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8/14/2019 Way Finding in Social Networks
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Wayfinding in
Social Networks
David Liben-Nowell
Carleton College
dlibenno@carleton.edu
Microsoft ResearchCambridge | 7 December 2007
Workshop on Online Social Networks
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Analyzing Social Networks, pre-1995
Social Network Analysis: Old Sool
social networks have been around for 100K+ years!
before the web, hard to acquire (surveys, interviews, ...).
but many interesting, relevant, generalizable observations!
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Zacharys Karate Club
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admin
[Zachary 1977]
Recorded interactions in a
karate club for 2 years.
During observation,
adminstrator/instructor
conflict developed broke into two clubs.
Who joins which club?
Split along
administrator/instructor
minimum cut (!)
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Zacharys Karate Club
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admin
[Zachary 1977]
Recorded interactions in a
karate club for 2 years.
During observation,
adminstrator/instructor
conflict developed
broke into two clubs.
Who joins which club?
Split along
administrator/instructor
minimum cut (!)
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Zacharys Karate Club
inst2
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admin
[Zachary 1977]
Recorded interactions in a
karate club for 2 years.
During observation,
adminstrator/instructor
conflict developed
broke into two clubs.
Who joins which club?
Split along
administrator/instructor
minimum cut (!)
5
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Zacharys Karate Club
inst2
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admin
[Zachary 1977]
Recorded interactions in a
karate club for 2 years.
During observation,
adminstrator/instructor
conflict developed
broke into two clubs.
Who joins which club?
Split along
administrator/instructor
minimum cut (!)
6
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Zacharys Karate Club
inst2
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admin
[Zachary 1977]
Recorded interactions in a
karate club for 2 years.
During observation,
adminstrator/instructor
conflict developed
broke into two clubs.
Who joins which club?
Split along
administrator/instructor
minimum cut (!)
7
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Why to think algorithmically about social networks
The small-world phenomenon
Models of the navigable small world
Some conclusions and musings
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Milgram: Six Degrees of Separation
Social Networks as Networks: [Milgram 1967]
People given letter, asked to forward to one friend.
Source: random residents of Omaha, NE;
Target: stockbroker near Boston, MA.
Of completed chains, averaged six hops to reach target.8
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Milgram: The Explanation?
the small-world problem
Why is a random Omahaian close to a Boston stockbroker?
Standard (pseudosociological, pseudomathematical) explanation:
(Erdos/Renyi) random graphs have small diameter.
Bogus! In fact, many bogosities:degree distribution
clustering coefficients
...
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High School Friendships
[Moody 2001]11
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Homophily
homophily: a person xs friends tend to be similar to x.
One explanation for high clustering: (semi)transitivity of similarity.
x, y both friends of u x and u similar; y and u similar
x and y similar
x and y friends12
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Navigability of Social Networks
[Kleinberg 2000]
Milgram experiment shows more than small diameter:
People can construct short paths!
Milgrams result is algorithmic, not existential.
Theorem [Kleinberg 2000]: No local-information algorithm
can find short paths in Watts/Strogatz networks.
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Homophily and Greedy Applications
homophily: a person xs friends tend to be similar to x.
Key idea: getting closer in similarity space
getting closer in graph-distance space
[Killworth Bernard 1978] (reverse small-world experiment)
[Dodds Muhamed Watts 2003]
In searching a social network for a target,
most people chose the next step because of
geographical proximity or similarity of occupation
(more geography early in chains; more occupation late.)
Suggests the greedy algorithm in social-network routing:
if aiming for target t, pick your friend whos most like t.
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Greedy Routing
Greedy algorithm:
if aiming for target t, pick your friend whos most like t.
Geography: greedily route based on distance to t.
Occupation: greedily route based on distance
in the (implicit) hierarchy of occupations.
Want Pr [u, v friends] to decay smoothly as d(u, v) increases.
(Need social cues to help narrow in on t.Not just homophily! Cant just have many disjoint cliques.)
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The LiveJournal Community
www.livejournal.com Baaaaah, says Frank.
Online blogging community.
Currently 14.3 million users; 1.3 million in February 2004.
LiveJournal users provide:
disturbingly detailed accounts of their personal lives.
profiles (birthday, hometown, explicit list of friends)
Yields a social network, with users geographic locations.
(500K people in the continental US.)
[DLN Novak Kumar Raghavan Tomkins 2005]16
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LiveJournal
0.1% of LJ friendships
[DLN Novak Kumar Raghavan Tomkins 2005]17
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Distance versus LJ link probability
1e-07
1e-06
1e-05
1e-04
1e-03
10 km 100 km 1000 km
link
probability
distance
[DLN Novak Kumar Raghavan Tomkins 2005]18
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Analyzing Social Networks, pre-1995
Social Network Analysis: Old Sool
social networks have been around for 100K+ years!
before the web, hard to acquire (surveys, interviews, ...).
but many interesting, relevant, generalizable observations!
Social Network Analysis: New Sch00l
automatically extract networks without having to ask.
phone calls, emails, online communities, ...
big! but are these really social networks?
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The Hewlett-Packard Email Community
[Adamic Adar 2005]
Corporate research community.
Captured email headers over 3 months.
Define friendship as 6 emails u v and 6 emails v u.
Yields a social network (n = 430),
with positions in the corporate hierarchy.
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Emails and the HP Corporate Hierarchy
black: HP corporate hierarchy gray: exchanged emails.
[Adamic Adar 2005]
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Emails and the HP Corporate Hierarchy
So what?
[Adamic Adar 2005]
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Why to think algorithmically about social networks
The small-world phenomenon
Models of the navigable small world
Some conclusions and musings
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Requisites for Navigability
1e-07
1e-06
1e-05
1e-04
1e-03
10 km 100 km 1000 km
linkprobability
distance
[Kleinberg 2000]:
for a social network to be navigable without global knowledge,
need well-scattered friends (to reach faraway targets)
need well-localized friends (to home in on nearby targets)
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Kleinberg: Navigable Social Networks
[Kleinberg 2000]
put n people on a k-dimensional grid
connect each to its immediate neighbors
add one long-range link per person;Pr
[u v]
1
d(u,v) .24
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Navigability of Social Networks
put n people on a k-dimensional grid
connect each to its immediate neighbors
add one long-range link per person; Pr[u v] 1d(u,v)
.
Theorem [Kleinberg 2000]: (short = polylog(n))
If = k
then no local-information algorithm can find short paths.
If = k
then people can find shortO(log2 n)paths using
the greedy algorithm.
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Geographys Role in LiveJournal
[DLN Novak Kumar Raghavan Tomkins 2005]
By simulating the Milgram experiment,
we find that LJ is navigable via geographically greedy routing.
By Kleinbergs theorem,
navigable 2D geographic mesh Pr[u v] d(u, v)2.
Original goal of this research:
verify that Pr[u v] d(u, v)2 in LiveJournal.
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The LiveJournal Odyssey
Dot shown for every
inhabited location
in LiveJournal network.
Circles are centered
on Ithaca, NY.
Each successive circles
population increases by 50,000.
Uniform population radii would decrease quadratically.
(actually mostly increase!)
People dont live on a uniform grid!
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Why does distance fail?
500 m
500 m
Population density varies widely across the US!
and : best friends in Minnesota, strangers in Manhattan.
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Rank-Based Friendship
How do we handle non-uniformly distributed populations?
Instead of distance, use rank as fundamental quantity.
rankA(B) := |{C : d(A, C) < d(A, B)}|
How many people live closer to A than B does?
Rank-Based Friendship : Pr[A is a friend of B] 1/rankA(B).
Probability of friendship 1/(number of closer candidates)
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Relating Rank and Distance
Rank-Based Friendship: Pr[A is a friend of B] 1/rankA(B).
Kleinberg (k-dim grid): Pr[A is a friend of B] 1/d(A, B)k.
Uniform k-dimensional grid:
radius-d ball volume dk1/rank 1/dk
For a uniform grid, rank-based friendship
has (essentially) same link probabilities as Kleinberg.
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l ti t k
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Population Networks
A rank-based population network consists of:
a k-dimensional grid L of locations.
a population P of people, living at points in L (n := |P|).
a set E P P of friendships:one edge from each person in each directionone edge from each person, chosen by rank-based friendship
e.g.,
locations rounded
to the nearestintegral point in
longitude/latitude.
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The Real Theorem
Theorem: For any n-person rank-based population network
in a k-dimensional grid, k = (1), for any source s Pand for a randomly chosen target t P,
the expected length (over t) of Greedy(s, loc(t)) is O(log3 n).
Intuition: difficulty of halving distance to isolated target t
is canceled by low probability of choosing t.
Real theorem: not just for grids.
(use doubling dimension of metric space instead of k).
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Sh t P th d R k B d F i d hi
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Short Paths and Rank-Based Friendships
Theorem: For any n-person population network in a k-dim grid,
for any source s P and a randomly chosen target t P,
the expected length (over t) of Greedy(s, t) is O(log3 n).
Theorem [Kleinberg 2000]: For any n-person uniform-density
population network, any source s, and any target t,
the length of Greedy(s, t) is O(log2 n) with high probability.
Lose: expectation (not whp).
Lose: another log factor.
Gain: arbitrary population densities.
Gain? holds in real networks?
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Ranks and Friendships in LiveJournal
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Ranks and Friendships in LiveJournal
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
linkprobability
rank
shows Pru[u is friends with the v : ranku(v) = r]
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Ranks and Friendships in LiveJournal
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Ranks and Friendships in LiveJournal
r1.15
r1.2 +
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
linkprobability
rank
shows Pru[u is friends with the v : ranku(v) = r]very close to 1/r, as required for rank-based friendship!
again, must correct for nongeographic friends.
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Ranks and Friendships in LiveJournal
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Ranks and Friendships in LiveJournal
r1.25
1e-11
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
linkprobability
minus
rank
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
linkprobability
rank
1e-11
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
linkprobabilityminus
rank
shows probability of rank-r friendship, less = 5.0 106
.
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Ranks and Friendships in LiveJournal
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Ranks and Friendships in LiveJournal
r1.25
1e-11
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
linkprobability
minus
rank
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
linkprobability
rank
1e-11
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
linkprobabilityminus
rank
shows probability of rank-r friendship, less = 5.0 106
.LJ location resolution is city-only.
average us ranks {r , . . . , r + 1300} are in the same city
well average probabilities over ranks {r , . . . , r + 1300}
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Ranks and Friendships in LiveJournal
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Ranks and Friendships in LiveJournal
r
1.30
1e-09
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
link
probabilityminus
,averaged
rank
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
linkprobability
rank
1e-11
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
linkprobabilityminus
rank
1e-09
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1 10 100 1000 10000 100000
linkprobabilityminus,averaged
rank
still very close to 1/r
(though not absolutely perfect)
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Why to think algorithmically about social networks
The small-world phenomenon
Models of the navigable small world
Some conclusions and musings
Geographic/Nongeographic Friendships
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Geographic/Nongeographic Friendships
1e-07
1e-06
1e-05
1e-04
1e-03
10 km 100 km 1000 km
linkprobability
distance
1e-07
1e-06
1e-05
1e-04
1e-03
10 km 100 km 1000 km
linkprobabilityminus
distance
good estimate of friendship probability:
or Pr[u v] + f(d(u, v)) for 5.0 106.
friends (nongeographic) f(d) friends (geographic).
LJ: E[number of us friends] = 500,000 2.5.
LJ: average degree 8.
5.5/8 66% of LJ friendships are geographic, 33% are not.
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A Puzzle to Ponder
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A Puzzle to Ponder
This talk:
66% of LJ friendships form through geographic processes.
Geography is very important, even in a virtual community.
The 5.5 rank-based geographic friends per personsuffice to account for the small world.
Not in this talk:Why should networking people care?
Also not in this talk:And what about the other 2.5 nongeographic friends?
But why does geography matter so much?
virtualized real-world friendships?
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Routing Choices
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Routing Choices
In real life, many ways to choose a next step when searching!
Geography: greedily route based on distance to t.
Occupation: greedily route based on distancein the (implicit) hierarchy of occupations.
Age, hobbies, alma mater, ...
Popularity: choose people with high outdegree.
[Adamic Lukose Puniyani Huberman 2001]
[Kim Yoon Han Jeong 2002] ...
What does closest mean in real life?
How do you weight various proximities?
minimum over all proximities? [Dodds Watts Newman 2002]
a more complicated combination?
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A Puzzle to Ponder
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A Puzzle to Ponder
This talk:
66% of LJ friendships form through geographic processes.
Geography is very important, even in a virtual community.
The 5.5 rank-based geographic friends per personsuffice to account for the small world.
Not in this talk:Why should networking people care?
Also not in this talk:And what about the other 2.5 nongeographic friends?
But why does geography matter so much?
virtualized real-world friendships?
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Open Directions
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Op
A half-sociological, half-computational question ...
Why should rank-based friendship hold, even approximately?
Are there natural processes that generate it?
E.g., a generative process based on geographic interests?
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Thank you!
David Liben-Nowell
dlibenno@carleton.edu
http://cs.carleton.edu/faculty/dlibenno
Some references
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Milgram 1967The Small World Problem. Psychology Today.
Kleinberg 2000The Small-World Phenomenon: An Algorithmic Perspective. STOC00.
Dodds Muhamad Watts 2003An Experimental Study of Search in Global Social Networks. Science.
Liben-Nowell Novak Kumar Raghavan Tomkins 2005Geographic Routing in Social Networks. Proc. Natl. Acad. Sci.
Kumar Liben-Nowell Tomkins 2006
Navigating Low-Dimensional and Hierarchical Population Networks. ESA06.
Adamic Adar 2005How to search a social network. Social Networks.
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