Information processing reveals how microscopic components affect the macroscopic system-state in complex networks Rick Quax, 1 Andrea Apolloni 2 , and Peter M.A. Sloot 1,3,4 1 Computational Science, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands 2 Institut des Systmes Complexes Rhone-Alpes (IXXI) and Laboratoire de Physique, cole Normale Suprieure de Lyon, 46 Alle d'Italie, 69007 Lyon, France 3 National Research University of Information Technologies, Mechanics and Optics (ITMO), Kronverkskiy 49, 197101 Saint Petersburg, Russia 4 Nanyang Technological University, 50 Nanyang Avenue, 639798 Singapore Nature processes information. We observe this from physical systems, which register information in the system state, transfer information through interactions, and lose information due to thermal noise. Being able to quantify this information processing could lead to a unifying framework for a better understanding of complex systems. In this letter we present a formalism to describe to what extent a macroscopic system is affected by the microstates of its constituents. We study this numerically for a heterogeneous network of Ising-spins, and suggest an answer to the unexplained phenomenon that real systems with a heterogeneous topology are mainly controlled by nodes with fewer connections [Liu, Y.-Y. et al., Nature 473, 7346 (2011)]. Counter to intuition we find that due to selective information dissipation, not the hubs but rather the intermediately connected nodes are remembered best by the system. Our study suggests that the concept of inherent information processing may lead to a better understanding of the emergent behavior of complex systems. I. INTRODUCTION Information theory, such as the one based on Shannon’ s entropy [1], is traditionally used for statistical inference, where an external observer attempts to describe the state of a system and its behavior [1–5]. In information processing, however, there is no external observer and no global frame of reference. Instead, each component (particle, agent, or element) can be considered as an observer which stores information about the state and behavior of other components in the system [6,7]. This information is transferred and copied through the system due to the interactions, and it is lost due to randomness. The concept of inherent information processing in nature is usually discussed in colloquial terms [4,8], or replaced by an epidemic spreading model [9–13]. Today it is still unknown how information is processed inside a system at a microscopic level. Nevertheless, here we show that we can study an important aspect, namely the time it takes for information about the state of a single node to be lost from the system state. In other words, how long does the network’s state remember the state of its constituting nodes? We will argue that this is a measure of the influence of each (microscopic) node to the (macroscopic) behavior of the network as a whole. The goal of this work is two-fold. Firstly, we present an outline of a theoretical framework that can quantify how information is stored and processed in networks of dynamical units. We believe that such a framework could be used as a unifying tool to describe the behavior of complex systems. Secondly, we use the framework to formulate a quantitative measure of influence of a single node