Measures in Social Networks: Description versus Prescription

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Measures in Social Measures in Social

Networks:Networks:

Description versus Description versus

PrescriptionPrescriptionJ. Todd Hamill, Major,

USAFDick Deckro

AFIT/ENS

The views expressed in this work are those of the author alone and do not represent the views of the United States

Air Force, the Department of Defense or the United States Government

22nd ISMOR / August 2005

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Overview

Problem StatementPerspectivesSNA Assumptions & Measures Implications of Non-cooperative NetworksModels

►Network Flow: Gains, Losses, & Thresholds►Flow Typology►Extensions of the Key Player Problem

The Way Ahead

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Problem Statement

Overall goal - ‘shaping intentions’ through influence►… in the context of military psychological operations

that strive to influence an adversary’s “… emotions, motives, reasoning, and ultimately, their behavior…” in order to achieve a given political goal. (JP 3-13, 1998:II-4)

Extend previous social network analysis (SNA) and operations research (OR) methodologies to generate and analyze courses of action applied to networks of individuals

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Overview



The Way Ahead

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Perspectives

Descriptive Models►A model that attempts

to describe the actual relationships and behavior of a system

►The “what is” question►For a decision problem,

such a model seeks to describe how individuals make decisions

Prescriptive Models►A model that attempts to

describe the best or optimal solution of a system

►The “what’s best” & “what if” questions

►For a decision problem, such a model is used as an aid in selecting the best alternative solution

Models never perform analysis. Analysts do analysis, aided by models where appropriate.

Provides insightPerhaps create requirements

Provides insightPerhaps create requirementsActionable Options Evaluations

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SNA vs. OR Analysis

SNA as conducted by the social scientists has, in general, been descriptive

OR has a history of both descriptive and prescriptive modeling►There is a wide array of network flow and

decision analysis models in OR

A key to utilizing this wealth of models is having good measures and metrics that are appropriate for the desired analyses AND appropriate for the mathematical models being used►GIGO remains a consideration for modeling

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Overview



The Way Ahead

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SNA Assumptions

Actors and their actions are viewed as interdependent rather than independent, autonomous units

Relational ties between actors are channels for transfer of “flow” of resources (either material or nonmaterial)

Network models focusing on individuals view the network structural environment as providing opportunities for or constraints on individual actions

Network models conceptualize structure (social, economic, political, and so forth) as lasting patterns of relations among actors

(Wasserman and Faust, 1994:4)

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SNA MeasuresLinks

(Brass, 1995:47)

Definition

Path between two actors is mediated by one or more others

AttributeIndirect LinksFrequencyStabilityMultiplexityStrengthDirectionSymmetry

1 2 3

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SNA MeasuresLinks

(Brass, 1995:47)

Definition

How many times or how often the link occurs (e.g. telephone calls, meetings, etc.)


1 2 3

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SNA MeasuresLinks

(Brass, 1995:47)

Definition

Existence of link over time


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SNA MeasuresLinks

(Brass, 1995:47)

Definition

Extent to which two actors are linked together by more than one relationship (linkages between two given actors occur within several contexts)


1 2 3

1 2 3

Co-workers

Friends

1 2 3 Either or both

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SNA MeasuresLinks

(Brass, 1995:47)

Definition

Amount of time, emotional intensity, intimacy, or reciprocal services (frequency or multiplexity often used as measure of strength of tie)


1 2 3

S(1,2) S(2,3) S(i, j) +

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SNA MeasuresLinks

(Brass, 1995:47)

Definition

Extent to which like is from one actor to another


1 2 3

1 2 3

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SNA MeasuresLinks

(Brass, 1995:47)

Definition

(Reciprocity) Extent to which relationship is bidirectional


1 2 3

1 2 3

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SNA MeasuresActors

(Brass, 1995:47)

AttributeCentralityPrestigeDegreeClosenessBetweenness

Definition

- Extent to which an actor is central to a network

- Various measures (including degree, closeness, and betweenness)

- Some measures of centrality weight an actor’s links to others by the centrality of those others

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SNA MeasuresActors

(Brass, 1995:47)


Definition

- Based on asymmetric relationships, prestigious actors are the object rather than the source of relations

- Measures similar to centrality are calculated by accounting for the direction of the relationship (i.e., in-degree)

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SNA MeasuresActors

(Brass, 1995:47)


Definition

Number of direct links with other actors- Undirected- In (to the actor)- Out (from the actor)

1 2 3

1 2 3

d2 = 2

di(2) = 2

do(2) = 0

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SNA MeasuresActors

(Brass, 1995:47)


Definition

- Extent to which an actor is close to, or can easily reach all the other actors in the network

- Usually measured by averaging geodesic distances to all other actors

- Same equation for directed and undirected networks

1

1

,g

C i i jj

C n d n n

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SNA MeasuresActors

(Brass, 1995:47)


Definition

- Extent to which an actor mediates, or falls between any other two actors on the shortest path between those two actors

- Usually averaged across all possible pairs in the network

- Same for directed/undirected networks

/B i jk i jkj k

C n g n g

number of geodesics between and jkg j k

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SNA MeasuresActor Roles

A B

Ego

AB

Ego

B

Ego

A

A B

Ego

B

Ego

A

A BEgo B

Ego

A

Ego

Liaison Representative Gatekeeper Coordinator

Itinerant Broker Bridge Isolate Star

(Degenne, 1999:129)

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SNA Measures Network

(Brass, 1995:47)

Definition

Number of actors in the network

AttributeSizeComponentConnectivityDensityCentralizationTransitivity

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(Brass, 1995:47)

Definition

- Largest connected subset of network nodes and links

- All nodes in the component are connected (either direct or indirect links) and no nodes have links to nodes outside the component


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(Brass, 1995:47)

Definition

- (Reachability)- Extent to which actors in the network

are linked to one another by direct or indirect ties

- Sometimes measured by the maximum, or average, path distance between any two actors in the network


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(Brass, 1995:47)

Definition

- Ratio of the number of actual links to the number of possible links in the network


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(Brass, 1995:47)

Definition

- Difference between the centrality scores of the most central actor and those of other actors in a network is calculated, and used to form ratio of the actual sum of the differences to the maximum sum of the differences


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(Brass, 1995:47)

Definition

- Three actors (A, B, C) are transitive if whenever A is linked to B and B is linked to C, then C is linked to A

- Transitivity is the number of transitive triples divided by the number of potential transitive triples (number of paths of length 2)


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Overview



The Way Ahead

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Non-Cooperative Networks

Terrorists are generally non-cooperative ►Al-Qaeda Training Manual (Post, 2005)► Inside Al Qaeda: How I Infiltrated the World's Deadliest Terrorist

Organization (Sifaoui, 2004) Intelligence databases are biased, large, incomplete, have

ambiguous boundaries, and are dynamic (Sparrow, 1991) Analysis of 9-11 terrorist network (Krebs, 2002)

►“Deep trusted ties not easily visible to outsiders”►“Trade efficiency for secrecy”

In general…►Secrecy improves terrorists’ probability of mission success►Trade organizational efficiency of communication for secrecy►Reliance upon observational data may lead to improper

conclusions with some centrality measures►Potential paradigm shift in SNA from prosecution to prevention

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Overview



The Way Ahead

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Social Closeness Terms Flow Model Properties

People or groupsNodes (sinks, sources, or transshipment)

Connectivity or affinityCapacitated arcs (or edges) between nodes

Social Closeness Capacity

Influence Commodity

Potential Influence Magnitude of flow

People or groups initiating influence in the network

Source(s)

Target people or groups to be influenced

Sink(s)

People or groups involved in influencing

Transshipment node(s)

Multi-Criteria within a shared contextMulti-Commodity, contexts share capacity

Multi-Context or Multi-Criteria in different contexts

Multiple independent single-commodity models for each context or criteria

Current Mappings

(Renfro, 2001:95)

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Gains, Losses, & Thresholds

Gains and losses represent predispositions, communication problems, and other similar factors based on the specific scenario under consideration.

Thresholds can also be set for cases where individuals or groups require a minimum level of influence before they take a specific course of action.

Requires Generalized Network Flow►Arcs may consume or generate flow►Seen in power networks, canals, transportation of

perishable commodities, and cash management (Ahuja, et al, 1993:8)

►Develop maximum flow and minimum cost, maximum flow approaches

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Gains, Losses, & Thresholds

Gains► Influence of significant others upon opinion shift (Friedkin

and Cook, 1990:130)► Interpersonal power (French, 1956:183-4)

Losses►Organizational structure and information flow (Lopez, et al,

2002) Thresholds

►Collective behavior and internal cost/benefit analysis; Number or proportion required at point where benefits exceed costs for that actor (Granovetter, 1978:1420)

►Applied to innovations, rumors and diseases, strikes, voting, educational attainment, leaving social occasions, migration, and experimental social psychology (Granovetter, 1978:1423-4)

►Recent model of innovation diffusion (Valente, 1996)

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Thresholds

Network Flow

“Outflow minus inflow must equal supply (or demand)”

Mass Balance Constraints (Three cases)Supply node: outflow > inflow outflow = inflow + bj

xjk – gijxij = bj

Demand node: outflow < inflow outflow = inflow – bj

xjk – gijxij = -bj

Transshipment node: outflow = inflow xjk – gijxij = 0

Amount of flow from node i to node j on arc (i, j) is xij

i j k

(Ahuja, Magnanti, and Orlin, 1993:5)

gij

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Network Flow

Given the following…►A network structure►Social closeness measures for all arcs (i, j)

The objectives…►Identify key actors that serve as ultimate

targets of influence►Identify actors that are accessible and likely

to propagate influence through the network►Identify the minimum amount of influence

required

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Underlying Assumptions

Amount of influence generated by COA is measurable► Interpretation of influence amount is inviolate among

individuals and their interactions Directed network mimics the anticipated operational

channels of communication►No discussion or interaction, as seen in traditional SNA

approaches, is modeled External Costs – Course of Action

►Represent risk friendly forces are subjected to when implementing the COA

Node “a” to all initial target nodes - execution Target nodes to “tgt” node - observation

Internal Costs – Propagation►Represent propagation risks perceived by individuals within

the network (Operational and Personal)

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Notional ExampleMinimum cost, maximum flow

atgt

{1}

{1}

{1}

1

4 52

37

6 11

10

9

8

2

½

4/3{1}

{1}

{1}

{2}

83

13

{1} 1

3

{1}

Solution (z* = 93.32)

jigij uij upper bound of flow on arc (i, j)

xij: amount of flow on arc (i, j)

Legend

{xij}

34

3

btgt = - 2ba = 3

b4 = - 1

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atgt

btgt = - tba = a (1

,6)

(1,8)

(1,7)

(1,9)

(1,10)

1

4 52

37

6 11

10

9

8

2

½

4/3

b4 = - 1

(2,8)

(3,6)

(1,2)

(7,9)(1

0,10

)

(5,6)(6,6)

(6,7

)

(6,5)(6,4)

(6,8)(6,4)

(6,9)

(6,20)

(6,24)

(6,15)

jigij

(uij, cij)

uij upper bound of flow on arc (i, j)cij: cost per unit flow on arc (i, j)gij: gain/loss factor for arc (i, j)

Legend

Post-Optimality Analysis

What if?...

gij

(uij, cij)

btgt = - tba = a

b4 = - 1

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Overview



The Way Ahead

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Flow Typology

“Most commonly used centrality measures are not appropriate for most of the flows we are routinely interested in.” (Borgatti, 2005)► Implicit assumptions regarding traffic flow► Paths Trails Walks► Serial or Parallel► Replication or Transference

Impact on network flow modeling► Dependent upon how commodity of influence is interpreted► (Generalized) network flow cannot replicate all of these

processes Paths only

► Path with potentially serial and/or parallel process May imply transference (serial) or replication (parallel)

Implications for network flow – Side constraints

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Overview



The Way Ahead

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Key Player Problem (KPP)

(KPP-1 or KPP-neg) Given a social network, find a set of k nodes (called a kp-set of order k) which, if removed, would maximally disrupt communication among the remaining nodes.► Would allow target selection in the classical sense► “Given a network of terrorists who must coordinate in order to

mount effective attacks, and given that only a small number can be intervened (e.g., by arresting or discrediting), which ones should be chosen in order to maximally disrupt the network?”

(KPP-2 or KPP-pos) Given a social network, find a kp-set of order k that is maximally connected to all other nodes.► The underlying premise is to find a set of actors that would

facilitate “the diffusion of practices or attitudes….”► “Translates to locating an efficient set of enemies to surveil,

turn (into double-agents), or feed misinformation to.”

(Borgatti, 2003:241)

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Key Player Problem

Rationale► Underlying motivation for current measures► Question of interest in sociological and military

Current Approach► New measures of “goodness”► Developed greedy heuristic

Alternative Approaches (OR techniques)► KPP-1 (Leinart, Deckro, and Kloeber, 2000)► KPP-2 (Set covering, and others…)

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KPP-2

The transpose of a modified version of the m-step reachability matrix is equivalent to the constraint matrix for both set covering and set partitioning approaches to KPP-2►Accounts for directional relationships►Math program guarantees optimal solution►Extensions of set partitioning can increase analytic

options►Measures of distance and ‘goodness’ developed by

Borgatti may serve as objective function coefficients►Small example…

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Notional ExampleKPP-2

1 1

0 1 0 0 0 0 1 1 0 0 0 0

1 0 0 1 0 0 1 1 0 1 0 0

0 0 0 1 0 0 0 0 1 1 0 0

0 1 1 0 1 0 0 1 1 1 1 0

0 0 0 1 0 1 0 0 0 1 1 1

0 0 0 0 1 0 0 0 0 0 1 1

R

X X I I 2 2

1 1 0 1 0 0

1 1 1 1 1 0

0 1 1 1 1 0

1 1 1 1 1 1

0 1 1 1 1 1

0 0 0 1 1 1

R

X X

(xi_m) Choose node i as initial target, relying upon reach of m steps or less

1 2

2

_1 1

11_1 2_1 _1 1_ 2 2 _ 2 _ 2

_

1

0,1

n

i mi m

nR Rn n

i m

Min x

ST x x x x x x

x

X X

1 3

2

4 5 61 3

2

4 5 6

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Overview



The Way Ahead

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Future Paradigm

Relationship

UniqueActor Affiliation

InfluenceAction

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Conclusions

Prescriptive approach to SNA SNA Assumptions & Measures Implications of Non-cooperative Networks Models

►Gains, Losses, & Thresholds►Flow Typology and Network Flow►Potential Extensions of the Key Player Problem

Attractive option to analyze, better understand, and predict behavior of non-cooperative networks in response to external influence

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Contact Information

J. Todd Hamill, Major, USAFCommercial:(937) 305-1662Email: [email protected]

Dr. Dick DeckroDSN: 785-6565 x 4325Commercial:(937) 255-6565 X4325DSN: 785-6565 x 4325Email: [email protected]

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Works Cited

Ahuja, R. K., Magnangti, T. L., & Orlin, J. B. (1993). Network flows: Theory, algorithms, and applications. Upper Saddle River: Prentice Hall.

Amblard, F., & Deffuant, G. (2004). The role of network topology on extremism propagation with the relative agreement opinion dynamics. Physica A, 343, 725-738.

Borgatti, S. P. (2005). Centrality and network flow. Social Networks, 27, 55-71. Borgatti, S. P. (Ed.). (2003). Identifying sets of key players in a network. international conference integration of

knowledge intensive multi-agent systems Brass, D. J. (1995). A social network perspective on human resources management. Research in Personnel and

Human Resources Management, 13, 39-79. Degenne, A., & Forsé, M. (Eds.). (1999). Introducing social networks (1st ed.). London: SAGE Publications Ltd. French, J. R. (1956). A formal theory of social power. Psychological Review, 63, 181-184. Friedkin, N. E., & Cook, K. S. (1990). Peer group influence. Sociological Methods & Research, 19(1), 122-143. Granovetter, M. S. (1973). The strength of weak ties. American Journal of Sociology, 78(6), 1360-1380. Krebs, V. E. (2002). Mapping networks of terrorist cells. Connections, 24(3), 43-52. Leenders, R. Th. A. J. (2002). Modeling social influence through network autocorrelation: Constructing the weight

matrix. Social Networks, 24, 21-47.James A. Leinart, Richard F. Deckro, Jack M. Kloeber, Jr. and Jack A. Jackson (2002). “A Network Disruption Modeling

Tool”, Military Operations Research, 7(1) 69- 77. Lopez, L., & Sanjuan, M. A. F. (2002). Relation between structure and size in social networks. Physical Review E, 65,

np. Post, J. M. (Ed.). (2005). Military studies in the jihad against the tyrants: The al-qaeda training manual. Maxwell Air

Force Base: USAF Counterproliferation Center. Renfro, R. S. (2001). Modeling and analysis of social networks. (Doctoral dissertation, Air Force Institute of

Technology).Sifaoui, M. (2004). Inside Al Qaeda: How I infiltrated the world’s deadliest terrorist orgainzation. New York:

Thunder’s Mouth Press. Sparrow, M. K. (1991). The application of network analysis to criminal intelligence: An assessment of the prospects.

Social Networks, 13, 251-274. U.S. Department of Defense (1998). Joint Publication 3-13: Joint Doctrine for Information Operations. Valente, T. W. (1996). Social network thresholds in the diffusion of innovations. Social Networks, 18, 69-89. Wasserman, S., & Faust, K. (1994). Social network analysis: Methods and applicationsCambridge University Press.

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Measures in Social Networks: Description versus Prescription

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