ESD.342 Random networks Lecture 13

Random Networks and Percolation

• • • • •

Percolation, cascades, pandemics Properties, Metrics of Random Networks Basic Theory of Random Networks and Cascades

Watts Cascades Analytic Model of Watts Cascades

2/16/2011

Random networks © Daniel E Whitney 1997-2010 and cascades

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Types of Percolation Models

• “Short loop” models (Stauffer, Grimmett, Morris) usually assume regular network structure or same nodal degree for all nodes • “Long loop/no loop” models (Newman, Calloway, Watts) usually assume random tree-like structure • “Collective action” models (Schelling, Granovetter) assume k = n (all nodes see all other nodes) • “Local action” models assume z K * 19/46

Example Percolation on a Tree

zi = excess degrees at step i ni = ith neighbors z1=3 z2=2

n1=3 n2=z2*n1=2*3=6

z3=2

n3=z3*n2=z3*z2*n1=2*2*3=12

For E - R, avg excess degree of a neighbor = z −1 2/16/2011


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Single Node Seed, No Threshold

Seed

First Step

Second Step

Number of nodes hit on first step =

number of edges out from seed = Sz

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Multi-Node Seed, Threshold

Number of nodes hit < number of edges out because some nodes are hit multiple times, allowing stable nodes to be flipped 2/16/2011


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Two Steps, Multi-Node Seed, Threshold

1 ≤ k ≤ K* : flip if ≥ 1 neighbor flips K * +1 ≤ k ≤ 2K* : flip if ≥ 2 neighbors flip

Seed

First Step

Second Step

K* = 4 2/16/2011


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Denser connections: better resistance to cascade

Watts’ Cascade Diagram

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Theoretical boundary: infinite nodes Simulation boundary: 10000 nodes Network is too densely connected Network is disconnected

K* = ⎣1/ φ⎦ Lower threshold: more likely to cascade Watts, PNAS April 2002


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Vulnerable Clusters in Finite E-R

Networks

pk

z k

K* 2K* 3K* 4K*

pk

z k

K* 2K* 3K* 4K*

pk

z k

K* 2K* 3K* 4K*

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Watts Theory and Simulations

• Theory – Global cascades region: “small” seed can start a global cascade – No global cascades region: “small” seed cannot start a global cascade • Simulations on network with 10000 nodes – Global cascades region: seed of one node can start a global cascade – No global cascades region: seed of one node cannot start a global cascade 2/16/2011


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Cascades in Finite E-R Networks Can

Happen in the No Global Cascades Region

D. E. Whitney, ŅDynamic theory of cascades on finite clustered random networks with a threshold ruleÓPhysical Review E. E 82, 066110 (2010)

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Simulations: Necessary Seed Sizes (n =

4500)

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Simulations: Threshold Seed Size - A

Phase Transition

Likelihood of TNC vs Size of Seed 100 90

n = 4500, z = 11.5, K* = 4

80 70 60 50 40 30

“Threshold” seed size = 215

20 10 0 150

170

190

210

230

250

270

290

Size of Seed

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Typical Cascade Trajectories Throughout

Transition Range of Seed Size

n = 4500, z = 11.5, K* = 4 350

S = 200 (no TNCs)

300

S = 210 (no TNCs) 250

S = 220 (30% TNCs)

200

S = 220 (70% no TNCs) S = 230 (90% TNCs)

150

S = 230 (90% TNCs) 100

S = 240 (100% TNCs) 50

S = 270 (100% TNCs) S = 300 (100% TNCs)

0 1

3

5

7

9

11

13

15

Step

“Near death” phenomenon 2/16/2011


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Theory k

pk = ∑ pS (i,S ) pnS (k − i,n − S ) i= 0

⎛ S⎞ i S−i ⎛ N −1 − S ⎞ k−i N −1−S−(k−i) pk = ∑ ⎜ ⎟ p (1 − p) ⎜ ⎟ p (1 − p) ⎝ k −i ⎠ i= 0 ⎝ i ⎠ k

Any unflipped node: n-S of them

Any unflipped node: n-S-F1 of them

Seed = S

Network = n

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First flipped set = F1

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Theory Step 2 k k −i ⎛ S ⎞

F1−i' S−i ⎛ F1⎞ i' p = ∑ ∑ ⎜ ⎟ p (1 − p) ⎜ ⎟ p 1− pF1 F1 k i i' ⎝ ⎠ ⎝ ⎠ i=0 i'=0 i

(

)

⎛n − S − F1 −1⎞ k−i−i' n−S−F1−1−(k−i−i') ×⎜ ⎟ pnSF1 (1− pnSF1) ⎝ k − i − i' ⎠ pF1 = z F1 /n reflects available edges from F1

pnSF1 reflects larger p of unflipped nodes

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Theory: Cascades in “Global Cascades”

Region - Seed = 1 Node

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Theory and Simulations: Cascade in “Global

Cascades” Region

n = 4500, z = 11.5, S = 1, K* = 9 Theory and Simulation (avg of 10 runs) 4000

T = theory S = simulation

3500 3000 2500

Flip Flip Flip Flip

2000 1500

by Step (T) Total (T) by Step (S) Total (S)

1000 500 0 1

3

5

7

9

11

13

15

Step

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Theory and Simulations: A Cascade in “No

Global Cascades” Region

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Theory: Ability to Predict Threshold Seed Size S Transition: Theory and Simulations n = 4500, z = 14.56, K* = 4

S Transition: Theory and Simulations n = 4500, z = 14.56, K* = 5 1

1

0.9

0.9

0.8

0.8

0.7

0.7 0.6

0.6 Simulations Theory

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0 120

Simulations Theory

0.5

130

140

150

160

170

0 240

180

250

260

270

280

290

300

310

Size of Seed, S

Size of Seed, S

S Transition: Theory and Simulations n = 4500, z = 11.5, K* = 5

S Transition: Theory and Simulations n = 4500, z = 11.5, K* = 4

1 1

0.9

0.9

0.8

0.8

0.7

0.7

0.6 0.5 0.4 0.3 0.2 0.1 0 2/16/2011 50 70

Simulations Theory

0.6 Simulations Theory

0.5 0.4 0.3 0.2 0.1

Random networks © Daniel E Whitney 1997-2010 90 110 130 150 Size of Seed, S and cascades 0 150

200

250

Size of Seed, S

300

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At S = 215, Failure Most of the Time

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Occasionally, Success: Why?

Wake-up “Near death” 2/16/2011


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Cause of Wake-up After Near Death

• Caused by a critical mass phenomenon • Near death only a few nodes flip on each step • At most they can hit one node each since, for so few nodes, the likelihood of multiple hits is about zero • So only nodes that are one hit short of flipping have any chance to flip during this phase • This chance is proportional to how many one-shorts there are on any step and how many net edges Fj has • This population is growing but at the same time the number of flippers is falling

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Derivation of Critical Mass

pr(a node in the network links to Fj) = # nodes with edges to Fj = Fj z Fj

Fj z Fj n

=# nodes that will be hit by Fj

fraction of these that will flip = N fractional representation of one short in the network = OS NOS Fj zFj n number of nodes that Fj flipped nodes will flip = n

If each node in Fj flips one node, the cascade is self - sustaining. NOS zFj So 1 = n or NOS = n /zFj or Fj * NOS = Fj * n /zFj 2/16/2011


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Theory and Simulations: Evolution of max

One Short Failures (avg of 20 runs)

If one short exceeds the bound, a TNC almost always occurs.

If one short does not exceed the bound, a TNC almost never occurs.

Variation in one short can cause a TNC when mean is below bound.

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When r > 0 Cascades Occur for Bigger z

Increasing r generates larger vulnerable clusters 2/16/2011


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References

• [Callaway, Newman, Strogatz, Watts] “Network Robustness and Fragility: Percolation on Random Graphs,” Phys Rev Lett (85) 25, Dec 18, 2000 pp 5468-5471 • [Dall’Asta] “Inhomogeneous Percolation Models for Spreading Phenomena in Random Graphs,” arXiv:cond-mat/0509020 1 Sept 2005 • [Newman] “Structure and Function of Complex Networks,” arXive:cond mat/0303516 v1 25 March 2003 pp 20-23 and 38-39 • [Newman] “Random Graphs as Models of Networks” Chapter 2 in Handbook of Graphs and Networks ed. Bornholdt and Schuster, New York: Wiley 2005 • [Newman, Strogatz, Watts] “Random Graphs with Arbitrary Degree Distributions and Their Applications” arXive:cond-mat/0007235 v2 7 May 2001 • [Watts] “A Simple Model of Global Cascades on Random Networks” PNAS April 30, 2002, pp 5766-5771 • [Wikipedia] http://en.wikipedia.org/wiki/Percolation_theory and

http://en.wikipedia.org/wiki/Percolation_threshold 2/16/2011


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More References

• D. E. Whitney, Dynamic theory of cascades on finite clustered random networks with a threshold rule, Physical Review E. E 82, 066110 (2010) • Granovetter, M., “Threshold Models of Collective Behavior,” (1978) Am J. Soc. 83, 1420-1443 • Lopez-Pintado, D., “Diffusion in Complex Social Networks,” Games and Economic Behavior, forthcoming • Morris, S. N., “Contagion,” Rev Econ Studies, (67) 57-78 (2000) • Rogers, E. M., Diffusion of Innovations, New York: Free Press, 2003 • Schelling, T.C., “A Study of Binary Choices with Externalities,” J. Conflict Resolution, 17 (3) 381- 428 (1973) • Valente, T. W. Network Models of the Diffusion of Innovations, Hampton 1995 2/16/2011 Random networks © DanielPress, E Whitney 1997-2010 44/46 and cascades

More References

• http://cnets.indiana.edu/tag/epidemic-modeling • Multiscale mobility networks and the spatial spreading of infectious diseases.�D.Balcan, V. Colizza, B. Gonsalves, H. Hu, J. J. Ramasco, A. Vespignani�Proc Natl Acad Sci U S A 106, 21484-21489 (2009). • Predictability and epidemic pathways in global outbreaks of infectious diseases: the SARS case study�V. Colizza, A. Barrat, M. Barthelemy, A. Vespignani�BMC Medicine 5, 34 (2007) • Seasonal transmission potential and activity peaks of the new influenza A(H1N1): a Monte Carlo likelihood analysis based on human mobility�D. Balcan, H. Hu, B. Goncalves, P. Bajardi, C. Poletto, J. J. Ramasco, D. Paolotti, N. Perra, M. Tizzoni, W. Van den Broeck, V. Colizza, A. Vespignani�, BMC Medicine 7, 45 (2009) 2/16/2011


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Backups

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Generalized Random Networks

• E-R random network has a Poisson degree distribution • Random networks can be built with arbitrary degree distributions, but software is required • Newman says that it is better to generate a specific degree sequence from the distribution and then generate a network with that degree sequence in order to guarantee that the software uses the same degree sequence all the way through the generating process

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Subgraph Shapes vs p p ~ n a , z ~ n a +1 z

0

n-1 n-1/2

n-1/3

n-1/4

1

n1/3

n1/2

p

0 n-2 n-3/2

n-4/3

n-5/4

n-1

n-2/3

n-1/2

0.1

0.178

a

z for n = 1000: 0 0.001 0.0316

1

10

31.6

The threshold probabilities at which different subgraphs appear in a random graph. For pn 3 / 2 → 0 the graph consists of isolated nodes. For p ~ n −3 / 2 trees of order 3 appear, while for p ~ n −4 / 3 trees of order 4 appear, but not many. At p ~ n −1 trees of all orders are present and at the same time cycles of all orders appear, but again, not many. The probability p ~ n −2 / 3 marks the appearance of complete subgraphs of order 4 and p ~ n −1/ 2 corresponds to complete subgraphs or order 5. As a approaches 0 the graph contains complete subgraphs of increasing order. 2/16/2011


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Site and Bond Percolation on Regular

Graphs

• “The most common percolation model is to take a regular lattice, like a square lattice, and make it into a random network by randomly ‘occupying’ sites (vertices) or bonds (edges) with a statistically independent probability p. At a critical threshold pc, long-range connectivity first appears, and this is called the percolation threshold.” [see wikipedia reference “percolation threshold”] • For a square grid, pc = 0.5 for bond percolation and pc = 0.59274621 for site percolation 2/16/2011


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Percolation Step by Step

Regular Networks: Occupancy Each node has 2 neighbors.

It must be linked to both for

there to be a chance of a

giant cluster. So pc = 1.

Each node has 4 neighbors. It must be linked to at least 2 for there to be a chance of a giant cluster. So pc = 0.5.

Random Networks: z

In a random network with n nodes each node has n neighbors. It must be linked to at least one for there to be a chance of a giant cluster. So pc = 1/n. But z = pn so this is the same as zc = 1. See Albert and Barabasi “Stat Mech of Complex Networks” for a detailed derivation

There is no proof or formula for pc when d > 2 except for d > 19 and some special cases. See Slade, Gordon, “Probabilistic Models of Critical Phenomena,”

Princeton Companion to Mathematics, edited by Timothy Gowers.

Scheduled for publication in 2007. for a detailed discussion

http://www.math.ubc.ca/~slade/

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Vulnerable Clusters Multiply the Seed’s

Search Efficiency and Effectiveness

By itself, a single seed cannot flip a stable node Other stable nodes Vulnerable cluster

Seed Vulnerable nodes have a few links to each other (average ~ 1.5) and more links to stable nodes outside their cluster. Working together, vulnerable nodes can flip stable nodes but most likely this happens only when vulnerable nodes co-exist in clusters 2/16/2011 Random networks © Daniel E Whitney 1997-2010 and cascades

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Cluster-Hopping Creates TNCs When

Vulnerable Clusters Are Small

Vulnerable cluster

Seed

Stable node Vulnerable node

Whitney, ICCS, 2007 2/16/2011


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Do Random Networks Have Cycles?

⎛ n⎞

⎜ ⎟ k! zk ⎝ k⎠ k Expected k − cycles = p ≈ for n >> k

2k 2k number of loops in random networks 1E+22 1E+20 1E+18 1E+16 1E+14 1E+12

loops z = 3

1E+10

loops z = 6

1E+08 1E+06 10000 100 1 0

10

20

30

40

50

60

n - number of nodes

Diestel, R., Graph Theory, 3rd edition online at page 298

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and cascades

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Percolation Theory for Random Graphs • Most theory assumes we are dealing with a sparse network that has few or no closed loops, especially no small closed loops • This is measured by the clustering coefficient, which is small for big random networks where the theory has been developed • If there is no clustering or small closed loops then it is easy to calculate how many neighbors, second neighbors, third neighbors, etc, a given node has because no node is its own third neighbor and the probability that a node is its own nth neighbor goes down as n goes up. • If there are more nth neighbors than (n-1)th neighbors for all n and the network is tree-like, then there is a giant cluster • The calculations can be done for any random graph whose degree distribution is known, not just E-R random graphs, as long as there is negligible clustering 2/16/2011


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Variants of the Theory • 1. The percolation (or cascade) proceeds when a link is established between two nodes. This is basic simple percolation described on the previous slide. • 2. The percolation proceeds if a link is established with a node that is “vulnerable” – A) Vulnerability can be a function of k or it can be the same for all nodes (some number 0 ≤ b ≤ 1) – B) Simple percolation has b = 1 for all nodes – C) Watts rumors cascade model has

b = 1 for k ≤ K *

b = 0 for k > K *

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Derivation of Cascade Conditions (Newman)

Its degree = 4 Its excess degree = 3 The neighbor The edge The node

Pick an edge leading from a node and follow it to a neighboring node. What is the (excess) degree distribution of this neighbor? If its degree = k, its edges are k-times more numerous than if its degree = 1 (think of the edge list) But the fraction of nodes with degree k is pk. So the likelihood of encountering a node of degree k by this process is proportional to kpk

Distribution of excess degrees of neighbor = qk−1 =

kpk

or

∞

∑ kp

qk =

(k + 1) pk +1 z

k

k= 0 ∞

∞

q = avg excess degree of neighbor = ∑ kqk = k= 0

2/16/2011

∑ (k + 1)kp

∞

k +1

k= 0

z

=

∑ k(k −1) p k= 0

z

k

< k 2 > −z z2 = = z z1

Random networks © Daniel E Whitney 1997-2010 57/46 Ref: Newman, “Random and Graphs as Models of Networks,” Ch 2 in Handbook of Graphs and Networks, ed Bornholdt and Schuster cascades

continuing

avg number of 2nd neighbors per 1st neighbor = z2 =< k 2 > −z avg number of 3rd neighbors per 2nd neighbor = z 3 = z2 =< k 2 > −z ⎡ z2 ⎤ Avg number of mth neighbors = z1⎢ ⎥ ⎣ z1 ⎦ ⎡ z2 ⎤ This diverges when ⎢ ⎥ = 1

⎣ z1 ⎦

m−1

(from prev slide)

∞

∑ k(k −1) p Generalizeable Cascade condition

k

k= 0

z or

∞

∑ k(k −1) p

k

k= 0

Molloy-Reed criterion

= 1

=z

For E - R k 2 = k

2

= z2

So, for E - R this is the same as z = 1

2/16/2011 Random networks © Daniel E Whitney 1997-2010 and cascades

(See notes)

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continuing For the case where all nodes have vulnerability = b: ∞

b∑ k(k −1) pk = z

See notes

k= 0

For the case where vulnerability is a function ρk of k ∞

∑ k(k −1)ρ

k pk = z

k= 0

Watts rumor cascade model :

ρk =

Supporting derivations of these typically use generating functions 2/16/2011


{

1 for k ≤ K * 0 for k > K *

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Rules for Simulating Cascades

• Build a random network with some value of z and set the value of K* • Choose a node at random (the seed) and flip it • Find its neighbors and flip all that are vulnerable • Find their neighbors and flip all that can be flipped – “Vulnerable” ones flip if one neighbor flipped – “First-order” stable ones flip if two neighbors flipped, etc • Keep going until all nodes have flipped that can • Use some criterion to say if a global cascade has happened or not • Watts made a new network each time but I reused the network to save time. This permitted me to examine it. 2/16/2011


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ESD.342 Network Representations of Complex Engineering Systems Spring 2010

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ESD.342 Random networks Lecture 13

ESD.342 Random networks Lecture 13

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