Abstract
It is notoriously difficult to predict the behaviour of a complex self-organizing system, where the interactions among dynamical units form a heterogeneous topology. Even if the dynamics of each microscopic unit is known, a real understanding of their contributions to the macroscopic system behaviour is still lacking. Here, we develop information-theoretical methods to distinguish the contribution of each individual unit to the collective out-of-equilibrium dynamics. We show that for a system of units connected by a network of interaction potentials with an arbitrary degree distribution, highly connected units have less impact on the system dynamics when compared with intermediately connected units. In an equilibrium setting, the hubs are often found to dictate the long-term behaviour. However, we find both analytically and experimentally that the instantaneous states of these units have a short-lasting effect on the state trajectory of the entire system. We present qualitative evidence of this phenomenon from empirical findings about a social network of product recommendations, a protein–protein interaction network and a neural network, suggesting that it might indeed be a widespread property in nature.
Highlights
Many non-equilibrium systems consist of dynamical units that interact through a network to produce complex behaviour as a whole
We show that for a system of units connected by a network of interaction potentials with an arbitrary degree distribution, highly connected units have less impact on the system dynamics when compared with intermediately connected units
We present qualitative evidence of this phenomenon from empirical findings about a social network of product recommendations, a protein–protein interaction network and a neural network, suggesting that it might be a widespread property in nature
Summary
Many non-equilibrium systems consist of dynamical units that interact through a network to produce complex behaviour as a whole. 1), St becomes more and more independent of sti0 until eventually the unit’s state provides zero information about St This mutual information integrated over time t is a generic measure of the extent that the system state trajectory is dictated by a unit. We show analytically that for this class of systems, the impact of a unit’s state on the short-term behaviour of the whole system is a decreasing function of the degree k of the unit for sufficiently high k. That is, it takes a relatively short time-period for the information about the instantaneous state of such a high-degree unit to be no longer present in the information stored by the system. We find further qualitative evidence in the empirical data of the dynamical importance of units as function of their degree in three different domains, namely viral marketing in social networks [39], evolutionary conservation of human proteins [40] and the transmission of a neuron’s activity in neural networks [41]
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