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Transcript of 1 Analyzing and Characterizing Small-World Graphs Van Nguyen and Chip Martel Computer Science, UC...
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Analyzing and Characterizing Small-World Graphs
Van Nguyen and Chip MartelComputer Science, UC Davis
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Contents
Small-world phenomenon & Models
The diameter of Kleinberg’s grid
A Framework for Small-world graphs
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Small-world phenomenon
Nebraska
BostonTwo strangers meet and discover that they are two ends of a short chain of acquaintances
Milgram’s pioneering work (1967): “six degrees of separation between any two Americans”
Source person in Nebraska, target at person in Boston Chained people supposed to forward to someone they knew
based on a first-name basis
Here, we often use `small-world graphs’ for graphs with small diameter (poly-log functions of size)
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Modeling Small-Worlds
Many networks are Small-Worlds (e.g. WWW, Internet AS)
Motivated models of small-worlds: (Watts-Strogatz, Kleinberg)
New Analysis and Algorithms Applications peer-to-peer systems gossip protocols secure distributed protocols
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Kleinberg’s Model
Based on an n by n, 2-D grid, where each node has 4 local undirected links
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Kleinberg’s Model
Based on an n by n, 2-D grid, where each node has 4 local undirected links
q=2
Add q directed random links per each node
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Kleinberg’s Model
Based on an n by n, 2-D grid, where each node has 4 local undirected links
Add q directed random links per each node whereDefine d(u,v): lattice
distance between u and v
u
v
d(u,v)=2+5=7
Now, u has a link to v with probability proportional to d -r(u,v). Parameter r determines crucial behaviors of the model.
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Kleinberg’s SW networkis Greedy Routable iff r=2
Greedy routing algorithmusing local information only,
find a short path from s to t
When u is the current node, choose next v: the closest to t (use lattice distance) with (u,v) a local or random edge.
s
u
t
v
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Kleinberg’s SW networkis Greedy Routable iff r=2
A greedy routing algorithmusing local information only,
find a short path from s to tu
t
v
s
This greedy routing achieves expected `delivery time’ of O(log2n),
i.e. the st paths have expected length O(log2n).
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Kleinberg’s SW networkis Greedy Routable iff r=2
A greedy routing algorithmusing local information only,
find a short path from s to tu
t
v
s
This greedy routing achieves expected `delivery time’ of O(log2n), i.e. the st paths have
expected length O(log2n). This does not work unless r=2 : for r2, >0 such that the
expected delivery time of any decentralized algorithm is (n).
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Our Results An analysis of the expected diameter of Kleinberg's setting. For a 2D-grid, One random link/node (q=1) If 0 r 2: diameter=(logn) –
PODC’04
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Our Results An analysis of the expected diameter of Kleinberg's setting. For a 2D-grid, One random link/node (q=1) If 0 r 2: diameter=(logn) – PODC’04 If 2< r <4: diameter < logcn for c>1 If 4< r: diameter > nc for 0<c<1
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Our Results An analysis of the expected diameter of Kleinberg's setting. For a 2D-grid, One random link/node (q=1) If 0 r 2: diameter=(logn) – PODC’04 If 2< r <4: diameter <logcn for c>1 If 4< r: diameter> nc for 0<c<1
Can be generalized for k-D grid, say if k< r <2k: diameter < logcn for c>1
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Our ResultsA framework to construct classes of random graphs with (logn) expected diameter We start with a general framework where
random arcs are added to a fixed base graph. Then we refine this setting adding additional properties.
A more refined class of random graphs where with local information only we find paths of expected poly-log length.
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Prior work on similar (diameter) problems
Diameter of a cycle plus a random matching: Bollobas & Chung, 88 Can be seen as a special case of
Kleinberg’s grid setting where: 1-D lattice, undirected graph, r=0 (random links are uniform)
Diameter of long-range percolation graphs Benjamini & Berger, 2001 Coppersmith et al., 2002 Biskup, 2004: similar to our approach
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The diameter of Kleinberg’s SW setting
For simplicity, use a 1-D setting
Define C(r,n) as an n-node cycle. Each node has 2 local links and One directed random-link: i is
connected to j i with
Pr[ij] ~ |i-j|-r
For 0 r 1, we showed the diameter is (logn) in PODC’04
Now consider r>1.
0 12
n-1.
..
...
i
j
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Upper bound for the diameter of C(r,n) when 1<r<2
We use a probabilistic recurrence approach Our approach is similar to Karp's
(STOC’91) We establish a (probabilistic) relation
between the diameter of a segment and that of a smaller one.
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Upper bound for the diameter of C(r,n) when 1<r<2
We use a (probabilistic) relation between the diameter of a segment and a sub-segment. We relate D(x) , the diameter of a segment of length x, to D(y) , where y=xa for some a(0,1).
Intuitively, w.h.p, D(x) is bounded by a constant multiple of D(y).
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Upper bound for the diameter of C(r,n) when 1<r<2
Iterating the relation, starting with x=n, standard recurrence techniques bound D(n) - the graph's expected diameter - based on D(x0) for some x0 small enough (a poly-log function of n).
D(n)D(na)
D(na2)
D(x0)
…
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Partitioning:A segment of length x is divided into multiplesub-segments of length
y=x a for a(0,1).
Partitioning Hierarchy
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A partition is complete when every pair of sub-segments has two random directed edges connecting one to the other.
Partitioning A segment of length x is
divided into multiple sub-segments of length y=x a for some a(0,1).
Partitioning Hierarchy
A B
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We iterate this partitioning from x=n to some small x0
(for fixed a). We need to specify y (or a) s.t. Small enough # iterations
is order loglog (n) Not too small Almost
surely, each phase’s partition is complete
Partitioning HierarchyD(n)D(na)
D(na2)
D(x0)
…
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Supporting Facts
Fact 1: For a fixed a s.t. r/2< a <1 and for x large enough, almost surely, all partitions of length x segments are complete Note: 0<r<1 and y=x a
Implies that all sub-segments are large enough so can get to another by one link.
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Fact 2: If a partition of a segment of length x is complete, then almost surely D(x) is at most twice the maximum diameter of a subsegment, plus 1. Basically, any shortest path st can be upper bounded by two shortest paths within a
sub-segment plus 1
length(st) length(sv)+length(wt)+1for (v,w) 2 max D(y) +1
Supporting Facts
A
u u+x-1s tv w* *
B
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Fact 2 still true if we redefine D(x) as the maximum value of the diameters of all segments of length x
Supporting Facts
Fact 2: If a partition of a segment of length x is complete, then almost surely D(x) is at most twice the maximum diameter of a sub-segment, plus 1.
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Poly-log diameter for 1r2
Consider the sequence of maximum diameter values in our partitioning hierarchy
D(n), D(na), … ,D(x0)Where almost surely, D(x) 2D(x a)+1
The # of terms is (loglog n) D(x0) x0, bounded by a poly-log(n) So, D(n)= O(logcn)
for c>0 depending on r only
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The diameter of C(r,n)
For r>2, C(r,n) is a ‘large’ world expected diameter (nc), c=r-1/r
Random links tend to go to close nodes Few long links
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Higher dimensions We generalize to k-dimensional grids If 0 r k: diameter=(logn) If k< r <2k: diameter < logcn , c>1
If 2k< r: diameter> nc for 0<c<1 The case r=2k is still open.
Also generalized for Growth Restricted Graphs (mention more later)
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Building Small-World Graphs
We start with a general framework where random arcs are added to a fixed base graph. Then we refine this setting adding additional properties.
Create Families of Random Graphs - FRG (H,): H: set of base graphs (e.g. grids) : a distribution for adding random links
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Building Small World Graphs
Based on a random assignment operation: For a given node u, make a random trial under
distribution to find another node v Each assignment performs an independent trial E.g. in Kleinberg’s grid setting,
Base graphs are grids
is defined as having uv with probability d-r (u,v) We want to characterize distributions so most shortest paths are (logn)
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Our small-world graphs: the distribution of random links
u
C
Neighbor sets should have exponential growth If a node u is surrounded
by a moderate size set of vertices C, a random link from u is likely to “escape” from C.
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Our small-world graphs: the distribution of random links
Neighbor sets should have exponential growth If a node u is surrounded
by a moderate size set of vertices C, a random link from u is likely to “escape” from C. diversity and fairness: no small
set takes most of chance to be hit “don't give too much to a small group“
u
C
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Our small-world graphs: the distribution of random links
Neighbor sets should have exponential growth If a node u is surrounded
by a moderate size set of vertices C, a random link from u is likely to “escape” from C.
u
C
We define (in paper) precisely the two parameters: the size of set C and the probability (for escaping from C)
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Our small-world graphs: the distribution of random links
Similar criterion for the `inverse direction’ If a node u is surrounded
by a moderate size set of vertices C, there likely exists a random link coming to u from outside of C.
u
C
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Expansion families
Expansion familiesA Random Family (H,) is an
Expansion Family if the distribution satisfies the two expansion criteria.
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Out-ExpansionEach node likely to have a random link out of a neighborhood of certain size
In-ExpansionEach node is likely to be reached by a random link from outside of a neighborhood of certain size
FRG (H,)From a base graph (of a collection H) generate independent random links,
using distribution
Expansion family
Refining for small-worlds
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Expansion family log n-neighbored base graphs
small-world with expected diameter =(logn)
Refining for small-worlds
Includes many well-known SW settings, such as Kleinberg’s grid and hierarchy model
Out-ExpansionEach node likely to have a random link out of a neighborhood of certain size
In-ExpansionEach node is likely to be reached by a random link from outside of a neighborhood of certain size
FRG (H,)From a base graph (of a collection H) generate independent random links,
using distribution
Expansion family
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Applications of the framework
To obtain diameter bounds for some small-world models, E.g. Kleinberg’s k-dimension grid model
for any k 1 (as in our earlier PODC’04 paper )
To augment certain settings to become graphs with small diameters Example is next on Kleinberg’s Tree-
based setting
Also more: show later
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Kleinberg’s Tree-based setting
Quite different to grid setting Nodes are leaves of a full b-ary tree T A distance measure: h(u,v) – the height of the
least common ancestor of u and v That tree T is only used for defining this distance
A random link from a node u can go to v with probability b-h(u,v).
No local links possibly unconnected
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Kleinberg’s Tree-based setting
Quite different to grid setting Nodes are leaves of a full b-ary tree T A distance measure: h(u,v) – the height of the least
common ancestor of u and v A random link from a node u can go to v with probability
b-h(u,v). No local links possibly unconnected
If there is at least 3 random links going out from each node, this setting is an Expansion Family If we add in local links to make an appropriate
base graph, then the graph becomes a small-world: A way to do so, say, ring the nodes within a sub-tree
of size logn
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More refined classes using distance measures
We add a general distance function d:V2R+ and hence, define our base graphs as growth restricted graphs, where the growth of a neighborhood (nodes within distance r from u) is (r). E.g. think of a -D grid but can be
any positive real value
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A phase transition on diameter
Class InvDist(,): We also add random links such that
Pr[uv] ~ d-(u,v) E.g. Kleinberg’s 2-D setting for
greedy-routing is InvDist(2,2) The diameter is poly-log(n) if <2, but changes to polynomial (n) for >2
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A Design for Greedy-like Routing
We further refining, adding = and some condition on the connectivity of small neighborhoods to gain a class of random graphs where Greedy-like Routing is possible: Each node doesn’t have the global topology,
but `knows’ a small neighborhood (i.e. knows the random links coming from there)Choose the random link which goes closest to the destination
All Kleinberg’s settings (grid, tree, group-induced) are (or after some easy augment) of this class
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ExpansionEach node has q (,)-EXP links, where q>1
InExpansionSimilar to Expansion but for incoming links
FRG (H,)From a base graph (of a collection H) generate independent random links,
using distribution
Expansion family
Exp-family with logn-neighbored base graph
InvDist(,)Growth restricted graphs degree +random links: Pr[uv] ~ d-
(u,v)
METR()
where = and some easy conditions
-symmetric InvDist with logn-neighbored base graphs
0: small-world with D=(logn)
<<2: SW, D=poly-log(n)
2<:`large’ world, D= poly(n)
Greedy-routable with short paths (log2n)
small-world with D=(logn)
FRG Hierarchy
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Future Work Many known Network graphs follow some `growth restricted’ rules. E.g. wireless networks can be modeled using the
unit disk graph (=2) The Internet network distance defined by round-
trip propagation and transmission delay forms growth restricted metrics (Ng&Zhang, SPAA’02)
Idea: Using our framework, consider adding long links to certain Network graphs to shrink these graphs (into small-worlds, ideally) E.g., how to add in long links (fixed long wire) to
a wireless network (unit disk) to best shrink the graph diameter