Effective spreading from multiple leaders identified by percolation in the susceptible-infected-recovered (SIR) model

We consider two social networks, namely the Facebook network and the Enron email network. Their statistical features are shown in table 1.

Table 1.  The basic characteristics of Facebook and Enron email networks. We denote by $| V|$ and $| E|$ the number of nodes and edges, respectively, C the clustering coefficient [26] and r the assortative coefficient [27]. We denote by $\langle k\rangle$ the average degree, $\langle d\rangle$ the average shortest distance, and H the degree heterogeneity, such that $H=\langle {k}^{2}\rangle /\langle k{\rangle }^{2}$.

Networks	$| V|$	$| E|$	C	r	$\langle k\rangle$	$\langle d\rangle$	H
Email	33696	180811	0.510	−0.117	10.732	4.025	13.266
Facebook	59691	1456818	0.228	0.175	24.4060	4.335	3.348

(i) Facebook: the friendship relations in the New Orleans Facebook social network. It is a directed network, consisting of nodes which correspond to Facebook users. Each directed edge represents a post or a comment, connecting the users who are writing the comment with the users whose wall the post is posted. Since users may write multiple comments on the same wall, the network allows multiple edges connecting a node pair. Since users may also comment on their own wall, the network contains self-loops. In our experiment, we consider the network as a typical undirected network by deleting self-loops and merging multiple links into a single link. We assume that two nodes are connected if there is at least one directed link between them. The data can be freely downloaded at http://levich.engr.ccny.cuny.edu/~hmakse/soft_data.html.

(ii) Enron email network [25]: Enron's email communication network covers roughly 0.5 million email communications between a group of users. This dataset was originally open to the public, and was posted on the internet by the Federal Energy Regulatory Commission during its investigation. Nodes of the network correspond to email addresses in the system, and if an address i sent at least one email to address j, there is an undirected edge between i and j. Note that non-Enron email addresses are considered as sources and sinks in the network, as we only observe their communications with the Enron email addresses, but not the communications between them. The data can be freely downloaded at http://snap.stanford.edu/data/email-Enron.html.

To quantify the performance of our method, we examine the spreadability, i.e. the propagation coverage of a piece of news from a set of W selected spreaders, by our method as well as other methods. We will use the SIR model to mimic the spreading of news, and the spreadability is defined as the ratio of recovered nodes, i.e. the size of outbreak, or the number of users who received the information to the total number of users. We remark that the transmissibility p adopted in the SIR model is the same as the probability p used to recover edges to identify the clusters in the segmented states. As a result, for a single spreader, the maximum size of the SIR outbreak triggered by this spreader is precisely the size of the cluster that it belongs to. Likewise, the maximum size of the SIR outbreak triggered by a group of spreaders in distinct clusters is the sum of the size of the clusters that these nodes belong to. For example, if we measure the maximum coverage of three selected nodes on the network with N nodes, and if the first two nodes belong to the cluster S_i, and the third one belongs to the cluster S_j, the maximum coverage of each individual node 1, 2, and 3 is respectively ${S}_{i}/N,{S}_{i}/N$ and ${S}_{j}/N$, while for the whole group, the maximum coverage is ${S}_{i}+{S}_{j}$, and the spreadability is $({S}_{i}+{S}_{j})/N$.

We first apply our method on the Facebook network with 59691 nodes. Figure 1(a) shows the coverage obtained from 4000 initial spreaders chosen by our percolation method, compared with a set of 4000 spreaders identified by four other methods, namely the k-shell decomposition, the betweenness centrality, the collective influence (CI) method [6], and the high degree adaptive method (HDA) where the degrees of nodes are recalculated according to the updated network (see the appendix for the definition of each of these methods). The percolation method yields the highest spreadability for an arbitrary transmissibility p as shown in figure 1(a), while comparisons with other centrality measures can be found in supplementary figure S2. We show in section 4 of the supplementary data that our method also outperforms the other methods on weighted networks in terms of spreadability.

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Figure 1(b) shows the degree distribution of the 4000 spreaders identified by the percolation method. When $p\lt {p}_{c}\approx 0.01$, the percolation method yields isolated clusters of similar size [21], and since the set of selected spreaders comes from different clusters, a wide range of degree is found among the spreaders as shown in figure 1(b). In this case, the percolation method is more likely to choose high-degree nodes, see supplementary figure S3(b), where the red stars represent the degree distribution of the 4000 selected nodes when $p=0.008$. When $p\gt {p}_{c}$, the distribution becomes narrower as p increases, see the blue squares in supplementary figure S3(b). In this case, the percolation method prefers low-degree spreaders. The average original degree (i.e. the degree in the original network before edge removal) of the 4000 selected spreaders by the percolation method when $p\lt {p}_{c}$ is higher than that of the nodes selected when $p\gt {p}_{c}$. This implies that if we promote and advertise a new niche product which is regarded as difficult to accept, one can draw an analogy with the case of small transmissibility p where high-degree initial spreaders are preferred. On the other hand, for popular items which are easy to accept, one can draw an analogy with the case of large p and low-degree initial spreaders are preferred.

We then examine the cost of initializing the spreading from the selected spreaders. Information propagation sometimes induces a cost, for instance, one may need to pay the star bloggers for posting and passing on an advertisement. We assume that the direct influence of a user is equal to the number of its nearest neighbors, i.e. the degree of the user, and the difficulty of finding a user with degree k is proportional to $1/p(k)$, which can be considered the scarcity of the user. Here $p(k)$ is the occurring frequency of nodes with degree equal to k. The cost to initialize (or hire) a spreader i is proportional to his/her impact as well as scarcity, and hence the cost is assumed to be ${k}_{i}/p({k}_{i})$. Obviously, if $p({k}_{i})$ follows the power-law distribution, namely $p({k}_{i})\sim {k}^{-\alpha }$, then cost is proportional to ${k}^{1+\alpha }$. The larger the degree of a user, the higher the cost to hire him/her. Figure 1(c) shows the dependence of the average cost of 4000 spreaders under the parameter p, i.e. $\tfrac{1}{W}{\sum }_{i=1}^{W}\tfrac{{k}_{i}}{p({k}_{i})}$. The cost decreases abruptly at the critical point p_c, suggesting a phenomenon resembling a phase transition. It implies that when p increases just beyond p_c, the cost can be reduced substantially. We have tested that if the cost is defined as ${k}^{1+\alpha }$, similar results are obtained.

Besides spreadability and cost, we also examine redundancy in coverage which quantifies the efficiency of propagation. Specifically, the redundancy of a node i is defined as the number of initial spreaders who have the potential to infect node i. A method is inefficient if the initial chosen spreaders pass the same information to the same group more than once. Averaging redundancy over all infected nodes, we obtain the redundancy of the set of initial spreaders. Figure 1(e) compares the spreading redundancy of our method with the other four methods (comparisons with other centrality measures can be found in supplementary figure S4). The highest redundancy is found in the methods of k-shell and degree centrality, followed by CI, and then by betweenness centrality. Our percolation method has the lowest redundancy compared to the other four methods, since the spreaders identified by our method are usually located in different regions of the network. We also checked the Enron email network and results similar to the Facebook network are obtained, see figure 2. We also show in section 4 of the supplementary data that our method outperforms the other methods on weighted networks to reduce the redundancy in coverage.

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To further examine redundancy in coverage, we applied the five methods to identify four initial spreaders on a simulated network with clear communities. There are three steps to generate a network with community structures. In our experiment, we consider a network with 2000 nodes which has four communities, each of which contains 500 nodes. First, we generate a random network of size 500 and with node degrees distributed in a power-law with exponent 2.2 using the configuration model [28]. The minimum degree is 1 and the maximum degree is $\sqrt{500}\approx 23$ [29]. Second, we repeat the above procedures to generate independently the other three networks. Finally, for each pair of sub-networks we randomly select a fraction of node pairs to connect them. As shown in table 2, the four spreaders identified by the percolation method are likely to be found in different communities. For the other methods, there are high probabilities that at least two initial spreaders are in the same community. This result is easy to understand as our method relies on the segmentation of networks into isolated clusters. In this case, the network separates into four communities and thus one spreader is found in each community.

Table 2.  Percentage of different distribution of the four spreaders in a model network with four communities. For instance, the four spreaders identified by our method are found in four different communities in $93.9 \%$ of the 1000 realizations. In the percolation method, we have set $p=0.28$ which is slightly higher than the critical point of the network ${p}_{c}\approx 0.2687\pm 0.0152$.

Distribution of the four selected spreaders (red nodes)
Percolation	93.9	6.1	0	0	0
K-shell	9.4	43.6	11.9	16.2	18.9
Betweenness	13.1	64.5	11.7	10.6	0.1
CI	10.3	61.5	12.2	15.2	0.8
HDA	11.7	60.7	13	14.2	0.4

Although the leader score is aggregated after M realizations, a different set of leaders may be generated if another set of M realizations of percolation is generated, since link recovery in our percolation-based method is stochastic in nature and the highest degree nodes in small clusters are not unique. While the other methods always suggest the same set of initial spreaders, different spreaders can be generated by reiterating our percolation procedures, especially for large values of p. Figures 3(a) and (b) show the average number of common nodes in two different solutions of the percolation method, i.e. n_c, on Facebook and Enron email networks, respectively. It is clear that when p increases, the number of common spreaders decreases, indicating that the solutions become more diverse. This result has practical significance; in cases when some initial spreaders are offline, we can use the next best candidates as back-up spreaders without extensively losing spreadability. Compared with the other four methods, the percolation method provides higher flexibility in the choice of spreaders. We further calculate the Shannon entropy of the obtained solutions, which is defined as

$\begin{eqnarray}&&{\rm{Entropy}}=-\displaystyle \sum _{i}^{N}{q}_{i}{{\rm{log}}{q}}_{i},\end{eqnarray} \tag{ 1 }$

where q_i is the percentage of realizations where node i is found. Figures 3(c) and (d) show the dependence of Shannon entropy on parameter p. A non-zero Shannon entropy also indicates that various solutions are found and different sets of nodes are identified as the initial spreaders. A more non-trivial trend of the Shannon entropy is observed compared to the entropy computed by merely the number of different solutions. For instance, a peak of Shannon entropy at the intermediate values of $p\gt {p}_{c}$ is observed. In this case, a giant component exists in the network together with many small clusters. Depending on the random recovery of links, the set of smallest clusters is different for different solutions. When the total number of clusters is roughly equal to the number of identified spreaders, one spreader is identified for each cluster, including the smallest clusters. The different sets of smallest clusters then contribute to the different sets of identified spreaders, and hence a peak in the Shannon entropy at intermediate values of p when the number of clusters is roughly equal to L.

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To further examine the difference between our method and the other methods, figure 4 shows the fraction of common spreaders with the percolation method, i.e. f. Comparison with other methods can be found in supplementary figure S5. The overlap between the percolation method and the degree centrality method reaches the highest value at the critical point ${p}_{c}=0.01$ and then sharply decreases to less than $5 \%$ when $p=0.03$. This is because when p increases, most high-degree nodes are replaced by nodes with smaller degree, and there are many sets of spreaders generated from different realizations of the percolation method as we have discussed in figure 3.

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We show in figures 5 and 6 the spreadability-cost profile of the group of top-W selected spreaders. Similar results are found for the two networks. Four cases are presented, namely $p=0.008\lt {p}_{c}$, $p=0.01={p}_{c}$, $p=0.012$, and $p=0.02\gt {p}_{c}$. As we can see, the percolation method is most cost effective in a large range of spreader cost, i.e. with the highest spreadability given a selected spreader of the same cost. On the other hand, betweenness identified low-cost spreaders with high spreadability, but their spreadability is limited to a small maximum value compared to the other periods. In this case, the spreader with maximum spreadability is always identified by the percolation method. Besides the two real networks, we also investigated scale-free networks. Similar results are found; see supplementary figures S6–S11.

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To identify the most influential spreaders, various centrality measures have been proposed. The simplest method is degree centrality, which we compare our result with. Degree centrality is a straightforward and efficient metric. It assumes that a node with more nearest neighbors has a higher influence. However, node degree can only reflect its direct influence and not the indirect influence triggered by its nearest neighbors. For example, a node of small degree, but with a few highly influential neighbors may be more influential than a node having a larger number of less influential neighbors. In this paper, we employed the adaptive version of degree centrality as one of the baseline methods, namely the high degree adaptive (HDA) approach which recalculates node degree after the removal of links in the network. We compare the high degree (HD) method with the high degree adaptive (HDA) method and find that the adaptive method performs slightly better than the static high degree strategy (see supplementary figure S2).

The second method we used for comparison is k-shell decomposition. Recent research shows that the location of a node in a network may play a more important role than its degree. A node located at the center of the network is more influential than a node having a larger number of less influential neighbors. Similar to this rationale, Kitsak et al [5] proposed a coarse-grained method by using the method of k-core decomposition to quantify the influence of a node based on the assumption that nodes in the same shell have similar influence, and nodes in higher-level shells are likely to infect more nodes.

The third method is betweenness which is one of the most popular geodesic-path-based ranking measures. It is defined as the fraction of shortest paths between all node pairs that pass through the node of interest. Betweenness is, in some sense, a measure of the influence of a node in terms of its role in spreading information [31, 32]. For a network $G=(V,E)$ with $n=| V|$ nodes and $m=| E|$ edges, the betweenness centrality of node v, denoted by $B(v)$ is [7, 33]

$\begin{eqnarray}&&B(v)={{\rm{\Sigma }}}_{s\ne v,s\ne t,v\ne t}\displaystyle \frac{{g}_{{st}}(v)}{{g}_{{st}}}\end{eqnarray} \tag{ 2 }$

where g_st is the number of shortest paths between nodes s and t, and ${g}_{{st}}(v)$ denotes the number of shortest paths between nodes s and t which pass through node v.

The last method we compare our method with is the 'collective influence' (CI) method proposed by Morone and Makse [6]. Define ${\rm{Ball}}(i,l)$ as the set of nodes inside a ball of radius l (defined as the shortest path) around node i, $\partial {\rm{Ball}}(i,l)$ is the frontier of the ball. Then the CI index of node i at level l is defined as

$\begin{eqnarray}&&{{CI}}_{l}(i)=({k}_{i}-1)\displaystyle \sum _{j\in \partial {\rm{Ball}}(i,l)}({k}_{j}-1),\end{eqnarray} \tag{ 3 }$

where k_i is the degree of node i. Here we set l = 3.

Given a network $G(V,E)$, there are four steps to find the W influential spreaders by the percolation method. Firstly, all the edges are first removed and then recovered with a probability p; we then obtain a new network $G^{\prime}$ in segmented state. The required computational complexity is $O(| E| )$. Secondly, we find the strongly connected components of $G^{\prime}$ using Tarjan's algorithm [34] which has a complexity of $O(| V| +| E| )$. Thirdly, we select one node with the highest degree in each of the L largest components and assign one score to the selected nodes. This complexity for the procedures is $O(L\times | V| )$. Repeating the above three steps for different realizations, we rank the nodes according to their scores in descending order, and the top-W nodes are chosen to be the most influential spreaders. The different realizations of the percolation process can be computed in parallel and the complexity of each implementation is $O(| E| +| V| +| E| +L\cdot | V| )$. Considering $\langle k\rangle =\tfrac{2| E| }{| V| }$, then the complexity is $O[(\langle k\rangle +L+1)\cdot | V| ]$. Since $\langle k\rangle \ll | V|$ in real networks, then we have $O[(\langle k\rangle +L+1)\cdot | V| ]\sim O(| V| )$, i.e. the complexity of one realization of our method grows linearly with system size. Assume there are M realizations, the total complexity would be $O(M| V| )$.