Optimal Networks, Congestion and Braess Paradox R.J. Mondrag on, - - PowerPoint PPT Presentation

Optimal Networks, Congestion and Braess’ Paradox

R.J. Mondrag´

• n, 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

• ptimal 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

• ptimality and adding links

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Robust Networks

Why?

◮ Increasing use of networks as

infrastructure (e–commerce)

◮ Increasing threat of disruption of

communications due to failure of links

• r 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.

◮ dmin: minimum degree in the graph. ◮ For any graph: κ ≤ η ≤ dmin.

Robust Networks (Dekker & Colbert): There are regular graphs (i.e with constant vertex degree) where κ = η = dmin = d.

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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)

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Other graphs!

Scale-free models Barabasi & Albert (1999) constructed scale-free networks with key aspects:

◮ Growth:Given m0 nodes;

at every time step, a new node is introduced and is connected to m ≤ m0 already-existing nodes.

◮ Preferential Attachment: the probability Pi that a new node

will be connected to node i (one of the m0 already-existing nodes) depends on the degree ki of node i, Pi =

ki

• j kj

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The Erdos-Renyi model

increasing probability p

◮ For example, Erdos and Renyi

proposed a simple method to generate random networks:

◮ Start with N nodes without any

connection.

◮ 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.

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Optimal Random Graphs

Theorem

(Bollabas - compare with robust regular graphs) For any randomly generated graph of size n, the probability that κ = η = dmin 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.

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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.

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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)

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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 = ρN

◮ Each traffic source generates, on

average, the same amount of traffic Λ

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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 = ρN

◮ 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 = ρN

◮ 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

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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 Λc at which the

time delivery of packets increases dramatically.

◮ Congestion has the features of criticality in a phase

transition.

Λ load packet delivery time uncongested transition

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Congestion with increasing load

Λ load throughpuit uncongested transition

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Queue distribution snapshot

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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 = CB(i) = number of shortest–paths that visit node i number of shortest paths between s and d 1 1 1/3 2/3 2/3 1/3 source (s) destination (d)

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Delay and Congestion

Total number of packets on the network, n(t): (from Little’s law) dn(t) dt = ΛN − n(t) ¯ τ . N = number of nodes, Λ = average traffic per node, ¯ τ= average delay. ΛN → traffic entering the network n(t)/¯ τ → traffic leaving the network For low load Λ << 1, ¯ τ ≈ average shortest–path - there are small queues.

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For larger loads ¯ τ is increased by the length of the queues Average arrivals Ai to the node i using the betweenness centrality Ai = ρΛS ¯ ℓ ˆ CB(i) = ΛCB(i) N − 1 (1) Average time Wi that a packet spends in the queue at node i (Poisson arrivals (λi) and exponential service time (µi)) is ¯ Wi = ρi/(1 − ρi)(1/µi), where ρi = λi/µi Evaluating ρi gives ρi = ΛCB(i) N − 1 /µi (2) and substituting gives ¯ Wi = ρi/(1 − ρi)(1/µi) = 1 µi

• ΛCB(i)

µi(N − 1) − ΛCB(i)

• Queen Mary University of London

21/33

The steady state solution is ¯ n =

N

• i=1

1 µi ρΛ(N − 1) µi(N − 1) − ρΛCB(i). where

◮ ρΛN is the number of packets generated per unit of time by

the whole network,

◮ ¯

ℓ is the average shortest path of the network to account for the average number of packets that were produced in the past and they are still in transit and,

◮ ˆ

CB(i) is the proportion of all the packets in transit that pass through the node i.

◮ (to simplify we use the relationship N i=1 CB(i) = N(N − 1)¯

ℓ, that is a consequence of how we defined betweenness).

◮ ˆ

CB(i) is the normalised betweenness CB(i)/ N

i=1 CB(i) and it

is used as a probability density.

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Rewiring the Network

◮ Given a load Λ ◮ the number of nodes N and links L ◮ find the network with minimum average delay (minimise ¯

n)

The rewiring is done using simulated annealing

Polarization =

ℓ∗ ℓ(Λ) − 1

ℓ∗ = the average shortest path for the largest congestion load, ℓ(Λ) = is the average shortest path depending on load Λ

Low load High load

1 2 3 4 5 6 7 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

"allResults.dat" u 1:(673/$2-1)

load polarisation

"star1-44.dat" "star3-45.dat" "star5-8.dat" "star7.dat"

45.5

48.5

number of nodes = number of links

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Some Properties

If N number of nodes is equal to L the number of links then

◮ we can evaluate analytically the betweenness, if the graph has

S ‘stars’ CB(ray) = N − 1 CB(centre) = CB(SK)N2 − NS + N2S + S2 − NS2 S2 where CB(SK) is the betweenness of the skeleton graph.

skeleton

◮ we can evaluate the optimal networks as a function of the load ◮ 3–star ⇒ 4–star ⇒ 5–star . . .

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From Stars to Regular Graphs

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

"allResults.dat" u 1:(89/$2-1)

load polarisation

(a)

girth=4 girth=5 girth=4

"allResults.dat" u 1:(89/$2-1) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 "AllResult.dat" u 1:((108-$2)/$2)

relative chagne of average shortest path load

2(number of nodes) = number of links

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Which Regular Graphs?

◮ These graphs are all regular

(a) (b) (c) (d) (e) (f) (a) (b) (c) (d) (e) (f)

◮ The sum of the betweenness is the same i CB(i) = 80 ◮ The average shortest path is the same. ◮ but the nodes congest at different loads ◮ the girths are different

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Entangled Networks and Synchronisation

Very briefly (from review by Donetti et al. ) dxi dt = F(xi) − σ

• j

LijH(xj) F(xi) describes the evolution, H(xj) the coupling between neighbours and σ is a constant. L = [Lij] is the Laplacian matrix where Lij =      −1 if there is a link between i and j ki if j=i, and ki is the degree of node i if there is no link between i and j

• L. Donetti et al. Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all
• that. arXiv:cond:mat/0605565

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Entangled Network and Synchronisation

The eigenvalues of L are 0 = λ1 < λ2 ≤ · · · ≤ λN. ‘Robust’ synchronised states exist if the ratio Q = λN/λ2 is as small as possible; (Barahona and Pecora, Wang and Chen) Given system restrictions on λN, the smallness of Q relates to a larger ”spectral gap” given by the value of λ2. Outcomes of optimization

◮ Homogeneous regular networks (entangled networks Donetti et

al).

◮ They are in fact Ramanujan graphs (known to be extremal for

spectral gaps)

◮ long loops (large girth)

Synchronisation is not necessarily a property wanted in communication networks (route flapping). Large girth means that if a link fails, there is a long detour for information delivery.

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Entangled Networks and Random Walks

◮ The transition probability on the link from node i to node j is

pij = Aij/degi, A = [Aij] is the adjacency matrix. This gives a transition probability matrix P = [Pij].

◮ The stationary distribution is given by the largest eigenvalue

• f P, i.e λN = 1. The convergence rate to the stationary

distribution is controlled by the second largest eigenvalue λ2

• f P, hence spectral gap is important again.

◮ The network average of first transition times between two

sites i and j for k regular graphs with N vertices is τ = 2k N N − 1

• l=2,...N

1 λ′

l

(3) The eigenvalues λ′

i are for the normalized Laplacian.

Random walks grow well in large spectral gap graphs.

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Adding Links

◮ In a rectangular–toroidal network the

addition of a small amount of random links reduces the value of the critical load in–spite of the increased connectivity between the nodes (Fk´

s and Lawniczak)

Braess’ paradox: Each user chooses to minimise their expected delay by choosing an ’optimal’ route. The addition of an extra link and hence route choice could reduce the delay. The paradox is that this is usually true for uncongested networks but it is less likely to be true for a congested network.

• D. Braess, Unternehmensfoschung, 12:258-268, 1968

Networks with Qs, Kelly et al, Calvert et al Queen Mary University of London 30/33

Adding Links and Braess’ Paradox

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

"allBraess.dat" u ($1/30*0.352):(35/$2) "allBraess.dat" u ($1/30*0.352):(35/$3) "allBraess.dat" u ($1/30*0.352):(35/($3-sqrt($4-$3*$3))) "allBraess.dat" u ($1/30*0.352):(35/($3+sqrt($4-$3*$3)))

load critical load

adding one link

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

"allBraess.dat" u ($1/30*0.352):(35/$2) "allBraess.dat" u ($1/30*0.352):(35/$3) "allBraess.dat" u ($1/30*0.352):(35/($3-sqrt($4-$3*$3))) "allBraess.dat" u ($1/30*0.352):(35/($3+sqrt($4-$3*$3)))

load critical load

• riginal
• riginal+1 link

adding one link and then

• ptimising.

The paradox occurs when the graph is regular, adding a link and

• ptimising reduces the congestion threshold substantially.

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More Questions and Some Conclusions

◮ Low loads: For simple networks (L = N) it can be solved, can

we use symmetry to obtain (an approximation of) the optimal solution?

◮ ‘Middle’ of the range loads: Look like entangled networks, are

they?

◮ desirable qualities: adding new links has a small effect on the

performance of the network (Braess).

◮ High loads: They seem to be node–regular networks (or even

symmetric).

◮ Girth changes with the load and is relatively large (perhaps not

a good characteristic in communication networks).

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Problem

The optimization techniques work but do not give the networks that are physically appearing in the evolution of real networks. So what should we be optimizing dynamically to get a good match between theory and experiment?

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