Partial balance in social networks with stubborn links

Structural balance theory affirms that in signed social networks with simultaneous
friendly/hostile interactions, there is a general tendency of evolving to reduce the
tensions. From this perspective, individuals iteratively invert their sentiments to reduce
the felt tensions induced by imbalance. Each agent in a signed network has a mixture
of positive and negative links representing friendly or antagonistic interactions and his
stubbornness about interactions. We define stubbornness as an extreme antagonistic
interaction that is resistant to change. In the current paper, we investigate if the presence
of stubborn links renders an impact on the balanced state of the network and whether or
not the degree of balance in a signed network depends on the location of stubborn links.
Our results show that a poorly balanced configuration consists of multiple antagonistic
groups. Both analytical and simulation results demonstrate that the global level of
balance of the network is more influenced by the locations of stubborn links in the
resulting network topology than by the fraction of stubborn links. This means that even
with a large fraction of stubborn links the network would evolve towards a balanced
state. On the other hand, if a small fraction of stubborn links is clustered in five
stubborn communities, the network evolves into an unbalanced state.
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