Hyper an fandfb-handout

Graph distance distribution
for social network mining

Paolo Boldi, Marco Rosa, Sebastiano Vigna
Laboratory for Web Algorithmics
Università degli Studi di Milano, Italy

Lars Backstrom, Johan Ugander
Facebook

Friday, September 7, 2012

Plan of the talk


Plan of the talk

• Computing distances in large graphs (using
HyperANF)


Plan of the talk

HyperANF)

• Running HyperANF on Facebook (the largest
Milgram-like experiment ever performed)


Plan of the talk

HyperANF)

• Running HyperANF on Facebook (the largest
Milgram-like experiment ever performed)

• Other uses of distances (in particular: robustness)


Prelude
Milgram’s experiment is 45


Where it all started...



• M. Kochen, I. de Sola Pool: Contacts and inﬂuences.
(Manuscript, early 50s)




• A. Rapoport, W.J. Horvath: A study of a large
sociogram. (Behav.Sci. 1961)




• A. Rapoport, W.J. Horvath: A study of a large
sociogram. (Behav.Sci. 1961)

• S. Milgram, An experimental study of the sma$ world
problem. (Sociometry, 1969)


Milgram’s experiment



• 300 people (starting population) are asked to
dispatch a parcel to a single individual (target)




• The target was a Boston stockbroker




• The starting population is selected as follows:




• 100 were random Boston inhabitants (group A)




• 100 were random Nebraska strockbrokers (group B)




• 100 were random Nebraska strockbrokers (group B)
• 100 were random Nebraska inhabitants (group C)



• Rules of the game:



• parcels could be directly sent only to someone
the sender knows personally



• parcels could be directly sent only to someone
the sender knows personally

• 453 intermediaries happened to be involved in
the experiments (besides the starting
population and the target)



• Questions Milgram wanted to answer:



• How many parcels will reach the target?



• What is the distribution of the number of hops
required to reach the target?



required to reach the target?

• Is this distribution diﬀerent for the three
starting subpopulations?



• Answers:



• Answers:
• How many parcels will reach the target? 29%



• Answers:
required to reach the target? Avg. was 5.2



• Answers:
required to reach the target? Avg. was 5.2

• Is this distribution diﬀerent for the three starting
subpopulations? Yes: avg. for groups A/B/C was
4.6/5.4/5.7, respectively


Chain lengths


Milgram’s popularity


• Six degrees of separation slipped away from the
scientiﬁc niche to enter the world of popular
immagination:


immagination:

• “Six degrees of separation” is a play by John
Guare...


immagination:

Guare...

• ...a movie by Fred Schepisi...


immagination:

Guare...

• ...a movie by Fred Schepisi...
• ...a song sung by dolls in their national costume
at Disneyland in a heart-warming exhibition
celebrating the connectedness of people all


Milgram’s criticisms



• “Could it be a big world after all? (The six-
degrees-of-separation myth)” (Judith S. Kleinfeld,
2002)



2002)

• The vast majority of chains were never
completed



2002)

• The vast majority of chains were never
completed

• Extremely diﬃcult to reproduce


Measuring what?


Measuring what?

• But what did Milgram’s experiment reveal, after
all?


Measuring what?

all?

i) That the world is small


Measuring what?

all?

i) That the world is small

ii)That people are able to exploit this smallness


HyperANF
A tool to compute distances in large graphs


Introduction


Introduction

• You want to study the properties of a huge graph
(typically: a social network)


Introduction


• You want to obtain some information about its global
structure (not simply triangle-counting/degree
distribution/etc.)


Introduction


• You want to obtain some information about its global
structure (not simply triangle-counting/degree
distribution/etc.)

• A natural candidate: distance distribution


Graph distances and
distribution


Graph distances and
distribution
• Given a graph, d(x,y) is the length of the shortest
path from x to y (∞ if one cannot go from x to y)


Graph distances and
distribution

• For undirected graphs, d(x,y)=d(y,x)


Graph distances and
distribution

• For every t, count the number of pairs (x,y) such
that d(x,y)=t


Graph distances and
distribution

• For every t, count the number of pairs (x,y) such
that d(x,y)=t

• The fraction of pairs at distance t is (the density
function of) a distribution


Exact computation


Exact computation
• How can one compute the distance distribution?


Exact computation
• Weighted graphs: Dijkstra (single-source: O(n2)),
Floyd-Warshall (all-pairs: O(n3))


Exact computation

• In the unweighted case:


Exact computation

• a single BFS solves the single-source version of
the problem: O(m)


Exact computation

• a single BFS solves the single-source version of
the problem: O(m)

• if we repeat it from every source: O(nm)


Sampling pairs


Sampling pairs

• Sample at random pairs of nodes (x,y)


Sampling pairs

• Compute d(x,y) with a BFS from x


Sampling pairs

• Compute d(x,y) with a BFS from x
• (Possibly: reject the pair if d(x,y) is inﬁnite)


Sampling pairs

• For every t, the fraction of sampled pairs that
were found at distance t are an estimator of the
value of the probability mass function


Sampling pairs

• For every t, the fraction of sampled pairs that
were found at distance t are an estimator of the
value of the probability mass function

• Takes a BFS for every pair O(m)


Sampling sources


Sampling sources

• Sample at random a source x


Sampling sources

• Sample at random a source x
• Compute a full BFS from x


Sampling sources

• It is an unbiased estimator only for undirected and
connected graphs


Sampling sources

connected graphs

• Uses anyway BFS...


Sampling sources

connected graphs

• ...not cache friendly


Sampling sources

connected graphs

• ...not cache friendly
• ...not compression friendly


Cohen’s sampling


Cohen’s sampling

• Edith Cohen [JCSS 1997] came out with a very
general framework for size estimation: powerful,
but doesn’t scale well, it is not easily parallelizable,
requires direct access


Alternative: Diﬀusion



• Basic idea: Palmer et. al, KDD ’02



• Let Bt(x) be the ball of radius t about x (the set of
nodes at distance ≤t from x)




• Clearly B0(x)={x}




• Moreover Bt+1(x)=∪x→yBt(y)∪{x}




• Moreover Bt+1(x)=∪x→yBt(y)∪{x}
• So computing Bt+1 starting from Bt one just need a
single (sequential) scan of the graph


Easy but costly


Easy but costly

• Every set requires O(n) bits, hence O(n2) bits
overall


Easy but costly

overall

• Too many!


Easy but costly

overall

• Too many!
• What about using approximated sets?


Easy but costly

overall

• Too many!
• We need probabilistic counters, with just two
primitives: add and size?


Easy but costly

overall

• Too many!
• We need probabilistic counters, with just two
primitives: add and size?

• Very small!


HyperANF


HyperANF

• We used HyperLogLog counters [Flajolet et al.,
2007]


HyperANF

2007]

• With 40 bits you can count up to 4 billion with a
standard deviation of 6%


HyperANF

2007]

• With 40 bits you can count up to 4 billion with a
standard deviation of 6%

• Remember: one set per node!


Observe that


Observe that

• Every single counter has a guaranteed relative
standard deviation (depending only on the number
of registers per counter)


Observe that


• This implies a guarantee on the summation of the
counters


Observe that


• This implies a guarantee on the summation of the
counters

• This gives in turn precision bounds on the
estimated distribution with respect to the real one


Other tricks


Other tricks

• We use broadword programming to compute eﬃciently
unions


Other tricks

unions

• Systolic computation for on-demand updates of
counters


Other tricks

unions

• Systolic computation for on-demand updates of
counters

• Exploited micropara$elization of multicore
architectures


Real speed?


Real speed?

• Small dimension: 1.8min vs. 4.6h on a graph with
7.4M nodes


Real speed?

• Small dimension: 1.8min vs. 4.6h on a graph with
7.4M nodes

• Large dimension: HADI [Kang et al., 2010] is a
Hadoop-conscious implementation of ANF. Takes 30
minutes on a 200K-node graph (on one of the 50
world largest supercomputers). HyperANF does the
same in 2.25min on our workstation (20 min on this
laptop).


Try it!


Try it!

• HyperANF is available within the webgraph
package


Try it!

package

• Download it from


Try it!

package

• http://webgraph.law.dsi.unimi.it/


Try it!

package

• http://webgraph.law.dsi.unimi.it/
• Or google for webgraph


Running it on Facebook!
[with Lars Backstrom and Johan Ugander]


Facebook


Facebook

• Facebook opened up to non-college students on
September 26, 2006


Facebook

• Facebook opened up to non-college students on
September 26, 2006

• So, between 1 Jan 2007 and 1 Jan 2008 the number
of users exploded


Experiments (time)

• We ran our experiments on snapshots of facebook
• Jan 1, 2007
• Jan 1, 2008 ...
• Jan 1, 2011
• [current] May, 2011


Experiments (dataset)

• We considered:
• -: the whole facebook
• it / se: only Italian / Swedish users
• it+se: only Italian & Swedish users
• us: only US users
• Based on users’ current geo-IP location


Active users

• We only considered active users (users who have
done some activity in the 28 days preceding 9 Jun
2011)

• So we are not considering “old” users that are not
active any more

• For - [current] we have about 750M nodes


Distance distribution ($)


fb 2007

0.7
0.6
0.5

●
0.4
% pairs

●
0.3
0.2
0.1

● ●

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


fb 2008

0.7
0.6
0.5

●
0.4
% pairs

0.3

●
0.2

●
0.1

●

● ●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


fb 2009

0.7
0.6
0.5

●
0.4
% pairs

0.3

●
0.2

●
0.1

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


fb 2010

0.7
0.6
0.5
0.4 ●
% pairs

0.3

●
0.2

●
0.1

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


fb 2011

0.7
0.6
0.5
0.4 ●
% pairs

0.3

●
0.2

●
0.1

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


fb current

0.7
0.6
●
0.5
0.4
% pairs

0.3

●
0.2

●
0.1

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


Distance distribution (it)


it 2007

0.7
0.6
0.5
0.4
% pairs

0.3
0.2
0.1

● ● ●
●
● ●
●
● ●
● ● ● ● ● ●
● ● ● ●
0.0

●

0 5 10 15

distance


it 2008

0.7
0.6
0.5
0.4
% pairs

0.3

●

●
0.2

●
0.1

● ●

●
● ●
●
0.0

● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


it 2009

0.7
0.6
0.5

●
0.4

●
% pairs

0.3
0.2
0.1

●

●
●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


it 2010

0.7
●

0.6
0.5
0.4
% pairs

0.3
0.2

●

●
0.1

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


it 2011

0.7
●

0.6
0.5
0.4
% pairs

0.3

●
0.2
0.1

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


it current

0.7
●

0.6
0.5
0.4
% pairs

0.3

●
0.2

●
0.1

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


Distance distribution (se)


se 2007

0.7
0.6
0.5
0.4
% pairs

0.3
0.2

● ●

● ●
0.1

●
●
●
● ●
●
0.0

● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


se 2008

0.7
0.6

●
0.5
0.4
% pairs

●
0.3
0.2
0.1

●

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


se 2009

0.7
0.6
●
0.5
0.4
% pairs

0.3

●
0.2

●
0.1

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


se 2010

0.7
0.6
●
0.5
0.4
% pairs

0.3
0.2

●
●
0.1

●
0.0

● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


se 2011

0.7
●
0.6
0.5
0.4
% pairs

0.3

●
0.2

●
0.1

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


se current

0.7
0.6
0.5
0.4 ●
% pairs

0.3

●
0.2

●
0.1

●
0.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0 5 10 15

distance


Average distance
it se

● ●

2008 curr

4.0 4.5 5.0 5.5
avg. distance

avg. distance
4 5 6 7 8 9

●

●
● ●
●

6.58 3.90
● ● ● ● ●

2007 2008 2009 2010 2011 current 2007 2008 2009 2010 2011 current
it
year year

4.33 3.89
itse us

● ● ●
se
9

4.7
avg. distance

avg. distance
8

●
7

4.5
6

4.9 4.16
●

it+se
5

● ● ●
●
4.3

● ● ●
4

2007 2008 2009 2010 2011 current 2007 2008 2009 2010 2011 current

year year

● ●
fb us 4.74 4.32
4.6 4.8 5.0 5.2
avg. distance

●

●
● ●
$ 5.28 4.74
2007 2008 2009 2010 2011 current

year


Average distance
it se

● ●

2008 curr

4.0 4.5 5.0 5.5
avg. distance

avg. distance
4 5 6 7 8 9

●

●
● ●
●

6.58 3.90
● ● ● ● ●

2007 2008 2009 2010 2011 current 2007 2008 2009 2010 2011 current
it
year year

4.33 3.89
itse us

● ● ●
se
9

4.7
avg. distance

avg. distance
8

●
7

4.5
6

4.9 4.16
●

it+se
5

● ● ●
●
4.3

● ● ●
4

2007 2008 2009 2010 2011 current 2007 2008 2009 2010 2011 current

year year

● ●
fb us 4.74 4.32
4.6 4.8 5.0 5.2
avg. distance

●

●
● ●
$ 5.28 4.74
2007 2008 2009 2010 2011 current

year
$ (current): 92% pairs
are reachable!

Eﬀective diameter (@ 90%)
effective diameters

● 2008 curr
8

●
●
●
●
● ●
it 9.0 5.2
●
● ●
●
6

● ●
●

5.9 5.3
● ●
●
se
effective diam.

●

●
4

it
*
*
*
it
se
itse
us
+se 6.8 5.8
*
* fb
2

us 6.5 5.8
0

2008 2009 2010 2011
$ 7.0 6.2
year


Harmonic diameter
harmonic diameters

2008 curr
30

●

23.7 3.4
25

it
* it
20

* se
* itse
se 4.5 4.0
harmonic diam.

* us
* fb
15

it
+se 5.8 3.8
10

●
●
● ● ● ●
us 4.6 4.4
5

●
● ●
● ●
● ●
● ●
●
●
0

2008 2009 2010 2011
$ 5.7 4.6
year


Average degree vs. density ($)

Avg. degree Density
2009 88.7 6.4 * 10-7
2010 113.0 3.4 * 10-7
2011 169.0 3.0 * 10-7
curr 190.4 2.6 * 10-7


Actual diameter

2008 curr
it >29 =25
se >16 =25
it+se >21 =27
us >17 =30
# >16 >58


Actual diameter
Used the fringe/double-sweep
technique for “=”

2008 curr
it >29 =25
se >16 =25
it+se >21 =27
us >17 =30
# >16 >58


Other applications
Spid, network robustness and more...


What are distances good for?



• Network models are usually studied on the base of
the local statistics they produce



• Network models are usually studied on the base of
the local statistics they produce

• Not diﬃcult to obtain models that behave correctly
locally (i.e., as far as degree distribution, assortativity,
clustering coeﬃcients... are concerned)


Global = more informative!


An application


An application

• An application: use the distance distribution as a
graph digest


An application

• An application: use the distance distribution as a
graph digest

• Typical example: if I modify the graph with a
certain criterion, how much does the distance
distribution change?


Node elimination


Node elimination

• Consider a certain ordering of the vertices of a graph


Node elimination

• Fix a threshold ϑ, delete all vertices (and all
incident arcs) in the speciﬁed order, until ϑm arcs
have been deleted


Node elimination

• Fix a threshold ϑ, delete all vertices (and all
incident arcs) in the speciﬁed order, until ϑm arcs
have been deleted

• Compute the “diﬀerence” between the graph you
obtained and the original one


Experiment

0.12

0.1

0.08
probability

0.06

0.04

0.02

0
1 10 100
length

0.00 0.05 0.15 0.30
0.01 0.10 0.20

Deleting nodes in order of (syntactic) depth

Experiment (cont.)

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.05 0.1 0.15 0.2 0.25 0.3
θ

Kullback-Leibler L1
δ-average distance L2

Distribution divergence (various measures)

Removal strategies
compared
1

0.8
δ-average distance

0.6

0.4

0.2

0

0 0.05 0.1 0.15 0.2 0.25 0.3
θ

random PR near-root
degree LP


Removal in social networks

1

0.8
δ-average distance

0.6

0.4

0.2

0

0 0.05 0.1 0.15 0.2 0.25 0.3
θ

random degree PR LP


Findings


Findings

• Depth-order, PR etc. are strongly disruptive on
web graphs


Findings

• Depth-order, PR etc. are strongly disruptive on
web graphs

• Proper social networks are much more robust, still
being similar to web graphs under many respects


Another application: Spid



• We propose to use spid (shortest-paths index of
dispersion), the ratio between variance and average in
the distance distribution




• When the dispersion index is <1, the distribution is
subdispersed; >1, is superdispersed




• When the dispersion index is <1, the distribution is
subdispersed; >1, is superdispersed

• Web graphs and social networks are different under
this viewpoint!


Spid plot
4

3.5

3

2.5
spid

2

1.5

1

0.5

0
10000 100000 1e+06 1e+07 1e+08 1e+09 1e+10
size

Spid conjecture


Spid conjecture

• We conjecture that spid is able to tell social
networks from web graphs


Spid conjecture


• Averag distance alone would not suﬃce: it is very
changeable and depends on the scale


Spid conjecture



• Spid, instead, seems to have a clear cutpoint at 1


Spid conjecture



• What is Facebook spid?


Spid conjecture



• What is Facebook spid? [Answer: 0.093]


Average distance∝ Eﬀective diameter
18

16

14
average distance

12

10

8

6

4

2
0 5 10 15 20 25 30
interpolated effective diameter


That’s all, folks!


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