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Complex Networks Advanced Computer Networks: Part1

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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

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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

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Introduction Partial map of the Internet based on the January 15, 2005 data found on opte.org. opte.org 4 Internet_map_1024.jpg Example

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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

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Introduction Network Models Regular Networks: chains, grids, lattices and fully- connected graphs Random network model by Erdős and Rényi: ER model Small-world phenomenon by Watts and Strogatz: WS model Scale-free network model by Barabási and Albert: BA model Evolution mechanism of network structures are very interested among many researchers not only engineering but also physics communities. 6

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Types of Network Models Regular Networks 1. Ring of ten nodes connected to their nearest neighbours. 2. Fully connected network of ten nodes 7

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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

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Types of Network Models Random Networks Erdös and Rényi 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

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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

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Types of Network Models Random Networks Gene networks Ecosystems Spread of infectious diseases Computer viruses 11

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

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Types of Network Models Small-World Networks the network is much more highly clustered than a random graph, if A is linked to B and B is linked to C, there is a greatly increased probability that A will also be linked to C two properties short paths and high clustering for many natural and technological networks Enhance signal propagation speed, synchronizability and computational power, as compared with regular lattices of the same size 13

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Types of Network Models Small-World Networks Starts with a ring of n nodes, each connected by undirected links to its nearest and next-nearest neighbours out to some range k. Shortcut links are then added rather than rewired between randomly selected pairs of nodes, with probability p per link on the underlying lattice; thus there are typically nkp shortcuts in the graph How many steps are required to go from one node to another along the shortest route? 14

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Types of Network Models Small-World Networks how to actually find a short chain of acquaintances linking yourself to a random target person search problems 15

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Types of Network Models Scale-Free Networks Some nodes are more highly connected than others are. To quantify this effect, let p denote the fraction of nodes that have k links. k is called the degree and p is the degree distribution. connectivity probability distribution P(k) of a node connecting to k other nodes is a power-law degree distribution, where k is the degree of a node and γ is a scalar exponent 16

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Types of Network Models Scale-Free Networks The probability of attachment is proportional to the degree of the target node; thus richly connected nodes tend to get richer, leading to the formation of hubs and a skewed degree distribution with a heavy tail. Red, k=33 links; blue, k=12; green, k=11. Here n=200 nodes, m=199 links.. 17

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Types of Network Models Scale-Free Networks Resistant to random failures because a few hubs dominate their topology 18

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Types of Network Models Most large networks have been demonstrated that they have scale-free features according to the BA network properties. There are two issues of realistic networks that are not related in both ER and WS network properties. The first issue is a network grows. Both network models start with a fixed number of nodes (size of network) without modifying it. It means the size of network is constant. 19

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Types of Network Models Most real networks are growing continuously; new nodes are added in the system in anytime World-Wide-Web network grows by increasing the new documents. The second issue is a connectivity probability. Two nodes are connected together with randomly selection in the random network. Most real networks illustrate a preferential connection. New documents in the World-Wide-Web network will link to popular documents with already high connectivity. 20

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Types of Network Models The BA properties can support these issues of realistic networks: The network expands continuously following a power law distribution. The new nodes are added and connected with existing nodes in the network. The new nodes are connected with the existing one based on a preferential attachment; Higher connectivity probability to a node that has a large number of connections. 21

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Dynamic Networks Dynamic network is the study of networks that change in time. Biology Economics Computer Science Graph Theory Nodes and Links can be added or removed anytime!! Random walk and diffusion processes on complex networks have been extensively investigated due to their basic dynamic properties and broad application. 22 Transition Probability

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23 Network Analysis Graph representation of network The network can be presented by a set V of nodes and a set E of edges, linking together as a graph denoted G=(V,E) Average path length The distance between two nodes (dij) is equal to the total number of edges that connect through the shortest linkages The average value of all distance over the network L is the average path length, N is the total number of nodes in the network

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24 Network Analysis Degree and Degree distribution Degree (undirected network): at node i, the number k i of edges connect to the k i edges of neighbor nodes The node of higher degree more significant influence than others in term of dynamics, information flows, data traffic Degree distribution: a probability of a randomly picked node that have degree k is a constant determined by the given network Note: power-law distribution (logarithmic curve)

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25 Network Analysis Betweenness centrality (BC) It is a centrality measure of a vertex within a graph The vertices that occur on many shortest paths between other vertices have higher betweenness value is the number of path between node i and j going through k is the number of path between node i and j

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Example: Routing Strategies Increasing importance of large communication networks such as the Internet, World-Wide-Web, and peer-to-peer networks To understand the traffic flow and find the optimal path for sending information is one of the essential issues and challenged on the complex networks. The effective routing strategies can enhance efficiency of the networks as well as save costs and times. 26

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Example: Routing Strategies 27

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Example: Routing Strategies A well-known traditional routing strategy is a shortest path (a global routing strategy). To obtain the shortest path between any nodes, the global topological information is needed. The shortest path finding between a source (S) and a destination node (D). Starting at S, find the available paths via its neighbors (node 2, 3, 4, 5, and 6) continuously until reach D. The selected path for sending information from S to D is the shortest available path, which is S-2-5-D. 28

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Example: Routing Strategies The needed whole network structure information, The shortest path routing strategies are not suitable for large networks and dynamic networks because of technical limitation of routers. The routers have to have huge memory to storage much information and frequently adapted when the structure of networks changed. They have to keep all the available shortest paths between any pair of nodes into their routing table. 29

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Example: Routing Strategies Limitations of the global routing strategy, Local routing strategy. Each node only has the topological information of its neighbors. 30

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Example: Routing Strategies C. Yin et al proposed an efficient local routing strategy on scale-free network. They employed a preferential selection among local neighbors of each node. Each node looks for the packets destination among its local neighbors. If the packets destination is found within the local search of node l, it deliveries the packets directly from l to its target, otherwise, it send the packets to its neighbor i according to the preferential probability. 31

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Example: Routing Strategies Where, the sum runs over the j neighbors of node l. α is a tunable parameter. k is the degree of node i. 32

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Example: Routing Strategies W. Wang et al proposed a traffic dynamics based on local routing protocol on a scale- free network. The node capacity is not only constant but also corresponded to its degree. 33

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Example: Routing Strategies Y. Li et al presented an effective local routing strategy on the BA network. According to the preferential probability, they proposed a preferential delivering exponent α. The preferential probability is used to select the next node as well as the node capacity is proportioned to its degree when the degree is lower than a cut-off value K, otherwise, infinite. 34

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Example: Routing Strategies The preferential probability is based on the degree of each node; higher connection means higher preferential probability. Nodes with higher degree or connection are more susceptible to traffic congestion than the lower connection. It means that if we select the next node to send the packets according to the preferential probability. The selected node is more sensitive to be congested than other local neighbors. 35

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Example: Routing Strategies W. Wang et al indicated integration between local static and dynamic information for routing traffic. Local static information is degree of each node and the number of packets in the queue of each node is local dynamic information. 36

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Example: Routing Strategies It means that the next node is chosen based on both its degree and the number of packets in its queue to guarantee that the nodes without packets have higher probability to receive packets. Where, the sum runs over the j neighbors of node l. β is a tunable parameter. and k and n are the degree and the number of packets in queue of node i, respectively. 37

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Example: Routing Strategies From the previous studies, Selected the node among local neighbors based on only the degree of each node, which is susceptible to traffic congestion. Chose the next node based on the degree but the capability of each node to generate or delivery packets is not constant, varied on its degree. Selected the node to transfer the packets based on its degree and its capability corresponding to both its degree and the threshold value. Used both degree of node and the number of packets in nodes queue to select the next node to transfer packets. 38

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Example: Routing Strategies There are two major ways to enhance the capability of network in the local routing strategy (based on preferential selection): Node capability should be varied Number of packets in queue should affect to the next node selection. 39

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Example: Routing Strategies Starting with a small number of nodes (m 0 ), At each time step, a new node with m links (m m 0 ) is added to the existing graph connected to the existing nodes according to the preferential attachment: The new node will be connected to the existing node i depends on the connectivity of node i. Then we model the traffic of packets on BA. 40

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Example: Routing Strategies At each time step, there are R packets generated in the system with randomly sources and destinations selection. Determine all nodes acting as both hosts and routers to generate and delivery packets. All nodes can forward packets to other nodes each time step. This is a node capability. 41

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Example: Routing Strategies To navigate packets, all nodes search among their neighbors as a local search. If the target node of packet is found within the local searched area, the packet is forwarded directly to its target. Otherwise, the packet is forwarded to a neighbor node according to the estimated waiting time of each node. 42

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Example: Routing Strategies Where, the sum runs over the j neighbors of node l. α is an adjustable parameter. k, n and c are the degree, the number of packets in queue, and the capability of node i, respectively. The estimated waiting time of packets in the queue of i is based on the capability of i. If the node i has high capability to deliver and generate packets (c), the estimated waiting time of packets will be low. 43

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Example: Routing Strategies The total number of existing packets in the network is called a load of network as well as each node knows the traffic load of its neighbors. When the delivered packet arrives to its destination, it will be removed from the system. 44

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Example: Routing Strategies For the queue length of each node, it is assumed as unlimited queue as well as First-In- First-Out discipline is applied at each queue. Appling path iteration avoidance (PIA) to improve the capacity of network. PIA identifies that a path between a pair of nodes cannot be visited by the same packet more than two times, which is according to the realistic traffic systems. 45

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Example: Routing Strategies 46

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References Steven H. Strogatz,.Exploring complex networks, Nature 410, (8 March 2001). S. Sodsee, P. Meesad, A Local Routing Strategy Based-on Estimated Waiting Time on BA Networks, ICISE2009, December 2009, Nanjing, China. B. J. Kim, Path finding strategies in scale-free networks, Phy. Rev. E 65, , C. Yin, B. Wang, W. Wang, T. Zhou, H. Yang, Efficient routing on scale-free networks based on local information, Phy. Lett. A 351, 220, B. Wang, T. Zhou, Traffic flow and efficient routing on scale-free networks: A survey, Journal of the Korean Physical Society, Vol. 50, No.1, W. Wang, B. Wang, C. Yin, Y. Xie, T. Zhou, Traffic dynamics based on local routing protocol on a scale-free network, Phy. Rev. E 73, , L. Zhao, Y. Lai, K. Park, N. Ye, Onset of traffic congestion in complex networks, Phy. Rev. E 71, , W. Wang, C. Yin, G. Yan, B. Wang, Integrating local static and dynamic information for routing traffic Phy. Rev. E 74, , G. Zhang, B. Yuan, Towards a comprehensive understanding of routing on complex networks, 3rd EuroNGI Conference on Next Generation Internet Networks, pp(s): , May Z. Wu, G. Peng, W. Wong, K. Yeung, Improved routing strategies for data traffic in scale-free networks, Journal of Statistical Mechanics: Theory and Experiment, Iss. 11, pp , A. Barabasi, R. Albert, H. Jeong, Mean-field theory for scale-free random networks, Physica A: Statistical Mechanics and its Applications, Vol. 272, Iss. 1-2, pp(s) Y. Li, Z. Xi, C. Yin, B. Wang, An effective local routing strategy on BA network, Available online at G. Yan, T. Zhou, B. Hu, Z. Fu, B. Wang, Efficient routing on complex networks, Phy. Rev. E 73, C. Yin, B. Wang, W. Wang, G. Yen, H. Yang, Traffic dynamics based on an efficient routing strategy on scale free networks, Eur. Phys. J. B 49, pp(s): , B. Wang, W. Wang, Routing strategies in traffic network and phase transition in network traffic flow, Pramana-Journal of physics, Vol. 71, No. 2, pp(s): , Z. Chen, X. Wang, Effects of network capacity under variations of network structure and routing strategy, Proceedings of the 2006 IEEE International Conference on Networking, Sensing and Control, pp(s):150 – 155,

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