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Complex Systems: From Nonlinear Dynamicsto Graphs via Time Series
Michael Small
School of Mathematics and StatisticsThe University of Western Australia
Small (UWA) Complex Systems 1 / 13
Random graphs
Erdos-Renyi random graphs
A random graph G (N,m)
Randomly (uniformly) distribute m edges between N nodes: for each edgepick two distinct nodes, such that all edges are unique (equivalently, put aedge between each of the 1
2N(N − 1) possible pairs of nodes withprobability p).
P. Erdos and A. Renyi. Publicationes Mathematicae 6 (1959) 290-297.
Small (UWA) Complex Systems 2 / 13
Random graphs
N = 1000
m = 499 m = 1000 m = 3484
For m < N−12 there is no giant component, for m > N−1
2 the largest
component scales with N23 . At m = 1
2 (N−1) logN there is a sharp transitionin connectivity of largest component.
Small (UWA) Complex Systems 3 / 13
Random graphs
Small-world networks
Erdos number
The length of the shortest chain of co-publication (papers with a sharedby-line) between an individual and Paul Erdos.
Examples
W. Gates & G. Papadimitriou — G. Papadimitriou & X.T. Deng —X.T. Deng & P. Hell — P. Hell & P.Erdos
M. Small & G. Chen — G. Chen & C.K.T. Chui —C.K.T. Chui & P. Erdos
M. Small & J. Lee — J. Lee & L. Caccetta — L. Caccetta & P. Erdos
M. Guidici & A. Seress — A. Seress & P. Erdos
Small (UWA) Complex Systems 4 / 13
Random graphs
Small-world networks
Erdos number
The length of the shortest chain of co-publication (papers with a sharedby-line) between an individual and Paul Erdos.
Examples
W. Gates & G. Papadimitriou — G. Papadimitriou & X.T. Deng —X.T. Deng & P. Hell — P. Hell & P.Erdos
M. Small & G. Chen — G. Chen & C.K.T. Chui —C.K.T. Chui & P. Erdos
M. Small & J. Lee — J. Lee & L. Caccetta — L. Caccetta & P. Erdos
M. Guidici & A. Seress — A. Seress & P. Erdos
Small (UWA) Complex Systems 4 / 13
Random graphs
Small-world networks
Erdos number
The length of the shortest chain of co-publication (papers with a sharedby-line) between an individual and Paul Erdos.
Examples
W. Gates & G. Papadimitriou — G. Papadimitriou & X.T. Deng —X.T. Deng & P. Hell — P. Hell & P.Erdos
M. Small & G. Chen — G. Chen & C.K.T. Chui —C.K.T. Chui & P. Erdos
M. Small & J. Lee — J. Lee & L. Caccetta — L. Caccetta & P. Erdos
M. Guidici & A. Seress — A. Seress & P. Erdos
Small (UWA) Complex Systems 4 / 13
Random graphs
Small-world networks
Erdos number
The length of the shortest chain of co-publication (papers with a sharedby-line) between an individual and Paul Erdos.
Examples
W. Gates & G. Papadimitriou — G. Papadimitriou & X.T. Deng —X.T. Deng & P. Hell — P. Hell & P.Erdos
M. Small & G. Chen — G. Chen & C.K.T. Chui —C.K.T. Chui & P. Erdos
M. Small & J. Lee — J. Lee & L. Caccetta — L. Caccetta & P. Erdos
M. Guidici & A. Seress — A. Seress & P. Erdos
Small (UWA) Complex Systems 4 / 13
Random graphs
Small-world network
A small world network is a graph with “high” clusteringa and “low”diameterb
alots of trianglesbaverage 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
alots of trianglesbaverage 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
alots of trianglesbaverage 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
alots of trianglesbaverage 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
Social dynamics of Australian mathematicians
rank publications name1 54 Teo, Kok Lay2 51 Zheng, Wei Xing3 46 Wu, Yonghong4 43 Praeger, Cheryl E5 35 Small, Michael6 35 Mengersen, Kerrie7 32 Li, Cai Heng8 30 Liu, Lishan9 30 Smyth, Gordon K10 29 Elliott, Robert J11 27 Gao, Junbin12 26 Tordesillas, Antoinette13 25 Zhao, Ming14 25 Chan, Derek Y C15 25 Wang, Shuaian16 24 Shao, Quanxi17 24 Hill, James M18 24 Campbell, S J19 24 Tang, Youhong20 23 Richardson, Anthony J
Publication co-authorship by Australian mathematicians since 2012 (15683 uniqueauthors, 5903 publications, 186314 co-authorships, 1045 components).
Small (UWA) Complex Systems 6 / 13
Random graphs
rank betweenness name1 0.38732 Richardson, Anthony J2 0.35333 Baddeley, Adrian3 0.35314 Lippmann, John4 0.34932 Price, Daniel J5 0.34903 Bate, Matthew R6 0.32503 Galloway, Duncan K7 0.30271 Gaensler, B M8 0.23505 Mengersen, Kerrie9 0.21663 Possingham, Hugh P10 0.2002 Froyland, Gary11 0.181 Martin, Tara G12 0.16407 Thompson, John F13 0.15225 Silburn, Peter A14 0.14683 Armstrong, Nicola J15 0.13784 Williams, A16 0.13304 Zheng, Wei Xing17 0.12935 Teo, Kok Lay18 0.12685 Ralph, Timothy C19 0.12669 Lam, Ping Koy20 0.12475 Craig, Vincent S J
Publication co-authorship by Australian mathematicians since 2012 (9007 authorsof the largest connected component).
Small (UWA) Complex Systems 7 / 13
Random graphs
prop. of triangles0 0.2 0.4 0.6 0.8 1
frequ
ency
0
1000
2000
3000
4000
5000
6000
7000clustering
D(ni,nj)0 10 20 30 40
prob
.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08path-length
ki
100 200 300 400
k j
50
100
150
200
250
300
350
400degree-degree scatter plot
k100 101 102
P(k)
10-3
10-2
10-1 degree distribution
Small (UWA) Complex Systems 8 / 13
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
ζ(γ)k−γ .
Examples
The Internet, the human brain, various cellular processes and patterns ofdisease transmission are all examples.
Small (UWA) Complex Systems 9 / 13
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
ζ(γ)k−γ .
Examples
The Internet, the human brain, various cellular processes and patterns ofdisease transmission are all examples.
Small (UWA) Complex Systems 9 / 13
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
ζ(γ)k−γ .
Examples
The Internet, the human brain, various cellular processes and patterns ofdisease transmission are all examples.
Small (UWA) Complex Systems 9 / 13
Random graphs
The Barabasi-Albert generative model
Preferential attachment (PA)
Add a new node to the network with m links connecting it to existingnodes with probability proportional to the existing nodes degree
Choice of m matters:
m = 1 m = 2 m = 3
2 5 10 20 50 100 200
10
100
1000
2 5 10 20 50 100 200
10
100
1000
5 10 20 50 100 200
10
100
1000
A. Barabasi and A. Reka. Science 286 (1999) 509-512.
Small (UWA) Complex Systems 10 / 13
Random graphs
The Barabasi-Albert generative model
Preferential attachment (PA)
Add a new node to the network with m links connecting it to existingnodes with probability proportional to the existing nodes degree
Choice of m matters:
m = 1 m = 2 m = 3
2 5 10 20 50 100 200
10
100
1000
2 5 10 20 50 100 200
10
100
1000
5 10 20 50 100 200
10
100
1000
A. Barabasi and A. Reka. Science 286 (1999) 509-512.
Small (UWA) Complex Systems 10 / 13
Random graphs
Recap
Erdos-Renyi random graphs
Emergent phenomenaCritical transitions
Small-world networks
Six-degrees of seperation/Erdos numbersWatts-Strogatz model
Scale-free networks
Barabasi-albert generative modelConfiguration modelsLikelihood models
Small (UWA) Complex Systems 11 / 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 anode (individual) has k links (friends) is p(k).
How many friends do I have?
On average, I expect to have E (k) = µk =∑∞
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 arandom link, not a random node!Suppose there are N nodes, then there will be Nµk
2 links, and there will be12kp(k)N links connected (on one end) to nodes of degree k .Hence, the probability of a node at the end of a randomly chosen link
having degree k is kp(k)NNµk
and the average is∑∞
k=1 k2p(k)
µk= E(k2)
µk=
σ2k+µ2
kµk
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 anode (individual) has k links (friends) is p(k).
How many friends do I have?
On average, I expect to have E (k) = µk =∑∞
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 arandom link, not a random node!Suppose there are N nodes, then there will be Nµk
2 links, and there will be12kp(k)N links connected (on one end) to nodes of degree k .Hence, the probability of a node at the end of a randomly chosen link
having degree k is kp(k)NNµk
and the average is∑∞
k=1 k2p(k)
µk= E(k2)
µk=
σ2k+µ2
kµk
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 anode (individual) has k links (friends) is p(k).
How many friends do I have?
On average, I expect to have E (k) = µk =∑∞
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 arandom link, not a random node!
Suppose there are N nodes, then there will be Nµk2 links, and there will be
12kp(k)N links connected (on one end) to nodes of degree k .Hence, the probability of a node at the end of a randomly chosen link
having degree k is kp(k)NNµk
and the average is∑∞
k=1 k2p(k)
µk= E(k2)
µk=
σ2k+µ2
kµk
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 anode (individual) has k links (friends) is p(k).
How many friends do I have?
On average, I expect to have E (k) = µk =∑∞
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 arandom link, not a random node!Suppose there are N nodes, then there will be Nµk
2 links, and there will be12kp(k)N links connected (on one end) to nodes of degree k .
Hence, the probability of a node at the end of a randomly chosen link
having degree k is kp(k)NNµk
and the average is∑∞
k=1 k2p(k)
µk= E(k2)
µk=
σ2k+µ2
kµk
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 anode (individual) has k links (friends) is p(k).
How many friends do I have?
On average, I expect to have E (k) = µk =∑∞
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 arandom link, not a random node!Suppose there are N nodes, then there will be Nµk
2 links, and there will be12kp(k)N links connected (on one end) to nodes of degree k .Hence, the probability of a node at the end of a randomly chosen link
having degree k is kp(k)NNµk
and the average is∑∞
k=1 k2p(k)
µk= E(k2)
µk=
σ2k+µ2
kµk
Small (UWA) Complex Systems 12 / 13
Random graphs
Why do my friends have more friends thanme?
I have (on average) E (k) =∑
kp(k) = µk friends. But, my friends have
on average E(k2)µk
=σ2k
µk+ 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 tochoose a random node, and then pick one of their friends — a simple wayto identify and immunise hub nodes (disease super-spreaders)
Exercise
Compute µk andσ2k
µk+ 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 andγ < 2.
Small (UWA) Complex Systems 13 / 13
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