## The Modelling of Large Complex Networks

Novel important discoveries in the field of complex networks are providing the grounds for building new tools for the analysis of large networks, which could also prove to be very useful in the near future for the study of Ara-bidopsis responses during stress. Large complex networks are ubiquitous in many disciplines, such as biology, computer science, and social sciences, to name a few. In recent years, topological data about large complex networks has become increasingly available, and as a consequence we have witnessed great advances towards understanding the organizing principles of such networks. Strikingly, we are finding that many of the architectural features of large complex networks are shared by systems as different as the World Wide Web, social networks, and many types of biological networks.

Until recently, such networks were modelled by the Random Graph Theory, an elegant mathematical theory developed by Erdos and Renyi starting from the 1960s (Erdos and Renyi 1960). According to this theory, a complex network was identified as a random graph, i.e., a graph built starting from a set of nodes and then randomly adding edges with a certain probability. However, with the availability of large amounts of data describing the topology of real networks it has become clear that the theory of random graph does not explain many of their topological properties. Probably the most noticeable of them is constituted by the distribution of the degrees of the nodes (the degree of a node is the number of edges connected to it): while the distribution predicted by the random graph theory follows a Poisson distribution, the degree distribution found for real networks follows a power law distribution of the type P(k) = a~k, where P(k) indicates the probability of having a node of degree k, and a is a real value which for many complex networks has been found to be close to 2 (Barabasi and Albert 1999). An important consequence of this is that while for a random graph the probability of a node having high degree is virtually zero, the power law distribution indicates that there is a large probability to have a few nodes with a very high degree; and these few nodes will therefore dominate the connectivity in the network. Such highly connected nodes are commonly called "hubs", and networks with a power law degree distribution were named "scale-free" (Barabasi and Albert 1999).

Most networks within the cell have been found to be scale-free. For example, metabolic networks (in which the nodes are metabolites and the links are biochemical reactions), protein-protein interaction networks (in which the nodes correspond to proteins and the links represent a physical interaction between them), co-expression networks (where nodes are the genes and the links represent the amount of co-expression between them, as measured by correlations computed from microarray experiments), have all been shown to be scale-free (Barabasi and Oltvai 2004). Currently, there is no reason to doubt that these results hold true for transcriptomic networks in Arabidopsis.

An important consequence of the power law distribution of the degree is that scale-free networks are extremely robust to random noise. In other words, no significant decrease in performance can be seen even when a high number of nodes, chosen at random, is deleted from the network. This is due to the fact that there is a relatively low probability to knock-out simultaneously many of the network's hubs, which are its crucial functional elements. Interestingly, it has been shown that proteins that are hubs in protein-protein interaction networks in yeast (S. cerevisiae) have a tendency to be essential genes (Jeong et al. 2001). In Arabidopsis the robustness of the ABA network against perturbation was tested through simulating gene disruption and pharmacological intervention (Li et al. 2006).

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