Transcript dist(i,s)+

Measuring and
Extracting Proximity
in Complex Networks
Emden Gansner, Yehuda Koren, Stephen North, Chris Volinsky
AT&T Labs Research
AT&T “Safe Harbor”
The following contains "forward-looking statements" which are
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large social networks
data source
|V|
|E|
co-authors
438K
31M
actor-actor
896K
1.1M
phone calls
300M
1000M
200M
800M
IM
18 node subgraph
Proximity: 1.35e+01
Captured: 1.31e+01
(97%)
Adam?
Glenn?
Emden?
95% of communication between…
5 node subgraph
Proximity: 7.10e+00
Captured: 6.74e+00
(95%(
Our goals
• Measure proximity between nodes.
• Explain proximity by extracting connection
subgraphs that are readily visualized.
What is proximity?
• proximity [prox·im·i·ty || prɑk'sɪmətɪ /prɒ-]n.
adjacency, nearness, closeness, vicinity
• Network proximity is an elusive notion!
• Let’s work by refining a series of definitions.
Measuring proximity
• Simplest approach – length of shortest path
Easily visualized
Measuring proximity
• Simplest approach – length of shortest path
Easily illustrated
Disregards alternative paths
Captures 56%
Captures 98%
Measuring proximity
• Simplest approach – length of shortest path
Easily visualized
Disregards alternative paths
Naïve calculation will be fooled by high
degrees
Example from a telephone call graph…
Which pair is closer?
Meaningful
connection
Suresh
Shankar
Random
connection?
Lefty
Stephen
• Both paths are 2-hops, about the same
lengths
• But when considering node-degrees…
Measuring proximity – 2nd try
• Net network flow between the nodes
Accounts for multiple paths
Distance indifferent – might favor long paths
High degree are still an issue
Measuring proximity – 3rd try
• Delivered electric current
(effective conductance)
• Resistor network model
Accounts for multiple paths
Penalizes long paths
• High degrees??
• Getting us closer…
• “intuitive”
Physical analogy is not
perfect!
1V
0V
edge weights
conductance,
inverse-resistance
When is the electrical current
analogy misleading?
Significant connection
Noise?
What does current flow mean?
When is the electric current
analogy misleading?
Significant connection
Noise?
• Same current flow in both cases!
• Degree-1 nodes are neutral (attract no flow)
• Degree-1 nodes are very common, due to
incomplete information
Augment network by a universal sink
[Faloutsos, McCurley & Tomkins, KDD 2004]
• Connect all nodes to a grounded universal sink (with 0V)
• Tax each node - deliver portion of the flow to the sink
No internal nodes of degree 1 (above problem solved)
Penalizes long paths
A new parameter to worry about:
Which tax system? Constant tax? Proportional tax? Tax brackets? How much?
• There is a worse problem…
Universal sink and (non-)monotonicity
Proximity
• In our previous notions of proximity, adding
nodes/edges to the network couldn’t decrease
proximity
• Hmmm…this “blind monotonicity” was part of
their shortcoming…
Network size
Universal sink and (non-)monotonicity
Proximity
• For all previous measures, adding nodes/edges
to the network couldn’t decrease proximity
• With universal sink – no monotonicity:
Larger network  proximity tends to zero, sink
attracts more flow
• Even adding s—t paths can decrease proximity!
Network size
Universal sink and (non-)monotonicity
• Problems with non-monotonicity:
Proximity
– Counter-intuitive and hard to use
– Size bias makes proximity-comparison across
different pairs completely unreliable
– Impossible to explain (size-dependent) proximity
using a connection subgraph
Network size
A random-walk perspective
• Current-flow model has a direct r.w. interpretation
• Reminder:
We defined proximity by “delivered current” or
“effective conductance”
• The escape probability, Pesc(st), is the
probability that a r.w. originating at s will reach t
before visiting s again
• Let Deg(s) be the number of r.w.’s originating at s
• The effective conductance between s and t, is
Pesc(st)*Deg(s)
• “Dead end” paths have no influence on
escape probability
• Both graphs have the same escapeprobability from red to green
Lower redgreen
escape probability
Higher redgreen
escape probability
In both cases higher effective conductance
by Rayleigh’s Monotonicity Law
Extending escape probability
• The escape probability, Pesc(st), is the
probability that a r.w. originating at s will reach t
before visiting s again
• The cycle-free escape probability, Pc.f.esc(st)
is the probability that a r.w. originating at s will
reach t without visiting any node more than once
• Multiply by degree to get an absolute quantity
(accounting for the number of "actually initiated"
r.w.'s):
The c.f. effective conductance between s and t is
Pc.f.esc(st)*Deg(s)
Higher redgreen
c.f. escape probability
Lower redgreen
c.f. escape probability
• The c.f. effective conductance is a good
candidate proximity measure:
Accounts for multiple paths
Favors short paths
Penalizes high-degree nodes
Penalizes dead-end paths
Parameter free
Has the “right” monotonicity
Accommodates edge directions
Has a natural extension to multiple endpoints
Computing c.f. escape probability
• Unlike previous measures, exact computation is
impossible
Pc.f.esc (s  t ) =

prob( p)
simple path p[ s t ]
• Practically, we can estimate it extremely well
• Probability of paths declines exponentially (e.g., 100th
path is x106 less probable than the first one.)
• Estimate using the most probable paths:
Pc.f.esc (s  t ) =

highly probable
simple path p[ s t ]
prob( p)
Finding k most probable paths
• For an edge u-v of weight w(u,v), define its length


w(u , v)
l (u , v) =  log 

 deg(u ) deg(v) 


•
•
•
•
Edge lengths are positive
Exp(-<length of path>) = Prob(path)
Short path
High-probable path
Compute k shortest simple paths in O(k|E|log|E|) time
[Katoh, Ibarki and Mine, 1982]
• Stop searching when probability drops below “10-6” of
first path
Extracting and explaining
proximity
Extracting proximity
• Cycle free effective conductance (CFEC) depends on the
full graph
• Find a small subgraph that captures the most proximity
• A tradeoff between “size” and “captured proximity”, can be
expressed in alternative ways:
– Extract a subgraph with at most B nodes that captures maximal
CFEC
• Maybe with B+1 nodes we can capture much more???
– Extract a minimal-sized subgraph that captures at least P% of
total CFEC
• Maybe we can capture (P-1)% of total CFEC with a much
smaller subgraph???
Extracting proximity
• Find a small subgraph that captures most proximity
• Achieve an efficient balance between “size” and
“proximity” by maximizing the ratio:
 CFEC(s  t ) 

subgraph
• Larger α  emphasize proximity  larger subgraph
– α=0  returns only the shortest path
– α=∞ return all paths
• Optionally, explicitly fix lower and upper bounds on
subgraph size
What solutions do we seek?
• Overlapping paths delivering the most flow
The path merger algorithm
• We already have a collection of paths
• Find the subset of the paths that maximizes
 CFEC(s  t ) 
subgraph

• Combine the selected paths
into a “proximity subgraph”
• Overlapping paths are
cheaper to add
• An NP-hard problem…
Optimal algorithm
• Scanning all subsets takes O(2k) time (can we
do better?)
• A branch-and-bound pruning significantly
reduces running time
• Huge deviations in path-quality make this
approach effective
e.g. often it is clear that the best-subset must
contain first path(s)
• Prematurely terminate exponential algorithm
after scanning “too many” subsets
Agglomerative algorithm
• If optimal algorithm couldn’t finish, improve
current result by an agglomerative algorithm
• Iteratively, merge the two subsets that
maximize the ratio
• Record the best subset discovered
Working with large graphs in
external storage
• Dealing with full graph is sometimes
infeasible and usually unnecessary
• Prior to running the algorithm, we
construct a candidate graph in main
memory
• We begin by growing increasing
neighborhoods around the endpoints
S
T
Dist(S,i)=2
S
Dist(T,i)=2
T
Dist(S,i)=3
S
Dist(T,i)=3
T
Dist(S,i)=4
S
Dist(T,i)=4
T
Dist(S,i)=5
Dist(T,i)=5
S
T
Shortest path of
length 10
Most probable path of length 10 was found
No use for lowprobability paths...
Paths longer than
“24” unneeded!
S
Dist(S,i)=12
T
Dist(T,i)=12
S
T
• Stop adding nodes
• Any s—t path through unscanned node must be longer
than “24”, thus useless
• Can we prune the resulting graph?
• Yes!
• From two circles into an ellipse…
Dist(S,i)=12
i
Dist(T,i)=12
Pruning the candidate graph
• We can safely prune a significant portion of the
candidate graph
• Use the fact:
dist(i,s)+dist(i,t)>L  all s—t paths going via i
are longer than L
• We ignore much less probable paths
Paths longer than “24” are not interesting
• Take only nodes within the ellipse defined by:
dist(i,s)+dist(i,t)<24
From 2-centers of circles to 2-foci
of ellipse
S
Dist(S,i)=12
T
Dist(T,i)=12
Some statistics…
Distribution of proximities in phone-call
network
Distribution of #hops in phone-call
network
Summary
• Proposed cycle free effective conductance
(CFEC) with a random walk interpretation
to measure “proximity” in social networks
and other ad-hoc networks
• Described a way of approximating CFEC
• Described a way of visualizing CFEC as a
subgraph
• Extended the method to external datasets
• Showed empirical evidence for its utility
Extensions
• Study proximity in other kinds of networks.
• Extend c.f. effective conductance to:
– Multiple endpoints (already demonstrated)
– Directed edges (future work – use k-shortest
paths in a directed graph, alg. due to
Hershberger et al ).