Dynamic NetworksDynamic Networkswithwith

Applications ApplicationsAnna Nagurney

John F. Smith Memorial ProfessorUniversity of Massachusetts - Amherst

Radcliffe Institute for Advanced Study November 9, 2005

The Virtual Center forSupernetworks

http://supernet.som.umass.edu

Funding for research provided by:

National Science Foundation

AT&T Foundation

John F. Smith MemorialFund - University ofMassachusetts at Amherst

Outline of Presentation:

• Background• Scientific Study of Networks• Interdisciplinary Impact of Networks• Characteristics of Networks Today• The Braess Paradox and a New Discovery• New Challenges and Opportunities:

Unification of EVIs and PDSs• Novel Applications

Networks are pervasive in ourdaily lives and essential to thefunctioning of our societies andeconomies.

Everywhere we look:in business, science,social systems, technology,and education,

networks provide the infrastructure forcommunication, production, andtransportation.

Throughout history, networks haveevolved to serve as the foundation forconnecting humans to one another andtheir activities.

Roads were laid, bridges built, andwaterways crossed so that humans, bethey on foot, on animal, or vehiclecould traverse physical distancethrough transportation.

Communications were conducted usingthe available means of the period, fromsmoke signals, drum beats, andpigeons, to the telegraph, telephone,and computer networks of today.

The study of the efficient operation ontransportation networks dates to ancientRome with a classical example being thepublicly provided Roman road networkand the time of day chariot policy,whereby chariots were banned from theancient city of Rome at particular timesof day.

We are in a New Era of Decision-Making Characterized by:

• complex interactions amongdecision-makers in organizations;

• alternative and at times conflictingcriteria used in decision-making;

• global reach of many decisions;• high impact of many decisions;• increasing risk and uncertainty, and• the importance of dynamics and

realizing a fast and sound responseto evolving events.

The New Era is Network-Based withthe Internet providing criticalinfrastructure along withtransportation/logistical networks as wellas other telecommunication networks andenergy networks.

No longer are networks independentof one another but critically linkedwith major questions arising regardingdecision-making tools.

Events such as the 2005 KatrinaHurricane disaster, the 2003 blackoutin North America, 9/11/2001, and thecomputer viruses and wormsdemonstrate irrevocably that we mustharness the best and mostpowerful methodologies for themodeling, analysis, and solutionof complex decision-makingproblems.

Examplesof

Physical Networks

Transportation Networks

provide us with the means to crossdistance in order to conduct ourwork, and to see our colleagues,students, friends, and familymembers.

They provide us with access to foodand consumer products.

MBTA Network

Major Highway and RailroadNetworks in the US

Water Freight TransportRoutes for the US

Communication Networks

allow us to communicate withinour own communities andacross regions and nationalboundaries,

and have transformed the waywe live, work, and conductbusiness.

..

Iridium SatelliteConstellation Network

Satellite and UnderseaCable Networks

Energy NetworksEnergy Networks

provide the energy for our homes,schools, and businesses, and torun our vehicles.

New England ElectricPower Network

Duke Energy GasPipeline Network

Components of SomeCommon Physical Networks

Network System Nodes Links Flows

Transportation Intersections,Homes,Workplaces,Airports,Railyards

Roads,Airline Routes,Railroad Track

Automobiles,Trains, andPlanes,

Manufacturingand logistics

Workstations,DistributionPoints

Processing,Shipment

Components,Finished Goods

Communication Computers,Satellites,TelephoneExchanges

Fiber OpticCablesRadio Links

Voice,Data,Video

Energy PumpingStations,Plants

Pipelines,TransmissionLines

Water,Gas,Oil

US Railroad Freight Flows

Internet Traffic FlowsOver One 2 Hour Period

from Stephen Eick, Visual Insights

The scientific study ofnetworks involves:

• how to model such applications asmathematical entities,

• how to study the modelsqualitatively,

• how to design algorithms to solvethe resulting models.

The basic componentsof networks are:

• Nodes

• Links or arcs

• Flows

Nodes Links Flows

12 13 14 15

7 9

3

16

4

1817 19 20

8

1 2

6

5

10

11

Brief History of the Science of Networks

1736 - Euler credited with the earliest paperon graph theory - Konigsberg bridgesproblem.

1758 - Quesnay in his Tableau Economiqueintroduced an abstract network in theform of a graph to depict the circular flowof financial funds in an economy.

1781 - Monge, who had worked underNapoleon Bonaparte in providinginfrastructure support for his army,publishes what is probably the first paperon transportation in minimizing the costassociated with backfilling n places fromm other places with surplus brash withcost being proportional to distance.

1838 - Cournot not only states that acompetitive price is determined by theintersection of supply and demand curvesbut does it in the context of spatiallyseparate markets in which transportationcosts are included.

1841 - Kohl considered a two node, tworoute transportation network problem.

1845 - Kirchhoff wrote Laws of ClosedElectric Circuits.

1920 - Pigou studied a transportationnetwork system of two routes andnoted that the decision-makingbehavior of the users on the networkwould result in different flow patterns.

1936 - Konig published the first book ongraph theory.

1939, 1941, 1947 - Kantorovich,Hitchcock, and Koopmans consideredthe network flow problem associatedwith the classical minimum costtransportation problem and providedinsights into the special networkstructure of these problems, whichyielded special-purpose algorithms.

1948, 1951 - Dantzig published thesimplex method for linear programmingand adapted it for the classicaltransportation problem.

1951 - Enke showed that spatial priceequilibrium problems can be solvedusing electronic circuits

1952 - Copeland in his book asked "Doesmoney flow like water or electricity?"

1952 - Samuelson gave a rigorousmathematical formulation of spatialprice equilibrium and emphasized thenetwork structure.

1956 - Beckmann, McGuire, and Winstenin their book, Studies in the Economicsof Transportation, provided a rigoroustreatment of congested urbantransportation systems under differentbehavioral mechanisms due to Wardrop(1952).

1969 - Dafermos and Sparrow coined theterms user-optimization and system-optimization and develop algorithms forthe computation of solutions thatexploit the network structure oftransportation problems.

In a basic networkIn a basic networkproblem domain:problem domain:

one wishes to move the flow fromone node to another in a waythat is as efficient as possible.

Classic Examples ofNetwork Problems Are:

•The Shortest Path Problem

•The Maximum Flow Problem

•The Minimum Cost FlowProblem.

The Shortest Path Problem

1

2

3

4

5

6

2

4

2 1

3

4

2

3

21

Consider a network with OriginNode 1 and a Destination Node 6.

What is the shortest path from 1 to 6?

Applications of theShortest Path Problem

Arise in transportation andtelecommunications.

Other applications include:• simple building evacuation models• DNA sequence alignment• dynamic lot-sizing with backorders• assembly line balancing• compact book storage in libraries.

The Maximum FlowProblem

Each link has a maximum capacity.

How does one Maximize the flow froms to t, subject to the link capacities?

s

1

2

t

10, 8

1

106

Applications of theApplications of theMaximum Flow ProblemMaximum Flow Problem

• machine scheduling

• network reliability testing

• building evacuation.

The Minimum Cost FlowProblem

1

2

3

4

$2 ,13

Each link has a linear cost and amaximum capacity.

How does one Minimize Cost for agiven flow from 1 to 4?

$3 ,12

$6,6

$6 ,15

$4 ,10

Applications of the MinimumCost Flow Problem

• warehousing and distribution

• vehicle fleet planning

• cash management

• automatic chromosomeclassification

• satellite scheduling.

Network problems arise in othersurprising and fascinating ways forproblems, which at first glance andon the surface, may not appear toinvolve networks at all.

The study of networks is not limitedto only physical networks but alsoto abstract networks in whichnodes do not coincide to locationsin space.

The advantages of a scientificnetwork formalism:

• many present-day problems areconcerned with flows (material,human, capital, informational,etc.) over space and time and,hence, ideally suited as anapplication domain for networktheory;

• provides a graphical or visualdepiction of different problems;

• helps to identify similarities and

differences in distinct problems

through their underlying

network structure;

• enables the application ofefficient network algorithms;

• allows for the study of disparateproblems through a unifyingmethodology.

One of the primary purposes ofscholarly and scientificinvestigation is to structure theworld around us and to discoverpatterns that cut acrossboundaries and, hence, help tounify diverse applications.

Network theory provides us with apowerful methodology toestablish connections withdifferent disciplines and to breakdown boundaries.

Interdisciplinary Impact ofNetworks

Networks

Energy

Manufacturing

Telecommunications

Transportation

Interregional Trade

General Equilibrium

Industrial Organization

Portfolio Optimization

Flow of FundsAccounting

Engineering

Computer Science

Routing Algorithms

Economics

Biology

DNA Sequencing

Targeted CancerTherapy

Sociology

Social Networks

OrganizationalTheory

Characteristics of Networks Today• large-scale nature and complexity of

network topology;• congestion;• alternative behavior of users of the

network, which may lead to paradoxicalphenomena;

• the interactions among networksthemselves such as in transportationversus telecommunications networks;

• policies surrounding networks today mayhave a major impact not onlyeconomically but also socially, politically,and security-wise.

• large-scale nature and complexityof network topology

– In Chicago’s Regional Network, thereare 12,982 nodes, 39,018 links, and2,297,945 O/D pairs.

–AT&T’s domestic network has100,000 O/D pairs. In their call detailgraph applications (nodes are phonenumbers, edges are calls) - 300million nodes and 4 billion edges

• congestion is playing an increasingrole in transportation networks:

In the United States alone, congestionresults in $100 billion in lostproductivity annually. In Europecongestion is estimated to be $150billion.

The number of cars is expected toincrease by 50% by 2010 and todouble by 2030.

Congestion

Courtesy: Pioneer Valley Planning Commission

Wasting Away in Traffic Annual delay hours/driver

Los Angeles 93San Francisco - Oakland 72Washington, DC 69Atlanta 67Houston 63Dallas 60Chicago 58Detroit 57Boston 51Miami - Hialeah 51

Source: Texas Transportation Institute 2003 Data

• alternative behaviors of theusers of the network

–system-optimized versus

–user-optimized (network equilibrium),

which may lead to

paradoxical phenomena.

The Braess’ ParadoxAssume a network witha single O/D pair (1,4).There are 2 pathsavailable to travelers:p1=(a,c) and p2=(b,d).For a travel demand of6, the equilibrium pathflows are xp1

* = xp2* = 3

andThe equilibrium pathtravel cost isCp1

= Cp2= 83.

32

1

4

a

c

b

d

ca(fa)=10 fa cb(fb) = fb+50

cc(fc) = fc+50 cd(fd) = 10 fd

Adding a LinkIncreased Travel Cost for All!

Adding a new link creates anew path p3=(a,e,d).The original flow distributionpattern is no longer anequilibrium pattern, since atthis level of flow the cost onpath p3, Cp3

=70.The new equilibrium flowpattern network is xp1

* = xp2* = xp3

*=2.The equilibrium path travelcosts: Cp1 = Cp2 = Cp3

= 92.

32

1

4

a

c

b

d

e

ce(fe) = fe + 10

This phenomenon is relevant totelecommunications networks andthe Internet which is anotherexample of a

noncooperative network.

The Price of Anarchy!!!

The 1968 Braess article has beentranslated from German to Englishand appears as

On a Paradox of Traffic Planning

by Braess, Nagurney, Wakolbinger

in the November 2005 issue ofTransportation Science.

Transportation science hashistorically been the disciplinethat has pushed the frontiers interms of methodologicaldevelopments for networkproblems beginning with thework of Beckmann, McGuire, andWinsten (1956).

Dafermos (1980) showed that thetraffic network equilibrium (alsoreferred to as user-optimization)conditions as formulated by Smith(1979) were a finite-dimensionalvariational inequality.

In 1993, Dupuis and Nagurney provedthat the set of solutions to avariational inequality problemcoincided with the set of solutions toa projected dynamical system in Rn.

In 2002, Cojocaru proved this result forHilbert Spaces.

EQUILIBRIA of PDS andVARIATIONAL INEQUALITIES

An important feature of any PDS is that itis intimately related to a variationalinequality problem (VI).

x0

A Geometric Interpretation of a

Variational Inequality and a

Projected Dynamical System

BellagioResearch

Team ResidencyMarch 2004

We are working with ProfessorsCojocaru and Daniele on infinite-dimensional projected dynamicalsystems and evolutionary variationalinequalities and their relationshipsand unification.

This allows us to model dynamicnetworks with:

• dynamic (time-dependent) suppliesand demands

• dynamic (time-dependent) capacities• structural changes in the networks

themselves.

A New Discovery throughthe Investigation ofNetwork Dynamics inthe Form of IncreasingTravel Demand

What happens if thedemand is varied in theBraess Network?

The answer lies in thesolution of anEvolutionary

(Time-Dependent)Variational Inequality.

What happens if the demandchanges

0%25%50%75%

100%1 4 7 10 13 16 19

Demand

Percentage of demand allocated to a path

Path 3Path 2Path 1

I II III

Braess

Example

0

5

10

0 10 20Demand(t) = t

Equi

libriu

m P

ath

Flow

Paths 1 and 2Path 3

I II III

The Solution of an Evolutionary (Time Dependent)

Variational Inequality

3.64 8.88

Braess Network withTime-DependentDemands

In Regime II, the Addition of aNew Road Makes Everyone Worse

Off!

0

40

80

120

160

0 5 10 15 20

Demand

Cos

t of U

sed

Path

s

Network 1

Network 2

I II III

The new road is NEVER usedafter a certain demand isreached even if the traveldemand approaches infinity.

Hence, in general, except for alimited range of travel demand,building the new road is acomplete waste!

If the demand is a step function, the solution to theEVI has the structure:

New Challengesand

Opportunities:The Unification

of EVIs and PDSs

Double-Layered Dynamics

The unification of EVIs and PDSs allowsthe modeling of dynamic networks overdifferent time scales.

Papers:Projected Dynamical Systems and Evolutionary

Variational Inequalities via Hilbert Spaces withApplications (Cojocaru, Daniele, and Nagurney),Journal of Optimization Theory and Applications, vol.127, no. 3, pp. 1-15, December 2005.

Double-Layered Dynamics: A Unified Theory of ProjectedDynamical Systems and Evolutionary VariationalInequalities (Cojocaru, Daniele, and Nagurney),European Journal of Operational Research, in press.

A Pictorial of theDouble-Layered Dynamics

PDSt1

PDSt2

EVI

In other words, x* is a globally finite-timeattractor for the unique solution of PDStstarting at x0

t and it will be reached in ltunits of time.

A Globally Finite-Time Attractor x*

A Dynamic Network Examplewith Time-Varying Demand and

Capacities

We consider a network consisting of asingle origin/destination pair of nodesand two paths connecting these nodes.

Let cost on path 1 be: 2x1(t)-1.5 andcost on path 2 be: x2(t)-1.

The demand is t in the interval [0,2].

Suppose that we also have capacities:(0,0) ≤ (x1(t), x2(t)) ≤ (t, 3/2 t).

With the help of PDS theory, we cancompute an approximate curve ofequilibrium by choosing

Using a simple MAPLE computation, weobtain that the equilibria are the points:

Interpolating these points, we obtain theapproximate curve of network equilibria:

The tools that we are using in ourdynamic network research include:

• network theory• optimization theory• game theory• evolutionary variational inequality

theory• projected dynamical systems theory• double layered dynamics theory• network visualization tools.

Novel Applicationsof

Dynamic Networks

Supernetworks: A New Paradigm

Applications of Supernetworks• Telecommuting/Commuting Decision-

Making

• Teleshopping/Shopping Decision-Making

• Supply Chain Networks with ElectronicCommerce

• Financial Networks with ElectronicTransactions

• Reverse Supply Chains with E-Cycling

• Knowledge Networks

• Energy Networks/Power Grids

A Supernetwork Conceptualization ofCommuting versus Telecommuting

A Supernetwork Framework forTeleshopping versus Shopping

The Supernetwork Structure ofa Supply Chain Network

International Financial Networkswith Electronic Transactions

The 4-Tiered E-Cycling Network

Supernetwork Structure:Integrated Financial/Social

Network System

The Electric PowerSupply Chain Network

The Transportation NetworkEquilibrium Reformulation

Electric Power Supply TransportationChain Network Network

Preliminary Results on SolvingTime-Dependent Electric Power

Supply Chain Networks asDynamic Transportation Problems

Flow Results

t=0t=1/2t=1

More Flow Results

t=0t=1/2t=1

Thank you!

For more information, seehttp://supernet.som.umass.edu

The Virtual Center for Supernetworks