Threshold networks

Networks are ubiquitous, and we experience them in our daily lives. Every phone call we make is routed through a telecommunications network, every product we buy has come through a complex manufacturing, supply and transport network. Our families, friends and colleagues form our social networks, and when we see a doctor, the conditions we discuss usually originate from interacting bio-molecular networks. An important property of networks is their structure, which provides information about the connectivity amongst the elements that form the network. In recent years it has become apparent that in addition to the network structure, the dynamics of the network elements is vital for our understanding of the emergent behaviour of real-world networks. In this proposal we will develop a novel mathematical framework that will allow us to analyse a large variety of dynamical networks across numerous disciplines.

Our theory will be based on two concepts. Firstly, we will approximate the complex (nonlinear) behaviour of network elements in the real world by an appropriately chosen set of simpler (piece-wise linear) models. This approach has proven extremely helpful in elucidating behaviours in engineered systems. Secondly, we will assume that network elements communicate with each other through events. For instance, in the telecommunications example above, the event corresponds to the situation where placing a phone call triggers activity in network routers.

While these two concepts allow us to describe numerous networks in a unifying language, existing mathematical techniques cannot be used to analyse them. This shortcoming mainly results from the fact that current mathematical approaches assume what is known as smooth dynamics. In contrast, we will deal with non-smooth systems. A main part of the proposal is to develop ideas from general non-smooth dynamical systems and make them fit for investigating event-driven dynamical networks. A common assumption in the study of networks is that elements communicate with each other through weak signals. This assumption is often made for mathematical convenience. We will go beyond weak signals and develop an approach for strong signals, which more closely resembles what is observed in real world networks.

Once the mathematical techniques are in place we will showcase the versatility of this new approach by studying three different applications. Firstly, we will investigate in more detail how a heartbeat is generated. What we perceive as a single heartbeat is the orchestrated action of millions of connected muscle cells. Within each muscle cell numerous bio-chemically coupled networks shape the single cell behaviour. A better understanding of the cardiac muscle network is crucial for treating and even preventing cardiac arrhythmias such as atrial fibrillation. As a second application we will study patterns of activity in the brain. Here, specialised neural cells form a complex network, and they communicate with each other through chemical and electrical pulses. We will be primarily interested in activity patterns when neurons suddenly stop communicating or become hyperactive, as these dynamics are often observed in medical conditions such as epilepsy. Thirdly, we will apply our technique to information spread on social networks. A special emphasis will be on when such information travels extremely quickly amongst a large number of people (so-called viral activity). By achieving a better understanding of how ideas, behaviours or styles become viral, and how that depends on network structure and characteristics of the network elements, we can develop means for intervention strategies (e.g. quelling London riots quickly) and for the spread of desirable behaviours (e.g. a healthy lifestyle amidst the obesity epidemic).

1) Networks are ubiquitous in numerous disciplines including engineering, biology, medicine, finance and politics, to name but a few. While these networks may describe vastly different phenomena ranging from gene expressions to market collapses and election decisions, their structure and dynamics often share remarkable similarities. As we will develop a mathematical framework with which we will analyse the network behaviour that emerges from these conserved core principles, we envisage impact across a broad variety of applications. For example, in the field of systems medicine, networks are at the centre of conditions such as epilepsy and atrial fibrillation. Both are widespread and very costly in terms of medical treatment and lost revenues to the economy. Cardiac and neuro-surgeons alike now appreciate that common procedures such as ablation only remove local node dynamics, but do not address the underlying network dynamics that are responsible for the conditions. A better understanding of the network dynamics and how to manipulate them will result in a higher quality of life, more accurate surgical procedures and higher economic productivity. The notion of increasing productivity is at the heart of supply chain management. Our findings will assist in optimising existing supply chain networks and in providing guidance on how to structure novel ones. For the latter, particular emphasis will be on the now common theme of networks of networks. In our digital age, cyber security has become one of the foremost concerns across a range of scales from individuals to multi-national corporations. Given that the security threads emerge from computers being connected, a better understanding of how to improve the node dynamics in these networks (i.e. the behaviour of individual computers) will directly lower the risk of cyber crime. Social networks have gained considerable attention in recent years, especially in how they shape a large number of decisions in e.g. elections or consumer behaviour. A better understanding of how these networks evolve and adapt is crucial for both companies and policy makers.

2) The project will also enhance the employability of the PDRA by increasing their scientific skill set and soft skills. At the same time, the PDRA will learn to become an independent researcher. This will help them to pursue a wide range of career choices, both inside and outside academia. The extended stays of the PDRA in the USA (at UCLA) will also allow them to explore employment opportunities on the American market, with valuable input provided by Professor Mason A Porter (UCLA).