Chapter 1 Emergent Phenomena on Complex Networks 1.0.1 More is different When many interacting elements give rise to a collective behavior that cannot be explained or predicted by considering them individually, we say that an emergent phenomenon is taking place. These phenomena emerge from the interactions, and are characterized by properties that are universal, i.e. common to many different interacting systems. Statistical mechanics, is a well developed theory that has given very precise meaning to these statements. Traditionally, statistical mechanics is a branch of physics develop to explain from the statistical, microscopic interactions between the elements of a thermodynamics system, as for example a gas, or magnetic system, the different emerging phases of matter as for example liquid/solid/gas, ferromagnetic/paramagnetic phases (see figure 1.4). Think for example of a large, thermodynamic set of water molecules, how can we explain from analyzing a single molecule the occurrence of such different phases of water, ice and vapor? Take a carbon molecule, how we can predict the different phases of diamond, graphite and graphene? The carbon molecules are always the same but emergent macroscopic differences are generated when we consider the interactions between the atoms! Emergent phenomena, are not exclusively occurring when we consider thermodynamic systems. In random graph theory, when we link two nodes, with probability p = c N, where N are the number of nodes in the network, we observe an emergent phenomena when c crosses the critical value c = 1. For c < 1, in fact the network is formed by disconnected, small, tree like clusters, when c > 1 one cluster becomes giant, i.e. it includes a finite fraction of all the nodes in the network, independently on the network size, changing dramatically the diffusion and connectivity properties of the network (see figure 1.7). In the way we add links, nothing change between c < 1 and c > 1 beside the probability of the links. Moreover, links are placed completely random, their are not corre- 1
2 CHAPTER 1. G. BIANCONI: PROCESSES ON NETWORKS Emergent phenomena: Phases of Ma2er Figure 1.1: The phases of matter are classical examples of emerging phenomena lated. Nevertheless the giant component emerge connecting a finite fraction of the nodes in the network. In epidemiology, epidemic spreading is an emergent phenomena. If the infection rate λ is greater that a critical value λ c, the epidemic threshold, then the epidemics spreads to a finite fraction of all network (see figure??). Emergent phenomena are everywhere in complex systems and they are fascinating phenomena. Let us just mention that the two biggest challenges in biology: life and cognition are marvelous examples of emergent phenomena. The life of the cell is and emergent phenomena coming from the chemical interactions between its molecules. Cognition is an emergent phenomena coming from the interactions between the 10 11 neurons in the brain. Our goal in this module will be to characterize the dynamical processes on networks, and in particular the emergent phenomena occurring by the interplay between the network structure (the pattern of interactions) and the dynamical process taking place on it. 1.0.2 Interplay between structure and dynamics on networks In order to shed light on emergent phenomena, dynamical processes defined on large networks should be investigated. Traditionally, in physics and in mathe-
3 Emergent phenomena in random graphs Emergence of the giant component c<1 c>1 Figure 1.2: Emergence of the giant component in random networks. matics the networks that have been considered are on one side regular lattices, on the other side random graphs. These are either completely regular and symmetric networks (lattices) or completely random structures (random graph). Nevertheless most of complex systems are described by complex networks, that are stochastic but not totally random as random graphs. The architecture of complexity is described by the network universalities, i.e. the small world, the scale free properties and the universal emergence of communities. Therefore is of crucial importance to understand the interplay between structure and dynamics in complex networks. In fact complex networks topologies strongly affect the dynamical properties of the network. For example the response of networks to random damage or to targeted attack is strongly dependent on the degree distribution of the networks. In fact scale-free networks are significantly more
4 CHAPTER 1. G. BIANCONI: PROCESSES ON NETWORKS Emergent phenomena Epidemic outbreaks Figure 1.3: The epidemic outbreak is a emergent phenomena. robust than regular networks to random perturbation. Due to the universality of scale-free degree distribution this result has wide applications. For example, it can explain at the same time the robustness of the scale-free airport network or of the scale-free protein interaction network in the cell. The robustness of networks is also affected by the interactions between different networks. In fact in the case that the networks are interdependent, i.e. the function of one network depend on the function on another network, such as it happens in complex infrastructures, then the whole system becomes significantly more fragile to perturbation, and can be affected by dramatic cascading events propagating back and forth in the different layers. Also the process of epidemic spreading is significantly affected by the topology of the underlying network structure on top of which the process take place. Consider for example the recorder data
5 COMPLEXITY:BETWEEN RANDOMNESS AND ORDER! LATTICES COMPLEX NETWORKS RANDOM GRAPHS Regular networks Symmetric Scale free networks Small world With communiges ENCODING INFORMATION IN THEIR STRUCTURE Totally random Binomial degree distribugon Figure 1.4: Complex networks describe the architecture of complex systems. They are not regular but neither totally random. They follow complex networks universalities (small world, scale-free properties and non trivial community structure) and encode information in their topology. on the spread of the Black Death in Europe. The epidemic spread at that time like a wave through the entire Europe. Nowadays, due to the global air transportation system the epidemic spreading does not follow anymore a spatial wave, but travels though hub locations and spread much faster around the globe. In the brain the structural brain networks formed by fibers connecting different brain regions is strongly connected with the functional brain networks describing the correlated activity of different regions of the brain. Therefore, understanding how the structure of the networks affects the network dynamics is of fundamental importance to understand emergent phenomena in complex networks.
6 CHAPTER 1. G. BIANCONI: PROCESSES ON NETWORKS INTERPLAY BETWEEN STRUCTURE AND FUNCTION: ROBUSTNESS! POWER OUTAGE US 2013 LETHALITY OF PROTEIN KNOCK- OUT POWER OUTAGE ITALY 2013 Figure 1.5: The robustness of complex networks to random damage is a general questions with wide applications ranging from the robustness of power-grids and infrastructure to the robustness of biological networks in the cell. In this case the topology of the networks, and the possible interdependencies between different networks, can significantly affect the response to random damage and targeted attack.
7 INTERPLAY BETWEEN STRUCTURE AND FUNCTION: EPIDEMIC SPREADING! H1N1 Black Death Thex Figure 1.6: The epidemic spreading is significantly affected by the structure of the underlying network on top of which the epidemic process takes place. For example, the global airline transportation systems has changed the spatial patterns of the epidemic spreading and the risk of global pandemics.
8 CHAPTER 1. G. BIANCONI: PROCESSES ON NETWORKS INTERPLAY BETWEEN STRUCTURE AND FUNCTION: THE BRAIN! SYNERGY BETWEEN STRUCTURE AND FUNCTION IN THE BRAIN Figure 1.7: In the brain their is a strong interaction between structural and functional brain networks.