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Measures in Social Networks: Description versus Prescription

Measures in Social Networks: Description versus Prescription. J. Todd Hamill, Major, USAF Dick Deckro AFIT/ENS.

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Measures in Social Networks: Description versus Prescription

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  1. 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 22nd ISMOR / August 2005

  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

  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

  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

  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 • Provides insight • Perhaps create requirements • Provides insight • Perhaps create requirements • Actionable Options Evaluations Models never perform analysis. Analysts do analysis, aided by models where appropriate.

  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

  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

  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)

  9. SNA MeasuresLinks Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry Definition Path between two actors is mediated by one or more others 1 2 3 (Brass, 1995:47)

  10. SNA MeasuresLinks Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry Definition How many times or how often the link occurs (e.g. telephone calls, meetings, etc.) 1 2 3 (Brass, 1995:47)

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

  12. SNA MeasuresLinks Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry 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 Co-workers 1 2 3 Friends 1 2 3 Either or both (Brass, 1995:47)

  13. SNA MeasuresLinks Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry 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) + (Brass, 1995:47)

  14. SNA MeasuresLinks Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry Definition Extent to which like is from one actor to another 1 2 3 1 2 3 (Brass, 1995:47)

  15. SNA MeasuresLinks Attribute Indirect Links Frequency Stability Multiplexity Strength Direction Symmetry Definition (Reciprocity) Extent to which relationship is bidirectional 1 2 3 1 2 3 (Brass, 1995:47)

  16. SNA MeasuresActors • 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 Attribute Centrality Prestige Degree Closeness Betweenness (Brass, 1995:47)

  17. SNA MeasuresActors • 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) Attribute Centrality Prestige Degree Closeness Betweenness (Brass, 1995:47)

  18. SNA MeasuresActors • Definition • Number of direct links with other actors • Undirected • In (to the actor) • Out (from the actor) Attribute Centrality Prestige Degree Closeness Betweenness 1 2 3 d2 = 2 di(2) = 2 1 2 3 do(2) = 0 (Brass, 1995:47)

  19. SNA MeasuresActors • 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 Attribute Centrality Prestige Degree Closeness Betweenness (Brass, 1995:47)

  20. SNA MeasuresActors • 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 Attribute Centrality Prestige Degree Closeness Betweenness (Brass, 1995:47)

  21. SNA MeasuresActor Roles A B B Ego Ego Ego Ego A A B B A Liaison Representative Gatekeeper Coordinator B A Ego Ego A B Ego A B Ego Itinerant Broker Bridge Isolate Star (Degenne, 1999:129)

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

  23. SNA MeasuresNetwork Attribute Size Component Connectivity Density Centralization Transitivity • 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 (Brass, 1995:47)

  24. SNA MeasuresNetwork Attribute Size Component Connectivity Density Centralization Transitivity • 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 (Brass, 1995:47)

  25. SNA MeasuresNetwork Attribute Size Component Connectivity Density Centralization Transitivity • Definition • Ratio of the number of actual links to the number of possible links in the network (Brass, 1995:47)

  26. SNA MeasuresNetwork Attribute Size Component Connectivity Density Centralization Transitivity • 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 (Brass, 1995:47)

  27. SNA MeasuresNetwork Attribute Size Component Connectivity Density Centralization Transitivity • 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) (Brass, 1995:47)

  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

  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

  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

  31. Current Mappings (Renfro, 2001:95)

  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

  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)

  34. Network Flow Thresholds “Outflow minus inflow must equal supply (or demand)” Amount of flow from node i to node j on arc (i, j) is xij gij i j k 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 (Ahuja, Magnanti, and Orlin, 1993:5)

  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

  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)

  37. Notional ExampleMinimum cost, maximum flow 1 6 11 2 ba = 3 {1} {1} 4/3 btgt = - 2 3 2 4 5 a 10 ½ {2} {1} {1} tgt {1} {1} 9 b4 = - 1 7 {1} {1} 3 8 Solution (z* = 93.32) Legend gij uij upper bound of flow on arc (i, j) xij: amount of flow on arc (i, j) i j {xij}

  38. Post-Optimality Analysis 1 6 11 (6,20) (6,5) 2 (6,4) (6,15) ba = a (1,6) (2,8) ba = a (6,6) (10,10) 4/3 btgt = - t btgt = - t 2 4 5 a What if?... 10 ½ (3,6) (7,9) (1,8) tgt (1,9) (5,6) (6,7) 9 b4 = - 1 (1,10) (6,24) (1,7) b4 = - 1 (1,2) 7 (6,9) 3 (6,8) (6,4) 8 Legend gij 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) i j (uij, cij) gij (uij, cij)

  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

  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

  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

  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)

  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…)

  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…

  45. Notional ExampleKPP-2 (xi_m) Choose node i as initial target, relying upon reach of m steps or less

  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

  47. Future Paradigm

  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

  49. Contact Information J. Todd Hamill, Major, USAF Commercial:(937) 305-1662 Email: jonathan.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.edu

  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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