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Python library (C++ backend) for degree-preserving network randomization
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XSwap: Fast degree-preserving network permutation

XSwap is an algorithm for degree-preserving network randomization (permutation) [1]. Permuted networks can be used for a number of purposes in network analysis, including for generating counterfactual distributions of features when only the network's degree sequence is maintained or for computing a prior probability of an edge given only the network's degree sequence. Overall, permuted networks allow one to quantify the effects of degree on analysis and prediction methods. Understanding this effect is useful when a network's degree sequence is subject to biases. This implementation is a modified version of the algorithm due to Hanhijärvi et al. with two additional parameters (allow_self_loops and allow_antiparallel), which enable greater generalizability to bipartite, directed, and undirected networks.

  1. Randomization Techniques for Graphs Sami Hanhijärvi, Gemma C. Garriga, Kai Puolamäki Proceedings of the 2009 SIAM International Conference on Data Mining (2009-04-30) DOI: 10.1137/1.9781611972795.67

Usage examples

Permuting an edge list

>>> edges = [(0, 1), (1, 0)]
>>> permuted_edges, permutation_statistics = xswap.permute_edge_list(
        edges, allow_self_loops=True, allow_antiparallel=True,
>>> permuted_edges
[(0, 0), (1, 1)]
>>> permutation_statistics
{'swap_attempts': 20, 'same_edge': 10, 'self_loop': 0, 'duplicate': 1,
 'undir_duplicate': 0, 'excluded': 0}

Computing degree-sequence based prior probabilities of edges existing

>>> edges = [(0, 1), (1, 0)]
>>> prior_prob_df = xswap.prior.compute_xswap_priors(
        edges, n_permutations=10000, shape=(2, 2), allow_self_loops=True,
>>> prior_prob_df
   source_id  target_id   edge  source_degree  target_degree  xswap_prior
0          0          0  False              1              1          0.5
1          0          1   True              1              1          0.5
2          1          0   True              1              1          0.5
3          1          1  False              1              1          0.5

Choice of parameters

Bipartite networks

Bipartite networks should be indexed using the bi-adjacency matrix, meaning that the edge (0, 0) is from source node 0 to target node 0, and is not a self-loop. Moreover, bipartite networks should be permuted using allow_self_loops=False and allow_antiparallel=True.

Directed and undirected networks

For non-bipartite networks, the decisions of allow_self_loops and allow_antiparallel are not always the same. For undirected networks, set allow_antiparallel=False, as otherwise the edges (1, 0) and (0, 1), which represent the same edge, will be treated as separate. Antiparallel edges may or may not be allowed for directed networks, depending on context. Similarly, self-loops may or may not be allowed for directed or undirected networks, depending on the specific network being permuted.


The XSwap library includes Roaring Bitmaps (, available under the Apache 2.0 license (


Development of this project has largely taken place in the Greene Lab at the University of Pennsylvania. However, as an open source project under the hetio organization, this repository is grateful for its community of maintainers, contributors, and users.

This work is funded in part by the Gordon and Betty Moore Foundation’s Data-Driven Discovery Initiative through Grants GBMF4552 to Casey Greene and GBMF4560 to Blair Sullivan.

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