Information Spreading on Activity-Driven Temporal Networks with Two-Step Memory

Zhong, Linfeng; Xue, Xiaoyu; Bai, Yu; Huang, Jin; Cheng, Qing; Huang, Longyang; Pan, Weijun

doi:https://doi.org/10.1155/2021/4506012

Discrete Dynamics in Nature and Society

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Abstract Introduction Model Results Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Dynamical Modeling, Analysis, and Control of Information Diffusion over Social Networks

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Research Article | Open Access

Volume 2021 | Article ID 4506012 | https://doi.org/10.1155/2021/4506012

Information Spreading on Activity-Driven Temporal Networks with Two-Step Memory

Linfeng Zhong,¹Xiaoyu Xue,²Yu Bai,¹Jin Huang,¹Qing Cheng,¹Longyang Huang,¹and Weijun Pan¹

Academic Editor: Chenquan Gan

Received31 Jul 2020

Revised16 Aug 2020

Accepted13 Nov 2020

Published16 Jan 2021

Abstract

Information spreading dynamics on the temporal network is a hot topic in the field of network science. In this paper, we propose an information spreading model on an activity-driven temporal network, in which a node is accepting the information dependents on the cumulatively received pieces of information in its recent two steps. With a generalized Markovian approach, we analyzed the information spreading size, and revealed that network temporality might suppress or promote the information spreading, which is determined by the information transmission probability. Besides, the system exists a critical mass, below which the information cannot globally outbreak, and above which the information outbreak size does not change with the initial seed size. Our theory can qualitatively well predict the numerical simulations.

1. Introduction

Information spreading on social networks is a hot topic in the fields of network science, computer science, and physics [1–9]. Research studies wish to know the evolutionary mechanisms and diffusion laws of the information, and further design some effective measure to control the information spreading. Using the massive real-data, researchers revealed many important evolution mechanisms, such as social reinforcement effect and memory effect [10–12]. To include those important evolution mechanisms into the information spreading dynamics, scholars proposed some successful models [13–18]. For instance, Watts [19] generalized the threshold model to complex networks, and revealed that the information spreading size first increases then decreases with the average degree of the network.

With these proposed mathematical models, the important question is how the network topologies affect the spreading dynamics. The studies in this topic can be divided into three aspects according to the complexity of the network. The first one is the effects of static networks. For static networks, research studies addressed the influences of the degree distribution, weight distribution, and community on the information spreading [20–26]. An important conclusion is that a small fraction of nodes with large degrees makes the information outbreak with any values of information transmission probability [27, 28]. In reality, individuals can transmit the information through more than one communication channel. Therefore, using the multiplex networks describes the real-world network more accurately [29–32], which is the second aspect. Research studies revealed that the network multiplexity could suppress or promote the information spreading, which depends on the interaction between two networks [33–36]. The third aspect is that the effects of temporal network [37–42] on information spreading, since the nodes and connections do not always exist. Scholtes et al. [43] revealed that the spreading dynamics might speed up and slow down the information spreading on non-Markovian temporal networks. Wang et al. [44] proposed a heuristic method immunization strategy for information spreading on the temporal network.

To our best knowledge, when including the reinforcement and memory effect into the information spreading model, the study about the effects of network temporality on information spreading dynamics is still lacking. To this end, we propose an information spreading model, in which a node is accepting the information when its cumulative received information in the recent two steps is more significant than a threshold in Section 2. With a generalized Markovian approach, we analyze the information spreading size in Section 3. Moreover, we perform extensive numerical simulations on activity-driven temporal networks in Section 4.

2. Model

In this section, we introduce the information spreading on temporal networks with two-step memory.

2.1. Activity-Driven Network

We first introduce the activity-driven temporal network which is proposed by Perra et al. [45–47], where is the maximum time step. We set in this paper. The activity-driven network is build according to the following three steps for a given network size . (i) For each node , a potential activity is assigned according to a given distribution . In this paper, we assume that follows a power-law distribution. Specifically, , where is an exponent of potential activity distribution, and , . In numerical simulations, we set . The larger the value of , the more homogeneous of potential activity, which induces the more homogeneous of degree distribution. (ii) At time step, we generate a temporal network . Each node becomes active with probability , where is a parameter. If node becomes active, it will connect nodes randomly. Otherwise, node can only receive other connections. (iii) At the end of time step , all edges are deleted. We repeated steps (i)–(iii) until . According to the above three steps, the average degree of network is . By adjusting the values of , , and , we can set the average degree of network .

2.2. Information Spreading Model

In this subsection, we introduce a novel information spreading model in which each node can remember the pieces of information it received in recent two steps. The information spreading dynamics is described by a generalized susceptible-infected-susceptible (SIS) model. A node in the susceptible state means that it had not accepted or believed the truth of the information but may accept it when its misgivings were eliminated. A node in the infected state means that it has accepted the information and is willing to share it with neighbors. To include the willingness to accept the information, we introduce an adoption threshold of ; the higher the value of , the less willing to accept the information. In what follows, we introduce how the information is spreading on activity-driven temporal networks.

Initially, we randomly select fraction of nodes in the infected state, and the remaining nodes in the susceptible state and every node do not obtain any pieces of information, i.e., . At each time step , every infected node tries to transmit the information to its every susceptible neighbor on network , e.g., node , with probability . If node received the information, its received accumulated pieces of information become . When node received enough pieces of information from infected neighbors in recent steps, i.e., , node becomes infected. Since every node can only remember the recent two-step memory about its accumulated, we here call our model as “two-step-memory” (TSM)-based information spreading. The infected node recovers with probability and returns to a susceptible state. Note that when , the next time step is . The information spreading dynamics evolves until there are no nodes in the infected state, or the time step reaches .

3. Theory

In this section, we propose a generalized discrete Markovian approach [44, 48–50] to steady the TSM-based information spreading dynamics on activity-driven temporal networks. In the theory, we assume that every node transmitting the information to susceptible is independent. As a result, the dynamical correlations among the state of neighbors are neglected.

The is denoted as the probability that node is in the infected state at time , and is denoted as the probability of node is in the susceptible state. For a susceptible node , it becomes infected state at time only when its received accumulated pieces of information in time steps and are larger than the threshold . is denoted as the probability of the node obtaining pieces of information from infected neighbors at time simultaneously on the network . To compute the value of , we should consider three aspects: (i) select neighbors from the neighbor set of node , and denote the subset as . (ii) Every node in transmits the information to node with probability . (iii) Each node in the set does not transmit the information to node . Combining the above three aspects, we obtain the expression of aswhere represents the number of elements in set . Once we know the expression of , we obtain the value of .

To compute the value of , we should consider two aspects. On the one hand, node is in the infected state at time but does not recover at time . The probability of this event is . On the other hand, node is in the susceptible state at time , and we obtained more than pieces of information at time steps and with probability:where represents that node received at most pieces of information from neighbors at time and . The expression of is

Considering the above two aspects, we obtain the evolution of as

Averaging all values of , we obtain the probability that a node is randomly selected in the infected state at time as

In the steady-state, the fraction of nodes in the infected state can be denoted as for simplicity.

According to equation (4), the nonlinearity of the system makes us hardly obtain an analytical threshold. Therefore, we locate the threshold of the system by using the following method: observing the peak of , which defines aswhere is the information outbreak size with information transmission probability , and is a small increment of . At the threshold point, reaches its maximum value [51].

The above theoretical derivation is for the information spreading dynamics on temporal networks. When studying the spreading dynamics on static networks, we only need to set for .

4. Results

In this section, we will perform extensive numerical simulations on the activity-driven temporal networks and their corresponding static networks. To build the static network, we only need to generate network and set the remaining temporal network equal to network . All numerical simulation results presented in this paper are averaged over 500 times.

In Figure 1, we first investigate the information spreading size on both temporal and static networks for a given average degree . We focus on the following three aspects. (i) How the temporal network affects the information spreading? We reveal two distinct results. For small values of , the temporal network structure suppresses the information spreading since the network connectivity is small than that in its corresponding static networks. However, when is very large, the temporal network structure promoting the information spreading since a node in a temporal network can connect more distinct nodes. We note that the results are not affected by the initial seed size of and . (ii) The phase transition of the system is the second point we will discuss. We find that the system always exhibits a hysteresis loop. Specifically, the spreading size depends on the initial seed size between the invasion threshold and persistence threshold [52]. The two threshold points can be located by using equation (6). (iii) The third point we investigate in Figure 1 is how affects . We find that increases with on both temporal and static networks. That is to say, the more homogeneous the degree distribution of the network, the less robust of the network for information spreading. The theoretical predictions agree well with the numerical simulation results. The differences between the theoretical and numerical predictions are induced by the strong dynamical correlations among the states among neighbors.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1

Information spreading on activity-driven networks and the corresponding static networks. The information spreading size in the steady-state versus the information transmission probability with potential distribution exponent: (a) , (b) , and (c) with adoption threshold . is a function of with (d) , (e) , and (f) with . Other parameters are set to be , , , and . The lines are theoretical predictions according to equations (1)–(5), and symbols are average values of numerical predictions. The shadow represents the standard deviations of numerical simulations. In each subfigure, the lines and symbols in the same color have the same parameter.

We further investigate the effects of initial seed size on the information spreading dynamics in Figure 2. For any values of and , there is a finite value of critical mass , below which the information cannot outbreak globally and above which the information outbreak size does not change with . For the effects of network temporality, there have two situations. When , the network temporality suppresses the information spreading for small values of , e.g., . However, when , the network temporality always promotes the information spreading regardless of the value of . Our suggested theory can qualitatively describe the above phenomena.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2

Information spreading on activity-driven networks and the corresponding static networks. The information spreading size versus initial seed size with potential distribution exponent (a) , (b) , and (c) with adoption threshold . versus with (d) , (e) , and (f) with . Other parameters are set to be , , , and . The lines are theoretical predictions, and symbols are average values of numerical predictions. The shadow represents the standard deviations of numerical simulations. In each subfigure, the lines and symbols in the same color have the same parameter.

5. Conclusions

In this paper, we study the information spreading dynamics on activity-driven temporal networks. To study the effects of network temporality on information spreading, we first proposed an information spreading model, which assumes that a node accepting the information depends on the cumulatively received pieces of information from neighbors in the recent two steps. Then, we developed a generalized Markovian approach to describe the information spreading dynamics, and gave the expression of the information outbreak size. By performing extensive numerical simulations, we found that network temporality may suppress and promote the spreading of information. Specifically, the network temporality suppresses the information spreading for small values of information transmission probability, while promoting the information spreading for large values of information transmission probability. Finally, we found that the system has a critical mass. When the initial seed size is smaller than the critical mass, the information cannot outbreak globally. When the initial size is larger than the critical mass, the information spreading size does not change with the initial seed size values. Our presented results help us understand the effects of network temporality on information spreading dynamics. And the results can also help us analyze the traffic of the complex dynamic aviation network.

Data Availability

The datasets used in the present study are available from the first author upon reasonable request ([email protected]).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant no. U1733203), the Safety Foundation of CAAC (no. AQ20200019), and the Foundation of CAFUC (no. J2020-084).

References

C. Castellano, S. Fortunato, and V. Loreto, “Statistical physics of social dynamics,” Reviews of Modern Physics, vol. 81, no. 2, p. 591, 2009.
View at: Publisher Site | Google Scholar
S. Boccaletti, G. Bianconi, R. Criado et al., “The structure and dynamics of multilayer networks,” Physics Reports, vol. 544, no. 1, pp. 1–122, 2014.
View at: Publisher Site | Google Scholar
Z.-K. Zhang, C. Liu, X.-X. Zhan, X. Lu, C.-X. Zhang, and Y.-C. Zhang, “Dynamics of information diffusion and its applications on complex networks,” Physics Reports, vol. 651, pp. 1–34, 2016.
View at: Publisher Site | Google Scholar
M. Kitsak, L. K. Gallos, S. Havlin et al., “Identification of influential spreaders in complex networks,” Nature Physics, vol. 6, no. 11, pp. 888–893, 2010.
View at: Publisher Site | Google Scholar
W. Wang, Q.-H. Liu, J. Liang, Y. Hu, and T. Zhou, “Coevolution spreading in complex networks,” Physics Reports, vol. 820, pp. 1–51, 2019.
View at: Publisher Site | Google Scholar
Y. Yi, Z. Zhang, L. T. Yang, C. Gan, X. Deng, and L. Yi, “Reemergence modeling of intelligent information diffusion in heterogeneous social networks: the dynamics pers,” IEEE Transactions on Network Science and Engineering, vol. 99, p. 1, 2020.
View at: Publisher Site | Google Scholar
J. Wang and J. Wang, “Cross-correlation complexity and synchronization of the financial time series on potts dynamics,” Physica A: Statistical Mechanics and Its Applications, vol. 541, Article ID 123286, 2020.
View at: Publisher Site | Google Scholar
J. Wang and J. Wang, “Measuring the correlation complexity between return series by multiscale complex analysis on potts dynamics,” Nonlinear Dynamics, vol. 89, no. 4, pp. 2703–2721, 2017.
View at: Publisher Site | Google Scholar
J. Wang, J. Wang, and H. E. Stanley, “Multiscale multifractal DCCA and complexity behaviors of return intervals for potts price model,” Physica A: Statistical Mechanics and Its Applications, vol. 492, pp. 889–902, 2018.
View at: Publisher Site | Google Scholar
H. P. Young, “The dynamics of social innovation,” Proceedings of the National Academy of Sciences, vol. 108, no. 4, pp. 21285–21291, 2011.
View at: Publisher Site | Google Scholar
D. Centola, “The spread of behavior in an online social network experiment,” Science, vol. 329, no. 5996, pp. 1194–1197, 2010.
View at: Publisher Site | Google Scholar
D. Centola, “Physician networks and the complex contagion of clinical treatment,” JAMA Network Open, vol. 3, no. 1, Article ID e1918585, 2020.
View at: Publisher Site | Google Scholar
D. Centola, How Behavior Spreads: The Science of Complex Contagions, vol. 3, Princeton University Press, Princeton, NJ, USA, 2018.
D. Guilbeault, J. Becker, and D. Centola, “Complex contagions: a decade in review,” in Complex Spreading Phenomena in Social Systems, Springer, Berlin, Germany, 2018.
View at: Google Scholar
D. Centola, “The social origins of networks and diffusion,” American Journal of Sociology, vol. 120, no. 5, pp. 1295–1338, 2015.
View at: Publisher Site | Google Scholar
M. Zheng, L. L. ü, M. Zhao et al., “Spreading in online social networks: the role of social reinforcement,” Physical Review E, vol. 88, no. 1, p. 12818, 2013.
View at: Publisher Site | Google Scholar
L. Lü, D.-B. Chen, and T. Zhou, “The small world yields the most effective information spreading,” New Journal of Physics, vol. 13, no. 12, Article ID 123005, 2011.
View at: Google Scholar
W. Wang, M. Tang, H.-F. Zhang, and Y.-C. Lai, “Dynamics of social contagions with memory of nonredundant information,” Physical Review E, vol. 92, no. 1, p. 12820, 2015.
View at: Publisher Site | Google Scholar
D. J. Watts, “A simple model of global cascades on random networks,” Proceedings of the National Academy of Sciences, vol. 99, no. 9, pp. 5766–5771, 2002.
View at: Publisher Site | Google Scholar
M. Del Vicario, A. Bessi, F. Zollo et al., “The spreading of misinformation online,” Proceedings of the National Academy of Sciences, vol. 113, no. 3, pp. 554–559, 2016.
View at: Publisher Site | Google Scholar
G. Miritello, E. Moro, and R. Lara, “Dynamical strength of social ties in information spreading,” Physical Review E, vol. 83, no. 4, p. 45102, 2011.
View at: Publisher Site | Google Scholar
Y. Sun, C. Liu, C.-X. Zhang, and Z.-K. Zhang, “Epidemic spreading on weighted complex networks,” Physics Letters A, vol. 378, no. 7-8, pp. 635–640, 2014.
View at: Publisher Site | Google Scholar
Y. Feng, L. Ding, Y.-H. Huang, and L. Zhang, “Epidemic spreading on weighted networks with adaptive topology based on infective information,” Physica A: Statistical Mechanics and Its Applications, vol. 463, pp. 493–502, 2016.
View at: Publisher Site | Google Scholar
X. Chu, J. Guan, Z. Zhang, and S. Zhou, “Epidemic spreading in weighted scale-free networks with community structure,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2009, no. 7, Article ID P07043, 2009.
View at: Publisher Site | Google Scholar
L.-F. Zhong, J.-G. Liu, and M.-S. Shang, “Iterative resource allocation based on propagation feature of node for identifying the influential nodes,” Physics Letters A, vol. 379, no. 38, pp. 2272–2276, 2015.
View at: Publisher Site | Google Scholar
L.-F. Zhong, Q.-H. Liu, W. Wang, and S.-M. Cai, “Comprehensive influence of local and global characteristics on identifying the influential nodes,” Physica A: Statistical Mechanics and Its Applications, vol. 511, pp. 78–84, 2018.
View at: Publisher Site | Google Scholar
R. Pastor-Satorras and A. Vespignani, “Epidemic spreading in scale-free networks,” Physical Review Letters, vol. 86, no. 14, p. 3200, 2001.
View at: Publisher Site | Google Scholar
R. Pastor-Satorras and A. Vespignani, “Epidemic dynamics and endemic states in complex networks,” Physical Review E, vol. 63, no. 6, p. 66117, 2001.
View at: Publisher Site | Google Scholar
M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and M. A. Porter, “Multilayer networks,” Journal of Complex Networks, vol. 2, no. 3, pp. 203–271, 2014.
View at: Publisher Site | Google Scholar
P. J. Mucha, T. Richardson, K. Macon, M. A. Porter, and J.-P. Onnela, “Community structure in time-dependent, multiscale, and multiplex networks,” Science, vol. 328, no. 5980, pp. 876–878, 2010.
View at: Publisher Site | Google Scholar
J. Gómez-Gardenes, I. Reinares, A. Arenas, and L. M. Floría, “Evolution of cooperation in multiplex networks,” Scientific Reports, vol. 2, p. 620, 2012.
View at: Publisher Site | Google Scholar
J. Y. Kim and K.-I. Goh, “Coevolution and correlated multiplexity in multiplex networks,” Physical Review Letters, vol. 111, no. 5, p. 58702, 2013.
View at: Publisher Site | Google Scholar
K.-M. Lee, C. D. Brummitt, and K.-I. Goh, “Threshold cascades with response heterogeneity in multiplex networks,” Physical Review E, vol. 90, no. 6, p. 62816, 2014.
View at: Publisher Site | Google Scholar
C. D. Brummitt, K.-M. Lee, and K.-I. Goh, “Multiplexity-facilitated cascades in networks,” Physical Review E, vol. 85, no. 4, p. 45102, 2012.
View at: Publisher Site | Google Scholar
Z. Li, F. Yan, and Y. Jiang, “Cross-layers cascade in multiplex networks,” Autonomous Agents and Multi-Agent Systems, vol. 29, no. 6, pp. 1186–1215, 2015.
View at: Publisher Site | Google Scholar
Q. Guo, X. Jiang, Y. Lei, M. Li, Y. Ma, and Z. Zheng, “Two-stage effects of awareness cascade on epidemic spreading in multiplex networks,” Physical Review E, vol. 91, no. 1, p. 12822, 2015.
View at: Publisher Site | Google Scholar
P. Holme, “Modern temporal network theory: a colloquium,” The European Physical Journal B, vol. 88, no. 9, p. 234, 2015.
View at: Publisher Site | Google Scholar
N. Masuda and P. Holme, Temporal Network Epidemiology, Springer, Berlin, Germany, 2017.
P. Holme and J. Saramäki, “Temporal networks,” Physics Reports, vol. 519, no. 3, pp. 97–125, 2012.
View at: Publisher Site | Google Scholar
S. Lee, L. E. Rocha, F. Liljeros, and P. Holme, “Exploiting temporal network structures of human interaction to effectively immunize populations,” PloS One, vol. 7, no. 5, Article ID e36439, 2012.
View at: Publisher Site | Google Scholar
J. Tang, M. Musolesi, C. Mascolo, and V. Latora, “Temporal distance metrics for social network analysis,” in Proceedings of the 2nd ACM Workshop on Online Social Networks, pp. 31–36, Barcelona, Spain, August 2009.
View at: Google Scholar
N. Masuda and P. Holme, “Predicting and controlling infectious disease epidemics using temporal networks,” F1000prime Reports, vol. 5, no. 6, 2013.
View at: Google Scholar
I. Scholtes, N. Wider, R. Pfitzner, A. Garas, C. J. Tessone, and F. Schweitzer, “Causality-driven slow-down and speed-up of diffusion in non-markovian temporal networks,” Nature Communications, vol. 5, no. 1, pp. 1–9, 2014.
View at: Publisher Site | Google Scholar
W. Wang, Y. Ma, T. Wu, Y. Dai, X. Chen, and L. A. Braunstein, “Containing misinformation spreading in temporal social networks,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 29, no. 12, Article ID 123131, 2019.
View at: Publisher Site | Google Scholar
N. Perra, B. Gonçalves, R. Pastor-Satorras, and A. Vespignani, “Activity driven modeling of time varying networks,” Scientific Reports, vol. 2, p. 469, 2012.
View at: Publisher Site | Google Scholar
M. Karsai, N. Perra, and A. Vespignani, “Time varying networks and the weakness of strong ties,” Scientific Reports, vol. 4, p. 4001, 2014.
View at: Publisher Site | Google Scholar
N. Perra, A. Baronchelli, D. Mocanu, B. Gonçalves, R. Pastor-Satorras, and A. Vespignani, “Random walks and search in time-varying networks,” Physical Review Letters, vol. 109, no. 23, Article ID 238701, 2012.
View at: Publisher Site | Google Scholar
S. Gómez, A. Arenas, J. Borge-Holthoefer, S. Meloni, and Y. Moreno, “Discrete-time Markov chain approach to contact-based disease spreading in complex networks,” EPL (Europhysics Letters), vol. 89, no. 3, p. 38009, 2010.
View at: Publisher Site | Google Scholar
L. Pan, W. Wang, S. Cai, and T. Zhou, “Optimal interlayer structure for promoting spreading of the susceptible-infected-susceptible model in two-layer networks,” Physical Review E, vol. 100, no. 2, p. 22316, 2019.
View at: Publisher Site | Google Scholar
E. Valdano, L. Ferreri, C. Poletto, and V. Colizza, “Analytical computation of the epidemic threshold on temporal networks,” Physical Review X, vol. 5, no. 2, p. 21005, 2015.
View at: Publisher Site | Google Scholar
X. Chen, W. Wang, S. Cai, H. E. Stanley, and L. A. Braunstein, “Optimal resource diffusion for suppressing disease spreading in multiplex networks,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2018, no. 5, Article ID 10.1088/1742-5468/aabfcc, p. 53501, 2018.
View at: Google Scholar
H. Yang, M. Tang, and T. Gross, “Large epidemic thresholds emerge in heterogeneous networks of heterogeneous nodes,” Scientific Reports, vol. 5, p. 13122, 2015.
View at: Publisher Site | Google Scholar

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Copyright © 2021 Linfeng Zhong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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