The network of collaborations in an open source project can reveal relevant emergent properties that influence its prospects of success. In this work, we analyze open source projects to determine whether they exhibit a rich-club behavior, i.e., a phenomenon where contributors with a high number of collaborations (i.e., strongly connected within the collaboration network) are likely to cooperate with other well-connected individuals. The presence or absence of a rich-club has an impact on the sustainability and robustness of the project. For this analysis, we build and study a dataset with the 100 most popular projects in GitHub, exploiting connectivity patterns in the graph structure of collaborations that arise from commits, issues and pull requests. Results show that rich-club behavior is present in all the projects, but only few of them have an evident club structure. We compute coefficients both for single source graphs and the overall interaction graph, showing that rich-club behavior varies across different layers of software development. We provide possible explanations of our results, as well as implications for further analysis.

1. Analyzing Rich-Club Behavior
in Open Source Projects
OpenSym 2019, the 15th International Symposium on Open Collaboration
Skövde, Sweden
Mattia Gasparini1, Javier Luis Cànovas Izquierdo2,
Robert Clarisò2, Marco Brambilla1, Jordi Cabot2
Politecnico di Milano1 Universitat Oberta de la Catalunya2

2. Introduction
• Git and Github data to analyze evolution,
success and management of Open Source
Software.
• Define developers behavioral patterns.
• Discover how collaborations between
developers work.
2

3. Problem
Statement
ANALYSIS OF
COLLABORATION
NETWORKS
COMMITS, ISSUES AND
PULL REQUESTS AS
SOURCES
DISCOVER PRESENCE OF
SPECIFIC COLLABORATION
STRUCTURES: RICH-CLUBS
3

4. Rich-club coefficient
• Graph structural property:
It represents the tendency of well-connected nodes (i.e.: hubs) to interact with other well-
connected nodes.
• Formulation:
𝜙 𝑘 =
2𝐸 𝑘
𝑁𝑘(𝑁𝑘 − 1)
𝜌 𝑘 =
𝜙(𝑘)
𝜙 𝑟𝑎𝑛𝑑𝑜𝑚(𝑘)
𝐸 𝑘: number of edges between nodes of degree greater or equal to 𝑘
𝑁𝑘: number of nodes with degree greater or equal to 𝑘
𝜙 𝑘 : rich-club coefficient
𝜌 𝑘 : normalized rich-club coefficient
4

5. Related Work
• Rich-club phenomenon for a specific project [2],
or for a single FLOSS community [3].
• Study of the presence of a rich-club effect
across the whole GitHub social network [4].
• Analysis on open source communities exploiting
email exchanges among participants [5].
5
[2] Weifeng Pan, Bing Li, Yutao Ma, and Jing Liu. 2011. Multi-granularity evolution analysis of software using complex network theory
[3] Guido Conaldi. 2010. Flat for the few, steep for the many: Structural cohesion and Rich-Club effect as measures of hierarchy and control in FLOSS communities
[4] Antonio Lima, Luca Rossi, and Mirco Musolesi. 2014. Coding Together at Scale: GitHub as a Collaborative Social Network
[5] Sergi Valverde and Ricard V. Solé. 2007. Self-organization versus hierarchy in open-source social networks

6. Case Study
6
Top-100 starred projects in 2016 on
GitHub
926K commits produced by 50K Git users
1.3M issues-related events generated by
118K GitHub users
280K pullrequest-related events
generated by 20K GitHub users

7. Analysis Pipeline
7

8. Data Collection &
Preprocessing
• Git repository cloning for
commits data using Gitana
• Github activities for issues
and PR activities querying
GHArchive
• Duplicity and clashing
problem
8

9. Graphs Construction
• Definition of 4 undirected graphs:
a. PR graph
b. Commits graph
c. Issues graph
d. Supergraph (a + b + c)
• Nodes: users
• Edges connect a pair of users if
they interacted on the same
element (issue, PR, file)
9

10. Graphs Example
Materialize PR graph (a) Materialize commits graph (b) Materialize issues graph (c) Materialize supergraph (d)
10

11. Rich-club Coefficient
Calculation
• Calculation using algorithm
implementation included in
NetworkX6
• Normalized coefficient
𝜌(𝑘): rich-club effect
relevant if 𝜌 𝑘 > 1
• Discard networks for which
randomization fails
11
[6] https://networkx.github.io/documentation/stable/reference/algorithms/rich_club.html

12. Rich-club Coefficient
Results
• 60 projects have a defined
coefficient for the
supergraph.
• Each graph presents a rich-
club effect, since 𝜌 𝑘 > 1
for some 𝑘

13. Materialize7:
Rich-Club
Supergraph
Coefficient
Maximum normalized coefficient (k =
49) corresponds to maximum club effect
with nodes of degree at least 49.
13[7] https://materializecss.com

14. Materialize:
Supergraph
14

15. Swift8:
Rich-Club
Supergraph
Coefficient
15[8] https://swift.org/

16. Swift:
Supergraph
16

17. Rich-club Coefficient Results
17

18. Maximum coefficient distribution
• Distribution of the maximum
rich-club coefficient for each
type of graph across the studied
projects.
• Mean value around 1 for issues
and commits graphs
coefficients: weak rich-club
presence.
• Mean value around 1.4 for PR
graphs coefficient: strong rich-
club presence.
Further insights
18

19. Multi-club users
• 25 over 60 projects present a set
of users belonging to multiple rich-
clubs.
• Distribution of multi-club users
across the 25 projects.
• Developers form community with
strong influence in each project
level.
Further insights
19

20. Conclusions
First systematic evaluation of the rich-club
behaviour on open source projects:
• 60% of projects shows rich-clubs in the
supergraph, mostly with a slight effect.
• Rich-club behavior could undermine the open
paradigma, but phenomeon requires further
analysis.
• Strong rich-club presence in PR graphs may
reside to criticality of the activity.
• 25 over 60 projects have users belonging to
multiple rich-clubs.
20

21. Future Work
Weighted rich-club
coefficient
Rich-club effect at module
and ecosystem level
Time dimension to
highlight temporal clubs
21

22. Questions?