Many real-world problems involve the identification of multiple influential nodes in complex networks, such as finding a few individuals who are critical to the spread of information on the internet, or who may speed up the transmission process of pestilence in crowds once infected [1]. The problem of identifying multiple influential nodes differs from that of discovering the most influential nodes. The latter refers to finding the k (k > 1) most influential spreaders, which is commonly addressed by ranking the influence of individual nodes. The former involves the identification of a set of k nodes with the maximum influence as a whole. That is, identifying multiple influential nodes should take into account the different roles that nodes play in the propagation process rather than just evaluating their individual influence [2].

Methods to identify multiple influential spreaders fall in three categories. The first regards this as an influence maximization (IM) problem. Some well-known methods include the greedy [3], new greedy [4], community-based greedy [5], k-medoid [6], two-phase influence maximization [7], and collective influence [8] algorithms. However, as the IM problem is NP-hard, these algorithms are challenged by increasing network sizes, and thus are not applicable to huge real networks.

Methods in the second category attempt to identify multiple influential nodes by ranking their influences, which are calculated according to various topology-based centrality measures: 1) classic topological centrality metrics, such as degree centrality [9], betweenness centrality [10], and closeness centrality [10]; 2) centrality measures that take into account multiple (global or local) network features, such as KED centrality [11], efficiency centrality (EC) [12], composite centrality based on analytic hierarchy process [13], and classified neighbors centrality [14]; and 3) local-information-based iterative algorithms such as PageRank [15], LeaderRank [16], and VoteRank [17]. However, the ranking approach may not always find a set of nodes with the maximum influence [18], possibly because they separately measure the influence of each node, and thus omit overlapping effects of topologically adjacent top-ranked nodes.

Algorithms in the third category consider the distance between nodes when evaluating node importance. For instance, the local index rank (LIR) algorithm [19] is based on the local index (LI) value of a node, which represents the number of neighbors whose degree exceeds that of the focus node. Spreaders are selected from nodes whose LI values are 0 (i.e., 0-LI nodes). However, the LIR method cannot avoid some adjacent 0-LI nodes, and sometimes there are not enough 0-LI nodes to be selected as spreaders. Another example is the independent set (IS) algorithm [20], which divides nodes into independent sets by the Welsh-Powell coloring algorithm and selects spreaders in the largest independent set to ensure that selected nodes are non-adjacent. However, special situations may occur, such as not enough spreaders in the largest independent set; meanwhile directly selecting rest spreaders in following independent sets may derogate the advantages brought by independent set.

We propose three methods with different degrees of dispersion to identify multiple spreaders. The first one is LIR-2 method which extends the concept of the local index to second-order neighbors and does not restrict the spreaders’ selection from the 0-LI nodes. By doing so, this method enlarges the distance between the 0-LI nodes and can guarantee to select enough spreaders. The second one is IS-LIR method which hybrids LIR and IS to ensure that nodes in the same independent set are non-adjacent. The third one is IS-LIR-2 method, which hybrids the improved second-order LIR method and IS method so that the selected spreaders are more dispersed. Comparing the proposed three methods with traditional methods for multiple spreader identification on 10 synthetic networks and five real networks based on the SIR propagation model, we find our methods more effective in maximizing the size of the spreading coverage, and that a higher dispersion of the selected multiple spreaders helps to amplify the spreading.

The rest of this paper is organized as follows.Sec. 2 introduces work relating to the identification of multiple influential spreaders. Sec. 3 formalizes the research problem and proposes our method. Sec. 4 describes our experiments, including baseline methods, the SIR propagation model, evaluation metrics, parameter settings for experiments, and datasets. Sec. 5 provides the experimental results and discusses why diversity should be considered when we select a set of influential spreaders. We summarize our work in Sec. 6.

Identifying a set of influential nodes in a network is important for designing network immunization [21], system control strategy [22] and improving the network robustness [23,24]. Work about multiple spreader identification falls in three categories. The first regards it as an influence maximization problem [3], and thus utilizes optimization algorithms to directly identify a set of spreaders. The greedy algorithm [3] is a classic example. These algorithms are accurate but time-consuming, and thus do not suit large-scale networks. Some researchers employ information about network structures to reduce the time complexity while maintaining the high accuracy of classic optimization algorithms. The NewGreedy algorithm [4] removes edges that do not contribute to propagation, so as to speed up the simulation process. The community-based greedy algorithm (CGA) [5] mines the top-k spreaders from detected communities so as to reduce the running time. Another algorithm [6] constructs an information transfer probability matrix and uses the k-medoid clustering algorithm to find the most centrally located nodes in clusters as spreaders. Two-phase influence maximization (TIM) [7] includes the phases of parameter estimation and node selection to reduce time complexity.

Methods in the second category select the top-ranked spreaders, whose influence is calculated based on network topological information. Classic indicators such as degree centrality [9], betweenness centrality [10], closeness centrality [10], and coreness centrality [25], have been utilized to estimate the influence of spreaders. Some researchers take into account multiple (global or local) network features when measuring the importance of spreaders [26]. For instance, KED centrality [11] combines the number and diversity of paths. Composite centrality based on the analytic hierarchy process (AHP) [13] combines degree, betweenness, and closeness centrality. Classified neighbors centrality (CNC) [14] classifies the neighbors of a focal node into four groups according to the removal order in the process of k-shell decomposition, weights each class differentially, and sums them to characterize the spreading capacity of the node. PageRank [15], LeaderRank [16], and VoteRank [17] all consider the importance of a node itself and its connections with other nodes to identify influential nodes. These rank-based algorithms often have simple forms and low time complexity and can effectively mine a single important node. However, they may not efficiently find multiple important spreaders because they seldom consider interactions between spreaders, i.e., they ignore the overlapping effects of top-ranked nodes if they are topologically adjacent.

Algorithms in the third category attempt to minimize the overlapping effects of spreaders during selection. The SuperNode algorithm [27] uses the Blondel community detection algorithm to get the community division in the network, and selects important nodes from the communities according to size so that the selected nodes have some distance. An independent set (IS)-based partitioned ranking algorithm [20] divides nodes into independent sets by the Welsh-Powell coloring algorithm, then selects the top-ranked nodes in the largest independent set based on certain centrality indicators. The local index rank (LIR) algorithm [19] selects spreaders from nodes with 0-LI values, i.e., those whose direct neighbors have lower degrees than themselves. However, there may not be enough 0-LI nodes to be selected as spreaders in some cases, and the selection of adjacent nodes cannot be avoided. We seek to overcome the above deficiencies by extending LIR methods to two-layer neighbors and integrating them with IS methods.

We formalize the problem of multiple influential spreader identification and propose the LIR-2, IS-LIR, and IS-LIR-2 algorithms, which consider the diversity of nodes to different degrees.

We introduce the classic SIR model, which will be utilized to simulate epidemic spreading, and present two evaluation metrics to compare the performance of the proposed methods with eight baseline methods: degree centrality ranking (DC) [9], LIR [19], degree centrality ranking based independent set (IS-DC) [20], eigenvector centrality ranking based independent set (IS-EV) [20], neighborhood centrality ranking based independent set (IS-ND) [20], and VoteRank [17]. We describe the synthetic and real networks used in our experiments, and discuss parameter settings.

We present the experimental results evaluating the proposed methods, and determine whether identified spreaders are effective while varying the infection probability λ. We show the relationships between the dispersion and the effectiveness of influential spreaders identified by our proposed methods and the baseline methods.

Figure 2 displays the final outbreak size F (t _end ) for different numbers of spreaders (denoted by the proportion of selected spreaders p) identified by different methods based on SIR simulations, and shows that our proposed methods generally outperform the baseline methods on synthetic and real networks. On WS networks, IS-LIR-2 has the largest final outbreak scale on WS1. IS-LIR-2, IS-LIR, and VoteRank perform similarly to or better than other algorithms on WS2 and WS3. On BA networks, the performance of IS-LIR-2 and LIR-2 is superior to the other methods, especially on BA2, BA3, and BA4. On LFR networks, IS-LIR performs better than the other methods, and IS-LIR-2 performs better on LFR2 but not so well on LFR3. In experiments on real networks, IS-LIR and IS-LIR2 could identify more influential spreaders in most cases on almost all five real networks. However, LIR-2 was not significantly superior on real networks except the arenas-email network. LIR-2, IS-LIR, and IS-LIR2 had obvious advantages selecting multiple spreaders in most cases. This implies that to take into account the dispersion of selected nodes can improve performance.

As the infected probability in the SIR model is a key parameter that may affect the final break size of infections, we explored the performance (represented by the final outbreak size proportion F (t _end )) of our proposed methods with different values of the infected probability λ. As shown in Figure 3, whether λ is small or large, IS-LIR and IS-LIR-2 have significant advantages over baseline methods on most of the synthetic and real networks. Specifically, on WS networks, as the infected rate increases, the performance of IS-LIR-2 increases significantly on WS1 and WS2, and IS-LIR performs best on WS2. On BA networks, IS-LIR-2 and LIR-2 are consistently superior to other algorithms on most BA networks. Focusing on LFR networks, we can see that IS-LIR is always better than the baseline methods except on the LFR2 network, where IS-LIR-2 performs better. On real networks, we can see that IS-LIR and IS-LIR-2 maintain their advantages whether λ is small or large.

Figure 4 presents the structural characteristics of influential nodes identified by LIR-2, IS-LIR, and IS-LIR2 and the baseline methods, and shows that spreaders identified by IS-LIR-2, IS-LIR, and LIR-2 have the largest harmonic mean of the average shortest path length L between any two nodes in most cases, except the LFR3 and football networks. On LFR3, multiple spreaders identified by IS-LIR, IS-ND, and IS-EV have the top three average shortest path lengths (as shown in Figure 4J). On the football network, vote-rank and IS-DC identified spreaders with larger average shortest path lengths than our proposed method in a few cases (as shown in Figure 4K). These results may explain why our proposed methods outperform the baseline methods in identifying multiple influential spreaders (as shown in Figure 2): if the identified spreaders have a larger mean shortest path length, they may result in a more heavier infection spreading. This implies that taking into account the dispersion of nodes can help find the most influential spreaders.

To effectively identify a set of influential spreaders is important in infectious disease prevention or information dissemination. To address this problem, inspired by the LIR method [19] and IS method [20], we proposed the LIR-2, IS-LIR, IS-LIR-2 algorithms, which take into account the dispersion of selected spreaders in different ways. In evaluation experiments on 10 synthetic networks and five real networks, our proposed methods, especially IS-LIR and IS-LIR-2, were more effective than six baseline methods at identifying more influential spreaders. One potential reason is that the spreaders found by our methods have a larger average shortest path length, i.e., the selected spreaders are more dispersed, so as to reduce the opportunity to infect the same nodes in the propagation process. IS-LIR, LIR-2, and IS-LIR-2 achieved a good balance between expanding the final spreading range of the spreaders on the SIR model and increasing the topological distance between them. However, we merely studied static, undirected, and unweighted networks. How to extend our methods to other types of networks, and how to investigate their sensitivity to specific network characteristics are two interesting questions to be addressed in future work.