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The authors have declared that no competing interests exist.

Conceived and designed the experiments: MK HL PH TS. Performed the experiments: MK. Analyzed the data: MK HL. Wrote the paper: MK HL FC PH TS.

Many networks exhibit time-dependent topologies, where an edge only exists during a certain period of time. The first measurements of such networks are very recent so that a profound theoretical understanding is still lacking. In this work, we focus on the propagation properties of infectious diseases in time-dependent networks. In particular, we analyze a dataset containing livestock trade movements. The corresponding networks are known to be a major route for the spread of animal diseases. In this context chronology is crucial. A disease can only spread if the temporal sequence of trade contacts forms a chain of causality. Therefore, the identification of relevant nodes under time-varying network topologies is of great interest for the implementation of counteractions.

We find that a time-aggregated approach might fail to identify epidemiologically relevant nodes. Hence, we explore the adaptability of the concept of centrality of nodes to temporal networks using a data-driven approach on the example of animal trade. We utilize the size of the in- and out-component of nodes as centrality measures. Both measures are refined to gain full awareness of the time-dependent topology and finite infectious periods. We show that the size of the components exhibit strong temporal heterogeneities. In particular, we find that the size of the components is overestimated in time-aggregated networks. For disease control, however, a risk assessment independent of time and specific disease properties is usually favored. We therefore explore the disease parameter range, in which a time-independent identification of central nodes remains possible.

We find a ranking of nodes according to their component sizes reasonably stable for a wide range of infectious periods. Samples based on this ranking are robust enough against varying disease parameters and hence are promising tools for disease control.

Animal trade represents an important economic sector. At the same time, it also provides a major route for economically most important infectious livestock diseases

Epidemiology has been influenced by network science in recent years

One way to meet this demand is the utilization of time-dependent networks, also known as temporal networks. In temporal networks an edge is represented by a triple

In their recent review on temporal networks, Holme and Saramäki

In fact, there has already been research on temporal networks in the context of epidemic spreading. Vernon et al.

Together with a better understanding of the initial spread of a disease

The term risk-based centrality is context-dependent and has at least a two-fold meaning. It can either characterize the potential of one node to infect other nodes or it can characterize the exposure of a node of being infected by others. Many of the well-established centrality measures for static networks have already been adapted for temporal networks

The difficulty to define a risk-based centrality is caused by the complexity of any centrality measure that considers finite dynamic timescales. Even worse, the centrality of a node will not only depend on the infectious period

To our knowledge, these dependencies have so far not been investigated systematically. This paper attempts to start filling this gap. To this end, we investigate the temporal robustness of two simple measures of centrality for an SIR-like disease spread on the German pig trade network.

Temporal robustness means that a centrality measure is insensitive to variations in the time of infection

Here, we try to answer the question to what extent this is still feasible in the context of temporal networks. Particularly we focus on the case of epidemiological relevant centrality in the context of network topologies significantly changing on the timescale of a typical infectious period.

A frequently used measure in epidemiology is the final size of an epidemic, which is the number of all infected individuals throughout an epidemic. In network terminology, this is equivalent to the number of nodes that can be reached from a primarily infected node, i.e. the size of its out-component, when a transmission probability of

Another measure of centrality is defined by its reversal, i.e. the number of nodes from which a particular node can be reached. This number is given by the size of its in-component and corresponds to a

Node components have already been used for risk assessments in static representations of animal trade networks

We make use of the out-component

In the following, we first present the data, on which the analysis is based, and then the algorithm used to calculate out- and in-components. We then investigate the dependence of both measures on the time of infection

We propose conditions under which centrality can be assigned independently of

The data used in this paper is an excerpt of HIT

Although the trade of live pigs is subject to seasonal variation and temporal irregularities, we found a period of one year sufficient to obtain a representative picture of the dynamic patterns of the network (see

In contrast to the temporal network, the static time-aggregated network, where

All other nodes, which cannot reach the giant strongly-connected component, have an out-component with a size three orders of magnitude smaller.

We are not aware of an efficient algorithm to determine the out-component

However, the determination of

In order to calculate

It is not clear in general, if a given observation period of a temporal network can capture an entire dynamic process on the network

The nodes visited by our algorithm are identified as the out-component

The in-component

The size of both measures can be conveniently normalized to the number of nodes

To investigate their temporal dependency and to gain an understanding of their robustness, we determine both measures for all nodes for infectious periods

To retain readability, we will restrict the detailed description of results to the analysis of the out-component. The results for the in-component show no conceptual differences and their main figures are replicated in

Before analyzing the robustness of the size of the out-component

In a time-aggregated representation of a network, any primary infection will cause secondary infections as long as there is at least one outgoing edge during the observation period. In temporal networks, the occurrence of secondary infections is more constrained, as the infectious period

Given an infectious period

Panel A: Outbreak probability

Another interesting measure is the average size of the out-component

Rocha et al.

Both observations support our argument that a temporal view on the network is essential to capture its dynamics fully. Calculations based on a time-aggregated network strongly overestimate the size and probability of an outbreak.

We explored the dependence of the out-component

For illustration purposes we began with an exemplary infectious period of

Panel A shows the size of the out-component for an exemplary node

The explanation lies in the temporal sparsity of edges in the network, as illustrated by the following: Let us assume that a node

Besides small fluctuations of

To allow for a more complete view on the network,

Most nodes exhibited vanishing out-components for almost all times of infection. This is indicated by the bright region in the lower half of

To allow for further investigation of the size of the out-component

One should recall that the values of any risk-based measure as such are usually of minor importance for disease control. In most cases it is sufficient to identify the nodes that exhibit the highest values with respect to a particular measure. In fact, these will be the ones where interventions are most promising. In order to locate these nodes, it is sufficient to order the nodes for each

Each curve corresponds to one node. The top hundred nodes with the largest out-components are shown. Curves representing nodes with higher ranking are darker than those with lower rankings. For illustration purposes an arbitrarily chosen node is displayed in red.

For the purpose of disease control, it is desirable to know the set of nodes

This sensitivity can be analyzed by investigating the intersection

The size of the intersection

A further reduction of the dimensionality is possible by recalling disease control requirements. Since one is primarily interested in the sensitivity of

Analogous averaging over all

Shown are the mean intersections

One can see that

To account for the high variability in the ranking

Finally we compared our proposed measure to centrality measures on a static network representation. We followed the approach described in the previous section. Accordingly we determined the intersection of two top samples, i.e. the highest ranked nodes. One top sample is based on the dynamic out-component, the other on a selection of static centrality measures. In detail we compared

The relative size of the intersection of the top nodes is based on their value of the dynamic out-component and on static measures of centrality. In the upper panel the comparison for a fixed infectious period of

We analyzed if time-independent determination of central nodes is possible even in a network with high temporal heterogeneity. The network under consideration is the German pig trade network. We investigated an epidemiological relevant centrality measure on this network with a topology that changes on the timescale of epidemic spreading. The spreading is described by a state-discrete SIR-like model. We focused on the out-component of a node as a measure of centrality for two reasons. First of all, an intuitive adaptability of the out-component exists in the time-aggregated case to finite infectious periods. Secondly, under the assumption of an infection probability

We found that the rapidly changing network topology, whose timescale is in the order of a typical infectious period, was reflected in the observed temporal heterogeneity of the out-component. We also demonstrated that the dynamic out-component only barely correlated with any static centrality measure. Therefore any static approximation should be used with caution. For the German pig trade network, however, a ranking based on the size of the out-component would be stable enough for disease control requirements.

Furthermore, the stable ranking allowed the sampling of nodes. We found such samples to be robust against variations in the length of the infectious period. For the German pig trade network, this enables the determination of disease-independent high-risk samples.

We emphasize that the results presented here are only valid for the specific network under consideration. Nevertheless, we expect similar results for other networks of animal trade, especially for pig trade networks due to the highly standardized and industrialized nature of these networks.

Our work contributes to improve surveillance and control of diseases, which propagate via trade of live animals. In the context of surveillance, one might argue that the in-component is a more suitable measure of centrality, but as shown in

This paper is based on three assumptions that are critical for the applicability of its results. First, the data used was collected during a disease-free period of the network. It is known, however, that the topology of animal trade networks changes significantly if a disease is detected

The second assumption is the homogeneity of the nodes. In reality, the nodes of an animal trade network exhibit different functionalities, e.g. breeders and slaughterhouses. This yields very different infection probabilities. This paper circumvents this problem by assuming an infection probability of

Finally, the analysis of the robustness is based on averaging the size of the out-component over several days of primary infection. This approach is supported by the likely unavailability of any information on the exact day of a primary infection in the case of disease surveillance. However, if this information had been available, averaging might represent an unnecessary limitation (see

An additional remark has to be made on the assumption that a transmission probability of

In conclusion, we showed that the notion of time-independent node centrality is critical in the context of temporal networks. However, stationary sampling of nodes remains still possible for the presented network.

Our findings can be applied in a more accurate risk assessment of a disease outbreak in the absence of counteractions. As a next step, the effect of vaccination protocols could also be taken into account as well as the implementation of a sophisticated surveillance system.

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We thank Dr. R. Carmanns, Bavarian State Ministry for Nutrition, Agriculture and Forestry for providing data. This work was supported by the Federal Ministry for Food, Agriculture and Consumer Protection: Forschungs-Sofort-Programm A/H1N1-1.3. PH acknowledges support of the German Academic Exchange Service (DAAD) via a postdoctoral fellowship. We also thank two anonymous reviewers for helpful comments.