Hello Alex. sorry for the delay!

I finally found some time to look into this problem. But as a word of caution: I am not sure whether I am understanding the respective publication correctly, and I do not have any experiences with graphs !!!

Here is what I tried:

ClearAll["Global`*"]
(* function for calculating the edge weight between two vertices v1,v2: *)
getEdgeWeight[v1_List, v2_List] := Module[{trans1, trans2},
  If[v1 == v2, Return[{Null, 0}]];
  {trans1, trans2} = Partition[#, 2, 1] & /@ {v1, v2};
  {UndirectedEdge[v1, v2], Total[Count[trans1, #] & /@ trans2]}
  ]

tt0 = WeatherData[Entity["City", {"Munich", "Bavaria", "Germany"}], 
   "MeanTemperature", {{2008}, {2018}, "Day"}];
temperatures = First@Normal@tt0["ValueList"];
temperatNums = Round@QuantityMagnitude[temperatures];
lenght = tt0["PathLength"];
vertexNameLength = Round[N@Sqrt[lenght]]; (* according to Ferreira et al *)
vertexList = Partition[temperatNums, vertexNameLength];
combs = Select[Flatten[Outer[getEdgeWeight, vertexList, vertexList, 1], 1], Last[#] != 0 &];
{conns, weigth} = Transpose[combs];
gr = Graph[conns, EdgeWeight -> weigth, ImageSize -> Large]

The resulting graph is highly connected, but the connections/edges are weighted; Mathematica can easily disentangle this:

CommunityGraphPlot[gr, ImageSize -> Large]

One still has to convince oneself of the fact that this has something to do with the periodicity of the data. So I plot the data with a colored background according to the graph communities:

grcomms = FindGraphCommunities[gr];
indx = Flatten /@ Map[Position[vertexList, #] &, grcomms, {2}];
colr = {Red, Yellow, Magenta};
prolog = MapIndexed[{colr[[First[#2]]], Rectangle[{1 + (#1 - 1) vertexNameLength, -14}, {#1 vertexNameLength, 30}]} &, indx, {2}];
ListLinePlot[temperatNums, Prolog -> prolog, ImageSize -> Large]

Does that help? In any case: You should check carefully what I did! Regards -- Henrik

Hi Alex,

Yes, strange, on Mathematica 11.3 I get the same results/graphs as in your notebook. For the above Mathematica 12.0 was used - maybe some algorithm for calculating graph communities has changed (i.e. improved).

Hi Henrik,

I found another paper. I thought it may be interesting to you, especially Fig3 and Fig4. Is it possible to investigate this approach to your city air temperature time series (for example).

Thanks, Alex

Attachments:

bracis14.pdf

Thank you Mohammad - but there is definitely no reason for any congratulations! I simply tried to follow the idea/concept in the above attached publication.

Dear Henrik,

I hope your Christmas is filled with joy this year!

I applied your program for the hourly river water temperature time series of the Skokomish river in the USA. If possible, please have a look at the notebook and let me know the output is correct or not?

Have a nice weekend

Attachments:

Henrik.nb

data.xlsx

Hi Henrik,

Please take a look at the attached manuscript. Studying periodicity using the complex network is a novel approach in water engineering, so I decided to apply this concept for 44 rain stations in the northern region of Algeria.If you remember, you have extracted a graph for one of the stations about two months ago (Figure 6). I need your help in providing graphs for all stations. We will write the section related to complex network and also the results and other details. May I send you data from all the stations?

It's my pleasure and honor to work with you.

Regards, Mohammad

Attachments:

Algeria_Manuscri...docx