Fig. 1: Examples of high and low polarization and its general connection to frequencies of balanced and unbalanced cycles.

The two networks depict bill co-sponsorship relations in the U.S. Senate. Positive ties (blue) link senators, who tended to promote the same bills together more often than by chance, and negative ties (red) correspond to those who collaborated less often than at random (see “Results: Polarization in the U.S. Congress” for the data description and a detailed analysis). The four triad diagrams depict four possible strongly (un)balanced undirected 3-cycles. Positive ties are marked with blue and negative with red dashed lines. a 96th Congress (1979–81, Carter administration) was a period of low polarization with frequent positive between-party and negative within-party links. b 114th Congress (2015–17, Obama administration) featured high polarization with in-group ties being almost exclusively positive and out-group ties negative. c Schematic depiction of the relationship between relative frequencies of (un)balanced cycles and low polarization, which is characterized by comparable frequencies, or more rarely by a majority, of unbalanced cycles (note that, even though only 3 cycles are depicted, the general relationship pertains to cycles of all lengths). d Relationship between (un)balanced cycles and high polarization, which implies that there is a clear majority of balanced cycles.