Cycles and Holes in the Big Tree

Question

Now that everyone (hopefully) is familiar with circles, let me try to blow your mind once again in this New Year with yet another circular concept : cycles.

You have certainly met cycles when building connections, in your CC7 or beyond. Cycles are just closed paths, starting from a profile, following connections and getting back from when you start. Small cycles are frequent in endogamic populations. 

And since our Big Family is just a larger endogamic population, cycles of all sizes are everywhere. Actually our Big Tree is not a tree at all, but rather a mesh or intertwined cycles. Secondary connections create new cycles, smaller and smaller, and that's how we all get closer and closer to each other.

Large cycles are like holes waiting to be mended. That could be a new kind of fun challenge for connectors. At the end of the page are some examples of quite large holes. Can you find largest ones? Can you help mending them?

Happy circular and cyclic 2024!

Comments

mdr Bernard!  My tree looks like spaghetti when I get into earlier ancestors.  

Answer 1

Today I studied what you wrote on the space page and it is quite complex, but nice idea. 

I was thinking if I could implement some functionality internally in ConnectionFinder and I guess I could but first I must know what to implement. 

Ideal approach would be to exclude only one connection, not the whole person, but that is not possible at the moment. That way I could get the smallest circle for each connection. For most relations it would return length 2 due to parents for brothers, children for spouses, the other parent for children,... actually due to the nuclear family. But on bigger circles there is most likely one connection with nuclear family of only 3 people. Actually this approach wouldn't work if all profiles in the circle have nuclear family bigger than 3.

Excluding all relations of a person will find more holes.

I did write on WT+ a function to check for a circle on each triplet on the path (excluding the middle person). 

And for each circle I checked the geodesic distances. Here I was thinking that there is only one check needed and that is for any other triplet (ignoring the middle person) on the circle not to return a shorter path. If the path is shorter, that means that ignored person in original check has a shorter connection to another profile in the circle.

I hope it is clear what I mean.

Here is the example of the WT+ function for the path from me to Bernard.

https://plus.wikitree.com/function/WTPathCycle/PathCycle.htm?WikiTreeID1=Vatant-1&WikiTreeID2=Trtnik-2

Comments

@Bernard, I have been thinking about cycles and holes as circles, since all are round, and you have spoken of the diameter.  So I think of the number of the nodes in the cycle as the circumference. And we could even think of any shorter distance found between nodes in a non-geodesic cycle as chords within the circle.  But I agree than finding non-ambiguous jargon can be difficult.

I wasn't sure why I was looking at cycles touching the path from you to me.  The people in the cycles near you meant nothing to me, and the people in the cycles near me probably meant nothing to you.

I agree that for performance reasons it would make sense to limit the tool to two profiles within 7 or fewer degrees of each other.

@Aleš: I re-ran the report for Bernard and myself this evening and there was no timeout this time:

Time taken: 25.3 s

As regards the cryptic text, maybe something like "0 to 15: 14" would be easier to understand than "0->15=14", or maybe it's worth having a wider column to include more explanatory text.

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Vocabulary, continued. Mind you, will never find me fed up of discussing terminology.

Diameter is defined for every subset E of a metric space as the maximal distance between two points of E. We use it for the whole of the Big Tree, see the definition in the Outer Rim page. https://www.wikitree.com/wiki/Space:The_outer_rim_of_the_global_tree#Distance.2C_eccentricity.2C_diameter.2C_radius.2C_and_central_nodes

Note that if diameter is intuitive for a cycle, its other parameters are not. In a geodesic cycle, all nodes have the same eccentricity (in the induced subgraph), so the radius of the cycle is equal to its diameter, and all nodes are central. This flies in the face of the usual definitions of radius and center, but it is the consequence of the formal definitions.

Chord is defined in graph theory and matches your intuitive definition; See https://mathworld.wolfram.com/CycleChord.html

See also the related definition of chordless cycle. Any geodesic cycle is chordless (but not the other way round).

I recommend this paper : https://sciendo.com/pdf/10.21307/joss-2020-002 which gives in its introduction a very clear definition of all those concepts, with examples

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q4: I corrected the column name (antipodal distance(s)) and the distance format to 0 to 3: 3

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That seems OK, thx!

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q1: I implemented caching and reduced the timeout to 10 seconds.

So if you get timeouts in the results, refresh the page. It will continue to query CF for additional 10 seconds. So for long paths you may need do do a few refreshes.

After all CF requests are cached, the results are shown instantly (0.6 seconds for path of length 30).

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q3: I was checking the cycles on the path from Bernard to Paddy and my assumption is not correct. Even if next triplet is part of the same geodesic cycle, it is not necessary the shortest cycle. See the 4th cycle (This cycle is not geodesic - Jestin-22 to Boucher-5204 excluding Jestin-37) on the path where the triplet is on the previous cycle, its cycle ignoring the middle profile is different.

So this isn't really an optimisation. It would just make improvements to the display of the results, but I don't really see a point in additional complications.

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Thx Aleš for all the fine tune-up of the function. Tell me when you consider it's stable and I can add it to the FSP with examples and usage recommendations.

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I have no other planned changes at the moment.

Here is the official entry point on WT+

https://plus.wikitree.com/default.htm?report=disp3

And here is the help. https://www.wikitree.com/wiki/Help:WikiTree_Plus#disp3 Once you write your help, I will copy it here.

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Added the WT+ section

https://www.wikitree.com/wiki/Space:Cycles_and_Holes_in_the_Big_Tree#Finding_cycles_with_WikiTree.2B

Do I need to add a screenshot?

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Made a new post announcing the WT+ implementation.

https://www.wikitree.com/g2g/1688632/have-you-checked-the-wikitree-function-to-find-cycles

Answer 2

That's a lot to get your head around at 11.30pm, I get the concept, but not the detail.

Comments

Try it in your own circles  - with your tomorrow morning coffee, which should help . 

Here is a path of length 8, from you to your paternal grandfather, skipping your father. It goes through your maternal grandmother and back.

https://www.wikitree.com/index.php?title=Special:Connection&action=connect&person1Name=Burgess-6632&person2Name=Burgess-3729&relation=0&ignoreIds=16308627

Adding back your father to close the path, you get a cycle of 10 profiles ... hum ... endogamic community, am I right?

I see 3 private profiles in the path, but you should see them in clear, I guess you are their PM.

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Reading now what you write on your profile about your Cornish relatives. "it is a web, not a family tree". Exactly spot on. You should find among them things similar to the mesh of my Brezhoneg relatives, their Celtic cousins. Although I have not yet found direct connection between Breizh and Cornwall - and common ancestors are too far away in the past, well beyond the documentary horizon!

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I think I might need several coffees before tackling that, but it will be interesting with the Cornish family, lots of cousin marriages, and other interlocking families, and Keast/Lancaster/Hoskins or Burgess/Shackel/Keast could be worth a try.

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Good luck with that. Say you come from down under to London flying through Shanghai and Dubai, and back through  San Francisco and Tokyo. That's a cycle. No more than a round trip.

Answer 3

My first try for a fairly big hole was to check te path between myself and another Swedish member for whom my shortest connection path for a long time went through America. By now we do have an all-Swedish connection path of 24 degrees. If I strike out his father I get another 24-degree-path - but the profiles from myself all the way to #17 are the same in both paths. Not much of a round trip.

If instead I strike out my father, I get a round trip - but this second leg of the trip is 32 degrees. Not very useful.

I guess this tells me that the network between my part of Sweden and his part of Sweden is still not very well filled in.

Comments

Heh, yes, it clearly shows I'm not quite understanding yet, but I will try again.

Am I still misunderstanding something when I present these two paths?
Path 1

Path 2

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Looks better, but some devilish details. I've been through all that with the first examples I made myself, and so was Eva, don't worry

Problem with those two alternative paths is that Ekberg-58 is the son of Ekberg-59, so there is a shortcut there already.

So I would skip Boström-67, and try those two paths :

https://www.wikitree.com/index.php?title=Special:Connection&action=connect&person1Name=Ekedahl-46&person2Name=Ekberg-59&relation=0&ignoreIds=

and, skipping Lundberg-268, an alternative one step longer

https://www.wikitree.com/index.php?title=Special:Connection&action=connect&person1Name=Ekedahl-46&person2Name=Ekberg-59&relation=0&ignoreIds=13684567

Putting those together could work. Too bad they make a odd cycle of length 41, so you have to check 41 antipodal pairs to make sure it is geodesic. Looks challenging. Let me check, stay tuned.

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Looks GOOD

I have not checked ALL the antipodes pairs, but more than half of them, I'm pretty confident the remaining ones are also at distance 20. But I let you the thrill of checking them.

Actually I conjecture that for an odd cycle, you have just to check half of antipodal distances (+1) to make sure the cycle is geodesic. I will be completely sure of that when I have written a formal proof, because graphs are treachery beasts. You can be quite sure of something until you enter the "Here be Monsters" territory.

P	AP	d
Ekedahl-46	Ekberg-59	20
Ekedahl-45	Ekberg-58	20
Ekedahl-44	Ekberg-57	20
Cederqvist-9	Samzelius-1	20
Melin-216	Lundholm-25	20
Melin-218	Fjaestad-2	20
Melin-220	Fjästad-5	20
Arosenius-4	Fjæstad-39	20
Arosenius-6	Christensen-3833	20
Arosenius-7	Christensen-4401	20
Arosenius-17	Fjaestad-28	20
Söderberg-359	Hallén-98	20
Bergström-779	Lundqvist-343	20
Bergström-778	Hallgren-133	20
Dahlgren-100	Hallgren-136	20
Leeb_Lundberg-2	Hallgren-134	20
Lundberg-332	Trägård-7	20
Lundberg-312	Trägård-5	20
Lundberg-308	Trägårdh-20	20
Lundberg-268	Trägårdh-19	20
Ekberg-59	Ekedahl-91	20
Ekberg-58	Ekedahl-46	20
Ekberg-57	Ekedahl-45	20
Samzelius-1	Ekedahl-44
Lundholm-25	Cederqvist-9
Fjaestad-2	Melin-216
Fjästad-5	Melin-218
Fjæstad-39	Melin-220
Christensen-3833	Arosenius-4
Christensen-4401	Arosenius-6
Fjaestad-28	Arosenius-7
Hallén-98	Arosenius-17
Lundqvist-343	Söderberg-359
Hallgren-133	Bergström-779
Hallgren-136	Bergström-778
Hallgren-134	Dahlgren-100
Trägård-7	Leeb_Lundberg-2
Trägård-5	Lundberg-332
Trägårdh-20	Lundberg-312
Trägårdh-19	Lundberg-308
Ekedahl-91	Lundberg-268

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I'm checking! On my phone!

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All remaining distances were 20. Yes, exciting.

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Wow!!

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Amazing! Such a large cycle in a quite dense part of the tree goes against all my expectations, like the largest ones found by Shawn Ligocki's program. More of the same will lead me to completely reconsider my views on the Life Cycle of Cycles.

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Actually I conjecture that for an odd cycle, you have just to check half of antipodal distances (+1) to make sure the cycle is geodesic.

Here is a counterexample : take a cycle of length 5, and add a single diagonal. Then 4 antipodal distances (out of 5) are equal to 2, but the cycle is not geodesic. This generalizes to any length (odd or even), so you always have to check all antipodal distances.

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Indeed Julien! I was about to post a similar example, a cycle of length 9 with one antipodal path of length 3. So my conjecture was wrong ... dommage. 

Note : to be added to the FSP.

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BV: "Such a large cycle in a quite dense part of the tree goes against all my expectations"

Well, Sweden just isn't as densely WikiTreed as you expected.

Answer 4

Answering just so i can find this later. Interesting.

Comments

Thx! Looking forward for your comments!

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I thought i would check the largest hole distance examples after making several interconnections of USBH branches as part of the 1880 census project. And making many interconnections during weekly challenges. But noted none of the largest holes appeared to be USBH examples. However, the top example in the largest hole table at 95 is down to 45. 61 Strudwick is down to 36. 54 Goodrich to Hayworth is down to 19. Just checked a few. Just an fyi.

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Thanks! I've been busy otherwise, so did not look back at this for a while. Certainly time to double-check, and update the table.

Answer 5

This is fascinating.

I did a quick look at my connections with my 2xggps, and if I look for a path that doesn't go through my grandparent, on my dad's side the next closest connections are 15 to 18 degrees, while on my mom's side they're 8 to 10. This makes sense, since my dad descends from widely scattered families, while most of my mom's ancestors from the 1750s on lived within a fifty-mile radius of my grandparents' hometown.

Comments

Indeed Sharon, you got an important point : the small size of cycles is an instant indicator of endogamic areas.

Answer 6

Thanks, Bernard. With the example of looking at how I'm connected to my grandfathers without going through my Mom and Dad, you may have given me a tool to figure out if/how the Hinson families on both my maternal and paternal sides are connected. Just in time for the Connect-a-Thon; down the rabbit hole I go

Comments

A cycle can indeed look like a (rabbit) hole. Never thought about that, good metaphor.

And the larger the hole, the more rabbits you are likely to find inside. Good hunt!

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Gave me also the idea to find a cool image to illustrate the page. Searching "hollow tree" yields a lot of those. I just asked the webmaster if I could use that one : https://futuretreehealth.com.au/wp-content/uploads/2023/03/hollow-tree.jpg

Future Tree Health writes on their web : "We are building knowledge network for all things trees."

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I LOVE that image.

Answer 7

I made a spreadsheet to create a .bat file to make the process of checking the distance between antipodes more efficient.

Each of the 17 lines in the .bat file is of the form

"C:\Program Files\Mozilla Firefox\firefox.exe" https://www.wikitree.com/index.php?title=Special:Connection&action=connect&person1Name=Knapp-181&person2Name=Waldron-202&relation=0&ignoreIds=

When I run the file, Firefox opens 17 new tabs, as expected.

However, all seventeen open at
https://www.wikitree.com/index.php?title=Special:Connection
apparently ignoring the first & in the URL and everything after that.  The form is pre-populated with my own WikiTree ID and a prompt to enter an "ID for Person 2".

Manually copying all 17 URLs from the .bat file to the Firefox address bar works perfectly, but is a lot more time-consuming.

Is this a flaw in my code, or some sort of security measure at the WikiTree server end?

Might it work differently with a different browser?

And is this an "Ask Aleš" question?!

Comments

I'm not the one able to answer that ... but a security measure blocking all kind of batch query from the same machine won't surprise me - I would do that if I was Aleš.

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I have recently been locked out of a MyHeritage account twice for similar efforts to save time, so I had better be careful! Short .bat files are OK at MyHeritage, but the very first line appeared to ring alarm bells at WikiTree.

I tried to put something like

timeout /t 10

between the command lines at MyHeritage, but it didn't work there and assuredly wont work here.

I'd better see what Aleš says before experimenting further! I've drawn his attention to this discussion.

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Yeah, better be careful. I remember being locked out from Wikipedia for similar attempts (don't remember exactly what I tried).That was years ago, playing with Omnimark, a really cool tool for data crunching. Unfortunately locked in an old machine which I don't dare re-starting (no more battery etc), and with a business licence I can't transfer to a new machine.

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I think it is a problem of the .bat file and ? or & in the parameters. Try putting the parameter in double quotes like

"C:\Program Files\Mozilla Firefox\firefox.exe" "https://www.wikitree.com/index.php?title=Special:Connection&action=connect&person1Name=Knapp-181&person2Name=Waldron-202&relation=0&ignoreIds="

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Thanks a million, Aleš.  That worked perfectly.

In case anyone else wants to try it, here is the Excel formula that worked:

=CONCATENATE(CHAR(34),"C:\Program Files\Mozilla Firefox\firefox.exe",CHAR(34)," ",CHAR(34),"https://www.wikitree.com/index.php?title=Special:Connection&action=connect&person1Name=",B2,"&person2Name=",C2,CHAR(34))

It was also interesting to note that out of 17 cases the connection finder eleven times found the clockwise path around the cycle as I had drawn it and six times found the anti-clockwise path.  I wonder what determines that ratio!

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Another thing I noticed when looking at these 17 cases is that the WikiTree Browser Extension added ": 4 branches (5-2-7-4)" or equivalent in just two of the 17 cases.  This WBE feature has always seemed to work only intermittently, with no obvious pattern as to when it does or does not work.

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I did discover that it works every time if you hit the Find Connection button on the Connection Finder page.

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I think you're correct, Tommy - but it vanishes again if I hit Ctrl-R to reload the page.  And it seems to work if I hit the <Enter> key with the cursor in the "ID for Person 2" box. But it almost never works if I click one of the "23 degrees from Theodore Roosevelt" links at the bottom of either the Connection Finder page or a profile page.

Answer 8

Fascinating stuff!

This post prompted me to look for the smallest cycle including both of my parents (ignoring their own descendants etc.).

I remember the first time that I met someone who is connected to both of my parents in this way.  Her husband is related to my mother and her uncle's wife was related to my father.  That gives me an even cycle of 18 in my offline tree, passing by 3 branches, and including both of my parents, but also including two living individuals and eight other individuals not yet on WikiTree.  Unfortunately, my offline software (Ancestral Quest) doesn't have a connection finder, so I cannot easily check the shortest distances between the antipodes.  In fact, I don't know of any desktop genealogy software with a connection finder, and I wish that Aleš would create such a standalone program that would search in GEDCOM files for shortest connections!

Next, I turned to WikiTree, to look for an alternative geodesic cycle including my parents.  I found this even cycle of 32, but it goes through four private people (none of them managed by me) so that again I cannot check all the shortest distances between the antipodes.  The hyperlink starts with my mother and goes through her father to her antipode.  X-ing out my mother's father goes round the circle in the reverse direction through my own father to the same antipode, 16 degrees in either direction.

My parents are the only overlap between the offline cycle of 18 and the online cycle of 32.

X-ing out the private people in the online cycle led me to a slightly larger even cycle of 34, where I could (with some difficulty) compare all the 17 pairs of antipodes, all confirming that 17 is the shortest distance.

So the length of this geodesic cycle is 34, passing by 5 branches (2-6-15-4-7), and including only deceased individuals.

My parents are the only overlap between the WikiTree cycle of 32 and the WikiTree cycle of 34.

The offline cycle and the WikiTree cycle of 34 have an overlap of 6 people.

I will be on the lookout for both smaller and larger cycles.

Comments

Thanks for this, Paddy. Looking forward for further results. Indeed private profiles along the line are an issue. I have the same one with the 22-cycle passing by my parents, but the PM is a cousin I am in contact with, so I could check those.

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I've introduced myself to the profile manager of the private profiles, with a link to this discussion, and asked if she will add me to the trusted lists.

Answer 9

I found only one non-trivial geodesic cycle passing through one of my main profiles (by this I mean profiles such that I manage all the profiles of a path from them to myself). It has length 10, and crosses several times the Atlantic between La Rochelle and Québec, around 1700. Start from this, then remove the middle profile Auboyneau-1. If I remember correctly, it is the result of a (short) re-connection made by Greg Lavoie.

Answer 10

I wish I understood this concept better. I never took Calculus in high school for a reason :-) However, I do understand that we need to increase the number of connection pathways, especially for profiles that have only one connection to the tree. For USBH, a lot of our profiles fit that description. But our lack of connection is due to lack of profiles. So we have to keep increasing the number of profiles while at the same time increasing our % of connection.

Comments

Emma, I know the problem with USBH profiles, we have already discussed that, but useful reminder in this context.

Regarding your love of maths, come on, this has nothing to do with Calculus! Graph theory at this level needs only basic arithmetic.

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Only basic arithmetic! Ha! 

Answer 11

Wow Bernard, and all the others who worked on this!

I read the whole thing, it will have to soak in, maybe forever... until I might understand a part of it,

I was looking for the definition of "hole" and finally believe it is just 'missing inner triangles', correct?

In all that, I noticed one typo: "goes through sapce-time regions of WikiTree" under the heading "A minimal cycle of length ... 95 !"

Thanks for all you do!

Comments

Hello Rick. Thanks for your comments. I have a parallel feedback from Shawn pointing at some potentially confusing things in the FSP.

The first example given (the double marriage) says : "this cycle cannot be split in smaller ones (triangles)". This is confusing. This cycle is of length 4, so any diagonal shortcut would indeed create two triangles.

In the example of length 7 given in a further section, the diagonal connection creates a triangle and a cycle of length 6. But it could as well have been, using another diagonal, a cycle of length 4 and a cycle of length 5.

So, it's not "no internal triangles" but more generally "no smaller internal cycle", or "no internal shortcut path" with the caveat that "internal" has to be defined clearly.

As shown by several further examples, in the FSP and in this discussion, checking if there is indeed no shortcut needs work for larger circles!

I have to introduce less particular examples to illustrate the definitions. All the comments here are really welcome, and I appreciate those, like yours, which genuinely point at unclear points.

And thanks for catching the typo. Will fix it.

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Bernard, I used your phrase "passing by 3 branches" above, but it confused me at first. This is probably your direct translation from French, but I think that English speakers might prefer to say "passing through 3 branches" or just "comprising 3 branches". The phrase "passing by" makes me think of a "by-pass", which doesn't go "through" anything! In fact, when checking the antipodes, we are effectively just looking for by-passes in this latter sense.

I wonder what others think about the terminology. You might want to bear this in mind when editing the otherwise excellent FSP.

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Good catch, Paddy. Yes, in French we would say "passant par" which has not the semantic overload "passing by" has in English.

I had been searching for neat vocabulary in graph theory literature, but even mathematicians are not completely consistent in their vocabulary use.

And I would like to introduce new terms I've not found at all in the literature, like how to name those "almost geodesic" cycles, which are just slightly "flattened", but have at least "antipodal distance" equal to its "diameter" ... a lot of slippery terms for a non-native speaker.

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Your English is better than that of many native speakers, myself included!  And infinitely better than my French!!

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Hi Bernard

I am still struggling with the intricacies of cycles.  On the FSP, you define "geodesic" as follows:

"A cycle is said "geodesic" if a shortest path between two profiles of the cycle is part of the cycle."

Should this definition be the other way round?  E.g.:

"A cycle is said to be "geodesic" if the shorter part of the cycle connecting any two profiles of the cycle is a shortest path between those two profiles."

Or shouldn't the definition at least include the word "any" (and the phrase "said to be")?

If there are two completely different shortest paths of equal length between two (non-antipodean) profiles in the cycle, is the cycle still geodesic?

For example, my grandfather Jack (Waldron-816) and his brother married my grandmother and her sister Lillie (McNamara-3846) respectively.  Hence there are two different paths of length 2 between Jack and Lillie, who is both his brother's wife and his wife's sister.  Does this mean that any cycle going through Jack and Lillie is not geodesic (apart from their own 4-cycle)?  Or does "a shortest path" in your definition mean "one of the shortest paths"?

I also came up with the following results, which I hope are correct:

* if a WikiTree profile has only one first-degree connection, then the only geodesic cycle containing that profile is the trivial 2-cycle with the first-degree connection;

* if a WikiTree profile's only first-degree connections are his or her two parents, and the parents are married, then the only geodesic cycle containing that profile is the 3-cycle with the two parents (this applies to my own profile);

* if a WikiTree profile's only first-degree connections are his or her children, then the only geodesic cycles containing that profile are the 3-cycles with any two of the children; and

* if a WikiTree profile is part of a geodesic cycle of length greater than 3, then he or she must have a first degree connection who is not one of his or her married parents or a first degree connection who is not one of his or her children.

Would checking these conditions initially for each profile on the path from you to me, for example, significantly speed up Aleš's new toy for finding geodesic cycles overlapping with that path?

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OK. I found several definitions of a geodesic cycle in the literature. Hopefully, they are all equivalent, and they are equivalent to the one used in the FSP page, which tries to look less "hard-math", but maybe at the price of being unclear.

"A finite cycle C in a graph G is called geodesic if, for any two vertices x, y ∈ C, the length of at least one of the two x–y arcs on C equals the distance between x and y in G."

"A cycle in a graph is geodesic if the distance of each pair of nodes on the cycle coincides with their distance restricted on the cycle"

Seems to me none of those is crystal-clear. If I was to put it in "hard-math" language, I would put it that way.

Let G be a graph and C a cycle in G. Let's call dG(x,y) the distance of two nodes in G, and dC(x,y) their distance in the induced subgraph G[C]. The induced subgraph is a graph containing all the nodes of C, and all the edges joining nodes of C in G.

C is geodesic if and only if for any pair (x,y) of nodes of C, dC(x,y)=dG(x,y)

See https://en.wikipedia.org/wiki/Induced_subgraph

If the cycle is (v1,v2, ...,vn) of length n, we can compute dC from the values of indices in the cycle.

dC(vj,vk) = Min( |k-j|, n-|k-j| )

We can go on frightening people away. The intuitive definition "a geodesic cycle is a cycle with no shortcut inside" seems more suitable for the casual WikiTreereader.

To your question : if there is an alternate path of same length between two antipodes, is the cycle still geodesic? I would say yes. Suppose you have a geodesic cycle of legnth 6 (v1,v2,v3,v4,v5,v6), and an alternate geodesic path (v1,w2,w3,v4) from v1 to its antipode v4. Then we have three distinct geodesic cycles with antipodes v1,v4, the original one plus (v1,v2,v3,v4,w3,w2) and (v1,w2,w3,v4,v5,v6)

For the rest : I've put the 2-cycles out of the definition, because the edges are not distinct. In other words, I want the cycles to be Eulerian (each edge is traversed only once).

The other propositions seem correct. Not sure checking what you propose would shorten of complexify the algorithm.

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Finally an answer I can comprehend - still leaves me wondering "Why" - do the fascinating discussions relate to the hidden workings of Connection Finder or is there some great mystery solved by this graph-based approach?

An aside - the many DNA matches for any (person's) test are often the only clue revealed about where person X may fit into the seeable/known tree of person Y  - I use that lateral scan (eyeball only as bulk data -sniffing blocked on DNA sites) to trace potential family groups, very inefficient and time-consuming as data transfer to a graphing tool is a bit like building sandcastles with a teaspoon...