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Identifying localized influential spreaders of information spreading
Accepted Manuscript Identifying localized influential spreaders of information spreading Xiang-Chun Liu, Xu-Zhen Zhu, Hui Tian, Zeng-Ping Zhang, Wei Wang
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S0378-4371(18)31477-8 https://doi.org/10.1016/j.physa.2018.11.045 PHYSA 20363
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Physica A
Received date : 2 August 2018 Revised date : 1 October 2018 Please cite this article as: X.-C. Liu, X.-Z. Zhu, H. Tian et al., Identifying localized influential spreaders of information spreading, Physica A (2018), https://doi.org/10.1016/j.physa.2018.11.045 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlights (1) Proposing a random walked based strategy for identifying the localized influential spreaders from a randomly selected initial-seed node. (2) Selecting large degree nodes preferentially is more likely to find the most influential spreaders.
*Manuscript Click here to view linked References
Identifying localized influential spreaders of information spreading Xiang-Chun Liu1,2 , Xu-Zhen Zhu1 , Hui Tian1∗ , Zeng-Ping Zhang 3 , Wei Wang4 1. State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, People’s Republic of China 2. School of Information Engineering, Minzu University of China, Beijing, 100081 People’s Republic of China 3. School of Computer & Information Management, Inner Mongolia University of Finance and Economics, Hohhot 010070, People’s Republic of China 4. Cybersecurity Research Institute, Sichuan University, Chengdu 610065, People’s Republic of China
Abstract Identifying the influential spreaders of information spreading dynamics is a hot topic in the field of network science. To identify the influential spreaders, most previous studies were based on the global information of the network. In this paper, we propose a strategy for identifying the influential spreaders from a randomly selected initial-seed node. The seeds are connected as a chain, and are localized to the initial-seed. In our proposed preferentially random walk based influential spreaders identifying strategy, the walker’s movement is adjusted by neighbors’ degrees. The seeds are those nodes that the walker ever visited. Through extensive numerical simulations on artificial networks and four real-world networks, we find that selecting large degree nodes preferentially is more likely to find the most influential spreaders. The outbreak threshold decreases when preferentially select hubs. Our results shed some light into identifying the most localized influential spreaders. Keywords: Complex networks, information spreading dynamics, localized influential spreaders PACS: 89.75.Hc, 87.19.X-, 87.23.Ge
∗
[email protected]
Preprint submitted to Physica A
October 1, 2018
1. Introduction Many real-world systems can be described by complex networks, the nodes represent the elements and edges stand for the relationships among different elements [1, 2, 3, 4, 5]. The diffusion of information, disease, healthy behavior, and innovation in social system can be mapped into investigating the spreading dynamics on complex networks, which have attracted much attention from the field of mathematics, physics, and network science [6, 7, 8, 9, 10, 11, 12, 13]. On the one hand, researchers investigated the interplay between the spreading dynamics and network topology, and obtained some interesting results. For instance, a few hubs (i.e., nodes with large degrees) induce the outbreak threshold vanishes for networks with scale-free degree distributions [14, 15, 16, 17, 18]. In addition, the heterogeneity of weight distribution will increase the value of outbreak threshold [19, 20, 21, 22]. Recently, some scholars revealed the mulplexity and temporal of networks markedly altered the spreading dynamics and outbreak threshold, and some novel theoretical approaches were developed [23, 24, 25, 26, 27]. On the other hand, some strategies have been developed to select some nodes to suppress or promote the spreading dynamics [28, 29, 30, 31, 32, 33, 34]. For the former case, researchers called it as network immunization. In this case, researcher usually propose some strategies to measure the importance of nodes, such as degrees, betweenness, eigenvector centrality and k-core, and immunize the nodes with high scores [35]. For the later case, it is called as influential maximization [36]. Generally speaking, we can use some some measurements, such as k-core, eigenvector centrality, H-index, and iteration algorithm, to locate the influential spreaders [37, 38, 39, 40, 41]. Recently, Morone and Makse studied the influence maximization in complex networks through optimal percolation, and found that some weakly connected nodes were also the optimal influencers [42]. By considering densities of loops and heterogeneous of degree for distinct nodes, Wu et al [41] proposed an effective influential identifying strategy based on the strategy proposed in Ref. [42]. Through extensive numerical simulations on realworld networks, We et al verified the effectiveness of their proposed strategy. Hu et al revealed that a node’s influence dependents on the characteristic number of nodes [43]. In reality, finding the most influential seeds is always high cost. For instance, when advertising for a product we should seek for some cities and stars to popularize it. For the popular stars, they are always expensive. In reality the capital budget is limited, how to maximize the influential in this situation is the purpose of this work. Inspired by Refs. [44, 45], we thus wish to find a convenient method 2
to locate the most influential spreaders from a randomly selected ‘initial-seed’. We propose a localized influential seeds selecting strategy. Through extensive numerical simulations on artificial and real-world networks, we find that such a strategy can effectively promote the information spreading dynamics when nodes with large degrees are preferentially selected. The outbreak threshold decreases if we preferentially select hubs. The paper is organized as follows. In section 2, we describe the spreading dynamics. In Sec. 3, we state the localized influential spreaders selecting strategy. We perform extensive numerical simulations in Sec. 4. We draw conclusions in section 5. 2. Model descriptions Denoting the network as G = (V, E), where V is the node set, and E is the edge set. The network size N is the number of elements of set V , i.e. N = |V |. Similarly, the number of edges is M = |E|. We use the susceptible-informedrecovered (SIR) model [46] to describe the information spreading dynamics on complex networks. At each time step, each node can only be one of susceptible, infected and recovered state. For the node in the susceptible state, it has not got any information from neighbors. The infected node represents that it has got the information and willing to share with others. A node in the recovered state stands for it has lose interest in the information. To trigger the spreading of information, we should choose some nodes as the seeds (spreaders) according to some strategies (to be defined later). Denoting the number of seeds as W . At each time step, each infected node will transmit the information to each susceptible neighbor with probability λ, and then recovers with probability µ. Without lose of generality, we set µ = 1 in this work. The information spreading dynamic terminates once there is no nodes in the infected state. In this case, we denote the information spreading size as R. For a given seed selecting strategy, the larger values of R the more effective of the strategy. 3. Localized influential spreaders selecting strategy In reality, it is difficult or expensive to obtain the global information about the network topology. By contrast, the local information about the network topology is easier to access in practice. Therefore, we choose the seeds from a randomly selected ‘initial-seed’ node in the network. Such a strategy is always needed when
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advertising for some small shops and enterprises for a given budget. To effectively promoting the information spreading dynamics, we propose an influential spreaders selection strategy according to a random walk based strategy as • (1) Initial the first ‘initial-seed’ node i randomly, and the number of spreader is set to be W . Put node i into the spreader set S. The walker initially set in the initial-seed node. • (2) The walker’s movement is adjusted by neighbors’ degrees following a family of function as kα (1) Θki = ∑ i α , j∈NS kj
where NS is the non-seed (i.e., nodes has not be selected as seed) neighbors of seed set S, ki is the degree of node i, −∞ < α < +∞ is an exponent of preferential selecting. For the case of α = 0, nodes are randomly selected. For the case of α > 0, hubs are more likely to be selected as spreaders. For the case of α < 0, nodes with small degrees are more likely to be selected as spreader. If the walker moves to node i, we set it as the influential spreader (i.e., seed), and put it into the spreader set S.
• (3) Repeating step (2) until W number of spreaders are selected. We illustrate the seed selecting strategy on network G with 8 nodes and 10 edges in Fig. 1. For the case of randomly selecting strategy, i.e., α = 0, all spreaders are randomly selected from the initial-seed 4, as shown in Fig. 1(a). For the case of α = −1 (α = 1), nodes with small (large) degrees are selected with a large probability, see Fig. 1(b) [Fig. 1 (c)]. 4. Numerical simulations In this section, we will perform extensive numerical simulations on artificial and real-world networks. We evaluate the effectiveness of the influential spreaders ∑ selecting strategy by studying the average information outbreak size 1 ⟨R⟩ = J Jj=1 Rj , where J is the number of numerical simulations, Rj is the outbreak size of the j-th simulation. The larger value of ⟨R⟩ the more effective of the strategy. To obtain ⟨R⟩, all the simulations are average over at least 5,000 times, i.e., J ≥ 5, 000. We first study the performances of our strategies on scale-free (SF) ∑ networks. −γD The degree distribution of SF network is P (k) = ζk , where ζ = 1/ k k −γD , 4
Figure 1: (Color online) Illustration of localized influential spreaders selecting strategy on network G with 8 nodes and 10 edges. (a) Randomly selecting (i.e., α = 0) 3 seed nodes from the initial-seed node 4. Selecting seed prefer to nodes with (b) small degrees (e.g., α = −1), and (c) large degrees (e.g., α = 1). γD is the degree exponent. Since in simulations we wish to check the effectiveness of the strategy on artificial networks with heterogeneous and homogeneous degree distribution, we here only choose two values of degree exponent, i.e., γD = 2.1 and 3.5. For a network with γD = 2.1, the degree distribution is heterogeneous; and for a network with γD = 3.5, the degree distribution is homogeneous. To locate the numerical critical point, we adopt the relative variance, which is defined as ⟨R(∞) − ⟨R(∞)⟩⟩2 χ= , (2) ⟨R(∞)⟩2
where ⟨·⟩ is the average value. The value of χ exhibits a peak [47] at the outbreak threshold. When the information transmission probability is lower than the outbreak threshold λc , i.e., λ ≤ λc , there are few nodes obtained the information; otherwise, when λ > λc , the global information outbreak becomes possible. In Fig. 2, we first investigate the information spreading on SF networks with strong degree heterogeneous distribution with degree exponent√γD = 2.5. We set the degree distribution follows a structural cutoff, i.e., kmax ∼ N . The network size is set to be N = 10, 000, and the seed size is W = 50. For any values of α, ⟨R⟩ increases gradually with λ. We find that preference to select nodes with large degrees as the influential spreaders is beneficial to the information spreading. Compared with the cases of α = 0 and α = −1, the values of ⟨R⟩ are larger when α = 1, as shown in Fig. 2(a). This phenomenon indicate that the localized influential spreaders are the hubs. Importantly, we find that ⟨R⟩ exponentially 5
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Figure 2: (Color online) Spreading dynamics on SF networks with degree exponent γD = 2.5. (a) The average outbreak size ⟨R⟩, and (b) relative variance χ versus the information transmission probability λ under different influential spreaders selecting strategies.
increases with λ when α = 1, i.e., ⟨R⟩ ∼ exp(µ0 λ), where µ0 is a constant. In Fig. 2(b), we exhibit the relative variance χ as a function of λ. The relative variance is relative large and the peak is higher when α = −1. The outbreak threshold decreases with α. In order to understand the effectiveness of different strategies, we investigate the cumulative degree distribution Q(k) of ∑ the seeds in Fig. 3. The cumulative degree distribution is computed as Q(k) = k′ >k G(k ′ ), where G(k ′ ) is the degree distribution of the selected seeds. With the increase of α, the heterogeneity of degree distribution decreases, and the hubs are more likely to be selected as seeds. The hubs have more neighbors to be informed, thus, the final informed size is larger near the outbreak threshold. We further investigate the performance of the influential spreaders selecting strategy on less heterogeneous networks with degree exponent γD = 3.5 as shown in Fig. 4. Similar to the phenomena presented in Fig. 2, if we selecting nodes with 6
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large degrees as the seeds, the final informed size ⟨R⟩ is larger near the outbreak threshold. Compared with the strategy of selecting small degrees, the outbreak threshold is smaller when selecting hubs. Finally we perform our localized seed selecting strategy on four real-world networks, including arXiv astro-ph [48], Facebook Friendships [49], Flickr [50], and Route views [48] networks, as shown in Fig. 5. More statistical characteristics of the real-world networks are exhibited in Table 1, including the network size (N ), number of edges (E), maximum degree (kmax ), first (⟨k⟩) and second moments (⟨k 2 ⟩) of degree distribution, degree-degree correlations (c) [51], clustering (c) [52, 53], modularity (Q) [54], and the theoretical threshold predicted by the dynamical message passing (DMP) method (λDMP ) [55, 56, 57]. The value of c DMP λc = 1/Λ(B), where B is the nonbacktracking matrix of the network. Since we wish to check the performance of our strategy near the outbreak threshold, we here consider the most accurate theoretical threshold λDMP = 1/Λ(B). For the c four real-world network, we find that the final informed size ⟨R⟩ is the largest for the case of α = 1, i.e., prefer to select nodes with large degrees is more likely to get the most influential spreaders. 5. Conclusions In reality, identifying the most influential spreaders are always cost. Many previous studies locating the most influential spreaders globally, which may greatly restrict its applications especially when the funding is limited. For instance, a s7
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Figure 4: (Color online) Spreading dynamics on SF networks with degree exponent γD = 3.5. The average outbreak size ⟨R⟩ versus the information transmission probability λ under different influential spreaders selecting strategies.
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Figure 5: (Color online) (Color online) Spreading dynamics on real-world networks. The average outbreak size ⟨R⟩ versus the information transmission probability λ on (a) arXiv astro-ph, (b) Facebook Friendships, (c) Flickr, and (d) Route views networks.
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Table 1: Statistical characteristics of the real-world networks, which including the network size (N ), number of edges (E), maximum degree (kmax ), first (⟨k⟩) and second moments (⟨k 2 ⟩) of degree distribution, degree-degree correlations (c), clustering (c), modularity (Q), and the theoretical threshold predicted by DMP method (λDMP ). c Networks arXiv astro-ph [48] Facebook Friendships [49] Flickr [50] Route views [48]
N 17903 63392 105722 6474
E 196972 816831 2316668 12572
kmax 504 1098 5425 1458
⟨k⟩ 22.004 25.771 43.826 3.884
⟨k2 ⟩ 1445.8 2268.9 15304 640.08
r 0.201 0.177 0.247 −0.182
c 0.318 0.148 0.402 0.01
Q 0.493 0.506 0.634 0.612
λDMP c 0.011 0.008 0.002 0.029
mall shop wish to advertise its commodity. Since its capital budget is limited, it always selects the advertiser from local spreaders. How to maximize the influential in this situation is the purpose of this work. We here proposed a simple localized influential spreaders identifying strategy, in which based on a preferentially random walk strategy. Specifically, the walker is given randomly initially, and then preferentially walks to a neighbor according to the degree of neighbors. The influential spreaders are those nodes that the walker ever visited. With this simple influential spreaders selecting strategy, we perform extensive numerical simulations on artificial networks and four real-world networks. We found that such a strategy can effectively identifying the most influential spreaders when the nodes with large degrees are preferentially visited. Our results may shed some light into proposing some more effective strategy for localized spreader selecting, such as including the k-core, H-index, eigenvector centrality of nodes. Another line for designing effective strategies should consider the global and local information of the network topology, such as first selecting multiple seeds and then preferentially selecting the neighbors of those seeds. 6. Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 61602048 and 61563038), and Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region under Grant NMGIRT-A1609. [1] M. Newman, Networks: an introduction, Oxford University Press, 2010. [2] R. Albert, A.-L. Barab´asi, Statistical mechanics of complex networks, Reviews of Modern Physics 74 (1) (2002) 47.
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Received date : 2 August 2018 Revised date : 1 October 2018 Please cite this article as: X.-C. Liu, X.-Z. Zhu, H. Tian et al., Identifying localized influential spreaders of information spreading, Physica A (2018), https://doi.org/10.1016/j.physa.2018.11.045 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlights (1) Proposing a random walked based strategy for identifying the localized influential spreaders from a randomly selected initial-seed node. (2) Selecting large degree nodes preferentially is more likely to find the most influential spreaders.
*Manuscript Click here to view linked References
Identifying localized influential spreaders of information spreading Xiang-Chun Liu1,2 , Xu-Zhen Zhu1 , Hui Tian1∗ , Zeng-Ping Zhang 3 , Wei Wang4 1. State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, People’s Republic of China 2. School of Information Engineering, Minzu University of China, Beijing, 100081 People’s Republic of China 3. School of Computer & Information Management, Inner Mongolia University of Finance and Economics, Hohhot 010070, People’s Republic of China 4. Cybersecurity Research Institute, Sichuan University, Chengdu 610065, People’s Republic of China
Abstract Identifying the influential spreaders of information spreading dynamics is a hot topic in the field of network science. To identify the influential spreaders, most previous studies were based on the global information of the network. In this paper, we propose a strategy for identifying the influential spreaders from a randomly selected initial-seed node. The seeds are connected as a chain, and are localized to the initial-seed. In our proposed preferentially random walk based influential spreaders identifying strategy, the walker’s movement is adjusted by neighbors’ degrees. The seeds are those nodes that the walker ever visited. Through extensive numerical simulations on artificial networks and four real-world networks, we find that selecting large degree nodes preferentially is more likely to find the most influential spreaders. The outbreak threshold decreases when preferentially select hubs. Our results shed some light into identifying the most localized influential spreaders. Keywords: Complex networks, information spreading dynamics, localized influential spreaders PACS: 89.75.Hc, 87.19.X-, 87.23.Ge
∗
[email protected]
Preprint submitted to Physica A
October 1, 2018
1. Introduction Many real-world systems can be described by complex networks, the nodes represent the elements and edges stand for the relationships among different elements [1, 2, 3, 4, 5]. The diffusion of information, disease, healthy behavior, and innovation in social system can be mapped into investigating the spreading dynamics on complex networks, which have attracted much attention from the field of mathematics, physics, and network science [6, 7, 8, 9, 10, 11, 12, 13]. On the one hand, researchers investigated the interplay between the spreading dynamics and network topology, and obtained some interesting results. For instance, a few hubs (i.e., nodes with large degrees) induce the outbreak threshold vanishes for networks with scale-free degree distributions [14, 15, 16, 17, 18]. In addition, the heterogeneity of weight distribution will increase the value of outbreak threshold [19, 20, 21, 22]. Recently, some scholars revealed the mulplexity and temporal of networks markedly altered the spreading dynamics and outbreak threshold, and some novel theoretical approaches were developed [23, 24, 25, 26, 27]. On the other hand, some strategies have been developed to select some nodes to suppress or promote the spreading dynamics [28, 29, 30, 31, 32, 33, 34]. For the former case, researchers called it as network immunization. In this case, researcher usually propose some strategies to measure the importance of nodes, such as degrees, betweenness, eigenvector centrality and k-core, and immunize the nodes with high scores [35]. For the later case, it is called as influential maximization [36]. Generally speaking, we can use some some measurements, such as k-core, eigenvector centrality, H-index, and iteration algorithm, to locate the influential spreaders [37, 38, 39, 40, 41]. Recently, Morone and Makse studied the influence maximization in complex networks through optimal percolation, and found that some weakly connected nodes were also the optimal influencers [42]. By considering densities of loops and heterogeneous of degree for distinct nodes, Wu et al [41] proposed an effective influential identifying strategy based on the strategy proposed in Ref. [42]. Through extensive numerical simulations on realworld networks, We et al verified the effectiveness of their proposed strategy. Hu et al revealed that a node’s influence dependents on the characteristic number of nodes [43]. In reality, finding the most influential seeds is always high cost. For instance, when advertising for a product we should seek for some cities and stars to popularize it. For the popular stars, they are always expensive. In reality the capital budget is limited, how to maximize the influential in this situation is the purpose of this work. Inspired by Refs. [44, 45], we thus wish to find a convenient method 2
to locate the most influential spreaders from a randomly selected ‘initial-seed’. We propose a localized influential seeds selecting strategy. Through extensive numerical simulations on artificial and real-world networks, we find that such a strategy can effectively promote the information spreading dynamics when nodes with large degrees are preferentially selected. The outbreak threshold decreases if we preferentially select hubs. The paper is organized as follows. In section 2, we describe the spreading dynamics. In Sec. 3, we state the localized influential spreaders selecting strategy. We perform extensive numerical simulations in Sec. 4. We draw conclusions in section 5. 2. Model descriptions Denoting the network as G = (V, E), where V is the node set, and E is the edge set. The network size N is the number of elements of set V , i.e. N = |V |. Similarly, the number of edges is M = |E|. We use the susceptible-informedrecovered (SIR) model [46] to describe the information spreading dynamics on complex networks. At each time step, each node can only be one of susceptible, infected and recovered state. For the node in the susceptible state, it has not got any information from neighbors. The infected node represents that it has got the information and willing to share with others. A node in the recovered state stands for it has lose interest in the information. To trigger the spreading of information, we should choose some nodes as the seeds (spreaders) according to some strategies (to be defined later). Denoting the number of seeds as W . At each time step, each infected node will transmit the information to each susceptible neighbor with probability λ, and then recovers with probability µ. Without lose of generality, we set µ = 1 in this work. The information spreading dynamic terminates once there is no nodes in the infected state. In this case, we denote the information spreading size as R. For a given seed selecting strategy, the larger values of R the more effective of the strategy. 3. Localized influential spreaders selecting strategy In reality, it is difficult or expensive to obtain the global information about the network topology. By contrast, the local information about the network topology is easier to access in practice. Therefore, we choose the seeds from a randomly selected ‘initial-seed’ node in the network. Such a strategy is always needed when
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advertising for some small shops and enterprises for a given budget. To effectively promoting the information spreading dynamics, we propose an influential spreaders selection strategy according to a random walk based strategy as • (1) Initial the first ‘initial-seed’ node i randomly, and the number of spreader is set to be W . Put node i into the spreader set S. The walker initially set in the initial-seed node. • (2) The walker’s movement is adjusted by neighbors’ degrees following a family of function as kα (1) Θki = ∑ i α , j∈NS kj
where NS is the non-seed (i.e., nodes has not be selected as seed) neighbors of seed set S, ki is the degree of node i, −∞ < α < +∞ is an exponent of preferential selecting. For the case of α = 0, nodes are randomly selected. For the case of α > 0, hubs are more likely to be selected as spreaders. For the case of α < 0, nodes with small degrees are more likely to be selected as spreader. If the walker moves to node i, we set it as the influential spreader (i.e., seed), and put it into the spreader set S.
• (3) Repeating step (2) until W number of spreaders are selected. We illustrate the seed selecting strategy on network G with 8 nodes and 10 edges in Fig. 1. For the case of randomly selecting strategy, i.e., α = 0, all spreaders are randomly selected from the initial-seed 4, as shown in Fig. 1(a). For the case of α = −1 (α = 1), nodes with small (large) degrees are selected with a large probability, see Fig. 1(b) [Fig. 1 (c)]. 4. Numerical simulations In this section, we will perform extensive numerical simulations on artificial and real-world networks. We evaluate the effectiveness of the influential spreaders ∑ selecting strategy by studying the average information outbreak size 1 ⟨R⟩ = J Jj=1 Rj , where J is the number of numerical simulations, Rj is the outbreak size of the j-th simulation. The larger value of ⟨R⟩ the more effective of the strategy. To obtain ⟨R⟩, all the simulations are average over at least 5,000 times, i.e., J ≥ 5, 000. We first study the performances of our strategies on scale-free (SF) ∑ networks. −γD The degree distribution of SF network is P (k) = ζk , where ζ = 1/ k k −γD , 4
Figure 1: (Color online) Illustration of localized influential spreaders selecting strategy on network G with 8 nodes and 10 edges. (a) Randomly selecting (i.e., α = 0) 3 seed nodes from the initial-seed node 4. Selecting seed prefer to nodes with (b) small degrees (e.g., α = −1), and (c) large degrees (e.g., α = 1). γD is the degree exponent. Since in simulations we wish to check the effectiveness of the strategy on artificial networks with heterogeneous and homogeneous degree distribution, we here only choose two values of degree exponent, i.e., γD = 2.1 and 3.5. For a network with γD = 2.1, the degree distribution is heterogeneous; and for a network with γD = 3.5, the degree distribution is homogeneous. To locate the numerical critical point, we adopt the relative variance, which is defined as ⟨R(∞) − ⟨R(∞)⟩⟩2 χ= , (2) ⟨R(∞)⟩2
where ⟨·⟩ is the average value. The value of χ exhibits a peak [47] at the outbreak threshold. When the information transmission probability is lower than the outbreak threshold λc , i.e., λ ≤ λc , there are few nodes obtained the information; otherwise, when λ > λc , the global information outbreak becomes possible. In Fig. 2, we first investigate the information spreading on SF networks with strong degree heterogeneous distribution with degree exponent√γD = 2.5. We set the degree distribution follows a structural cutoff, i.e., kmax ∼ N . The network size is set to be N = 10, 000, and the seed size is W = 50. For any values of α, ⟨R⟩ increases gradually with λ. We find that preference to select nodes with large degrees as the influential spreaders is beneficial to the information spreading. Compared with the cases of α = 0 and α = −1, the values of ⟨R⟩ are larger when α = 1, as shown in Fig. 2(a). This phenomenon indicate that the localized influential spreaders are the hubs. Importantly, we find that ⟨R⟩ exponentially 5
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Figure 2: (Color online) Spreading dynamics on SF networks with degree exponent γD = 2.5. (a) The average outbreak size ⟨R⟩, and (b) relative variance χ versus the information transmission probability λ under different influential spreaders selecting strategies.
increases with λ when α = 1, i.e., ⟨R⟩ ∼ exp(µ0 λ), where µ0 is a constant. In Fig. 2(b), we exhibit the relative variance χ as a function of λ. The relative variance is relative large and the peak is higher when α = −1. The outbreak threshold decreases with α. In order to understand the effectiveness of different strategies, we investigate the cumulative degree distribution Q(k) of ∑ the seeds in Fig. 3. The cumulative degree distribution is computed as Q(k) = k′ >k G(k ′ ), where G(k ′ ) is the degree distribution of the selected seeds. With the increase of α, the heterogeneity of degree distribution decreases, and the hubs are more likely to be selected as seeds. The hubs have more neighbors to be informed, thus, the final informed size is larger near the outbreak threshold. We further investigate the performance of the influential spreaders selecting strategy on less heterogeneous networks with degree exponent γD = 3.5 as shown in Fig. 4. Similar to the phenomena presented in Fig. 2, if we selecting nodes with 6
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Figure 3: (Color online) The cumulative degree distribution Q(k) of different localized influential spreaders selecting strategies.
large degrees as the seeds, the final informed size ⟨R⟩ is larger near the outbreak threshold. Compared with the strategy of selecting small degrees, the outbreak threshold is smaller when selecting hubs. Finally we perform our localized seed selecting strategy on four real-world networks, including arXiv astro-ph [48], Facebook Friendships [49], Flickr [50], and Route views [48] networks, as shown in Fig. 5. More statistical characteristics of the real-world networks are exhibited in Table 1, including the network size (N ), number of edges (E), maximum degree (kmax ), first (⟨k⟩) and second moments (⟨k 2 ⟩) of degree distribution, degree-degree correlations (c) [51], clustering (c) [52, 53], modularity (Q) [54], and the theoretical threshold predicted by the dynamical message passing (DMP) method (λDMP ) [55, 56, 57]. The value of c DMP λc = 1/Λ(B), where B is the nonbacktracking matrix of the network. Since we wish to check the performance of our strategy near the outbreak threshold, we here consider the most accurate theoretical threshold λDMP = 1/Λ(B). For the c four real-world network, we find that the final informed size ⟨R⟩ is the largest for the case of α = 1, i.e., prefer to select nodes with large degrees is more likely to get the most influential spreaders. 5. Conclusions In reality, identifying the most influential spreaders are always cost. Many previous studies locating the most influential spreaders globally, which may greatly restrict its applications especially when the funding is limited. For instance, a s7
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Figure 4: (Color online) Spreading dynamics on SF networks with degree exponent γD = 3.5. The average outbreak size ⟨R⟩ versus the information transmission probability λ under different influential spreaders selecting strategies.
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Figure 5: (Color online) (Color online) Spreading dynamics on real-world networks. The average outbreak size ⟨R⟩ versus the information transmission probability λ on (a) arXiv astro-ph, (b) Facebook Friendships, (c) Flickr, and (d) Route views networks.
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Table 1: Statistical characteristics of the real-world networks, which including the network size (N ), number of edges (E), maximum degree (kmax ), first (⟨k⟩) and second moments (⟨k 2 ⟩) of degree distribution, degree-degree correlations (c), clustering (c), modularity (Q), and the theoretical threshold predicted by DMP method (λDMP ). c Networks arXiv astro-ph [48] Facebook Friendships [49] Flickr [50] Route views [48]
N 17903 63392 105722 6474
E 196972 816831 2316668 12572
kmax 504 1098 5425 1458
⟨k⟩ 22.004 25.771 43.826 3.884
⟨k2 ⟩ 1445.8 2268.9 15304 640.08
r 0.201 0.177 0.247 −0.182
c 0.318 0.148 0.402 0.01
Q 0.493 0.506 0.634 0.612
λDMP c 0.011 0.008 0.002 0.029
mall shop wish to advertise its commodity. Since its capital budget is limited, it always selects the advertiser from local spreaders. How to maximize the influential in this situation is the purpose of this work. We here proposed a simple localized influential spreaders identifying strategy, in which based on a preferentially random walk strategy. Specifically, the walker is given randomly initially, and then preferentially walks to a neighbor according to the degree of neighbors. The influential spreaders are those nodes that the walker ever visited. With this simple influential spreaders selecting strategy, we perform extensive numerical simulations on artificial networks and four real-world networks. We found that such a strategy can effectively identifying the most influential spreaders when the nodes with large degrees are preferentially visited. Our results may shed some light into proposing some more effective strategy for localized spreader selecting, such as including the k-core, H-index, eigenvector centrality of nodes. Another line for designing effective strategies should consider the global and local information of the network topology, such as first selecting multiple seeds and then preferentially selecting the neighbors of those seeds. 6. Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 61602048 and 61563038), and Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region under Grant NMGIRT-A1609. [1] M. Newman, Networks: an introduction, Oxford University Press, 2010. [2] R. Albert, A.-L. Barab´asi, Statistical mechanics of complex networks, Reviews of Modern Physics 74 (1) (2002) 47.
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