In this lecture, I'm going to expand on the rich get richer and small world network concepts. And discuss other models to generate networks that are similar to the networks that we see in complex systems around us and in human and mouse cells. Another concept that is also very important, and other concepts that came after the realization of the small world and cell free network is the concept of medical motifs that we're going to further discuss in the next two lectures. Let's go back to our discussion of complex systems. And one of the most important elements of the evolution of agents in those complex systems and that concept is the survival of the fittest. So, an agent fitness is what would determine the agent's ability to proliferate, evolve, and survive in a complex system. And the fitness of an agent is dependent upon many of its properties such as its ability to live long independent. How many connections it has with other agents? The availability of resources in the environment. The innovativeness of the agents. The ability of the agents to self-repair itself. The ability of the agents to reproduce how energy efficient is the agent. How adaptable the agent is, how robust the agent is when he is losing one of its components. At this is some of going back to some of the concepts that we discussed when we talked about scale free networks. What is the overall strength of the agents? Ability to woe and ability to sense the environment. So in markets and ecosystems there is a fluctuation demand for specific types of agents. Throughout evolution the number of copies of each agent is constantly fluctuating. When we discuss the scale free evolutionary process, we made assumptions that hubs have higher fitness. So, the hubs in complex systems are in general the most fit agents but they're not necessarily have to be the best Innovators. They just have to be good listeners of innovations, and this one example is the way Microsoft identified new products, and then adapt them, modify them, and then have an edge on the market. So it is not too critical to be an originator of an innovation. But those who have the ability to listen to innovations and then copy and mutate products end up reaping the most benefits and surviving the observation that its hard to get things started. But once the process begins it takes less energy to keep it going. So this leads us to this idea of duplication divergence. In complex systems growth by duplication, divergence is very common. And it's an obvious concept in biology. But it's also happening in technological systems and economical systems. Some examples from technological and economic systems are various models of cars. Or for example, the different search engines that appear on the web, and the browser wars between Firefox and Chrome. And those are just some examples of this concept of duplication divergence. So a few years after the rich get richer and the small world models were introduced and created big buzz in the community of network analysis and complex systems. Several papers came out with an alternative model that can generate, scale free networks, in concept of is more relevant to biology. And this is the duplication divergence network growth model. This is one of the publications that describe this model. And in this publication they fitted the model to the protein interactions that was published at the time from a profile name yeast protein using the yeast two hybrid method. So this model is similar to the rich get richer in a sense that it starts with a seed random network of very few nodes and then you keep adding nodes to the system. But here the addition of nodes to the system is not based on the connectivity degree. It is by duplicating a node that already exists in the network. So in example three, you can see that the known number six was duplicated, and then the duplication is done with all of the links of the initial original node that was duplicated. Then using sample abilities that can be tuned, you can remove some of those links that were copied from the original node. So in our case, we duplicated node number one and then created node number six, who is a child of node number one. Another publication that received a lot of attention that came out after the big buzz of the '98 and '99 papers by Barabasi and Albert, and Watts and Strogatz, was a paper by Milo from Uri Alon's lab in the Weizmann Institute. What they've done, they took the real-world networks. Mostly biological networks such as the gene regulatory network of E. coli and yeast and neuronal connectivity map. The same neuronal connectivity map that was analyzed by Watson [FOREIGN]. And they counted recurring network motifs in those networks. So one thing to keep in mind about those particular networks, that they analyze those networks and directed networks, so for directed graphs. The initial networks that were analyzed by [INAUDIBLE] and Albert. And protein, protein interactions networks in East are undirected networks. So we don't have an hour that connects pairs of loan it's rather just an edge with no direction. So what they've done, they've created randomized networks, or [INAUDIBLE] network. And counted the number of times they see short circuits that connect several nodes into in a specific way. And we identify that some of those short circuits, those circuits that are made of three and four nodes are much higher in their count. Compare to those network motifs found in randomized networks. And two of those motifs, for example in this table show that it's that feed-forward loop and the bi-fan motif. If you think about the bi-fan motif, it's probably originated from the concept of duplication divergence. The fit for loop motif and the bi-fan motif are likely to be a result of duplication divergence, where nodes X and Y public originated by a duplication. And then nodes Z and W are also originated by duplication event in the network evolution. There is also a possibility to identify network motifs in undirected networks, or networks of protein protein interactions. And here, in this publication, those network motifs which are called graphlets, were enlisted in the yeast protein protein interactions. And then, they compare to those network motifs that we would random or shuffle networks.
