In many complex systems, not only there are positive interactions promoting coordinated actions and activating collective behaviors but also prevalent are the negative interactions between agents suppressing and antagonizing them. For instance, in social networks people can be friends or foes. Such systems can best be modeled as “signed” networks. There had been considerable effort devoted to such signed interactions in social network literature; however, quantitative understanding of the generic impact of such signed interactions on network dynamics is still lacking.
Towards this goal here we investigate a simple dynamical process on signed network. Specifically we study a threshold cascade dynamics model based on Watts model [PNAS 99, 5766 (2002)] by generalizing it for signed networks. The model incorporates the effect of negative links and is used to apply various network-theoretical concepts and tools to quantitatively understand the signed network dynamics. We formulate the signed network in terms of a two-layer multiplex network with a positive-link layer (mean degree z+) and a negative-link layer (mean degree z−). In such signed multiplex network, the threshold cascade dynamics rule is formulated as follows (Fig. 1). Activations: an inactive node gets activated if i) the fraction of active nodes among its neighbors in the positive-link layer exceeds the prescribed threshold r, and at the same time, ii) there is no active neighboring node in the negative-link layer. Deactivations: an active non-seed node can be deactivated if either the fraction of active nodes among its neighbors in the positive layer drops to or below the threshold r or it encounters active neighbors in the negative-link layer.
Primary results of numerical simulations are displayed in Fig. 1. Firstly, the presence of the negative inter- actions (encoded by the negative-link mean degree z−) is found very effective in suppressing global cascade. As the negative links are introduced in the network, the cascade size ρ decreases promptly. While the cascade covers lesser fraction, it takes longer time to proceed, up to the point by which the depletion effect gets saturated at around z− ≈ z+. We will discuss in more detail other nontrivial effects of the existence of negative interactions to show the additional facets of complexity such as degenerate stationary states. We also address the effect of the degree correlations widespread in real-world signed networks to find their observable impact on cascade outcomes: For instance, the positive correlations between positive and negative degrees suppresses the global cascades compared to its uncorrelated counterparts, while the negative correlations tend to promote the global cascades. We anticipate this work could prompt the community to the study of dynamic processes on signed networks at large, which we believe will prove itself a fertile ground for yet another layer of complexity in networked complex systems.