Cellular automata Pattern formation Swarm intelligence • Three network metrics • Random & regular networks Complex networks: • Small-world & scale-free networks Spatial communities • Case studies Structured morphogenesis
3.
Commonalities
4.
NetLogo Tutorial
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2. A Complex Systems Sampler d. Complex networks 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
network anatomy is important to characterize because structure affects function (and vice-versa) ex: structure of social networks prevent spread of diseases control spread of information (marketing, fads, rumors, etc.)
ex: structure of power grid / Internet understand robustness and stability of power / data transmission Fall 2015
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2. A Complex Systems Sampler d. Complex networks – Three metrics: average path length 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
the average path length of a network over all pairs of N nodes is L = l(A, B) = 2/N(N–1)
The path length between A and B is 3
A,B l(A, B)
the network diameter is the maximal path length between two nodes: D = max l(A, B) property: 1 L D N–1
Fall 2015
René Doursat: "Complex Systems Made Simple"
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2. A Complex Systems Sampler d. Complex networks – Three metrics: degree distribution the degree of a node A is the number of its connections (or neighbors), kA
A
the average degree of a network is
k = 1/N A kA
The degree of A is 5
number of nodes
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 Fall 2015
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2. A Complex Systems Sampler d. Complex networks – Three metrics: clustering coefficient
A
B’
the neighborhood of a node A is the set of kA nodes at distance 1 from A given the number of pairs of neighbors: FA = B,B’ 1
B
= kA (kA –1) / 2 and the number of pairs of neighbors that are also connected to each other: EA = BB’ 1 the clustering coefficient of A is CA = EA / FA 1
The clustering coefficient of A is 0.6
and the network clustering coefficient:
C = 1/N A CA 1
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2. A Complex Systems Sampler d. Complex networks – Regular networks: fully connected in a fully (globally) connected network, each node is connected to all other nodes fully connected networks have the LOWEST path length and diameter: L=D=1 the HIGHEST clustering coefficient: C=1
A fully connected network
and a PEAK degree distribution (at the largest possible constant): kA = N–1, P(k) = (k – N+1) also the highest number of edges: 2
E = N(N–1) / 2 ~ N Fall 2015
René Doursat: "Complex Systems Made Simple"
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2. A Complex Systems Sampler d. Complex networks – Regular networks: lattice
A
a lattice network is generally structured against a geometric 2-D or 3-D background
r
for example, each node is connected to its nearest neighbors depending on the Euclidean distance: A B d(A, B) r 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
Fall 2015
René Doursat: "Complex Systems Made Simple"
D >> 1
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2. A Complex Systems Sampler d. Complex networks – Regular networks: lattice: ring world 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) 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) PEAK degree distribution (low value): kA = K, P(k) = (k – K)
Fall 2015
René Doursat: "Complex Systems Made Simple"
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2. A Complex Systems Sampler d. Complex networks – Random networks in a random graph each pair of nodes is connected with probability p LOW average path length: L lnN / lnk ~ lnN for N >> 1 (because the entire network can be L covered in about L steps: N ~ k ) LOW clustering coefficient (if sparse): C = p = k / N > 1 and the HIGH clustering coefficient of regular lattices: C 0.75 for K >> 1
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2. A Complex Systems Sampler d. Complex networks – Small-world networks Ring Lattice large world well clustered
Watts-Strogatz (1998) small world well clustered
temperature differential. â¢observed at multiple scales, whether frying pan or geo/astrophysical systems. â¢spontaneous symmetry- breaking of a homogeneous.
Common global properties of complex systems. 4. NetLogo ..... some have different definitions across disciplines; no global agreement ... broken symmetry.
This course will explore canonical examples of complex systems through agent-based modeling and ... Complex systems are characterized by a large number of.
a great number of new patterns ... how beneficial or nefarious activity/failures spread over a network: â« diseases .... ex: cognition and consciousness emerging from neurons .... hierarchy levels of science show qualitative leaps (new properties).
introducing practical complex systems modeling and simulation. â from a computational ... fully solvable and regular trajectories for inverse-square force laws ... ex: crystal and gas (covalent bonds or electrostatic forces). â either highly ...
Introduction. 2. A Complex Systems Sampler. 3. Commonalities. 4. NetLogo Tutorial a. What is NetLogo? b. Graphical interface c. Programming concepts d.
Aug 9, 2008 - âemergent engineeringâ will be less about direct design and more ... most pervasive, efficient and robust type of systemsâ¯maybe, in fact, the ..... opposed to many âlight-weightâ (few rules), highly âsocialâ, simple agents
simple and disordered, âmore is differentâ, adaptation & evolution. â by interactive ... introducing practical complex systems modeling and simulation. â from a ...
number of nodes node degree. P(k). 0 ⤠ãkã ⤠Nâ1. 2. A Complex Systems Sampler d. Complex networks â Three metrics: degree distribution ...
Sep 22, 2015 - explaining your modeling choices in the âInformationâ tab, and add your ... European Conference on Artificial Life (ECAL 2007), F. Almeida e ...
DNA binding), and to biology (bacteria, flocks, ... c) decentralized dynamics: no master blueprint or grand .... Artificial Life, Evolutionary Robotics, Adaptive.
Jan 23, 2006 - Examples of complex systems. Pattern formation â Physical: convection cells. Concepts collected from this example. ⢠large number of ...
â¢with enough food, they grow and divide independently. â¢under starvation, they ... â¢A is faster than B, but B is autocatalytic. â¢when A runs out of reactants, ...
Jan 20, 2005 - ... or accident). Traffic jam. (Department of Physics, University of Illinois at Urbana-Champaign) ..... transition from randomness or chaos to order ...
as a pattern formation process in complex ..... harmonic oscillations. Wang, DeLiang .... spatiotemporal patterns of activity â yet, not a main field of research.
May 18, 2010 - physicalist/symbolic nature of cognition, remains today a key component .... and still producing great work in the DEVO project (p21); Jean- ...... In the twentieth century, his work ... âsolution-richâ space needed for successful