[Retracted] Information Spreading on Memory Activity-Driven Temporal Networks

Zhong, Linfeng; Bai, Yu; Liu, Changjiang; Du, Juan; Pan, Weijun

doi:https://doi.org/10.1155/2021/8015191

Complexity

On this page

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

Research Article Retraction

!

This article has been Retracted. To view the article details, please click the ‘Retraction’ tab above.

Special Issue

Epidemic Spreading Dynamics on Temporal Networks

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 8015191 | https://doi.org/10.1155/2021/8015191

[Retracted] Information Spreading on Memory Activity-Driven Temporal Networks

Linfeng Zhong,¹Yu Bai,¹Changjiang Liu,²Juan Du,³and Weijun Pan¹

Academic Editor: Giovanni Petri

Received29 Apr 2021

Revised29 Jun 2021

Accepted15 Jul 2021

Published27 Jul 2021

Abstract

Information spreading dynamics on temporal networks have attracted significant attention in the field of network science. Extensive real-data analyses revealed that network memory widely exists in the temporal network. This paper proposes a mathematical model to describe the information spreading dynamics with the network memory effect. We develop a Markovian approach to describe the model. Using the Monte Carlo simulation method, we find that network memory may suppress and promote the information spreading dynamics, which depends on the degree heterogeneity and fraction of bigots. The network memory effect suppresses the information spreading for small information transmission probability. The opposite situation happens for large value of information transmission probability. Moreover, network memory effect may benefit the information spreading, which depends on the degree heterogeneity of the activity-driven network. Our results presented in this paper help us understand the spreading dynamics on temporal networks.

1. Introduction

Extensive real-data analyses revealed that social network exhibits strong temporal properties [1–3], i.e., the edges and nodes do not always exist at any time, and may vary with time. For instance, in scientist collaborative networks, two researchers may collaborate to publish a paper but rarely collaborate at every time step [4–9]. Besides, two researchers may build their first collaboration. Another example is that in the transportation network, two cities may build expressways and high-speed trains. Thus, a new edge is added. The emergence and disappearance of edges and nodes widely exist for the online social network due to the login or log out of the online platforms. Therefore, the temporal network is a widely used method to describe the social network, in which nodes represent individuals and edges stand for their relationships.

For the information spreading on temporal networks, researchers from different disciplines made great contributions [10–19]. In what follows, we first review the progress of information spreading on social network. Different from the static networks, i.e., the network topology does not change with time, researchers found some important results [15, 20–23]. When the information is spreading on the static networks, scholars found that the existence of some hubs may eliminate the threshold point [24, 25]. Specifically, any values of information transmission probability can trigger the information spreading on social networks. Based on these results, we can understand why information can always spread on social platforms. Researchers further revealed that the network community, clustering, and degree-degree correlations could alter the spreading dynamics of information [26, 27]. In reality, sharing a piece of information is risky, and thus affirming its reality and reliability is fatal. Researchers used the threshold-based model to include this factor in the spreading dynamics, such as the Watts threshold model and other generalized models. For that threshold-based information spreading model, the phase transition of the dynamical system is always discontinuous, i.e., first-order phase transition [28]. Wang et al. [29, 30] proposed nonredundant information spreading dynamics and revealed a transition between the continuous and discontinuous transition in the system.

When researchers studied the information spreading dynamics on temporal networks, scholars found that network temporality can suppress or promote the information spreading [1, 20, 21, 31–37]. Xue et al. proposed a mathematical model to describe a heterogeneous population, where a fraction of nodes adopts a complex contagion. They used a Markovian approach to describe the spreading dynamics. They found that the promotion or suppression of the network temporality is determined by the heterogeneities of population and degree distribution. Scholtes et al. [38] found that the non-Markovian temporal network may speed up or slow down the information spreading. Wang et al. [39] proposed a heuristic immunization strategy for information spreading and demonstrated the effectiveness of this strategy in real-world data.

For the temporal network, an essential factor is the network memory [38, 40–42], which means that edges that existed in the current time step may already occur in previous time steps. Sun et al. [43] revealed that network memory inhibits the spreading process for SIR models, in which the epidemic threshold is enlarged while the spreading size decreases. How memory affects the information spreading dynamics is an important question. To address this, we propose a mathematical model on temporal networks with memory. Then, we develop a Markovian theory for the dynamical model. Through extensive Monte Carlo simulations, we systematically investigate the dynamics. Finally, we conclude.

2. Information Spreading Model

In this section, we introduce the information spreading dynamics on temporal networks with network memory.

2.1. Activity-Driven Network with Memory

Mathematically, the temporal network can be described as , where represents the temporal network at time step . For network , we use the temporal adjacency matrix to represent the topology of . If , there is an edge between nodes and at time . Otherwise, .

To build the activity-driven network with memory effect, we generalize the activity-driven model proposed in [44–46]. We build memory activity-driven network as follows.(i)Assign value for network size and potential activity according to a given distribution . In this paper, we assume follows a power-law distribution. That is to say, , where is the potential activity distribution exponent, , , and . Mathematically, the larger the value of , the more homogeneous the degree distribution of the temporal network. Therefore, we can change to investigate the degree heterogeneity of the temporal network.(ii)Generating temporal network : for each node , there are two possible ways to build edges. If node becomes active with probability , where is a parameter, node forwardly connects to randomly selected nodes. If node is inactive, it can only receive the connections from other active nodes.(iii)At the end of the time step , we delete every edge with probability . Therefore, the memory effect is induced. The higher the value of , the more substantial the memory effect of the temporal network. For the case of , the temporal network is memoryless. When , the network is static.

According to the above steps, we know that the average degree of network is .

2.2. Information Spreading Model

We here adopt the information spreading model proposed in [47]. This model uses a generalized susceptible-infected-susceptible (SIS) model to describe the dynamics of information spreading. The susceptible nodes mean that they do not receive any information but may receive the information. The infected nodes represent that they have obtained the information and willing to share it with neighbors. In this model, we assume that there are two types of nodes, i.e., activists and bigots. The activists are willing to share the information with friends. Thus, we assume that they have a smaller adoption threshold . The bigots are less likely to accept the information. Therefore, we set those bigots with a higher adoption threshold . In this model, we randomly select a fraction of nodes as the bigots and the remaining nodes as activists.

The information spreading dynamics evolve as follows:(i)Randomly selecting fraction of nodes to receive the information and setting the rest as susceptible nodes.(ii)At every time step, every infected node tries to transmit the information to its very susceptible neighbor with probability . If node succeeds in receiving a piece of information, it would be represented by the mark +1 and the like. In this situation, we consider the state of node . If node is an activist, it becomes infected. If node is a bigot, it becomes infected only when its received information is larger than .(iii)Every infected node becomes a susceptible state with probability .

When the infected nodes no longer exist or the spreading dynamics has run 10000 times, the dynamics ends.

3. Theoretical Analyses

To obtain mathematical analysis results on the spreading dynamics, we use a generalized discrete Markovian approach, which is inspired by Refs. [39, 48–50]. In theory, we assume that there are no dynamical correlations among the state of neighbors. That is to say, the infection probabilities of a susceptible node from informed neighbors are dependent. At any time step , node can only be susceptible or infected state. Define when node i is in the infected state and when in susceptible state. The probability of node in the susceptible and infected states is and , respectively. For the sake of simplicity, we denote and as and , respectively.

For the evolution of , we should consider two situations: the evolutions of bigots and activists, which are denoted as and , respectively. Therefore, we have

In the following, we study the evolutions of and . For an activist, such as node , the evolution of includes two situations. On the one hand, node is in the infected state at time step and does not recover to susceptible state with probability . On the other hand, node is susceptible at time step and gets infected by neighbors at this time step with probability , whererepresents the probability of node remaining susceptible at time step . The evolution of is

For the bigots, the evolution of is more complex. The bigots will only be infected if the information received is greater than the threshold . Therefore, we should first compute the number of information received by node i and denote as n pieces of information received by node at time step . Using the results presented in Ref. [47], we have

Until now, we have presented the expression of the evolution equations of the information spreading dynamics. In the steady state, i.e., , we have . We denote the fraction of nodes in the infected state in the steady state as , which can be computed as

We use as the order parameter of the system in the following numerical studies.

4. Results

In this section, we use the Monte Carlo method to study the information spreading dynamics on temporal network with memory effect. When generating the temporal network, we set the average degree of every temporal network with average degree . In the dynamical system, we set the order parameter aswhere and . We set . To numerically locate the steady state of the dynamical system, we compute the average values of . When the average value of variance is smaller than , the steady state of the system is reached.

In Figure 1, we first investigate the information spreading dynamics without bigots (i.e., ). For different strengths of the memory effect, we find two distinct regions. When the information transmission probability is small, a strong memory effect suppresses the information spreading dynamics (see and ). That is to say, the temporal network is not beneficial in information transmission. However, for larger values of , the temporal network is beneficial for information spreading. For static networks, the network topology does change with time. The information can be accessed by more susceptible nodes for small values of since the giant connected cluster is larger. However, more different nodes will be connected to the temporal giant connected cluster for temporal networks and promote the information spreading for larger . Once the network memory effect is not strong enough, compared with the information spreading on static and temporal networks, the information is suppressed regardless of the information transmission probability. Thus, we know that the middle memory effect can suppress the information spreading.

(a)

(b)

(c)

We further investigate the information spreading on a given strength of memory effect for different values of bigots in Figure 2. We note that both the fraction of bigots and memory affect the information spreading. For a given network topology, we find that increases with since bigots are not beneficial in transmitting the information. Besides, we note that the increase pattern of versus is different for distinct values of . When , increases continuously with . However, discontinuously increases with when is large (e.g., in Figures 2(b)–2(d) and 2(f)).

(a)

(b)

(c)

(d)

(e)

(f)

The network memory effect may benefit the information spreading, which depends on the degree heterogeneity of the activity-driven network. When the potential exponent is and 3.5, the network memory effect suppresses the information spreading (Figures 2(a), 2(c), 2(d), and 2(f)). However, when the potential exponent is , the network memory effect promotes the information spreading, as shown in Figures 2(b) and 2(e).

Finally, we investigate the phase transition of the system with different initial seed sizes in Figure 3. Generally speaking, the phenomena are the same as those stated in Figure 2. We find that the system always has a hysteresis loop no matter what value and is assigned. In more detail, the final information spreading size depends on . The larger , the higher in the hysteresis loop region. We also note that the larger , the smaller . That is to say, the bigots hinder the information spreading dynamics.

(a)

(b)

(c)

(d)

5. Conclusions

In this paper, we proposed a mathematical model to investigate the effects of network memory on the information spreading dynamics on temporal networks. We first proposed a dynamical model in which the temporal network has a memory effect. Specifically, the current temporal network can remember the previous interconnections. Furthermore, an information diffusion model is developed on this type of temporal network. Then, we used a Markovian approach to describe the information spreading dynamics. Finally, we used the Monte Carlo simulation method to study the information spreading model numerically and found that network memory may promote and suppress the dynamics. The effects depend on the heterogeneous degree distribution and the fraction of bigots in the populations.

The results presented in this paper may shed some light into investigating the dynamics of information on temporal networks. On the one hand, the Markovian theory may be used to study other dynamics on temporal networks. On the other hand, the memory effect of temporal network should be included when studying other dynamics. Finally, some further studies about memory of temporal network should be investigated, for instance, developing more accurate theory and designing more realistic models to describe the spreading dynamics.

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 (no. U1733203), the Safety Foundation of CAAC (no. AQ20200019), and the Foundation of CAFUC (no. J2021-072).

References

N. Masuda and P. Holme, Temporal Network Epidemiology, Springer, Berlin, Germany, 2017.
P. Holme and J. Saramäki, Temporal Network Theory, Springer, Berlin, Germany, 2019.
P. Holme, “Temporal network structures controlling disease spreading,” Physical Review E, vol. 94, no. 2, Article ID 022305, 2016.
View at: Publisher Site | Google Scholar
L. M. Camarinha-Matos and H. Afsarmanesh, “Collaborative networks,” in Proceedings of the International Conference on Programming Languages for Manufacturing, pp. 26–40, Springer, Berlin, Germany, May 2006.
View at: Google Scholar
K.-H. Tsai, “Collaborative networks and product innovation performance: toward a contingency perspective,” Research Policy, vol. 38, no. 5, pp. 765–778, 2009.
View at: Publisher Site | Google Scholar
M. Mandell, R. Keast, and D. Chamberlain, “Collaborative networks and the need for a new management language,” Public Management Review, vol. 19, no. 3, pp. 326–341, 2017.
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
C. Castellano, S. Fortunato, and V. Loreto, “Statistical physics of social dynamics,” Reviews of Modern Physics, vol. 81, no. 2, pp. 591–646, 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 perspective,” IEEE Transactions on Network Science and Engineering, vol. 8, pp. 828–840, 2020.
View at: Google Scholar
L. Pan, W. Wang, L. Tian, and Y.-C. Lai, “Optimal networks for dynamical spreading,” Physical Review E, vol. 103, no. 1, Article ID 012302, 2021.
View at: Publisher Site | Google Scholar
P. Holme and J. Saramäki, “Temporal networks,” Physics Reports, vol. 519, no. 3, pp. 97–125, 2012.
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
W. Wang, M. Tang, H. Eugene Stanley, and L. A. Braunstein, “Unification of theoretical approaches for epidemic spreading on complex networks,” Reports on Progress in Physics, vol. 80, no. 3, Article ID 036603, 2017.
View at: Publisher Site | Google Scholar
G. F. De Arruda, F. A. Rodrigues, and Y. Moreno, “Fundamentals of spreading processes in single and multilayer complex networks,” Physics Reports, vol. 756, pp. 1–59, 2018.
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, Article ID 066117, 2001.
View at: Publisher Site | Google Scholar
R. Pastor-Satorras and A. Vespignani, “Epidemic dynamics in finite size scale-free networks,” Physical Review E, vol. 65, no. 3, Article ID 035108, 2002.
View at: Publisher Site | Google Scholar
R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, “Epidemic processes in complex networks,” Reviews of Modern Physics, vol. 87, no. 3, pp. 925–979, 2015.
View at: Publisher Site | Google Scholar
W. Wang, M. Tang, H. Yang, Y. Younghae Do, Y.-C. Lai, and G. Lee, “Asymmetrically interacting spreading dynamics on complex layered networks,” Scientific Reports, vol. 4, no. 1, p. 5097, 2014.
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
W. Wang, M. Tang, P. Shu, and Z. Wang, “Dynamics of social contagions with heterogeneous adoption thresholds: crossover phenomena in phase transition,” New Journal of Physics, vol. 18, no. 1, Article ID 013029, 2016.
View at: Publisher Site | Google Scholar
H. Peng, W. Peng, D. Zhao, and W. Wang, “Impact of the heterogeneity of adoption thresholds on behavior spreading in complex networks,” Applied Mathematics and Computation, vol. 386, Article ID 125504, 2020.
View at: Publisher Site | Google Scholar
S. Lee, L. E. C. 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, New York, NY, USA, August 2009.
View at: Publisher Site | Google Scholar
N. Masuda and P. Holme, “Predicting and controlling infectious disease epidemics using temporal networks,” F1000 Prime Reports, vol. 99, p. 5, 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, Article ID 062816, 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, Article ID 045102, 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, Article ID 012822, 2015.
View at: Publisher Site | 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. 5024–5029, 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. Gour, J.-P. Ranjeva, M. Ceccaldi et al., “Basal functional connectivity within the anterior temporal network is associated with performance on declarative memory tasks,” Neuroimage, vol. 58, no. 2, pp. 687–697, 2011.
View at: Publisher Site | Google Scholar
S. Unicomb, G. Iñiguez, J. P. Gleeson, and M. Karsai, “Dynamics of cascades on burstiness-controlled temporal networks,” Nature Communications, vol. 12, no. 1, pp. 133–210, 2021.
View at: Publisher Site | Google Scholar
J. P. Gleeson and R. Durrett, “Temporal profiles of avalanches on networks,” Nature Communications, vol. 8, no. 1, pp. 1227–1313, 2017.
View at: Publisher Site | Google Scholar
K. Sun, A. Baronchelli, and N. Perra, “Contrasting effects of strong ties on sir and sis processes in temporal networks,” The European Physical Journal B, vol. 88, no. 12, pp. 1–8, 2015.
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, no. 1, 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, no. 1, 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
X. Xue, L. Pan, M. Zheng, and W. Wang, “Network temporality can promote and suppress information spreading,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 30, no. 11, Article ID 113136, 2020.
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,” Europhysics Letters, vol. 89, no. 3, Article ID 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, Article ID 022316, 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, Article ID 021005, 2015.
View at: Publisher Site | Google Scholar

Copyright

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.

PDF Download Citation

Download other formats

Order printed copies

Views

364

Downloads

881

Citations