Way Finding in Social Networks

8/14/2019 Way Finding in Social Networks

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Wayfinding in

Social Networks

David Liben-Nowell

Carleton College

[email protected]

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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admin

[Zachary 1977]



During observation,


conflict developed

broke into two clubs.


Split along


minimum cut (!)

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admin

[Zachary 1977]



During observation,


conflict developed



Split along


minimum cut (!)

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admin

[Zachary 1977]



During observation,


conflict developed



Split along


minimum cut (!)

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admin

[Zachary 1977]



During observation,


conflict developed



Split along


minimum cut (!)

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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



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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



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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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Requisites for Navigability

1e-07

1e-06

1e-05

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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]

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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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1e-07

1e-06

1e-05

1e-04

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1e-02

1e-01

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linkprobability

rank

shows Pru[u is friends with the v : ranku(v) = r]

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r1.15

r1.2 +

1e-07

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1e-03

1e-02

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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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r1.25

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linkprobability

minus

rank

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linkprobability

rank

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linkprobabilityminus

rank

shows probability of rank-r friendship, less = 5.0 106

.

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r1.25

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linkprobability

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linkprobability

rank

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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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r

1.30

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link

probabilityminus

,averaged

rank

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1 10 100 1000 10000 100000

linkprobability

rank

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rank

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linkprobabilityminus,averaged

rank

still very close to 1/r

(though not absolutely perfect)

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Geographic/Nongeographic Friendships


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Geographic/Nongeographic Friendships

1e-07

1e-06

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10 km 100 km 1000 km

linkprobability

distance

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1e-06

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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

[email protected]

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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Way Finding in Social Networks