d-k-Anonymity

This repository contains code to measure anonymity in complex networks as proposed in [1] and used in [2]. Using this measure, vertices are partitioned into equivalence classes. Vertices are equivalent if they have the same structural position in their $d$-neighbourhood; parameter d can be used to set the radius of the neighbourhood. This framework uses Nauty [3].

The framework

• src: Directory containing source code
  • anonymity.cpp: Contains the main function and code
  • util.h: Includes the required libraries
  • graph: Contains files related to graphs
    • graphgen.cpp: Code to generate graphs and read graphs from input
    • graphutil.cpp: Functions useful for working with graphs such as get_neighbourhood or get_degree_distribution
    • twinnode.cpp: Functions to find twin nodes and add them correctly to equivalence classes
  • Measure: Contains files related to the anonymity measures
    • dk-anonymity.cpp: Code to partition vertices of a (possibly directed) graph based on d-k-anonymity
• examples: Directory containing example graphs

Installation

Before using this code, the Nauty framework should be downloaded from: https://users.cecs.anu.edu.au/~bdm/nauty/

• Move this git-directory (dkAnonymity) in the downloaded nauty directory (nauty27r3)
• Run the following commands:

./configure
make

Last tested with nauty version nauty27r3.

Compilation

In teh git-directory (dkAnonymity) run the following command:

make anonymity 

How to run

To run the code, at least one input graph should be given. Command line arguments are optional. Use the following command to run the code:

./bin/anon ./examples/example2 [-arguments]

For example:

./bin/anon ./examples/example2 -dir 1 -d 6

Command line arguments

Various command line arguments can be used to adjust the settings of the algorithm.

• -dir : Set graph mode to directed (note: cache is not implemented for directed)
• -d 5 : Set the distance (default 5)
• -h 3 : Set the heuristic / algorithm choice
  • 0 - Naive (slowest)
  • 1 - Iterative
  • 2 - Iterative + number of vertices and edges as heuristic (fastest, default)
  • 3 - Iterative + degree distribution heuristic (fastest)
  • 4 - Iterative + equivalence class distribution (not implemented for directed)
• -c : Turn off twin node check (default on)
• -cs 5 : Select maximum number of neighbour to check for twin nodes (default 5)
• -eq: Choose to print equivalence classes per iteration (default: off)
• -s 1: Choose which statistics to print
  • 0 - Print final statistics only
  • 2 - Print statistics per iteration (default)
  • 3 - Print per class split (Preceded by /, debug mode)
  • 4 - Print all statistics, including progress updates (Preceded by //, debug mode)
• -t: Print runtime for each canonical labeling computation (default: off)

Input graphs

Input graph files should be using the .dre format. Some examples are included in directory 'examples' Start with !n=number_vertices, then for each vertex 0 to n-1 vertex: edge1, target2, ...;. For the last vertex, end with . instead of ;.

Example example2:

!n=6
0: 3 4 5;
1: 3 4 5;
2: 3 4 5;
3: 0 1 2;
4: 0 1 2;
5: 0 1 2.

Examples

Folder examples contains some small toy examples to test the d-k-anonymity measure. For each test case "x" there are 3 files:

• testx: The edgelist that can be read by Nauty
• testx.group: The grouping that d-k-anonymity will make if d = diameter(G). Vertices on the same line are in the same group
• testx.png: A visualization of the graph with each node coloured according to the group it belongs in. Generated with Networkx in visualize.py

Note that test1 is an empty graph: networkx does not show these nodes. Therefore test1 is empty Examples can be generated with visualize.py, code to generate the examples are given in examples.sh

References

[1] de Jong, Rachel G., Mark P. J. van der Loo, and Frank W. Takes. "Algorithms for Efficiently Computing Structural Anonymity in Complex Networks." ACM Journal of Experimental Algorithmics (2023).
[2] de Jong, Rachel G., Mark P. J. van der Loo, and Frank W. Takes. "The effect of distant connections on node anonymity in complex networks." Scientific Reports 14.1 (2024): 1156.
[3] B. D. McKay and A. Piperno, “Practical graph isomorphism”, Journal of Symbolic Computa-tion, vol. 60, no. 0, pp. 94–112, 2014.