Mathematical models of networks - Leonid Zhukov · Lecture outline 1 Random graphs Erdos-Renyi...
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Mathematical models of networks
Leonid E. Zhukov
School of Data Analysis and Artificial IntelligenceDepartment of Computer Science
National Research University Higher School of Economics
Social Network AnalysisMAGoLEGO course
Leonid E. Zhukov (HSE) Lecture 3 22.04.2016 1 / 29
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Lecture outline
1 Random graphsErdos-Renyi model
2 Preferential attachmentBarabasi-Albert model
3 Small world modelWatts-Strogatz model
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Network models
Empirical network features:
Power-law (heavy-tailed) degree distribution
Small average distance (graph diameter)
Large clustering coefficient (transitivity)
Giant connected component
Generative models:
Random graph model (Erdos & Renyi, 1959)
Preferential attachement model (Barabasi & Albert, 1999)
Small world model (Watts & Strogatz, 1998)
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Random graph model
Graph G{E ,V }, nodes n = |V |, edges m = |E |Erdos and Renyi, 1959.
Gn,p - each pair out of N = n(n−1)2 is connected with probability p,
number of edges m - random number
〈m〉 = pn(n − 1)
2
〈k〉 =1
n
∑i
ki =2〈m〉n
= p (n − 1) ≈ pn
ρ =〈m〉
n(n − 1)/2= p
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Node degree distribution
Poisson Distribution
P(ki = k) =λke−λ
k!, λ = pn = 〈k〉
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Random graph
〈k〉 = pn > 1
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Phase transition
Consider Gn,p as a function of p
p = 0, empty graph
p = 1, complete (full) graph
There are exist critical pc , structural changes from p < pc to p > pc
Gigantic connected component appears at p > pc
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Random graph model
p < pc p = pc p > pc
igraph:erdos.renyi.game()
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Phase transition
Graph G (n, p), for n→∞, critical value pc = 1/n
when p < pc , (〈k〉 < 1) there is no components with more thanO(ln n) nodes, largest component is a tree
when p = pc , (〈k〉 = 1) the largest component has O(n2/3) nodes
when p > pc , (〈k〉 > 1) gigantic component has all O(n) nodes
Critical value: 〈k〉 = pcn = 1- on average one neighbor for a node
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Connected component
〈k〉 = pn
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Clustering coefficient
Clustering coefficient
C (k) =#of links between NN
#max number of links NN=
pk(k − 1)/2
k(k − 1)/2= p
C = p =〈k〉n
when n→∞, C → 0
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Graph diameter
G (n, p) is locally tree-like (GCC) (no loops; low clustering coefficient)
on average, the number of nodes d steps away from a node 〈k〉d
in GCC, around pc , 〈k〉d ∼ n,
d ∼ ln n
ln〈k〉
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Random graph model
Node degree distribution function - Poisson:
P(k) =λke−λ
k!, λ = pn = 〈k〉
Average path length:
〈L〉 ∼ log(N)/ log〈k〉
Clustering coefficient:
C =〈k〉n
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Preferential attachment model
Barabasi and Albert, 1999Dynamical growth model
t = 0, n0 nodes
growth: on every step add a node with m0 edges (m0 ≤ n0),ki (i) = m0
Preferential attachment: probability of linking to existing node isproportional to the node degree ki
Π(ki ) =ki∑i ki
=ki
2m0t
after t steps: n0 + t nodes, m0t edges
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Preferential attachment model
Barabasi, 1999
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Preferential attachement
ki (t) = m0
( ti
)1/2
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Preferential attachement
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Preferential attachment
Time evolution of a node degree
ki (t) = m0
( ti
)1/2
Degree distribution function:
P(k) =2m2
0
k3
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Preferential attachement
Preferential attachment vs random graph
BA : P(k) =2m2
0
k3, ER : P(k) =
〈k〉ke−〈k〉
k!, 〈k〉 = pn
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Preferential attachement
Preferential attachment vs random graph
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Preferential attachment model
m0 = 1 m0 = 2 m0 = 3
igraph: barabasi.game()
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Preferential attachment model
Node degree distribution - power law):
P(k) =2m2
0
k3
Average path length :
〈L〉 ∼ log(N)/ log(log(N))
Clustering coefficient (numerical result):
C ∼ N−0.75
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Small world
Motivation: keep high clustering, get small diameter
Clustering coefficient C = 1/2Graph diameter d = 8
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Small world
Watts and Strogatz, 1998
Single parameter model, interpolation between regular lattice and randomgraph
start with regular lattice with n nodes, k edges per vertex (nodedegree), k << n
randomly connect with other nodes with probability p, forms pnk/2”long distance” connections from total of nk/2 edges
p = 0 regular lattice, p = 1 random graph
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Small world
Watts, 1998
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Small world model
20% rewiring:ave. path length = 3.58 → ave. path length = 2.32clust. coeff = 0.49 → clust. coeff = 0.19
igraph:watts.strogatz.game()
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Small world model
Node degree distribution function - Poisson like (numerical result)
Average path length (analytical result) :
〈L〉 ∼ log(N)
Clustering coefficientC = const
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Model comparison
Random BA model WS model Empirical networks
P(k) λke−λ
k! k−3 poisson like power lawC 〈k〉/N N−0.75 const large
〈L〉 log(N)log(〈k〉)
log(N)log log(N) log(N) small
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References
On random graphs I, P. Erdos and A. Renyi, PublicationesMathematicae 6, 290297 (1959).
On the evolution of random graphs, P. Erdos and A. Renyi,Publicaton of the Mathematical Institute of the Hungarian Academyof Sciences, 17-61 (1960)
Collective dynamics of small-world networks. Duncan J. Watts andSteven H. Strogatz. Nature 393 (6684): 440-442, 1998
Emergence of Scaling in Random Networks, A.L. Barabasi and R.Albert, Science 286, 509-512, 1999
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