often suffer from computational limitations. It is difficult to even approximate a suitable $k$ set for a huge network. The usual workaround is to use proxy measures, that is, network centralities that were designed to find influential spreaders. For instance, we may calculate the average PageRank value of each agent in a town (for definition and discussion of network centralities, see Section 6). Table 1 contains the PageRank values as well as the Harmonic centrality of each node. It also lists the average centrality values for each town.

Table 1. The PageRank values and Harmonic centrality of the social network depicted in Figure 1. Agents are referred as $x_i$ , where $x$ denotes the first letter of the towns’ color (red, orange, blue, green, pink,) and $i$ denotes the agents location in the town (top, middle, bottom). Bold numbers represent the maximum value among towns

PageRank correctly predicts the potential of the towns, in the sense, that larger PageRank values correspond to greater spreading capabilities. Harmonic centrality is far less effective, as it ranks the town with the best audience as the second worst.

From this small example, we can deduce a couple of interesting points.

• • Influence Maximization techniques are ineffective in comparing sets of nodes, because the most influential agents might be spread over different sets.

• • Although some sets contain less influential agents (top and bottom green agents are less influential than any of the red ones), as a group they might be more effective than any other.

• • Aggregating network centralities at a set level can be a good predictor of the sets’ spreading potential.

• • Some centrality measures are more effective than others. Their performance might also vary depending on the underlying graph structure and the applied diffusion model.

Based on these observations, we can formulate a hypothesis. We expect to observe a discrepancy between the rankings of the best Influence Maximization proxies and the best algorithms for the Audience Selection problem. In the next section, we will describe the ranking frameworks and present our hypothesis formally. To avoid inferring too much from a small example, we test our hypothesis on two real-life social networks.

Note that the best way to uncover the potential of an audience is by simulation. However, this might be prohibitive for various reasons. Running diffusion simulations are costly, especially since we have to do it many times—often on a very large network too—to get a reliable estimate of the set’s spreading potential.

But the real difficulty comes from the cardinality of the sets rather than the running time of one simulation. If a marketing campaign consists of a series of events, each tied to a different location, the number of sets we need to assess can grow excessively. For instance, if we have to pick five towns among 100, that already means more than 75 million combinations that each need an evaluation by simulation. Calculating influence proxy measures, that is, network centralities that were specifically designed with the underlying problem in mind, is a much cheaper and sometimes the only feasible solution. In the example at hand, the analyst would select the five towns with the highest average node centrality.

5. Diffusion models

A central observation of diffusion theory is that the line that separates a failed cascade from a successful one is very thin. Granovetter (Reference Granovetter1978) was the first to offer an explanation for this phenomenon. He assumed that each agent has a threshold value, and an agent becomes an adopter only if the amount of influence it receives exceeds the threshold. He uses an illuminating example of how a peaceful protest becomes a riot. A violent action triggers other agents who in turn also become agitated and a domino effect ensues. However, much depends on the threshold distribution. Despite the similarities, network diffusion can substantially differ depending on the type of contagion we deal with. A basic difference stems from the possible adopting behaviors. In most models, agents can adopt two or three distinct states: susceptible, infected, and recovered. Consequently, the literature distinguishes between SI, SIS, and SIR types of models.

In the conventional Influence Maximization setup, a seed set of users are fixed as adopters (or are infected), then the results of diffusion are observed. In the SI model, agents who become adopters stay so until the end of simulation (Carnes et al., Reference Carnes, Nagarajan, Wild and van Zuylen2007; Chen et al., Reference Chen, Wang and Yang2009; Kim et al., Reference Kim, Lee, Park, Lee, Hong, Meng, Chen, Winiwarter and Song2013). A real-life example of such a diffusion can be a profound innovation that eventually conquers the whole network (e.g., mobile phones, internet). In SIS models, agents can change from susceptible to adopter and then back to susceptible again (Kitsak et al., Reference Kitsak, Gallos, Havlin, Liljeros, Muchnik, Stanley and Makse2010; Barbillon et al., Reference Barbillon, Schwaller, Robin, Flachs and Stone2020). For example, an innovation or rumor that can die out behaves this way. Another example would be of a service that agents can unsubscribe from and subscribe again if they want. In SIR models, agents can get “cured” and switch from adopter to recovered status (Yang et al., Reference Yang, Wang, Ren, Bai, Shi, Wang and Zhou2007; Gong and Kang, Reference Gong and Kang2018; Bucur and Holme, Reference Bucur and Holme2020). Recovered agents enjoy temporary or lasting immunity to addictive misbehavior (e.g., to online gaming). Other models allow users to self-activate themselves through spontaneous user adoption (Sun et al., Reference Sun, Chen, Yu and Chen2020).

In this paper, we feature a classical SI diffusion model: the Linear Threshold (LT) framework. We represent our network $G$ with a graph $(V,E)$ , where $V$ is the set of nodes and $E$ is the set of arcs (directed edges) between the nodes. In LT, each node, $\mathbf{v}$ chooses a threshold value $t_{\mathbf{v}}$ uniform at random from the interval $[0,1]$ . Similarly, each edge $e$ is assigned a weight $w_e$ from $[0,1]$ , such that, for each node the sum of weights of the entering arcs add up to 1, formally

\begin{equation*}\sum _{\{e\in E| e=(\mathbf {u},\mathbf {v}), \mathbf {u}\in V\}}w_e=1. \quad \quad \mbox { for all } \mathbf {v}\in V.\end{equation*}

One round of simulation for a sample set $S_i$ took the following steps.

1. 1. Generate random node thresholds and arc weights for each $\mathbf{v}\in V$ and $e \in E$ .

2. 2. Activate each node in $V$ that belongs to $S_i$ .

3. 3. Mark each leaving arc of the active nodes.

4. 4. For each $\mathbf{u}\in V\setminus S_i$ sum up the weights of the marked entering arcs. If the sum exceeds, the node’s threshold activate the node.

5. 5. If there were new activations go back to step 3.

6. 6. Output the spread, i.e., the percentage of active nodes.

We ran 5000 simulations and took the average of spread, that is, the expected percentage of individuals that the campaign will reach if $S_i$ is chosen as initial spreaders.

In this setup, we can think of node thresholds as a measure of innovativeness or risk attitude of the users, while edge weights represent how much influence they bear on each other. Randomization of the parameters is useful in two ways. First, it is difficult to measure peer influence and individual thresholds. There are much more data available, since Kempe et al. (Reference Kempe, Kleinberg and Tardos2003) introduced this diffusion model, so there is room for potential improvements in this aspect. For instance, a strong argument can be made that the parameters for innovators and early adopters should be chosen from different intervals, as they play a key role in the diffusion (Sziklai and Lengyel, Reference Sziklai and Lengyel2022). Nevertheless, randomization is also essential for obtaining multiple observations needed for statistical inference. Perturbing parameter values is a standard technique to achieve this.

6. Influence Maximization proxies

In this section, we give a brief overview of the measures we employed in our analysis.

Among the classical centrality measures, we included Degree and Harmonic Centrality. The former is a self-explanatory benchmark, and the latter is a distance-based measure proposed by Marchiori and Latora (Reference Marchiori and Latora2000). Harmonic centrality of a node, $\mathbf{v}$ is the sum of the reciprocal of distances between $\mathbf{v}$ and every other node in the network. For disconnected node pairs, the distance is infinite; thus, the reciprocal is defined as zero. A peripheral node lies far away from most of the nodes. Thus, the reciprocal of the distances will be small which yields a small centrality value.

PageRank, introduced by Page et al. (Reference Page, Brin, Motwani and Winograd1999), is a spectral measure and a close relative of Eigenvector centrality (Bonacich, Reference Bonacich1972). Eigenvector centrality may lead to misleading valuations if the underlying graph is not strongly connected. PageRank overcomes this difficulty by (i) connecting sink nodes (i.e., nodes with no leaving arc) with every other node through a link and (ii) redistributing some value uniformly among the nodes. The latter is parameterized by the so-called damping factor, $\alpha \in (0,1)$ . The method is best described as a stochastic process. Suppose we start a random walk from an arbitrary node of the network. If anytime we hit a sink node, we restart the walk by choosing a node uniform at random from the node set. After each step, we have a $(1-\alpha )$ probability to teleport to a random node. The probability that we occupy node $\mathbf{v}$ as the number of steps tends to infinity is the PageRank value of node $\mathbf{v}$ . The idea was to model random surfing on the World Wide Web. PageRank is a core element of Google’s search engine, but the algorithm is used in a wide variety of applications. The damping value is most commonly chosen from the interval $(0.7,0.9)$ , here we opted for $\alpha =0.8$ . For an axiomatic characterization of PageRank, see Was and Skibski (Reference Was and Skibski2023).

Generalized Degree Discount (GDD) introduced by Wang et al. (Reference Wang, Zhang, Zhao and Yi2016) is a suggested improvement on Degree Discount (Chen et al., Reference Chen, Wang and Yang2009) which was developed specifically for the independent cascade model. The latter is an SI diffusion model where each active node has a single chance to infect its neighbors, transmission occurring with the probability specified by the arc weights. Discount Degree constructs a spreader group of size $q$ starting from the empty set and adding nodes one by one using a simple heuristic. It primarily looks at the degree of the nodes but also takes into account how many of their neighbors are spreaders. GDD takes this idea one step further and also considers how many of the neighbors’ neighbors are spreaders. The spreading parameter of the algorithm was chosen to be $0.05$ .

$k$ -core, sometimes referred to as, $k$ -shell exposes the onion-like structure of the network (Seidman, Reference Seidman1983; Kitsak et al., Reference Kitsak, Gallos, Havlin, Liljeros, Muchnik, Stanley and Makse2010). First, it successively removes nodes with only one neighbor from the graph. These are assigned a $k$ -core value of 1. Then it removes nodes with two or less neighbors and labels them with a $k$ -core value of 2. The process is continued in the same manner until every node is classified. In this way, every node of a path or a star graph is assigned a $k$ -core value of 1, while nodes of a cycle will have a $k$ -core value of 2.

Linear Threshold Centrality (LTC) was, as the name suggests, designed for the Linear Threshold model (Riquelme et al., Reference Riquelme, Gonzalez-Cantergiani, Molinero and Serna2018). Given a network, $G$ with node thresholds and arc weights, LTC of a node $\mathbf{v}$ represents the fraction of nodes that $\mathbf{v}$ and its neighbors would infect under the Linear Threshold model. In the simulation, we derived the node and arc weights that is needed for the LTC calculation from a simple heuristics: each arc’s weight is defined as 1 and node thresholds were defined as $0.7$ times the node degree. Note that we ran LTC and GDD under various parameters and chose the one which performed the best during the CRRN test (see Fig. A1 in the Appendix).

Suri and Narahari (Reference Suri and Narahari2008) define a cooperative game on graphs and derive node centrality by computing the Shapley value. Nodes are considered players who cooperate to compose the best $k$ -element set, for some given $k$ . The Shapley value is the expected marginal contribution of each player when players are added to the set one by one and each order of the players is equally likely. Marginal contribution of a node $\mathbf{u}$ is just the difference between the value of the node set with and without $\mathbf{u}$ . Depending on how we define the value of a node set, we can derive different cooperative games. Here we use the G1 game variant proposed by Michalak et al. (Reference Michalak, Aadithya, Szczepański, Ravindran and Jennings2013) who also gave a polynomial-time algorithm to compute the corresponding Shapley(G1)-value. In G1, the characteristic function value of a node set $C$ , is the number of nodes in $C$ plus the number of distinct neighbors of $C$ .

Top candidate (TC) algorithm originates from the field of social choice. It is a group identification method designed to find experts on recommendation networks (Sziklai, Reference Sziklai2018, Reference Sziklai2021). The algorithm takes a graph as an input and outputs a list of experts. The size of the output can be adjusted with a parameter, $\alpha \in [0,1]$ . The smallest set $(\alpha =0)$ corresponds to the elite, while the largest set $(\alpha =1)$ is composed by anyone who could conceivably considered as expert. The algorithm works as follows. In the beginning, each node is considered as an expert and asked to nominate $\alpha$ fraction of its most popular neighbors. Popularity is measured by the number of recommendations the node received, that is, in the number of entering arcs. The nodes that receive no nominations are surely not experts; hence, they are removed from the expert set and their nominations are withdrawn. This in turn might leave some other nodes without nominations. The removal of experts continues until the set stabilizes. The $\alpha$ parameter is what makes TC suitable as a centrality index for strongly connected graphs. We run the algorithm multiple times, incrementing the $\alpha$ parameter by 0.01. Each node is assigned the largest $1-\alpha$ value for which the node first becomes expert. This can be considered as a measure of exclusivity. For strongly connected graphs, every node becomes an expert at $\alpha =1$ the latest, but those nodes will have an exclusivity value of 0. Note that social networks consisting of one component become strongly connected when the undirected link between the users (the “friendship”) is converted into two opposing directed edges.

This is by no means a comprehensive list, there are other proxy and centrality measures, e.g., HITS (Kleinberg, Reference Kleinberg1999), LeaderRank (Lü et al., Reference Lü, Zhang, Yeung and Zhou2011), DegreeDistance (Sheikhahmadi et al., Reference Sheikhahmadi, Nematbakhsh and Shokrollahi2015), IRIE (Jung et al., Reference Jung, Heo and Chen2012), GroupPR (Liu et al., Reference Liu, Xiang, Chen, Xiong, Tang and Yu2014), Attachment centrality (Skibski et al., Reference Skibski, Rahwan, Michalak and Yokoo2019) for more see the survey of Li et al. (Reference Li, Fan, Wang and Tan2018). Announcing a clear winner falls outside the scope of this manuscript. A real ranking analysis should always consider the diffusion model and a network characteristics carefully, as different algorithms will thrive in different environments.

7. Ranking analysis

The performance of the sample sets according to the various influence measures is displayed in Tables A1 (for iWiW) and A3 (for Pokec). The last column shows the percentage of nodes that the samples managed to activate on average in the diffusion simulation. The ranking matrix together with the SRD values is displayed in Tables A2 and A4. The tables can be found in the Appendix. The SRD score of a centrality is the distance between the ranking induced by the measure and the reference ranking, that is, the last column. In the CRRN test (Comparison of Ranks with Random Numbers), we compare the SRD scores with those of random rankings. Figures 2 and 3 show the result of the CRRN test. The SRD distribution is generated by observing the distances of random rankings from the reference. We accept a solution if the distance between the solution’s ranking and the reference ranking falls left to the 5% significance thresholds.

For both networks, all of the methods pass the test, that is, they rank the sample sets more or less correctly. The performances vary, although degree and PageRank rank high, while $k$ -core and Harmonic centrality rank low on both networks. The first places of degree should be taken with a grain of salt. A possible reason behind its exceptional performance might come from the fact that sample sets were created by random sampling. It remains to be seen how degree performs compared to other measures when sets are generated in a more realistic way, e.g., based on location data.

Figure 2. iWiW CRRN test. The order of the centrality measures in the legend (from top to bottom) follows the same order as the colored bars in the figure (from left to right). The bars’ height is equal to their normalized SRD values. The black curve is a continuous approximation of the cumulative distribution function of the random SRD values. All (normalized) SRD values fall outside the 5% threshold (XX1: 5% threshold, Med: Median, XX19: 95% threshold).

Figure 3. Pokec CRRN test. The order of the centrality measures in the legend (from top to bottom) follows the same order as the colored bars in the figure (from left to right). The bars’ height is equal to their normalized SRD values. The black curve is a continuous approximation of the cumulative distribution function of the random SRD values. All (normalized) SRD values fall outside the 5% threshold (XX1: 5% threshold, Med: Median, XX19: 95% threshold).

Cross-validation reveals how the methods are grouped. Multiple SRD scores are generated by sampling the data and recalculating the rankings. We create 8-folds, by repeatedly removing three random rows (sample sets) and recalculating the rankings. Then, an 8-fold Wilcoxon test compares the median values. Figures 4 and 5 show the boxplots of the attained SRD values. On the iWiW network, there is no significant difference between Harmonic centrality, Shapley G1 and GDD. On the Pokec network, Linear Threshold Centrality and Shapley G1 are tied for the 4th place.

Figure 4. Cross-validation on iWiW data. The boxplot shows the median (black diamond), Q1/Q3 (blue box), and min/max values. The measures are ranked from left to right by the median values. The ‘ $\sim$ ’ sign between two neighboring measures indicates that the Wilcoxon test found no significant difference between the rankings induced by the measures.

Figure 5. Cross-validation of Pokec data. The boxplot shows the median (black diamond), Q1/Q3 (blue box), and min/max values. The measures are ranked from left to right by the median values. The ‘ $\sim$ ’ sign between two neighboring measures indicates that the Wilcoxon test found no significant difference between the rankings induced by the measures.

Our main goal was to show that Audience Selection and Influence Maximization are two different problems. So let us take a look at how the top choices of these measures perform in the simulation. In case of iWiW, we took the 500 highest-ranked agents for each measure. If the 500th and 501st agents tied with each other, we discarded agents with the same score one by one randomly until the size of the set became 500. We ran 5000 simulations to obtain the average spread of the measures. The same procedure was repeated on the Pokec data, but there the top 2000 nodes were considered. Tables 3 and 4 display the results.

In case of iWiW, there is a significant disparity in the performances of GDD under the Influence Maximization and Audience Selection problems. In Influence Maximization, GDD is the best-performing method, while in Audience Selection, it performs poorly compared to other methods. In case of Pokec, the rank difference between methods is not as massive as in iWiW. However, there are many smaller differences in the performance of methods.

Are these differences statistically significant? Luckily, the same CRRN test, that we used before can be applied, but now our input data is different. When we compared centralities, we looked at how they rank the sample sets. Now we want to compare Influence Maximization with Audience Selection, therefore we look at how they rank centralities. We can set either the ranking of IM or that of AS as the reference and compare their distance to the empirical SRD distribution of size $n=8$ (since there are 8 centrality measures).

In case of iWiW, the distance between the IM and the AS ranking equals to a normalized SRD value of 0.375. In the SRD distribution, this corresponds to a $p$ -value of 0.0685. If we consider ties the distance remains the same, however, the $p$ -value becomes 0.07104 due to the change in the distribution. Assuming the usual 5% significance level, these two $p$ -values mean that the AS ranking is indistinguishable from a random ranking compared to the IM ranking. In other words, the rankings significantly differ from each other and Hypothesis 1 holds.

Table 3. Comparing the rankings of centralities induced by the Audience Selection (AS) and Influence Maximization (IM) problem on iWiW. Ranks in brackets show the tied ranks according to the cross-validation

Table 4. Comparing the rankings of centralities induced by the Audience Selection (AS) and Influence Maximization (IM) problem on Pokec. Ranks in brackets show the tied ranks according to the cross-validation

In case of Pokec, the SRD values are 0.375 (in the strict ranking case) and 0.34375 (when ties are considered). The first value corresponds to a $p$ -value of 0.0688, while the second to $p=0.0380$ . In other words, statistical significance depends on whether we allow ties or not. Nevertheless, an SRD score of 0.34375 is substantial even if it is not statistically significant. Considering that both Influence Maximization and Audience Selection are optimization problems, the difference is too big to treat them the same—especially, since they do not agree on the best method.

In conclusion, both our example and simulations on real data point toward that Hypothesis 1 holds: different methods perform well under the Influence Maximization model and the Audience Selection model.