SupCovidMobility

Effects of mobility restrictions during COVID19 in Italy

Alex Smolyak1,∗, Giovanni Bonaccorsi2, Andrea Flori2, Fabio Pammolli2,3 Shlomo
Havlin1

1 Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel;
2 Impact, Department of Management, Economics and Industrial Engineering,
Politecnico di Milano
3 SIT, Schaffhausen Institute of Technology, Schaffhausen
∗ Correspondence to<EMAIL_ADDRESS>

###### Abstract

To reduce the spread and the effect of the COVID-19 global pandemic, non-
pharmaceutical interventions (NPIs) have been adopted on multiple occasions by
governments. In particular lockdown policies, i.e., generalized mobility
restrictions, have been employed to fight the first wave of the pandemic. We
analyze data reflecting mobility levels over time in Italy before, during and
after the national lockdown, in order to assess some direct and indirect
effects. By applying methodologies based on percolation and network science
approaches, we find that the typical network characteristics, while very
revealing, do not tell the whole story. In particular, the Italian mobility
network during lockdown has been damaged much more than node- and edge-level
metrics indicate. Additionally, many of the main Provinces of Italy are
affected by the lockdown in a surprisingly similar fashion, despite their
geographical and economic dissimilarity. Based on our findings we offer an
approach to estimate unavailable high-resolution economic dimensions, such as
real time Province-level GDP, based on easily measurable mobility information.

Among non-pharmaceutical interventions (NPIs), limitations to mobility of
various degree have proven to be a successful strategy towards mitigating the
spread of COVID-19 in populations across the world [1, 2, 3]. In particular,
starting with the first epidemic cluster in Wuhan, China, lockdown
restrictions, i.e. full limitations of mobility on an entire territory, have
been widely and effectively adopted [4, 5].

Lockdowns are particularly effective since they incorporate a wide range of
more targeted interventions, among which two are worth mentioning here - the
limitation on large gatherings, and limitation on long range travel [3]. After
the initial widespread adoption of lockdown restrictions, several studies [2,
6, 7, 8] have focused on disentangling the specific effect of each NPI,
fostering the debate on the appropriateness and effectiveness of lockdowns as
policy interventions.

In fact, in addition to the direct effect of aiding in the containment of
COVID-19, one major side effect of the limitations was a severe economic
impact induced by the lockdown that has been felt globally. While online
commerce bloomed [9], traditional brick-and-mortar businesses took a severe
hit [10]. As a consequence, governments have been criticized both for
implementing lockdown too fast [11, 12] and too slow [13], with restrictions
too harsh and too lax.

Early studies on the economic effect of lockdown restrictions have highlighted
their impact on consumption [14, 15, 16], on the supply chain [17], on
inequality [18, 19] and on the overall economy [20]. The totality of the
effect of lockdown restrictions on the economy, however, is still difficult to
ascertain for several reasons. For one, the extent of the economic impact on
various layers of the population is unknown at this point as the crisis is
still ongoing. For another, different countries implement various economic
stimuli to offset and ease the economic downturn, both on the personal and
business levels. Lastly, with systems being as interconnected as they are,
efficient trade-off evaluation of all possible effects (businesses closing,
impact on education of all levels from primary school through higher education
and research, strain on healthcare systems and the higher-order impact due to
lack of doctors or equipment) is almost impossible in the short term.
Network Science [21, 22, 23] had frequently been employed over the past two
decades to investigate the behaviour of complex systems in multiple domains,
ranging from ecosystems [24, 25, 26] and biology [27, 28], through engineering
[29], urban traffic [30] and computers [31] to economics and finance [32, 33,
34, 35]. Network tools have also proven useful in analyzing the spreading of
the current pandemic in different contexts: from the direct estimation of the
epidemiological evolution of the pandemic [36, 37], to the development of new
modeling approaches [38, 39], to the analysis of mobility evolution [1, 40]
and finally in relation to economic conditions and backlashes [19, 18, 41].

Following this approach, in this work we develop a framework based on
methodologies from network science and percolation theory [22, 42] to explore
the effects of the COVID-19 pandemic on the network structure of human
mobility in Italy before and during the epidemic of 2020, from its very
beginning to the second wave. We employ rich mobility data of Italy provided
through Facebook’s Data for Good program (available at
https://dataforgood.fb.com/tools/disease-prevention-maps/) capturing movements
of more than 3 millions of Italian individuals at a sub-municipal level and on
a daily basis.

We use mobility data to construct the network of movements of individuals
across territories, measuring how lockdown has affected the number and the
intensity of connections among different areas of the country. In so doing, we
aim to investigate the evolution of the Italian mobility network, measure the
extent, beyond the immediately apparent one, of the mobility reduction after
lockdown and draw conclusions regarding the existence and significance of an
impact on the economy of lockdown policies.

Our hypothesis is that lockdown restrictions have increased the fragility of
the Italian mobility network. What we want to further investigate is if this
process has affected not only the network in general, which is plausible, but,
more importantly, also the core of the network (i.e. the Giant Connected
Component (GCC)). Furthermore we want to assess how long the disruptive
effects of lockdown on the Italian mobility network have lasted even after
restrictions were lifted. With respect to these objectives, the percolation
methodology represents the ideal tool because it allows us to focus on the
resilience profile of the network distinguishing between low impact and high
impact effects.

Finally, we investigate the economic consequences of mobility restrictions.
Using official statistics about the geographical distribution of income per
capita, we are able to observe a significant correlation between mobility and
levels of income, suggesting that one can be used as a proxy of the other.
Moreover, the percolation analysis allows us to investigate if the most
resilient part of the network, after lockdown restrictions, is also the
richest one. By observing that the average income per capita in the GCC has
increased after lockdown, we are able to uncover that restrictions have
heightened the economic segregation of the country, leaving poorer territories
disconnected from the mobility network.

While this work is not designed to make causal claims regarding the actual
economic effect of lockdown restrictions, which, as we said, need to be
disentangled from several concurrent phenomena, we believe that our analysis
represents a useful tool to assess the potential economic impact of disruptive
events such as the recent COVID-19 pandemic.

##  1 Network structure before and under lockdown

A network consists of elements, or nodes, connected to each other via links.
In our case, the fundamental units (nodes) are rectangular tiles of areas in
Italy (see Materials and Methods). We set an edge between two tiles if a
sufficient number of people travel from one tile to another in a given period
of time. The smallest window of time in our data is 8 hours, however we
aggregate them at the daily, weekly and monthly levels when necessary. Because
tiles typically do not represent any particular entities in the real world, we
apply a grouping procedure to them, aggregating multiple tiles to match them
with their corresponding administrative units. In the case of Italy three such
groupings exist: Municipality-, Province- and Region-scale, where the former
is a small geographic area (of the order of a township), the second is a
larger area aggregating several Municipalities and the latter is a relatively
large one. Thus, Italy consists of about 8000 Municipalities, over 100
Provinces and 20 Regions. Since the number of tiles in our data set is of the
order of the number of Municipalities, we do not analyze that administrative
division and the most granular analysis is conducted directly on the tiles.
The left panels of Fig. 1 show the connectivity of Italy’s Regions (the
coarsest level of aggregation) before (Fig. 1a) and during (Fig. 1b) the
lockdown, while the right panel shows tile-level network characteristics.
Throughout the analysis we identify four distinct periods, corresponding to
different behavior of the metrics under analysis, as will be detailed below.
Those are:

  1. 1.

Pre-lockdown, baseline period, starting end of February, 2020 until March 8th.

  2. 2.

Full lockdown, starting March 9th and lasting through May 18th

  3. 3.

Partial restriction, whereby some limitations remained in place but others
were lifted, May 19th till June 3rd

  4. 4.

Removal of all limitations, June 4th until Mid July.

One fundamental characterization of a network is its degree distribution,
P(k), where k, the degree of a node, is the number of other nodes it is
connected with. The degree distribution shows the probability P to have a node
of degree k. For random graphs, known also as Erdos-Renyi (ER) graphs, in
which edges between nodes exist with some fixed probability p, this
distribution follows the Poisson one, $P(K)=\frac{(np)^{k}e^{-np}}{k!}$ where
n is the number of nodes in the network which is assumed large [43]. In that
case, the product np is the average degree of the nodes in the network.
Importantly, the distribution of degree in ER graphs is relatively narrow
around the mean and most nodes have a degree close to the mean, i.e., large
deviations are exceedingly rare. Many real-world networks exhibit instead much
broader degree distributions, often characterized by a power law, $P(k)\propto
k^{-\beta}$, with or without exponential truncation to accommodate cutoffs,
finite size effects and other features which separate them from random graphs
[44]. In such a distribution, the average degree $\langle k\rangle$ is not
representative of the entire network degree distribution, due to the presence
of relatively few hubs that are connected to many more nodes than the typical
node.

Edges connecting pairs of nodes need not be identical and may carry
information with them. This information is typically regarded as an edge
“weight” and may be useful in determining shortest paths, estimating travel
time and assess the fragility of the network. In our case, edges carry two
types of information. One is geographical, namely the geometric distance
between two connected nodes, and the other is demographic, specifically, the
number of people traveling along the edge. When edges weights are available,
the simple degree may be generalized and augmented by a weighted degree, where
the sum of edges weights incident on a node is taken, rather than the number
of edges. Below, when discussing weighted degrees we refer to aggregating all
people traveling to and from a certain node. As discussed in Materials and
Methods, our edges are undirected.

We explore the global connectivity (i.e., the possibility to get from one node
to another via existing edges) through the functional and the pure network
perspectives (Fig. 1). From the functional perspective, we note the
connectivity of the largest-level aggregation of our data, namely, Italian
Regions. This view allows us to appreciate the significant drop in the
connectivity between large parts of the country. The simple unweighted degree
of the Regions (as seen in the colored maps in panels LABEL:sub@1a and
LABEL:sub@1b of Fig. 1) can be viewed as a qualitative effect of the mobility
restrictions. During lockdown connectivity dropped across the country such
that interactions between different Regions (the coarsest aggregation level)
almost disappeared. Advancing to more quantitative characteristics of the
mobility network, we note the drop in the average daily weighted degree, shown
by the color of the links, which measures the daily number of people moving
between locations.

Another important feature of a network is the extent to which nodes are
connected globally to each other, which can be investigated by measuring the
size of the network components. A component of a network is a set of nodes
reachable from one another. Among components it is particularly informative to
analyze the largest one, typically called the Giant Connected Component (GCC)
[23]. This is because in complex networks above a certain degree of
connectivity (an average degree of 1 in the case of ER networks, known as the
percolation threshold) most nodes of the network are reachable from any other
node, that is, most of the network belongs to the GCC. When the network
becomes fragmented (in our case via the removal of edges due to mobility
restrictions), less nodes tend to belong to the GCC and the number of smaller
components increases, as may be observed in Fig. 1c and 1d. Interestingly,
comparing the behavior of the mean weighted degree with that of the GCC of the
network, we find a striking similarity, which can also be seen in the
component sizes (SI) and will surface in other aspects of our analysis
(resilience). There is no particular mechanism driving that - and yet the drop
in average weighted degree traces very closely that of the component size
(Fig. 1c). As the GCC decreases, the network becomes more fragmented, leading
to an increase in number of components. We report values for the numbers of
non-isolated components and components larger than five nodes (Fig. 1d), as
well as their sizes (Fig. 1e). Finally, we measure the reachability within the
changing GCC. Interestingly, as the GCC shrinks during lockdown, the different
nodes effectively become more distant as the average path length and diameter
of the component increase (Fig. 1f).

LABEL:sub@1a

(a)

LABEL:sub@1b

(b)

LABEL:sub@1c

(c)

LABEL:sub@1d

(d)

LABEL:sub@1e

(e)

LABEL:sub@1f

(f)

Figure 1: Network characteristics of traffic before and during lockdown for
the coarsest (left) and finest (right) resolutions. (a) Region-level mobility
before lockdown. Region color shows the daily average simple degree for
business days before lockdown measure were implemented. The edge colors show
the log of the weighted link, i.e. the daily number of people traversing each
edge. (b) Mobility during lockdown, color scales matching (a). Many of the
existing edges have failed, and those that remain are much weaker. (c) Average
weighted degree (teal) and size of the Giant Connected Component (GCC, red)
averaged over a week, error bars showing the standard deviation for the
period. (d) Number of non-isolated components (teal) and number of components
with more than 5 nodes in them (red). (e) Average size of: non-isolated
components (teal), and components larger than 5 nodes (red). (f) Average
shortest path length (teal) and the diameter of the GCC (red).

##  2 Scaling

(a)

(b)

(c)

(d)

Figure 2: Comparison between pre-lockdown and lockdown of (a) weighted
degrees, (b) Unweighted degree (for the degree distribution we also show a
stretched exponential fit in the SI), (c) edge weights and (d) distances. Teal
and dark red lines show the Complimentary Cumulative Distribution function
(CCDF) of the weighted degrees, weights and distances before and during the
lockdown, while the dashed lines show the stretched exponential (top) and
power law fits (bottom) to the data.

Next, we examine how some of our tile-level network’s characteristics behave
before vs during lockdown. We analyze the PDFs of some network characteristics
and find that they can be approximated by power-laws [45] (see Sec.
LABEL:sec:theofit in the SI for additional possible fits). The probability
distribution of the data, together with the fitted power-law functions is
shown in Fig. 2. Starting with the degree distributions, we note the larger
exponents during lockdown (Fig. 2a and 2b). Both the weighted and simple
degree distribution contract which means less people (weighted degree) travel
to fewer places. That is backed up by examining the distributions of the edge
weights (Fig. 2c) and distances traveled (Fig. 2d).

Weight distribution tells probably the most important story of the positive
and negative aspects of the lockdown. The reduction of people traveling along
edges assists in reducing contagion [1, 46, 47, 48] but also induces a
significant economic impact [10, 41, 18, 20]. We note here the entire
distribution shrinks, both the typical and extreme values. The relative
ubiquity of the impact is elaborated in Sec. 3. The extent to which reduction
in mobility affects the economy as a whole is discussed in Sec. 5.

To conclude our exploration of scaling behavior we revisit an often-measured
characteristic of human mobility, namely, laws governing distances traveled
[49, 50, 51, 52]. Two points are worth noting here. The exponents observed
here are noticeably larger than the findings on human traveling in Ref. [50].
Several sources of the difference may be identified. One is the specific
geometry of Italy, characterized mostly by relatively short distances. The
typical width of the country is about 300km, while the distance between Naples
and Milan, two economic centers relatively far from one another, is close to
900km. Due to that, we do not expect to find enough distances much greater
than 300km. Indeed, we can see in Fig. 2d that before the lockdown the slope
changes and become larger at about 200km. Another important factor comes from
our data limitations relating to our sampling period being 8 hours and the
lower bound on number of at least 10 people to establish an edge between two
tiles (see Materials and Methods for details). Taking those limitations into
account, however, the scale-free behavior of human mobility is approximately
reproduced in our analysis. More importantly, we find that under the lockdown
restrictions, the scaling laws for weights and distances still hold, while the
exponents increase further, shrinking the covered distances. For the
distances, as opposed to weights, where the entire distribution shrank, the
range of the distribution does not change much, with the farthest points
observed before the lockdown being reachable during lockdown as well.
Moreover, the most prominent change takes place in the medium distances,
between roughly twenty and two hundred kilometers, the distances that cover
(with some excess) what would typically be the daily commute. Workers related
to essential activities, however, are probably still present and travel to
distances comparable to those before the lockdown [41, 53].

##  3 Local Impact and Recovery

(a)

(b)

(c)

(d)

Figure 3: Clockwise from top-left: Baseline map of Italian Provinces;
Reduction in mobility during Lockdown; Partial lift of limitations; All
limitations lifted. The color bar indicates the reduction in weighted degree
for each Province, with darker blue meaning stronger reduction. The lower
panels show the gradual “return to normal”, with the majority of Provinces
behaving similarly, but a few noticeable outliers as described in the main
text and Fig. 4

Previous section analyzes mobility between individual tiles, whereas here, in
order to take a step toward assessing potential economic implications, we
aggregate mobility across Provinces - a more granular perspective than the
Regions in Fig. 1 but containing economic meaning as compared to the
individual tiles. The lockdown (top-right) panel of Fig. 3 shows the full
impact of the restrictions. For non-zero distances (i.e., excluding people who
stay within the same tile for the measurement period, as explained in
Materials and Methods), mobility decreased dramatically, with Provinces
dropping between 50% and 90%. The lower two panels, perhaps, hold the more
interesting and unexpected effects of the lockdown. While some Provinces (as
will be detailed in Fig. 4) rebound relatively quickly, even when only some of
the limitations are lifted, other Provinces remain far below their initial
levels throughout the time frame shown.

Indeed, Fig. 4a highlights the relative similarity of the impact suffered by
the most mobile Provinces, those with the highest weighted degree. Those
Provinces show a consistent, close to 60%, reduction of mobility during
lockdown and an equally consistent recovery when restrictions are gradually
lifted. This is contrasted by the higher variation seen in the lower panels of
Fig 3 and detailed by Fig. 4 for Provinces with lower initial mobility
(weighted degree). A noticeable difference exists in the level of recovery,
with some of the most impacted Provinces, such as the southern Province of
Cosenza and the northern Aosta Valley, remaining well below initial levels
after restrictions are lifted, while others, such as the coastal Provinces of
Genoa and Olbia-Tempio recovering close to, and above, their initial levels.

(a)

(b)

Figure 4: (Top) Most mobile - Provinces with the highest weighted degree, and
(Bottom) most impacted Provinces and their recovery through partial
restrictions to complete lift of limitations.

To generalize those statements we compare mobility levels at the three stages,
lockdown, partial restrictions, and lifting of restrictions to the initial
levels (Fig. 5). The level to which the relations are linear over the various
phases of mobility limitations is very surprising. This evidence will be
exploited in Sec. 5 to assess the economic impact of mobility restrictions.
Moreover, we note that overall mobility had not returned to pre-lockdown
levels in the post-lockdown phase, but roughly to 87% of those levels, which
may thus suggest the presence of long-lasting negative effects in the
structure of the Italian mobility network persisting the removal of mobility
restrictions.

(a)

(b)

(c)

(d)

Figure 5: Uniformity in damage. (a) A linear relation between the lockdown
levels of mobility and the initial ones shows a 60% drop, as seen via the
regression coefficient, consistent with 4a. (b), (c) show the relations remain
very closely linear and uniform in the following stages of partial (b) and
full (c) lift of limitations. We note that following the removal of all
limitations, overall mobility is still about 15% below its initial levels.
Zooming into the smaller Provinces we see the variation mentioned in Fig. 4b
while still largely following the overall trend (d) traces the slope of impact
vs. the averages pre-lockdown levels over time. We see again the shape
encountered in Figs. 1c and 1f. Insert - the distributions of impact
magnitude, i.e. weighted degree drop per time frame.

##  4 A Resilience Perspective

Figs. 1c and 3 may lead us to assume things were, to some extent, almost back
to normal when restrictions were lifted. We wish, however, to employ a
methodology taken from traffic jam and service quality domain [42] to show
this is not, in fact, the case. For that we return briefly to the tile
perspective taken in Sec. 2. For every edge between two tiles we calculate the
distribution of its weight across time. We then determine a maximal value for
that weight, $W_{max}$, as each edge’s 95th percentile. Now we introduce a
parameter $q$, the fraction of that maximal weight $q=W_{q}/W_{max}$ and we
state that for a given level of $q$, an edge whose current weight is above
$W_{q}$ is considered functional and below it - the edge does not exist. This
may be motivated from several perspectives. The obvious approach is to say
that if an edge exists but it is only a small fraction of what that edge had
been previously - we cannot state that connectivity between the two nodes the
edge joins had really been restored. An argument closer to our economic
analysis in the previous and following sections may say that there exists a
non-linear relation between the number of people traveling along an edge and
the value the edge carries. That relation may come from a skewed distribution
of people’s income, the non-uniformity of economic activity of the travelers
or other similar notions. Because of that relation we want to see not only
which edges remain, but which edges carry sufficient weight to carry economic
value.

Armed with this approach we now modify the parameter $q$ and analyze the
changes in network connectivity. We start by asking whether or not the edges
that fail under our $q$-perturbation differ during changing conditions. The
result is shown in Fig. 6a. We find that initially (before the lockdown) the
network is insensitive to the removal of low-value edges. That is, only at
high levels of $q$ do significant edges fail. During lockdown, however, the
picture changes. Now we see that many edges are at 20%-60% of their maximal
levels and the weight they carry is very substantial. Thus the network is much
more brittle than it was initially and edges carry only a small fraction of
their maximum historic capacity. We can ask, then, how is this under-the-
surface fragmentation reflect on the GCC, which is often considered the main
functioning component of the network. The standard presentation [42] relates
the level at which the GCC fragments, termed $q_{c}$, to network strength and
resilience, showing that high $q_{c}$ corresponds to a robust network, while
low levels of $q_{c}$ characterize a weak, fragile network. Fig. 6b shows at
which value of $q=q_{c}$ the GCC breaks down, i.e. split into noticeably
smaller components. We note, again, that before the lockdown it happens only
close to $q=1$, that is, when the edges are removed only when they are
marginally below their maximal value. What this means, simply, is that before
the lockdown, the edge weights are relatively stable and close to their
typical peaks. During lockdown the network becomes much more fragile and
breaks down at much lower levels, at $q_{c}$ close to 0.2. That means the
edges that are present - are nearly five times weaker than in normal times. On
one hand, that is to be expected. We claimed throughout the analysis mobility
had been highly impacted. On the other hand, we saw in Fig. 5a that node
weighs dropped by about 60% on average - but the surprising detail here is
that the global mobility is actually more fragile, there are many critical
edges that break under very light pressure. But perhaps most interesting is
the situation post lockdown, when everything, presumably, goes back to normal.
Looking at nominal edges, and even to some extent node degrees and
distributions, it may be assumed that this is indeed the case. However, under
the magnifying glass of $q$-analysis we see that picture is not so clear.
Below we explore the extent to which the network remains damaged.

In order to further understand the lasting impact of the lockdown, we compare
the nodes that form the GCC at a given $q$-level over time. To this end we
calculate what is known as the Jaccard coefficient, which is a straight-
forward metric to measure node similarity. It is defined as the size of
intersection of two sets divided by its union:

$J(q)=\frac{GCC_{1}(q)\cap GCC_{2}(q)}{GCC_{1}(q)\cup GCC_{2}(q)}$ (1)

We perform this calculation for a range of $q$ values and discover (Fig. 6c)
that the GCC, as constructed at various levels of $q$, 40% and up, retains
only about 50%-70% of the initial nodes. Moreover, during the lockdown itself,
when the simple intersection shows a decrease of 60%-70% as seen by other
metrics in Figs. 1\- 5, even for very low levels of $q$, the GCC is almost
entirely gone, bearing only about 10% similarity to the initial structure. We
now want to use this understanding to tie together the levels of mobility and
observed decline of network functionality to the actual levels of economic
activity.

(a)

(b)

(c)

Figure 6: Network under Q-Resilience. (a) The distinct characteristics of
failing edges during lockdown vs the other periods. In particular, during the
lockdown the network is particularly brittle, and moreover, the edges lost at
relatively low Q’s carry comparatively high weights, reflecting the stronger
impact, (b) Fracturing of the GCC at various q-levels over time. As can be
seen, during, but more importantly, after the lockdown the network remains
susceptible to failure for only very slight perturbations - the links that are
present are roughly 20% of the original weights. The GCC is in effect very
weakly connected. (c) Jaccard similarity for different levels of Q. Here the
fragility of the network is even more prominent - if we ask when the edges
regain their original strength - the answer is they do not, and many of them
remain at around 60% of their initial levels

##  5 Economic Impact

Reliable and accurate data on economic activity from official sources takes a
long time to measure and evaluate. There are many reasons for that, some
include multiple sources of information that need to be gathered and unified.
Others just due to the fact that economic and financial information is
reported long after the occurrence of the phenomenon. For instance, a company
would file its fiscal year reports only towards the end of the following
year’s first quarter, and similarly national accounts are typically released
with some months of delay.

However, mobility patterns can be readily observed in near-real-time via
various methods. One such method is employed in this manuscript, relying on
cellphone data. Other providers of such data exist and often share or sell
this data so it can be observed in very high spatio-temporal resolution. Other
measures include public transportation usage, sensors that count numbers of
vehicles on the roads and more. In short, mobility is relatively easy to
observe.

Here we propose the use of mobility data as a proxy for the economic
performance. As an example, Fig. 7a shows the correlation between Italian
Provinces’ Gross Domestic Product (GDP) and their mobility-based weighted
degree. The official economic data used in this figure is only available until
2017. Newer data is available only for the regional level, for 2018, while if
we are interested in quarterly data of 2019 and onward, we can only look at
the country level. This creates a difficult situation for policy makers and
stakeholders when economic impact or distribution of financial aid need to be
determined at a granular spatial level of intervention.

Thus, we come to the concluding section of our analysis, bringing together the
various network insights gathered so far into an effective fine-grained
decision-aiding tool (see Materials and Methods for further details). This
tool is made available due to two interesting facts. One, as seen in Fig. 7a,
there is a close relation between a Province’s mobility performance and its
GDP. In fact, the levels of mobility correlate with the GDP more than 60%:
$corr(GDP,\langle K_{w}\rangle)=0.64$ (we present additional metrics tying
together GDP per capita to average degree, as well as “normalized” average
degree, in the SI). This means that mobility is a reasonable proxy for the
economy. We next compare OECD data which calculates economic activity based on
multiple attainable sources on the national level [54] to produce an estimated
GDP (EGDP) with our mobility metrics. While this is an approximate metric, it
analyzes various Internet search patterns related to economic activity and
shows a close prediction (to within its margin of error) to the officially
published quarterly national GDP [55]. Fig. 7b shows how closely the average
mobility weighted degree (teal, left axis) follows the proposed economic
metric (red, right axis). The correlation here is even higher,
$\rho(EGDP,\langle K_{w,p}\rangle)=0.89$, and the weekly updates $\Delta
EGDP=EGDP(i)-EGDP(i-1),\Delta\langle K_{w}\rangle=\langle
K_{w}(i)\rangle-\langle K_{w}(i-i)\rangle$, are also very closely tied,
$\rho(\Delta EGDP,\Delta\langle K_{w}\rangle)=0.58$.

While the causal relations between affected mobility and economic impact may
be far from trivial (although the intuition that people, when traveling,
conduct various business activities, is straightforward), we now have a means
to offer both a national and a regional or local estimate of impact to the
economy. Specifically, the remarkably linear relations of Fig. 5 can help us
give a rough estimate of a uniform impact to Provinces’ GDPs. However, the
deviations from that linear relationship, as seen from the mobility patterns
of each individual Province, as shown above (Sec. 3), together with the close
relations between GDP and mobility as shown here, can allow for a finer-
grained, Province-specific analysis, and give in-depth expectations about
heterogeneous local economic impact from the observed dynamics of mobility.
Figs. 7c-7f show a sample of such forecasts. The process to obtain the
estimates is detailed in Materials and Methods.

Finally, in order to uncover heterogeneous responses of Italian territories to
mobility restrictions which may signal competing economic trajectories of
local economic systems, we show in Fig. 8 how the network of Italian mobility
relates to average income levels. We consider the network of mobility
constructed at the Municipality level and income per capita referred to 2019,
hence before the outbreak of the pandemic. The red line in Fig. 8 shows the
dynamics of GCC, while in green we report its number of nodes. The Italian
mobility network is composed by a main connected group of Municipalities which
are on average associated with higher levels of income per capita than those
more peripheral in the network. Interestingly, after the deployment of
lockdown restrictions we note a remarkable reduction in the size of GCC, while
the economic well-being of the remaining population, as seen through the
average income per capita, increases. Hence, a relevant effect of mobility
restrictions put in place to contain the spread of the virus is the emergence
of an enforced economic segregation, where Municipalities with high-income
population tend to be connected while Municipalities with poorer economic
conditions of their population are excluded from the main component of the
Italian mobility network. This effect vanishes once mobility restrictions are
lifted, with a rapid reversal to the pre-lockdown configuration. In addition,
the percolation process is informative to identify those Municipalities which
may exit GCC when the network starts deteriorating. As shown in Fig. 6,
lockdown restrictions determine a sharp decrease of the parameter $q$. As a
consequence, Fig. 8 highlights how during the lockdown on average the
configuration reached before the dissipation of GCC is very similar to the one
at $q=0$. Nevertheless, during business as usual periods (pre-lockdown phase)
or after the removal of lockdown restrictions, we notice that progressively
nodes that are removed have lower average income per capita than those that
remain in the GCC. Once again, the core of the GCC seems to point to the
presence of highly connected nodes that have on average also better economic
conditions of their population.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: Inferring the economy from mobility. (a) The correlation between the
GDP levels of individual Provinces for 2017 and the weighted Province degree.
Log of values is shown to highlight the relation holds regardless the
strongest Province economies. Inset - shifted correlation to validate the
significance of the relationship (b) The Average weighted degree, teal, left
axis; and the EGDP estimates, red, right axis. Here, as in the right panel of
Fig. 1 we see the similarity in dynamics. (c)-(f) The Province-based GDP
forecast based on mobility data for Turin, Padua, Como and Pisa Provinces. The
thick line is the average forecast, with the shaded area showing the 25-75
percentile range. (a)  Figure 8: Average income per capita and total number of
nodes in the GCC for every week of the period of observation. The GCC is
either calculated at the start of the percolation process, i. e. without
removing any nodes (GCC at $q=0$, red and green lines), or it is obtained by
removing edges incrementally by weight until the optimal $q$ is reached but
before the GCC is dissolved (GCC at $q_{c}$, blue and yellow lines). The lines
represent the mean value of income per capita of the nodes or the total number
of nodes in the GGC, averaged over one week of observations.

##  6 Conclusion

We analyze in detail the changes in mobility patterns in Italy induced by the
lockdown measures. We show the fragmentation and disconnection they brought.
While it is without a doubt the intention of those measures, we show also,
using concepts borrowed from percolation theory, that the impact goes much
deeper than is seen superficially. In fact, the network after the removal of
limitations is still far more fragile than it was initially. We estimate the
overall impact and the deviations from that uniformity for various Regions.
Combining those deviations with close relations between mobility and economic
conditions, we present a novel approach to estimate hard-to-observe economic
activity from easily observable mobility patterns. These perspectives on the
network resilience and local economic impact of lockdown measures may be used
by policy makers to estimate more accurately and promptly the costs and
benefits of potential additional restrictive measures.

## Materials and Methods

### Data

The data used for the analysis is provided under academic license agreement
with Facebook through the “Data for Good” program and is available at
https://dataforgood.fb.com/tools/disease-prevention-maps/. Economic data is
taken from the Istituto Nazionale di Statistica (ISTAT) http://dati.istat.it/
and the Ministero dell’Economia e delle Finanze (MEF)
https://www.mef.gov.it/documenti-pubblicazioni/open-data/index.html.

### Network structure

As described in Sec. 1, we construct our network by observing people move
between various locations. For ease of processing, continuous space is divided
by Facebook into tiles, approximately 7x7km each, covering the entire country.
When calculating overall statistics, such as degree distributions, component
sizes and shortest paths we analyze the network using those elements. However,
in order to make our research more meaningful from a geographic and economic
perspective, we also map those tiles to Italy’s Regions and Provinces based on
the the location of a tile’s center. For that we aggregate all relevant node
information into a higher-level node (Province or Region) by summing all the
edge weights incident on the appropriate nodes.

Edges represent the number of people traveling within an eight-hour period
between two nodes. That is, if over a period of eight hours, $N>10$ people
were located at nodes u and v then and edge will exist between the two nodes
and its weight will be N. The reason for this limitation lies in privacy
restrictions, where a smaller number of people may make it possible to
identify specific individuals, so information is only exposed where such
deanonymization is unlikely.

Those two limitations carry noteworthy meaning to our analysis. For one, the
relatively short time period of eight hours is shorter than the typical time
frames in human mobility research. That leads directly to shorter typical
distances traveled. Secondly, the limitation on quantity can also directly
lead to the rarer but longer edges being omitted from our data set.

Self-loops reflect a situation where, throughout the measurement period, a
person is not recorded outside their initial tile. Those events are frequent
in our analysis, more so at the large aggregation levels. Those loops
effectively mean people stay in the initial location, whether a tile or a
Region. Since we are interested in connectivity between various locations,
those loops are removed from our analysis.

Information is only available about the presence of people in two locations,
without time ordering, so our edges are undirected. This precludes us from
performing Origin-Destination analysis, or to follow explicitly the direction
of flow of people

### Percolation and Resilience analysis

Percolation takes its origins in the analysis of disordered systems, where
matter is modeled as sites that may or may not be occupied with probability
$p_{n}$ (node percolation) or sites between which edges are present or absent
with probability $p_{e}$. The question is then asked whether or not a Giant
Connected Component exists, that is, whether the largest component is
extensive, i.e., proportional to the original system size. Originally
developed around regular systems, such as two-dimensional or three-dimensional
lattices, network theory expanded the notion to random networks, where each
“site” has potentially a random degree taken from a given degree distribution.
Specifically, important differences appear when the degrees are drawn from a
Poisson distribution (Erdos-Renyi, (ER) networks, where every two nodes are
connected by an edge with probability $p$) or a scale free (SF) degree
distribution, where node degrees are distributed according to a power law,
$P(x)\propto x^{-\beta}$ with usually $2<\beta<3$. In the former case the
network experiences a percolation transition when the average degreee,
$\langle k\rangle$ crosses the threshold of 1, where below that value the GCC
increases slowly with network size, $N$, and its relative fraction of the
entire network goes to zero as $N$ goes to infinity [21, 23]. SF networks, on
the other hand, do not experience such a transition, and rather have a non-
zero GCC for any nonzero average degree. It is often assumed for real-world
networks that in order for a node to be functional, whether the node is a
financial institution, a geographical location or an infrastructure component
such as a power plant [29], it needs to be part of the GCC. While we do not
suggest Regions, Provinces and Municipalities disconnected from the GCC cease
to function, we do suggest their participation in economic activity is
impaired. Since there can only be one GCC in a network [21, 23], we assume
only nodes connected to it operate at their fullest capacity. Thus, as
described in Sec. 4, we define a critical level of $q$ at which the reduction
of the GCC is largest, thereby reflecting the most significant impact to the
economic activity.

### Province-based GDP forecast

To generate GDP forecasts per Province we combine the close relations between
mobility and GDP, both statically (Fig. 7a) and dynamically (Fig. 7b). For
that we rewrite the GDP over time for a Region as

$GDP_{t}=GDP_{0}+\sum_{i=1}^{t}\Delta GDP_{i}$ (2)

where $GDP_{0}$ is the initial level (here taken as the value at 2017 as the
latest published data point for Provinces, but may be replaced with a more
recent estimate), and the weekly updates $GDP_{i}$ are estimated using the
dynamic relations. We may also rewrite the process as a multiplication of
percentage changes, more in keeping with economic literature, but no
additional insight is gained by it and the equations become more cumbersome.
We now regress the weekly GDP changes against the mobility updates as

$\Delta GDP_{i}=\alpha\cdot\Delta\langle K_{i}\rangle+\beta$ (3)

for the initial estimate. Then, based on the correlation between $GDP_{i}$ and
$\Delta\langle K_{i}\rangle$ we generate correlated random variables from the
latter and substitute 3 into 2. The last part is done so that statistics can
be generated around the average levels, that may be derived directly by
plugging the relevant Province’s $\langle K_{i}\rangle$ into 3. We now have
all the components to calculate both the expected $GDP_{t}$ and the range it
may take.

## Acknowledgements

We thank the Israel Science Foundation, the Binational Israel-China Science
Foundation (Grant No. 3132/19), the NSF-BSF (Grant No. 2019740), the EU H2020
project RISE (Project No. 821115), the EU H2020 DIT4TRAM, and DTRA (Grant No.
HDTRA-1-19-1- 0016) for financial support.

## Author contribution

All authors conceived and designed the research. A.S. and G.B. carried out the
numerical experiments and analysed them. A.S., G.B., A.F and S.H. interpreted
the simulation results. All authors wrote the manuscript.

## Competing interests

The authors declare no competing interests

## References

  * [1] Frank Schlosser et al.  “COVID-19 lockdown induces disease-mitigating structural changes in mobility networks”  In _Proceedings of the National Academy of Sciences_ 117.52 National Acad Sciences, 2020, pp. 32883–32890
  * [2] Nils Haug et al.  “Ranking the effectiveness of worldwide COVID-19 government interventions”  In _Nature human behaviour_ Nature Publishing Group, 2020, pp. 1–10
  * [3] Bnaya Gross et al.  “Spatio-temporal propagation of COVID-19 epidemics”  In _Europhysics Letters 131, 58003-58008_ , 2020
  * [4] Moritz U. G. Kraemer et al.  “The effect of human mobility and control measures on the COVID-19 epidemic in China”  In _Science_ American Association for the Advancement of Science, 2020  DOI: 10.1126/science.abb4218
  * [5] Matteo Chinazzi et al.  “The effect of travel restrictions on the spread of the 2019 novel coronavirus (COVID-19) outbreak”  In _Science_ American Association for the Advancement of Science, 2020  DOI: 10.1126/science.aba9757
  * [6] You Li et al.  “The temporal association of introducing and lifting non-pharmaceutical interventions with the time-varying reproduction number (R) of SARS-CoV-2: a modelling study across 131 countries”  In _The Lancet Infectious Diseases_ 21.2 Elsevier, 2021, pp. 193–202
  * [7] Jonas Dehning et al.  “Inferring Change Points in the Spread of COVID-19 Reveals the Effectiveness of Interventions”  In _Science_ 369.6500 American Association for the Advancement of Science, 2020  DOI: 10.1126/science.abb9789
  * [8] Jan M. Brauner et al.  “Inferring the Effectiveness of Government Interventions against COVID-19”  In _Science_ 371.6531, 2020, pp. eabd9338  DOI: 10.1126/science.abd9338
  * [9] OECD  “E-commerce in the time of COVID-19”, 2020  URL: http://www.oecd.org/coronavirus/policy-responses/e-commerce-in-the-time-of-covid-19-3a2b78e8/
  * [10] Alexander W. Bartik et al.  “The impact of COVID-19 on small business outcomes and expectations”  In _Proceedings of the National Academy of Sciences_ 117.30 National Academy of Sciences, 2020, pp. 17656–17666  DOI: 10.1073/pnas.2006991117
  * [11] Rohan Venkataramakrishnan  “Coronavirus: Did India rush into a full lockdown without planning? Or did it have little choice?”, 2020  URL: https://scroll.in/article/957101/coronavirus-did-india-rush-into-a-lockdown-without-planning-or-did-it-have-little-choice
  * [12] Raphael Ahren  “Complete lockdown would cause more harm than good, dissenting top doctor claims”, 2020  URL: https://www.timesofisrael.com/complete-lockdown-would-cause-more-harm-than-good-dissenting-top-doctor-claims/
  * [13] Helen Ward  “We scientists said lock down. But UK politicians refused to listen”, 2020  URL: https://www.theguardian.com/commentisfree/2020/apr/15/uk-government-coronavirus-science-who-advice
  * [14] Vasco M Carvalho et al.  “Tracking the COVID-19 Crisis with High-Resolution Transaction Data”, 2020, pp. 49
  * [15] Natalie Cox et al.  “Initial Impacts of the Pandemic on Consumer Behavior: Evidence from Linked Income, Spending, and Savings Data”, pp. 37
  * [16] Amory Martin, Maryia Markhvida, Stéphane Hallegatte and Brian Walsh  “Socio-economic impacts of COVID-19 on household consumption and poverty”  In _Economics of disasters and climate change_ 4.3 Springer, 2020, pp. 453–479
  * [17] Dabo Guan et al.  “Global Supply-Chain Effects of COVID-19 Control Measures”  In _Nature Human Behaviour_ 4.6, 2020, pp. 577–587  DOI: 10.1038/s41562-020-0896-8
  * [18] Serina Chang et al.  “Mobility Network Models of COVID-19 Explain Inequities and Inform Reopening”  In _Nature_ , 2020  DOI: 10.1038/s41586-020-2923-3
  * [19] Giovanni Bonaccorsi et al.  “Economic and social consequences of human mobility restrictions under COVID-19”  In _Proceedings of the National Academy of Sciences_ 117.27 National Acad Sciences, 2020, pp. 15530–15535
  * [20] Raj Chetty, John N Friedman, Nathaniel Hendren and Michael Stepner  “The Economic Impacts of COVID-19: Evidence from a New Public Database Built Using Private Sector Data”, 2020, pp. 116
  * [21] Mark Newman  “Networks”  Oxford university press, 2018
  * [22] Albert-László Barabási  “Network science”  Cambridge university press, 2016
  * [23] Reuven Cohen and Shlomo Havlin  “Complex networks: structure, robustness and function”  Cambridge university press, 2010
  * [24] Jordi Bascompte, Pedro Jordano, Carlos J Melián and Jens M Olesen  “The nested assembly of plant–animal mutualistic networks”  In _Proceedings of the National Academy of Sciences_ 100.16 National Acad Sciences, 2003, pp. 9383–9387
  * [25] Richard J Williams and Neo D Martinez  “Simple rules yield complex food webs”  In _Nature_ 404.6774 Nature Publishing Group, 2000, pp. 180–183
  * [26] Jianxi Gao, Baruch Barzel and Albert-László Barabási  “Universal resilience patterns in complex networks”  In _Nature_ 530.7590 Nature Publishing Group, 2016, pp. 307–312
  * [27] Hawoong Jeong et al.  “The large-scale organization of metabolic networks”  In _Nature_ 407.6804 Nature Publishing Group, 2000, pp. 651–654
  * [28] Albert-Laszlo Barabasi and Zoltan N Oltvai  “Network biology: understanding the cell’s functional organization”  In _Nature reviews genetics_ 5.2 Nature Publishing Group, 2004, pp. 101–113
  * [29] Sergey V Buldyrev et al.  “Catastrophic cascade of failures in interdependent networks”  In _Nature_ 464.7291 Nature Publishing Group, 2010, pp. 1025
  * [30] Daqing Li et al.  “Percolation transition in dynamical traffic network with evolving critical bottlenecks”  In _Proceedings of the National Academy of Sciences_ 112.3 National Acad Sciences, 2015, pp. 669–672
  * [31] Shuai Shao, Xuqing Huang, H Eugene Stanley and Shlomo Havlin  “Percolation of localized attack on complex networks”  In _New Journal of Physics_ 17.2 IOP Publishing, 2015, pp. 023049
  * [32] Prasanna Gai and Sujit Kapadia  “Contagion in financial networks”  In _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 466.2120 The Royal Society Publishing, 2010, pp. 2401–2423
  * [33] Stefano Battiston et al.  “Debtrank: Too central to fail? financial networks, the fed and systemic risk”  In _Scientific Reports_ 2 Nature Publishing Group, 2012, pp. 541
  * [34] SV Buldyrev, Fabio Pammolli, Massimo Riccaboni and Harry Eugene Stanley  “The rise and fall of business firms: A stochastic framework on innovation, creative destruction and growth”  Cambridge University Press, 2020
  * [35] Alex Smolyak et al.  “Mitigation of cascading failures in complex networks”  In _Scientific reports_ 10.1 Nature Publishing Group, 2020, pp. 1–12
  * [36] Alessandro Vespignani et al.  “Modelling COVID-19”  In _Nature Reviews Physics_ Nature Publishing Group, 2020, pp. 1–3
  * [37] Congying Liu et al.  “A new SAIR model on complex networks for analysing the 2019 novel coronavirus (COVID-19)”  In _Nonlinear Dynamics_ 101.3 Springer, 2020, pp. 1777–1787
  * [38] Alberto Aleta et al.  “Modelling the Impact of Testing, Contact Tracing and Household Quarantine on Second Waves of COVID-19”  In _Nature Human Behaviour_ 4.9 Nature Publishing Group, 2020, pp. 964–971  DOI: 10.1038/s41562-020-0931-9
  * [39] Per Block et al.  “Social network-based distancing strategies to flatten the COVID-19 curve in a post-lockdown world”  In _Nature Human Behaviour_ 4.6 Nature Publishing Group, 2020, pp. 588–596
  * [40] Giulia Pullano et al.  “Evaluating the effect of demographic factors, socioeconomic factors, and risk aversion on mobility during the COVID-19 epidemic in France under lockdown: a population-based study”  In _The Lancet Digital Health_ 0.0 Elsevier, 2020  DOI: 10.1016/S2589-7500(20)30243-0
  * [41] Alessandro Spelta et al.  “After the lockdown: simulating mobility, public health and economic recovery scenarios”  In _Scientific Reports_ 10.1 Nature Publishing Group, 2020, pp. 1–13
  * [42] Limiao Zhang et al.  “Scale-free resilience of real traffic jams”  In _Proceedings of the National Academy of Sciences_ 116.18 National Academy of Sciences, 2019, pp. 8673–8678  DOI: 10.1073/pnas.1814982116
  * [43] Béla Bollobás  “Modern graph theory”  Springer Science & Business Media, 2013
  * [44] Réka Albert and Albert-László Barabási  “Statistical mechanics of complex networks”  In _Reviews of modern physics_ 74.1 APS, 2002, pp. 47
  * [45] Jeff Alstott, Ed Bullmore and Dietmar Plenz  “powerlaw: a Python package for analysis of heavy-tailed distributions”  In _PloS one_ 9.1 Public Library of Science, 2014, pp. e85777
  * [46] Jayson S. Jia et al.  “Population flow drives spatio-temporal distribution of COVID-19 in China”  In _Nature_ 582.78127812 Nature Publishing Group, 2020, pp. 389–394  DOI: 10.1038/s41586-020-2284-y
  * [47] Stephen M. Kissler et al.  “Reductions in commuting mobility correlate with geographic differences in SARS-CoV-2 prevalence in New York City”  In _Nature Communications_ 11.1, 2020, pp. 4674  DOI: 10.1038/s41467-020-18271-5
  * [48] Pierre Nouvellet et al.  “Reduction in mobility and COVID-19 transmission”  In _Nature Communications_ 12.1, 2021, pp. 1090  DOI: 10.1038/s41467-021-21358-2
  * [49] Dirk Brockmann, Lars Hufnagel and Theo Geisel  “The scaling laws of human travel”  In _Nature_ 439.7075 Nature Publishing Group, 2006, pp. 462–465
  * [50] Marta C Gonzalez, Cesar A Hidalgo and Albert-Laszlo Barabasi  “Understanding individual human mobility patterns”  In _nature_ 453.7196 Nature publishing group, 2008, pp. 779–782
  * [51] Xiao Liang et al.  “The scaling of human mobility by taxis is exponential”  In _Physica A: Statistical Mechanics and its Applications_ 391.5 Elsevier, 2012, pp. 2135–2144
  * [52] Laura Alessandretti, Ulf Aslak and Sune Lehmann  “The scales of human mobility”  In _Nature_ 587.7834 Nature Publishing Group, 2020, pp. 402–407
  * [53] Edoardo Porto, Paolo Naticchioni and Vincenzo Scrutinio  “Partial Lockdown and the Spread of Covid-19: Lessons from the Italian Case”, pp. 27
  * [54] Nicolas Woloszko  “Tracking activity in real time with Google Trends”, 2020  DOI: https://doi.org/https://doi.org/10.1787/6b9c7518-en
  * [55] OECD  “Tracking GDP growth in real time”, 2020  URL: http://www.oecd.org/economy/weekly-tracker-of-gdp-growth/


# Calibration of the Timing Performance of GECAM-C

Shuo Xiao  School of Physics and Electronic Science, Guizhou Normal
University, Guiyang 550001, People’s Republic of China;  Guizhou Provincial
Key Laboratory of Radio Astronomy and Data Processing, Guizhou Normal
University, Guiyang 550001, People’s Republic of China  Ya-Qing Liu  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China  Ke Gong  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  Zheng-Hua An  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China  Shao-Lin Xiong  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China  Xin-Qiao Li  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  Xiang-Yang Wen  Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China  Wen-
Xi Peng  Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  Da-Li Zhang  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China  You-Li Tuo  Institut für
Astronomie und Astrophysik, University of Tübingen, Sand 1, 72076 Tübingen,
Germany  Shi-Jie Zheng  Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China  Li-
Ming Song  Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  Ping Wang  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China  Xiao-Yun Zhao  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  Yue Huang  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China  Xiang Ma  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  Xiao-Jing Liu  Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China  Rui
Qiao  Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  Yan-Bing Xu  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China  Sheng Yang  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  Fan Zhang  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China  Yue Wang  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Yan-Qiu Zhang  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Wang-Chen Xue  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Jia-Cong Liu  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Chao Zheng  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Chen-Wei Wang  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Wen-Jun Tan  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China  Ce Cai  College of Physics, Hebei Normal University, 20
South Erhuan Road, Shijiazhuang, 050024, China  Qi-Bin Yi  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  Key Laboratory of Stellar and Interstellar
Physics and Department of Physics, Xiangtan University, 411105 Xiangtan, Hunan
Province, China  Peng Zhang  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  College of Electronic and Information Engineering, Tongji University,
Shanghai 201804, China  Xi-Hong Luo  School of Physics and Electronic Science,
Guizhou Normal University, Guiyang 550001, People’s Republic of China;
Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing,
Guizhou Normal University, Guiyang 550001, People’s Republic of China  Jiao-
Jiao Yang  School of Physics and Electronic Science, Guizhou Normal
University, Guiyang 550001, People’s Republic of China;  Guizhou Provincial
Key Laboratory of Radio Astronomy and Data Processing, Guizhou Normal
University, Guiyang 550001, People’s Republic of China  Qi-Jun Zhi  School of
Physics and Electronic Science, Guizhou Normal University, Guiyang 550001,
People’s Republic of China;  Guizhou Provincial Key Laboratory of Radio
Astronomy and Data Processing, Guizhou Normal University, Guiyang 550001,
People’s Republic of China  Ai-Jun Dong  School of Physics and Electronic
Science, Guizhou Normal University, Guiyang 550001, People’s Republic of
China;  Guizhou Provincial Key Laboratory of Radio Astronomy and Data
Processing, Guizhou Normal University, Guiyang 550001, People’s Republic of
China  Shi-Jun Dang  School of Physics and Electronic Science, Guizhou Normal
University, Guiyang 550001, People’s Republic of China;  Guizhou Provincial
Key Laboratory of Radio Astronomy and Data Processing, Guizhou Normal
University, Guiyang 550001, People’s Republic of China  Lun-Hua Shang  School
of Physics and Electronic Science, Guizhou Normal University, Guiyang 550001,
People’s Republic of China;  Guizhou Provincial Key Laboratory of Radio
Astronomy and Data Processing, Guizhou Normal University, Guiyang 550001,
People’s Republic of China  Shuang-Nan Zhang  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China  University of Chinese Academy of Sciences, Chinese
Academy of Sciences, Beijing 100049, China

###### Abstract

As a new member of the Gravitational wave high-energy Electromagnetic
Counterpart All-sky Monitor (GECAM) after GECAM-A and GECAM-B, GECAM-C
(originally called HEBS), which was launched on board the SATech-01 satellite
on July 27, 2022, aims to monitor and localize X-ray and gamma-ray transients
from $\sim$ 6 keV to 6 MeV. GECAM-C utilizes a similar design to GECAM but
operates in a more complex orbital environment. In this work, we utilize the
secondary particles simultaneously produced by the cosmic-ray events on orbit
and recorded by multiple detectors, to calibrate the relative timing accuracy
between all detectors of GECAM-C. We find the result is 0.1 $\mu\rm s$, which
is the highest time resolution among all GRB detectors ever flown and very
helpful in timing analyses such as minimum variable timescale and spectral
lags, as well as in time delay localization. Besides, we calibrate the
absolute time accuracy using the one-year Crab pulsar data observed by GECAM-C
and Fermi/GBM, as well as GECAM-C and GECAM-B. The results are $2.02\pm 2.26\
\mu\rm s$ and $5.82\pm 3.59\ \mu\rm s$, respectively. Finally, we investigate
the spectral lag between the different energy bands of Crab pulsar observed by
GECAM and GBM, which is $\sim-0.2\ {\rm\mu s\ keV^{-1}}$.

methods: data analysis – instrumentation: detectors

1.5in

##  1 Introduction

Since the launch on December 10, 2020, the GECAM mission (Xiong, 2020) has
observed hundreds of GRBs including the Brightest-Of-All-Time (BOAT) GRB
221009A (An et al., 2023) and the second brightest GRB 230307A (Xiong et al.,
2023), as well as X-ray bursts from magnetar such as SGR J1935+2154. Thanks to
the excellent energy resolution ($\sim$ 16% at 59.5 keV), anti-saturation
design at high count rates ($<$ 4$\times 10^{5}$ counts per second) (Zhang et
al., 2019; Chen et al., 2021; Li et al., 2021) and the highest time resolution
(0.1 $\mu\rm s$) among all GRB satellites ever flown (Xiao et al., 2022a),
GECAM has advantages in energy spectral and timing analysis (Xiao et al.,
2022b), such as spectral lag (Xiao et al., 2022c, 2023a), minimum variable
timescale (Xiao et al., 2023b) and quasiperiodic oscillation (Xiao et al.,
2023c), as well as time delay localization for burst (Xiao et al., 2021) and
pulsar or magnetar navigation experiment (Luo et al., 2023).

Figure 1: An illustration of the secondary particles produced by cosmic rays
hitting the satellite and observed by all 14 detectors on board GECAM-C
simultaneously at UTC 2022-10-06T06:16:01.84390998 (red dotted line). The
vertical coordinate represents the detector serial number, from 1 to 12
represent GRD, whereas 13 and 14 represent CPD. The different colors represent
different energies, note that there are 8 GRDs recorded with energies higher
than 10,000 keV due to the very high energy of the cosmic ray. Note their time
difference is less than 0.3 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$.

To achieve all-sky monitoring, joint data analysis, localization of bursts, as
well as the study of self-organized criticality (Bak et al., 1987) for a more
comprehensive sample, especially since GECAM-A is currently not stable for
operation, GECAM-C (i.e. the High Energy Burst Searcher, HEBS) is designed as
a new member of GECAM on board the SATech-01 satellite
111https://www.globaltimes.cn/page/202301/1283692.shtml launched on July 27,
2022. GECAM-C (Zhang et al., 2023) consists of two dome modules which are
installed on the top and bottom side of the satellite (see Figure 1 in Zheng
et al. (2023)), each dome module has one data acquisition board (DAQ) to read
out data from six Gamma-Ray Detectors (GRDs) and one Charged Particle Detector
(CPD). An encapsulated crystal box placed NaI(Tl) or LaBr3 (Ce/Ce+Sr) crystal
and a read out by a SiPM array are utilized in each GRD design, the
preamplifier of each GRD has two channels (high gain and low gain), except for
GRD06 and GRD12 detectors, which are readout by a single channel. Therefore,
the detectors installation positions and detectors design are not the same as
in GECAM-A or GECAM-B. The detection range of GECAM-C is about 6 keV–6 MeV
with an energy resolution of 18% at 59.5 keV (Zheng et al., 2023). The dead
time for a normal event (i.e. that with energy within the dynamic range of GRD
or CPD readout) is 4 $\mu\rm s$ and 4.8 $\mu\rm s$ for GRD and CPD,
respectively.

The time system of GECAM-C is the same as that of GECAM-B (see Xiao et al.
(2022a) for details), except that the former has only two DAQs and the time
accuracy of GNSS receiver and hardware delay is about +/-400 nanoseconds
(3$\sigma$). The timing performance, including relative time accuracy between
all detectors of GECAM-C and absolute time accuracy, is one of the core
metrics of GRB satellites. In the previous calibration of the timing
performance of GECAM-B, we proposed using cosmic ray events for relative time
accuracy calibration and obtain an accuracy of 0.12 $\mu\rm s$ (1$\sigma$),
and the data of Crab pulsar observed by GECAM-B and Fermi/GBM from 2020
December to 2021 August is utilized to calibrate the absolute time accuracy
and obtain an accuracy of $3.06\pm 6.04\ \mu\rm s$ (1$\sigma$). Additionally,
we introduced a new method for absolute time calibration using bursts from a
magnetar, such as SGR J1935+2154 (Xiao et al., 2022a). Although GECAM and GBM
have similar responses and been chosen with the same energy range, we may be
unable to completely avoid systematic error due to the spectral lags between
different energy bands in Crab pulsar with $t_{\rm lag}(E)=-0.27(E/{\rm
keV})+C$ ($\mu\rm s$) (i.e. low-energy photons arrive before high-energy
photons.) (Molkov et al., 2009; Tuo et al., 2022). Fortunately, for the very
similarly designed GECAM-B and GECAM-C, we have the valuable opportunity to
perform time calibration through the Crab pulsar data observed by them.

In this work, the calibration of the relative time accuracy of 12 GRDs and 2
CPDs onboard GECAM-C using cosmic rays observed on orbit is presented in
Section 2. We perform the absolute time calibration in Section 3 using the
data of Crab pulsar observed by GECAM-C and Fermi/GBM, as well as GECAM-C and
GECAM-B. A summary is given in Section 4.

##  2 RELATIVE TIME CALIBRATION

When a cosmic ray (mostly proton) hits a satellite, it may produce many
secondary particles through hadron shower or other processes observed by
multiple detectors almost simultaneously. Figure 1 shows a time vs. detector
serial number scatter plot for a very high energy cosmic ray event, whose
secondary particles are observed by all 14 detectors at UTC
2022-10-06T06:16:01.84390998, note that there are 8 GRDs recorded with
energies higher than 10,000 keV. All 14 detectors detected within 0.3 $\mu\rm
s$ of each other, thus, similarly to GECAM-B, we initially set up that if more
than 1 detector observes events on a time window of +-0.3
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, they are considered simultaneous
events produced by a cosmic ray event. The left panel in Figure 2 shows the
distribution of the number of detectors observed cosmic ray events (i.e.
marked as simultaneous events) in an hour (from UTC 2022-10-06T06:00:00 to
2022-10-06T06:59:59). It is worth noting here that unlike GECAM-A or GECAM-B
where all detectors are installed on a dome module, here each group of the 7
detectors of GECAM-C in each dome at each side of the satellite point in
opposite directions relative to he other group, thus the number of detectors
with more than 7 observing the same cosmic ray events suddenly declines.

However, if the relative time accuracy exceeds 0.3
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, there might be events observed by
some detectors not marked as simultaneous events according to our set
threshold. To check this, we analyze 13353 cosmic ray events in which at least
five detectors are marked as observing simultaneous events during this hour,
to avoid the coincident by unrelated individual events as much as possible.
Then we search all events recorded by all 14 detectors in a time window of
$\pm$1 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ before and after the
recording time of the simultaneous events, and calculating the maximum time
difference among them. The distribution of the maximum time difference is
shown in the right panel of Figure 2, where we find that most of them (97.0%)
are distributed within less than 0.3
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ and the others are approximately
uniformly distributed after more than 0.3
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$; the latter is produced by not
simultaneous events (i.e. coincident by unrelated individual events) and
indicates that this result is not affected by the time window (i.e. $\pm$1
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$) we set for this analysis. Besides,
about 76% of the maximum time difference is less than or equal to 0.1
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, thus the relative time accuracy
between the detectors onboard GECAM-C is $\sim$ 0.1
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$.

Furthermore, we employ the procedure above to analyze the cosmic ray events in
each month from 2022 August to 2023 July to test whether the relative time
accuracy between the detectors changes over time. As shown in Figure 3, the
percentages of the maximum time differences of less than or equal to 0.1
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ for simultaneous events observed by
multiple detectors obtained in most months are greater than 68%, which
demonstrates the stability of the time resolution. In addition, we note a
tendency for the percentage to first increase and then decrease with time, for
which we have not yet identified a reason, but have ruled out temperature and
attitude changes. However, this does not affect the conclusion of 0.1
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ time resolution.

We also respectively investigated the time accuracy between detectors in the
same DAQ (i.e. DAQ#1 or DAQ#2), as well as between detectors in different
DAQs. As shown in Figure 4, there are 98.2%, 98.6% and 92.3% of the events
with maximum time difference of less than 0.3
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, respectively. Besides, there are
98.6%, 99.0% and 96.2% of the events with maximum time difference of less than
0.2 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ in the events of less than 0.3
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, respectively. Therefore, the time
accuracy between detectors in two DAQs is slightly worse than that of
detectors in the same DAQ. It’s worth noting here that both DAQs observed
significantly fewer cosmic ray events than a single DAQ due to they are
mounted at each side of the satellite and pointing in opposite directions
relative to the other group.

Figure 2: Left panel: the distribution of the number of detectors marked as
observed simultaneous events in an hour (from UTC 2022-10-06T06:00:00 to
2022-10-06T06:59:59), the number of detectors with more than seven observing
cosmic ray events is suddenly much fewer since each group of 7 detectors on
board GECAM-C are mounted at each side of the satellite and pointing in
opposite directions relative to the other group (i.e. higher-energy cosmic
rays are needed to be observed by more than seven detectors). Right panel: the
distribution of the maximum time difference for 13353 cosmic ray events
observed by multiple detectors, most of them (97.0%) are distributed within
less than 0.3 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ and are approximately
uniformly distributed after more than 0.3
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, which is produced by unrelated
individual events. Figure 3: The percentage of maximum time differences of
less than or equal to 0.1 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ for
simultaneous events observed by multiple detectors in each month from 2022
August to 2023 July (i.e. vertical coordinate). The red dotted line represents
the 1 $\sigma$ in the Gaussian distribution. Therefore, the time resolution of
GECAM-C is better than 0.1 $\mu\rm s$ (1 $\sigma$) due to the fact that the
percentage in most months greater than 68%. Figure 4: The distribution of the
maximum time difference for 13353 cosmic ray events observed by multiple
detectors in DAQ#1 (top panel), DAQ#2 (middle panel), as well as DAQ#1 and
DAQ#2 (i.e. both DAQs need at least one detector detects the event) (bottom
panel), respectively. 98.2%, 98.6% and 92.3% of the events are distributed
within less than 0.3 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, respectively.
Besides, there are 98.6%, 99.0% and 96.2% of the events with maximum time
difference of less than 0.2 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ in the
events of less than 0.3 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$,
respectively. Therefore, the time accuracy between detectors in two DAQs is
slightly worse than that of detectors in the same DAQ.

Figure 5: The Crab pulse profiles in the 20–500 keV energy band observed
GECAM-C (red), GECAM-B (blue) and GBM (black) in each month from August 2022
to July 2023. Scale represents the coefficient by which its vertical
coordinate is multiplied, and we subtract the lowest value of the pulse
profile, respectively. Figure 6: The time difference of observed Crab pulse
profiles between GECAM-C and GBM, between GECAM-C and GECAM-B, as well as
between GECAM-B and GBM in different months, respectively. The orange shaded
regions are the 1$\sigma$ ranges (i.e. 2.02$\pm$ 3.02
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, -5.82$\pm$ 4.11
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ and 4.98$\pm$ 2.19
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$) obtained by fitting. The
reduced-$\chi^{2}$ is 1.05, 0.94 and 0.86, respectively, which indicates their
well fit goodness of fit.

Figure 7: Left panel: the black dots are the MCCF versus time difference of
the between GECAM-C and GBM observations of Crab pulsar profiles in one year,
the red dotted line represents the best time difference. Right panel:
distribution of time differences obtained by MC, the red line is obtained by
fitting a Gaussian function. Therefore, the time difference is $3.37\pm 3.27\
\mu\mathrm{s}$, which is consistent with the result obtained by fitting each
month (i.e. $2.02\pm 3.02\ \mu\mathrm{s}$) within the error. Figure 8:
Spectral lags of Crab profiles at different energies observed by GECAM-C
(red), GECAM-B (blue) and GBM (black), respectively. Note that for GECAM-C and
GBM the lag is compared to 8-14 keV and for GECAM-B the lag is compared to
14-23 keV. The dotted lines represent the results obtained by fitting them
with a linear function, respectively.

##  3 ABSOLUTE TIME CALIBRATION

###  3.1 Between GECAM-C and Fermi/GBM

Crab pulsar (PSR J0534+2200, RA = $\rm 05^{h}34^{m}31.972^{s}$, Dec = $\rm
22^{\circ}0^{\prime}52".07$) (Lyne et al., 1993) is widely used in previous
absolute time accuracy calibrations of almost all astronomical satellites
(Kuiper et al., 2003; Rots et al., 2004; Kirsch et al., 2004; Edwards et al.,
2006; Molkov et al., 2009; Terada et al., 2008; Cusumano et al., 2012; Basu et
al., 2018; Bachetti et al., 2021; Tuo et al., 2022; Xiao et al., 2022a) due to
its relatively high brightness and stable evolutionary period. Since Fermi/GBM
has the time accuracy of about 10 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$
and the time resolution of 2 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$
(Meegan et al., 2009; Paciesas et al., 2012), as well as an energy response of
the NaI detector similar to that of GECAM-C, we utilize the Crab pulsar data
observed by GECAM-C and GBM from August 1, 2022 to July 31, 2023 for absolute
time calibration. Besides, to improve the statistics, only those detectors on
board GECAM-C or GBM with incidence angles less than 70 degrees from the Crab
pulsar and only the time intervals when the Crab pulsar is not blocked by the
Earth are selected, as well as counts in the deposition energy range of 20-500
keV (the energy responses of GBM and GECAM are similar in this energy range)
are used to analysis.

To calculate the phase in Crab pulsar profile of each event observed GECAM-C
and GBM, the arrive times of them are corrected to the solar system barycenter
(DE200) using the $\rm package$ $tat-pulsar$ 222https://github.com/tuoyl/tat-
pulsar developed by us (Tuo et al., 2022), then folded with the radio
ephemeris 333http://www.jb.man.ac.uk/ pulsar/crab.html (Lyne et al., 1993),
respectively. We then utilize the Li-CCF method (Li et al., 2004; Xiao et al.,
2021), which can obtain a more accurate result using as much information in
the data as possible, to calculate the time difference between the Crab pulse
profiles observed by GECAM-C and GBM, respectively, which is taken as the
GECAM-C absolute time accuracy; the detailed procedures can be found in our
previous work (Xiao et al., 2022a).

Figure 5 shows the pulse profiles observed by GECAM-C, GECAM-B, and Fermi/GBM
in each month from August 2022 through July 2023, which have very similar
profiles. We calculate the time difference between the pulsar profiles
observed by GECAM-C and GBM in each month using Li-CCF and the results are
shown in Table 1. Since the results in different months are roughly consistent
within the errors (see the top panel in Figure 6), we obtain the absolute time
accuracy by fitting with a constant, which is 2.02$\pm$ 3.02
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ (reduced-$\chi^{2}$=1.05). Note
that if we do not take into account the error introduced by the time
resolutions of GBM (i.e. 2 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$) and
GECAM-C (i.e. 0.1 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$), the statistical
uncertainty on the absolute time accuracy is
$\sqrt{3.02^{2}-2^{2}-0.1^{2}}=2.26\
\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, i.e., the final absolute time
accuracy is 2.02$\pm$ 2.26 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, which
is consistent with 0 within 0.9 $\sigma$ error.

In addition, we directly calculated the time difference between the Crab
pulsar profiles in one year observed by GECAM-C and GBM. The left panel in
Figure 7 shows the MCCF versus time difference, the time difference
corresponding to the maximum value of MCCF is the best value (i.e. 3.37
$\rm\mu\mathrm{s}$). The right panel in Figure 7 shows the distribution of
time differences obtained by a Monte Carlo (MC) simulation, since its
approximation follows a Gaussian distribution, the standard deviation can be
taken as the 1 $\sigma$ error. Therefore, the result is $3.37\pm 3.27\
\mu\mathrm{s}$, which is consistent with the result obtained by fitting each
month (i.e. $2.02\pm 3.02\ \mu\mathrm{s}$) within the error.

###  3.2 Between GECAM-B and Fermi/GBM

Meanwhile, we can also further calibrate the absolute time accuracy of
GECAM-B. The time difference between the pulsar profiles observed by GECAM-C
and GBM in each month. The results are shown in Table 1 and the final absolute
time accuracy is 4.98$\pm$ 2.19 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$
(reduced-$\chi^{2}$=0.94) through fitting (see the middle panel in Figure 6).
If we do not take into account the error introduced by the time resolution of
GBM (i.e. 2 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$) and GECAM-B (i.e. 0.1
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$), the final absolute time accuracy
is $\sqrt{2.19^{2}-2^{2}-0.1^{2}}=0.89\
\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, which is consistent with 0 within
5.6 $\sigma$ error.

###  3.3 Between GECAM-C and GECAM-B

Crab pulsar have a negative lag between different energy bands (i.e. high-
energy photons follow low-energy photons) with $t_{\rm lag}(E)=-A(E/{\rm
keV})+C$ ($A\approx 0.27\ {\rm\mu s/keV}$) (Molkov et al., 2009; Tuo et al.,
2022), which can be explained as having two independent and different spectral
and phase distribution components (Massaro et al., 2000). Since the detector
design and energy response of GECAM-C and GBM are not identical, systematic
error may be introduced in the time calibration. Therefore, we utilize the
Crab pulsar data observed by GECAM-C and the very similar GECAM-B for time
calibration. The time difference between the pulsar profiles observed by
GECAM-C and GECAM-B in each month and the results are shown in Table 1; the
final absolute time accuracy is -5.82$\pm$ 4.11
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ (reduced-$\chi^{2}$=0.86) through
fitting (see the bottom panel in Figure 6). Therefore, there is no significant
difference in the time difference between GECAM and GBM versus GECAM-C and
GECAM-B. If we do not take into account the error introduced by the time
resolution of GECAM-C (i.e. 0.1 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$)
and GECAM-B (i.e. 0.1 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$), the final
absolute time accuracy is $\sqrt{4.11^{2}-0.1^{2}-0.1^{2}}=3.59\
\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, which is consistent with 0 within
1.6 $\sigma$ error.

In addition, we can calculate the time difference between GECAM-B and GECAM-C
by using the time difference between GECAM-C and GBM, and between GECAM-B and
GBM, and the error is obtained by error transfer. The result is $-2.96\pm
2.42\ \rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$, which is consistent with the
time difference obtained directly calculation (i.e. $-5.82\pm 3.59\
\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$) within the error.

Table 1: Time difference between Crab pulse profiles observed by GECAM-C and GBM, GECAM-C and GECAM-B, as well as GECAM-B and GBM in each month. Time (month) |  Time difference Between GECAM-C and GBM ($\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$) |  Time difference Between GECAM-C and GECAM-B ($\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$) |  Time difference Between GECAM-B and GBM ($\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$)
---|---|---|---
202208 | $22.30\pm 14.36$ | $-24.33\pm 22.43$ | $8.11\pm 7.94$
202209 | $-22.30\pm 11.43$ | $36.49\pm 22.09$ | $-4.05\pm 13.60$
202210 | $2.03\pm 14.79$ | $-20.27\pm 26.71$ | $8.11\pm 19.02$
202211 | $10.14\pm 17.70$ | $-18.25\pm 15.05$ | $4.05\pm 7.66$
202212 | $-4.05\pm 9.63$ | $0.00\pm 10.96$ | $6.08\pm 5.96$
202301 | $2.03\pm 5.75$ | $6.08\pm 12.42$ | $-12.16\pm 9.95$
202302 | $14.19\pm 7.05$ | $-24.33\pm 16.22$ | $14.19\pm 12.31$
202303 | $-4.05\pm 12.33$ | $8.11\pm 14.05$ | $-4.05\pm 8.52$
202304 | $-6.08\pm 9.01$ | $-16.22\pm 13.24$ | $18.25\pm 8.34$
202305 | $8.11\pm 13.23$ | $-4.05\pm 16.93$ | $6.08\pm 9.69$
202306 | $8.11\pm 11.57$ | $-12.16\pm 16.86$ | $8.11\pm 8.00$
202307 | $-6.08\pm 11.76$ | $-10.14\pm 13.11$ | $2.03\pm 8.72$
Table 2: Summary of the time calibration results of GECAM. Detectors | Time accuracy (1$\sigma$)
---|---
Detectors on GECAM-C (time resolution) | 0.1 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$
GECAM-C and GBM | 2.02$\pm 2.26$ $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$
GECAM-C and GECAM-B | -5.82$\pm 3.59$ $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$
GECAM-B and GBM | 4.98$\pm 0.89$ $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$

##  4 Conclusion

In this work, we performed the relative time calibration of the detectors on
board GECAM-C and the absolute time calibration using the one-year
observational data from its launch in 2022 Aug to 2023 July. The results are
summarized in Table 2.

The relative time accuracy of the detector on board GECAM-C is calibrated to
$\sim$ 0.1 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ (1$\sigma$) using
observed secondary particles produced almost simultaneously by cosmic rays
hitting the satellite, which is consistent with the result of GECAM-B. The
time accuracy between detectors in two DAQs is slightly worse than that of
detectors in the same DAQ. Besides, we further verify the accuracy is stable
by analyzing data in each month, which also indicates that GECAM-C, as same as
GECAM-B, has the highest time resolution among all GRB missions ever flown,
such as 2 $\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ for Fermi/GBM and HXMT/HE
(Meegan et al., 2009;Liu et al., 2020), 10
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ for AstroSAT (Agrawal, 2006), 100
$\rm\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ for Swift/BAT (Sakamoto et al.,
2008), 2 ms for Konus-Wind (Aptekar et al., 1995) and 50 ms for SPI-ACS (von
Kienlin et al., 2003).

The absolute time accuracy of GECAM-C is calibrated to $2.02\pm 2.26\ \mu\rm
s$ by the one-year Crab pulsar data observed jointly with Fermi/GBM, and we
also investigate the accuracy in each month and confirm that it is stable
(consistent within the error). Besides, to investigate the time-calibration
systematic errors that may result from the Crab pulsar spectral lags in
different energies (since different instruments have different energy
responses), we calibrated the time accuracy between GECAM-C and GECAM-B, as
well as between GECAM-B and GBM, and the results are $-5.82\pm 3.59\ \mu\rm s$
and $4.98\pm 0.89\ \mu\rm s$, respectively. Since GECAM and GBM have similar
energy responses, there is no significant difference in the time difference
between GECAM and GBM versus GECAM-C and GECAM-B. However, the time difference
between GECAM-B and GBM is consistent with 0 within 5.6 $\sigma$ error, and
the difference in energy response may be one of the reasons. If we consider
the error introduced by the time resolution, the time difference between
GECAM-C and GBM, GECAM-C and GECAM-B, as well as GECAM-B and GBM are $2.02\pm
3.02\ \mu\rm s$, $-5.82\pm 4.11\ \mu\rm s$ and $4.98\pm 2.19\ \mu\rm s$,
respectively, which are all consistent with 0 within 3 $\sigma$.

For the high time resolution and absolute time accuracy, GECAM not only can
identify cosmic ray events with very few false positives, but also has
advantages as well as potential in timing analysis, for example,
(1) Time delay localization is a well-known method of localizing bursts by
calculating the time difference between signals received by satellites at
different positions (i.e. requires high time resolution and absolute time
accuracy) (e.g. Hurley et al., 2011; Pal’Shin et al., 2013; Xiao et al.,
2021), and the precise location of GRB and soft gamma repeater (SGR) is
crucial in multimessenger and multiwavelength astronomy for guiding follow-up
observational, as well as identifying associated transients, neutrinos and
gravitational waves. The absolute time accuracy 3 $\mu\rm s$ corresponds to
only about 0.01 degrees in the time delay localization in low Earth orbits
satellites. More importantly, the high time resolution of 0.1 $\mu\rm s$
combined with the Li-CCF method can give more accurate locations (Li et al.,
2004; Xiao et al., 2021).
(2) Timing analysis is an important approach to revealing radiation mechanisms
and progenitors, such as spectral lag between different energy bands (e.g.
Zhang et al., 2009; Ukwatta et al., 2012; Bernardini et al., 2014) and minimum
variation timescale (i.e. requires high time resolution) (e.g. Schmidt, 1978;
Fenimore et al., 1993; Titarchuk et al., 2007; Ackermann et al., 2014; Golkhou
& Butler, 2014; Golkhou et al., 2015). Meanwhile, in the future, the
sensitivity of the detector should be considered to be improved to obtain
signals with higher signal-to-noise ratios in order to best utilize the
advantages of high time resolution.
(3) The quasiperiodic oscillation (i.e. requires high time resolution for
high-frequency searches) of magnetar (e.g. Huppenkothen et al., 2014; Xiao et
al., 2022d, e, 2023c; Roberts et al., 2023) can be used to constrain both the
equation of state and the interior magnetic field of the neutron star via
asteroseismology. Besides, it is worth noting that the impact of dead time on
the search for high-frequency QPOs are significant, and thus further
decreasing the dead time should be considered.
(4) Cosmic rays hit a detector or the satellite structure can simultaneously
produce many secondary particles recorded by multiple GRDs and CPDs nearly
simultaneously (i.e. requires high relative time accuracy to identify); for
the time resolution 0.1 $\mu\rm s$, both detectors have only $\sim$ 0.2 false
positives per second if they both have a count rate of 1000. Besides, relative
time accuracy is also critical when performing joint analysis using multiple
detectors.
(5) Pulsar navigation experiment is important for both orbit estimation of
Earth satellites and deep space navigation for spacecraft (e.g. Witze, 2018;
Zheng et al., 2019; Luo et al., 2023), which needs that the arrival times of
events observed are corrected to the solar system barycenter (DE200) (i.e.
requires high time resolution and absolute time accuracy); the absolute time
accuracy of 3 $\mu\rm s$ corresponds to a light travel distance of only about
900 meters.

Finally, an interesting by-product of this work is the spectral lag between
different energy bands of the Crab pulsar observed by GECAM and GBM. As shown
in Figure 8, the energy and spectral lag relations of the Crab pulsar observed
by GECAM-C, GECAM-B, and GBM can be fitted with a linear function with slopes
$-0.17\pm 0.10\ \mu\mathrm{s}\ \rm keV^{-1}$, $-0.20\pm 0.12\ \mu\mathrm{s}\
\rm keV^{-1}$ and $-0.19\pm 0.03\ \mu\mathrm{s}\ \rm keV^{-1}$, respectively.
These results are consistent within 3 $\sigma$ error with those previously
obtained through RXTE/PCA, HEXTE, and INTEGRAL/ SPI ($-0.6\pm 0.2\
\mu\mathrm{s}\ \rm keV^{-1}$), as well as NICER and HXMT ($-0.26\pm 0.02\
\mu\mathrm{s}\ \rm keV^{-1}$), however, the rates of decrease we measured are
slightly smaller.

## Acknowledgments

We acknowledge the public data from Fermi/GBM. This work is supported by the
National Key R&D Program of China 2022YFF0711404, the Scientific Research
Project of the Guizhou Provincial Education (Nos. KY[2022]123, KY[2022]132,
KY[2022]137, KY[2021]303, KY[2020]003 and KY[2023]059). The authors also thank
supports from the Strategic Priority Research Program on Space Science, the
Chinese Academy of Sciences (Grant No. XDA15010100, XDA15360100, XDA15360102,
XDA15360300, XDA15052700), the National Natural Science Foundation of China
(Projects: 12273042, 12061131007, Grant No. 12173038, Nos. 12273008 and
12103013), the Foundation of Education Bureau of Guizhou Province, China
(Grant No. KY (2020) 003), Guizhou Provincial Science and Technology
Foundation (Nos. ZK[2022]304, [2021]023, [2023]024).

## References

  * Ackermann et al. (2014) Ackermann, M., Ajello, M., Asano, K., et al. 2014, Science, 343, 42
  * Agrawal (2006) Agrawal, P. 2006, Advances in Space Research, 38, 2989
  * An et al. (2023) An, Z.-H., Antier, S., Bi, X.-Z., et al. 2023, arXiv preprint arXiv:2303.01203
  * Aptekar et al. (1995) Aptekar, R., Frederiks, D., Golenetskii, S., et al. 1995, Space Science Reviews, 71, 265
  * Bachetti et al. (2021) Bachetti, M., Markwardt, C. B., Grefenstette, B. W., et al. 2021, The Astrophysical Journal, 908, 184
  * Bak et al. (1987) Bak, P., Tang, C., & Wiesenfeld, K. 1987, Physical review letters, 59, 381
  * Basu et al. (2018) Basu, A., Joshi, B. C., Bhattacharya, D., et al. 2018, Astronomy & Astrophysics, 617, A22
  * Bernardini et al. (2014) Bernardini, M. G., Ghirlanda, G., Campana, S., et al. 2014, Monthly Notices of the Royal Astronomical Society, 446, 1129
  * Chen et al. (2021) Chen, C., Xiao, S., Xiong, S., et al. 2021, Experimental Astronomy, 1
  * Cusumano et al. (2012) Cusumano, G., La Parola, V., Capalbi, M., et al. 2012, Astronomy & Astrophysics, 548, A28
  * Edwards et al. (2006) Edwards, R. T., Hobbs, G., & Manchester, R. 2006, Monthly Notices of the Royal Astronomical Society, 372, 1549
  * Fenimore et al. (1993) Fenimore, E., Epstein, R., & Ho, C. 1993, Astronomy and Astrophysics Supplement Series, 97, 59
  * Golkhou & Butler (2014) Golkhou, V. Z., & Butler, N. R. 2014, The Astrophysical Journal, 787, 90
  * Golkhou et al. (2015) Golkhou, V. Z., Butler, N. R., & Littlejohns, O. M. 2015, The Astrophysical Journal, 811, 93
  * Huppenkothen et al. (2014) Huppenkothen, D., Heil, L., Watts, A., & Göğüş, E. 2014, The Astrophysical Journal, 795, 114
  * Hurley et al. (2011) Hurley, K., Briggs, M., Kippen, R., et al. 2011, The Astrophysical Journal Supplement Series, 196, 1
  * Kirsch et al. (2004) Kirsch, M. G., Becker, W., Benlloch-Garcia, S., et al. 2004, in X-Ray and Gamma-Ray Instrumentation for Astronomy XIII, Vol. 5165, International Society for Optics and Photonics, 85–95
  * Kuiper et al. (2003) Kuiper, L., Hermsen, W., Walter, R., & Foschini, L. 2003, A&A, 411, L31, doi: 10.1051/0004-6361:20031353
  * Li et al. (2004) Li, T.-P., Qu, J.-L., Feng, H., et al. 2004, Chinese Journal of Astronomy and Astrophysics, 4, 583
  * Li et al. (2021) Li, X., Wen, X., Xiong, S., et al. 2021, arXiv preprint arXiv:2112.04772
  * Liu et al. (2020) Liu, C., Zhang, Y., Li, X., et al. 2020, SCIENCE CHINA Physics, Mechanics & Astronomy, 63, 1
  * Luo et al. (2023) Luo, X.-H., Xiao, S., Zheng, S.-J., et al. 2023, The Astrophysical Journal Supplement Series, 266, 16
  * Lyne et al. (1993) Lyne, A., Pritchard, R., & Graham Smith, F. 1993, Monthly Notices of the Royal Astronomical Society, 265, 1003
  * Massaro et al. (2000) Massaro, E., Cusumano, G., Litterio, M., & Mineo, T. 2000, arXiv preprint astro-ph/0006064
  * Meegan et al. (2009) Meegan, C., Lichti, G., Bhat, P. N., et al. 2009, ApJ, 702, 791, doi: 10.1088/0004-637X/702/1/791
  * Molkov et al. (2009) Molkov, S., Jourdain, E., & Roques, J. 2009, The Astrophysical Journal, 708, 403
  * Paciesas et al. (2012) Paciesas, W. S., Meegan, C. A., von Kienlin, A., et al. 2012, The Astrophysical Journal Supplement Series, 199, 18
  * Pal’Shin et al. (2013) Pal’Shin, V., Hurley, K., Svinkin, D., et al. 2013, The Astrophysical Journal Supplement Series, 207, 38
  * Roberts et al. (2023) Roberts, O. J., Baring, M. G., Huppenkothen, D., et al. 2023, arXiv preprint arXiv:2306.08130
  * Rots et al. (2004) Rots, A. H., Jahoda, K., & Lyne, A. G. 2004, The Astrophysical Journal Letters, 605, L129
  * Sakamoto et al. (2008) Sakamoto, T., Barthelmy, S., Barbier, L., et al. 2008, The Astrophysical Journal Supplement Series, 175, 179
  * Schmidt (1978) Schmidt, W. K. H. 1978, Nature, 271, 525, doi: 10.1038/271525a0
  * Terada et al. (2008) Terada, Y., Enoto, T., Miyawaki, R., et al. 2008, Publications of the Astronomical Society of Japan, 60, S25
  * Titarchuk et al. (2007) Titarchuk, L., Shaposhnikov, N., & Arefiev, V. 2007, ApJ, 660, 556, doi: 10.1086/512027
  * Tuo et al. (2022) Tuo, Y., Li, X., Ge, M., et al. 2022, The Astrophysical Journal Supplement Series, 259, 14
  * Ukwatta et al. (2012) Ukwatta, T., Dhuga, K., Stamatikos, M., et al. 2012, Monthly Notices of the Royal Astronomical Society, 419, 614
  * von Kienlin et al. (2003) von Kienlin, A., Beckmann, V., Rau, A., et al. 2003, Astronomy & Astrophysics, 411, L299
  * Witze (2018) Witze, A. 2018, Nature, 553, 261
  * Xiao et al. (2021) Xiao, S., Xiong, S., Zhang, S., et al. 2021, The Astrophysical Journal, 920, 43
  * Xiao et al. (2022a) Xiao, S., Liu, Y., Peng, W., et al. 2022a, Monthly Notices of the Royal Astronomical Society, 511, 964
  * Xiao et al. (2022b) Xiao, S., Xiong, S.-L., Cai, C., et al. 2022b, Monthly Notices of the Royal Astronomical Society, 514, 2397
  * Xiao et al. (2022c) Xiao, S., Xiong, S.-L., Wang, Y., et al. 2022c, The Astrophysical Journal Letters, 924, L29
  * Xiao et al. (2022d) Xiao, S., Zhang, Y.-Q., Zhu, Z.-P., et al. 2022d, arXiv preprint arXiv:2205.02186
  * Xiao et al. (2022e) Xiao, S., Peng, W.-X., Zhang, S.-N., et al. 2022e, The Astrophysical Journal, 941, 166
  * Xiao et al. (2023a) Xiao, S., Tuo, Y.-L., Zhang, S.-N., et al. 2023a, MNRAS, 521, 5308, doi: 10.1093/mnras/stad885
  * Xiao et al. (2023b) Xiao, S., Yang, J.-J., Luo, X.-H., et al. 2023b.  https://arxiv.org/abs/2307.07079
  * Xiao et al. (2023c) Xiao, S., Li, X.-B., Xue, W.-C., et al. 2023c, arXiv preprint arXiv:2307.14884
  * Xiong (2020) Xiong, S. 2020, SCIENTIA SINICA Physica, Mechanica & Astronomica, 50, 129501
  * Xiong et al. (2023) Xiong, S., Wang, C., & Huang, Y. 2023, GRB Coordinates Network, 33406, 1
  * Zhang et al. (2009) Zhang, B., Zhang, B.-B., Virgili, F. J., et al. 2009, The Astrophysical Journal, 703, 1696
  * Zhang et al. (2019) Zhang, D., Li, X., Xiong, S., et al. 2019, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 921, 8
  * Zhang et al. (2023) Zhang, D., Zheng, C., Liu, J., et al. 2023, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 1056, 168586, doi: https://doi.org/10.1016/j.nima.2023.168586
  * Zheng et al. (2023) Zheng, C., An, Z.-H., Peng, W.-X., et al. 2023, arXiv preprint arXiv:2303.00687
  * Zheng et al. (2019) Zheng, S., Zhang, S., Lu, F., et al. 2019, The Astrophysical Journal Supplement Series, 244, 1