Fig 1.
a, one user’s trajectories in two consecutive days. We compute the median of the longitudinal records as the prime location and geotag to the corresponding county. b, we iterate such process for all users and construct the bipartite graphs based on collective mobility. c, by comparison of two consecutive days’ bipartite graphs, we aggregate the inter-county travel information to a weighted and directed graph. The weights are the numbers of inter-county travelers. d, the undirected graph with weight as the sum between each pair of counties.
Fig 2.
Inter-county mobility patterns reflect social distancing and travel restrictions.
a, weekly average flux from Feb. 10, to Feb. 17, 2020. b, weekly average directed graph from Mar. 30, to Apr. 6, 2020. The mobility connections weakened after the National Emergency Declaration that took place on Mar. 13, 2020. Also, both long-distance and short-distance travel (i.e., links) have reduced. c, the distributions of edge weights for the two weeks. Both distributions follow the truncated power-law distributions with α = 1.93 for the one before and 1.75 for after, indicating that people’s movements have been disrupted significantly even though the fundamental patterns remained the same. d, the spatial distances of the unweighted links of the two weeks. The proportion of short-distance links increased from 0.4% to 2.1% while the middle range link’s proportion rose from 74.4% to 88.0%. In contrast, the proportion of edges crossing long distances dropped significantly from 25.3% to 9.9%. e, spatial distances of the weighted links of the two weeks. The proportion of short-distance travel increases significantly to 37.5% after the social distancing rules. The fractions of mid and long-distance travel experienced about a 9.9% and 7.3% decrease, respectively. f-h, temporal changes of the inter-county mobility network. They are the daily percentage change of the total number of devices in each node (i.e., county) (f), node in-degree (g), and total inter-county mobility flux (h). Red lines are the day of the national emergency declaration. While the data only lost a small number of devices, the two measures both plummeted more than 50% within two weeks before recovering.
Fig 3.
Percolation of mobility networks.
a shows the sizes of the largest two components with the change of q in the week of Feb. 10 to Feb. 17, 2020. There are two critical thresholds: qc at which SSCC experienced the largest size increase and qc2 at which the size of SSCC reached the second largest peak. The networks at two thresholds are shown in b and c respectively. We show the top three sub-components, size-wise, which are in blue (largest), purple (second-largest) and grey (fourth-largest) at both thresholds if they exist. d, e and f are the percolation directed networks in the week of Mar. 30 to Apr. 6, 2020. g-i demonstrate similar percolation patterns for the same week of data before National Emergency using undirected graphs where g highlights the similar percolation phenomenon; The undirected graph results after the mobility perturbation are shown in j-l and we can observe that across all scenarios at qc2, a large sub-component on the west coast states detached from the network while at qc at least three major sub-components are separated. These large clusters are similar despite the decrease of the value of qc after Mid March.
Fig 4.
The changes of the percolation metrics using a 7-day moving window to smooth out the weekend/weekday fluctuations.
a and b, the changes of qc with time and the median edge weights of the largest component which reflects the connection strength. c, the changes in the sizes of the largest strongly connected component for undirected graphs. d,e and f show the directed graph’s scenario for qc, median edge weight and largest component node size. Grey lines indicate the time series of the undirected graphs while blue ones are for directed graphs. The red vertical line highlights the time of the national emergency declaration (March 1, 2020) while the blue and red horizontal lines indicate the mean value of each feature before and after the declaration.
Fig 5.
Recurrent critical links detected at different stages.
a, b, c, d, components and recurrent critical links before the national emergency declaration (i.e., Stage 0), after the declaration from Mar. 13 to Mar. 27, 2020 (i.e., Stage 1), from Mar. 28 to Apr. 10, 2020 (i.e., Stage 2), and from Apr. 11 to Apr. 25, 2020 (i.e., Stage 3), respectively. The recurrent critical bridges in various periods are highlighted in red. These links are the edges of which the weights are just below the threshold of qc or qc2. The removal of the edge between two nodes will disintegrate the functional components. The critical links were located near the borderlines between various sub-components. The heat map shows the average daily new infection case per county during the period on logarithmic scales.
Fig 6.
Results from randomization and their comparison to the original mobility data.
a A sub-network of the original mobility network with their weights illustrated. The sub-network’s percolation process is shown in e and its largest components at qc are shown in i. We applied three types of randomization to the original networks: randomly assigning weights with distribution unchanged (b), randomly shuffling the weights while each node’s in-degree remain unchanged (c), and randomly shuffle the weights while each node’s out-degree remain unchanged (d). Examples of the percolation processes of the three types are shown in f, g, and h respectively, and their largest components at qc are shown in j, k, and l.
Fig 7.
Temporal changes of randomized percolation metrics.
a, the distribution of daily qc from 1,000 iterations of randomization that preserve network’s weights. b, the distribution of daily qc from 1,000 iterations of randomization that preserve the indegrees of the network. The black dots represent the daily median value and the shaded areas indicate the interquartile range (IQR).