Selection of top-K influential users based on radius-neighborhood degree, multi-hops distance and selection threshold

The last decade has seen a broad research focus on studying and analysis of social network phenomena especially the influence maximization that received a significant interest and attention due to its various application in various situations such as advertisement, rumor control, spread of epidemic, understanding the collective behavior of users in online systems by observing users behaviors across product and contents. A lot of work has been made followed up the work of Domingos [1] and Kemp et al. [2] that present a greedy algorithm which provide the highest influence coverage while suffer from scalability issue, which pushed a lot of researchers to investigate in the improvement of time complexity of the original greedy algorithm [3–5]. But despite the huge work dedicated to reduce the runtime complexity, the majority if not all based greedy takes a lot of time to complete the selection of seed set. This motivates to develop methods based heuristics and as stated by Kemp [2] and Chen et al. [6] that degree gives acceptable influence spread in a low time. Selection of most influential users has become a vital task in the field of social network analysis. Identification of such influential users permits summarizing the underlying interactions of the network by observing how such set of selected nodes may influence a lot of users and how it can incite the spread of certain cascade behavior over the network, which helps in profoundly understanding and discovering interesting and favorable properties shared by important users in the network.

Various algorithms have been developed to tackle the problem of influence maximization. Most of the algorithms are based on either heuristics that detect most influential nodes according to the score value of introduced metrics or algorithms [7, 8], or greedy algorithm. However, the major issue with most algorithms based heuristics is that it did not take into account separating seed set from each other and little works have been addressed this problem. By this way, how we can identify efficiently the identification of such seed set in a way the promoted product, will be maximized?

So, in this paper, we try to combine designing an efficient selection method based on degree over radius hops and separating the selected seed by a number of hops. Another challenge that faces methods based heuristics is that those algorithms may perform better on some graph while providing a little performance on other graphs, we dealt with this issue by determining a selection threshold and multi-hops distance value that depend on each graph data in order to keep good influence coverage while maintaining acceptable time complexity. And that the methods based greedy algorithm suffer from high time complexity and as known, social networks have an increasing scale and are sparse in nature; which makes its applicability impractical.

In this paper, we propose two extended algorithms from our previous work [9]. The proposed algorithms come with the improvement of the seed set selection process by designing a selection threshold value for each graph data for the purpose of preventing the choice of the node with low influence that have a low power in other users to adopt the promoted behavior. The selection threshold value is based on structural properties that vary with the graph. As well as, we tried to fix neighborhood hops at radius minus 1, this that when we compute the neighborhood of radius is as computing the neighborhood of entire graph and this is applied for each candidate seed to be selected. This helps to select the node that has an influence on a different range of users. Thus we extended our algorithms to a directed graph and then test the selected seed set on independent cascade and linear threshold models. The aim is to identify the most influential users that are close to touching a large number of users within the real social networks. At a glance, the proposed algorithms identify the most influential users in the network that when measuring the influence of diffusion models showed its performance against most well-known approaches in the literature.

In this section, we attempt to provide a comprehensible overview of related work on a selection of top-K influential users, by providing a literature review that tackled the select of seed set based on multi-hops distance and then we exposed works that addressed the problem of influence maximization from neighborhood perspective and other centrality measures. The reason for selection of such works comes from the design of our extension of [9] that uses both multi-hops distance and neighborhood constraints to select top-K influential users. We note that the purpose is to highlight the general ideas of each related paper and the difference between the existing approaches and our proposed approach.

Wang et al. [10] proposed an efficient algorithm for distance-aware influence maximization problem, that takes into account the distance in which a query was promoted. As consequence, the top-K influential users are different depending on the query promotion locations. For this purpose, they developed two approaches maximum influence arborescence-Distance aware (MIA-DA) that uses the maximum influence arborescence model (MIA) to compute the influence spread and reverse influence sampling-Distance aware (RIS-DA) based on information of pre-sampled query locations. They also tried to improve the RIS-DA approach that returns a good approximation solution for any query. In the same direction as [10], Wang et al. [11] proposed a priority-based algorithm which seeks to find users based on their influence level, by using the distance-aware to select top-K users that are more close to the promotion locations. Thereafter, they calculated the influence coverage based on MIA model. Then, they proposed a novel index that was used as bounds in pruning strategies. They adopted as well various pruning strategies that exclude low ranked nodes from evaluation to accelerate the search of most influential users. In the same direction as [10, 11], Nguyen et al. [12] propose a heuristic algorithm that takes into account propagation probabilities of nodes in the network. Then they estimate the optimal number of hops between neighbors for nodes selections. This work is the closest work to ours in term of considering the optimal number of hops between neighbors to boost the influence spread. Liu et al. [13] propose a neighborhood centrality measure that takes into account node neighbors and neighbors of neighbors that was called in their paper neighborhood centrality. The neighborhood centrality relies on immediate neighbors and 2-hops neighbors and as argued by authors that increasing the neighborhood hops decrease the influence spread. Similarly, Bae et al. [14] propose a new coreness centrality measure using the k-shell indices that were considered more performant than a degree and other centrality measures of its neighbors to compute the influence coverage of a node in a network using the k-shell indices of its neighbors. However, their approach is limited to test on the scale-free network, which limits its applicability and its efficiency on the large-scale real-world network. Likewise, Ruan et al. [15] present an improved coreness measure by decreasing the impact of densely local connections and taking into consideration the effect of the connections between nodes on the nodes’ spreading capability.

Zhang et al. [16] proposed a VoteRank algorithm to select a set of decentralized spreaders that did not fall within the same influence range with the highest spread capability. Their approach relies on all nodes voting in a spreader in each step, and the voting capability of neighbors of chosen spreader will be decreased in next step. The susceptible, infected, and resistant (SIR) model is used as diffusion model to calculate the influence coverage.

Zhang et al. [17] proposed two greedy algorithms namely greedy and greedy++ based 2- players coordination game (CG). They incorporate their CG into the general diffusion process that is considered as voter model and linear threshold model. They prove that the objective function of the CG is monotone and submodular and then they accelerate the computation of spread coverage by using two heuristics Lazy forward and StaticGreedy. Their greedy ++ algorithm is faster by three orders of magnitude. But still, need improvement in term of running time. Analysis of equilibria is missing and it would be interesting to study the existence of pure and mixed Nash equilibrium in which users have no incentive to deviate by choosing other strategies (i.e users are all satisfied by the current profit).

L et al. [18] present a review the state of the art of the influence maximization and outline concepts and metrics used to identify vital nodes. Then, they conducted extensive empirical analysis over existing approaches to real-world datasets, and point out future research directions.

Radicchi et al. [19] present a method that investigates the detection of superblockers that seeks to minimize the spread of influence and superspreaders that when selecting the influence coverage is maximized. And then mention that recently is was argued that the identification of superblockers and superspreaders are equivalent. They conduct extensive analysis over a big real-world network in the purpose to identify if there is a similarity between superblockers and of superspreaders. They found that the two problems are not equivalent and that superblockers did not act optimally as superspreaders. Namtirtha et al. [20] conduct a study of the analysis of k-shell most influential node and found that even core node in k-shell decomposition method is not the most influential spreader and that lower core node may be the most influential. As to deal with this issue, they propose an indexing method using nodes k-shell, degree, contact distance and neighbors influence potential to increase the spread of influence under SIR model.

Alshahrani et al. [21] proposed a new algorithm PrKatz for selection of top-K influential users based on Katz centrality and the propagation probability threshold that permits to compute the influence over all the paths and select the one that maximizes the influence. The algorithm PrKatz relies on the use of a combination of Katz centrality and propagation probability threshold tested over each edge for each user in the network. Then top-K influential users are extracted in a decreasing order following the new formulated Katz centrality. Their algorithm outperforms the state of the art algorithms in term of influence coverage.

Alshahrani et al. [9] proposed two novel algorithms namely ’‘RND d-hops” and “CPRND d-hops” considering the results of [6]. These proposed algorithms are for selecting Top-K propagators based on degree centrality heuristic and distance that separate the selection of each seed node from another within the network. This study considers the good performance of degree centrality measure in terms of influence achieved and low runtime complexity. Furthermore, the in-depth investigation has been performed on the usefulness of this metric.

Due to the good performance in term of influence coverage and running time of algorithm “RND d-hops” [9]. We extended the results of RND d-hops algorithm in term of controlling the identification of seed set based on a precomputed selection threshold value depending on the structure of each data and by selecting each consecutive seed set far away r−1 hops to overcome the selection of nodes in the same range of influence coverage. We extended also, our analysis to the directed graph under the Independent cascade model and linear threshold model. As well as, we discussed complexity analysis of each proposed algorithm.

We compared as well as our algorithms to the state of the art approaches under the IC model and LT model including: simple heuristics including degree, Page Rank and sophisticated heuristic such as degree discount heuristic, BCT, TIM+ and IMM:

• Degree: The degree centrality that selects Top-K propagators with the highest degree centrality. This baseline heuristic was used for comparison purpose in various research work such as [2] and [6].

• Page Rank: It is used by Google search engine. Page Rank proceeds by counting the number and quality of links of a node to all other nodes in such a way to determine how important is the node. Each node depends on the PageRank of all other nodes [22].

• Degree discount heuristic: It was introduced in [6] selects seed set based on the degree centrality score and discount the edge that bond with the next selected seed from the nodes degree computation.

Approximation algorithms

• BCT algorithm: It was proposed in [23] to find the most cost-effective seed users who can influence the most relevant users to the advertisement.

• TIM+ algorithm: It was proposed in [24] to improve the scalability of time complexity while providing low coverage spread of influence.

• IMM algorithm: It was proposed in [25] and provides the same worst-case guarantees as existing approaches but tries to improve its efficiency. This enables IMM to support a larger class of diffusion models than existing algorithms such TIM, TIM+ [24]. IMM compute the influence spread on large graph in less time than TIM and TIM+, since it try to reduce the time complexity by excluding unnecessary computation routine in Reverse-Reachable set. In spite of good results in term of speeding up the running time, the IMM algorithm suffers from difficulty of estimation of maximum influence and that taking all possible seeds set guarantee only one seed set as output.

To compare our proposed algorithms against existing algorithms related to our proposed approach listed above [6, 22–25], we run all algorithms under the IC model with \(p_{u,v}=0.1\) and LT model with a uniform probability distribution and we set the seed set size K from 10 to 50. The results of our algorithms to the result of discrete influence maximization. For \(d_s-d_I=r-1\), which portrays the multi-hops distance in which each consecutively selected seed separated with next seed by radius minus the graph center and by excluding only immediate selected seed neighbors. The selection threshold value of each graph depends on graph diameter as depicted in formula (1). We performed experiments on directed and undirected graphs under two diffusion models namely IC model and LT model.

Figures 5, 6 shows influence coverage on twitter data under the IC model and LT model respectively. We can notice that our algorithm outperform existing approaches in term of influence spread under the IC while has a low influence spread on LT model. This could be justified that our seed selection method suit more the IC model since our approach tries to select users that are not close to each other and that the LT need a fraction of neighbors to be infected to infect the current user. This results are the same as for Eu-Email data as depicted in Figs. 7, 8. So, according to experiments results our approach gives good performance under the IC model and is not good for LT model.

Tables 7, 8, 9 and 10 represent the running time of our approaches on the directed graph and LT model and IC. We can observe clearly that the LT model for our approach is faster than other approaches such as degree discount heuristics and that even the running time under IC for our algorithm still acceptable to be applied on large scale graph.

To sum up, according to experiments on large scale graph with different topology and size, we can notice that our approach perform very well on undirected graph under the IC model, which is illustrated in Fig. 1 and Table 2. This can be justified by score value of neighborhood computed from active node till its radius, this helps to identify the node with most direct and indirect relationship. This permits to quantify in an efficient manner the potential users to be targeted and that adding the constraint of a selection threshold value for each node score favorize node with a certain efficiency and that separate each consecutively selected node by a certain number of hops enables targeting nodes that have an influence on users from a distinct region of influence. This computation of neighborhood number of score value for each node makes our approach a little bit time consuming but still perform well on large scale graph and therefore adequate for the real-world application. And as another observation for results of our approach under the LT model for undirected and directed setting, the one can observe that our approach gives lower results in term of influence spread and as discussed before that separating the selection of seed set by a certain hops reduces the influence of LT model, since this model needs a number of users to be activated in order that it can activate new nodes in its immediate neighborhood, this could be improved further by controlling the multi-hops distance in which we separate the selected nodes. However, the approach still doing well in term of time complexity and improved significantly compared to our previous approaches [9, 21].

In directed setting, we can notice clearly from our experiment on two large-scale data, that it performs well in term of influence coverage and surpass well-known algorithms from state of the art literature and that this is thanks to our followed methodology by providing reward of each gained indegree node and discount each outdegree node \(+1\), to prevent the zero in the denominator. This methodology is very efficient in a way that sometimes users gain followers only as a feedback of following back, so this measure will measure how much effort made by each node versus how much followers it receives and that despite sometimes proposed score may give a little bit lower results in term of influence but it will perform better on real-world situation. Since as known always people follow known people who have a lot of connections and that known people rarely follow back modest people(i.e nodes that have few connections), so this metric will certainly work very well in real-world application. Consequently, as our approach computes for each node the corresponding indegree and outdegree over radius hops, this will be more time consuming but as stated before, the results will be accurate and near to real-world situations.

In this paper, we proposed new algorithms for maximizing the influence spread by the selection of top-K influential users within the network. Specifically, we presented two efficient algorithms for directed and undirected graph namely “DERND D-hops”, “UERND D-hops” respectively under the IC model and LT model. The proposed algorithms try to improve the results of the undirected graph of our previous algorithm RND d-hops [9] and extend this algorithm to the directed graph. As observed through experiments and as argued by previous works that methods based heuristics may perform better on some graph while not be good on other graphs. For this purpose we dealt with this issue by introducing structural characteristics for each graph data, through a selection threshold value that permits to improve the selection of seed set in both directed and undirected graph and by using a predefined multi-hops distance for the selection of consecutive seed set depending on graph radius. This permitted the selection in a moderately large region to pick the most suitable nodes as the seed set. We computed the worst case time complexity and we demonstrate that our proposed algorithms is better than the state of the art approach especially for undirected graph in term of influence coverage and we reduced the time complexity compared with [9]. As a future extension, an investigation and more in-depth study should be performed regarding the multi-hops distance that separates all seed set in each data and not relies only on consecutive seed set. As well as, the IC model and LT model should be generalized to a more accurate and performant model that take into account not just propagation probability threshold but as well the type of shared content and that in real-world scenario companies may promote different products and that the extent of adoption of such product requires the study of users behavior and which product is more likely to be interested by a specific user. So, such study is very interesting besides including the structural properties of each data to boost the influence spread in a real-world scenario.