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On Infinity, Continuity a nd Time-Varying Graphs

On Infinity, Continuity a nd Time-Varying Graphs. !!!. Nicola Santoro Carleton University. Distributed Computing. Network of computational entities communicating by message transmission. Information Diffusion, Agreement, Distributed Data, …. Distributed Computing.

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On Infinity, Continuity a nd Time-Varying Graphs

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  1. On Infinity, Continuity and Time-Varying Graphs !!! Nicola Santoro Carleton University Nicola Santoro - Patras 2019

  2. Distributed Computing Network ofcomputationalentitiescommunicating by message transmission Information Diffusion, Agreement, Distributed Data, … Nicola Santoro - Patras 2019

  3. Distributed Computing System of mobilecomputational entitiesoperating in a network Search, Exploration, Monitoring, Gathering, … Nicola Santoro - Patras 2019

  4. Distributed Computing Networkof computational entities communicating by message transmission System of mobile computational entities operating in a network Nicola Santoro - Patras 2019

  5. Dynamic Networks - Dynamic Graphs topology is subject to changes • Faults and Failures • Link/Node deletion ===> disactivation • Link/Node addition ===> (re)activation -- small scale phenomena (few, localized) -- fault-tolerance Nicola Santoro - Patras 2019

  6. Dynamic Networks topology is subject to extensive changes Self Stabilization instability instability stability Nicola Santoro - Patras 2019

  7. Dynamic Networks Changes are continuous, possibly extensive , and never stop instability Nicola Santoro - Patras 2019

  8. Dynamic Networks Changes are continuous, possibly extensive , and never stop Changes are not anomaliesbut ratherintegral part of thenatureof the system. instability Nicola Santoro - Patras 2019

  9. Dynamic Networks : WIRELESS MOBILE mobile ad hoc networks - topology may change dramatically over time due to the movement of the nodes vehicular networks - topology changes continuously as vehicles move Nicola Santoro - Patras 2019

  10. Dynamic Networks : WIRELESS MOBILE Nicola Santoro - Patras 2019

  11. Dynamic Networks : WIRELESS MOBILE half communication range Nicola Santoro - Patras 2019

  12. Dynamic Networks : WIRELESS MOBILE Communication Graph Nicola Santoro - Patras 2019

  13. Dynamic Networks : WIRELESS MOBILE Communication Graph Nicola Santoro - Patras 2019

  14. Dynamic Networks : WIRELESS MOBILE Nicola Santoro - Patras 2019

  15. Dynamic Networks : TEMPORALLY CONNECTED end-to-end connectivitydoes notnecessarilyhold the network might bealwaysdisconnected still … communication may beavailableover time and space, making computations feasible Nicola Santoro - Patras 2019

  16. Dynamic Networks : PERIODIC Public Transports with fixed timetable Moving entities creating the network are periodic Monitoring/Guarding tours Low-earth orbiting satellite (LEO) systems NYC transit (MTA) Nicola Santoro - Patras 2019

  17. Dynamic Networks : PEER-TO-PEER OVERLAY networks Nicola Santoro - Patras 2019

  18. Dynamic Networks : SOCIAL NETWORKS/WEB GRAPHS FACEBOOK Nicola Santoro - Patras 2019

  19. Dynamic Networks : SOCIAL NETWORKS/WEB GRAPHS LINKEDIN SANTORO Nicola Santoro - Patras 2019

  20. Dynamic Networks Live System lifetime of system Unlimited (infinite) Computinginlive systems unknown beginning Nicola Santoro - Patras 2019

  21. Dynamic Networks Dead System lifetime of system Finite Post-mortem analysis known beginning end Collected (eg Social) Network Data need to be analyzed to understand patterns structures Nicola Santoro - Patras 2019

  22. Dynamic Networks G G Live System Dead System Infinite Finite G On-Line Off-line compute compute Decentralized Centralized The computation takes place in the evolving system The finite system data is the input of the computation Nicola Santoro - Patras 2019

  23. Dynamic Networks • Formalism to describe/represent dynamic networks • and their evolution for (post-mortem or live) analysis graphs • natural means to represent and analyze static networks time-varying graphs • natural means to represent and analyze dynamic networks Nicola Santoro - Patras 2019

  24. TVG A generalmathematical formalism that describes many different types of dynamic networks Includes existing formalisms as special cases Allows representation and analysis of live systemswithout restrictive assumptions Casteigts, Flocchini, Quattrociocchi, Santoro. Time-varying graphs and dynamic networks. IJPEDS, 2012. Nicola Santoro - Patras 2019

  25. TVG G = (N, E, T, ψ, ρ, ζ ) Nicola Santoro - Patras 2019

  26. TVG G = (N, E, T, ψ, ρ, ζ ) TVG-entities (nodes, vehicles, sensors, web sites, … ) Nicola Santoro - Patras 2019

  27. TVG G = (N, E, T, ψ, ρ, ζ ) E ⊆ N × N connections between pairs of TVG-entities (edges, links, interactions, contacts … ) Nicola Santoro - Patras 2019

  28. TVG G = (N, E, T, ψ, ρ, ζ ) E ⊆ N × N ( × L ) connections between pairs of TVG-entities (edges, links, interactions, contacts … ) with (domain- specific multi-valued) labels e.g. <satellite; bandwidth 4 MHz; encryption available;...> Nicola Santoro - Patras 2019

  29. TVG G = (N, E, T, ψ, ρ, ζ ) lifetime of system R Limited (finite) Nicola Santoro - Patras 2019

  30. TVG G = (N, E, T, ψ, ρ, ζ ) lifetime of system Unlimited (infinite) Nicola Santoro - Patras 2019

  31. TVG G = (N, E, T, ψ, ρ, ζ ) lifetime of system 0 Unlimited (infinite) beginning Nicola Santoro - Patras 2019

  32. TVG G = (N, E, T, ψ, ρ, ζ ) node presencefunction ψ : N × T → {0, 1} edge presencefunction ρ : E × T → {0, 1} ψ(x,t)=1 iff x is in present at time t ρ(e,t)=1 iff e is present at time t Nicola Santoro - Patras 2019

  33. TVG G = (N, E, T, ψ, ρ, ζ ) latency function ζ : E × T → T {^} ˛Ç ζ((x,y), t) = d message from x to y, if sent at time t, will arrive at time t+d traversal from x to y, if started at time t, ends at time t+d Nicola Santoro - Patras 2019

  34. TVG G = (N, E, T, ψ, ρ, ζ ) latency function ζ : E × T → T {^} ˛Ç ζ((x,y), t) = ^ message from x to y, if sent at time t, will not arrive traversal from x to y, if started at time t, cannot be done Nicola Santoro - Patras 2019

  35. TVG G = (N, E, T, ψ, ρ, ζ ) Nicola Santoro - Patras 2019

  36. TVG G = (N, E, T, ψ, ρ, ζ ) G(t) = (N(t) , E(t)) SNAPSHOT at time t ∈ T N(t) = { x ∈ N : ψ(x, t)=1 } E(t) = { e ∈ E : ρ(e, t)=1 } G = (N , E) FOOTPRINT Nicola Santoro - Patras 2019

  37. TVG journey G = (N, E, T, ψ, ρ, ζ ) JOURNEY “walk over time” < (e1 , t1 ), (e2, t2 ), . . . , (ek, tk) , (ek+1, tk+1), . . . > • <e1 , e2 ,. . . , ek, ek+1 ,. . . > is a walk in G • - ∀i, ρ(ei , ti) = 1 AND ti+1≥ ti + ζ (ei , ti) Nicola Santoro - Patras 2019

  38. TVG journey ``distance” from node x to node y at time t topological measure: minimum length among all the journeys from x to y starting at or after time t shortest journey temporal measure: minimum durationamong all the journeys from x to y starting at or after time t fastest journey temporal measure: earliest arrivalamong all the journeys from x to y starting at or after time t foremostjourney Nicola Santoro - Patras 2019

  39. journey TVG E F 18:00 12:10 D C 9:30 19:15 H 22:10 G 9:00 22:15 22:00 B A 20:30 9:00 I min # hops (2 hops) shortest A,I,B foremostA,C,D,E,F earliest arrival (19:15) fastest A,G,H,B smallest duration (15 minutes) NicolaSantoro - Patras 2019

  40. TVG Live System Unlimited known unknown 0 t past future present Nicola Santoro - Patras 2019

  41. TVG Dead System Finite study the past Discrete • emergence of special structures • forecast the future known 0 t past future present Nicola Santoro - Patras 2019

  42. TVG Live System Infinite Computinginlive systems unknown 0 t past future present Nicola Santoro - Patras 2019

  43. TVG Computingindynamic systems unknown present Nicola Santoro - Patras 2019

  44. TVG Computingindynamic systems ASSUMPTIONS a-priori knowledge oracle something must be known present Nicola Santoro - Patras 2019

  45. TVG BASIC ASSUMPTIONS The number of nodes (thus, the footprint) is finite Number of events in any limited interval of time is finite not enough ! 0 Nicola Santoro - Patras 2019

  46. TVG EDGE x y transient $ t ∀t’>t : ψ ((x,y), t’) = 0 recurrent ∀t $ t’>t : ψ ((x,y),t’) = 1 LOCAL FAIRNESS (e.g., Population Protocols) Nicola Santoro - Patras 2019

  47. TVG EDGE ? x y transient undecidable recurrent Nicola Santoro - Patras 2019

  48. TVG If N is finite from a certain time on there will be only recurrent edges undecidable ? t transient & recurrent only recurrent 0 Nicola Santoro - Patras 2019

  49. TVG EDGE x y recurrent bounded recurrent $ B (∀t ψ((x,y),t)=1 ($ t’, t <t’ £ t+B, (ψ((x,y),t’)=1))) periodic $ P (∀t ψ((x,y),t)=1 ( ψ((x,y),t’) = 1)) Nicola Santoro - Patras 2019

  50. TVG periodic ? EDGE x y undecidable Nicola Santoro - Patras 2019

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