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Generalized Firefighting on the 2 dimensional infinite grid

and. Centrality measures in social networks. Generalized Firefighting on the 2 dimensional infinite grid. Kah Loon Ng DIMACS. Containing fires in infinite grids L d. Fire starts at only one vertex: d= 1: Trivial. d = 2: Impossible to contain the fire with 1 firefighter per time step.

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Generalized Firefighting on the 2 dimensional infinite grid

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  1. and Centrality measures in social networks Generalized Firefighting on the 2 dimensional infinite grid Kah Loon Ng DIMACS

  2. Containing fires in infinite grids Ld Fire starts at only one vertex: d= 1: Trivial. d = 2: Impossible to contain the fire with 1 firefighter per time step

  3. 8 time steps 18 burnt vertices Containing fires in infinite grids Ld d = 2: Two firefighters per time step needed to contain the fire.

  4. Containing fires in infinite grids L2 • We assume that the number of firefighters available for deployment is given by a function that is periodic. Justification for considering periodic functions: 1 1 2 1 1 2

  5. For example, = period of mod 3 if mod 3 if mod 3 if We also write Containing fires in infinite grids L2 Given a periodic function , let

  6. Given positive integers and , we define a periodic function Given two periodic functions and with periods and respectively, we say Containing fires in infinite grids L2

  7. For example, if Note that and then Containing fires in infinite grids L2 If is a periodic function and is any positive integer, define the translate function by

  8. Given a positive integer , let be the function with period defined by where Note that is chosen to be the smallest integer such that . Define to be the periodic function of period Containing fires in infinite grids L2

  9. Main Theorem: If the number of firefighters available for deployment per time step can be represented by a periodic function such , then any fire that breaks out at a single vertex in can be contained after some finite time . Lemma 0: Suppose and are two periodic functions such that . If any fire breaking out at a single vertex in can be contained “using ” , then it can also be contained “using ”. Containing fires in infinite grids L2

  10. Let . Case 1: If for all , we are done. Case 2: If for some . Let be the smallest such . Since , we have . Time 1 to : Deploy firefighters as in , and accumulate at least “spare” firefighters. Containing fires in infinite grids L2 Proof of Lemma 0:

  11. Given any periodic function with “Rearrange” to get that is non decreasing in its period. Lemma 1: Containing fires in infinite grids L2 Outline proof of Main Theorem:

  12. Lemma 4: 35 1,1,1,1,5, 1,1,1,1,5, ……… 1,1,1,1,5, 1,1,1,1,5, 1,1,1,1,5,…… 1,1,1,1,1,………………………….1,1,2,2,…………………. 36 Containing fires in infinite grids L2 Outline proof of Main Theorem:

  13. 1, 1, 2, 2, 3, 1, 1, 2, 2, 3,… 2, 2, 3, 1, 1, 2, 2, 3, 1, 1,… Gain Loss Containing fires in infinite grids L2 Outline proof of Main Theorem: Lemma 4: Lemma 2: If is a periodic function that is non decreasing over its period, then for any positive integer , .

  14. Lemma 3: If is a periodic function that is non decreasing over its period, then for any periodic function with satisfying we have . for , In other words, for , 1,1,1,1,1,5,…..…….1,1,1,1,1,5, 1,1,1,1,1,5,………..1,5,1,1,1,.. 1,1,………………………….,1, 2,2,…………………,2,2,1,1,.. 36 37 Containing fires in infinite grids L2 Outline proof of Main Theorem: Lemma 4:

  15. The strategy using End of phase 2 End of phase 1 End of phase 3 Advance firefighters Retreat firefighters Containing fires in infinite grids L2

  16. 4 8 7 6 9 2 8 7 5 3 1 9 11 10 10 12 11 12 Containing fires in infinite grids L2 The strategy using : Completing phase 1:

  17. Containing fires in infinite grids L2 The strategy using : “Delayed response” – Phase 1 can still be completed.

  18. (0,0) (0,0) Containing fires in infinite grids L2 The strategy using : Completing phase 2 after phase 1 has been completed:

  19. (0,0) (0,0) Containing fires in infinite grids L2 The strategy using : Completing phase 3 after phase 2 has been completed:

  20. Finishing the job: (0,0) (0,0) 2 Retreat firefighters Active vertices 4 1 3 Containing fires in infinite grids L2 The strategy using :

  21. (0,0) Containing fires in infinite grids L2 A few points to note: Our theorem says nothing about periodic functions where . In fact, we can easily construct a function where and still is sufficient to contain the outbreak.

  22. Obviously, is sufficient to contain the outbreak but we do not consider such (cheating! ) situations. The exact time required to contain the outbreak (and thus the number of burnt vertices) can be explicitly computed as a function of . However, our strategy does not guarantee that the number of burnt vertices is minimized. Containing fires in infinite grids L2 A few points to note: Our theorem says nothing about periodic functions where . In fact, we can easily construct a function where and still is sufficient to contain the outbreak.

  23. Centrality Measures in Graphs (or Social Networks) • Centrality = Importance = Prominence? • 3 types of centrality indices: • degree • closeness • betweeness

  24. Centrality Measures in Graphs (or Social Networks) • Centrality indices can be computed for each vertex or a group of vertices. • Majority of centrality concepts are based on non-directed and dichotomous relations • However, for some specific purposes (for example, measuring prestige) directed graphs or valued relations might need to be used.

  25. = D, degree measure = C, closeness measure = B, betweeness measure Centrality Measures in Graphs (or Social Networks) = centrality index for vertex under measure

  26. = deg = degree of vertex = deg Centrality Measures in Graphs (or Social Networks) (degree centrality)

  27. Centrality Measures in Graphs (or Social Networks) (closeness centrality)

  28. Jordan centers of a graph (subset of with the smallest eccentricities) • Centroid of a graph (subset of with the smallest “weight”) Centrality Measures in Graphs (or Social Networks) (closeness centrality) • Closeness centrality measures are related to: • One shortcoming of closeness measures is that it cannot be used for disconnected graphs.

  29. Let = number of distinct shortest paths between and = number of distinct shortest paths between and that passes through For each , maximum value is 1. Centrality Measures in Graphs (or Social Networks) (betweeness centrality)

  30. We assume here that if some information (or disease) is passed from to , each of the shortest paths is equally likely to be chosen. • If we sum over all , we obtain measures of the pair-dependency of vertex on vertex . These values can also be viewed as indices of how much “gate keeping” does for . • seems to be an improvement over and but there are still inadequacies. Centrality Measures in Graphs (or Social Networks) (betweeness centrality)

  31. LAMBERTESCHI GUADAGNI GINORI BISCHERI STROZZI CASTELLANI PERUZZI BARBADORI SALVATI ACCIAIUOLI PAZZI TORNABUONI MEDICI RIDOLFI ALBIZZI Centrality Measures in Graphs (or Social Networks)

  32. Acciaiuoli 0.071 0.000 0.368 0.214 0.483 0.212 Albizzi 0.438 0.143 0.093 Barbadori 0.104 0.214 0.400 Bischeri 0.214 0.389 0.055 Castellani 0.333 0.071 0.000 Ginori 0.467 0.286 0.255 Guadagni 0.071 0.000 0.326 Lamberteschi 0.429 0.522 0.560 Medici 0.071 0.000 0.286 Pazzi 0.214 0.022 0.368 Peruzzi 0.114 0.500 0.214 Ridolfi 0.143 0.389 0.143 Salvati Strozzi 0.438 0.103 0.286 Torabuoni 0.092 0.483 0.214 Centrality Measures in Graphs (or Social Networks)

  33. Acciaiuoli 0.368 0.071 0.000 0.214 0.483 0.212 Albizzi 0.093 0.143 0.438 Barbadori 0.104 0.214 0.400 Bischeri 0.055 0.389 0.214 Castellani 0.071 0.000 0.333 Ginori 0.467 0.255 0.286 Guadagni 0.000 0.326 0.071 Lamberteschi 0.429 0.522 0.560 Medici 0.000 0.071 0.286 Pazzi 0.214 0.368 0.022 Peruzzi 0.214 0.500 0.114 Ridolfi 0.143 0.389 0.143 Salvati Strozzi 0.286 0.438 0.103 Torabuoni 0.092 0.214 0.483 Centrality Measures in Graphs (or Social Networks) Some observations Strozzi family has high degree centrality but low closeness centrality.

  34. Acciaiuoli 0.368 0.071 0.000 0.214 0.483 0.212 Albizzi 0.093 0.143 0.438 Barbadori 0.104 0.214 0.400 Bischeri 0.055 0.389 0.214 Castellani 0.071 0.000 0.333 Ginori 0.467 0.255 0.286 Guadagni 0.000 0.326 0.071 Lamberteschi 0.429 0.522 0.560 Medici 0.000 0.071 0.286 Pazzi 0.214 0.368 0.022 Peruzzi 0.214 0.500 0.114 Ridolfi 0.143 0.389 0.143 Salvati Strozzi 0.286 0.438 0.103 Torabuoni 0.092 0.214 0.483 Centrality Measures in Graphs (or Social Networks) Some observations Tornabouni family has high closeness centrality but low betweeness centrality.

  35. Acciaiuoli 0.368 0.071 0.000 0.483 0.214 0.212 Albizzi 0.438 0.143 0.093 Barbadori 0.104 0.400 0.214 Bischeri 0.389 0.214 0.055 Castellani 0.333 0.071 0.000 Ginori 0.286 0.467 0.255 Guadagni 0.000 0.071 0.326 Lamberteschi 0.522 0.429 0.560 Medici 0.071 0.000 0.286 Pazzi 0.368 0.214 0.022 Peruzzi 0.214 0.114 0.500 Ridolfi 0.389 0.143 0.143 Salvati Strozzi 0.286 0.103 0.438 Torabuoni 0.092 0.483 0.214 Centrality Measures in Graphs (or Social Networks) Some observations

  36. Centrality Measures in Graphs (or Social Networks) Eigenvector centrality “The centrality of a vertex does not depend on the number of vertices it is adjacent to, but also these vertices’ centrality.” Bonacich (1972) defines (eigenvector) centrality as a positive scalar multiple of the sum of “adjacent” centralities: In matrix form, if we have (Perron-Frobenius) Since is nonnegative, there exists an eigenvector of the maximal eigenvalue with only nonnegative entries.

  37. Information centrality Adopts the idea that information is passed along the network along all possible paths, not necessary the shortest one. Flow betweenes centrality The flow betweeness of vertex is defined as the amount of flow through vertex when the maximum flow is transmitted from to , averaged over all and . Random walk betweenes centrality The random walk betweeness of a vertex is the number of times that a random walk starting at and ending at passes through along the way. Centrality Measures in Graphs (or Social Networks)

  38. Centrality Measures in Graphs (or Social Networks) Different measures for different types of flow • Used goods (eg books) – consider trails in graphs? • Money (eg a dollar note) – consider walks in graphs? Markov process? • Attitude/Belief – Influence process • Gossip and infection – similarities and differences • Package delivery – known destination? Shortest route? Paths vs. Walks vs. Trails Transfer vs. Duplication

  39. Centrality Measures in Graphs (or Social Networks) Some references: • Freeman L.C. (1979): Centrality in social networks: Conceptual clarification. Social Networks 1, 215-239. • Bonacich P. (1991): Simultaneous group and individual centralities. Social Networks 13, 155-168 • Friedkin N. E. (1991): Theoretical foundations for centrality measures. Amer. Journal of Sociology 96, 1478-1504 • Stephenson K., Zelen M. (1989): Rethinking centrality:methods and examples. Social Networks 11, 1-37. • Borgatti S.P. (2005): Centrality and network flow. Social Networks 27, 55-71. • Ruhnau B. (2000): Eigenvector centrality – a node centrality? Social Networks 22, 357-365 • Faust K. (1997): Centrality in affiliation networks. Social Networks 19, 157-191. • Newman M. (2005): A measure of betweeness centrality based on random walks. Social Networks 27, 39-54. • Bell D., Atkinson J., Carlson J. (1999): Centrality measures for disease transmission networks. Social Networks 21, 1-21.

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