Abstract
Human mobility networks are widely used for diverse studies in geography, sociology, and economics. In these networks, nodes usually represent places or regions and links refer to movement between them. They become essential when studying the spread of a virus, the planning of transit, or society’s local and global structures. Therefore, the construction and analysis of human mobility networks are crucial for a vast number of reallife applications. This work presents a collection of networks that describe the human travel patterns between municipalities in Mexico in the 2020–2021 period. Using anonymized mobile location data, we constructed directed, weighted networks representing the volume of travels between municipalities. We analysed changes in global, local, and mesoscale network features. We observe that changes in these features are associated with factors such as COVID19 restrictions and population size. In general, the implementation of restrictions at the start of the COVID19 pandemic in early 2020, induced more intense changes in network features than later events, which had a less notable impact in network features. These networks will result very useful for researchers and decisionmakers in the areas of transportation, infrastructure planning, epidemic control and network science at large.
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Introduction
Intermunicipal mobility is a type of medium and large scale human mobility within a country where millions of individuals travel daily from one county or municipality to another, either going from home to work, shopping, accessing public services, cargo loading, vacation, etc. These movements and travels generate complex structures and dynamics of socioeconomic interactions between different areas both at regional and national levels.
Given the nature of mobility systems, complex networks have been widely adopted to model commuting phenomena^{1,2,3,4}^{5}. Characterizing and understanding the properties of mobility networks is crucial for decisionmaking, urban planning, traffic engineering, and, as has become clear with the COVID19 pandemic, designing, implementing and evaluating mobility restrictions and lockdowns to contain or control the epidemic spread^{6,7,8,9,10,11,12}. Therefore, there is a need for public mobility network datasets aimed at researchers and decisionmakers in the areas of geography, urban planning, epidemic control, etc. Traditionally these networks had been obtained from mobility surveys. However, the emergence of cell phone and GPS data in recent years has facilitated the acquisitions of accurate and large sets of human mobility data, thus allowing the construction and characterization of large and detailed human mobility networks.
The construction of origindestination networks aiming to describe mobility patterns, measure the volume of public transportation and to plan public transportation has already been assessed. These mobility patterns include largescale and longrange commuting patterns^{13,14}, spatiotemporal patterns for different socioeconomic strata^{4}, patterns in bikesharing systems^{15}, etc. Origindestination matrices have also been used to measure the volume of use of public transport^{16}^{17,18} and public transport planning^{19,20}. Some approaches for modeling and generating origindestination matrices include gravity models^{21,22,23}, Bayesian models^{24} ,^{25,26}, linear assignment matrix approximation^{27}, Principal Components Analysis^{28} and gradient approximation method^{29}, among others.
Origindestination networks have been elaborated from different types of data. Most reports use data from the different public transport systems or road side interviews^{21}. Alsger et al.^{16} and Munizaga and Palma^{18} used smart card fare data to estimate origindestination networks. Lotero et al.^{4} considered data from bikeshared systems for these matrices. Wang and Mirchandani^{17} conducted experiments on two datasets generated by ridehailing applications. Toledo and Kolechkina^{27} estimated the matrices using data from traffic counts on the network links. Recently, other works have attempted to utilize social network data from Twitter or Facebook^{30,31,32} ,^{33}^{34,35} or data set from mobile phone data such ACAPS dataset or SafeGraph^{36} ,^{37,38}. In particular, Edsberg et al.^{34} show that origindestination networks based on data from cell phones and social networks present highquality results.
Reconstructing mobility patterns is important in order to understand and model the spread of infectious diseases^{39}. By analyzing the movements of individuals and populations, researchers can gain insights into how diseases may spread from one location to another. This information can be used to inform public health interventions and policies aimed at preventing the spread of disease. In particular, during the COVID19 pandemic, the use of mobility data has been especially promising for incorporating realistic mobility flows into epidemiological models, helping to predict hospital admissions, and to assess the impact of mitigation policies on collective behavior. Additionally, the data can help refine interventions by providing near realtime information about changes in patterns of human movement, which can inform policy and messaging around social distancing and other interventions. Specific applications using many of the aforementioned data sources, from social networks^{40} to cellphone location data^{41}.
Human mobility exhibits different behaviours across scales. At small spatial scales, such as within buildings or neighborhoods, people tend to move more randomly and frequently. At larger scales, such as between cities or countries, mobility is more likely to be directed and purposeful, with longer travel times and distances. These differences in scale are important for understanding various aspects of human behavior, such as socioeconomic interactions and cultural dynamics. Recent work^{42} suggests that while human mobility may appear scalefree when looking at the overall distribution of displacements, there are meaningful scales corresponding to spatial containers that restrict mobility behavior. As such, specific approaches are needed to understand different scales of human mobility through empirical data.
Here we introduce a new public dataset of daily intermunicipal origindestination networks in Mexico for 2020 and 2021, directly constructed from large datasets of geolocation data. This dataset will contribute to research and decisionmaking communities from diverse interests, from pure network theory to those studying human mobility, urban planning, nationalscale social and economic relations, epidemic control.
This article is organized as follows: in the Results section, we present and describe the collection of mobility networks and show an analysis of changes in global (sum of weights), local (centrality measures), and mesoscale (community structure) network features. In Discussion, we argue how when initial COVID19 restrictions were implemented in early 2020, more intense changes in network features were induced than in later events, which had a less notable impact on network features. In Methods, we describe the methodology we utilized to collect data and show the algorithm for network construction.
Results
A travel network dataset
We release a public dataset of 731 intermunicipal origindestination networks in Mexico. These networks were constructed from a large and anonymized mobile location dataset (see^{43} for more information about this dataset). Each network is the intermunicipal origindestination network in Mexico for each day during the 2020–2021 period. Nodes represent municipalities (thirdlevel administrative division) or official metropolitan zones (see Methods section below). These are weighted and directed networks, where the weight of edge (i, j) is equal to the total number of observed travels from node i to node j normalized by the different number of mobile devices we recorded on that day. The data set with these 731 networks are freely available in an OSF repository http://dx.doi.org/10.17605/OSF.IO/42XQZ.
For analysis and visualization purposes, we chose nine representative dates capturing different important events during the evolution of the pandemic in Mexico; these are shown in Table 1. As a first visualization, we show in Fig. 1 mobility networks over the Mexico map for this set of dates, drawing only the 1\(\%\) of edges with the highest weight.
Changes in global network movement were observed throughout the first 2 years of the pandemic
In order to quantify the total observed movement in each network of the collection, we consider the total sum of weights in the network \(S_{G} = \sum w_i\), where i runs over all edges of the network. In our context, a higher value of \(S_{G}\) can be understood as higher mobility between municipalities in the country, which, in turn, is associated with people’s decisions to move outside their locality.
Figure 2 shows the time series related to the \(S_{G}\) metric, also indicating the set of dates described in Table 1 and official school vacation periods (summer breaks, winter breaks and Easter holidays, shaded in blue). The observed decay in mobility that starts in February 2020, is probably due to the postholiday season, and is prolonged after the start of the lockdown, reaching a local minimum point shortly before the beginning of the summer holiday season. Following the summer break, mobility continued to decay, reaching its lowest point again shortly before the beginning of winter break, when there was a pronounced mobility rise. For the first half of 2021 we observe a sustained rise in mobility until July 2021, when it reaches a relatively high plateau.
Node centrality measures highlight the dynamics of municipality movement
Since networks exhibit a different mobility pattern each day, centrality measures associated with each node (locality) also change. In this section we explore the variability of centrality measures over the period we studied. We choose three different centrality measures:

Degree centrality: the number of inbound, outbound, or total adjacent edges to a given node. In this network, a single detected travel between two municipalities is enough to establish a link or edge between them.

Strength centrality: the sum of the weights of inbound, outbound, or total adjacent edges to a given node. For an interpretation of the weight in these networks, see methods note 2.

Betweenness centrality: a measure of the number of shortest paths that pass through a given node, indicating its role as a bridge between regions of a network.
For this section, variability is measured using the coefficient of variation (cv).
Figure 3 shows the variation of node strength for nine different nodes (municipalities or metropolitan zones). These nodes were chosen to illustrate different behaviours across the studied period. Figures corresponding to degree and betweenness centralities can be found in Supplementary Material Fig. 1 and Fig. 2.
The regions of Valle de México, Guadalajara, Monterrey and PueblaTlaxcala correspond to densely populated regions. In these areas, the coefficients of variation (100 \(\times\) standard deviation/mean) for the three different centrality measures (degree, strength, and betweenness, respectively) are: (30.0, 53.1, 9.7) in Valle de México, (24.3, 50.3, 17.8) in Guadalajara, (31.3, 49.2, 25.9) in Monterrey, and (29.0, 44.0, 34.7) in PueblaTlaxcala. Morelia, a city in westcentral Mexico, is included because it maintains a highly stable mobility pattern with coefficients of variation 24.0, 40.3 and 16.0 for the considered centrality measures. Meanwhile, places like San Pedro Topiltepec exhibit higher variability in their degree centrality; this is due to its having a mostly low strength throughout the analysis period, but exhibiting a higher strength during summer 2021; the coefficients of variation for this location are 320.4, 410.1 and 285.8. Tijuana is an important border city, so changes in centrality were expected whenever there were changes in the regulations on the border with the USA; the coefficients of variation are 34.9, 41.5 and 89.6. Finally, Acapulco and Cancún are cities that represent popular tourist beach destinations; the coefficients of variation for these cities are (41.4, 41.1, 60.2) and (33.3, 41.8, 62.1), respectively.
In order to explore the variability observed in node centrality measures with the population of the area they represent, the relationship between the three considered centrality measures and the population reported in the 2020 national census was analyzed as shown in Fig. 4 (census data taken from https://www.inegi.org.mx/programas/ccpv/2020/default.html#Datos_abiertos). Table 1 in Supplementary Material shows coefficients of determination when fitting linear regression models.
While the actual centrality values of each node may vary over time, it is important to notice that their overall ranking remains relatively constant; that is, regardless of the actual values at any given date, the ranking structure remains relatively stable. In Supplementary Material Fig. 3, we show the rank turnover time series; rank turnover is a recently proposed measure^{44} that quantifies the stability of a ranked list over time; we observe that the top 10, top 50, and top 100 nodes by either degree, strength or betweenness centrality are quite stable, compared to a null model of random daily rankings.
Node strengths show asymmetric heavytailed distributions.
We performed several tests to study the distribution function of node strengths for the networks. ShapiroWilk test for the logarithm of node strength gives pvalues below 0.01 for all 731 networks, where the largest pvalue is equal to 0.004 for January 3rd, 2020. This value indicates that node strength distributions are not well fitted by a lognormal distribution. We show an example for a particular date in Fig. 5A, where we observe deviations from normality at both tails. Therefore, we calculated the excess kurtosis, which is a standard measure for tail extremity or the tendency to produce outliers^{45}. We show in Fig. 5C a time series of excess kurtosis, where we observe that it takes positive values at all times.
It is worth noticing that the histogram of node strength in logscale is not symmetric around the peak; an example of this can be seen in Fig. 5B. Consequently, we have applied the Beta Rank Function (BRF), which is a ranksize function, and a family of probability distributions that exhibits Paretian behavior in both tails, with different exponents^{46}. BRF has the form of \(X \sim (1r)^b/r^a\), where r is the rank of the observation of size X, and a, b are free parameters. Unlike doublePareto distribution, BRF has a smooth peak, and has been proved to be useful for fitting and modelling asymmetric heavytailed size distributions^{47}. We can see how both tails decay from the peak with different rates, each one controlled by an independent parameter of the BRF. That BRF is a better model for our network node strength distribution is confirmed in ranksize plots shown in Fig.5D. Similar results hold for the rest of the networks. Analogous analysis for degree and betweenness distributions are shown in Supplementary Material Fig. 4 and Fig. 5.
Formation and evolution of communities in the network.
Recalling that a community in a network is a set of nodes with a larger density of connections between them than external to the set, communities in these mobility networks correspond to groups of municipalities or metropolitan zones with a high internal mobility and a relative low external mobility (from the region to the outside or the other way around). Community detection algorithms in mobility or commuting networks have been widely used to detect or delimit geographical functional regions^{48}^{49}^{50}^{51}^{52}.
As a proof of concept, we have applied the labelpropagation algorithm^{53} to detect communities in our networks. In Fig. 6, panel A, we show an alluvial plot that schematically shows the time evolution of community structure on the networks. In this diagram each line represents a municipality in Mexico grouped according to the network community they belong to. For the sake of visualization, the groupings shown correspond to the previously established representative dates. We observe that the number of communities increases during the lockdown period with respect to the prelockdown period; later, it shows a steep increase surrounding the start of the third COVID19 wave  in summer 2021.
A question that may be asked in these networks is whether communities capture phenomena associated to geographic divisions; for instance, whether municipalities from the same state organize within single or many communities, and whether is common that these communities cross state lines. In Fig. 6, panel A, the color of each line represents the state to which that municipality belongs; we may observe that some communities are exclusively composed by municipalities of the same state, while others do cross state lines; in fact, we observe that interstate communities were more common just after the first national COVID19 cases peak (in summer 2020, with about 50% of communities crossing state lines). Furthermore, we hypothesized that different states would have different communityforming behaviors; In Fig. 6, panel B, we show the alluvial plots for the subset of municipalities from four different states: Campeche, Nuevo León, Oaxaca, and Chiapas. Curiously, we observe that the first two states tend to have more stable communities that encompass a larger fraction of the state’s municipalities than Oaxaca and Chiapas.
At this stage, establishing a mechanism that describes why different municipalities behave differently in terms of community formation would be beyond the scope of this work; we should highlight, however, that we chose these four states as they are the two highest and lowest ranked states in terms of their GDP. Furthermore, the characterization of community structures in networks is not trivial, from the selection of the community detection algorithm forward; as such, we consider that the findings presented here serve mostly as a proof of concept of the type of analyses that this dataset may enable.
Discussion
Analysis of global features, centrality measures and community formation in human mobility networks allow us to inquire about different kinds of effects that the pandemic may have had on intermunicipal mobility.
The sum of total weights over all edges of these mobility networks serves as an indicator of the total intermunicipal mobility that we observe on each day. We can observe in Fig. 2 a prepandemic peak at late February  early March 2020, since the first confirmed case in Mexico was reported on February 24, 2020. This peak is followed by a persistent decay and it continues to decrease from the previous maximum value observed in late February. In fact, this decay continues until JuneJuly of 2020, when we observe a peak that coincides with the official summer break. After the summer break we still see a downward trend until the winter break and, again, we see a peak that coincides with an official vacation period. In late February 2021 we observe the lowest point of mobility and from here there is a sustained upward trend, apparently unaffected by Easter holidays until it reaches a new maximum that is even higher than the prepandemic peak. At this point mobility reaches a high plateau and, even though mobility still fluctuates, these fluctuations do not seem to be correlated neither with the pandemic third wave nor with vacation periods. From July 2021 total intermunicipal mobility stays at high levels, comparable with prepandemic levels. This is consistent with what we observe on the maps of Fig. 1; notice that, associated with the beginning of the trafficlight system and official lockdowns less mobility is visually observed (July and September 2020 show less mobility than March 2020) and how mobility increases on August 2021, coincident with the third peak of daily contagions.
Centrality measures evolve over time (Figs. 1 and 2). How they change seem to be weakly associated to the evolution of the pandemic. We also observe that not all nodes exhibit the same behavior. While some nodes, such as Guadalajara, PueblaTlaxcala and Valle de Mexico show larger fluctuations in their degree centrality, other nodes, such as Morelia exhibit more stable time series. Changes in betweenness centrality for the Valle de Mexico are not similar to those observed for other nodes. Evolution for degree and strength in Valle de Mexico seem to be correlated, but they differ from evolution of betweenness.
A relatively simple intuition is that municipalities with larger populations will have higher values of centrality measures regardless of time. We do see that all of the analysed centrality measures are correlated with population size. However, some of these metrics show a better linear correlation with population on linear scale while some other show a better linear correlation on logarithmic scale (see Table 1 in the Supplementary Material for model fits).
A related question is whether variations in these centrality measures are also correlated with population size. We observe a negative correlation between the coefficient of variation and population, showing that centrality variation is higher in smaller municipalities. Again, the best fit for this correlation is not necessarily observed in the linear scale. In any case, it should be noted that while population size is clearly a relevant factor, the dispersion of the point cloud in the range of midsized municipalities indicates other factors that may be in play to explain centrality measures in these networks fully.
Distributions of node strengths seem to be heavy tailed (Fig. 5). In light of ShapiroWilk results and the observation of the large values for the excess kurtosis, we can interpret this as a tendency to produce more outliers than the normal distribution. This means that a very few hubs in the network (nodes with very large strength) capture a very large portion of the total intermunicipal traffic. We observe a larger excess kurtosis during 2021 than during 2020 and a sustained upward trend with a few periods of very rapid increase. This translates into growing differences between large and small nodes in terms of strength. Regardless of how many people moved each year, this indicates that the way in which people travelled between municipalities exhibited changes through time. In addition to this, strength distributions are not symmetric around the peak, as a normal or lognormal distribution would be. This asymmetry  which deviates these distributions from a power law  suggests that these are not scalefree networks and that there exist at least two different regimes: one for large nodes and another for small nodes. Considering that these larger nodes have been identified to be larger metropolitan areas, whereas the smaller nodes tend to be smaller populations, this result may be related to the differential contribution of larger areas to intermunicipal mobility, whereas smaller towns contribute only to regional mobility.
Regarding formation of communities in the network, we could identify that for each day in this analysis there is a “giant” community that includes about half or more of the nodes in the network and that this structure is preserved for all dates. Therefore, if there are changes in the community structure of intermunicipal mobility, these have to occur at more local levels. In Fig. 6, each thread, which refers to a municipality in the network, is colored according to their state. For example, all lightblue threads starting at the bottomleft corner of the diagram correspond to Oaxaca municipalities, while green threads just above are Puebla municipalities. Notice here how all nodes in Puebla tend to stay inside the largest community, but municipalities in Oaxaca move to different communities, indicating a change in the structure of intermunicipal level mobility within this state.
When we wonder how communities form within different states we also see different behaviors (6B). Campeche and Nuevo León are two examples of states with a somewhat steady community structure, meaning that the majority of municipalities tend to stay in the same national community and that these community do not change much with time. In contrast, Oaxaca and Chiapas are states where municipalities fall in many different communities and they change from one community to another with time. This suggests the formation of local dynamics, where inhabitants move around small and specific groups of neighboring municipalities in a more considerable measure than people in Campeche and Nuevo León do.
It should be noted that the structure of this network is capturing a largescale mobility behaviour. As previously mentioned, human mobility exhibits different characteristics at different scales^{42}. In recent work, it has been shown that the granularity in which mobility is analyzed using digital data captures does capture some of the structural differences across scales^{40}. Our research group’s work on studying these differences across scales is ongoing^{43}.
It would be beneficial to have an external dataset for validating the mobility patterns observed in our study. In our previous work, we have already validated intrametropolitan mobility using official origindestination networks^{54}. Interestingly, the modular structure obtained using device mobility is similar to the one obtained from the official data. However, for intermunicipal travel, we were unable to obtain origindestination data from the Mexican authorities, which hinders our ability to follow a similar validation strategy. Our future work aims to address this issue by exploring alternative data sources such as online social networks.
In summary, we present a set of daily intermunicipal origindestination networks in Mexico for 2020 and 2021. These networks were constructed from geolocation information from mobile devices, using a large volume of data points. Making this dataset available to the community contributes in at least two important ways. First, it releases information that, by its nature and cost, can be difficult to access and does so with great detail and accuracy. Second, integrating location information into a network model eliminates the need to deal directly with point data and simplifies the task of exploring different dynamics between locations in Mexico.
Furthermore, the network collection was designed to include pandemic and prepandemic periods. This opens the possibility of using the set to explore changes in mobility patterns derived from the emergency, government regulations, and the decision of the general population to move between localities at different times during the health crisis.
To illustrate the potential use of network collection, we conducted several exploratory analysis focused on describing global and local changes in the networks and community detection. These tasks invite the formulation of questions that explore indepth the latent patterns in the displacement dynamics captured by the networks.
We hope that making publicly the full dataset available will foster exchange among network scholars, be helpful to anyone interested in the dynamics that the country’s population creates between specific areas, and enrich the perspective on how such dynamics may be related to more complex processes.
Methods
Mobile device location data
We used mobile device location data for the time period between 20200101 and 20211231 within Mexican territory provided by Veraset, a company that aggregates anonymized mobile device location data. This source dataset is provided as a table in which each record (called a ping) contains the position (latitude and longitude) of a given (anonymized) device for a given timestamp (with temporal resolution up to seconds). The set of all unique device ids for a given day is called the device panel.
Intermunicipal travel network construction
We define an Intermunicipal Travel Network (IMTN) as a directed, weighted graph G(V, E), for a given day, such that:

Nodes represent localities (either municipalities or metropolitan zones, as defined by the national geographic agency; see Note 1)

Links represent mobility from the source node to the target node, defined by observing at least one device that moved from node i to node j.

Link weights represent the total fraction of observed devices that moved from node i to node j, out of the total number of observed devices; this acts as a normalized measure of flow between nodes (see Note 2).
Complexity for network construction algorithm
Building intermunicipal travel networks requires large amounts of data and computation.The pseudocode in Algorithm 1 shows the algorithm used for network construction, starting with the daily mobile device location data. Complexity to compute these networks is \(O\left( nm + l^2\right)\), where n is the number of pings, m is the number of municipalities and l is the number of devices. To build the networks, an average of 103,890,203 records per day were processed for 2 years and the average of devices per day was 3,239,070.
Note 1: on the aggregation of metropolitan area nodes
The political division of Mexico has municipalities as the smallest unit. Generally, a population center is contained within a municipality; however, there are large urban areas in which a single population center extends through many different municipalities, such that the movement between municipal boundaries is capturing the urban mobility and not travel between different locations. The National Geography and Statistics Institute (INEGI) defines 74 metropolitan areas in Mexico, based on measurements from 2015 https://www.inegi.org.mx/contenido/productos/prod_serv/contenidos/espanol/bvinegi/productos/nueva_estruc/702825006792.png.
Since intracity mobility is beyond the scope of this manuscript, and would greatly skew the mobility metrics, as the volume of intracity mobility is way larger than that of true travels between different locations, we decided to aggregate the municipalities that form these metropolitan areas into single nodes in the network.
Note 2: on the interpretation of edge weights in the network
We defined edge weights in the network as follows:
where \(D_{ij}\) is the number of devices that were observed to move from i to j, that is, were observed in i and their next immediate ping was in j; and D is the total number of devices in the day’s device panel. In this way, the weight represents a normalized measure of flow between regions; we may observe that in limit cases, the weight will be zero when there is no movement observed from one municipality to the other, and the weight would be 1 if all observed devices within the country travelled from region i to region j, which would be a virtually impossible scenario. An advantage of using this approach is that it controls variability in the number of observed devices each day, allowing for comparisons between days.
Network analysis
Networks were analyzed using the igraph library, version 1.2.7^{55}, for the R programming language, version 4.1.0. Rank turnover was calculated according to the definition found in^{44}.
Data availability
The collection of 731 intermunicipal networks is publicly available on a OSF repository http://dx.doi.org/10.17605/OSF.IO/42XQZ.
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Aknowledgements
This project was supported by the Fondo Conjunto de Cooperacio ́n M ́exicoUruguay from the Agencia Uruguaya de Cooperacio ́n Internacional and the Agencia Mex icana de Cooperacio ́n Internacional para el Desarrollo. This project was supported by CONACYT Ciencia Básica y/o Ciencia de Frontera. Modalidad: Paradigmas y Controversias de la Ciencia 2022 Project: 320557.
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Data engineering: P.G., A.M.V., A.H.Ã. Network reconstruction algorithms: G.A.J., P.G. Data Analysis: O.F., M.F.G., M.H.R., G.A.J. Project design: M.H.R., G.A.J. Figure preparation: O.F., G.O.Ã., M.H.R., G.A.J. Manuscript writing: O.F., M.F.G., G.O.Ã., M.H.R., G.A.J. All authors reviewed the manuscript.
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Fontanelli, O., Guzmán, P., MenesesViveros, A. et al. Intermunicipal travel networks of Mexico during the COVID19 pandemic. Sci Rep 13, 8566 (2023). https://doi.org/10.1038/s41598023355425
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DOI: https://doi.org/10.1038/s41598023355425
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