SNA 6: processes on networks Lada Adamic Processes on networks - - PowerPoint PPT Presentation

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SNA 6: processes on networks Lada Adamic Processes on networks - - PowerPoint PPT Presentation

Jun 24, 2023 103 likes •549 views

SNA 6: processes on networks Lada Adamic Processes on networks Diffusion (simple) ER graphs Scale-free graphs Small-world topologies Complex contagion/thresholds Collective action Innovation Problem

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SNA 6: processes on networks

Lada Adamic

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Processes on networks

¤ Diffusion (simple)

¤ ER graphs ¤ Scale-free graphs ¤ Small-world topologies

¤ Complex contagion/thresholds ¤ Collective action ¤ Innovation ¤ Problem solving

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Diffusion in networks: ER graphs

¤ review: diffusion in ER graphs

http://www.ladamic.com/netlearn/NetLogo501/ERDiffusion.html

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ER graphs: connectivity and density

average degree = 2.5 average degree = 10 nodes infected after 10 steps, infection rate = 0.15

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Quiz Q:

¤ When the density of the network increases, diffusion in the network is

¤ faster ¤ slower ¤ unaffected

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Diffusion in “grown networks”

¤ nodes infected after 4 steps, infection rate = 1

http://www.ladamic.com/netlearn/NetLogo501/BADiffusion.html preferential attachment non-preferential growth

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Quiz Q:

¤ When nodes preferentially attach to high degree nodes, the diffusion over the network is

¤ faster ¤ slower ¤ unaffected

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Diffusion in small worlds

¤ What is the role of the long-range links in diffusion over small world topologies?

http://www.ladamic.com/netlearn/NetLogo4/SmallWorldDiffusionSIS.html

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Quiz Q:

¤ As the probability of rewiring increases, the speed with which the infection spreads

¤ increases ¤ decreases ¤ remains the same

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Simple vs. complex contagion

¤ Simple contagion: each friend infects you with some probability for each unit

  • f time

¤ Complex contagion: you will only take action if a certain number or fraction of your neighbors do

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What is the role of the shortcuts?

¤ long range links unlikely to coincide in influence

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Quiz Q:

¤ Relative to the simple contagion process the complex contagion process:

¤ is better able to use shortcuts ¤ advances more rapidly through the network ¤ infects a greater number of nodes

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networked coordination game

¤ choice between two things, A and B (e.g. basketball and soccer) ¤ if friends choose A, they get payoff a ¤ if friends choose B, they get payoff b ¤ if one chooses A while the other chooses B, their payoff is 0

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coordinating with one’s friends

Let A = basketball, B = soccer. Which one should you learn to play? fraction p = 3/5 play basketball fraction p = 2/5 play soccer

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which choice has higher payoff?

¤ d neighbors ¤ p fraction play basketball (A) ¤ (1-p) fraction play soccer (B) ¤ if choose A, get payoff p * d *a ¤ if choose B, get payoff (1-p) * d * b ¤ so should choose A if

¤ p d a ≥ (1-p) d b ¤ or ¤ p ≥ b / (a + b)

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two equilibria

¤ everyone adopts A ¤ everyone adopts B

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what happens in between?

¤ What if two nodes switch at random? Will a cascade occur? ¤ example:

¤ a = 3, b = 2 ¤ payoff for nodes interaction using behavior A is 3/2 as large as what they get if they both choose B ¤ nodes will switch from B to A if at least q = 2/(3+2) = 2/5 of their neighbors are using A

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how does a cascade occur

¤ suppose 2 nodes start playing basketball due to external factors (e.g. they are bribed with a free pair of shoes by some devious corporation)

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Quiz Q:

Which node(s) will switch to playing basketball next?

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the complete cascade

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you pick the initial 2 nodes

¤ A larger example (Easley/Kleinberg Ch. 19) ¤ does the cascade spread throughout the network?

http://www.ladamic.com/netlearn/NetLogo412/CascadeModel.html

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implications for viral marketing

¤ if you could pay a small number of individuals to use your product, which individuals would you pick?

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try it on Lada’s Facebook network

¤ you can play with a partner ¤ each person gets to pick 2 nodes

¤ first person picks one blue ¤ second person picks one red ¤ first person picks an additional blue ¤ second person picks an additional red

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Quiz question:

¤ What is the role of communities in complex contagion

¤ enabling ideas to spread in the presence of thresholds ¤ creating isolated pockets impervious to outside ideas ¤ allowing different opinions to take hold in different parts of the network

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bilingual nodes

¤ so far nodes could only choose between A and B ¤ what if you can play both A and B, but pay an additional cost c?

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try it on a line

¤ Increase the cost of being bilingual so that no node chooses to do so. Let the cascade run ¤ Now lower the cost.

¤ What happens?

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Quiz Q:

¤ The presence of bilingual nodes

¤ helps the superior solution to spread throughout the network ¤ helps inferior options to persist in the network ¤ causes everyone in the network to become bilingual

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knowledge, thresholds, and collective action ¤ nodes need to coordinate across a network, but have limited horizons

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can individuals coordinate?

¤ each node will act if at least x people (including itself) mobilize

nodes will not mobilize

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mobilization

¤ there will be some turnout

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Quiz Q:

¤ will this network mobilize (at least some fraction of the nodes will protest)?

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innovation in networks

¤ network topology influences who talks to whom ¤ who talks to whom has important implications for innovation and learning

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better to innovate or imitate?

brainstorming: more minds together, but also danger of groupthink working in isolation: more independence slower progress

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in a network context

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modeling the problem space

¤ Kauffman’s NK model ¤ N dimensional problem space

¤ N bits, each can be 0 or 1

¤ K describes the smoothness of the fitness landscape

¤ how similar is the fitness of sequences with

  • nly 1-2 bits flipped (K = 0, no similarity, K

large, smooth fitness)

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Kauffman’s NK model

distance fitness K large K medium K small

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Update rules

¤ As a node, you start out with a random bit string ¤ At each iteration

¤ If one of your neighbors has a solution that is more fit than yours, imitate (copy their solution) ¤ Otherwise innovate by flipping one of your bits

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Quiz Q:

¤ Relative to the regular lattice, the network with many additional, random connections has on average:

¤ slower convergence to a local optimum ¤ smaller improvement in the best solution relative to the initial maximum ¤ more oscillations between solutions

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Coordination: graph coloring

¤ Application: coloring a map: limited set

  • f colors, no two adjacent countries

should have the same color

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graph coloring on a network

¤ Each node is a human subject. Different experimental conditions:

¤ knowledge of neighbors’ color ¤ knowledge of entire network

¤ Compare:

¤ regular ring lattice ¤ small-world topology ¤ scale-free networks

Kearns et al., ‘An Experimental Study of the Coloring Problem on Human Subject Networks’, Science, 313(5788), pp. 824-827, 2006

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simulation

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Quiz Q:

¤ As the rewiring probability is increased from 0 to 1 the following happens:

¤ the solution time decreases ¤ the solution time increases ¤ the solution time initially decreases then increases again

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recap

¤ network topology influences processes

  • ccurring on networks

¤ what state the nodes converge to ¤ how quickly they get there

¤ process mechanism matters:

¤ simple vs. complex contagion ¤ coordination ¤ learning