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As we discussed in Fig. The reason for this peak has already been explained in ref. Here, the interesting phenomenon happens when different similarity methods are compared. In this case, many links need to be randomly selected from a large number of candidates, resulting in a low reconstruction precision. These similarity metrics overwhelmingly suppress high degree nodes, so that the links are mostly connected to the nodes that are supposed to have low degree.

During the news propagation process, the time stamp when the news reaches each node is recorded. We thus used the temporal information of the news propagation to improve the existing similarity methods see the Methods section. Here, we present the advantage of these temporal similarity methods in Fig. In Fig. S5 in the SI. This is because the news proposed by every node can reach a large part of the networks, so that the news coverage can no longer reflect the topology information of the network. These results indicate that the temporal information is crucial to the network reconstruction from the propagation processes.

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This is easy to understand as the degree correlation is mainly determined by the normalization factor of the similarity methods. Therefore, when selecting the temporal similarity method, one still needs to be very careful, as an inappropriate method may still result in a negative degree correlation and very low reconstruction accuracy. As shown above, the different similarity metrics yield very different results when varying the spreading parameters. In practice, one needs to estimate the spreading parameter before selecting the most appropriate similarity metrics to reconstruct the network.

These three values are normally publicly accessible in real online systems. We further apply the methods to the real networks. Firstly, the methods are applied to real undirected networks. We consider nine empirical networks including both social networks and nonsocial networks: i Dolphin: an undirected social network of frequent associations between 62 dolphins in a community living off Doubtful Sound, New Zealand It includes bands that performed between and , with most of the bands from to We only take into account the giant component of these networks.

This is because a pair of nodes located in two disconnected components, their similarity scores will be zero according to CN and its variants. The results of the similarity methods on these networks are reported in Table 2 in detail. Consistent with the results in the artificial networks, the temporal similarity methods significantly outperform the classic similarity methods not necessarily in degree correlation. The special range is also observed when LHN methods is applied to real networks. However, we also observe that Jac no longer leads to the sudden drop of correlation and precision in the real networks we considered.

Comparing all the methods, the TRA method generally enjoys the highest accuracy. The methods are also applied to real directed networks. We considered several real directed networks to validate our methods. Marks FW food web in St. Like the undirected networks, the temporal similarity methods have a much higher AUC and precision than the classic similarity methods. However, one can also see that AUC and Precision in directed networks are on average lower than the undirected networks. This indicates that it is generally more difficult to reconstruct directed networks via similarity metrics.

We remark that the results on other networks are similar. We select real networks from diverse backgrounds in order to study the performance of the similarity methods in different situations. Table 2 and 3 show that the method with the highest accuracy is almost unchanged in different networks. This means that the performance of similarity methods with respect to the accuracy is robust.

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However, when the degree correlation is measured, the results depend more on the networks, as shown in Table 2 and 3. The degree correlation measures whether the node degree in the reconstructed network is correlated with the node degree in the real network. In this case, a method that performs well in one type of networks is not guaranteed to perform well in other types of networks. If the degree distribution is homogeneous, Jac or LHN similarity measures may outperform CN in degree correlation due to the higher accuracy.

For each method, we also study its temporal version. The description of these methods and their results are presented in SI see Fig. S10 and Table S5, S6, S7. We study the influence of different parameters i. When N increases, the precision of both traditional similarity metrics and temporal similarity metrics tend to decrease. Therefore, it is better to use the temporal similarity metrics to reconstruct networks. This is because the latter group of metrics all has some form of punishment based on node degree.

In LHN, the drop of precision in the special range is most significant. This is because the degree punishment is most severe in LHN. We then compare the results of different metrics on SW and BA networks. However, TRA method reaches the highest value later i. If the time information of the spreading is available, it is better to use THPI to reconstruct the network as it works similar to other metrics in heterogeneous networks and it works best in homogeneous networks.

In this paper, we applied several standard similarity metrics to reconstruct the propagation network based on the observed spreading results. We find that even though some similarity methods such as Jaccard and LHN perform well in link prediction, they may cause problems when they are used to reconstruct networks, as they punish too much the nodes received many news and assign a large number of links to the nodes that supposed to have low degree.

We find that the resource allocation method not only has high reconstruction accuracy, but also results in similar network structural properties as the original network. Finally, we take into account the temporal information of the propagation process, and we find that such information can significantly improve the reconstruction accuracy of the existing similarity methods, especially when the infection rate is large.

The special range cannot be observed if one uses AUC to assess the network inference results. It can only be seen when one picks up the top ranking predicted links and uses them to reconstruct the network. This is also an important message for the link prediction research in which AUC is usually adopted as the only metric to evaluate the prediction results. Some problems still remain unsolved.

For example, our methods now require full time information. When only partial time information is available, the temporal similarity methods must be modified. In addition, our work only considers the simplest epidemic spreading model. Other more realistic models describing the disease contagion and information propagation need to be examined Furthermore, similar problems in other fields also need to be addressed.

For instance, most link prediction methods are based on the observed network topology. When the time information of the observed links is available, the similarity methods should be modified accordingly to incorporate the temporal information of the network. Node similarity is also a basic network feature for community detection.

Improving the community detection accuracy with the time information could be important problem. We believe that our work will inspire possible solutions to the above mentioned problems in the near future. The original similarity methods and the improved ones based on time information are listed below.

The formula reads. It can prevent the large degree nodes from having too high similarity with other nodes. The index is defined as. It is defined as. In this case, i is definitely not the node that passes the news to j , so i and j are unlikely to be connected in the networks. We pose this setting as it applies to our step-by-step spreading model.

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How to cite this article : Liao, H. Reconstructing propagation networks with temporal similarity. Newman, M. The structure and function of complex networks. SIAM Rev.

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Zeng, A. Buy Softcover. FAQ Policy. About this book Many decision problems in Operations Research are defined on temporal networks, that is, workflows of time-consuming tasks whose processing order is constrained by precedence relations. Show all. Background Theory Pages Wiesemann, Wolfram.

Show next xx. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. National Center for Biotechnology Information , U. PLoS One. Published online Apr 1. Sokolov 4. Hartmut H. Igor M. Author information Article notes Copyright and License information Disclaimer. Competing Interests: The authors have declared that no competing interests exist. Received Oct 1; Accepted Feb This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

This article has been cited by other articles in PMC. Abstract We extend the concept of accessibility in temporal networks to model infections with a finite infectious period such as the susceptible-infected-recovered SIR model. Introduction Networks are one of the most important ways to represent a finite set of elements with complex interaction patterns. Open in a separate window. Fig 1. Transitivity is not assured in temporal networks. Materials and Methods Epidemiological Model: State Vector Formalism Since the beginning of modern epidemiological modelling [ 24 ] a Markovian dynamics was typically assumed.

Tracking All Initial Conditions: Matrix Formalism It is often important to evaluate the role of single nodes in a contact network with respect to their influence on the outbreak dynamics. Application to Empirical Contact Networks Social Contacts Network As a first example, we consider a social contact graph [ 26 ] that has been recorded during a three-days conference.

Fig 2. Prevalence, incidence and cumulative incidence for the social contacts network. Table 1 Comparison between the temporal networks and the time aggregated networks. Sexual Contacts network The result is rather different, if we analyse the dynamics on a sexual contacts network from a Brazilian escort website [ 14 ] see Fig 3. Fig 3. Prevalence, incidence and cumulative incidence for the sexual contacts network.

## Optimization of Temporal Networks under Uncertainty - PDF Free Download

Livestock-Trade Network Finally, we apply our formalism to an excerpt of the national German livestock database HI-Tier [ 35 ], which has been established according to EU legislation. Fig 4. Prevalence, incidence and cumulative incidence for the livestock-trade network. Fig 5. Estimating the critical infectious period for the social A , the sexual B and the livestock-trade network C.

Conclusion We have introduced a formalism to calculate paths for infections in temporal networks, which is based on elementary operations from linear algebra and Boolean arithmetic.

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