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UNCLASSIFIED Measures in Social Networks: Description versus Prescription J. Todd Hamill, Major, USAF Dick 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 22 nd ISMOR / August 2005

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UNCLASSIFIED 2 Overview Problem Statement Perspectives SNA Assumptions & Measures Implications of Non-cooperative Networks Models ► Network Flow: Gains, Losses, & Thresholds ► Flow Typology ► Extensions of the Key Player Problem The Way Ahead

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UNCLASSIFIED 3 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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UNCLASSIFIED 4 Overview Problem Statement Perspectives SNA Assumptions & Measures Implications of Non-cooperative Networks Models ► Network Flow: Gains, Losses, & Thresholds ► Flow Typology ► Extensions of the Key Player Problem The Way Ahead

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UNCLASSIFIED 5 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 insight Perhaps create requirements Provides insight Perhaps create requirements Actionable Options Evaluations

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UNCLASSIFIED 6 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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UNCLASSIFIED 7 Overview Problem Statement Perspectives SNA Assumptions & Measures Implications of Non-cooperative Networks Models ► Network Flow: Gains, Losses, & Thresholds ► Flow Typology ► Extensions of the Key Player Problem The Way Ahead

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UNCLASSIFIED 8 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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UNCLASSIFIED 9 SNA Measures Links (Brass, 1995:47) Definition Path between two actors is mediated by one or more others Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry 123

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UNCLASSIFIED 10 SNA Measures Links (Brass, 1995:47) Definition How many times or how often the link occurs (e.g. telephone calls, meetings, etc.) Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry 123

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UNCLASSIFIED 11 SNA Measures Links (Brass, 1995:47) Definition Existence of link over time Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry

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UNCLASSIFIED 12 SNA Measures Links (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) Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry 123 123 Co-workers Friends 123 Either or both

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UNCLASSIFIED 13 SNA Measures Links (Brass, 1995:47) Definition Amount of time, emotional intensity, intimacy, or reciprocal services (frequency or multiplexity often used as measure of strength of tie) Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry 123 S(1,2)S(2,3) S(i, j) +

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UNCLASSIFIED 14 SNA Measures Links (Brass, 1995:47) Definition Extent to which like is from one actor to another Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry 123 123

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UNCLASSIFIED 15 SNA Measures Links (Brass, 1995:47) Definition (Reciprocity) Extent to which relationship is bidirectional Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry 123 123

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UNCLASSIFIED 16 SNA Measures Actors (Brass, 1995:47) Attribute Centrality Prestige Degree Closeness Betweenness 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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UNCLASSIFIED 17 SNA Measures Actors (Brass, 1995:47) Attribute Centrality Prestige Degree Closeness Betweenness 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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UNCLASSIFIED 18 SNA Measures Actors (Brass, 1995:47) Attribute Centrality Prestige Degree Closeness Betweenness Definition Number of direct links with other actors - Undirected - In (to the actor) - Out (from the actor) 123 123 d 2 = 2 d i (2) = 2 d o (2) = 0

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UNCLASSIFIED 19 SNA Measures Actors (Brass, 1995:47) Attribute Centrality Prestige Degree Closeness Betweenness 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

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UNCLASSIFIED 20 SNA Measures Actors (Brass, 1995:47) Attribute Centrality Prestige Degree Closeness Betweenness 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

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UNCLASSIFIED 21 SNA Measures Actor Roles AB Ego A B B A AB B A AB B A LiaisonRepresentativeGatekeeperCoordinator Itinerant BrokerBridgeIsolateStar (Degenne, 1999:129)

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UNCLASSIFIED 22 SNA Measures Network (Brass, 1995:47) Definition Number of actors in the network Attribute Size Component Connectivity Density Centralization Transitivity

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UNCLASSIFIED 23 SNA Measures Network (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 Attribute Size Component Connectivity Density Centralization Transitivity

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UNCLASSIFIED 24 SNA Measures Network (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 Attribute Size Component Connectivity Density Centralization Transitivity

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UNCLASSIFIED 25 SNA Measures Network (Brass, 1995:47) Definition -Ratio of the number of actual links to the number of possible links in the network Attribute Size Component Connectivity Density Centralization Transitivity

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UNCLASSIFIED 26 SNA Measures Network (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 Attribute Size Component Connectivity Density Centralization Transitivity

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UNCLASSIFIED 27 SNA Measures Network (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) Attribute Size Component Connectivity Density Centralization Transitivity

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UNCLASSIFIED 28 Overview Problem Statement Perspectives SNA Assumptions & Measures Implications of Non-cooperative Networks Models ► Network Flow: Gains, Losses, & Thresholds ► Flow Typology ► Extensions of the Key Player Problem The Way Ahead

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UNCLASSIFIED 29 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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UNCLASSIFIED 30 Overview Problem Statement Perspectives SNA Assumptions & Measures Implications of Non-cooperative Networks Models ► Network Flow: Gains, Losses, & Thresholds ► Flow Typology ► Extensions of the Key Player Problem The Way Ahead

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UNCLASSIFIED 31 Social Closeness TermsFlow Model Properties People or groupsNodes (sinks, sources, or transshipment) Connectivity or affinity Capacitated arcs (or edges) between nodes Social ClosenessCapacity InfluenceCommodity Potential InfluenceMagnitude of flow People or groups initiating influence in the network Source(s) Target people or groups to be influencedSink(s) People or groups involved in influencingTransshipment 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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UNCLASSIFIED 32 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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UNCLASSIFIED 33 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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UNCLASSIFIED 34 Threshold s Network Flow “Outflow minus inflow must equal supply (or demand)” Mass Balance Constraints (Three cases) Supply node: outflow > inflow outflow = inflow + b j x jk – g ij x ij = b j Demand node: outflow < inflow outflow = inflow – b j x jk – g ij x ij = -b j Transshipment node: outflow = inflow x jk – g ij x ij = 0 Amount of flow from node i to node j on arc ( i, j ) is x ij ijk (Ahuja, Magnanti, and Orlin, 1993:5) g ij

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UNCLASSIFIED 35 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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UNCLASSIFIED 36 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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UNCLASSIFIED 37 Notional Example Minimum cost, maximum flow a tgt {1} 1 452 3 7 611 10 9 8 2 ½ 4/3 {1} {2} {1} Solution (z* = 93.32) ji g ij u ij upper bound of flow on arc ( i, j ) x ij : amount of flow on arc ( i, j ) Legend {x ij } 3 b tgt = - 2 b a = 3 b 4 = - 1

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UNCLASSIFIED 38 a tgt b tgt = - t b a = a (1,6) (1,8) (1,7) (1,9) (1,10) 1 452 3 7 611 10 9 8 2 ½ 4/3 b 4 = - 1 (2,8) (3,6) (1,2) (7,9) (10,10) (5,6) (6,6) (6,7) (6,5) (6,4) (6,8) (6,4) (6,9) (6,20) (6,24) (6,15) ji g ij ( u ij, c ij ) u ij upper bound of flow on arc ( i, j ) c ij : cost per unit flow on arc ( i, j ) g ij : gain/loss factor for arc ( i, j ) Legend Post-Optimality Analysis What if?... g ij ( u ij, c ij ) b tgt = - t b a = a b 4 = - 1

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UNCLASSIFIED 39 Overview Problem Statement Perspectives SNA Assumptions & Measures Implications of Non-cooperative Networks Models ► Network Flow: Gains, Losses, & Thresholds ► Flow Typology ► Extensions of the Key Player Problem The Way Ahead

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UNCLASSIFIED 40 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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UNCLASSIFIED 41 Overview Problem Statement Perspectives SNA Assumptions & Measures Implications of Non-cooperative Networks Models ► Network Flow: Gains, Losses, & Thresholds ► Flow Typology ► Extensions of the Key Player Problem The Way Ahead

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UNCLASSIFIED 42 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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UNCLASSIFIED 43 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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UNCLASSIFIED 44 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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UNCLASSIFIED 45 Notional Example KPP-2 (x i_m ) Choose node i as initial target, relying upon reach of m steps or less

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UNCLASSIFIED 46 Overview Problem Statement Perspectives SNA Assumptions & Measures Implications of Non-cooperative Networks Models ► Network Flow: Gains, Losses, & Thresholds ► Flow Typology ► Extensions of the Key Player Problem The Way Ahead

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UNCLASSIFIED 47 Future Paradigm

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UNCLASSIFIED 48 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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UNCLASSIFIED 49 Contact Information J. Todd Hamill, Major, USAF Commercial:(937) 305-1662 Email: jonathan.hamill@afit.edujonathan.hamill@afit.edu Dr. Dick Deckro DSN: 785-6565 x 4325 Commercial:(937) 255-6565 X4325 DSN: 785-6565 x 4325 Email: richard.deckro@afit.edurichard.deckro@afit.edu

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UNCLASSIFIED 50 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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