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									Gossip Algorithms and
Emergent Shape

                 Ken Birman




Leiden; Dec 06   Gossip-Based Networking Workshop   1
 On Gossip and Shape
     Why is gossip interesting?
          Scalability….
                Protocols are highly symmetric
                Although latency often forces us to bias them
          Powerful convergence properties…
                Especially in support of epidemics
          New forms of consistency…
                Probabilistic… but this is often adequate

Leiden; Dec 06           Gossip-Based Networking Workshop        2
 Consistency
     Captures intuition that if A and B compare
      their states, no contradiction is evident
          In systems with “logical” consistency, we say
           things like “A’s history and B’s are closed under
           causality, and A is a prefix of B”
          With probabilistic systems we seek rapidly
           decreasing probability (as time elapses) that A
           knows “x” but B doesn’t
  …. Probabilistic convergent consistency

Leiden; Dec 06        Gossip-Based Networking Workshop         3
 Exponential convergence
     A subclass of convergence behaviors
     Not all gossip protocols offer
      exponential convergence
          But epidemic protocols do have this
           property… and many gossip protocols
           implement epidemics



Leiden; Dec 06     Gossip-Based Networking Workshop   4
 Value of exponential convergence
     An exponentially convergent protocol
      overwhelms mishaps and even attacks
          Requires that new information reach relevant
           nodes with at most log(N) delay
          Can talk of “probability 1.0” outcomes
     Even model simplifications (such as idealized
      network) are washed away!
          Predictions rarely “off” by more than a constant
          A rarity: a theory relevant to practice!

Leiden; Dec 06        Gossip-Based Networking Workshop        5
 Convergent consistency
     To illustrate our point, contrast Cornell’s
      Kelips system with MIT’s Chord
     Kelips is convergent. Chord isn’t




Leiden; Dec 06   Gossip-Based Networking Workshop   6
 Kelips (Linga, Gupta, Birman)

        Take a a collection
            of “nodes”
                                        110



                                       230          202




                                        30




Leiden; Dec 06           Gossip-Based Networking Workshop   7
 Kelips
                                                     Affinity Groups:
                                                  peer membership thru
          Map nodes to                               consistent hash
          affinity groups
                                            0    1      2            N -1
                                           110


                                                                              N
                                          230          202
                                                                            members
                                                                            per affinity
                                                                            group
                                           30




Leiden; Dec 06              Gossip-Based Networking Workshop                         8
      Kelips                                             110 knows about
                                                         other members –
                                                             230, 30…
                                                 Affinity Groups:
Affinity group view                           peer membership thru
                                                 consistent hash
 id   hbeat    rtt
 30    234    90ms
                                        0     1    2             N -1
230    322    30ms                      110


                                                                          N
                                       230        202
                                                                        members
                                                                        per affinity
                                                                        group
                                        30

                      Affinity group
                      pointers




  Leiden; Dec 06      Gossip-Based Networking Workshop                           9
      Kelips
                                                         202 is a
                                                    Affinity Groups:
Affinity group view                                   “contact” for
                                                 peer membership thru
                                                     110 in group
                                                    consistent hash 2
 id      hbeat       rtt
 30      234      90ms
                                           0    1      2                 N -1
230      322      30ms                    110


      Contacts                           230          202
                                                                                  N
                                                                                members
 group    contactNode                                                           per affinity
  …              …                                                              group
                                          30
  2              202



                                                              Contact
                                                              pointers



  Leiden; Dec 06           Gossip-Based Networking Workshop                             10
      Kelips                                    “cnn.com” maps to group 2.
                                                   So 110 tells group 2 to
                                                   “route” inquiries about
                                                       Affinity Groups:
                                                       cnn.com to it.
Affinity group view                                peer membership thru
                                                      consistent hash
 id       hbeat       rtt
 30       234      90ms
                                            0      1        2            N -1
230       322      30ms                    110


      Contacts                            230               202
                                                                                  N
                                                                                members
 group      contactNode                                                         per affinity
  …               …                                                             group
                                           30
  2               202


 Resource Tuples
                                                       Gossip protocol
                                                       replicates data
 resource          info
                                                          cheaply
      …            …
 cnn.com      110
   Leiden; Dec 06           Gossip-Based Networking Workshop                            11
 How it works
     Kelips is entirely gossip based!
          Gossip about membership
          Gossip to replicate and repair data
          Gossip about “last heard from” time used
           to discard failed nodes
     Gossip “channel” uses fixed bandwidth
          … fixed rate, packets of limited size


Leiden; Dec 06      Gossip-Based Networking Workshop   12
                                                            Node 175 is a    175

 How it works
                                                          contact for Node
                                                             102 in some
                                                 Hmm…Node 19 looks like
                                                            affinity group
                                                 a much better contact in
                                                     affinity group 2

                 Node 102


                                                                      19




                             Gossip data stream

     Heuristic: periodically ping contacts to check liveness,
      RTT… swap so-so ones for better ones.

Leiden; Dec 06              Gossip-Based Networking Workshop                 13
 Work in progress…
     Prakash Linga is extending Kelips to
      support multi-dimensional indexing,
      range queries, self-rebalancing
     Kelips has limited incoming “info rate”
          Behavior when the limit is continuously
           exceeded is not well understood.
          Will also study this phenomenon


Leiden; Dec 06      Gossip-Based Networking Workshop   14
 Replication makes it robust
     Kelips should work even during
      disruptive episodes
          After all, tuples are replicated to N nodes
          Query k nodes concurrently to overcome
           isolated crashes, also reduces risk that very
           recent data could be missed
     … we often overlook importance of
      showing that systems work while
      recovering from a disruption
Leiden; Dec 06      Gossip-Based Networking Workshop   15
    Chord (MIT group)
   The MacDonald’s of DHTs
   A data structure mapped to a network
       Ring of nodes (hashed id’s)
       Superimposed binary lookup trees
       Other cached “hints” for fast lookups
   Chord is not convergently consistent


Leiden; Dec 06     Gossip-Based Networking Workshop   16
 Chord picture
                                          0
                             255
                                              Finger        30
                       248                     links



                 241                                              64

            202


                       199                                  108
                             177
                                              123
Leiden; Dec 06                Gossip-Based Networking Workshop         17
          Chord picture
                                         Transient Network                  USA
Europe                                        Partition


                                            0                         0
                                  255                           255
                                                       30 248               30
                            248

                          241                           241                      64
                                                          64
                          202                          202

                                199                    108 199              108
                                   177                        177     123
                                             123



         Leiden; Dec 06            Gossip-Based Networking Workshop               18
 … so, who cares?
     Chord lookups can fail… and it suffers
      from high overheads when nodes churn
          Loads surge just when things are already
           disrupted… quite often, because of loads
          And can’t predict how long Chord might
           remain disrupted once it gets that way
     Worst case scenario: Chord can become
      inconsistent and stay that way
Leiden; Dec 06      Gossip-Based Networking Workshop   19
 Saved by gossip!
     Epidemic gossip: remedy for what ails
      Chord!
          c.f. Epichord (Liskov), Bambou
     Key insight:
          Gossip based DHTs, if correctly designed,
           are self-stabilizing!



Leiden; Dec 06      Gossip-Based Networking Workshop   20
 Connection to self-stabilization
     Self-stabilization theory
          Describe a system and a desired property
          Assume a failure in which code remains
           correct but node states are corrupted
          Proof obligation: show that property is
           reestablished within bounded time
     But doesn’t bound badness when
      transient disruption is occuring
Leiden; Dec 06      Gossip-Based Networking Workshop   21
 Beyond self-stabilization
     Tardos poses a related problem
          Consider behavior of the system while an endless
           sequence of disruptive events occurs
          System never reaches a quiescent state
          Under what conditions will it still behave correctly?
     Results of form “if disruptions satisfy  then
      correctness property is continuously satisfied”
     Hypothesis: with convergent consistency we
      may be able to develop a proof framework
      for systems that are continuously safe.
Leiden; Dec 06        Gossip-Based Networking Workshop       22
 Let’s look at a second example
     Astrolabe system uses a different
      emergent data structure – a tree
     Nodes are given an initial location –
      each knows its “leaf domain”
     Inner nodes are elected using gossip
      and aggregation


Leiden; Dec 06   Gossip-Based Networking Workshop   23
Astrolabe
 Intended as help for
   applications adrift in
                            Astrolabe
   a sea of information
 Structure emerges
   from a randomized
   gossip protocol
 This approach is
   robust and scalable
   even under stress
   that cripples
   traditional systems

Developed at RNS,
  Cornell
 By Robbert van
  Renesse, with many
  others helping…
 Today used
  extensively within
  Amazon.com
          Astrolabe is a flexible monitoring
          overlay


                             Name      Time    Load    Weblogic?    SMTP?      Word
                                                                               Versi
                                                                                on

                             swift     2271
                                       2011    1.8
                                               2.0         0             1      6.2

                            falcon     1971    1.5         1             0      4.1

                            cardinal   2004    4.5         1             0      6.0




swift.cs.cornell.edu
                          Periodically, pull data from monitored systems

                             Name       Time    Load   Weblogic    SMTP?      Word
                                                          ?                  Version

                             swift      2003     .67      0          1        6.2

                            falcon      1976     2.7      1          0        4.1

                            cardinal    2231
                                        2201     3.5
                                                 1.7      1          1        6.0




cardinal.cs.cornell.edu
        Leiden; Dec 06         Gossip-Based Networking Workshop                        25
 Astrolabe in a single domain
     Each node owns a single tuple, like the
      management information base (MIB)
     Nodes discover one-another through a
      simple broadcast scheme (“anyone out
      there?”) and gossip about membership
          Nodes also keep replicas of one-another’s
           rows
          Periodically (uniformly at random) merge
           your state with some else…
Leiden; Dec 06      Gossip-Based Networking Workshop   26
          State Merge: Core of Astrolabe epidemic


                           Name      Time    Load    Weblogic?    SMTP?      Word
                                                                             Versi
                                                                              on

                           swift     2011    2.0         0             1      6.2

                          falcon     1971    1.5         1             0      4.1

                          cardinal   2004    4.5         1             0      6.0




swift.cs.cornell.edu



                           Name       Time    Load   Weblogic    SMTP?      Word
                                                        ?                  Version

                           swift      2003     .67      0          1        6.2

                          falcon      1976     2.7      1          0        4.1

                          cardinal    2201     3.5      1          1        6.0




cardinal.cs.cornell.edu
        Leiden; Dec 06       Gossip-Based Networking Workshop                        27
          State Merge: Core of Astrolabe epidemic


                           Name      Time    Load    Weblogic?    SMTP?      Word
                                                                             Versi
                                                                              on

                           swift     2011    2.0         0             1      6.2

                          falcon     1971    1.5         1             0      4.1

                          cardinal   2004    4.5         1             0      6.0




swift.cs.cornell.edu
                                                                                      swift     2011    2.0


                                                                                     cardinal    2201    3.5
                           Name       Time    Load   Weblogic    SMTP?      Word
                                                        ?                  Version

                           swift      2003     .67      0          1        6.2

                          falcon      1976     2.7      1          0        4.1

                          cardinal    2201     3.5      1          1        6.0




cardinal.cs.cornell.edu
        Leiden; Dec 06       Gossip-Based Networking Workshop                                            28
          State Merge: Core of Astrolabe epidemic


                           Name      Time    Load    Weblogic?    SMTP?      Word
                                                                             Versi
                                                                              on

                           swift     2011    2.0         0             1      6.2

                          falcon     1971    1.5         1             0      4.1

                          cardinal   2201    3.5         1             0      6.0




swift.cs.cornell.edu



                           Name       Time    Load   Weblogic    SMTP?      Word
                                                        ?                  Version

                           swift      2011     2.0      0          1        6.2

                          falcon      1976     2.7      1          0        4.1

                          cardinal    2201     3.5      1          1        6.0




cardinal.cs.cornell.edu
        Leiden; Dec 06       Gossip-Based Networking Workshop                        29
 Observations
     Merge protocol has constant cost
          One message sent, received (on avg) per
           unit time.
          The data changes slowly, so no need to
           run it quickly – we usually run it every five
           seconds or so
          Information spreads in O(log N) time
     But this assumes bounded region size
          In Astrolabe, we limit them to 50-100 rows
Leiden; Dec 06       Gossip-Based Networking Workshop   30
 Big systems…


     A big system could have many regions
          Looks like a pile of spreadsheets
          A node only replicates data from its
           neighbors within its own region




Leiden; Dec 06      Gossip-Based Networking Workshop   31
          Scaling up… and up…

             With a stack of domains, we don’t want
              every system to “see” every domain
                   Cost would be huge
             So instead, we’ll see a summary
                           Name             Time           Load         Weblogic         SMTP?       Word
                               Name             Time           Load          ?
                                                                            Weblogic        SMTP? VersionWord
                                   Name 2011 Time 2.0 Load                       ?                     Version
                                                                                Weblogic 1 SMTP? 6.2 Word
                            swift                                            0
                                                                                     ?                     Version
                                swift Name 2011 Time 2.0 Load                    0 Weblogic 1 SMTP? 6.2 Word
                           falcon           1976            2.7              1          ? 0           4.1      Version
                                    swift Name 2011 Time 2.0 Load                    0 Weblogic 1 SMTP? 6.2 Word
                               falcon           1976            2.7              1         ? 0            4.1      Version
                          cardinal      swift Name 2011 Time 2.0 1 Load
                                            2201            3.5                         0 Weblogic 1 6.0
                                                                                           1            SMTP? 6.2 Word
                                   falcon           1976            2.7              1        ? 0             4.1      Version
                              cardinal      swift Name 2011 Time 2.0 1 Load
                                                2201            3.5                        0 Weblogic 1 6.0
                                                                                              1             SMTP? 6.2 Word
                                       falcon          1976             2.7             1        ? 0              4.1      Version
                                  cardinal          2201
                                                swift         2011  3.5         2.0 1         0 1          1 6.0          6.2
                                           falcon          1976             2.7            1           0              4.1
                                      cardinal         2201
                                                    swift         2011  3.5         2.0 1       0 1            1 6.0          6.2
                                               falcon         1976              2.7           1            0              4.1
                                          cardinal         2201             3.5            1           1              6.0
                                                   falcon         1976              2.7         1              0              4.1
                                              cardinal        2201              3.5           1            1              6.0
                                                  cardinal        2201              3.5         1              1              6.0


cardinal.cs.cornell.edu
        Leiden; Dec 06                   Gossip-Based Networking Workshop                                                            32
           Astrolabe builds a hierarchy using a P2P
           protocol that “assembles the puzzle” without
           any servers

    Dynamically changing
    query output is visible
    system-wide                         Name           Avg
                                                       Load
                                                              WL contact    SMTP contact                        SQL query
                                         SF

                                         NJ
                                                       2.6
                                                       2.2

                                                       1.8
                                                       1.6
                                                              123.45.61.3

                                                              127.16.77.6
                                                                             123.45.61.17

                                                                             127.16.77.11
                                                                                                             “summarizes”
                                        Paris          2.7
                                                       3.1    14.66.71.8     14.66.71.12                             data

 Name       Load    Weblogic?   SMTP?            Word         …             Name       Load      Weblogic?    SMTP?    Word     …
                                                Version                                                               Version

 swift      2.0
            1.7        0          1              6.2                        gazelle        1.7
                                                                                           4.1      0           0      4.5

falcon      1.5
            2.1        1          0              4.1                        zebra          3.2
                                                                                           0.9      0           1      6.2

cardinal    4.5
            3.9        1          0              6.0                         gnu            .5
                                                                                           2.2      1           0      6.2




                  San Francisco                                                            New Jersey

    Leiden; Dec 06                              Gossip-Based Networking Workshop                                                    33
 Large scale: “fake” regions
     These are
          Computed by queries that summarize a
           whole region as a single row
          Gossiped in a read-only manner within a
           leaf region
     But who runs the gossip?
          Each region elects “k” members to run
           gossip at the next level up.
          Can play with selection criteria and “k”
Leiden; Dec 06      Gossip-Based Networking Workshop   34
                                                replicated
           Hierarchy is virtual… data is “sees” its neighbors and
                               Yellow leaf node
                                                                             the domains on the path to the root.



                                         Name           Avg    WL contact     SMTP contact
                                                        Load

                                          SF            2.6    123.45.61.3     123.45.61.17

                                          NJ            1.8    127.16.77.6     127.16.77.11        Gnu runs level 2 epidemic
                                         Paris          3.1    14.66.71.8      14.66.71.12
                                                                                                   because it has lowest load
                       Falcon runs level 2 epidemic
                        because it has lowest load
 Name       Load     Weblogic?   SMTP?            Word         …              Name       Load      Weblogic?   SMTP?    Word     …
                                                 Version                                                               Version

 swift       2.0        0          1              6.2                         gazelle        1.7      0          0      4.5

falcon       1.5        1          0              4.1                         zebra          3.2      0          1      6.2

cardinal     4.5        1          0              6.0                          gnu            .5      1          0      6.2




                   San Francisco                                                             New Jersey

    Leiden; Dec 06                               Gossip-Based Networking Workshop                                                    35
                                         is replicated
           Hierarchy is virtual… datasees different leaf domain but
                              Green node
                                                                       has a consistent view of the inner domain



                                         Name           Avg    WL contact    SMTP contact
                                                        Load

                                          SF            2.6    123.45.61.3    123.45.61.17

                                          NJ            1.8    127.16.77.6    127.16.77.11

                                         Paris          3.1    14.66.71.8     14.66.71.12




 Name       Load     Weblogic?   SMTP?            Word         …             Name       Load      Weblogic?   SMTP?    Word     …
                                                 Version                                                              Version

 swift       2.0        0          1              6.2                        gazelle        1.7      0          0      4.5

falcon       1.5        1          0              4.1                        zebra          3.2      0          1      6.2

cardinal     4.5        1          0              6.0                         gnu            .5      1          0      6.2




                   San Francisco                                                            New Jersey

    Leiden; Dec 06                               Gossip-Based Networking Workshop                                                   36
 Worst case load?
     A small number of nodes end up
      participating in O(logfanoutN) epidemics
          Here the fanout is something like 50
          In each epidemic, a message is sent and
           received roughly every 5 seconds
     We limit message size so even during
      periods of turbulence, no message can
      become huge.
Leiden; Dec 06      Gossip-Based Networking Workshop   37
 Self-stabilization?
     Like Kelips, it seems that Astrolabe
          Is convergently consistent, self-stabilizing
          And would “ride out” a large class of
           possible failures
     But Astrolabe would be disrupted by
          Incorrect aggregation (Byzantine faults)
          Correlated failure of all representatives of
           some portion of the tree
Leiden; Dec 06       Gossip-Based Networking Workshop   38
 Focus on emergent shape
     Kelips: Nodes start with a-priori
      assignment to affinity groups, end up
      with a superimposed pointer structure
     Astrolabe: Nodes start with a-priori leaf
      domain assignments, build the tree
     What other data structures can be
      constructed with emergent protocols?

Leiden; Dec 06   Gossip-Based Networking Workshop   39
 Emergent shape
     We know a lot about a related question
          Given a connected graph, cost function
          Nodes have bounded degree
          Use a gossip protocol to swap links until
                 desired graph framework of
           some Example: The “Anthill” emerges Alberto Montresor,
                  Ozalp Babaoglu, Hein Meling and Francesco Russo
     Another related question
          Given a gossip overlay, improve it by
           selecting “better” links (usually, lower RTT)
Leiden; Dec 06         Gossip-Based Networking Workshop             40
 Example of an open problem
     Given a description of a data structure
      (for example, a balanced tree)
          … design a gossip protocol such that the
           system will rapidly converge towards that
           structure even if disrupted
          Do it with bounced per-node message
           rates, sizes (network load less important)
     Use aggregation to test tree quality?
Leiden; Dec 06      Gossip-Based Networking Workshop   41
 Van Renesse’s dreadful aggregation tree
                                            
                     D                                                 L


        B                     F                        J                     N



   A        C            E        G             I      K                   M     O
                                           An event
                                   G gossips with e                   P learns O(N)
                                          occurs
                                   H and learns e at H              time units later!

  A    B C       D       E    F G     H          I    J K       L       M    N O P




Leiden; Dec 06               Gossip-Based Networking Workshop                           42
 What went wrong?
     In Robbert’s horrendous tree, each
      node has equal “work to do” but the
      information-space diameter is larger!
     Astrolabe benefits from “instant”
      knowledge because the epidemic at
      each level is run by someone elected
      from the level below

Leiden; Dec 06   Gossip-Based Networking Workshop   43
 Insight: Two kinds of shape
     We’ve focused on the aggregation tree
     But in fact should also think about the
      information flow tree




Leiden; Dec 06   Gossip-Based Networking Workshop   44
                        Information space perspective
                                   Bad aggregation graph: diameter O(n)
                                    
                D                                            L


    B                   F                       J                    N

                                                                             H–G–E–F–B–A–C–D–L–K–I–J–N–M–O–P
A       C           E       G           I            K           M       O



A   B C     D       E   F G     H       I       J K      L   M       N O P



                                   Astrolabe version: diameterO(log(n))
                A                                            I


    A                   E                        I                   M
                                                                                             C–D
                                                                                                   A–B
                                                                                 G–H
                                                                                       E–F




A       C           E       G               I        K           M       O




                                                                                                                      K–L
                                                                                                                I–J



                                                                                                                            M–N
                                                                                                                                  O–P
A   B C     D       E   F G     H       I       J K      L       M   N O P




                    Leiden; Dec 06                                           Gossip-Based Networking Workshop                           45
 Gossip and bias
       Often useful to “bias” gossip,
        particularly if some links are fast and
        others are very slow
                                   Roughly adjust
                     Demers: Shows how tohalf theprobabilities to even the load.
    A                              gossip that must         X
                       Ziao later showed will cross also fine-tune gossip rate
                 C                    this link!                   Y
B

        D
                                                                Z
  E          F



Leiden; Dec 06         Gossip-Based Networking Workshop                       46
 Gossip and bias
     Idea: “shaped” gossip probabilities
     Gravitational Gossip (Jenkins)
          Groups carry multicast event streams
           reporting evolution of a continuous function
           (like electric power loads)
          Some nodes want to watch “closely” while
           others only need part of the data


Leiden; Dec 06      Gossip-Based Networking Workshop   47
 Gravitational GossipWhen a gossips totopic
                  Jenkins:
                 includes information about
                                            b,

                                          t in a way weighted by b’s level
                                                 of interest in topic t

                                     b



   Middle ring: Nodes want               Inner ring: Nodes want
      75% of the traffic                   100% of the traffic

                         a                   c


                                                 Outer ring: Nodes want
                                                   20% of the traffic




Leiden; Dec 06         Gossip-Based Networking Workshop                      48
 Gravitational Gossip




Leiden; Dec 06   Gossip-Based Networking Workshop   49
 How does bias
 impact information-flow graph?
     Earlier, all links were the same
     Now, some links carry
          Less information
          And may have longer delays
          (Bounded capacity messages: “similar” to
           long links?)
     Question: Model biased information
      flow graphs and explore implications

Leiden; Dec 06      Gossip-Based Networking Workshop   50
 Questions about bias
   When does the biasing of gossip target
    selection break analytic results?
        Example: Alves and Hopcroft show that with
         fanout too small, gossip epidemics can die
         out, logically partitioning a system
   Question: Can we relate the question to
    flooding on an expander graph?


Leiden; Dec 06     Gossip-Based Networking Workshop   51
 Can an overlay learn its shape?
     The unifying link across the many
      examples we’ve cited
     And needed in real world overlays when
      these adapt to underlying topology…
          For example, to pick “super-peers”
          Hints of a theory of adaptive tuning?
                Related to “sketches” in databases


Leiden; Dec 06          Gossip-Based Networking Workshop   52
 Can an overlay learn its shape?
     Suppose we wanted to use gossip to
      implement decentralized balanced
      overlay trees (nodes “know” locations)
          Build some sort of overlay
          Randomly select initial upper-level nodes
          Jiggle until they are nicely spaced
     Instance of “facility location” problem

Leiden; Dec 06      Gossip-Based Networking Workshop   53
 Emergent landmarks




Leiden; Dec 06   Gossip-Based Networking Workshop   54
 Emergent landmarks
                 Too
                 many




                                                           Too
                                                           few




Leiden; Dec 06          Gossip-Based Networking Workshop         55
 Emergent landmarks




Leiden; Dec 06   Gossip-Based Networking Workshop   56
 Emergent landmarks




Leiden; Dec 06   Gossip-Based Networking Workshop   57
 Emergent landmarks
                 One per
                 cluster




                                                              Centroi
                                                                d




Leiden; Dec 06             Gossip-Based Networking Workshop             58
       2-D scenario: Gossip variant of Euclidian
       k-medians (a popular STOC topic)

           Nodes know position on a line, can gossip
           In log(N) time seek k equally spaced nodes
                Later, after disruption, reconverge in log(N) time

K=3



           NB: The facility placement problem is just the
            d-dimensional k-centroids problem

      Leiden; Dec 06        Gossip-Based Networking Workshop      59
 Why is this hard?
     After k rounds, each node has only
      communicated with 2k other nodes
     In log(N) time, aggregate information
      has time to transit the whole graph, but
      with bounded bandwidth we don’t have
      time to send all data


Leiden; Dec 06   Gossip-Based Networking Workshop   60
 Emergent shape problem
     Can a gossip system learn its own shape
      in a convergently consistent way?
          Start with centralized solutions draw from
           the optimization community.
          Then seek decentralized gossip variants that
           achieve convergently consistent
           approximations, (re)converge in log(N)
           rounds, and use bounded bandwidth

Leiden; Dec 06      Gossip-Based Networking Workshop   61
                                                   Optional content (time
                                                        permitting)

 … more questions
     Notice that Astrolabe forces participants
      to agree on what the aggregation
      hierarchy should contain
          In effect, we need to “share interest” in
           the aggregation hierarchy
     This allows us to bound the size of
      messages (expected constant) and the
      rate (expected constant per epidemic)
Leiden; Dec 06      Gossip-Based Networking Workshop                    62
                                                   Optional content (time
                                                        permitting)

 The question
     Could we design a gossip-based system
      for “self-centered” state monitoring?
     Each node poses a query, Astrolabe
      style, on the state of the system
          We dynamically construct an overlay for
           each of these queries
          The “system” is the union of these overlays


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                                                Optional content (time
                                                     permitting)

 Self-centered monitoring




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                                                    Optional content (time
                                                         permitting)

 Self-centered queries…
     Offhand, looks like a bad idea
          If everyone has an independent query
          And everyone is iid in all the obvious ways
          Than everyone must invest work proportional to
           the number of nodes monitored by each query
     If O(n) queries touch O(n) nodes, workload
      grows as O(n) and global load as O(n2)



Leiden; Dec 06       Gossip-Based Networking Workshop                    65
                                                    Optional content (time
                                                         permitting)

 Aggregation
     … but in practice, it seems unlikely that
      queries would look this way
     More plausible is something Zipf-like
          A few queries look at broad state of system
          Most look at relatively few nodes
          And a small set of aggregates might be shared by
           the majority of queries
     Assuming this is so, can one build a scalable
      gossip overlay / monitoring infrastructure?
Leiden; Dec 06       Gossip-Based Networking Workshop                    66
 Summary of possible topics…
     Consistency models               Self-centered
     Necessary conditions              aggregation
      for convergent                   Bandwidth-limited
      consistency                       systems; “sketches”
     Self-stabilization               Using aggregation to
     Implications of bias              fine-tune a shape
     Emergent structures               with constant costs



Leiden; Dec 06   Gossip-Based Networking Workshop        67

								
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