Complex Systems: From Nonlinear Dynamics
to Graphs via Time Series
Michael Small
School of Mathematics and Statistics
The University of Western Australia
Small (UWA)
Complex Systems
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Random graphs
Erdös-Rényi random graphs
A random graph G (N, m)
Randomly (uniformly) distribute m edges between N nodes: for each edge
pick two distinct nodes, such that all edges are unique (equivalently, put a
edge between each of the 12 N(N − 1) possible pairs of nodes with
probability p).
P. Erdös and A. Rényi. Publicationes Mathematicae 6 (1959) 290-297.
Small (UWA)
Complex Systems
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Random graphs
N = 1000
m = 499
m = 1000
m = 3484
For m < N−1
there is no giant component, for m > N−1
the largest
2
2
2
1
component scales with N 3 . At m = 2 (N −1) log N there is a sharp transition
in connectivity of largest component.
Small (UWA)
Complex Systems
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Random graphs
Small-world networks
Erdös number
The length of the shortest chain of co-publication (papers with a shared
by-line) between an individual and Paul Erdös.
Examples
W. Gates & G. Papadimitriou — G. Papadimitriou & X.T. Deng —
X.T. Deng & P. Hell — P. Hell & P.Erdös
Small (UWA)
Complex Systems
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Random graphs
Small-world networks
Erdös number
The length of the shortest chain of co-publication (papers with a shared
by-line) between an individual and Paul Erdös.
Examples
W. Gates & G. Papadimitriou — G. Papadimitriou & X.T. Deng —
X.T. Deng & P. Hell — P. Hell & P.Erdös
M. Small & G. Chen — G. Chen & C.K.T. Chui —
C.K.T. Chui & P. Erdös
Small (UWA)
Complex Systems
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Random graphs
Small-world networks
Erdös number
The length of the shortest chain of co-publication (papers with a shared
by-line) between an individual and Paul Erdös.
Examples
W. Gates & G. Papadimitriou — G. Papadimitriou & X.T. Deng —
X.T. Deng & P. Hell — P. Hell & P.Erdös
M. Small & G. Chen — G. Chen & C.K.T. Chui —
C.K.T. Chui & P. Erdös
M. Small & J. Lee — J. Lee & L. Caccetta — L. Caccetta & P. Erdös
Small (UWA)
Complex Systems
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Random graphs
Small-world networks
Erdös number
The length of the shortest chain of co-publication (papers with a shared
by-line) between an individual and Paul Erdös.
Examples
W. Gates & G. Papadimitriou — G. Papadimitriou & X.T. Deng —
X.T. Deng & P. Hell — P. Hell & P.Erdös
M. Small & G. Chen — G. Chen & C.K.T. Chui —
C.K.T. Chui & P. Erdös
M. Small & J. Lee — J. Lee & L. Caccetta — L. Caccetta & P. Erdös
M. Guidici & A. Seress — A. Seress & P. Erdös
Small (UWA)
Complex Systems
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Random graphs
Small-world network
A small world network is a graph with “high” clusteringa and “low”
diameterb
a
b
lots of triangles
average distance between random nodes
A constructive definition (Watts-Strogatz)
Start with a lattice and gradually add (or rewire) random links.
Small (UWA)
Complex Systems
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Random graphs
Small-world network
A small world network is a graph with “high” clusteringa and “low”
diameterb
a
b
lots of triangles
average distance between random nodes
A constructive definition (Watts-Strogatz)
Start with a lattice and gradually add (or rewire) random links.
Small (UWA)
Complex Systems
5 / 13
Random graphs
Small-world network
A small world network is a graph with “high” clusteringa and “low”
diameterb
a
b
lots of triangles
average distance between random nodes
A constructive definition (Watts-Strogatz)
Start with a lattice and gradually add (or rewire) random links.
→
Small (UWA)
Complex Systems
5 / 13
Random graphs
Small-world network
A small world network is a graph with “high” clusteringa and “low”
diameterb
a
b
lots of triangles
average distance between random nodes
A constructive definition (Watts-Strogatz)
Start with a lattice and gradually add (or rewire) random links.
→
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→
Complex Systems
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Random graphs
Social dynamics of Australian mathematicians
rank
1
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3
4
5
6
7
8
9
10
11
12
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18
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publications
54
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46
43
35
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32
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27
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name
Teo, Kok Lay
Zheng, Wei Xing
Wu, Yonghong
Praeger, Cheryl E
Small, Michael
Mengersen, Kerrie
Li, Cai Heng
Liu, Lishan
Smyth, Gordon K
Elliott, Robert J
Gao, Junbin
Tordesillas, Antoinette
Zhao, Ming
Chan, Derek Y C
Wang, Shuaian
Shao, Quanxi
Hill, James M
Campbell, S J
Tang, Youhong
Richardson, Anthony J
Publication co-authorship by Australian mathematicians since 2012 (15683 unique
authors, 5903 publications, 186314 co-authorships, 1045 components).
Small (UWA)
Complex Systems
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Random graphs
rank
1
2
3
4
5
6
7
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9
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17
18
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betweenness
0.38732
0.35333
0.35314
0.34932
0.34903
0.32503
0.30271
0.23505
0.21663
0.2002
0.181
0.16407
0.15225
0.14683
0.13784
0.13304
0.12935
0.12685
0.12669
0.12475
name
Richardson, Anthony J
Baddeley, Adrian
Lippmann, John
Price, Daniel J
Bate, Matthew R
Galloway, Duncan K
Gaensler, B M
Mengersen, Kerrie
Possingham, Hugh P
Froyland, Gary
Martin, Tara G
Thompson, John F
Silburn, Peter A
Armstrong, Nicola J
Williams, A
Zheng, Wei Xing
Teo, Kok Lay
Ralph, Timothy C
Lam, Ping Koy
Craig, Vincent S J
Publication co-authorship by Australian mathematicians since 2012 (9007 authors
of the largest connected component).
Small (UWA)
Complex Systems
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Random graphs
clustering
7000
0.07
0.06
0.05
4000
prob.
frequency
5000
3000
0.02
1000
400
0.04
0.03
2000
0
path-length
0.08
6000
0.01
0
0.2
0.4
0.6
prop. of triangles
0.8
0
1
degree-degree scatter plot
0
10 -1
10
20
D(ni,n j)
30
40
degree distribution
350
300
10 -2
P(k)
kj
250
200
150
10 -3
100
50
100
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200
ki
300
Complex Systems
400
10 0
10 1
10 2
k
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Random graphs
A scale-free primer
Scale-free network
A scale-free network is a graph with a power-law degree distribution
p(k) =
Small (UWA)
1 −γ
k .
ζ(γ)
Complex Systems
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Random graphs
A scale-free primer
Scale-free network
A scale-free network is a graph with a power-law degree distribution
p(k) =
Small (UWA)
1 −γ
k .
ζ(γ)
Complex Systems
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Random graphs
A scale-free primer
Scale-free network
A scale-free network is a graph with a power-law degree distribution
p(k) =
1 −γ
k .
ζ(γ)
Examples
The Internet, the human brain, various cellular processes and patterns of
disease transmission are all examples.
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Complex Systems
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Random graphs
The Barabási-Albert generative model
Preferential attachment (PA)
Add a new node to the network with m links connecting it to existing
nodes with probability proportional to the existing nodes degree
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Complex Systems
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Random graphs
The Barabási-Albert generative model
Preferential attachment (PA)
Add a new node to the network with m links connecting it to existing
nodes with probability proportional to the existing nodes degree
Choice of m matters:
m=1
m=2
m=3
1000
1000
1000
100
100
100
10
10
10
2
5
10
20
50
100
200
2
5
10
20
50
100
200
5
10
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50
100
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A. Barabási and A. Réka. Science 286 (1999) 509-512.
Small (UWA)
Complex Systems
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Random graphs
Recap
Erdös-Renyi random graphs
Emergent phenomena
Critical transitions
Small-world networks
Six-degrees of seperation/Erdös numbers
Watts-Strogatz model
Scale-free networks
Barabási-albert generative model
Configuration models
Likelihood models
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Complex Systems
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Random graphs
How many friends do I have?
Consider a social network — nodes are people and links denote friendship.
Suppose the degree distribution is p(k). That is, the probability that a
node (individual) has k links (friends) is p(k).
How many friends do I have?
Small (UWA)
Complex Systems
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Random graphs
How many friends do I have?
Consider a social network — nodes are people and links denote friendship.
Suppose the degree distribution is p(k). That is, the probability that a
node (individual) has k links (friends) is p(k).
How many friends do I have?
On average, I expect to have E (k) = µk =
P∞
k=1 kp(k)
friends
How many friends do my friends have?
Small (UWA)
Complex Systems
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Random graphs
How many friends do I have?
Consider a social network — nodes are people and links denote friendship.
Suppose the degree distribution is p(k). That is, the probability that a
node (individual) has k links (friends) is p(k).
How many friends do I have?
On average, I expect to have E (k) = µk =
P∞
k=1 kp(k)
friends
How many friends do my friends have?
This is a different question since by choosing a friend, we are choosing a
random link, not a random node!
Small (UWA)
Complex Systems
12 / 13
Random graphs
How many friends do I have?
Consider a social network — nodes are people and links denote friendship.
Suppose the degree distribution is p(k). That is, the probability that a
node (individual) has k links (friends) is p(k).
How many friends do I have?
On average, I expect to have E (k) = µk =
P∞
k=1 kp(k)
friends
How many friends do my friends have?
This is a different question since by choosing a friend, we are choosing a
random link, not a random node!
k
Suppose there are N nodes, then there will be Nµ
2 links, and there will be
1
2 kp(k)N links connected (on one end) to nodes of degree k.
Small (UWA)
Complex Systems
12 / 13
Random graphs
How many friends do I have?
Consider a social network — nodes are people and links denote friendship.
Suppose the degree distribution is p(k). That is, the probability that a
node (individual) has k links (friends) is p(k).
How many friends do I have?
On average, I expect to have E (k) = µk =
P∞
k=1 kp(k)
friends
How many friends do my friends have?
This is a different question since by choosing a friend, we are choosing a
random link, not a random node!
k
Suppose there are N nodes, then there will be Nµ
2 links, and there will be
1
2 kp(k)N links connected (on one end) to nodes of degree k.
Hence, the probability of a node at the end ofPa randomly chosen link
∞
2
2
σ 2 +µ2
k=1 k p(k)
having degree k is kp(k)N
= E µ(kk ) = kµk k
Nµk and the average is
µk
Small (UWA)
Complex Systems
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Random graphs
Why do my friends have more friends than
me?
I have (on average) E (k) =
P
kp(k) = µk friends. But, my friends have
σk2
µk
E (k 2 )
µk
on average
=
+ µk friends.
Nodes with large numbers of links are more likely to be linked.
Hence, to find a node with high degree, the easiest (cheap/best) way is to
choose a random node, and then pick one of their friends — a simple way
to identify and immunise hub nodes (disease super-spreaders)
Exercise
σ2
Compute µk and µkk + µk for a scale free network (i.e. p(k) = k −γ for
some positive constant γ). Comment on what you observed for γ < 3 and
γ < 2.
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Complex Systems
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