Optimal Networks, Congestion and Braess’ Paradox R.J. Mondrag´ on, Dept of Electronic Engineering and D.K. Arrowsmith School of Mathematical Sciences Queen Mary, University of London IWCSN - 2007 Guilin, CHINA
Structure and Function of Networks The problem ◮ Interest in and the investigation of dynamic networks has developed rapidly in recent years ◮ An outcome has been that the interplay between the structure and the function of a network is very important ◮ The problem can be easily described where introducing links in a network clearly creates short-cuts ◮ However, the existence of the short cuts can lead to more localized traffic and thereby local congestion ◮ This can have a profound effect on the efficiency of the network. Queen Mary University of London 2/33
Optimal Network We are interested in how to deliver efficiently information in a communications network (e.g. minimising the transit time of the information packets) Possible Approaches ◮ Given a network ‘find’ an algorithm to optimise the delivery of packets ◮ Given a packet delivering algorithm ‘build’ a network that is optimal for this algorithm R. Guimer` a et al. , Optimal network topologies for local search with congestion, Physical Review Letters, 89 , 2002 Queen Mary University of London 3/33
Outline Networks - robustconnectivity � Regular graphs � Non-regular graphs � Symmetric graphs Communication dynamics � packet traffic � load and congestion Optimization problems � star graphs and their evolution � linked systems Braess paradox � optimality and adding links Queen Mary University of London 4/33
Robust Networks Why? ◮ Increasing use of networks as infrastructure (e–commerce) ◮ Increasing threat of disruption of communications due to failure of links or nodes or attacks (lack of robustness). Solution All the nodes ‘look’ the same so no node is special. i.e. a regular graph and multiple edge connectivity Removing one node will not disrupt the ‘flow of information’. A. H. Dekker & B. D. Colbert, Network Robustness and Graph Topology, 27th Australasian Computer Science Conference, 2004 Queen Mary University of London 5/33
Networks robust to change ◮ Node connectivity = κ : Minimum number of nodes needed to be removed to obtain a disconnected network ◮ Link connectivity = η : Minimum number of links needed to be removed to obtain a disconnected network. ◮ degree of a node is the number of links(edges) attached to the vertex. ◮ d min : minimum degree in the graph. ◮ For any graph: κ ≤ η ≤ d min . Robust Networks ( Dekker & Colbert ): There are regular graphs (i.e with constant vertex degree) where κ = η = d min = d . Queen Mary University of London 6/33
Regular and Symmetric Graphs Theorem (Mader & Watkins) For any connected node–similar graph of degree d: 1. η = d (link connectivity = degree) 2. κ ≥ 2 / 3( d + 1) (node connectivity has a lower bound) 3. if d ≤ 4 , then κ = d (node connectivity = degree) 4. if the graph is symmetric, then κ = d ( Dekker & Colbert ) regular → node–similar → symmetric(edge-node similar) Queen Mary University of London 7/33
Other graphs! Scale-free models Barabasi & Albert (1999) constructed scale-free networks with key aspects: ◮ Growth :Given m 0 nodes; at every time step, a new node is introduced and is connected to m ≤ m 0 already-existing nodes. ◮ Preferential Attachment : the probability P i that a new node will be connected to node i (one of the m 0 already-existing k i nodes) depends on the degree k i of node i , P i = � j k j Queen Mary University of London 8/33
The Erdos-Renyi model ◮ For example, Erdos and Renyi proposed a simple method to generate random networks: ◮ Start with N nodes without any connection. increasing probability p ◮ Select pairs of nodes with uniform probability p ◮ Make a link between the selected vertices unless they are already connected or a self-edge is generated ◮ Repeat until E edges are present in the network. ◮ The resulting degree distribution can be shown to be Poissonian. Queen Mary University of London 9/33
Optimal Random Graphs Theorem (Bollabas - compare with robust regular graphs) For any randomly generated graph of size n, the probability that κ = η = d min approaches 1 as n → ∞ 1. Erdos-Renyi generation of graphs 2. convergence is rapid - sample of 200,000 random graphs 3. vertex size n ranging from 7 to 30 4. percentage of optimally connected graphs rise from 94.8% to 99.98%. 5. random graphs are not node-similar, but nodes are equally important in a statistical sense 6. links in the E-R algorithm are placed randomly so that no node is special by design. Queen Mary University of London 10/33
The dynamics overlay for a communications network ◮ The underlying structure is a network of nodes and links. ◮ Packets are issued by certain nodes and have to traverse the network to a destination. ◮ Each node contains a queue where packets can be stored in transit if a node has more than one packet to transmit ◮ Each node is a source of traffic ◮ The nodes produces packets at fixed rates (in space and time) ◮ The packets are sent through the shortest or least busy route. ◮ There is interest in the average delivery time of the packets and also the throughput, the percentage delivered from those sourced. Queen Mary University of London 11/33
Traffic properties on network ◮ Pictorially, as a simple example, network with N nodes (diagram is assumed on a torus) ◮ Each node contains a queue where packets can be stored in transit (if the node is busy) Queen Mary University of London 12/33
Traffic properties on network ◮ Pictorially, as a simple example, network with N nodes (diagram is assumed on a torus) ◮ Each node contains a queue where packets can be stored in transit (if the node is busy) ◮ The proportion of sources/sinks of traffic is ρ ∈ (0 , 1], i.e.# sources = ρ N ◮ Each traffic source generates, on average, the same amount of traffic Λ Queen Mary University of London 13/33
Traffic properties on network ◮ Pictorially, as a simple example, network with N nodes (diagram is assumed on a torus) ◮ Each node contains a queue where packets can be stored in transit (if the node is busy) ◮ The proportion of sources/sinks of traffic is ρ ∈ (0 , 1], i.e.# sources = ρ N ◮ Each traffic source generates, on average, the same amount of traffic Λ ◮ The packets are sent through the shortest and/or less busy route Queen Mary University of London 14/33
Traffic properties on network ◮ Pictorially, as a simple example, network with N nodes (diagram is assumed on a torus) ◮ Each node contains a queue where packets can be stored in transit (if the node is busy) ◮ The proportion of sources/sinks of traffic is ρ ∈ (0 , 1], i.e.# sources = ρ N ◮ Each traffic source generates, on average, the same amount of traffic Λ ◮ The packets are sent through the shortest and/or less busy route ◮ If one node is busy (queue busy), then another route is chosen Queen Mary University of London 15/33
Congestion with increasing load ◮ As the load Λ increases, queues start to grow ◮ Queues start to form because the nodes are receiving more packets than they can distribute ◮ The time of delivery takes longer ◮ There is often a critical zone or load Λ c at which the time delivery of packets increases dramatically. ◮ Congestion has the features of criticality in a phase transition. packet delivery time uncongested load transition Λ Queen Mary University of London 16/33
Congestion with increasing load uncongested throughpuit load transition Λ Queen Mary University of London 17/33
Queue distribution snapshot Queen Mary University of London 18/33
The network and busy nodes ◮ Fixed number of nodes and links. ◮ Each node is a source of traffic and has a queue ( M / M / 1). ◮ Each node produces the same amount of traffic. ◮ Estimate the traffic load at node i using the Betweenness Centrality (assumption: routing using shortest–path) Betweenness = C B ( i ) = number of shortest–paths that visit node i number of shortest paths between s and d 1/3 2/3 1 1 source (s) destination (d) 2/3 1/3 Queen Mary University of London 19/33
Delay and Congestion Total number of packets on the network, n ( t ): ( from Little’s law ) dn ( t ) = Λ N − n ( t ) . dt ¯ τ N = number of nodes, Λ = average traffic per node, ¯ τ = average delay. Λ N → traffic entering the network n ( t ) / ¯ τ → traffic leaving the network For low load Λ << 1, ¯ τ ≈ average shortest–path - there are small queues. Queen Mary University of London 20/33
For larger loads ¯ τ is increased by the length of the queues Average arrivals A i to the node i using the betweenness centrality C B ( i ) = Λ C B ( i ) A i = ρ Λ S ¯ ℓ ˆ (1) N − 1 Average time W i that a packet spends in the queue at node i (Poisson arrivals ( λ i ) and exponential service time ( µ i )) is ¯ W i = ρ i / (1 − ρ i )(1 /µ i ) , where ρ i = λ i /µ i Evaluating ρ i gives ρ i = Λ C B ( i ) N − 1 /µ i (2) and substituting gives � � W i = ρ i / (1 − ρ i )(1 /µ i ) = 1 Λ C B ( i ) ¯ µ i µ i ( N − 1) − Λ C B ( i ) Queen Mary University of London 21/33
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