Divergent opinions resulting from polarization are widespread across various fields, including economics, technology, and politics, and are often considered the genesis of disagreement among people. Numerous studies were therefore devoted to achieving complete consensus. However, as we show here, polarization is inevitable when individuals exhibit the self-confidence effect in interpreting social pressure, a psychological mechanism that drives adaptive consolidation of biased opinions. We also demonstrate that polarization, which merely reflects opinion distribution, does not necessarily cause high-level connected disagreement, which is highly related to the random walk normalized Laplacian (RWNL) of a network. By developing a networked dynamical model incorporating the self-confidence effect and analyzing the boundaries of opinion patterns, we find that polarization and its connected disagreement have different formation mechanisms. The level of connected disagreement intensifies as the number of unstable eigenmodes of the RWNL increases, a process greatly influenced by network topology. This finding helps us elucidate how connected disagreement evolves across different social networks and, more importantly, provides insights into developing effective modulation strategies to mitigate the level of connected disagreement when eliminating polarization is difficult.

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• Emergence of patterns
• Patterns in complex systems
• Social dynamics
• Social networks
• Bifurcation analysis
• Collective behavior models