Topology and Dynamics of Complex Networks

Feb 15, 2005 - Introduction. Elementary features – Node diversity & dynamics. Network ..... each node in the network obey a differential equation: ➢ generally ...
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CS 790R Seminar Modeling & Simulation

Topology and Dynamics of Complex Networks ~ Lecture 3: Review based on Strogatz (2001), Barabási & Bonabeau (2003), Wang, X. F. (2002) ~

René Doursat Department of Computer Science & Engineering University of Nevada, Reno Spring 2005

Topology and Dynamics of Complex Networks • Introduction • Three structural metrics • Four structural models • Structural case studies • Node dynamics and self-organization • Bibliography

2/15/2005

CS 790R - Topology and Dynamics of Complex Networks

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Topology and Dynamics of Complex Networks • Introduction – Examples of complex networks – Elementary features – Motivations

• Three structural metrics • Four structural models • Structural case studies • Node dynamics and self-organization • Bibliography

2/15/2005

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Introduction Examples of complex networks – Geometric, regular Network

Nodes

Edges

BZ reaction

molecules

collisions

slime mold

amoebae

cAMP

animal coats

cells

morphogens

insect colonies

ants, termites

pheromone

flocking, traffic

animals, cars

perception

fireflies

photons ± long-range

swarm sync 2/15/2005

¾ interactions inside a local neighborhood in 2-D or 3-D geometric space ¾ limited “visibility” within Euclidean distance

CS 790R - Topology and Dynamics of Complex Networks

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Introduction Examples of complex networks – Semi-geometric, irregular Network

Nodes

Edges

Internet

routers

wires

brain

neurons

synapses

WWW

pages

hyperlinks

Hollywood

actors

gene regulation proteins ecology web 2/15/2005

species

movies

¾ local neighborhoods (also) contain “long-range” links: ƒ either “element” nodes located in space ƒ or “categorical” nodes not located in space

binding sites competition

¾ still limited “visibility”, but not according to distance

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Introduction Elementary features – Node diversity & dynamics Network

Node diversity

Node state/ dynamics

Internet

routers, PCs, switches ...

routing state/ algorithm

brain

sensory, inter, electrical motor neuron potentials

WWW

commercial, popularity, educational ... num. of visits

Hollywood

celebrity level, traits, talent ... contracts

gene regulation

protein type, DNA sites ...

boundness, concentration

ecology web

species traits (diet, reprod.)

fitness, density

2/15/2005

¾ nodes can be of different subtypes: , , ... ¾ nodes have variable states of activity:

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Introduction Elementary features – Edge diversity & dynamics Network

Edge diversity

Internet

bandwidth -(DSL, cable)...

brain

excit., inhib. synapses ...

synap. weight, learning

WWW

--

--

Hollywood

theater movie, partnerships TV series ...

gene regulation

enhancing, blocking ...

mutations, evolution

ecology web

predation, cooperation

evolution, selection

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Edge state/ dynamics

¾ edges can be of different subtypes: , , ... ¾ edges can also have variable weights:

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Introduction Elementary features – Network evolution

¾ the state of a network generally evolves on two time-scales: ƒ fast time scale: node activities ƒ slow time scale: connection weights

¾ examples:

¾ the structural complexity of a network can also evolve by adding or removing nodes and edges ¾ examples:

ƒ neural networks: activities & learning ƒ gene networks: expression & mutations 2/15/2005

CS 790R - Topology and Dynamics of Complex Networks

ƒ Internet, WWW, actors. ecology, etc. 8

Introduction Motivations 9 complex networks are the backbone of complex systems ƒ every complex system is a network of interaction among numerous smaller elements ƒ some networks are geometric or regular in 2-D or 3-D space ƒ other contain “long-range” connections or are not spatial at all ƒ understanding a complex system = break down into parts + reassemble

9 network anatomy is important to characterize because structure affects function (and vice-versa) 9 ex: structure of social networks ƒ prevent spread of diseases ƒ control spread of information (marketing, fads, rumors, etc.)

9 ex: structure of power grid / Internet ƒ understand robustness and stability of power / data transmission 2/15/2005

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Topology and Dynamics of Complex Networks • Introduction • Three structural metrics – Average path length – Degree distribution (connectivity) – Clustering coefficient

• Four structural models • Structural case studies • Node dynamics and self-organization • Bibliography

2/15/2005

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Three structural metrics Average path length ¾ the path length between two nodes A and B is the smallest number of edges connecting them:

A

l(A, B) = min l(A, Ai, ... An, B)

B

¾ the average path length of a network over all pairs of N nodes is L = 〈l(A, B)〉

∑A,B l(A, B)

= 2/N(N–1)

The path length between A and B is 3

¾ the network diameter is the maximal path length between two nodes: D = max l(A, B) ¾ property: 1 ≤ L ≤ D ≤ N–1

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Three structural metrics Degree distribution (connectivity) ¾ the degree of a node A is the number of its connections (or neighbors), kA

A

¾ the average degree of a network is

〈k〉 = 1/N ∑A kA

The degree of A is 5

number of nodes

¾ the degree distribution function P(k) is the histogram (or probability) of the node degrees: it shows their spread around the average value 0 ≤ 〈k〉 ≤ N–1 P(k)

node degree 2/15/2005

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Three structural metrics Clustering coefficient

A

B’

¾ the neighborhood of a node A is the set of kA nodes at distance 1 from A ¾ given the number of pairs of neighbors: FA = ∑B,B’ 1

B

= kA (kA –1) / 2 ¾ and the number of pairs of neighbors that are also connected to each other: EA = ∑B↔B’ 1 ¾ the clustering coefficient of A is The clustering coefficient of A is 0.6

CA = EA / FA ≤ 1 ¾ and the network clustering coefficient:

〈C〉 = 1/N ∑A CA ≤ 1

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Topology and Dynamics of Complex Networks • Introduction • Three structural metrics • Four structural models – – – –

Regular networks Random networks Small-world networks Scale-free networks

• Structural case studies • Node dynamics and self-organization • Bibliography

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Four structural models Regular networks – Fully connected ¾ in a fully (globally) connected network, each node is connected to all other nodes ¾ fully connected networks have the LOWEST path length and diameter: L=D=1 ¾ the HIGHEST clustering coefficient: C=1

A fully connected network

¾ and a PEAK degree distribution (at the largest possible constant): kA = N–1, P(k) = δ(k – N+1) ¾ also the highest number of edges: 2

E = N(N–1) / 2 ~ N 2/15/2005

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Four structural models Regular networks – Lattice

A

¾ a lattice network is generally structured against a geometric 2-D or 3-D background

r

¾ for example, each node is connected to its nearest neighbors depending on the Euclidean distance: A ↔ B ⇐⇒ d(A, B) ≤ r ¾ the radius r should be sufficiently small to remain far from a fully connected network, i.e., keep a large diameter: A 2-D lattice network

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CS 790R - Topology and Dynamics of Complex Networks

D >> 1

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Four structural models Regular networks – Lattice: ring world ¾ in a ring lattice, nodes are laid out on a circle and connected to their K nearest neighbors, with K > 1 (mean between closest node l = 1 and antipode node l = N / K) ¾ HIGH clustering coefficient: C ≈ 0.75 for K >> 1 A ring lattice with K = 4

(mean between center with K edges and farthest neighbors with K/2 edges) ¾ PEAK degree distribution (low value): kA = K, P(k) = δ(k – K)

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Four structural models Random networks ¾ in a random graph each pair of nodes is connected with probability p ¾ LOW average path length: L ≈ lnN / ln〈k〉 ~ lnN for N >> 1 (because the entire network can be L covered in about 〈k〉 steps: N ~ 〈k〉 ) ¾ LOW clustering coefficient (if sparse): C = p = 〈k〉 / N > 1 ¾ and the HIGH clustering coefficient of regular lattices: C ≈ 0.75 for K >> 1

CS 790R - Topology and Dynamics of Complex Networks

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Four structural models Small-world networks Ring Lattice ƒ large world ƒ well clustered

Watts-Strogatz (1998) ƒ small world ƒ well clustered

Random graph ƒ small world ƒ poorly clustered

p = 0 (order)

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