Complex systems: Definition [1]

Lets start by defining some of the most important characteristics that are common in all complex systems.

1. They are composed by many interacting parts.
2. Each component has its own internal structure and each one perform an specific behavior or function.
3. The behavior of a small part of the system affects in a non linear way the whole system. Complex systems present emergent behavior in such a way that any property of the system is not the simple sum of its parts. For example, the mass of a system composed by many particles is the sum of the masses of its components, whereas the organization of the system can not be obtained from the sum of individual organization degrees.

As a typical example of a complex system we can consider a cell, obviously, a cell is formed by many parts (ribosomes, nucleus, membrane, DNA, RNA, etc.), each part of a cell has its own specific function. The components of the cell respond in a non linear way when an external perturbation is done. For example sometimes a DNA mutation can be fatal, falciform anemia is a disease produced by one single mutation from the 600 aminoacids that built the beta-globin protein, that carried out oxygen absorption.

Complex Networks

The complex networks are a set of many connected nodes that interact in different ways. Is common that the nodes in an netrwork are also called vertices or elements, mathematically those are represented by the symbols $v_1,v_2,\cdots v_N$, where $N$ is the total number of nodes in the network.

In nature, we can find many different networks, meaning different kinds of nodes and connections. For example in a social network the nodes are people and the connections can be friendship relations. In the same society, we can define conecctions in a different way, for example: two people are connected if they are siblings. Obviously the network defined through the friendship relationship is different than the one defined through the sibling bond, because if two people are friends that doesn't imply that they are siblings and viceversa, however the nodes in each network can be the same. The last example is to note that with the same set of nodes we can define different networks depending on how we define the connections, that, of course depends on the phenomenon that we want to study. In the following tables are shown some of the networks and the interactions that we can define for complex networks in nature.

Social Networks
 Friendship Two people are connected if they are friends Scientific Two scientists are connected if they have been coauthors in any paper Family Two people are connected if they belong to the same close family

Information Networks
 Internet Two computers are connected if they are in the same domain WWW Two web pages are connected if there is a link from one to the other Words Two words are connected if they are synonims

Biological Networks
 Proteic Two proteins are connected if they participate in the same metabolic path Genetic Two genes are connected if one regulates the expression of the other Ecologic Two species are connected if they have a predator-prey relationship

 In the last examples there are directed networks and non-directed networks, the family network is not directed because if A is relative of B, obviously, B is relative of A. On the other hand the defined WWW network is a directed network because there can be a link from page A to page B, but not necessarily from page B to page A. Another important concept is an island (or subnetwork) in a netwok. The islands in a network can have different sizes, ranging from the ones with one node which is not connected with any other, to the islands that are the complete network (all the nodes are connected among them). It is important to mention that no matter whether an island is not connected to the main body in the network, that doesn't mean that the island doesn't belong to the network. The network is not only composed by the conecctions but also by the nodes that are part of the system.A network can be composed by many islands, as shown in the figure in the bottom

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