Quantifying collective intelligence and behaviours of SARS-CoV-2 via environmental resources from virus' perspectives.

Collective intelligence of viruses is witnessed in many research articles. Most of the researches focus on the qualitative properties or observations. In this research, we model the behaviours and collective intelligence of SARS-CoV-2 by minimal spanning trees (MSTs), which specify the underlying mechanisms of resource allocation in the viral colony. The vertices of the trees are 50 states, DC and NYC in the USA. The weights of the edges are assigned by the reciprocal of the sum of cases or deaths of COVID-19. The types of trees are decided by the chosen 18 factors. We sample 304 time-series data and compute their MST-based auto-correlations for stability analysis. Then we perform correlated analysis and comparative analysis on these stable factors. Our results show MST approach fits the collective intelligence modelling very well; the total cases and total deaths over areas are highly correlated in terms of MSTs; and these stable factors have little to do with the geographical distance. The results also indicate the colonisation of SARS-CoV-2 is pretty mature and organised. Based on the results, for environmental or health policies, we should also turn our attention to the transmission routes that are independent of or far away from human population or densities. The viruses' colonies might already exist in the wild in a large scale, not only in the populated or polluted cities. We shall build or conduct a monitoring system of their colonisation and survival techniques, in order to terminate, contain or live with their communities.

Introduction

Though the origins and ways of diffusion and transmission of COVID1-19 regarding environmental factor-to-human or human-to-human at country level is still in debate and research (Bontempi, 2020a, 2020b; Chen, 2021a), there are some observations and tracking of the development regarding different waves of infection (Bontempi, 2021; bib_Coccia_2021Coccia, 2021a,b). Like ZIKA viruses (Liu et al., 2021), SARS-CoV-2 has mutated in various forms across the globe (Wise, 2021; Wang, 2021). It is vital the evolutionary mechanisms and behaviours of SARS-CoV-2, and their interaction with human and environment could be fully explored and revealed in order to terminate, contain or live with these viruses. So far, there are many approaches to explore the properties of viruses via genomics, epidemiology, immunology, virology, biology, biochemistry, environmental science, mathematical models, etc. In essence, they focus on two main levels: (1) micro-level (or reductionism), for example genetic decoding, modelling dynamical viral diffusion (Kucharski et al., 2020; Coccia, 2020c); molecular biology (Deichmann, 2017) and dynamics (He et al., 2020); and (2) macro-level (or holism), in particular the interaction or observation from the environmental factors: air pollution, climate change, etc.; demographic factors or the mixed factors - like Index c (Coccia, 2020a), which takes various factors into consideration to yield an index for the prediction of infectious diseases by studying the correlation between these factors and deaths of COVID-19 (Coccia, 2020b). There are some practical methods to tackle these issues: the study of multi-scale model and confinement strategies (Bellomo et al., 2020); the study of collective transmission (Liu et al., 2021); and the mechanisms of collective infectious units (CIUs) (Asher et al., 2019).

The behaviours of viruses are studied in various research literature. For some viruses, it has already shown that they have some cooperative and altruistic behaviours at the population level to weaken and attack our immune systems (Dolgin, 2019; Bunimovich et al., 2020). They also employ some social tricks. For example, phages could actively discuss and make decisions about their replicating and breaking strategies. This shows they have some degree of collective intelligence (CI). For survival, viruses would adapt to environmental changes in order to spread the viral infections via adaptive variants (Domingo et al., 1997; Miyashita et al., 2015). These variants are also witnessed by SARS-CoV-2 in recent development (CDC, 2020). However, most of these researches are focusing on the micro-scale modelling, in particular the structures of genetic parts (Chen, 2020a, 2021b, 2021b; Zhao, 2020) - even some are tackling the interactions between human body systems and viruses’ invasion (Mohammed, 1993; Smith, 2012). These researches focus more on the qualitative and static aspects and observations of the collective intelligence or behaviours in the viral colonies. In this article, we delve into the mechanisms of CI of SARS-CoV-2 via identifying the minimum spanning trees (MSTs) of macro-scale networks with respect to (w.r.t.) 18 different chosen factors. MST (Prim, 1957; Dijkstra, 1959; Chen, 2020b) is a way (or MST principle) to effectively allocate resources in a colony.

Collectively, SARS-CoV-2 is a colony with an aim to effectively build its kingdom. The underlying behaviours of their transmission and communication could be captured by MST. To begin with, the (52) vertices of the trees (or networks) are identified with the 50 USA states, the District of Columbia (DC) and New York City (NYC). The edges are represented by pairs of vertices. The weights assigned to the edges are decided by the reciprocal of the sum of magnitude of the values (COVID-19 cases and deaths). These weights basically witness the minimal paths or traces the viruses spread through. Up to this stage, we have constructed trees (or networks). Then we calculate and find the MSTs of these networks. These MSTs are then converted into incidence matrices. Then we calculate the Hamming distances between these incidence matrices to measure the distance between MSTs, whose trees are formed by the 18 factors. Based on this, we perform stability analysis, which finds the auto-correlation and tells whether the factors create a series of stable MSTs respectively. The stable factors are then chosen for our further analytical processes: correlated analysis, which measures the cross-correlation among all the stable factors, and comparative analysis, which measures the cross-relation between the stables factors and the physical-distance factor. Our findings show that the stable factors exist (there are seven factors out of the 18 ones) - this indicates the suitability of MST principle for modelling the collective intelligence and behaviours of SARS-CoV-2. They also show that a very strong correlation between total cases of COVID-19 over areas of states and total deaths of COVID-19 over areas of states. The findings also show there is little correlation between physical distance and the chosen factors in terms of MST. It shows the viruses’ density is the main factor that leads to the correlation between confirmed cases and deaths. A complete presentation of the results of this study is listed in section Results 3 and further explored for their environmental, health and social policy making in Section Conclusion 4.

Materials and methods

Research setting and sources

Minimal Spanning Tree (MST)

MST is a concise method to trace the behaviours of SARS-CoV-2. The idea is these viruses exist for survival by their collective decisions in choosing the shortest paths to expand their kingdom. In order to find out their collective intelligence and decision processes behind COVID-19 pandemic, we locate (based on the data we could obtain) 18 factors (or MST factors) to pin down their surviving and expanding strategies. These factors would look into the viruses’ strategies to survive and how they evolve. Here we choose 50 USA states, DC and NYC as our research targets (or vertices).

SARS-CoV-2's perspectives of environmental factors

There are two macro-factors to look at: total cases and new cases. They indicate the statuses of viruses' colonisation. The total cases/deaths indicate the status of total population of viruses, while the new cases/deaths indicate the status of the speed of generation of their offspring. There are also macro-environmental factors (or resources), from the viruses' point of views, to be delved into: the size of the areas, human's population, human's population densities. The set of main factors we are exploring would be .

total cases of COVID-19 (or factor 1):

new cases of COVID-19 (or factor 2);

total cases COVID-19 over areas of the states (or factor 3);

new cases of COVID-19 over areas of states (or factor 4);

total cases of COVID-19 over populations of the states (or factor 5);

new cases of COVID-19 over populations of the states (or factor 6)

total cases of COVID-19 over population densities of the states (or factor 7);

new cases of COVID-19 over population densities of the states (or factor 8)

total deaths of COVID-19 (or factor 9);

new deaths of COVID-19 (or factor 10);

total deaths of COVID-19 over areas of the states (or factor 11);

new deaths of COVID-19 over areas of the states (or factor 12);

total deaths of COVID-19 over populations of the states (or factor 13);

new deaths of COVID1-9 over populations of the states (or factor 14);

total deaths of COVID-19 over total cases of COVID-19 (or factor 15);

new deaths of COVID-19 over new cases of COVID-19 (or factor 16);

total deaths of COVID-19 over population densities of the states (or factor 17);

new deaths of COVID-19 over population densities of the states (or factor 18)

These factors could be divided into three main types of categorised properties:

Colonisation without taking environmental factors into consideration: factor 1, factor 2, factor 9, factor 10;

Colonisation strategies: factor 15, factor 16;

Viruses' densities with respect to environmental factors: factor 3, factor 4, factor 5, factor 6, factor 7, factor 8, factor 11, factor 12, factor 13, factor 14, factor 17, factor 18.

tree formalisation and processing

Firstly, we have to specify the trees. The vertices of the trees are the 50 states, DC and NYC. The weights of the edges, which are specified by the pairwise vertices. Weights are the distances from the viruses' points of view. They were the traces left by the viruses when they conquered the environment. For each pair of factors, if the sum of the associated values for the factors is high, if some sense, the viruses have already found the shorter paths to propagate. Hence the weights could be regarded as the degree of concentration for the viruses’ propagation. This concept is further embodied by the following indicators:

if the edges refer to the COVID-19 cases or deaths, then the weights are decided by the reciprocal of 1 plus the sum of the values (cases or deaths) for the two vertices, where 1 is added to the denominator to avoid 0;

if the edges refer to the physical/geographical distances (this will be applied for our latter comparative analysis), then the geographical distances should simply serve the weights.

The idea is the viruses find the MST by their collective intelligence - which is witnessed by the cases and deaths they cause. Secondly, the processing is also specified in the following. By assuming the viruses survive based on the MST principle, we feed the 18 factors to tell apart which of them are stable being MST, i.e., the channels have formed w.r.t. the factor. This is done by separately analyse each factor and its time-series results. We exploit adjacency matrices to represent their MSTs and compute their Hamming distances. By observing these distances, one locates the stable ones (w.r.t. time) and performs further analyses on their relations between the stable ones with similar procedures, but the MSTs are computed for pairing factors. This would reveal the underlying similarities of the factors from the point of MST. Then we obtain the geographical distances between states (a network) to yield their MST, which is time-invariant, and calculate the distances between its MST and the ones of all the stable factors via Hamming distancing on their adjacency matrices.

Measures of variables

Based on the methodology mentioned in Section 2, we devise a set of procedures to implement the methodology. According to our analytical purposes, the implementations are divided into three parts: stability (or auto-correlation), correlation (or cross-correlation), and comparative implementations. The results of the implementations would be presented in Section 2.3.

stability implementation

This implementation is to locate the factors that form the MST is a stable and consistent way. There are several steps in the implementation.

Download the data related to daily COVID-19 cases and deaths, the data related to the area and populations for the states, DC and NYC - these will serve as the vertices of the tree (or network) and the fundamental bases for weights of edges. Let us name the downloaded data DATA.

Filter out the non-essential data, add and extract new added factors shown in Section 2; the sampled data lie between March 16, 2020 and January 13, 2021 (in total, 304 days). These data would serve our source data. Let us name the data DATAcomb.

Choose the factor(s) one is interested in and extract them to a set named DAY.

Assign the weights to the edges between the vertices (in our case 50 states, DC and NYC) and save them as .

Find the MST of and save them as .

Relate each MST to an incidence matrix and save it as . Collectively they are saved as .

Calculate the average Hamming distances between the MSTs for the consecutive days and save them as .

correlation implementation

Now we have located the factors whose formed networks (dynamical behaviours) largely or strictly follow the MST principle. Next, let us find the correlations between all the following stable factors:

Factor 1: total cases;

Factor 3: total cases over areas;

Factor 9: total deaths;

Factor 11: total deaths over areas;

Factor 13: total deaths over populations;

Factor 15: total deaths over total cases;

Factor 17: total deaths over population densities.

This is done by comparing the pairwise MSTs for the pairs of the factors. To implement it, the MSTs are converted into adjacency matrices and the Hamming distances are applied to measure their relations.

comparative implementation

Finally, we want to find the relation between the stable factors and the MST of geographical distances to see whether they are correlated. Again, to implement it, we convert the MSTs into adjacency matrices and apply the Hamming distance on them.

Model and data analysis

This section implements the processes in Section 2.2. The results are produced from the databases or websites (Area, 2021a; Area, 2021b; Population, 2021; NYC, 2021), including Google Map Api (Google, 2021) - for the physical distances between states, DC and NYC - fetched and run by R program 4.0.3. The nominal numbers (or vertices) are the states (including DC and NYC) presented in Table 1 , where the acronyms for the states could refer to Appendix A 7.1.

Table 1

Numbering the 50 acronymous states, District of Columbia and New York City.

1	2	3	4	5	6	7	8	9	10	11
AK	AL	AR	AZ	CA	CO	CT	DC	DE	FL	GA
12	13	14	15	16	17	18	19	20	21	22
HI	IA	ID	IL	IN	KS	KY	LA	MA	MD	ME
23	24	25	26	27	28	29	30	31	32	33
MI	MN	MO	MS	MT	NC	ND	NE	NH	NJ	NM
34	35	36	37	38	39	40	41	42	43	44
NV	NY	NYC	OH	OK	OR	PA	RI	SC	SD	TN
45	46	47	48	49	50	51	52
TX	UT	VA	VT	WA	WI	WV	WY

In the following analysis, we simply call the 50 states, DC and NYC 52 vertexes. The raw data are presented in blocks chronically, where each block consists of 52 rows - each of which is a vector, with length 18, of values for the 18 factors. An extract of the raw data is presented at Table 5 (or DATAcomb) in Appendix 7.2. The data are sampled from March 16th, 2020 to January 13th, 2021.

Based on Table 5 and the method of assigning weights to edges in Section 2, we construct the lengths for the edges. The results are presented at Table 6 in Appendix 7.3. The table is separated by 18 factors (from left to right) and 304 days (from top to bottom), i.e., 18 times 304 blocks - each block contains a 52-by-52 matrix whose elements are the weights for the edges between the 52 vertexes. For example, for factor 1 at day 1 (i.e., March 16th, 2020), the weight assigned to the edge between state AK (or 1 by nominal ordering) and AL (or 2 by nominal ordering) is ; and for factor 18 at day 304 (i.e., January 13th, 2021), the weight between Alabama (or 2 by nominal ordering) and West Virginia (or 51 by nominal ordering) is.

stability analysis

Based on the constructed daily network with respect to factors in Table 6, we run R programs to find their MSTs, which are represented by incidence matrices (due to space limitation, the visualised MSTs and incidence matrices are not presented explicitly here). Then we calculate their Hamming distances between these incidence matrices. The results are listed in Figs. 1, 2 and 3. In total, there are 18 plots spread over the three figures - each plot is a time series of Hamming distances between MST structures of the 52 vertexes - there are 303 pairs of consecutive MST structures. There plots show the auto-correlation of the given factor. One would expect that when a factor is stable, the time series of auto-correlation of the MST structures should substantially converge, which is at least witnessed by a negative trend of the time series. Stability analysis is intended to find the stable variables in terms of MST principle. It serves two purposes: to pin down those variables that are stable and their dynamics are self-explanatory by MST principle; and to filter out those inexplicable by MST principle. Stability analysis also reveals the coordinated behaviours or collective intelligence (or strategies) among the viruses. The results are further analysed in Section 3.

correlated analysis

From Figs. 1, 2 and 3, we could then locate the factors that (almost) strictly follow MST principle (to allocate their resources of the SARS-CoV-2 colony). The stable and consistent factors of the 18 ones are located and collected in the set: factor 1, factor 3, factor 9, factor 11, factor 13, factor 15, factor 17. Then we compute their pairwise Hamming distances of the converted MSTs (or incidence matrices). The results are presented in Figs. 4, 5, and 6. There are plots spread over these figures. Each plot is a time series of Hamming distance between factor-based structures. It shows the cross-correlation between two factors. This indicator shows the similarities between allocating strategies across different factors. Furthermore, the average Hamming distances between the factors are calculated and presented in Table 2 . Each value in the cell is the sum of all the 304 Hamming distances of the corresponding factors, and then divided by 304 to yield the average Hamming distance. It represents the closeness between different evolutionary or propagating strategies of the viruses. The dynamical parts of propagating behaviours are revealed via MST modelling.

Table 2

Cross-Hamming distance for MSTs of stable factors and geographical MST.

	factor 1	factor 3	factor 9	factor 11	factor 13	factor 15	factor 17
factor 1	0	67.77	67.86	67.78	69.67	75.31	87.82
factor 3	67.77	0	15.68	0.07	5.37	9.17	101.22
factor 9	67.86	15.68	0	15.69	13.44	21.47	90.19
factor 11	67.78	0.07	15.69	0	5.38	9.17	101.20
factor 13	69.67	5.37	13.44	5.38	0	10.38	98.61
factor 15	75.31	9.17	21.47	9.17	10.38	0	100.20
factor 17	87.82	101.22	90.19	101.20	98.61	100.20	0
GeoDis	99.53	101.22	90.19	101.20	101.90	100.59	100.02

comparative analysis

We use Google Map Api, together with R program, to fetch and process the physical distances between states, DC and NYC. Then the minimal spanning tree is computed and listed in Table 3 . This table could be further converted into an incidence matrix. This incidence matrix is then used to compare with all the incidence matrices of all the factors, and all the 304 days, derived from MST principle. Such comparison is performed by measuring their Hamming distances. Again one could average out all the 304 Hamming distance for each factor. The result is presented at the last row in Table 2. These values show the relation of the viruses propagation and geographical distance. The low values will indicate that the propagation route complies with the geographical distance; on the other hand, the higher values would indicate that the viruses’ propagation is not based on the minimal geographical distance, in other words, there exists other media that shall help the propagation.

Table 3

MST for physical distances between states.

2−>26	3−>25	3−>26	4−>1	5−>34	6−>33	6−>46
7−>35	8−>47	11−>10	11−>42	12−>8	13−>15	14−>27
16−>17	17−>30	18−>44	19−>5	20−>41	21−>9	21−>12
21−>40	23−>50	24−>29	25−>15	25−>16	28−>47	30−>6
31−>20	31−>48	32−>9	33−>4	34−>46	35−>32	36−>32
37−>18	38−>16	39−>14	41−>7	42−>28	43−>29	43−>30
44−>2	45−>38	47−>51	48−>22	49−>39	50−>15	51−>37
52−>6	52−>27

The average Hamming distance between MSTs of the stable factors and MST of physical distance is computed and appended in the last row of Table 2. One could observe that there is little correlation between the stable factors and the physical-distance factor.

Results

This research focuses on understanding the behaviours of SARS-CoV-2 via delving into their collective intelligence. By identifying their CI via MSTs, we obtain a series of main results as follows:

From Figs. 1–3, the auto-correlated stability analysis for the time series of the structures shows that the factor total cases or deaths are overwhelmingly stabler than the ones of new cases/deaths. This indicates the viruses' colonisation is already in a mature stage - the totality of their colonisation or kingdom is almost complete - despite the fact that their attacking structures or methods are still dynamically evolving and expanding. This might also suggest that they probably would coexist with human begins and compete with us for quite a long time.

From Figs. 1–3, one also observes that the dynamical structures for new cases over areas and over human population are already in the polar stages, this might indicate their attacking strategies on based on some learning processes; moreover, one inspects the new cases over human population densities and finds out there is a high volatility of the dynamical structures - this might indicate the colonisation is still interacting and competing with with the environmental factor: human population densities.

From Fig. 2, the structures for total deaths over total cases is stabler than the ones for new deaths over new cases - this indicates the long-term strategy for viruses' colonisation is well regulated and managed, but the short-term strategy is to keep on exploring various possibilities for transmission.

From Figs. 4–6 or Table 2, one finds the best correlated pair of factors w.r.t. MST is “total cases COVID-19 over areas of states” and “total deaths of COVID-19 over areas of states”. This indicates the viruses' density is the top priority for their collective decision making in constructing their viruses' kingdom. It also shows the underlying colony of the viruses is already very mature in forming their attacking and propagating techniques;

From the last row of Table 2, it has shown the transmitted paths does not follow the MST of physical distance. It shall indicate they spreading is not conducted in a rippling way, but with a widespread transmission - an indication that other environmental factors like wind or climate change might contribute to the widespread (Coccia, 2020d, 2020e, 2021b). But our results probably hint a step further that the widespread might also happen in some less polluted cities. This shall declare that a sole physical lockdown policy or social distancing might produce little effect on the transmission of SARS-CoV-2.

Conclusion

By observing viruses' collective intelligence through MST with various environmental factors from the viruses' point of views, we obtain a series of results regarding their correlation and propagating behaviours. These findings should help us adopt some environmental polices, health polices and social policies in decision making. For environmental policies: The finding also suggests the viruses' transmission has little to do with geographical distance, i.e., the spread of viruses might resort to other media, not mainly through human-to-human or distance-related pollution. This might indicate there might exist other environments with sparse human population that incubate the viruses colonies. This is also indirectly witnessed by the fact that the confirmed cases and deaths are highly correlated via the environmental factor area”. This indicates the densities of viruses matter, not the density of human population. This might imply they already form their communities and colonies that are largely independent of interaction with human environment. This might also hint we should terminate the viruses largely in the environment, i.e., their existence might be accompanied by other media or carriers. For health policies: From result One, there is an evidence that the viruses' colonisation is pretty stable. This shall indicate that the viruses might coexist with us for some years to come and might evolve or mutate by its own approaches. In our future health policy, no matter at an individual's level or a governmental level, one should keep track of their colonisation, their identities, and their communities via genetic decoding or multi-scale environmental research. We might need to find a way to live with them. For social policies: It might indicate a sole lockdown policy could not deter the spread of the viruses (Chen, 2021a). We probably also have to find out other hosts, other than human beings, that are also the carriers or the spreaders of the viruses. On the other hand, we probably should not rule out the concept of herd immunity when the vaccination backfires or fails to deliver what we expect.

For the future research, one could combine these findings with the micro-scale modelling of the virus behaviour to complete a whole dynamical picture of how SARS-CoV-2 survives, evolves and expands its kingdom. This might provide a better solution to contain or terminate their expansion. One could also include other environmental factors or combine our approach with other existing risk models for detecting the evolutionary behaviours or surviving decision making of SARS-CoV-2.

Lastly, we specify some limitations of this study. The weights, which indicates the allocation of resources, are susceptible to subjective assignment. For example, instead of assigning weights for the edges (states) by the reciprocal of sum of the associated cases or deaths for the vertexes, one could adopt the reciprocal of the absolute difference, which symbolises the potential energy for the viruses' spread, between the associated cases or deaths for the vertexes. Other forms of reasonable weight assigning approaches could also be included. The optimal or suitable weight assigning could be largely resolved via machine learning techniques or other criteria. Moreover, we simply use the macro-indicator MST to represent the viruses' behaviours and intelligence - this simplification should be refined when the coordinating behaviours or hierarchies of the viruses’ colonisation are revealed or explored in latter research. Another limitation is the follow-up work. Since this research is performed when the global vaccination is still underway, a follow-up research should be also performed to compare the shift of their MST structures after the majority of people are vaccinated.

Author contribution

The author declares that he is the sole author of this manuscript and he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Table 4

Full Names for Acronyms

AK	AL	AR	AZ	CA	CO	CT
Alaska	Alabama	Arkansas	Arizona	California	Colorado	Connecticut
DC	DE	FL	GA	HI	IA	ID
District of Columbia	Delaware	Florida	Georgia	Hawaii	Iowa	Idaho
IL	IN	KS	KY	LA	MA	MD
Illinois	Indiana	Kansas	Kentucky	Louisiana	Massachusetts	Maryland
ME	MI	MN	MO	MS	MT	NC
Maine	Michigan	Minnesota	Missouri	Mississippi	Montana	North Carolina
ND	NE	NH	NJ	NM	NV	NY
North Dakota	Nebraska	New Hampshire	New Jersey	New Mexico	Nevada	New York
NYC	OH	OK	OR	PA	RI	SC
New York City	Ohio	Oklahoma	Oregon	Pennsylvania	Rhode Island	South Carolina
SD	TN	TX	UT	VA	VT	WA
South Dakota	Tennessee	Texas	Utah	Virginia	Vermont	Washington
WI	WV	WY
Wisconsin	West Virginia	Wyoming

Table 5

304 batches of daily raw data of 52 vertexes (states) against 18 factors

no.	date	state	factor 1	factor 2	⋯	factor 17	factor 18
1	03/16/2020	AK	1	0	⋯	0.00	0.00
2	03/16/2020	AL	29	7	⋯	0.00	0.00
3	03/16/2020	AR	17	1	⋯	0.00	0.00
⋮	⋮	⋮	⋮	⋮	⋮	⋯	⋮
50	03/16/2020	WI	53	26	⋯	0.00	0.00
51	03/16/2020	WV	0	0	⋯	0.00	0.00
52	03/16/2020	WY	3	0	⋯	0.00	0.00
1	03/17/2020	AK	3	2	⋯	0.00	0.00
2	03/17/2020	AL	39	10	⋯	0.00	0.00
3	03/17/2020	AR	24	7	⋯	0.00	0.00
⋮	⋮	⋮	⋮	⋮	⋮	⋯	⋮
50	03/17/2020	WI	88	35	⋯	0.00	0.00
51	03/17/2020	WV	1	1	⋯	0.00	0.00
52	03/17/2020	WY	11	8	⋯	0.00	0.00
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
1	01/13/2021	AK	49,203	406	⋯	545.95	4.85
2	01/13/2021	AL	410,995	3147	⋯	163.61	5.31
3	01/13/2021	AR	262,020	2467	⋯	197.72	3.07
⋮	⋮	⋮	⋮	⋮	⋮	⋯	⋮
50	01/13/2021	WI	558,020	2771	⋯	169.40	1.37
51	01/13/2021	WV	104,392	1189	⋯	56.59	1.25
52	01/13/2021	WY	48,289	217	⋯	234.62	0.00

Table 6

weights of edges for 52 vertexes (states) per day per factor

factor 1					⋯	factor 18
1	2	⋯	51	52	⋯	1	2	⋯	51	52
0.333	0.032	⋯	0.500	0.200	⋯	1.000	1.000	⋯	1.000	1.000
0.032	0.017	⋯	0.033	0.030	⋯	1.000	1.000	⋯	1.000	1.000
⋮	⋮	⋯	⋮	⋮	⋯	⋮	⋮	⋯	⋯	⋮
0.500	0.033	⋯	1.000	0.250	⋯	1.000	1.000	⋯	1.000	1.000
0.200	0.030	⋯	0.250	0.143	⋯	1.000	1.000	⋯	1.000	1.000
0.143	0.023	⋯	0.200	0.067	⋯	1.000	1.000	⋯	1.000	1.000
0.023	0.013	⋯	0.024	0.020	⋯	1.000	1.000	⋯	1.000	1.000
⋮	⋮	⋯	⋮	⋮	⋯	⋮	⋮	⋯	⋯	⋮
0.200	0.024	⋯	0.333	0.077	⋯	1.000	1.000	⋯	1.000	1.000
0.067	0.020	⋯	0.077	0.043	⋯	1.000	1.000	⋯	1.000	1.000
⋮	⋮	⋯	⋮	⋮	⋯	⋮	⋮	⋯	⋮	⋮
0.000	0.000	⋯	0.000	0.000	⋯	1.000	0.135	⋯	0.425	0.063
0.000	0.000	⋯	0.000	0.000	⋯	0.135	0.072	⋯	0.114	0.045
⋮	⋮	⋯	⋮	⋮	⋯	⋮	⋮	⋯	⋯	⋮
0.000	0.000	⋯	0.000	0.000	⋯	0.425	0.114	⋯	0.270	0.058
0.000	0.000	⋯	0.000	0.000	⋯	0.063	0.045	⋯	0.058	0.033
0.000	0.000	⋯	0.000	0.000	⋯	0.093	0.090	⋯	0.141	0.171
0.000	0.000	⋯	0.000	0.000	⋯	0.090	0.086	⋯	0.132	0.158
⋮	⋮	⋯	⋮	⋮	⋯	⋮	⋮	⋯	⋯	⋮
0.000	0.000	⋯	0.000	0.000	⋯	0.141	0.132	⋯	0.285	0.444
0.000	0.000	⋯	0.000	0.000	⋯	0.171	0.158	⋯	0.444	1.000