So, in this lecture I'm going to further discuss the concept of network motifs. I'm going to talk about projects and applications that we have worked on in my lab. One thing that we wanted to do after the first few network motifs paper publication was to identify larger sized network motifs. In various networks including the cell signally network that we created for our literature. However we face with some problems of finding large size network qualities. One of the issues was that finding network motif in large size networks, is very and demanding. Another issue was how many possibility for motifs of size five or more. As the size of the motif the number of nodes that can participate in a specific motif grows there is exponential increase in the number of types of motifs that you can identify. So, if you want to start bending those motifs into types, you have to do some simplification. So, the first thing that we study with our collaborators at IBM was to identify the sequence of the number of motifs that you can have For various types of network as you increase the size of the the motifs. We looked at undirected networks, directed networks, and mixed networks. In mixed networks you can have both directed and undirected mix. And the cell signaling that we cared about had mix directed and undirected mix. So after working out the Sequence of the number of motifs of various size as the size of the network motif grows, we were able to submit that sequence into the encyclopedia of integer sequences. So, if you are faced with a sequence of integers you can submit it to this website, see whether you identified a novel sequence. So, the sequence of network motif possibilities in mixed graph Was not there yet in the data base. So, we were able to contribute that novel sequence into this database. And that happened already in 2006. So, this work with our collaborators at IBM resulted in a publication in PNAS And when we dealt with this problem of having many motifs, we decided just to look on cycles or loops in those graphs. And then what we've done, we looked at the nodes within each loop and classified those nodes into three types, sink nodes, those nodes have two coming into them, source nodes those nodes have two arrows coming out of them. Pass-through nodes, those nodes have one link coming into them, and one link leaving them. And neutral nodes, nodes that have at least one mutual link or undirect link. So, now I would like you to guess What would be the distribution of nodes, pass-throughs versus sources and sinks, in cycles in random directed networks? So, this is an application of the method to nine directed and directed and mixed networks, including six biological networks. And three technological complex systems and what you see that in biologic networks you have a lot more sources and sink nodes in cycles compared to what you would expect randomly which is what the FAA Network follows the FA network has airplanes coming in and out of airports. So, there is an even probability for source and syncs versus pass through nodes. Almost in all biological networks, there is always more sources and syncs compared Will pass through nodes. One of the interesting features of some of those networks is that as the loop size changes, the propensity to have more sources and sinks changes. For example, if you look at the neuronal connectivity Map of the [INAUDIBLE], as the increase in size, you'd get an increased number of sources and sinks, an increasing matter of pass-through nodes and a decreasing number of neutral nodes. Running dynamical simulations to assess the stability of the real networks we showed in that paper That real network topologies, and here is an example of the cell-signaling network, are more stable than similar networks with links. Such stability is due to many sources and syncs found in the cycles and the loops. And in the paths within the topology of the real networks. In a follow up study, he studied with a network, and started swapping links to make the network rich in sources and syncs. Using a set of simple rules That couple the swapping of links towards increasing sources and syncs to the dynamical stability.,We are able to obtain networks that display of the properties observed in real world biochemical regulatory networks. The resultant networks are scale free with hubs created and destroyed over time. While the scale free topologies, small world clustering, network motifs and few feedback loops in the topology remain to be in network. So, in this magnetization model, we start with a random directed network. We then pick an arrow which is then directed edge randomly and place it somewhere else so it connects to other nodes. We then compute the magnetization for the arrow at its new location. We want to see wether the new location contributes to an increase in the number or sources and syncs in the network. If the swap increases the number of sources and syncs We accept this to our [INAUDIBLE]. After we finish the swap, we check whether the network is stable. And this can be determined by computing the adjacent symmetric eigen values. In case the network is stable we accept swaps regardless if they contribute to the global magnetization. Otherwise you rejects the swap that does not contribute to the global magnetization. If you repeat the process eventually we get a scale free network and that connectivity distribution resembles the topology of Many real world networks. And on the right panel, we look at the connectivity degree of the hubs in the network. So, as you can see, hubs rise and fall while the evolutionary process is going. The connectivity degree remains. This model's also recapitulates the high clustering coefficient, the small world topology, as well as the net motives that are identified in many real world directed networks. In this publication that came out in 2006, the idea was to grow networks but to have an evolutionary growth model of a network, while the network tried to adapt to the extracellular environment and have inputs and outputs. So, if you count the number of pathways from receptors to effectors in a cell signaling network from literature, we can see that pathways to effectors are branched as a power-law. This means that from some receptors, there are many branches. To reach the effectors like the transcription factors. While from other receptors there are only a few known pathways. This distribution of pathways is reminiscent of a classifier system. That is developed typically by machine learning algorithms such as artificial neuronal networks where the structure of the environment is transformed into the structure of the system. So, if we summarize some of those evolutionary models to generate complex networks that resemble The topology or the skeleton of natural complex systems from biology or complex systems from technological systems. We already looked at five different models. The scale-free model by Barabasi and Albert, the small-world model My watts and the duplication divergence model, the magnetization model and this model that tries to grow a network or train a network as it's interacting with its environment and having to perform A specific task. For your assignment, I would like you to try to think of other potential models that can capture the topology of real-world networks. As well as identify in the literature other models that people came up with. [MUSIC]