The Evolution and Integration ofSocial and Financial Networks
with ApplicationsAnna Nagurney
John F. Smith Memorial ProfessorUniversity of Massachusetts - Amherst
Cambridge Colloquium on Complexity and SocialNetworks
Kennedy School of Government December 12, 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 Discovery• Supernetworks - A New Paradigm with
Some Applications• Supernetworks Integrating Social and
Financial Networks• Social Networks and Supply Chains
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.
• Physical networks– Internet–Transportation/logistical networks–Energy/Power networks…
• Abstract networks–Social networks–Knowledge networks
No longer are networks independent of oneanother but critically linked with majorquestions arising regarding decision-making and appropriate managementtools.
Network-Based New Era
Moreover, interactions betweendecision-makers and individualscan be modeled as networks anddecision-making processes as well!
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 algorithmsfor the computation of solutions thatexploit the network structure oftransportation problems.
In a basic network problemIn a basic network problemdomain: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?
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
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
Network problems arise in othersurprising and fascinating ways forproblems, which at first glance and onthe surface, may not appear to involvenetworks at all.
The study of networks is not limited toonly physical networks but also toabstract networks in which nodes donot coincide to locations in space.
The advantages of a scientificnetwork formalism:
• many present-day problems are concernedwith flows (material, human, capital,informational, etc.) over space and timeand, hence, ideally suited as an applicationdomain for network theory;
• provides a graphical or visual depiction ofdifferent problems;
• helps to identify similarities and
differences in distinct problems through
their underlying network structure;
• enables the application of efficientnetwork algorithms;
• allows for the study of disparateproblems through a unifyingmethodology.
One of the primary purposes of scholarlyand scientific investigation is tostructure the world around us and todiscover patterns that cut acrossboundaries and, hence, help to unifydiverse applications.
Network theory provides us with apowerful methodology to establishconnections with different disciplinesand to break down 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.
• alternative behaviors of the users of thenetwork
–system-optimized versus
–user-optimized (network equilibrium),
which may lead to
paradoxical phenomena.
The Braess’ Paradox
Assume 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 Link IncreasedTravel 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.
The tools that we are using in ourdynamic network research include:
• network theory• optimization theory• game theory• variational inequality theory including
evolutionary• projected dynamical systems theory• double layered dynamics theory• network visualization tools.
x0
A Geometric Interpretation of a
Variational Inequality and a
Projected Dynamical System
EQUILIBRIA of PDS andVARIATIONAL INEQUALITIES
An important feature of any PDS is that itis intimately related to a variationalinequality problem (VI).
A Discovery through theInvestigation of NetworkDynamics in the Form ofIncreasing Travel Demand
What happens if thedemand is varied in theBraess Network?
Behaviorand
Induced Flows Matter!!!
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 used after acertain demand is reached even ifthe travel demand approachesinfinity.
Hence, in general, except for a limitedrange of travel demand, building thenew road is a complete waste!
Supernetworks may be comprised of suchnetworks as transportation,telecommunication, logistical, and/orfinancial networks.
They may be multilevel as when theyformalize the study of supply chainnetworks or multitiered as in the case offinancial networks with intermediation.
Decision-makers may be faced with multiplecriteria; thus, the study of supernetworksalso includes the study of multicriteriadecision-making.
Supernetworks:A New Paradigm
Supernetworks: A New Paradigm
SupernetworksSupernetworks
Computer ScienceComputer Science
ManagementManagementScienceScience
EngineeringEngineering
EconomicsEconomicsand Financeand Finance
A Multidisciplinary Approach
Applications of Supernetworks• Telecommuting/Commuting Decision-
Making
• Teleshopping/Shopping Decision-Making
• Supply Chain Networks with ElectronicCommerce
• Reverse Supply Chains with E-Cycling
• Knowledge Networks
• Energy Networks/Power Grids
• Financial Networks with ElectronicTransactions
A Supernetwork Conceptualization ofCommuting versus Telecommuting
A Supernetwork Framework forTeleshopping versus Shopping
The Supernetwork Structure ofa Supply Chain Network
The 4-Tiered E-Cycling Network
The Electric PowerSupply Chain Network
International Financial Networkswith Electronic Transactions
Research MotivationCan Social Networks and Financial Networks be
Unified?
Especially given the importance ofelectronic financial transactions:
– In 2001 15 million Americans paidtheir bills online with up to 46 millionexpected by 2005.
–$160 billion in mortgages were takenout online in the US (cf. Mullaney andLittle (2002)).
Strong importance of personalrelationships in financial transactions.
Definition of a Social Network
A social network is a set of actorsthat may have relationships withone another. Networks can havefew or many actors (nodes), andone or more kinds of relations(edges) between pairs of actors(Hannemann (2001)).
Roles of Social Networks inEconomic Transactions
Examples from Sociology:Granovetter (1985)Uzzi (1996)
Examples from Economics:Williamson (1983)Joskow (1988)Crawford (1990)Vickers and Waterson (1991)Muthoo (1998)
Roles of Social Networksin Economic Transactions
Examples from Marketing:Ganesan (1994)Bagozzi (1995)
Importance of Relationshipsin Financial Transactions
• Examples in the context of micro-financing–Ghatak (2002), Anthony (1997)
• Examples in the context of lending–Sharpe (1990), Petersen and Rajan
(1994, 1995), Berger and Udell (1995),Uzzi (1997, 1999), DiMaggio and Louch(1998), Arrow (1998), Wilner (2000),Burt (2000), Boot and Thakor (2000)
.Some of the RelatedFinancial Network Literature
Related Literature– Nagurney, A. and Ke, K. (2001), “Financial
Networks with Intermediation,”Quantitative Finance 1, 441-451.
– Nagurney, A. and Ke, K. (2003), “FinancialNetworks with Electronic Transactions:Modeling, Analysis, and Computations,”Quantitative Finance 3, 71-87.
– Nagurney, A. and Cruz, J. (2003),“International Financial Networks withElectronic Transactions,” in Innovations inFinancial and Economic Networks, EdwardElgar Publishers, Cheltenham, England.
– Nagurney, A. and Cruz, J. (2004),“Dynamics of International FinancialNetworks with Risk Management,”Quantitative Finance 4, 276-291.
– Nagurney, A., Wakolbinger, T., and L.Zhao, “The Evolution and Emergence ofIntegrated Social and Financial Networkswith Electronic Transactions: A DynamicSupernetwork Theory for the Modeling,Analysis, and Computation of FinancialFlows and Relationship Levels,” toappear in Computational Economics(2006)
More Related Literature
Supernetwork IntegratingSocial Networks with Financial
Networks• Models the interaction of financial and
social networks• Captures interactions among individual
sectors• Includes electronic transactions• Allows for non-investment• Incorporates transaction costs and risk• Shows the dynamic evolution of
–Financial flows and associated prices onthe financial network withintermediation
–Relationship levels on the socialnetwork
• 3 tiers of decision-makers: source agents,intermediaries and demand markets
• Source agents can transact eitherphysically or electronically with theintermediaries
• Source agents can transact directly withthe demand markets via internet links
• Intermediaries can transact eitherphysically or electronically with thedemand markets
Model Assumptions
Multicriteria Decision-Makers
Source agents and intermediaries:
• Maximize net revenue• Minimize risk• Maximize relationship value• Individual weights assigned to the
different criteria.
Role of Relationships
• Decision-makers in the network candecide about the relationship levels [0,1]that they want to establish.
• Establishing relationship levels incurssome costs.
• Higher relationship levels–Reduce transaction costs–Reduce risk–Have some additional value.
Novelty of Our Research
Supernetworks show the dynamic co-evolution of financial (financial product,price and even informational) flows andthe social network structure.
Financial flows and social networkstructure are interrelated.
Network of relationships has ameasurable economic value.
Supernetwork Structure:Integrated Financial/
Social Network System
A Source Agent’s MulticriteriaDecision-Making Problem
Optimality Condition of Source Agents
A Financial Intermediary’sMulticriteria Decision-Making
Problem
Optimality Conditions ofIntermediaries
Equilibrium Conditions for theDemand Markets
VI Formulation of theEquilibrium Conditions for the
Demand Markets
The Equilibrium StateDefinition 1: The equilibrium state of the
supernetwork integrating the financialnetwork with the social network is onewhere the financial flows and relationshiplevels between the tiers of the networkcoincide and the financial flows,relationship levels, and prices satisfy thesum of the two sets of optimalityconditions and the demand marketequilibrium conditions.
The equilibrium state is equivalent to a VI ofthe form:
Projected DynamicalSystem
The dynamic models can be rewritten as aprojected dynamical system defined bythe following initial value problem:
The set of stationary points of the projecteddynamical system coincides with the set ofsolutions of the variational inequalityproblem.
The Disequilibrium DynamicsThe trajectory of the PDS describes the
dynamic evolution of:• the financial product transactions on the
financial network• the relationship levels on the social
network• the demand market prices• the Lagrange multipliers or shadow
prices associated with theintermediaries.
The projection operation guarantees thatthe constraints underlying thesupernetwork system are not violated.
Dynamics of Demand MarketPrices
The demand market prices evolveaccording to the difference between thedemand at the market (as a function ofthe prices at the demand markets at thattime) and the amount of the financialproduct transactions.
The projection operator guarantees thatthe prices do not take on negativevalues.
Dynamics of Shadow Prices
The Lagrange multipliers/shadow pricesassociated with the intermediaries evolveaccording to the difference between thesum of the financial product transactedwith the demand markets and thatobtained from the source agents.
The projection operator guarantees thatthese prices do not become negative.
Dynamics of RelationshipLevels
The relationship levels evolve on the socialnetwork links of the supernetworkaccording to the difference between thecorresponding weighted relationshipvalue, the sum of the various marginalcosts and weighted marginal risks.
The relationship levels are guaranteed toremain within the range zero to one.
Dynamics of Financial ProductTransactions
The financial product transactions evolve onthe financial network links according tothe difference between the characteristicprice and various marginal and unit costsplus the weighted marginal risks.
These flows are guaranteed to not assumenegative values due to the projectionoperation.
Qualitative Properties
We have established:
• Existence of a solution to the VI• Uniqueness of a solution to the VI• Conditions for the existence of a
unique trajectory to the projecteddynamical system
• Convergence of the Euler method.
Computational Procedure:The Euler Method
Supernetwork Structure ofthe Numerical Examples
Financial NetworkNumerical Examples
• 2 source agents, 2 intermediaries, 2demand markets
• No electronic transactions• Transactions only between source
agents and intermediaries and betweenintermediaries and demand markets
• Financial holdings of each source agentare 20
• Variance-covariance matrices are equalto identity matrices
Financial NetworkNumerical Examples
Transaction cost functions of sourceagents
Handling cost functions of intermediaries
Transaction cost functions ofintermediaries
Financial NetworkNumerical Examples
• Demand functions
• Transaction cost functions fordemand markets
• Relationship value functions
• Relationship cost functions
Differences among FinancialNetwork Examples
• Example 1–The weight for risk and relationship
value is equal to 1.• Example 2
–The weight for relationship value forthe two source agents increased from1 to 10.
• Example 3–Like Example 2 but demand function
changed to
Financial Network Example 1Discussion
We set the weights associated with the riskfunctions and the relationship values to 1.
The financial flow on each link was equal to1. There was slack associated with thesource agent's financial transactions and,in fact, 18 units of financial flows were notallocated to any financial intermediaryfrom each source agent.
The equilibrium relationship levels were allequal to 0.
Financial Network Example 2Discussion
Now the relationship levels associated withthe source agents' transactions increasedfrom their values of 0 in Example 1 to newequilibrium levels of 1 but the financialflows stayed the same.
Hence, whereas before there were norelationships and, in effect, the socialnetwork component of the supernetworkcould be entirely eliminated, therelationship levels between the sourceagents and the financial intermediarieswere at their highest possible levels.
Financial Network Example 3Discussion
The relationship levels remained as inExample 2.
It is worth noting that in this, as in thepreceding examples, the budgetconstraint did not hold tightly for eachsource agent, that is, not all thefinancial holdings were allocated.
The top tier financial flows increased asdid the flows to the first demandmarket; the others decreased.
International Financial/Social Network System
Supply Chain/Social NetworkSystem
Characteristics of theSupply Chain Numerical
Examples
• 2 manufacturers• 2 retailers• 2 demand markets• Physical and electronic transactions
between manufacturers and retailers• Electronic transactions between
manufacturers and demand markets• Physical transactions between
retailers and demand markets
Network Structure of theSupply Chain Numerical
Examples
Supply Chain Examples: 1-3Manufacturer Information2 manufacturers
Production cost functions
Transaction cost functions
Supply Chain Examples 1-3:Retailer Information
2 retailersHandling cost functions
Transaction cost functions
Supply Chain Examples: 1-3Demand Market Information
2 demand marketsDemand functions
Transaction cost functions
Supply Chain Examples: 1-3Relationship Functions
Relationship value functions
Relationship cost functions
Differences among theSupply Chain Examples
Example 1All weights for relationship values
are equal to 1.Example 2
The weights for relationship valuesfor the two manufacturersincreased from 1 to 10.
Example 3The weights for relationship values
for the two manufacturersincreased from 10 to 20.
Supply Chain Example 1Discussion
The relationship levels were all equal to0 except for the relationship levelsbetween manufacturers and theretailers transacting via the Internet,whose relationship levels were thestrongest, i.e., equal to 1.
Hence, the supernetwork in equilibriumconsists of the supply chain networkand the links on the social networkjoining the manufacturers with theretailers through the Internet.
In Example 2 we increased the weightassociated with the relationship valuesassociated with the manufacturers.
Supply Chain Example 2Discussion
With the increase in weights associatedwith the manufacturers' relationshiplevels, the relationship levels betweenmanufacturers and the retailers for bothmodes of transaction were at thehighest levels, that is, all were equal to1.
In addition, the relationship levels betweenretailers and the demand marketsincreased. This may be due to the fact thatsince the product transactions increased itmade sense for the retailers to increasetheir relationship levels since, in view, ofthe transaction cost functions (which aredecreasing in the relationship levels),these costs would be reduced.
All the product transactions increased(relative to those obtained in Example 1),except for the transactions associatedwith B2C commerce.
Hence, the social network component (inequilibrium) in Supply Chain Example 2is much denser than that in Example 1.
We now have positive equilibriumrelationship levels not only on theInternet links between manufacturersand retailers but also on the physicallinks between manufacturers andretailers, as well as on the links on thesocial network representing retailerstransacting with the demand markets.
Supply Chain Example 3Discussion
Since the weights associated with therelationships at the manufacturersfurther relative to the weights inExample 2, the relationship levels thatwere already at level 1 could notincrease more (since they are already attheir upper bounds) even with anincrease in weight.
The network topology of the supernetworkin equilibrium for this example was thatobtained for Example 2.
Types of Simulations thatcan be Performed
We can simulate:
• Changes in production, transaction,handling, and relationship production costfunctions
• Changes in demand and risk functions• Changes in weights for relationship value
and risk• Addition and removal of actors• Addition and removal of multiple
transaction modes.
Summary
We modeled the behavior of the decision-makers, their interactions, and thedynamic evolution of the associatedvariables.
We studied the problems qualitatively aswell as computationally.
We developed algorithms, implementedthem, and established conditions forconvergence.
BellagioResearch
Team ResidencyMarch 2004
Present and Future Work
We are working on infinite-dimensionalprojected dynamical systems andevolutionary variational inequalitiesand their relationships and unification.
This allows us to model dynamicnetworks with:
• dynamic (time-dependent) suppliesand demands
• dynamic (time-dependent) capacities• structural changes in the networks
themselves.
Thank you!
For more information, seehttp://supernet.som.umass.edu
The Virtual Center for Supernetworks
