Embedding networks with the random walk first return time distribution
This work provides a new embedding method for network analysis, which is incremental as it builds on random walk theory to improve structural similarity capture.
The paper tackles the problem of node embedding in complex networks by proposing the first return time distribution (FRTD) of a random walk as an interpretable and mathematically grounded method, and it shows that this embedding outperforms manually designed graph metrics in network alignment tasks.
We propose the first return time distribution (FRTD) of a random walk as an interpretable and mathematically grounded node embedding. The FRTD assigns a probability mass function to each node, allowing us to define a distance between any pair of nodes using standard metrics for discrete distributions. We present several arguments to motivate the FRTD embedding. First, we show that FRTDs are strictly more informative than eigenvalue spectra, yet insufficient for complete graph identification, thus placing FRTD equivalence between cospectrality and isomorphism. Second, we argue that FRTD equivalence between nodes captures structural similarity. Third, we empirically demonstrate that the FRTD embedding outperforms manually designed graph metrics in network alignment tasks. Finally, we show that random networks that approximately match the FRTD of a desired target also preserve other salient features. Together these results demonstrate the FRTD as a simple and mathematically principled embedding for complex networks.