On local behavior and global structures in the evolution of complex networks

ing from these approaches, a system can be said to evolve if it is dynamic, i.e., changing its characteristics, according to the following basic principles: 1. The system generates offspring with mainly the same characteristics as itself (Inheritance). 2. The offspring show variations of the parental characteristics (Mutation). 3. The system’s ability to survive the selection is based on its characteristics (Fitness). 4. There is a selection mechanism based on the fitness of the network (Selection). For every evolving system there is a virtual space consisting of all possible system states. If there is a fitness function that assigns a value to each of the possible states, this space is also called the fitness landscape of the system [169]. For biological networks like protein-protein interaction networks [199, 219, 232], or metabolic networks [116, 201] it is quite clear that they have evolved within this framework. Both types of networks are part of an organism (the system) and their characteristics are encoded in the genetic code of this organism. Thus, they can be inherited by the offspring of the organism whereby they may suffer some mutation. Of course, the quality of these networks does not totally determine the fitness of the organism, but it does at least influence it. One simple example of the influence that metabolism has on the fitness of an organism is an enzyme called Gulonolactonoxidase that catalyzes the synthesis of vitamin C. This enzyme was lost in humans and guinea pigs, making vitamin C an essential nutrient for these organisms to survive and reducing fitness in those environments where vitamin C is scarce [162]. While the evolution of biological networks can thus be modeled meaningfully as a series of mutation and selection, evolution of other networks such as the social network in a city or the network of streets in a country is not determined by selection since there is no alternative network with which these singular networks have to compete. In analogy to the above given properties of an evolving system we will show why it is still meaningful to speak of the evolution of these networks: 7 Interestingly, the authors do not discuss their usage of the term evolution as opposed to dynamic. 6. The Principle of Locality in the Evolution of Complex Networks 116 1. Let Gi and Gi+1 describe the same network at two subsequent time steps . As long as edges are costly to build and delete we can assume that only a few changes have been made to the adjacency matrix of Gi to result in Gi+1. Thus, most of the structural properties of Gi will be preserved in Gi+1 (Inheritance). 2. The changes can be seen as a kind of mutation. 3. Under the assumption that edges are costly to build, to maintain, and to delete, it is obvious that no real network is built without the wish to use it for something, e.g., a means of transport, information exchange, or creation of knowledge or wealth. For example, in recent years larger and larger teams and collaborative networks in science have been built to achieve new insights, and seemingly they were built efficient enough to do so [19]. In summary, networks are built with a purpose and their structural properties determine their fitness for the given purpose. 4. Depending on the fitness of a given network it can be selected in two different meanings of the word. If the network is used by intelligent agents they can decide whether they want to use it or not. For example, every country has different transportation networks: the network of streets and highways, train networks, and airplane routes. A traveler will make his choice among them depending on the distance, the maximal amount of travel time he wants to spend, money, and the structural properties of these networks. On a second level, the participants or the owner of a network can decide whether they want to leave the network structure as it is in the next time step or whether they want to change it because they are unsatisfied with the performance of the network. Thus on the first level, if there are different singular networks with the same purpose (transport, communication, etc.), agents can decide which one to use. On the second level, agents can decide to change the network’s structure according to the fitness of the current structure. This latter aspect gives rise to an interesting feedback mechanism that is typical for complex adaptive systems [233] and can either make it very difficult to control them or give us a tool by which the system can be made to evolve towards an attractor, a stationary state as in chaotic systems [226]. Note that a very detailed modeling of evolutionary processes that incorporates the ideas of chromosomes, cross-over, or genes as proposed in early genetic algorithms by Holland [107, 169] is not necessary to call a model an evolutionary model. Interestingly, the life–long aim of paleontologist Stephen Jay Gould was to decipher the ”Structure of Evolutionary Theory” [98], and he comes to the conclusion that natural selection (together with heredity, variation, and overproduction of offspring) is just the sheer mechanism of evolution, but that a theory must contain, explain, and be based on the following three principles in order to be called an evolutionary theory (at least for the advance of biological organisms): 1. Agency: Natural selection aims at the level of organisms. 2. Efficacy: Natural selection is able to create new solutions to a problem. 3. Scope: Natural selection is able to create every new solution in organisms. In order to design an evolutionary model of complex networks it seems at first glance that there are two different agencies for selection: One on the level of the whole network and one on the level 8 The length of the time interval between these snapshots has to be determined meaningfully in every network or is given by the data at hand. 6. The Principle of Locality in the Evolution of Complex Networks 117 of single elements of the network. As depicted above, the first type of selection occurs if there is more than one network for the same purpose, allowing people to choose which one to take. The second type of selection occurs whenever the structure of a network needs revision. If an edge contributes to the functionality of a network, it is likely to stay in the network; if a new edge would improve the functionality of the network, it is likely to be built. In essence, the first type of selection will finally also lead to a selection of the elements of the network: If a transport network is not used by the customers, the owner of the network will select those edges to build or to destroy that optimize the functionality. Thus, in order to model the evolution of a complex network, we can disregard the first type of selection and concentrate on the second. Furthermore, we will only concentrate on those models where the central mechanism of selection is able to create new—and all new—structures of the network. 6.2. 2 Dynamic vs. Evolutionary Network Models Summarizing the ideas sketched above on the evolution of biological organisms, we will denote as a dynamic network every network that changes its adjacency matrix over time. A network is said to evolve if the building and deleting of edges is costly and if, therefore, we can assume that the network is built with a purpose. From this it follows that the network’s structure at least partly determines the fitness of the network with respect to the purpose. Furthermore there must be a selection mechanism that makes it more probable that a functional network structure is stabilized and that a less successful network structure is changed in a way that explores the set of all possible graphs for a better solution. With this definition, the small-world model and the preferential attachment model are definitely dynamic network models, but they are not evolutionary network models. The ’purpose’, so to speak, of the small–world model is to build small networks. But if the first rewiring steps fail to build a small network by chance, there is no mechanism to rewire these edges again to build a smaller network in a second try. Similarly with the preferential attachment model, where the ’purpose’ can be described as building a scale-free degree distribution. But if the network fails to build one there is no mechanism that allows for a rewiring and subsequent selection of those rewiring steps that lead the network towards a more scale-free degree distribution. Why should one be interested in an evolutionary network model in addition to a pure dynamic network model? The main difference between these types of network models is the following: in a dynamic network model we aim for a certain structure and try to find a changing rule that will result in this network structure. To stay with the picture of the meta-graph G(G): A dynamic network model chooses a narrow trajectory through the meta-graph such that the attractor of this model shows the desired structural property. In such a model, no selection mechanism is needed to guide the trajectory of the system since the changing rule will most likely only explore a small part of the possible search space, as in the small–world model by Watts and Strogatz where the rewiring mechanism results in a small network with very high probability and maintains the high clustering coefficient with high probability for an appropriately chosen rewiring probability prew [242]. In an evolutionary network model, we aim for a global structural property and allow the network to try every solution it can find in G(G) , i.e., for a small–world model the model would fix a wanted average clustering coefficient and a wanted average distance. As long as these criteria are not met, the agents would apply a changing rule that changes the network’s current structure until they have concertedly achieved a satisfying network structure. Thus, the second type of model is not restricted to a certain part of the fit