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
CS 790R - Topology and Dynamics of Complex Networks
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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: either “element” nodes located in space 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:
CS 790R - Topology and Dynamics of Complex Networks
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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: fast time scale: node activities 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:
neural networks: activities & learning gene networks: expression & mutations 2/15/2005
CS 790R - Topology and Dynamics of Complex Networks
Internet, WWW, actors. ecology, etc. 8
Introduction Motivations 9 complex networks are the backbone of complex systems every complex system is a network of interaction among numerous smaller elements some networks are geometric or regular in 2-D or 3-D space other contain “long-range” connections or are not spatial at all 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 prevent spread of diseases control spread of information (marketing, fads, rumors, etc.)
9 ex: structure of power grid / Internet 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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CS 790R - Topology and Dynamics of Complex Networks
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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
2/15/2005
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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
CS 790R - Topology and Dynamics of Complex Networks
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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
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Four structural models Small-world networks Ring Lattice large world well clustered
Watts-Strogatz (1998) small world well clustered
Random graph small world poorly clustered
p = 0 (order)
0