100 Circles: A Geometry of The Tree

Profile managers: Eva Ekeblad [send private message] and Bernard Vatant [send private message]

Introduction : The Circles of The Queen

This is a family everyone should be somehow familiar with, namely the British Royal Family of the late Queen Elizabeth II Windsor (1926-2022) (hereafter QEII, with due respect).

Most Royal Family close members have a profile in WikiTree, and the application My Connections is showing them, ordered by their distance in "degrees" to QEII. Each "degree" in the "My Connections" page is what we will call here a "circle" of the Queen. The first circle, or degree, gathers 8 people around her : 2 parents, 1 sister, 1 spouse and 4 children. The second circle gathers the parents, spouses, siblings and children of members of the first circle, 41 people altogether. Grandparents, aunts and uncles, nieces and nephews, grandchildren, and in-laws of the initial profile. My Connections is limited to 1000 results and/or ten circles, a limit reached in the middle of the 5th circle for QEII. The WikiTree Browser extension installed allows to break through this limit. The more recent CC7 application (available from the Tree Apps menu) will list the seven first circles in a table.

Following degrees/circles are computed the same way. Those profiles who were already counted in the previous circles should not be counted twice. It needs the smart algorithm of the Connection Finder to achieve that. Computing the shortest path from QEII to any of the 33+ million (as of June 2024) connected profiles allows to place each of them in one single circle, in any given state of the data base. For example former US President Barack Obama currently belongs to the 19st circle (June 2024).

Counting the population of all circles beyond the seven first circles is possible, thanks to a query specially developed by Aleš Trtnik in late 2020 for the 100 Circles project provided we use it sparingly (it's quite heavy on the server).

Do we really get as far as 100 Circles? Almost ... as of Feb 2024, the furthest connected profiles are 87 degrees from QEII, but actually less than 5000 profiles (the so-called "Outer Rim") are further than 60 degrees. Over 99.5% of connected profiles are within 40 degrees, and over 50% within 20 degrees. The following diagram is showing the distribution of circles population as of February 2024.

Several comments on this figure

• The logarithmic scale allows to see details of the first and last circles.
• The growth in first circles is quite regular, with a very slight inflexion around C15.
• The decrease after the peak (most populated circle is C19) is also very regular.
• After C60, the erratic aspect of the curve corresponds to a very few branches.

In focus : more women !

We are comparing in this section the distribution of circles population for some other reference profiles. The list has changed over time, and in February 2024 women have taken over, famous ones or less so, spanning nine centuries :

• Aliénor (Eleanor) d'Aquitaine (abt.1124-1204), Queen of France and England, with a huge descendancy.
• Mary (Stewart) Stuart Queen of Scots (1542-1587) has been a longstanding "central node", with all connected profiles at less than 80 degrees from her. See Outer Rim page for more details.
• Alice (Bradford) Fitch (abt.1659-1745) a New England settler, member of the Mayflower families, is one of the two top connected profiles in WikiTree, along with her brother Joseph Bradford (1675-1747)
• Geneviève Roy (1697-1771) is the top connected woman among Nouvelle-France early settlers.
• Johanna Adriana (de Beer) Pretorius (1751-1797) is the top connected woman among South African profiles.
• Anna Larsdotter (1793-1875), from Sweden, is taking over her husband Olof Andersson (1793-1860), who has been featured in this page for years.
• Marie Anne Lannéval (1815-1883) was a miller in Gourin (Bretagne, France). With 17 siblings, 2 spouses and 16 children, her circles have a strong start with a C1 at 37.
• Emma (Davis) Schipp (1862-1949), Australia, has a remarkable symmetry in her distribution, probably due to the geographical diversity in her first circles.
• Jeanne Suzanne Gardahaut (1900-1992) sailed from France in 1920 to meet her fiancé in Seattle ...
• Creola Katherine (Coleman) Johnson (1918-2020), a USBH profile, was an amazing computer and important contributor in the first US spatial missions.
• Elizabeth Alexandra Mary Windsor (1926-2022), Queen of the United Kingdom of Great Britain and Northern Ireland for over 70 years
• Patty (Luker) LaPlante (b. 1950s) , USA, currently the second "most connected" WikiTreer. See Most-Connected Members.
• Léa Haupaix (b.1990s), active WikiTreer in the France Project.

Columns CC1 to CC20 give the cumulative population of circles. For example the column CC7 gives the total population of circles up to the 7th, in other words the number of profiles at a maximal distance of 7 degrees from the Focus. This is the same CC7 number as the Connection Counts appearing on member profiles.

Peak value is the distance of the most populated circle. In standard statistics vocabulary, it's called "Mode" of the distribution.

Mean value is the average of all distances to the Focus. Due to the "long tail" of the distribution, the mean value is always slightly greater than the peak value. As a reference, the average distance between two random connected profiles, based on samples constructed using Jamie Nelson's very cool application, can be assessed, as of July 2022, to be in the interval [22-25] with a confidence of 95%. See G2G discussion for more details.

e is the eccentricity of the profile, in other words the distance of the furthest profiles, or the radius of the greatest non-empty circle.

Former featured profiles included :

• Samuel Lothrop Esq (1622-1700), member of the Great Puritan Migration, was known before the project for having one of the smallest mean distance to every connected profile, and an impressively sharp growth of his first circles (over one million in the 10 first circles).
• Mary (Allerton) Cushman (abt.1617-1699), a Mayflower passenger, with a distribution quite similar to Samuel Lothrop.
• François du Toit (1664-1731) French Huguenot, one of the founder patriarchs of the Cape Colony (South Africa). His distribution presents the typical "South Africa Bump".
• Jean-Joseph Marie Vatant (1804-1875) was the original focus of the research, along with his Swedish counterpart Olof Andersson (1793-1860). They are respective ancestors of the two co-authors of this page. Similar in many ways, native from very endogamic rural areas, with large families.
• Ernest Banks (1931-2015), famous baseball player, selection of the USBH Project.
• Kevin Bacon, our CC7 reference profile.

Other profiles have been previously included at the beginning of the project in 2021 : Marie Mars (1689-1776), Adaline (Carlton) Van Wye (1826-1897), Hermann Alexander von Keyserling (1880-1946) chosen by their respective PM, involved in the original project team.

Shapes of the circles distribution

Are circles distributions really as different as they seem from the above table? The differences are driven by different parameters at different scales. The sum of the populations of all the circles will be the same for all profiles, because it is the total population of the connected Tree. (This is exactly true only if data are retrieved the same day, the Tree is growing by about 10k profiles every day).

• In the closest circles, the differences are mainly linked to the care taken of the Focus and close relatives by the local PMs, and of course the size of first few circles' families. Large families have potentially over 100 profiles in C2, between 300 and 400 in C3, and over 1,000 in C4. One important part of the work in progress here is to ensure that those first circles are as complete as possible. For our reference Focus Mary Stuart, like for many "Euro Aristos" such a work has already mostly been achieved, hence the great values for C4 and C5. Similar values are expected to be reached for the (far less notable!) Vatant-5 and Andersson-5056, for whom a systematic completion is under way.
• Differences in C10 and C20 are linked to the distance of the Focus to the (mostly American) bulk of WikiTree. For profiles such as Jean-Joseph Vatant, it takes up to the 15th circle to see a significant growth of the circles population, and the peak circle is currently C28. Shortcuts to the bulk are certainly yet to be discovered. (One important parameter in this case is the very low WikiTree adoption rate in France)
• In all cases, the population of the peak circle is about the same, a little less than 10% of the total Tree population. When put together on the same diagram, most distributions look globally quite the same, simply more or less shifted to higher distances, and the peak more or less sharp.

Nevertheless, differences appear when looking more closely at a variety of cases. For most profiles under study, the growth to the peak is steeper than the decline to the long tail. In many cases, a secondary "bump" appears beyond the peak, as for Patty (Luker) LaPlante below.

Some cases show at the opposite a remarkable symmetry of the distribution on each side of the peak, such as Emma (Davis) Schipp. The reasons of such differences are not known at this stage.

South African profiles typically show a quick start, up to 100k around C8, then a dip between C10 and C15. This is due to a heavily connected local cluster.

Outer Rim profiles, generally connected through a long and tortuous path and several bottlenecks, display a chaotic distribution over many circles. Of course, such a distribution is likely to collapse drastically whenever a shorcut is found.

• More hill shapes on space page Shapes in the Tree

How the circles grow

It's raining where it's wet

With WikiTree growth, the circles population is of course changing. The overall population of The Tree is growing at a rate of over 3 million per year. How is this growth distributed over the 100 circles?

The first few circles growth is of course driven by the activity of the local PMs. If they have done a good job, up to the second and third circles can be complete after months of work, and are not subject to significative changes afterwards.

In the "far circles" (C15 and beyond), growth is driven by the global, somehow random, activity of WikiTree, independent of local growth. Since it's always raining where it's wet, most of the growth happens in the already most populated circles. Moreover, new reconnections keep bringing back profiles down to lower circles, making the overall distribution sharper and sharper with time.

As an illustration, the following plot compares the circles population for Jean-Joseph Vatant in November 2020 and November 2022.

In this global picture the work done during the same period to systematically populate the first circles (up to C4) is just invisible. Actually this work had practically no impact on the global distribution, given the strong endogamy of those first circles. The few reconnections with a visible global impact have happened beyond C10.

Seven circles ... and beyond

Léa Haupaix, featured as our last reference profile, has joined WikiTree in January 2021 and her branch was first connected to the Big Tree a few months later. As for many French people, finding connection paths is not obvious, and the first ones are generally going through tortuous bottlenecks. With time, and a hard work on the first circles to grow her CC7, more connections have been found, and Léa's mean distance which was over 38 in 2021 has fallen below 35. Still far above the average, but following the good direction!

The following plot compares Léa's circles population since July 2021. The progress on first circles is well visible thanks to the logarithmic scale, as well as the progressive reduction of the gap around the 10th circle, and the global shift of the curve towards smaller values. Note also the chaotic behaviour of the "long tail" of profiles beyond C70, which would look the same for any reference profile, see The Outer Rim of the Global Tree for more on this.

For other examples, see also the 2020 page Bridges from Sweden, which turned out as a comparison between Olof Andersson and four other profiles, has been updated to show the distribution changes in the circle populations of the same five profiles - showing the contrast between profiles affected only by the general growth of the Big Tree and profiles where "reconnections" have been actively worked on.

A slow but steady "collapse" of circles

For all studied reference profiles the peak of the distance distribution is getting sharper and shifted to lower values. The following table shows the changes in mean distance over 3 years for a few reference profiles.

There is of course a limit to the growth of each circle, hence to the "collapse" of the distribution towards lower circles, but this limit is far from reached, for three main reasons :

• Most connected profiles have not completed even their first circle : parents, siblings, spouse(s) and children are still missing.
• Each new reconnection pulls profiles to lower circles, and the whole distribution to lower distances. Such collapse events can be spectacular for profiles still at far distances from the bulk of the Tree.
• Many people in the first circles are yet to be born, or not yet included because they are too recent, for privacy reasons.

This last point is fascinating to consider. For example the 12 great-grandchildren of QEII are not yet in WikiTree, and when they are they will add to her C3. Later on, hopefully, their children will add to C4 by 2040, and their grandchildren to C5 by 2070, the sixth generation will be born somewhere around the turn of next century. With a mean of 3 generations by century, C9 members will be added by 2200, etc. Even Mary Stuart, who might look quite ancient to most of us, has still more than two centuries to wait to see the last members of her 20th circle come to life. QEII belongs to her 12th circle only ...

The shape of things to come : Billion Profiles Single Tree

Based on current trends, one can conjecture what the circles distribution would look like when the Single Tree has passed the billion threshold. Such a figure seems a bold hypothesis, but keeping the current annual growth rate of about 15%, this would be achieved in less than 30 years! In the Billion Profiles Single Tree, it can be conjectured that for most connected profiles, the distribution would converge towards having a similar shape, with a peak in the 15-20 range, and a mean distance below 20. Differences would simply came from the population of the very first circles, Samuel Lothrop case would certainly stay as a reference of sharp growth. One can wonder how such figures are possible, but various estimations of the issue, based on the mean size of families, indicate a ballpark estimation of tenfold growth every 2 circles in the ascending part of this asymptotic distribution, once the 10,000 threshold is passed, somewhere around the 6th circle. A tenfold growth every 2 circles would give this kind of figures for cumulated population :

• CC6 : 10,000
• CC8 : 100,000
• CC10 : 1M (figure already passed by Samuel Lothrop)
• CC12 : 10M
• CC14 : 100M ...

... reaching half a billion around C15, which would be the peak circle (as it already is in Lothrop distribution)

All this is of course highly conjectural, but indicates that the shape of things to come is somehow already visible in the current state of affairs. But let the Tree grow, and let's keep gathering data!

Bottom line

One can ask what all this can bring to the WikiTree experience, the genealogist work and everyday life. Here come a few answers (non-exhaustive list).

• Shifting viewpoint from "my ancestors", "my family", "my blood" to a more peripheral vision, towards an attention to all four directions of kinship (parents, children, siblings, spouses).
• Systematic exploration and expansion of the first circles is likely to reach and go over your zone of genealogical comfort, discovering unexpected alliances and filiations, traveling further across time, space and social boundaries.
• As the original Jean-Joseph Marie Vatant and Olof Andersson cases show, even very endogamic first circles eventually expand towards every corner of The Tree. Checking how and when this happens is a good lesson in history of migrations and exogamy.

See also

Free-space pages:

• Cycles and Holes in the Big Tree
• Seven Circles Magic
• Beyond CC7
• Connection Facts and Figures
• The outer rim of the global tree
• Olof Andersson in the global tree
• Circles and Bridges of Suzanne Gardahaut (1900-1992)
• The 100 Circles of the philosopher Hermann Graf von Keyserling (space page in progress)
• Families cluster (About connections within family clusters)
• Bridges from Sweden - (About how to get from here to there)
• The long and winding path - About profiles with a long distance to the "center"
• 100 Circles of Marie Mars
• Closer - closer - closest - About profiles with short mean distance to the rest of the tree
• Paule, An XII de la République - a deep dive in rural endogamy (in French)
• The Ancestry of Europe's Royal and Princely Houses - Work in Progress
• Ten circles meet each other - a conjecture based on figures observed in the current distribution and growth of circles : average WikiTree profiles should be less than 20 degrees from each other.
• 100 Circles - more discussion - a discussion page on miscellaneous related issues .

G2G discussions:

• Follow the conversation on G2G under the tag "100_circles".
• Mathematical graph structure of global tree (Shawn Ligocki post, G2G May 2018)
• Who is the main or center profile of the global one world tree, anyway? (G2G May 2018)
• Marketing the 19th century (G2G October 2019) While a main purpose of the post was to suggest that most potential WikiTree members could find at least one of their 19th-century ancestors on WikiTree, a second conclusion that can be drawn from the statistics is that WikiTree is heavily weighted towards 19th-century profiles, and this has consequences for the location of the current global tree's center
• Scrap closeness: what is your most distant connection in the global tree? (G2G September 2020)
• Number of profiles at N degrees from a profile P0 (October 2020)
• 100 Circles: A Geometry of the Tree (November 2020)
• What shall we do with those Outer Rim branches? (December 2020)
• Dr Gates and WikiTree Challenge made us closer to each other (February 2021)
• One million more connected, how far are you from them? (February 2021)
• Reconnection challenge: Aberle, Württemberg, Germany, 19th century (March 2021)
• The Ancestry of Europe's Royal and Princely Houses - March 2021 Update
• Ten circles meet each other: a conjecture - May 2021
• Circles of HM Queen Elizabeth II A. Windsor - June 2021
• WikiTree and Network Theory - June 2021
• WikiTree Network Defined - June 2021
• Multiple paths to ancestors - June 2021

External links:

• WikiTree and Network Theory - Shawn Ligocki
• Shapes in the Great Tree - Bernard Vatant
• The original publication by Bernard Vatant under the title Cent cercles de Jean-Joseph (in French).