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Consider a graph with vertices being people (in some region), and make an edge if one person pass another closer than say 1.5 meter during say one week. (Such a graph might be thought a kind of useful for modelling epidemics on graphs (papers, igraph) ). It is quite a similar to much studied "social networks". Of course, such a graph would much depend on the region and probably time period, but still we might hope for some universality and graphs for big cities and "normal" time periods, might be similar.

Question 0: Have such or similar graphs been studied ?

Question 1: What is rough estimate of an average vertex degree of such a graph ? I.e. how many persons pass your 1.5 meter neighbourhood during 1 week ? Any ideas are welcome.

Pursuing an analogy with other social networks, one may think that degree distribution has power form: "that is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as P(k) = C*k^l+... ". I.e. graph is scale-free network, which seems was rather unexpected feature of such graphs uncovered in 90-ies (it is not true for Erdos-Renyi model). It somehow reflects that there are quite many vertices with extremely large number of edges, which might be true for our situation also, because imagine supermarket cashier - thousands persons pass nearby him, or public person who shake many hands. For example, for citation graph exponent estimate is 1.7 and 2.1 for social graphs (see MO302559 ).

Question 3: If the power law is true what can be the estimate of the exponent ? I.e. what is "l" in P(k) = C*k^l+... ?

Question 4: What can be the mathematical random graph model for such graph - Barabasi-Albert, Watts-Strogatz, etc... ?

PS

Bonus Question: If confinement-lockdown-quarantine happens, what happens with the graph above ?

The only thing I was able to google which at least somehow close the question is the following claim: "According to a new U.K. study, you will shake 15,000 hands in your lifetime. " (link).

PSPS

Technical off-topic question: What are current technical abilities to get such information ? Does "Google/Apple" or whatever have such an information ?

Operators of mobile network clearly have information on clients positions, but the cell-size is about 500-1.5km in cities, so such an information would not be precise enough. Some people have tracking their positions turned on, for example, my daughter's cell phone provides me information on her position, via google service "family link", so in principle such information might be partially available to Google/Apple. It is was widely discussed that Israel's parliament allowed temporary mobile phone tracking of infected people, probably similar things has been done on China and South Korea. I vaguely remember there has been some volunteer projects where such information has been collected by volunteers, I cannot a provide a link at the moment. It seems having such information (made anonimized) would be helpful for modeling epidemics.

PSPSPS

See also:

Suggestions for reducing the transmission rate?

Relevant mathematics to the recent coronavirus outbreak

https://stats.stackexchange.com/search?q=covid-19

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• Concerning the first of the two questions in the title, "How many persons pass your 1.5 meter neighbourhood during 1 week?"
Here is a graph from Mixing patterns between age groups in social networks, showing daily average number of contacts per person in each age group. The data is based on the EpiSims social contact network, which forms the basis of most studies in the literature on this topic. A "contact" happens when two persons are in the same "room", where the size of the room was constructed such that the individuals would come close enough to transmit a disease. The 1.5 meter distance seems a reasonable proxy for this. If I multiply the number of contacts by 7 (to convert from daily to weekly average) I would estimate that 140 is a representative number in the 20-50 age range. For the elderly it is about half that number.

• Concerning the second question in the title: For the EpiSims social contact network the power law exponent of the degree distribution is 2.8.

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    $\begingroup$ Thank you very much for your answer ! It is unexpected for me. I might a little remark that multiplying by 7 may not always be fully correct since we meet partly same persons everyday - family members, work-colleagues, but, of course, I do not think , it will change answer much. $\endgroup$
    – Alexander Chervov
    Commented Mar 27, 2020 at 20:48
  • $\begingroup$ If you would have some time please take a look on the question: cstheory.stackexchange.com/questions/47957/… may be you know something about it or can suggest something ... $\endgroup$
    – Alexander Chervov
    Commented Dec 2, 2020 at 21:40

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