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Small World Networks
Jean VaucherIft6802 - Avril 2005
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Contents
Pertinence of topic Characterization of networks
Regular, Random or Natural Properties of networks
Diameter, clustering coefficient Watt’s network models (alpha & beta) Power Law networks
Clustered networks with short paths
Can these short paths be found ?
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Duncan J. Watts
Six degrees - the science of a connected age, 2003, W.W. Norton.
I read somewhere that everybody on this planet is separated by only six other people. Six degrees of separation between us and everybody on this planet. Six degrees of separation by John Guare
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Networks
Networks are everywhere Internet Neurons is brains Social networks Transportation
Networks have been studied long time Euler (1736): Bridges of Königsberg theory of graphs,
which is now a major (and difficult! – or almost obvious) branch in mathematics
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So what is new?
Global interconnections Internet Power grids Mass travel, mass culture
FAILURES Computer Viruses Power Blackouts Epidemics
Modeling & analysis
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Milgram’s Experiment
Found short chains of acquaintances linking pairs of people in USA who didn’t know each other;
Source person in Nebraska Target person in Massachusetts. Sends message by forwarding to people they knew personally
(who should be closer to target) Average length of the chains that were completed was
between 5 and 6 steps “Six degrees of separation” principle
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Correct question
WHY are there short chains of acquaintances linking together arbitrary pairs of strangers???
Or
Why is this surprising
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Random networks
In a random network, if everybody has 100 friends distributed randomly in the world population, this isn’t strange
In 6 hops, you can reach 1006 people - a million million > 6,000 million (world pop.)
BUT: our social networks tend to be clustered.
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Social networks
Not random But Clustered Most of our friends come from our
geographical or professional neighbourhood.
Our friends tend to have the same friendsBUT In spite of having clustered social networks,
there seem to exist short paths between any random nodes.
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Social network research
Devise various classes of networks
Study their properties
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Network parameters
Network type Regular Random Natural
Size: # of nodes Number of connexions:
average & distribution Selection of neighbours
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STAR TREE
GRID
BUS RING
REGULAR Network Topologies
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Connectivity in Random graphs
Nodes connected by links in a purely random fashion
How large is the largest connected component? (as a fraction of all nodes) Depends on the number of links per
node
(Erdös, Rényi 1959)
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Connecting Nodes
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Random Network (1)
• add random paths
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• paths
• trees
Random Network (2)
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• paths
• trees
• networks
Random Network (3)
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• paths
• trees
• networks …..
Random Network (3+)
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• paths
• trees
• networks
• fully connected
Network Connectivity (4)
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Connectivity of a random graph
1
1
Average number oflinks per node
Fract
ion
of
all
nod
es
in larg
est
com
pon
en
t
0
Dis
con
nect
ed
ph
ase
Con
ect
ed
ph
ase
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Regular or Ordered Network
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Network measures
Connectivity is not main measure. Characteristic Path Length (L) :
the average length of the shortest path connecting each pair of agents (nodes).
Clustering Coefficient (C) is a measure of local interconnection if agent i has ki immediate neighbors, Ci, is the
fraction of the total possible ki*(ki-1) / 2 connections that are realized between i's neighbors. C, is just the average of the Ci's.
Diameter: maximum value of path length
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Regular vs Random Networks
Average number ofconnections/node
Diameter
Number of connectionsneeded to fully connect
few, clustered
Random Regular
fewer, spread
large moderate
many fewer (<2/3)
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Natural networks
Between regular grids and totally random graphs
Need for parametrized models: Regular -> natural -> random
Watts Alpha model ( not intuitive) Beta rewiring model
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Clustering Clustering measures the fraction of neighbors of a node that are
connected themselves Regular Graphs have a high clustering coefficient
but also a high diameter Random Graphs have a low clustering coefficient
but a low diameter Both models do match the properties expected from real networks!
Random Graph (k=4)
Short path length L~logkN
Almost no clustering C~k/n
Regular Graph (k=4)
Long paths L ~ n/(2k)
Highly clustered C~3/4
Base metwork is circle
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Small-World Networks
Random rewiring of regular graph (by Watts and Strogatz) With probability p (or ) rewire each link in a regular graph to a
randomly selected node Resulting graph has properties, both of regular and random
graphs High clustering and short path length
FreeNet has been shown to result in small world graphs
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Example: 4096 node ring
Regular graph:
n nodes, k nearest neighbors
path length ~ n/2k
4096/16 = 256
Random graph:
path length ~ log (n)/log(k)
~ 4
Rewired graph (1% of nodes):
path length ~ random graph
clustering ~ regular graph
Small World Graph
K=4
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Small-worldnetworks
Beta network
Rewiring probability
0 10
1
L
C
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More exactly …. (p = )
Small world behaviour
C
L
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Effect of short-cuts
Huge effect of just a few short-cuts. First 5 rewirings reduces the path
length by half, regardless of size of network
Further 50% gain requires 50 more short-cuts
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The strength of weak ties
Granovetter (1973): effective social coordination does not arise from densely interlocking strong ties, but derives from the occasional weak ties this is because valuable information
comes from these relations (it is valuable if/because it is not available to other individuals in your immediate network)
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Two ways of constructing
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Alpha model
Watts’ first Model (1999) Inspired by Asimov’s “I, Robot”
novels R. Daneel Olivaw Elijah Baley
Caves of Steel (Earth) Solaria
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Two extreme types of social networks
Caveman’s world people live in isolated communities probability meeting a random person is high if
you have mutual friends and very low if you don’t
Solaria people live isolated from each other but with
supreme communication capabilities your social history is irrelevant to your future
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Alpha network
Alpha () distance parameter
=0 : if A and B have a friend in common, they know each other (Caveman world)
=∞ : A & B don’t know each other, no matter how many common friends they have (Solarian world)
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Number of mutual friendsshared by A and B
Like
lihood
th
at
A m
eets
B
Caveman world
Solaria world
=0
=
=1
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Fragmentednetworks
Small-worldnet- works
Alpha network
Path
len
gth
L
critical
Clu
steri
ng
coeffi
cien
t C
L drops because we only count nodes that are connected
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How about real networks
All nodes in alpha and beta networks are equal in the sense that the number of connections each nodes has is not very far from the average
Watts and Strogatz had used normal distribution
Real world is not like that Sizes of cities, Wealth of individuals in USA, Hubs in
transportation systems Barabási and Albert (1999)
Scale-free networks, whose connectivity is defined by a power-law distribution
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Random Networks
Each node is connected toa few other nodes.
The number of connectionsper node forms a Poisson distribution, with a small average of number of connections per node.
This & three following graphics from:Linked: The New Science of Networksby Albert-Laszlo Barabasi; 2002
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Scale-Free Networks
Each node is connected toat least one other; most areconnected to only one, whilea few are connected to many.
The number of connectionsper node forms a hyperbolic distribution, with no meaningfulaverage number of connectionsper node.
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Random Scale-Free
Scale-free networks are associated with networks that grow by “natural” processesin which the number of nodes increases with time not just the number of connections.
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Power law phenomena
Average & median are far apart Whales and minnows
Average from a few large nodes Median governed by majority of small
nodes
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Performance
Real power law networks also have short distances
Existence of central backbone of highly connected HUBS nodes
Similar phenomena noted in linguistics and economics Zipf Pareto
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Zipf's law - linguistics
Zipf, a Harvard linguistics professor, sought to determine the frequency of use of the 3rd or 8th or 100th most common words in English text.
Zipf's law states that the frequency y is inversely proportional to it's rank r:
Y ~ r -b, with b close to unity.
Zipf Presentations
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The Pareto Income Distribution
The Pareto distribution gives the probability that a person's income is greater than or equal to x and is expressed as
[ ] ( )
parameter shape is
income minimum is
,0,0 ,/
k
m
mxkmxmxXP k ≥>>=≥
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Vilfredo Pareto, 1848-1923
Italian economist Born in Paris Polytechnic Institute in Turin in 1869, Worked for the railroads. Pareto did not study economics seriously
until he was 42. In 1893 he succeeded his mentor, Walras,
as chair of economics at the University of Lausanne.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
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Pareto’s contributions
Pareto optimality. A Pareto-optimal allocation of resources
is achieved when it is not possible to make anyone better off without making someone else worse off.
Pareto's law of income distribution. In 1906, Italian economist Vilfredo Pareto
created a mathematical formula to describe the unequal distribution of wealth in his country, observing that 20% of the people owned 80% of the wealth.
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0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
10000 60000 110000 160000 210000
x
p(X>=x)
Pareto distribution, m=10000, k=1
0,01
0,1
1
10000 100000 1000000
x
p(X>=x)
log-log plot
Pareto distribution issaid to be scale-free becauseit lacks a characteristic lengthscale
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Building Power-law networks
It is easy to create PL networks
Build network node by node Connect new node to an existing
node Probability of connection proportional
to its number of links The rich get richer The poor get poorer
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Structure and dynamics
The case of centrality centers are in networks
by design (central control, dictatorship) by non-design (unnoticed critical resources,
informal groups) or they emerge as a consequence of
certain events ”he was at the right place at a right time” clapping in unison
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Further applications
Search in networks Short paths are not enough
Epidemics: medical & software Danger of short-cuts Paths + infectiousness
Infection by ideas Fads & Economic Bubbles Individual rationality Peer pressure
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Getting practical: search in networks
A node may be linked to another node via a short path but what does it matter if you cannot find the path?
In alpha and beta networks there is no notion of distance, therefore directed searches cannot recognize shortcuts
Kleinberg’s (gamma) networks (2000)
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Kleinberg’s Small-World Model
Embed the graph into an r-dimensional grid (2D in examples) constant number p of short range links (neighborhood) q long range links: choose long-range links such that the probability to have
a long range contact is proportional to 1/dr
Importance of r ! Decentralized (greedy) routing performs best iff. r = dimension of space
(here=2)
r = 2
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Influence of “r” (1)
• Each peer u has link to the peer v with probability proportional to where d(u,v) is the distance between u and v.
• Optimal value: r = dim = dimension of the space• If r < dim we tend to choose more far away neighbors (decentralized
algorithm can quickly approach the neighborhood of target, but then slows down till finally reaches target itself).
• If r > dim we tend to choose more close neighbors (algorithm finds quickly target in it’s neighborhood, but reaches it slowly if it is far away).
• When r = 0 – long range contacts are chosen uniformly. Random graph theory proves that there exist short paths between every pair of vertices, BUT there is no decentralized algorithm capable finding these paths
rvud ),(1
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r (log scale)
p(r)(log scale)
increasing
=0
Typic
al le
ngth
of
dir
ect
ed s
earc
h
2
short paths cannot be found
no short paths
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Influence of “r” (or ) Given node u if we can partition the remaining peers into sets A1,
A2, A3, … , AlogN , where Ai, consists of all nodes whose distance from u is between 2i and 2i+1, i=0..logN-1.
Then given r = dim each long range contact of u is nearly equally likely to belong to any of the sets Ai
A4
A3
A2
A1
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The New Yorker View
When gamma is at its critical value two, the resulting network has the peculiar property that nodes possess the same number of ties at all length scales (in 2D world)
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DHTs (distributed hash tables)and Kleinberg model
P-Grid’s model
Kleinberg’s model
Balanced n-ary search
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More hierarchy
Kleinberg’s model has only one distance measure, geographical (2D)
In human society the social distance is multidimensional
if A is close to B and C is close to B but in different dimension then A and C can be very far from each other ”violation of the triangle inequality” but multidimensionality may enable messages
to be transmitted in networks very efficiently
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Watts et al (2002) search in social networks
Searchablenetworks
H1 10
0
6
Kleinbergcondition
= homophily, the tendency of like toassociate with like
H=number of dimensionsalong which individualsmeasure similarity
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Small Worlds
& Epidemic diseases
Nodes are living entities Link is contact 3 States
Uninfected Infected Recovered (or dead)
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Epidemic diseases
Level of infectiousness needed to start an epidemic varies with presence of shortcuts
In regular grid, disease may die out due to lack of victims In small world, pandemics are facilitated
SRAS Mad cow disease in England
0Fraction of random shortcuts1Threshold infectiousness
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Failures in networks
Fault propagation or viruses Scale-free networks are far more resistant
to random failures than ordinary random networks because of most nodes are leaves
But failure of hubs can be catastrophic vulnerable or targets of deliberate attacks which may make scale-free networks more
vulnerable to deliberate attacks Cascades of failures
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Back to Social Networks
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Spread of ideas
Messages in social networks Fads & fashions
Body piercing, baseball caps Harry Potter, Amélie Poulin
Innovation, scientific revolutions Solar-centric universe Plate tectonics
Is it like the spread of disease ?
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Effect of peers & pundits
People’s decisions are affected by what others do and think Presure to conform ?
Efficient strategy when insufficient knowledge or expertise Ex: picking a restaurant
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Economic models
Selfish agents Individual rationality Markets
Equilibrium ??? Many agents are trend followers Speculation crashes
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Social Experiments
Factors which affect decisions Milgram Asch
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Stanley Milgram (1933-1984)
Controversial social psychologist Yale & Harvard Small world experiment, 1967
6 degrees of separation Obedience to authority - 1963
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Validity of Milgram’s experiment
Global connectivity ? US: Omaha Boston stockbroker Only 96 valid subjects (out of 300)
100 from Boston 100 big investors 96 picked at random in Nebraska
Success? 18 out of 96 Other experiments:
3 out of 60 Worse….
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Conformity
Other presentation
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Threshold models of decisions
Number of infected neighbors
1
Pro
bab
ility
of
infe
ctio
n
0
Fraction of neighborschoosing A over B
1
Pro
bab
ility
of
choosi
ng
op
tion
A
0 CriticalThreshold
Standard disease spreadingmodel
Social decision making
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Global Cascades
Idea catches on….
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Fin
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