optimal networks, congestion and braess’ paradoxcktse.eie.polyu.edu.hk/csn2007/presentation...

Optimal Networks, Congestion

and

Braess’ Paradox

R.J. Mondragon, Dept of Electronic Engineering

and

D.K. Arrowsmith School of Mathematical Sciences

Queen Mary, University of London

IWCSN - 2007Guilin, CHINA

Structure and Function of Networks

The problem

I Interest in and the investigation of dynamic networks hasdeveloped rapidly in recent years

I An outcome has been that the interplay between the structureand the function of a network is very important

I The problem can be easily described where introducing linksin a network clearly creates short-cuts

I However, the existence of the short cuts can lead to morelocalized traffic and thereby local congestion

I This can have a profound effect on the efficiency of thenetwork.

Queen Mary University of London 2/33

Optimal Network

We are interested in how to deliver efficiently information in acommunications network (e.g. minimising the transit time of theinformation packets)

Possible Approaches

I Given a network ‘find’ an algorithm to optimise the delivery ofpackets

I Given a packet delivering algorithm ‘build’ a network that isoptimal for this algorithm

R. Guimera et al., Optimal network topologies for local search with congestion, Physical Review Letters,89, 2002


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


Robust NetworksWhy?

I Increasing use of networks asinfrastructure (e–commerce)

I Increasing threat of disruption ofcommunications due to failure of linksor nodes or attacks (lack ofrobustness).

SolutionAll the nodes ‘look’ the same so no node isspecial. i.e. a regular graph and multipleedge 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


Networks robust to change

I Node connectivity = κ: Minimum number ofnodes needed to be removed to obtain adisconnected network

I Link connectivity = η: Minimum number oflinks needed to be removed to obtain adisconnected network.

I degree of a node is the number oflinks(edges) attached to the vertex.

I dmin: minimum degree in the graph.

I For any graph: κ ≤ η ≤ dmin.

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


Regular and Symmetric Graphs

Theorem(Mader & Watkins) For any connected node–similar graph ofdegree 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)interfaces, circuit boards, etc. would primarily putlinks out of action. On the other hand, when mod-elling network robustness of military networks in theface of combat (and indeed also of civilian networksin the face of terrorist activity), the major threat isthe destruction of entire nodes (usually by explosivemeans). In this case, we would expect node connec-tivity to be the most useful in modelling robustness.In related work (Dekker 2004) we describe a simplecombat simulation experiment which shows that thisis, in fact, the case.

The following well-known theorem, due to Menger,provides an alternative formulation of node and linkconnectivity:

Theorem 2.2 For any graph:

(i) the node connectivity ! is the smallest number ofnode-distinct paths between any two nodes.

(ii) the link connectivity " is the smallest number oflink-distinct paths between any two nodes.

Proof. Corollaries 4.2 and 4.3 of Gibbons (1985) !

These connectivity measures can be calculated us-ing the maximum-flow algorithm (Gibbons 1985), andwe have developed a network design and analysis toolcalled CAVALIER which incorporates these calcula-tions.

The CAVALIER tool also performs statistical andgraph-theoretical network analyses, 2-dimensionaland 3-dimensional visualisation (Dekker 2001), andhas a simulation capability to assess network perfor-mance (Dekker 2003b). All the figures in this paperwere produced using the CAVALIER tool.

There are well-known bounds on ! and ":

Theorem 2.3 For any graph, ! ! " ! dmin.

Proof. Due to Whitney: see Theorem 5.1 of Harary(1969) or Theorem 2.9 of Gibbons (1985). !

If ! = " = dmin for some graph, we say that thegraph is optimally connected, since the node and linkconnectivities are as high as possible, i.e. the networkis as robust as it could be, given the value of dmin. InSection 3, we consider several strategies for designingoptimally connected graphs.

3 Optimal Connectivity

Having discussed the importance of connectivity inmodelling robustness of a network topology, we nowreview some results from the graph-theoretic litera-ture relating connectivity of a graph to the degree ofsymmetry, and we describe several cases of optimallyconnected graphs, i.e. graphs with ! = " = dmin.

Definition 3.1 The following concepts of symmetrywill be used:

(i) We say that a graph is regular if every node hasthe same degree; in this case we also speak of thedegree d of the graph.

(ii) An automorphism of a graph is a permutation #of the nodes which preserves links, i.e. a — b is alink if and only if #a — #b is a link.

(iii) A graph is node-similar (more usually, vertex-transitive) if for any two nodes a and b there isan automorphism # such that #a = b.

(iv) A graph is symmetric if for any two links a — band x— y there is an automorphism # such that#a = x and #b = y.

(b)(a)

Figure 1: Two Regular Graphs

Our CAVALIER tool includes a module to enumer-ate all the automorphisms of a graph, and to checkthe conditions of node-similarity and symmetry (forsmall or sparse graphs, where this is feasible).

Essentially the condition of node-similarity saysthat all nodes “look the same,” while symmetry saysthat all links “look the same.” Clearly a symmetricgraph must be node-similar, and a node-similar graphmust be regular, but the converses of these implica-tions do not hold. Figure 1(a) shows a graph which isregular but not node-similar. The “soccer-ball” graphin Figure 1(b) is node-similar (every node is the in-tersection of a pentagon and two hexagons) but notsymmetric (some links join a pentagon and a hexagon,while others join two hexagons). The “soccer-ball”is one of the 13 semi-regular polyhedra described byArchimedes and Kepler, all of which are node-similar(essentially by definition).

Examples of symmetric graphs include the regularpolyhedra: tetrahedron (d = 3), cube (d = 3), octahe-dron (d = 4), dodecahedron (d = 3), and icosahedron(d = 5). The q-dimensional hypercube (with n = 2q

and d = q) is also symmetric, and has been used ina highly parallel computing (van de Goor 1989). The“Connection Machine” (Hillis 1985) was a supercom-puter constructed as a 12-dimensional hypercube.

The torus (a rectangular p " q grid with p, q # 3,where the top edge is connected to the bottom, andthe left to the right) is node-similar, and symmetric ifp = q. The torus has also been used in highly parallelcomputing, and has the advantage of requiring fewerlinks than the hypercube.

The property of being node-similar is of value inits own right for parallel computing, since it tendsto facilitate routing and load-balancing (van de Goor1989). For communications network design, node-similarity also facilitates routing, and ensures that theimpact of losing a node does not depend on whichnode is lost. The property of symmetry is also ofvalue, because when links “look the same,” tra!c oneach link is likely to be approximately equal.

Node-similar networks are particularly appro-priate for decentralised Network Centric Warfare(Dekker 2003a), since all nodes are of equal im-portance. As military forces move from “platform-centric” to “network-centric” organisational struc-tures, decentralised architectures become more im-portant, since they provide no high-value targetsto the enemy. On the other hand, decentralisedforces can still focus their attention on a given point,through “swarming” (Arquilla & Ronfeldt 2000) orself-synchronisation (Alberts et al. 1999) behaviour.

Within the civilian telecommunications infrastruc-ture, making the major communications backbonenode-similar would also have the advantage of pro-viding no high-value targets to terrorist attackers.

If a network is both node-similar and optimallyconnected, then it provides maximum resistance to

regular → node–similar → symmetric(edge-node similar)


Other graphs!

Scale-free modelsBarabasi & Albert (1999) constructed scale-free networks with keyaspects:

I Growth:Given m0 nodes;at every time step, a new node is introduced and is connectedto m ≤ m0 already-existing nodes.

I Preferential Attachment: the probability Pi that a new nodewill be connected to node i (one of the m0 already-existingnodes) depends on the degree ki of node i , Pi = ki∑

j kj


The Erdos-Renyi modelin

crea

sing

pro

babi

lity

p

I For example, Erdos and Renyiproposed a simple method to generaterandom networks:

I Start with N nodes without anyconnection.

I Select pairs of nodes with uniformprobability p

I Make a link between the selectedvertices unless they are alreadyconnected or a self-edge is generated

I Repeat until E edges are present inthe network.

I The resulting degree distribution canbe shown to be Poissonian.


Optimal Random Graphs

Theorem(Bollabas - compare with robust regular graphs) For any randomlygenerated 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% to99.98%.

5. random graphs are not node-similar, but nodes are equallyimportant in a statistical sense

6. links in the E-R algorithm are placed randomly so that nonode is special by design.


The dynamics overlay for a communications network

I The underlying structure is a network of nodes and links.

I Packets are issued by certain nodes and have to traverse thenetwork to a destination.

I Each node contains a queue where packets can be stored intransit if a node has more than one packet to transmit

I Each node is a source of traffic

I The nodes produces packets at fixed rates (in space and time)

I The packets are sent through the shortest or least busy route.

I There is interest in the average delivery time of the packetsand also the throughput, the percentage delivered from thosesourced.


Traffic properties on network

I Pictorially, as a simple example,network with N nodes (diagram isassumed on a torus)

I Each node contains a queue wherepackets can be stored in transit (if thenode is busy)

I The proportion of sources/sinks oftraffic is ρ ∈ (0, 1], i.e.#sources = ρN

I Each traffic source generates, onaverage, the same amount of traffic Λ

I The packets are sent through theshortest and/or less busy route

I If one node is busy (queue busy), thenanother route is chosen


Congestion with increasing loadI As the load Λ increases, queues start to grow

I Queues start to form because the nodes are receivingmore packets than they can distribute

I The time of delivery takes longer

I There is often a critical zone or load Λc at which thetime delivery of packets increases dramatically.

I Congestion has the features of criticality in a phasetransition.

Λload

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et d

eliv

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time

uncongested

transitionQueen Mary University of London 16/33

Congestion with increasing load

Λload

thro

ughp

uit uncongested

transition


Queue distribution snapshot


The network and busy nodes

I Fixed number of nodes and links.

I Each node is a source of traffic and has a queue (M/M/1).

I Each node produces the same amount of traffic.

I Estimate the traffic load at node i using the BetweennessCentrality (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)


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= averagedelay.

ΛN → traffic entering the network

n(t)/τ → traffic leaving the network

For low load Λ << 1, τ ≈ average shortest–path - there are smallqueues.


For larger loads τ is increased by the length of the queuesAverage arrivals Ai to the node i using the betweenness centrality

Ai = ρΛS ¯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

I ρΛN is the number of packets generated per unit of time bythe whole network,

I ¯ is the average shortest path of the network to account forthe average number of packets that were produced in the pastand they are still in transit and,

I CB(i) is the proportion of all the packets in transit that passthrough the node i .

I (to simplify we use the relationship∑N

i=1 CB(i) = N(N − 1)¯,that is a consequence of how we defined betweenness).

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

i=1 CB(i) and itis used as a probability density.


Rewiring the Network

I Given a load Λ

I the number of nodes N and links L

I find the network with minimum average delay (minimise n)The rewiring is done using simulated annealing

Polarization = `∗

`(Λ) − 1

`∗ = the average shortestpath for the largest congestionload,`(Λ) = is the average shortestpath depending on load Λ

Low load High load

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

load

polarisation

(a) (b)

(c) (d)

Figure 2: The polarisation of the network decreasesas the load is increased. For low load ! = 0.01 net-work is a (a) 1–star network as the load increases itchanges to a (b) 3–star network (! = 0.015), to a (c)5–star network. The transitions are marked witha vertical dashed line. As the load increases thenetwork changes topologies to networks with morestars. For values greater than ! ! 0.1 (d) the net-work has the topology of a ring. The network has45 nodes and 45 links.

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in tne n

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0.006 0.008 0.01 0.012 0.014

0.016

f2(x)/xf3(x)/x

f4(x)/x

!1,2" !1,3

" !2,3"

Figure 3: Total number of packets for di!erenttopologies as a function of the load. The 1–star–like(solid line), 3–star (long dashed line), 5–star (shortdashed line) and 7–star network (dotted line). Theinset shows that total number of packets for the 2–star network is always greater than total number ofpackets for the 1–star (solid line) or 3–star networks(dashed line), implying that the 2–star network isnot an optimal solution.

Figure 3 shows the total number of packets in the networkas a function of the load for the 1–star–like (solid line), 3–star–like (long dashed line), 5–star–like (short dashed line)and 7–star–like network (dotted line). The change from onetopology to another are marked with vertical dashed lines.Notice that not all star–like networks are optimal and thereare values of the load where there are are least two possibleoptimal solutions. The inset in figure 3 is an amplification ofthe behaviour of the 1–, 2– and 3– star–like networks for lowload. The 2–star–like network is not an optimal solution.

The case that the network has S nodes and L = 2S linksis interesting because it includes the case of the Manhattannetwork which has been considered by many researchers [19,11, 24] when looking at the transition to congestion. TheManhattan network is a toroidal regular symmetric meshnetwork with shortest path " =

"S/2 and constant between-

ness CB ="

S(S# 1)/2. Figure 4 shows the transition fromstar networks to regular networks for the case S = 36 andL = 72. As previously observed [12] there is a large drop onthe polarisation value as the load is increased. What we no-ticed is that the networks do not change abruptly from star–like to regular networks. As the load increases the networkchanges from star–like networks with increasing number ofcentres. There is an interval where the networks are notstar–like or regular networks (the interval 0.05 < ! < 0.014marked on the figure). In this interval the degree of thenode fluctuates closely to the average degree 2L/S = 4. Asthe load increases the topology changes to a regular graph.To check if the network is node–regular or symmetric, weevaluated the node and link connectivity of the graph. Af-ter ! > 0.014, the graphs are all regular symmetric but thisoptimal solution does not give a unique topology. In thefigure we have marked the girth of the regular symmetricnetworks. Notice that even that the girth is relatively largethis does not imply that the optimal network is an entangled

number of nodes = number of links


Some Properties

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

I we can evaluate analytically the betweenness, if the graph hasS ‘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

I we can evaluate the optimal networks as a function of the load

I 3–star ⇒ 4–star ⇒ 5–star . . .


From Stars to Regular Graphsnetwork as defined by Donetti et al. [7]. Also it is not clearif the synchronisation of the individual dynamical process ina communications network is a desirable property. An ex-ample of synchronised but unwanted dynamical behaviour isroute flapping. In this case a router alternatively forwardstra!c to a given destination via one route then another. Inthis case the router has to re–evaluate a new routing ta-ble and can create alternated congestion points between thedi"erent routes.

We also observed that, on average, the centres of the starstend to have a degree equal to number of links L divided bythe number of centres. This observation is interesting fortwo reasons. It can give us an indication when the star–likenetwork disappear and also that the star–like networks looklike a bipartite graph (see figure 4c).

As the load increases, the number of centres c in the star–like networks increases. As the total number of links is fixed,the degree of the centres decreases and the degree of the raysincreases. The stars disappear when the degree of the centreand the rays are the same. The average degree of the nodesare L/c and the average degree for the rays is L/(S ! c).The star–like networks will disappear when the degree of itscentres is the same as the degree of the rays, which happenswhen half of the nodes are centres. We were able to verifythat this is the case by observing the appearance of starsas the load increases for the cases S = 32, L = 72, 108;S = 64,L = 201.

The colouring of graphs has important applications incommunications networks. For example in the e!cient allo-cation of frequencies in wireless networks [13] or circuits inoptical networks [18]. As the node and link colouring prob-lem of bipartite graphs is well understood [23] we verifiedif, in general, a bipartite graph can be an optimal network.To do so we try a network with S = 32, L = 108. Withthis number of nodes and links it is possible to construct abipartite graph where one set of nodes corresponds to ninenodes with 24 links each, and the other set of nodes is 27nodes with 4 links each. Using the optimisation algorithmwe found out that bipartite graphs are not optimal networks.

A remark about our naming of the nodes in the star–like networks. For low loads, we called the nodes with thehighest degree the centres of the star–like network. Noticethat for the case where the number on links is equal to thenumber of nodes, our definition of a “centre” corresponds thedefinition of a centre of a graph (that is the node v with theproperty that the maximum shortest–path between v andany other node is as small as possible). However, we cannotgeneralise this observation to optimal networks where thenumber of nodes and links is not the same.

Distance to congestionIn normal operation the nodes produce time varying loads

#(t) which if large enough can cause congestion. To char-acterise how far is an optimal network from congestion wemeasured its distance to congestion #c ! # and its relativedistance to congestion

dc =#c ! #

#c, (20)

where # is the load for which the network is optimal and #c

is the load for which the optimal network is congested. Theresults are shown in figure 5. For low load #, the optimalnetwork operates far from its congestion point. For high #,as expected, the network will operate closer to its congestion

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girth=4 girth=5 girth=4

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load

polarisation

(b)

(c)

Figure 4: (a) Polarisation for a network with 36nodes and 72 links. From 0 < # < 0.05 the net-works are star–like, from 0.05 < # < 0.145 the degreeof nodes fluctuate closely to the average degree L/Swhere L. From 0.05 < # < 0.38 the optimal solutionare regular networks. (b) Change of the polarisationfor low loads. (c) A five star–like network.

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"AllResult.dat" u 1:((108-$2)/$2)

rela

tive

chag

ne o

f ave

rage

shor

test

pat

hload

2(number of nodes) = number of links


Which Regular Graphs?

I These graphs are all regular

(a) (b) (c)

(d) (e) (f)

(a) (b) (c)

(d) (e) (f)

I The sum of the betweenness is the same∑

i CB(i) = 80

I The average shortest path is the same.

I but the nodes congest at different loads

I the girths are different


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 betweenneighbours 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

0 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


Entangled Network and SynchronisationThe eigenvalues of L are 0 = λ1 < λ2 ≤ · · · ≤ λN .‘Robust’ synchronised states exist if the ratio Q = λN/λ2 is assmall as possible; (Barahona and Pecora, Wang and Chen)

Given system restrictions on λN , the smallness of Q relates to alarger ”spectral gap” given by the value of λ2.Outcomes of optimization

I Homogeneous regular networks (entangled networks Donetti et

al).

I They are in fact Ramanujan graphs (known to be extremal forspectral gaps)

I long loops (large girth)

Synchronisation is not necessarily a property wanted incommunication networks (route flapping).

Large girth means that if a link fails, there is a long detour forinformation delivery.


Entangled Networks and Random Walks

I The transition probability on the link from node i to node j ispij = Aij/degi , A = [Aij ] is the adjacency matrix. This gives atransition probability matrix P = [Pij ].

I The stationary distribution is given by the largest eigenvalueof P, i.e λN = 1. The convergence rate to the stationarydistribution is controlled by the second largest eigenvalue λ2

of P, hence spectral gap is important again.

I The network average of first transition times between twosites i and j for k regular graphs with N vertices is

τ = 2kN

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.


Adding Links

I In a rectangular–toroidal network theaddition of a small amount of randomlinks reduces the value of the criticalload in–spite of the increasedconnectivity between the nodes (Fks

and Lawniczak)

Braess’ paradox: Each user chooses to minimise their expecteddelay by choosing an ’optimal’ route. The addition of an extra linkand hence route choice could reduce the delay.The paradox is that this is usually true for uncongested networksbut 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


Adding Links and Braess’ Paradox

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"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)))

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oad

adding one link

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"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

criti

cal l

oad

originaloriginal+1 link

adding one link and thenoptimising.

The paradox occurs when the graph is regular, adding a link andoptimising reduces the congestion threshold substantially.


More Questions and Some Conclusions

I Low loads: For simple networks (L = N) it can be solved, canwe use symmetry to obtain (an approximation of) the optimalsolution?

I ‘Middle’ of the range loads: Look like entangled networks, arethey?

I desirable qualities: adding new links has a small effect on theperformance of the network (Braess).

I High loads: They seem to be node–regular networks (or evensymmetric).

I Girth changes with the load and is relatively large (perhaps nota good characteristic in communication networks).


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


optimal networks, congestion and braess’ paradoxcktse.eie.polyu.edu.hk/csn2007/presentation...

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