Source-linked AI summary
Who is the best player ever? A complex network analysis of the history of professional tennis
Filippo Radicchi
TL;DR
The paper asks how professional tennis players can be ranked from the history of their matches without relying on arbitrary external criteria. It constructs a directed, weighted contact network and applies a diffusion-based prestige score, identifying Jimmy Connors as the best player in its overall ranking while also evaluating years and playing surfaces.
Problem
The paper addresses the need to assess professional tennis players and tournaments without relying on externally introduced quality criteria.
Method
The authors represent matches as weighted directed contacts and rank players using a PageRank-like diffusion process over the resulting network.
Results
Jimmy Connors is identified as the best player in the overall ranking, and the method evaluates leading players by year, decade, and playing surface.
Takeaways & Limitations
The prestige score provides a network-based assessment whose importance is determined by the competitive contact structure rather than external criteria.
Abstract
from arXiv · showhide
We consider all matches played by professional tennis players between 1968 and 2010, and, on the basis of this data set, construct a directed and weighted network of contacts. The resulting graph shows complex features, typical of many real networked systems studied in literature. We develop a diffusion algorithm and apply it to the tennis contact network in order to rank professional players. Jimmy Connors is identified as the best player of the history of tennis according to our ranking procedure. We perform a complete analysis by determining the best players on specific playing surfaces as well as the best ones in each of the years covered by the data set. The results of our technique are compared to those of two other well established methods. In general, we observe that our ranking method performs better: it has a higher predictive power and does not require the arbitrary introduction of external criteria for the correct assessment of the quality of players. The present work provides a novel evidence of the utility of tools and methods of network theory in real applications.
I. INTRODUCTION
The paper applies network theory to professional tennis by representing matches as a contact network and ranking players through diffusion. It argues that the resulting prestige score has higher predictive power than established methods and avoids externally imposed quality criteria.
- Diffusion algorithms rank network vertices according to their centrality in a process occurring on the graph.
- The study extends network-based ranking to professional tennis using all matches played from 1968 to 2010.
- The tennis prestige score is more accurate and has higher predictive power than established professional-tennis ranking schemes.
- The method assesses player and tournament importance from the competitive contact network rather than introducing external criteria.
A. Data set
The dataset contains professional matches collected from the ATP website, covering 3,640 tournaments and 133,261 matches between January 1968 and October 2010. Tournament participation became relatively stable after the early years, averaging about 300 players with small fluctuations.
- 3,640 tournaments and 133,261 matches were collected from ATP records covering January 1968 to October 2010.
B. Network representation
The paper models each tennis match as a directed, weighted contact between opponents. The resulting network supports flexible analysis across periods and surfaces and exhibits complex structural features associated with social networks.
- A match in which player i defeats player j creates a directed edge j → i.
- The edge weight w_ji equals the number of times player j loses to player i.
- The network can be restricted to selected players, periods, or playing surfaces for different analyses.
- Top players tend to beat many weaker opponents while also playing many matches against other top players, consistent with a Matthew effect in career longevity.
C. Prestige score
The prestige score treats ranking as diffusion through the weighted player network. Player scores combine weighted redistribution of prestige, uniform redistribution, and a correction for dangling nodes, with scores normalized across all players.
- Each player starts with one unit of tennis prestige, which flows through weighted network connections.
- The prestige score P_i is the fraction of total prestige located on player i at the diffusion process’s steady state.
- The weighted term redistributes each vertex’s credit to neighboring nodes in proportion to connection weights.
- The parameter q controls the relative contribution of the score equation’s terms, including uniform redistribution among all players.
- A correction term prevents dangling nodes with zero out-strength from becoming sinks in the diffusion process.
1. Single tournament
For a single tournament represented as a binary rooted tree, prestige depends on victories and the diffusion parameter q, with closed-form solutions for the resulting scores. Low q strongly favors tournament winners, while q=1 distributes prestige uniformly.
- Network structure and score: A single tournament forms a binary rooted tree with 2^ℓ players, and a player's prestige score P_r depends on the number r of matches won.The score equations can be solved recursively and normalized using the number of players winning each possible number of matches.
- Network structure and score: The score P_r combines an equal contribution shared by all players with prestige accrued from matches won, scaled by q.The recursive relation is P_r=(2−q)^rP_0, with P_0 fixed by normalization.
- Limiting cases: For q=1, prestige is uniformly distributed as P_r=2^-ℓ for every number of victories r.At q=1, diffusion is absent and every player receives the same score.
- Limiting cases: For sufficiently low q, the winner's score is about two orders of magnitude larger than the score of first-round losers.The winner's score grows exponentially with tournament length ℓ.
- Tournament length: With ℓ=7, Grand Slam tournaments receive relative importance two or four times larger than ATP tournaments typically having ℓ=6 or ℓ=5.This difference follows from the exponential dependence of the winner's score on tournament length.
III. RESULTS
The prestige-based ranking identifies Jimmy Connors as the best player in tennis history and produces rankings for years, decades, and playing surfaces. Compared with victory counts and established rankings, it accounts for career context and anticipates subsequent annual leaders.
- Overall ranking: Jimmy Connors is identified as the best player in tennis history by the prestige ranking.The top 10 includes nine players who reached number one in the ATP ranking.
- Overall ranking: Rafael Nadal ranks 24th by prestige but 40th by victories, while Björn Borg enters the overall top 10 despite ranking 17th by victories.The prestige score elevates players whose career importance is not fully represented by accumulated victories, including Nadal’s recent-career relevance and Borg’s shorter career.
- Comparison with victories: Prestige rankings compare overall rank with victory-based rank for the top 30 players in either list across matches from 1968 to 2010.The comparison is shown in Figure 4, panel a.
- Comparison with victories: Prestige ranking is strongly correlated with victories, but incomplete careers penalize active players because they have not yet played all career matches.The bias can be reduced by constructing networks from matches played within the same year.
- Annual ranking: Prestige ranking identifies Rod Laver as the best player from 1968 to 1971 and highlights dominance periods for Ivan Lendl, Pete Sampras, and Roger Federer.The ranking is compared with ATP and ITF annual lists.
- Decades and surfaces: The best players by decade are Jimmy Connors, Ivan Lendl, Pete Sampras, and Roger Federer, while Guillermo Vilas, Jimmy Connors, and Andre Agassi lead clay, grass, and hard courts.These results come from decade-specific and surface-specific contact networks.
IV. DISCUSSION
The paper situates diffusion-based ranking within network science and discusses both its broader applicability and limits. It presents prestige rank as a complementary method rather than a replacement for established tennis rankings.
- Network-based ranking: Diffusion rankings use network topology rather than only local properties or predetermined criteria, extending methods such as PageRank beyond web pages.The paper presents citation networks as another application of diffusion-based ranking.
- Limitations of diffusion ranking: Citation-network rankings can be biased by disciplinary differences and by the one-way temporal structure of citations.Publication and citation averages depend on research-topic popularity, while papers cite older work but not future work.
- Limitations of diffusion ranking: Corrections for citation-network biases, such as age-decaying citation weights, may be reasonable but remain ad hoc.The paper identifies these adjustments as potentially arbitrary.
- Role of prestige rank: Prestige rank is intended to corroborate established ranking techniques rather than replace methods optimized over many years.The paper describes it as a novel method with a different spirit.