complex systems: from nonlinear dynamics to graphs … · complex systems: from nonlinear dynamics...

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.


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.


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


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.

→ →


Random graphs

Small-world network





→

→


Random graphs

Small-world network





→ →


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).


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).


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


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.


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.


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


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).


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


Random graphs





k=1 kp(k) friends


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



k=1 k2p(k)

µk= E(k2)

µk=

σ2k+µ2

kµk


Random graphs





k=1 kp(k) friends



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



k=1 k2p(k)

µk= E(k2)

µk=

σ2k+µ2

kµk


Random graphs





k=1 kp(k) friends



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



k=1 k2p(k)

µk= E(k2)

µk=

σ2k+µ2

kµk


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.


complex systems: from nonlinear dynamics to graphs … · complex systems: from nonlinear dynamics...

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