Symmetrydriven network reconstruction through pseudobalanced coloring optimization
Abstract
Symmetries found through automorphisms or graph fibrations provide important insights in network analysis. Symmetries identify clusters of robust synchronization in the network which improves the understanding of the functionality of complex biological systems. Network symmetries can be determined by finding a balanced coloring of the graph, which is a node partition in which each cluster of nodes receives the same information (color) from the rest of the graph. In recent work we saw that biological networks such as gene regulatory networks, metabolic networks and neural networks in organisms ranging from bacteria to yeast and humans are rich in fibration symmetries related to the graph balanced coloring. Networks based on real systems, however, are built on experimental data which are inherently incomplete, due to missing links, collection errors, and natural variations within specimens of the same biological species. Therefore, it is fair to assume that some of the existing symmetries were not detected in our analysis. For that reason, a method to find pseudosymmetries and repair networks based on those symmetries is important when analyzing real world networks. In this paper we introduce the pseudobalanced coloring (PBCIP) problem, and provide an integer programming formulation which (a) calculates a PBCIP of the graph taking into account the missing data, and (b) optimally repairs the graph with the minimal number of added/removed edges to maximize the symmetry of the graph. We apply our formulation to the C. elegans connectome to find pseudocoloring and the optimal graph repair. Our solution compares well with a manually curated groundtruth C. elegans graph as well as solutions generated by other methods of missing link prediction. Furthermore, we provide an extension of the algorithm using Bender's decomposition that allows our formulation to be applied to larger networks.
 Publication:

Journal of Statistical Mechanics: Theory and Experiment
 Pub Date:
 July 2022
 DOI:
 10.1088/17425468/ac7a26
 arXiv:
 arXiv:2111.07821
 Bibcode:
 2022JSMTE2022g3403L
 Keywords:

 network reconstruction;
 optimization over networks;
 network dynamics;
 Quantitative Biology  Quantitative Methods;
 Mathematics  Optimization and Control;
 Physics  Biological Physics;
 Physics  Data Analysis;
 Statistics and Probability
 EPrint:
 doi:10.1088/17425468/ac7a26