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Complex Networks Advanced Computer Networks: Part1 Slide 2 2 Agenda Introduction Types of Topologies How to create them? Network Analysis Degree and degree of distribution Average path length Betweeness Centrality Example: Routing Strategies Slide 3 Introduction Various nature and society systems can be described as complex networks such as social systems, biological systems, and communication systems. By presented as a graph, vertices (nodes) represent individuals or organizations and edges (links) represent interaction among them the World-Wide-Web, nodes are documents, web pages and hyperlinks are links linking to other documents. 3 Slide 4 Introduction Partial map of the Internet based on the January 15, 2005 data found on opte.org. opte.org 4 http://en.wikipedia.org/wiki/File: Internet_map_1024.jpg Example Slide 5 Introduction Why is network anatomy so important to characterize? Because structure always affects function. For instance, the topology of social networks affects the spread of information and disease, and the topology of the power grid affects the robustness and stability of power transmission. 5 Slide 6 Introduction Network Models Regular Networks: chains, grids, lattices and fully- connected graphs Random network model by Erds and Rnyi: ER model Small-world phenomenon by Watts and Strogatz: WS model Scale-free network model by Barabsi and Albert: BA model Evolution mechanism of network structures are very interested among many researchers not only engineering but also physics communities. 6 Slide 7 Types of Network Models Regular Networks 1. Ring of ten nodes connected to their nearest neighbours. 2. Fully connected network of ten nodes 7 Slide 8 Types of Network Models Random Networks placing n nodes on a plane, joining pairs of them together at random until m links are used. Nodes may be chosen more than once, or not at all. 8 Slide 9 Types of Network Models Random Networks Erds and Rnyi studied how the expected topology of this random graph changes as a function of m. When m is small, the graph is likely to be fragmented into many small clusters of nodes, called components. As m increases, the components grow, at first by linking to isolated nodes and later by coalescing with other components. 9 Slide 10 Types of Network Models Random Networks A phase transition occurs at m = n/2, where many clusters crosslink spontaneously to form a single giant component. For m > n/2, this giant component contains on the order of n nodes (its size scales linearly with n), while its closest rival contains only about log n nodes. All nodes in the giant component are connected to each other by short paths: the maximum number of 'degrees of separation' between any two nodes grows slowly, like log n 10 Slide 11 Types of Network Models Random Networks Gene networks Ecosystems Spread of infectious diseases Computer viruses 11 Slide 12 Types of Network Models Small-World Networks Watts and Strogatz studied a simple model that can be tuned through this middle ground: a regular lattice where the original links are replaced by random ones with some probability 0