The Martialis Index: methodology

The Martialis Index gives each person in the martial.is database a score for their influence as a teacher. It uses only the recorded, sourced teacher–student relationships. This page explains the full method, so that you can check it.

The index is recomputed as the database grows. Scores, ranks and the figures that describe them change with it; the method on this page does not.

The index measures influence through documented teaching. It does not measure skill, fame, rank, or historical importance in general.

The method in five steps

1. Build the lineage. Each recorded teacher–student relationship is a link. Each link has a confidence level from its sources (see “Confidence” below).
2. Students give credit to their teachers. Each student gives half of themselves to their teachers and keeps the other half as their own. When a student has more than one teacher, the teachers share that half in proportion to confidence.
3. Credit flows up the lineage. A teacher also gets half of the credit that their students got from their students, and so on. Each generation counts for half as much as the one before. The total that a person gets is their reach.
4. Breadth. Reach that comes through many different students counts for more than the same reach through one student. We measure the effective number of branches that carry a person's reach, and multiply reach by its square root. The result is the raw score.
5. Publish. The raw score goes on a 0–100 log scale, where 100 is the highest raw score in the database (see “Score, rank and tier” below).

A worked example

A teaches B and C. B teaches D. All the relationships are confirmed.

• D gives ½ to B. B's reach is 0.5.
• B gives A one half of itself and one half of what it received: ½ × (1 + 0.5) = 0.75.
• C gives ½ to A.
• A's reach is 0.75 + 0.5 = 1.25. Of this, 60% comes through B and 40% through C.
• A's effective number of branches is exp(−(0.6 ln 0.6 + 0.4 ln 0.4)) = 1.96. Two branches of equal size would give exactly 2.
• A's raw score is 1.25 × √1.96 = 1.75. B's raw score is 0.5 × √1 = 0.5.

Confidence

Each relationship in the database has one of five confidence levels. The index treats each level as the probability that the relationship is real. The values come from Sherman Kent's scale of estimative probability (Kent, 1964):

A student's teachers share the student's credit in proportion to these weights. When the weights add up to less than 1, the teachers do not get the full share: the rest stays with the student. For example, a student whose only recorded teacher is doubtful gives that teacher 0.3 × ½, not ½. A refuted relationship gives no credit: the index leaves it out.

The type of source (primary or secondary) does not change the weight. Older lineages often have only secondary sources. A source-type weight would penalize them for how they were recorded, not for how they taught.

Why each part is there

Credit halves at each generation

A student is partly the product of their teachers, and partly their own work. We give these two parts equal weight: γ = ½. If a teacher got their students' full credit (γ = 1), credit would accumulate without limit at the top of every long chain. With a lower value, only direct students would count. γ is one of two free parameters. The “Stability” section shows how the ranking changes when it changes.

The branch correction

Without it, a person known through one famous student gets half of that student's whole reach, and can rank near that student. With it, such a teacher ranks below people who taught several lines. The correction does not say that these teachers were unimportant. It says that the recorded lineage passes through one person, and that person gets the credit for the spread that followed.

The effective number of branches is the Hill number of order 1 (Hill, 1973): the exponential of the Shannon entropy of the shares in the worked example. It is the standard way to say “how many equal-sized parts this is equal to”. It is the same idea as the effective number of parties in political science (Laakso and Taagepera, 1979). The branch exponent (½) is the second free parameter. It sets how much breadth counts compared with reach.

Relation to standard measures

Reach is Katz centrality (Katz, 1953) on the credit-sharing links. Katz centrality counts all paths into a node, and each extra step counts for less by a fixed factor (here γ = ½).

Reach is also very close to PageRank (Brin and Page, 1998) with a damping factor of 0.5, run from students towards teachers. The two rankings agree almost entirely. The small difference comes from weakly evidenced relationships, where PageRank always passes on the full share. The usual PageRank damping factor is 0.85. With that value, credit accumulates at the top of long chains, and teachers with only one recorded student rise near the top. This is why we use a lower value, and why we add the branch correction.

Earlier work that follows influence down mentorship lineages:

• Marmulla, Dickson, Wäsche and Brandes (2023): coaching lineages in Australian football, where a coach keeps part of their influence as their own. This is the same idea as γ.
• Rossi, Freire and Mena-Chalco (2017): a “genealogical index” for academic advisor–student trees.
• Malmgren, Ottino and Amaral (2010): how mentorship affects the later output of scientists.

Stability

A ranking is only useful if small changes do not overturn it. We test this in two ways.

Parameters. We compute the ranking for every combination of γ ∈ {0.35, 0.5, 0.7} and branch exponent ∈ {0, 0.5, 1}, and compare each pair with Kendall's τ (Kendall, 1938). The rankings agree closely across the whole grid, and the top of the ranking changes very little.

Data. We remove a random 10% of the relationships and compute again, 20 times, and count how much of the top 20 stays in the top 20. The current value is in the footnote of the rankings page.

Limits and known bias

• Only recorded relationships count. A famous practitioner with no recorded students has no score. The score shows what this database documents, not fame or popularity. Well-known people can score low, and less-known teachers with well-documented lineages can score high.
• The database is not uniform. Some arts and countries are researched in more depth than others. More recorded students give a higher score.
• Earlier figures score higher. Their lineages have had more generations to grow. The rankings page also shows each era separately.

What is not in the score

We tested these signals and decided not to use them:

• Number of different arts. A tradition with many named sub-styles would score higher than an equally large but unified one. The arts that a lineage reaches are shown on each profile, but they do not change the score.
• Depth of documentation. This measures how well a person is researched, not how influential they were. It is shown separately as the coverage dots (see below).
• Ranks and honours given to others. Few teachers have a recorded rank that they gave, and formal grading began only in the late 19th century.
• Possible identities. When two entries may be the same person, we score each separately and never merge them. Some of these hypotheses are disputed.

Score, rank and tier

The published score is 100 × log(1 + raw) / log(1 + max_raw). The highest-scoring person always has 100. The log scale makes differences between smaller lineages visible. The scale is relative, because the data is incomplete: an absolute number would suggest a precision that we do not have.

People with the same raw score share the same rank. A person with no recorded students has no score and no rank. Their profile says that no students are recorded yet: this is a gap in the database, not a judgement of the person.

One new relationship can move a person several places. Therefore the highest scores also have a tier: Foundational (85 and higher), Major (65 to under 85) or Notable (40 to under 65). The tier changes less often than the rank. Below 40 there is no tier: a label there would describe the person, while the score only describes their recorded lineage. Read the rank as approximate.

Data coverage (the 0–5 dots)

Each profile shows five dots for how complete the person's record is. Coverage is not part of the score. Thus a well-researched minor figure does not rank above a major figure with few records.

The dots are the points earned out of the points that apply, scaled to 0–5:

A living person has no death date, so the death-date check does not apply to them. We treat a person as plausibly alive when no death is recorded and they were born less than 110 years ago. When the record has neither a birth date nor a death, we cannot tell, so the check does not apply either. A recorded death without a date counts as a missing death date.

Coverage counts citations but does not judge their quality: three links to Wikipedia count the same as three primary archival sources. It is also not the same as the database's own quality checks.

Reproducing the index

The index is computed in the martialis-db repository, by pipeline/scripts/index.py (AGPL-3.0-or-later), each time the database changes. It needs only Python 3. In a copy of the repository, make index recomputes it, make index-report prints the top of the ranking, and make index-sensitivity repeats the stability checks above. The result, export/martialis-index.json, is published with each version of the database, and this site always shows the scores computed from the version that it displays.

References

• Brin, S. and Page, L. (1998). The anatomy of a large-scale hypertextual web search engine. Computer Networks and ISDN Systems, 30(1–7), 107–117. doi:10.1016/S0169-7552(98)00110-X
• Hill, M. O. (1973). Diversity and evenness: a unifying notation and its consequences. Ecology, 54(2), 427–432. doi:10.2307/1934352
• Katz, L. (1953). A new status index derived from sociometric analysis. Psychometrika, 18(1), 39–43. doi:10.1007/BF02289026
• Kendall, M. G. (1938). A new measure of rank correlation. Biometrika, 30(1–2), 81–93. doi:10.1093/biomet/30.1-2.81
• Kent, S. (1964). Words of estimative probability. Studies in Intelligence, 8(4). cia.gov
• Laakso, M. and Taagepera, R. (1979). “Effective” number of parties: a measure with application to West Europe. Comparative Political Studies, 12(1), 3–27. doi:10.1177/001041407901200101
• Malmgren, R. D., Ottino, J. M. and Amaral, L. A. N. (2010). The role of mentorship in protégé performance. Nature, 465, 622–626. doi:10.1038/nature09040
• Marmulla, G., Dickson, G., Wäsche, H. and Brandes, U. (2023). Coaching legacies: influence propagation through temporal social networks in the Australian Football League. Frontiers in Sports and Active Living, 5, 1172264. doi:10.3389/fspor.2023.1172264
• Rossi, L., Freire, I. L. and Mena-Chalco, J. P. (2017). Genealogical index: a metric to analyze advisor–advisee relationships. Journal of Informetrics, 11(2), 564–582. doi:10.1016/j.joi.2017.04.001