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SCIENTIA SINICA Informationis, Volume 49, Issue 10: 1333-1342(2019) https://doi.org/10.1360/N112019-00041

A novel method to identify multiple influential nodes in complex networks

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  • ReceivedFeb 21, 2019
  • AcceptedApr 25, 2019
  • PublishedOct 17, 2019

Abstract

In the spread of information, a small fraction of nodes play an important role in the dynamics; therefore, detecting these influential nodes helps control the spread of information. However, classical methods can choose a single influential node yet fail in the case of multiple influential nodes. In this paper, we analyze the collective influence and overlapping influence of multiple spreaders. Further, based on the overlapping influence between spreaders, we clarify the problem of why multiple influential spreaders may have a low collective influence. Finally, a new algorithm is proposed to choose multiple spreaders, and the performance of the algorithm is validated on real networks.


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  • Figure 1

    (Color online) Comparison of the information spread in networks with loops. (a) An artificial network; (b) the back tracking edge (red) that is neglected in Eq. (7); (c) the back path of the information spread in networks with loops.

  • Figure 2

    (Color online) The principle eigenvalues of the remaining networks that are obtained by removing the edges attached to the influential nodes. (a) Airtraffic network; (b) Bitcoin network; (c) ca-HepTh network; (d) Reactome network.

  • Figure 3

    (Color online) The evolving paths of the spread as a function of time. In the experiments, every node has a probability of 0.001 to be an infected node initially. The results are the average of 50 independent simulations. (a) Airtraffic network; (b) Bitcoin network; (c) ca-HepTh network; (d) Reactome network.

  • Figure 4

    (Color online) The average distance between influential nodes. (a) Airtraffic network; (b) Bitcoin network;protect łinebreak (c) ca-HepTh network; (d) Reactome network.

  • Table 1   Structural properties of the different real networks, including network size ($N$), link number ($E$),degree heterogeneity ($H~=~\langle~k^2~\rangle/\langle~k~\rangle^2$), degree assortativity ($r$), average clustering coefficient ($\langle~C~\rangle$), average shortest path length ($\langle~d~\rangle$) and sparsity
    Network $N$ $E$ $H$ $r$ $\langle~C~\rangle$ $\langle~d~\rangle$ Sparsity
    Airtraffic 1225 2399 1.87 $-$0.0177 0.011 5.93 $3.2\times10^{-3}$
    Bitcoin 5880 21228 10.89 $-$0.163 0.022 3.59 $1.2\times10^{-3}$
    ca-HepTh 5039 11346 2.03 0.195 0.101 6.54 $8.9\times10^{-4}$
    Reactome 3851 140106 1.99 0.316 0.508 0.508 $1.9\times10^{-2}$
  •   

    Algorithm 1 The pseudo code to compute the set of influential nodes

    Require:The adjacent matrix $A=(a_{ij})_{N\times~N}$ and the size of influential nodes $L$.

    Data preparation. Extract the giant component of the original network and remove the isolate nodes and other small nodes clusters because nodes located in distinct clusters could not exchange information.

    Initialize the algorithm and calculate the importance of nodes based on Eq. (7).

    Rank nodes based on the importance of nodes and choose the particular node with the largest importance score.

    If the size of influential nodes is smaller than $L$, remove the edges attached to the influential nodes and go to step 2; otherwise go to step 5.

    Return the index of the chosen influential nodes.Output: The index of the selected influential nodes.

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