mint seminar 15 10 2010 monsuur

Standard of BehaviourNetwork Dynamics

Information and Situation AwarenessMulti-layer Networks and Stochastic Behaviour of Actors

Networks: Linking Actors’ Incentives, Informationand Influence

Herman Monsuur

Netherlands Defence Academy

presentation at MINT seminarCentre d’Economie de la Sorbonne

15 October, 2010

Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence



An examplevon Neumann-Morgenstern stable setsEffective coalitions and non-enforcementBehavioural characterisation of socially stable set

A new cap for the Ministry of Defence:White, Red or Black





von Neumann-Morgenstern stable sets

A set S ⊂ X is called a vNM stable set if

inner stabilityno y ∈ S is dominated by an x ∈ S

external stabilityevery y /∈ S is dominated by somex ∈ S





Effective coalition

Let X be the set of alternatives and let A be the set of agents ormembers of the organisation or society. R is the dominancerelation on X that is generated by effective coalitions: If there is atleast one effective coalition generating the domination of x over y ,then (x , y) ∈ R.

An effective coalition will be inclined to apply its binarydominance, but, in the larger context of the dominance relation, itmay have strategic reasons for not exercising its power.





Non-enforcement

This phenomenon of non-enforcement occurs in two instances. Adominance will not be enforced by any of its effective coalitions if

the preferred alternative is already surpressed by at least oneother effective coalitions which does enforce its preference.Such a surpressed alternative we call subdued.

effective coalitions are motivated by mutual interest:dominations are along circular patterns within the standard ofbehaviour.





Behavioural postulate 1 and 2

A solution concept φ assigns to a relation R on X in its domain asubset of 2X \ ∅. Elements of φ(R) are interpreted as stable sets.

Elements of C (R) are undominated. Therefore, for each x ∈ C (R)and (x , y) ∈ R, the effective coalitions for this dominance will feelno restraint in propagandising x to the detriment of y . Wetherefore require both

core primacyIf S ∈ φ(R), then C (R) ⊂ S .

core subductionIf S ∈ φ(R), x ∈ C (R) and (x , y) ∈ R, then y /∈ S .





Behavioural postulate 3

We consider it undesirable if a stable set S would change when theeffective coalitions, finding that the alternative x they propagandisedoes not lie in S , give up or dissolve. This property we call :

independence of non-enforced dominationsIf S ∈ φ(R) and (x , y) ∈ R, x /∈ S then S ∈ φ(R),where R ′ = R \ (x , y)





A theorem for acyclical relations

Theorem

Let the domain of a solution concept φ be the set of acyclicalrelations. Then φ satisfies core primacy, core subduction andindependence of non-enforced dominations, if and only ifφ(R) = φvNM(R) for all acyclical relations R.





Socially stable sets

A set S ⊂ X is called a socially stable set if

generalised inner stabilitya(R|S)cl = ∅external stabilityif y /∈ S , then there is an x ∈ S , such that (x , y) ∈ R

Here, a(R|S)cl is the asymmetric part of the transitive closure ofR|S . The generalised inner stability means that for x , y ⊂ S , weallow x to dominate y if, in turn, y dominates x directly orindirectly within R|S .





Examples of socally stable sets





Behavioural postulate 4

Given S ⊂ X , we introduce the S-equalised dominance relationdenoted by R⊗S :R⊗S = (x , y) ∈ R : (x , y) does not lie on a cycle in R|S.

The fourth behavioural postulate is

independence of stable cyclesIf S ∈ φ(R) then S ∈ φ(R⊗S).





Characterisation of socially stable sets

Theorem (Delver & Monsuur, 2001, SCW)

If φ satisfies core primacy, core subduction, independence ofnon-enforced domination, and independence of stable cycles,then φ(R) ⊂ φsoc(R), for all R in the domain of φ.




IntroductionUncovered setChanging the network structure

Network Evolution

Local, binary decisions shape global network structures. Weintroduce a mechanism that formalizes a possible incentive thatguides nodes in constructing their local network structure, usingthe notion of the covering relation.

Let a, b be nodes in V , a 6= b. Then a covers b in G = (V ,E ) if

for all x ∈ V \ a, (x , b) ∈ E implies (x , a) ∈ E , and

there exists at least one node c /∈ a, b ⊂ V such that(c , a) ∈ E while (c , b) /∈ E .

This means that node a covers node b if all nodes linked to b arealso linked to a and node a has at least one extra link. Intuitivelyspeaking, a outperforms b.





Uncovered set

We let U or U(G ) be the uncovered set: U = v ∈ V : there is nonode w ∈ V that covers v in the network G.





Axioms

A center φ assigns to any network G = (V ,E ) a non-empty subsetφ(G ) ⊂ V . The center φuc assigns to a network G the set ofuncovered nodes.We consider the following axiom for a center φ:

A center φ has the mediator property if for each pair ofdistinct nodes a and b, there is a shortest path connectingthese nodes, such that any node in between a and b on thispath is in φ(G ).





Characterisation of uncovered set

Theorem (Monsuur & Storcken, 2004, Operations Research)

The center set φuc is the only inclusion minimal set of nodes thatis compatible with structural equivalence, has the mediatorproperty and is stable.





Local, binary decisions

Dichotomy: A node is either covered or it is uncovered. We furtherassume that the status ‘uncovered’ is ranked higher than the status‘covered’.

The mechanism. Each step consists of taking, randomly, twodistinct nodes a and b from V . Then for the link (a, b):

if (a, b) ∈ E , it is deleted by a if the network remainsconnected and the status of node a does not decrease,

if (a, b) /∈ E , it is added if both a and b achieve a higherstatus.





Example of successor networks





Pairwise Stable Networks

Theorem (Monsuur, 2007, EJOR)

Let G be a network with U 6= X. Then there exists a sequence ofsuccessor networks that transforms G into one of the followingpairwise stable networks

a ring-network,

a uni-polar network,

a bi-polar network.





van Klaveren, Monsuur, Janssen, Schut & Eiben, 2009.Proceedings BNAIC.




Networked operationsInformation sharingStochastic behaviour of nodesA Markov model

Networks, Information and Choice. Janssen & Monsuur,2010. In: Collective Decision Making.

Networked operations offer decisive advantage through the timelyprovision and exploitation of (feedback) information andintelligence to enable effective decision-making and agile actions.

We focus on the aspect of information sharing in collaborationnetworks and discuss a feedback model for situational awareness,that combines exogenously given characteristics of nodes with theirpositioning within the (social, information or physical) networktopology.





Information feedback

We let A be the adjacency matrix, where aij ∈ [0, 1] is the extentto which value from node j is usable or transferable to node iregarding the improvement of i ’s situational awareness. We takeaii = 0 for each node i .





Feedback of operational links

We combine the transferred situational awareness, which dependson the network, with exogeneous values.

Combining operational feedback links with exogenous value.Given a scalar α and a vector of exogenous characteristics b,the value v is the unique solution of the equation:

v = αAv + b.

We say that v is ‘confirmed’ by the network structure and b.





Iteration of updating of situational awareness

Updating information in m steps yields the situation awareness vm,which for m ≥ 1 is defined recursively as follows:

v0 = b; v1 = b + αAv0 . . . vm = b + αAvm−1

Taking the limit of m to infinity, we get

v = limm→∞

vm = limm→∞

m∑k=0

αkAkb = (I − αA)−1 b

By iteration, nodes also receive information from nodes which arenot adjacent, but are two, three, or more steps away.





Network performance metric

We introduce a network performance metric which combines givencharacteristics of the nodes with the network topology. This canbe used to compare different network configurations. We define

NTb =eT v

eTb=

eT (I − αA)−1 b

eTb,

where e is a vector of 1′s. This means that we take the quotient ofthe total situational awareness after updating, and the total valueof exogenously given characteristics as expressed in the vector b.





Uncertain behaviour

At each stage k, k ≥ 1, of the process of updating informationwithin the network, the uncertain behaviour of the nodes ismodelled by a collection of iid random variables εk,ij : Ω → [0, 1],1 ≤ i , j ≤ n, such that εk,ij = 1 if an information flow betweennode j and i is possible and εk,ij = 0 otherwise.

For a fixed outcome ω in the sample space Ω the process ofupdating information in m steps yields the situation awarenessvm(ω), which is defined recursively by

vm(ω) = b + αAm(ω)vm−1(ω)

where the matrices Ak have the entries aijεk,ij .





The network metric

The network performance metric that combines the givencharacteristics of the nodes with the network topology is defined by

NTbm =E(eT vm

)eTb

= 1 +m∑

k=1

αkE(eT(∏k−1

s=0 Am−s

)b)

eTb

where E (·) denotes the expectation and e is a vector of 1′s.





Infinite m

For infinite m, the network performance metric becomes

NTb =

∫eT x dµ(x)

eTb

Here µ is the unique probability measure which satisfies theequation

µ =N∑

m=1

P (A1 = Dm) µ f −1m ,

where fm is the affine mapping fm : x 7→ b + αDmx andD1, . . . ,DN is the finite collection of all outcomes of A1(ω).





Computing the network metric

Theorem

Fix a sequence of matrices Ak(ω)k≥1 for some outcome ω in thesample space Ω. Let the orbit xn∞n=0 be defined by x0 = b andxn+1 = b + αAn+1xn. Then with probability one

NTb = limn→∞

1

n + 1

(1 +

n∑k=1

eT xk

eTb

)





Taking into account the behaviour of other nodes

Fix an arbitrary node i and an adjacent node of this node i , saynode j . From the point of view of a receiving node i , we assumethat at each stage k, k ≥ 1, node j behaves in the following way.

with probability pij(1) node j sends information to node i

with probability pij(0) it does not send information to node i ,

independently of the state of the system at stage k − 1,

andfurther

with probability 1− pij(1)− pij(0) the decision of node j tosend information depends on the outcomes of the 0-1 randomvariables εk−1,ij , 1 ≤ i , j ≤ n and εk−1,jt , t ∈ Nj .





Taking into account the behaviour of other nodes

Fix an arbitrary node i and an adjacent node of this node i , saynode j . From the point of view of a receiving node i , we assumethat at each stage k, k ≥ 1, node j behaves in the following way.

with probability pij(1) node j sends information to node i

with probability pij(0) it does not send information to node i ,

independently of the state of the system at stage k − 1, andfurther

with probability 1− pij(1)− pij(0) the decision of node j tosend information depends on the outcomes of the 0-1 randomvariables εk−1,ij , 1 ≤ i , j ≤ n and εk−1,jt , t ∈ Nj .





The Markov Model

We suggest the following (Markov) model which takes intoaccount these factors:

P (εk,ij = 1|εk−1,ij = βij , εk−1,jt = βjt , t ∈ Nj) =

pij(1)+(1− pij(1)− pij(0))1

γ + (1− γ)|Nj |

γβij + (1− γ)∑t∈Nj

βjt




Monsuur, Grant & Janssen, 2011, to appear

A node typically is not just part of one type of network, butsimultaneously belongs to multiple networks.




The modelling approach

To model behaviour of actors in interwoven networks, we use anagent-based approach:

Networks are replaced by multi-layered networks that influenceeach other.

To each actor in a network an objective function is assigned,incorporating endogenous network statistics and exogenouscovariates.


mint seminar 15 10 2010 monsuur

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