algorithmic and economic aspects of networks nicole immorlica
TRANSCRIPT
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Algorithmic and Economic Aspects of Networks
Nicole Immorlica
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Random Graphs
What is a random graph?
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Erdos-Renyi Random Graphs
Specify number of vertices nedge probability p
For each pair of vertices i < j, create edge (i,j) w/prob. p
G(n,p)
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Erdos-Renyi Random Graphs
What does random graph G(n,p) look like?
(as a function of p)
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Random Graph Demo
http://ccl.northwestern.edu/netlogo/models/GiantComponent
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Properties of G(n,p)
p < 1/n disconnected, small tree-like components
p > 1/n a giant component emerges containing const. frac. of nodes
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Proof Sketch
1. Percolation2. Branching processes3. Growing spanning trees
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Percolation
1. Infinite graph 2. Distinguished node i3. Probability p
Each link gets ``open’’ with probability p
Q. What is size of component of i?
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Percolation Demo
http://ccl.northwestern.edu/netlogo/models/Percolation
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Percolation on Binary Trees
V = {0,1}*E = (x,y) s.t. y = x0 or y = x1distinguished node ф
Def. Let £(p) = Pr[comp(ф) is infinite]. The critical threshold is pc = sup { p | £(p) = 0}.
0 100
100111
ф
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Critical Threshold
Def. Let £(p) = Pr[comp(ф) is infinite]. The critical threshold is pc = sup { p | £(p) = 0}.
Thm. Critical threshold of binary trees is pc = ½.
Prf. On board.
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Critical Threshold
Thm. Critical threshold of k-ary trees is pc = 1/k.
1/k p
1
0
£(p)
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Branching Processes
Node i has Xi children distributed as B(n,p):
Pr[Xi = k] = (n choose k) pk (1-p)(n-k)
Q. What is probability species goes extinct? A. By percolation, if p > (1+²)/n, live forever.
Note extinction Exists i, X1 + … + Xi < i.
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Erdos-Renyi Random Graphs
We will prove (on board)(1) If p = (1-²)/n, then there exists c1 s.t. Pr[G(n,p) has comp > c1 log n] goes to zero
(2) If p = (1+2²)/n, then there exists c2 s.t. Pr[G(n,p) has comp > c2 n] goes to one
First show (on board)(3) If p = (1+2²)/n, then there exists c2, c3 s.t. Pr[G(n,p) has comp > c2 n] > c3
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Emergence of Giant Component
Theorem. Let np = c < 1. For G ∈ G(n, p), w.h.p. the size of the largest connected component is O(log n).
Theorem. Let np = c > 1. For G ∈ G(n, p), w.h.p. G has a giant connected component of size (β + o(n))n for constant β = βc; w.h.p, the remaining components have size O(log n).
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Application
Suppose …the world is connected by G(n,p)someone gets sick with a deadly
diseaseall neighbors get infected unless
immunea person is immune with
probability q
Q. How many people will die?
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Analysis
1. Generate G(n,p)2. Delete qn nodes uniformly at
random3. Identify component of initially
infected individual
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Analysis
Equivalently,1. Generate G((1-q)n, p)2. Identify component of initially
infected individual
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Analysis
By giant component threshold,• p(1-q)n < 1 disease dies• p(1-q)n > 1 we die
E.g., if everyone has 50 friends on average, need prob. of immunity = 49/50 to survive!
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Summary
Random graphs G(n, c/n) for c > 1 have …
unique giant component small (logarithmic) diameterlow clustering coefficient (= p)Bernoulli degree distribution
A model that better mimics reality?
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In real life
Friends come and go over time.
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Growing Random Graphs
On the first day, God createdm+1 nodes who were all friends
And on the (m+i)’th day, He createda new node (m+i) with m random friends
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Mean Field Approximation
Estimate distribution of random variables by distribution of
expectations.
E.g., degree dist. of growing random graph?
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Degree Distribution
Ft(d) = 1 – exp[ -(d – m)/m ]
(on board)
This is exponential, but social networks tend to look more like power-law deg.
distributions…
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In real life
The rich get richer
… much faster than the poor.
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Preferential Attachment
Start: m+1 nodes all connected
Time t > m: a new node t with m friends distributed according to degree
Pr[link to j] = m x deg(j) / deg(.)
= m x deg(j) / (2mt)= deg(j) / (2t)
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Degree Distribution
Cumulative dist.: Ft(d) = 1 – m2/d2
Density function: ft(d) = 2m2/d3
(heuristic analysis on board, for precise analysis, see Bollobas et al)
A power-law!
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Assignment:
• Readings:– Social and Economic Networks, Chapters 4 & 5– M. Mitzenmacher. A brief history of generative
models for power law and lognormal distributions. Internet Mathematics 1, No 2, 226-251, 2005.
– D.J. Watts, and S.H. Strogatz. Collective dynamics of small-world networks. Nature 393, 440-442, 1998.
• Reactions:– Reaction paper to one of research papers, or a
research paper of your choice
• Presentation volunteer?