Currently, there is considerable research activity in network dynamics. This is clearly motivated by the wide expansion of communication media (mobile phones, Internet, multimedia, etc.), but also by the growing interest in network modelling of biological processes (neural networks, genetic networks, ecological networks, social networks ...). A large part of these studies focuses on topological properties of the underlying graph. However, in many cases, the nodes of the networks are units behaving in a non linear way. For example, in a communication network a relay regenerates (amplifies) weak signals, but it has a finite capacity and saturates if too many signals arrive simultaneously. A neuron has a non linear response to an input current, a gene expression is determined by a non linear function of the regulatory proteins concentration, etc.. These non-linearities might modify the network abilities in a drastic way. For example, a relay may have a high graph connectivity (``hub’’), but the dynamics drives it close to its saturation point, so that it has a weak reactivity to the changes in the inputs coming from the other units and a poor capacity to transmit information. Consequently, the information is transmitted via other units, possibly weaker links, and, in this regime, these units become temporary ``hubs’’ though they may have a low graph connectivity, while the main hub is delested. In biological networks similar effects may arise. For example, the capacity of a neuron to transmit a specific excitation strongly depends on its state, determined itself by the overall currents coming from afferent neurons. This suggests that the mere study of the graph topological structure of a network with non linear units is not sufficient to capture the full dynamical behaviour. However, there are relatively few studies which analyse the joint effect of topology of the network and non-linearity. A careful investigation of these problems requires therefore a proper analysis of the dynamical system corresponding to the network currently used and may require the development of new tools and methods.
We have developed an original approach based on a linear response theory proposed by Ruelle for dissipative dynamical systems. We have shown in a series of papers [Cessac & Sepulchre 2004, Cessac & Sepulchre 2005, Cessac & Sepulchre 2006] that this approach can be efficiently used to characterize the behaviour of a network where the nodes have non-linear transfer functions. On theoretical grounds we have shown the existence of a new type of resonances predicted by Ruelle but never observed before. These resonances only exist in systems where the volume is dynamically contracted in the phase space and they are not observed in the power spectrum. (In the networks we are interested in this contraction came from saturation effects in the transfer functions). We have also shown that linear response does not obey the fluctuation-dissipation relations in dissipative systems : it provides more information than the study of correlations functions especially concerning the effective, causal action from one node to the other, and the identification of characteristic times such as the time of relaxation to a stationary regime. More recently, one of us [Cessac, 2006] has applied the method we have developed on simple examples of non uniform hyperbolic systems like the Hénon map, and exhibited the existence of a pole in the upper half complex plane. This result present s a strong interest for the community studying non uniformly hyperbolic dynamical systems and may have an impact on the type of analysis we are performing on networks.
On practical grounds, we have predicted and evidenced non-intuitive and unexpected effects in networks with chaotic dynamics. We have shown that it is possible to transmit and recover a signal in a chaotic system. We also analysed how the dynamics interfere with the graph topology to produce an effective transmission network, whose topology depends on the signal and cannot be directly read on the ``wired’’ network. Moreover, with a suitable choice of the resonance frequency, one can transmit a signal from a node to another one by amplitude modulation, in spite of the presence of chaos. In addition, a signal, transmitted to any node via different paths, will be recovered only at some specific nodes.
These results open up important perspectives for applications. In this spirit we have contacted several groups and private companies to develop these applications in different fields. Currently, several collaborations are going on, where applications of linear response theory are one of the scope. These collaborations are Dynamique de réseaux de neurones à spikes ; Apprentissage dans les SysTèmes bIologiques COmplexes (ASTICO) ; Comportement chaotique dans les réseaux de données (CAOREDO) (see the related sections).