Mint Seminar 15 10 2010 Monsuur

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Mint Seminar 15 10 2010 Monsuur

  1. 1. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networks: Linking Actors’ Incentives, Information and Influence Herman Monsuur Netherlands Defence Academy presentation at MINT seminar Centre d’Economie de la Sorbonne 15 October, 2010 Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  2. 2. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set A new cap for the Ministry of Defence: White, Red or Black Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  3. 3. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set A new cap for the Ministry of Defence: White, Red or Black Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  4. 4. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set A new cap for the Ministry of Defence: White, Red or Black Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  5. 5. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set A new cap for the Ministry of Defence: White, Red or Black Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  6. 6. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set A new cap for the Ministry of Defence: White, Red or Black Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  7. 7. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set von Neumann-Morgenstern stable sets A set S ⊂ X is called a vNM stable set if inner stability no y ∈ S is dominated by an x ∈ S external stability every y /∈ S is dominated by some x ∈ S Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  8. 8. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set von Neumann-Morgenstern stable sets A set S ⊂ X is called a vNM stable set if inner stability no y ∈ S is dominated by an x ∈ S external stability every y /∈ S is dominated by some x ∈ S Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  9. 9. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set Effective coalition Let X be the set of alternatives and let A be the set of agents or members of the organisation or society. R is the dominance relation on X that is generated by effective coalitions: If there is at least one effective coalition generating the domination of x over y, then (x, y) ∈ R. An effective coalition will be inclined to apply its binary dominance, but, in the larger context of the dominance relation, it may have strategic reasons for not exercising its power. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  10. 10. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set Non-enforcement This phenomenon of non-enforcement occurs in two instances. A dominance will not be enforced by any of its effective coalitions if the preferred alternative is already surpressed by at least one other 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 of behaviour. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  11. 11. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set Behavioural postulate 1 and 2 A solution concept φ assigns to a relation R on X in its domain a subset of 2X ∅. Elements of φ(R) are interpreted as stable sets. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  12. 12. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set Behavioural postulate 1 and 2 A solution concept φ assigns to a relation R on X in its domain a subset 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 feel no restraint in propagandising x to the detriment of y. We therefore require both core primacy If S ∈ φ(R), then C(R) ⊂ S. core subduction If S ∈ φ(R), x ∈ C(R) and (x, y) ∈ R, then y /∈ S. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  13. 13. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set Behavioural postulate 3 We consider it undesirable if a stable set S would change when the effective coalitions, finding that the alternative x they propagandise does not lie in S, give up or dissolve. This property we call : independence of non-enforced dominations If S ∈ φ(R) and (x, y) ∈ R, x /∈ S then S ∈ φ(R), where R = R {(x, y)} Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  14. 14. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set A theorem for acyclical relations Theorem Let the domain of a solution concept φ be the set of acyclical relations. Then φ satisfies core primacy, core subduction and independence of non-enforced dominations, if and only if φ(R) = φvNM(R) for all acyclical relations R. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  15. 15. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set Socially stable sets A set S ⊂ X is called a socially stable set if generalised inner stability a(R|S )cl = ∅ external stability if 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 of R|S . The generalised inner stability means that for {x, y} ⊂ S, we allow x to dominate y if, in turn, y dominates x directly or indirectly within R|S . Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  16. 16. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set Examples of socally stable sets Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  17. 17. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set Examples of socally stable sets Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  18. 18. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set Examples of socally stable sets Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  19. 19. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set Behavioural postulate 4 Given S ⊂ X, we introduce the S-equalised dominance relation denoted 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 cycles If S ∈ φ(R) then S ∈ φ(R⊗S ). Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  20. 20. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors An example von Neumann-Morgenstern stable sets Effective coalitions and non-enforcement Behavioural characterisation of socially stable set Characterisation of socially stable sets Theorem (Delver & Monsuur, 2001, SCW) If φ satisfies core primacy, core subduction, independence of non-enforced domination, and independence of stable cycles, then φ(R) ⊂ φsoc(R), for all R in the domain of φ. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  21. 21. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure Network Evolution Local, binary decisions shape global network structures. We introduce a mechanism that formalizes a possible incentive that guides nodes in constructing their local network structure, using the notion of the covering relation. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  22. 22. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure Network Evolution Local, binary decisions shape global network structures. We introduce a mechanism that formalizes a possible incentive that guides nodes in constructing their local network structure, using the notion of the covering relation. Let a, b be nodes in V , a = 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 are also linked to a and node a has at least one extra link. Intuitively speaking, a outperforms b. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  23. 23. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure Uncovered set We let U or U(G) be the uncovered set: U = {v ∈ V : there is no node w ∈ V that covers v in the network G}. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  24. 24. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure Axioms A center φ assigns to any network G = (V , E) a non-empty subset φ(G) ⊂ V . The center φuc assigns to a network G the set of uncovered nodes. We consider the following axiom for a center φ: A center φ has the mediator property if for each pair of distinct nodes a and b, there is a shortest path connecting these nodes, such that any node in between a and b on this path is in φ(G). Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  25. 25. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure Axioms A center φ assigns to any network G = (V , E) a non-empty subset φ(G) ⊂ V . The center φuc assigns to a network G the set of uncovered nodes. We consider the following axiom for a center φ: A center φ has the mediator property if for each pair of distinct nodes a and b, there is a shortest path connecting these nodes, such that any node in between a and b on this path is in φ(G). Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  26. 26. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure Characterisation of uncovered set Theorem (Monsuur & Storcken, 2004, Operations Research) The center set φuc is the only inclusion minimal set of nodes that is compatible with structural equivalence, has the mediator property and is stable. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  27. 27. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure Local, binary decisions Dichotomy: A node is either covered or it is uncovered. We further assume that the status ‘uncovered’ is ranked higher than the status ‘covered’. The mechanism. Each step consists of taking, randomly, two distinct nodes a and b from V . Then for the link (a, b): if (a, b) ∈ E, it is deleted by a if the network remains connected and the status of node a does not decrease, if (a, b) /∈ E, it is added if both a and b achieve a higher status. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  28. 28. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure Example of successor networks Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  29. 29. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure Example of successor networks Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  30. 30. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure Example of successor networks Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  31. 31. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure Pairwise Stable Networks Theorem (Monsuur, 2007, EJOR) Let G be a network with U = X. Then there exists a sequence of successor networks that transforms G into one of the following pairwise stable networks a ring-network, a uni-polar network, a bi-polar network. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  32. 32. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Introduction Uncovered set Changing the network structure van Klaveren, Monsuur, Janssen, Schut & Eiben, 2009. Proceedings BNAIC. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  33. 33. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model Networks, Information and Choice. Janssen & Monsuur, 2010. In: Collective Decision Making. Networked operations offer decisive advantage through the timely provision and exploitation of (feedback) information and intelligence to enable effective decision-making and agile actions. We focus on the aspect of information sharing in collaboration networks and discuss a feedback model for situational awareness, that combines exogenously given characteristics of nodes with their positioning within the (social, information or physical) network topology. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  34. 34. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model Information feedback We let A be the adjacency matrix, where aij ∈ [0, 1] is the extent to which value from node j is usable or transferable to node i regarding the improvement of i’s situational awareness. We take aii = 0 for each node i. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  35. 35. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model Feedback of operational links We combine the transferred situational awareness, which depends on the network, with exogeneous values. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  36. 36. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model Feedback of operational links We combine the transferred situational awareness, which depends on 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. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  37. 37. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model 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 = lim m→∞ vm = lim m→∞ m k=0 αk Ak b = (I − αA)−1 b By iteration, nodes also receive information from nodes which are not adjacent, but are two, three, or more steps away. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  38. 38. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model Network performance metric We introduce a network performance metric which combines given characteristics of the nodes with the network topology. This can be used to compare different network configurations. We define NTb = eT v eT b = eT (I − αA)−1 b eT b , where e is a vector of 1 s. This means that we take the quotient of the total situational awareness after updating, and the total value of exogenously given characteristics as expressed in the vector b. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  39. 39. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model Uncertain behaviour At each stage k, k ≥ 1, of the process of updating information within the network, the uncertain behaviour of the nodes is modelled by a collection of iid random variables k,ij : Ω → [0, 1], 1 ≤ i, j ≤ n, such that k,ij = 1 if an information flow between node j and i is possible and k,ij = 0 otherwise. For a fixed outcome ω in the sample space Ω the process of updating information in m steps yields the situation awareness vm(ω), which is defined recursively by vm(ω) = b + αAm(ω)vm−1(ω) where the matrices Ak have the entries aij k,ij . Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  40. 40. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model The network metric The network performance metric that combines the given characteristics of the nodes with the network topology is defined by NTbm = E eT vm eT b = 1 + m k=1 αk E eT k−1 s=0 Am−s b eT b where E (·) denotes the expectation and e is a vector of 1 s. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  41. 41. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model Infinite m For infinite m, the network performance metric becomes NTb = eT x dµ(x) eT b Here µ is the unique probability measure which satisfies the equation µ = N m=1 P (A1 = Dm) µ ◦ f −1 m , where fm is the affine mapping fm : x → b + αDmx and {D1, . . . , DN} is the finite collection of all outcomes of A1(ω). Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  42. 42. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model Computing the network metric Theorem Fix a sequence of matrices {Ak(ω)}k≥1 for some outcome ω in the sample space Ω. Let the orbit {xn}∞ n=0 be defined by x0 = b and xn+1 = b + αAn+1xn. Then with probability one NTb = lim n→∞ 1 n + 1 1 + n k=1 eT xk eT b Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  43. 43. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model Taking into account the behaviour of other nodes Fix an arbitrary node i and an adjacent node of this node i, say node j. From the point of view of a receiving node i, we assume that 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, Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  44. 44. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model Taking into account the behaviour of other nodes Fix an arbitrary node i and an adjacent node of this node i, say node j. From the point of view of a receiving node i, we assume that 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, and further with probability 1 − pij (1) − pij (0) the decision of node j to send information depends on the outcomes of the 0-1 random variables k−1,ij , 1 ≤ i, j ≤ n and k−1,jt, t ∈ Nj . Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  45. 45. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Networked operations Information sharing Stochastic behaviour of nodes A Markov model The Markov Model We suggest the following (Markov) model which takes into account these factors: P ( k,ij = 1| k−1,ij = βij , k−1,jt = βjt, t ∈ Nj ) = pij (1)+(1 − pij (1) − pij (0)) 1 γ + (1 − γ)|Nj |  γβij + (1 − γ) t∈Nj βjt   Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  46. 46. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors Monsuur, Grant & Janssen, 2011, to appear A node typically is not just part of one type of network, but simultaneously belongs to multiple networks. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence
  47. 47. Standard of Behaviour Network Dynamics Information and Situation Awareness Multi-layer Networks and Stochastic Behaviour of Actors The modelling approach To model behaviour of actors in interwoven networks, we use an agent-based approach: Networks are replaced by multi-layered networks that influence each other. To each actor in a network an objective function is assigned, incorporating endogenous network statistics and exogenous covariates. Herman Monsuur Networks: Linking Actors’ Incentives, Information and Influence

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