Where is This Leading Me: Stationary Point and Equilibrium Analysis for Self-Modeling Network Models

In this chapter, analysis methods for the dynamics of self-modeling network models in relation to their network structure are presented. In particular, stationary points and equilibria are addressed and related to the network structure. It is shown how such analyses can be used for verification purposes: to verify whether an implemented network model used for simulation is correct with respect to the design description of the network’s structure. An always applicable method is presented first. It is based on substitution of state values from simulations in stationary point or equilibrium equations, which can always be done. In addition, methods are presented that are applicable for certain groups of network models, where the aggregation is specified by combination functions for which equilibrium equations can be solved symbolically. As shown, these methods cover cases of self-model states for adaptation principles such as Hebbian learning for mental networks and Bonding based on homophily for social networks. In addition, such methods are shown to cover cases where the combination functions for aggregation satisfy certain properties such as being monotonically increasing, scalar-free, and normalised. The analysis for this class of functions used for aggregation also takes into account the network’s connectivity in terms of its strongly connected components. This provides a class of functions which includes nonlinear functions but in contrast to often held beliefs, still enables analysis of the emerging network dynamics as well as linear functions do. Within this class, two specific subclasses of nonlinear functions (weighted Euclidean functions and weighted geometric functions) are addressed. Focusing on them in particular, it is illustrated in detail how methods for equilibrium analysis as normally only used for linear functions (based on a symbolic linear equation solver), can be applied to predict the state values in equilibria for such nonlinear cases as well. Finally, it shown how a stratified form of the condensation graph based on a network's strongly connected components can be used in equilibrium analysis.