“Explosive percolation and modeling interactingnetworks”

Lecture 16, Mar 7 2011

The past decade, a “Science of Networks”:(Physical, Biological, Social)

• Geometric versus virtual (Internet versus WWW).

• Natural /spontaneously arising versus engineered /built.

• Each network may optimize something unique.

• Fundamental similarities and differencesto guide design/understanding/control.

• Interplay of topology and function ?

• Up until now, studied largelyas individual networks in isolation .

NRC, 2005

SAMSI, 2010-2011 Program on Complex Networks:http://www.samsi.info/programs/2010-11-program-complex-

networks

• Network modeling and inference

• flows on networks

• network models for disease transmission

• dynamics of networks

“Dynamic Networks: Modeling and inference of networks inthe context of dynamical systems evolving in time, such as time-varying gene regulatory interactions or email social networks.”

Dynamics of networks – A key distinction

• Dynamical process taking place on a network substrate– Information flows– Epidemic spreading– Synchronization, etc.

• Dynamics of the nodes and edges constituting the network– Network growth/decline; edge intermittency; etc.

Key issue: separation of timescales.

i.e., when is the network substrate essentially static over thetimescale of the dynamical process? .... (e.g web crawling;routing tables in mobile ad hoc networks; epidemic spreading(Volz and Meyers Proc. Royal Society, B. 2007; Mucha et. al. Science 328, 2010.)

Dynamical processes on (static) networks(Durrett, Random Graph Dynamics, Cambridge Univ. Press (2006))

(Barrat, Barthelemy, Vespignani Dyn. Proc. on Complex Networks, Cambridge (2008))

• Diffusion

• Spreading / Percolation / Contact processes / Epidemics

• Synchronization – fireflies, Josephson junctions, ad hocsensor networks ...

• Searching on networks (WWW, P2P, sensor nets, social nets)

• Flows: data, materials, transportation, biochemical (on manylevels – genes, proteins, metabolites)

• Traffic, congestion, jamming

• Cascades / Avalanches / Sandpiles

Dynamics of the underlying network

• Network growth (long list here.... see next slide)

• Vulnerability to node and edge deletion

• Network shrinking (nodes – banking, garment industry)

• Online Social Networks: shrinking diameter and densification

• Link intermittency (Internet)

• Shifting centrality (AS level internet)

• Ad hoc / MANETS (wireless radio communications)(self-org algorithms, capacity, scaling, energy conservation)

• Coevolution:(socio-technical – land use/population growth; task oriented social networks(OSS))(biological – gene regulatory/ppi (conserved motifs) / ecosystems )

Dynamics of the network: growth

• Random graphs (Erdos-Renyi 1959, 1960; Gilbert 1959; Bollobas 1985)

• Configuration models (Bollobas 1980, Molloy and Reed 1995)

• Randomly grown graphs (CHKNS 2001)

• Growth by Preferential Attachment (Polya 1923; Yule 1925; Zipf 1949;Simon 1955; Price 1976; Barabasi-Albert 1999)

• Growth by copying (WWW inspired) (Kumar et. al. FOCS 2000); Copyingwith mutation (bio inspired) (e.g., DMC – Vazquez et. al. 2003))

• Non-linear PA (Krapivsky-Redner 2000, 2001); Geometric PA (Flaxman-Frieze-Vera 2006); PA from optimization (DBCBK 2007)

• Growth with feedback (White-Kejzar-Tsallis-Farmer 2006; R.D.-Roy 2008)

• Growth with choice (Bohman-Frieze 2001....)

Dynamics of the network: shrinking, densification, etc

• S. Saavedra, F. Reed-Tsochas, and B. Uzzi, “Asymmetric disassembly androbustness in declining networks”, PNAS 105 (43), 2008.

• J. Leskovec, D. Chakrabarti, J. Kleinberg, C. Faloutsos, Z. Ghahramani,“Kronecker Graphs: An Approach to Modeling Networks”, The Journal ofMachine Learning Research 11, 985–1042, 2010.

• P.M. Weichsel, “The Kronecker product of graphs”, Proceedings of theAmerican Mathematical Society, 1962.

Kronecker graphs – can we represent dynamic multi-type, multi-plexnetworks with these structures?

Beyond topology: The Flows are the network!(e.g., Matisziw, Grubesic (2010))

• Networks across space and time / Structure and function.

• Internet2 flows over one day show large variation.

• “Vulnerable” nodes and edges depends on time of day.

Extreme sensitivity to details of flows!! (Need heuristic models of flows.)

Caveats: 1) Braess’ Paradox; 2) Kurant and Thiran PRL 89 (2006).

“Vulnerability” – is network connectivity a good thing?

(The Percolation phase transition)Emergence of giant connected component in a random network

• Communications, Transportation, Synchronization, ... versus

• Spread of human or computer viruses

Connectivity in random networks• Solomonoff, Rapoport, “Connectivity of random nets”, Bull. of Math. Biophysics, 1951.• P. Erdos and A. Renyi, “On random graphs”, Publ. Math. Debrecen. 1959.• P. Erdos and A. Renyi, “On the evolution of random graphs”,

Publ. Math. Inst. Hungar. Acad. Sci. 1960.• E. N. Gilbert, “Random graphs”, Annals of Mathematical Statistics, 1959.

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• Start with N isolated vertices.

• Add random edges one-at-a-time.E = N(N − 1)/2 total edges possible.

• After E edges, probability p of anyedge is p = E/E = 2E/N(N − 1)

What does the resulting graph look like?(Typical member of the ensemble)

N=300

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p = 1/400 = 0.0025 p = 1/200 = 0.005

Emergence of a “giant component”Phase transition in connectivity

• pc = 1/N .

• p < pc, Cmax ∼ log(N)

• p > pc, Cmax ∼ A ·N

(Ave node degree t = pN

so tc = 1.)

A continuous (smooth) transition at pc.

Can any limited perturbation change the phase transition?[Bohman, Frieze, RSA 19, 2001]

[Achlioptas, D’Souza, Spencer, 2009]

• Possible to Enhance or Delay the onset?

• The “Product Rule” (An “Achlioptas Process”)– Choose two edges at random each step.– Add only the desirable edge and discard the other.

(Enhance) (Delay)

• The Power of Two ChoicesAzar; Broder; Mitzenmacher; Upfal; Karlin;

ProdRule: Explicit example

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(A) (B)

e1

e2

• Prod e1 = (7)× (2) = 14

• Prod e2 = (4)× (4) = 16

• To enhance choose e2. To delay choose e1.

Product Rule

• Enhance – similar to ERbut with earlier onset.

• Delay –changes fromcontinuous todiscontinuoustransition!

The window ∆ from Cmax = n1/2 to Cmax = 0.5n

• Let e0 denote the last edge added for which Cmax < n1/2.(Note ER has n2/3 at pc.)

• Let e1 denote the first edge added for which Cmax > 0.5n.

• Let ∆ = e1 − e0.

0e+00 2e+07 4e+07 6e+07

1.0

1.1

1.2

1.3

1.4

1.5

Δ/n

2/3

n0e+00 3e+07 6e+07

0.19

0.21

o+

ERBF

Δ

n

PR ∆ ∼ n2/3 ER (and BF) ∆ ∼ n.

∆ for PR is sublinear in n.

Delayed Product Rule: Discontinuous change

• Define “time”, t = e/n. (Find ec = 0.888n as n→∞).

• For t < tc, Cmax < n1/2, where tc→ 0.888... from below.

• For t > tc, Cmax > 0.5n, where tc→ 0.888... from above.

1e+06 5e+06 2e+07 1e+08

0.87

60.

880

0.88

40.

888

n

t/n

e0/n

= 0.88814 – 2.18n-0.38

+ e1/n

= 0.88809+ 0.015n-0.24

t

Jumps “instantaneously” from Cmax = n1/2 to 0.5n.

“Explosive Percolation in Random Networks”From nγ to greater than 0.6n “instantaneously”

Cmax jumps from sublinear nγ Nontrivial Scaling behaviorsto≥ 0.5n innβ edges, with β, γ < 1 . γ+ 1.2β = 1.3 for A ∈ [0.1, 0.6]

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0.1 0.2 0.3 0.4 0.5 0.614 15 16 17

0.6

0.7

0.8

0.9

γ

β

n

0.6

0.7

0.8

0.9

0.2

Rel Err o > 0.02o < 0.02o < 0.015o < 0.01o < 0.005

1x106 4x106 8x106 32x106

0.4 0.6

Achlioptas, D’Souza, Spencer, Science, 323 (5920), 2009

Accompanying commentary:Can the evolution of real-world networks be accurately modeled?

(See also Borgatti Science 323, 2009. Social networks vs random graphs)

Explosive percolation in other models ...

• R. Ziff, Phys. Rev. Lett. 103, 045701 (2009).“Explosive Growth in Biased Dynamic Percolation on Two-DimensionalRegular Lattice Networks”

• Y. S. Cho, J. S. Kim, J. Park, B. Kahng, D. Kim, Phys. Rev. Lett. 103,135702 (2009). “Percolation Transitions in Scale-Free Networks under theAchlioptas Process”(Chung-Lu weighted node power law growth model)• Discontinuous, but more over reduces the sensitivity to epidemicoutbreaks:• pc > 0 for γ > 2.3 or 2.4(Recall various derivations that showed limn→∞ pc = 0 for 2 < γ < 3.)

• F. Radicchi, S. Fortunato Phys. Rev. Lett. 103, 168701 (2009).“Explosive percolation in scale-free networks”(Configuration model power law – slightly different result)• pc > 0 for γ > 2.2, discontinuous for γ > 3.

Explosive percolation now observed in ...

• E. J. Friedman, A. S. Landsberg Phys. Rev. Let. 103, 255701 (2009).“Construction and Analysis of Random Networkswith Explosive Percolation” ... the “powder keg”

• Y.S. Cho, B. Kahng, D. Kim; Phys. Rev. E 81, 030103(R), (2010).“Cluster aggregation model for discontinuous percolation transition”

• Rozenfeld, Gallos, Makse; Eur. Phys. J. B, 75, 305-310 (2010).“Explosive Percolation in the Human Protein Homology Network”

• Growth of Wikipedias? (Bounova, personal communication.)

• J. Gomez-Gardenes, S. Gomez, A. Arenas, Y. Moreno; Phys. Rev. Let.(2011). Explosive synchronization Transitions in Scale-free Networks

• Various others ... (see arxiv)

Beyond “Product Rule”

• In general any rule that keeps components similar in size insubcritical regime should be explosive:– “Powder Keg” of Friedman and Landsberg PRL (2009).– Starting ER from proper initial state; Cho, Khang, Kim PRE (2010).

● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

1 5 10 50 500

2e−

051e

−04

5e−

045e

−03

Rank

Siz

e/N

+ ++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Rank−size top 1000 at t=t_c

o+

PRER

Local Cluster Aggregation Modelswith Explosive Percolation.... Master Equation

(R.D. and M. Mitzenmacher, Phys. Rev. Lett., 104, 2010.)

●

●●●

●

●

●

●

●

●●

●

●●

●●

●●●

●

●

●●

●●

●●

TR

AE

2e+05 1e+06 5e+06 5e+07

0.0020.005

0.020

n

Δ/n

t

C /n 1 -0.323

-0.367

Δ AE ο TR

• “Adjacent Edge (AE)” – 2 candidate edges share a common vertex.

• “Triangle Rule” (TR) – choose 3 vertices at random, 3 candidate edges.

Locality: more physical and simplifies analytic treatment

Evolution equations for the bounded size AE rule:

• Let xi denote fraction of nodes in components of size i,for 1 ≤ i ≤ K.

• Let Si =∑∞j=i xj

(the weight in the tail starting at size i)(SK+1 = 1−∑K

j=1 xj is useful)

• Probability first node is in component of size i is xi.

• Probability the smaller of the two additional components hassize j is S2

j − S2j+1.

• Expected evolution of xi’s:

dxidt = −ixi − i(S2

i − S2i+1) + i

∑j+k=i xj(S

2k − S2

k+1)

●

●●●

●

●

●

●

●

●●

●

●●

●●

●●●

●

●

●●

●●

●●

TR

AE

“Susceptibility”: W =∑∞i=1 ixi =

∑∞i=1 i

2ni

(Expected size of component to which arbitrary vertex belongs)

dW

dt=

K∑

j=1

K∑

k=1

2jkxj(s2k − s2k+1) +K∑

j=1

2jW ∗xjsK+1

+K∑

k=1

2kW ∗(s2k − s2k+1) + 2(W ∗)2sK+1.

• Where W ∗ = W −∑Ki=1 ixi

(contributions from comps of size greater than K).

• W → ∞ at the phase transition. Simple Euler’s methodnumerics yields tc = 0.796 (with K = 600).

(Evolution eqns. and direct simulation yield outstandingagreement.)

Investigating the mechanisms leading to explosivepercolation – Puzzles

• Scaling behaviors (usually only seen in continuous transitions)– Component density ni ∼ i−τ– Susceptibility W ∼ |t− tc|−α– 2nd largest C2 ∼ |t− tc|−µ

0.70 0.75 0.80 0.85

1e-05

1e-03

1e-01

t

W

exp=

-1.113

0.001 0.005 0.0502e-06

2e-05

2e-04

t - tc0.01 0.1

• Decrease in jump with n “weakly discontinuous”– J. Nagler, A. Levina, and M. Timme,Nature Phys. 7 (2011).

– S. S. Manna and Arnab Chatterjee.Physica A, 390, 177-182 (2011).

Author's personal copy

180 S.S. Manna, A. Chatterjee / Physica A 390 (2011) 177–182

101

102

103

104

105

106

107

108

N

0.20

0.30

0.12

0.14

0.16

0.18

0.220.240.260.28

0.32

g(N

)

N-0.06475

0.0

0.1

0.2

0.3

0.4

g(N

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

a b

Fig. 4. (Color online) (a) For AP on random graph, the maximal jump g(N) in the order parameter has been plotted on a double logarithmic scale withgraph size N . The slope is −0.06475. (b) The maximal jump g(N) is then plotted against N−0.06475 on a linear scale. The continuous line is a straight line fitof this data which is then extrapolated to N → ∞ to meet the g(N) axis at −0.000516.

Fig. 5. (Color online) (a) Order parameterC(r, α)with link density r forα = 1/2, 1/4, 0, −0.1, −1/4, −1/2, −1, −2, −3, −4 and−5,α values decreasingfrom left to right, for a random graph of N = 4096. (b) The asymptotic values of the gap δ(α), the percolation threshold rc(α) − 1/2 and the largest jumpg(α) of the order parameter plotted with α.

For our problem the asymptotic jump g(α) = limN→∞ ∆Cm(α,N) for different α are plotted in Fig. 3(b). For ordinarybond percolation g(α = 0,N) → 0 as Ndf /2−1, with df = 91/48, the fractal dimension of the incipient infinite percolationcluster [3]. However for α < 0 the g(α) jumps to 0.16 at α = −0.05 and then gradually increases to 1/2 as α → −∞. Thepercolation thresholds pc(N) are estimated by the average values of rm and t1 giving approximately the same asymptoticvalue for pc(α). The pc(0) = limN→∞ pc(0,N) → 1/2 limit is approached as N−1/2ν , ν = 4/3 [3] being the correlationlength exponent for the ordinary percolation. On the other hand for α < 0, pc(α) increased continuously with α (Fig. 3(b))and tends to unity as α → −∞.

In Fig. 5(a) we show C(r, α) vs. r for a random graph. The last five curves for α ≤ −1 almost coincide with the verticalline at r = 1 implying first order transition with rc = 1 in this range. Curves are smoother for −1 < α < 0. In Fig. 5(b)we show that rc(α) tends to 1/2 rapidly as α → 0. The jump g(α) is almost zero for −1/4 ≤ α ≤ 0 but then it rapidlyincreases and tends to 1/2 as α → −∞. The gap δ(α) is almost zero for α ≤ −1/2 and then slowly increases to ≈ 0.193for a random graph. This result indicates that for random graph αc = −1/2.

The approach to first order transition in our system is completely different from AP. Comparison is made of the tendencyof suppressing the growth of large clusters and enhancing the growth of small clusters as reflected in the following threequantities. In Fig. 6(a) we show the scaled cluster size distribution P(s, p) for square lattice at α = −1. The individualdistributions have strong dependence on ∆p = pc − p but not significantly on L. The best scaling form is P(s, p)∆p−η ∼G(s∆pζ ) with η = ζ = 1.77(5). The scaling function fits to the Gamma distribution G(x) ∼ xa exp(−bx) with a = 0.93(10)and b = 4.09(10). P(s, p) for α < −1 also fits to Gamma distributions with different values of a and b. In addition wehave checked the finite size scaling analysis as well. Right at the percolation point the cluster size distribution P(s, pc, L)L0.9scales excellently with s/L0.9 and the scaling function fits very well to a Gamma distribution. Here the percolation point isdetermined by the maximal jump in the order parameter.

Edge competition models are weakly discontinuous

• Weakly discontinuous versus strongly discontinuous– Weakly: discontinuity decreases with system size→ 0 as n→ 0.– Strongly: size of discontinuity (jump) independent of system size.

• J. Nagler, A. Levina, and M. Timme, Nature Phys. 7 (2011).

• R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, J. F. F.Mendes Phys. Rev. Lett. 105, 255701 (2010).

• Rigorous proof (Feb 25, 2011) O. Riordan, L. Warnke,arXiv:1102.5306.... All edge competition models continuousif n truly infinite.

• BUT, any finite system still shows a discontinuous jump: daCosta et al, system as large as 1018 shows jump ∆ = 0.1n.

• How large can the largest social network be? The largestnumber of routers possible? ....

Finite size systems:Mathematics versus physics versus engineering

• Mathematical results, asymptotic: n→∞.

• Physics deals with Avogadro’s number: n ∼ 1023.

• Engineered systems deal with 100 < n < 109.

Bootstrap percolation a cautionary tale!

(Once rigorous results where finally established (Holroyd, Prob. Theory, 2003)shown to be considerably off from predictions of simulation with n ∼ 108.)

MANETs for military applications “symptotics”

The search for the “most explosive” model(Friedman, Landsberg PRL (2009)), (Rozenfeld, et. al. EPJB (2010)),

Nagler et al Nature Phys (2011))

(a) (b)

(d)

(c)

(e)

FIG. 1: Deterministic model leading to explosive percolation. In (a) we start with N nodes without

links. In (b) we connect nodes in pairs. In (c) and (d) we iteratively connect each pair of clusters

with one link to form larger components. This mechanism continues recursively by joining smaller

clusters into larger ones, and the spanning component only emerges at the end of the process, when

the entire network is connected, as in (e).

half the number of components of step t ! 1, and therefore nt = 2m!t, for 0 " t < m.

Moreover, the number of nodes in each component is St = 2t (all components at a given

step t have the same number of nodes). The number of new links at step t is nt!1/2, and

therefore the total number of links in the network up to step t is

Mt = Nt!

i=1

2!i = N(1! 2!t). (1)

The resulting network is a tree, so that the total number of possible links in the network

is M # N ! 1. Consequently, the fraction of links added to the network up to time t is

p # Mt/M = Mt/(N ! 1) $ 1! 2!t, for N >> 1. Therefore, the dependence of the largest

component size St on the fraction of added links p follows

St =1

1! p, (2)

which exhibits a singular point at p = 1. This singularity is the hallmark of a discontinuous

or first-order percolation phase transition, similarly to the reported result of Ref. [9]. In

Fig. 2 we show the size of the largest component as a function of the fraction of links

added to the network, and we compare the PR process to the above presented deterministic

model for the same network size. Both cases exhibit explosive percolation, but obviously the

deterministic process leads to a much sharper transition, even for the small finite size that is

considered here (N = 32768 nodes). This model can be considered as an optimized (albeit

4

(From Rozenfeld, et. al.)

• (a) Phase k = 2, merge allisolated nodes into pairs.

• (b) Phase k = 4, merge pairsinto size 4 components.

• (c) Phase k = 8, merge pairsof 4’s into 8’s.

• etc.

• At edge e = n (time t = 1) one giant of size n emergest = 1, C1 = n

t = 1− 1n, C1 = n/2

t = 1− 2n, C1 = n/4

...

(Giant emerges when only one component remains)Strongly discontinuous, ∆ = 0.5

Re-visiting the Bohman Frieze Wormald model (BFW)(Random Structures & Algorithms, 25(4):432-449, (2004))

• Also start with n isolated vertices, and stage k = 2.

• Sample edges uniformly at random from the complete graph on n nodes.

• Can reject edges but in phase k must accept at least g(k) = 1/2 + (2k)−1/2

fraction of sampled edges.

• For each sampled edge:– if the resulting component size ≤ k, accept;– otherwise reject that edge if possible;– else augment k → k + 1 until either:

(a) k increases sufficiently that candidate edge accommodated(b) g(k) decreases sufficiently that candidate edge rejected

• (Asymptotically accept half of all edges, limk→∞ g(k) = 1/2.)

• Rigorous proof: (bounded-size differential eqns)– No component of size greater than 200 when e = 0.96689n edges added.– Giant component must exist once e = cn edges, with c ∈ [0.9792, 0.9793].

BFW – simultaneous emergence of multiple stable giants(Wei Chen and R.D. Phys Rev Lett in press)

• Two stable giants! (With C1 = 0.570, C2 = 0.405.)– Fraction of internal cluster edges > 1/2.– Even if restrict to sampling edges not yet present.– (If restrict to sampling only edges thatspan clusters, only one giant.)

• Now let g(k) = α+ (2k)−1/2

then α determines number of stable giants.

Applications for multiple giants? (Communications, epidemiology, ...)

BFW – Strongly discontinuous

104

106

108

105

107

0.1

0.2

0.3

0.40.5

n

∆C

max

(a)

∆ Cmax

∼ n−0.065

∆ Cmax

∼ n−0.043

restricted BFW

unrestricted BFW

MR(m=2)

PR

100

102

104

106

10−6

10−4

10−2

100

sn

(s)

(b) t/n=0.95

t/n=0.96

t/n=0.97

critical point

In reality a collection of interacting networks:(multi-type nodes, multi-plex edges)

Networks:

TransportationNetworks/Power grid(distribution/collection networks)

Biological networks- protein interaction- genetic regulation- drug design

Computernetworks

Social networks- Immunology- Information- Commerce

• E-commerce→WWW→ Internet→ Power grid→ River networks.

• Biological virus → Social contact network → Transportation networks →Communication networks→ Power grid→ River networks.

Coupling of like scales across space and time

Example real system 1 – Socio-technical co-evolution

• Structure and evolution of networks in Open Source Software.– A proxy for general task oriented social groups– Response to endogenous stresses (e.g., illness of key developer)– Response to exogenous stresses (e.g., release dates)

Example real system 2 – The Power GridThe “grid” is a collection of independently owned and operatedplants and transmission lines.

• Does increasingly distributed nature stabilize or destabilize?

• Load shedding strategies

• Economic considerations (Game theory)

Timescales in a Power Grid(M. Amin, “National Infrastructures as Complex Interactive Networks” (2000))

M. Amin/ Automation, Control, and Complexity: An Integrated Approach, Samad & Weyrauch (Eds.), John Wiley and Sons, pp. 263-286, 2000

spatial and energy levels, power systems are also multi-scaled in the time domain, from nanoseconds to decades, as

shown in Table 1:

Table 14.1 Multi-scale time hierarchy of power systems

Action/operation Time frame

Wave effects (fast dynamics, lightning caused overvoltages) Microseconds to milliseconds

Switching overvoltages Milliseconds

Fault protection 100 milliseconds or a few cycles

Electromagnetic effects in machine windings Milliseconds to seconds

Stability 60 cycles or 1 second

Stability Augmentation Seconds

Electromechanical effects of oscillations in motors & generators Milliseconds to minutes

Tie line load frequency control 1 to 10 seconds; ongoing

Economic load dispatch 10 seconds to 1 hour; ongoing

Thermodynamic changes from boiler control action (slow dynamics) Seconds to hours

System structure monit oring (what is energized & what is not) Steady state; on-going

System state measurement and estimation Steady state; on-going

System security monitoring Steady state; on-going

Load Management, load forecasting, generation scheduling 1 hour to 1 day or longer; ongoing.

Maintenance scheduling Months to 1 year; ongoing.

Expansion planning Years; ongoing

Power plant site selection, design, construction, environmental impact, etc. 10 years or longer

To make adaptive self-healing practical in agent-based, distributed control of an electric power system will require

the development, implementation and widespread local installation of Intelligent Electronic Devices (IED)

combining the functions of sensors, computers, telecommunication units, and actuators. Several current

technological advances in power developments can provide these necessary capabilities when combined in an

intelligent system design. Among them are:

• Flexible AC Transmission System (FACTS), high-voltage electronic controllers that increase power carrying

capacity of transmission lines (already fielded by American Electric Power).

• Unified Power Flow Controller (UPFC), a third generation FACTS device, that uses solid-state electronics to

direct the flow of power from one line to another to reduce overloads and improve reliability.

• Fault Current Limiters (FCL), that absorb the shock of short circuits for a few cycles to provide adequate time for

a breaker to trip.

• Wide Area Measurement System (WAMS) based on satellite communication and time stamping using GPS,

which can detect and report angle swings and other transmission system changes (in limited use within the

Western Area Power Administration).

• Several innovations in the areas of materials science and high temperature superconductors, including the use of

ceramic oxides instead of metals, oxide-power-in-tube (OPIT) wire technology, wide bandgap semiconductors,

and superconducting cables.

• Distributed resources such as small combustion turbines, Solid-Oxide Fuel Cells (SOFC), photovoltaics,

Superconducting Magnetic Energy Storage (SMES), Transportable Battery Energy Storage System (TBESS), etc.

• Information systems and on-line data processing tools such as: Open Access Same-time Information System

(OASIS) and Transfer Capability Evaluation (TRACE). The latter software determines the total transfer

capability for each transmission path posted on the OASIS network, while taking into account the thermal,

voltage and interface limits. (OASIS, Phase 1, is now in operation over the Internet).

Increased use of electronic automation raises significant issues regarding the adequacy of operational security: (1)

reduced personnel at remote sites makes them more vulnerable to hostile threats; (2) interconnection of automation

Connectivity in interacting (or modular) networks

Theory of multi-type, multiplicative branching (Galton-Watson) processes:• R. Otter, “The multiplicative process” Ann. Math. Stat. 20, (1949).

• T.E. Harris, The Theory of Branching Processes, Springer-Verlag, (1963).

Inverting self-consistency equations of multi-type branching processes:• I.J. Good, “Generalization to several variables of Lagrange’s expansion,

with applications to stochastic processes”. Proc. Cambridge Philos. Soc.56, pp. 367–380 (1960).

• Bollobas, Janson, and Riordan. “The phase transition in inhomogeneousrandom graphs”. Random Structures and Algorithms, 31:3122, (2007).

• Ostilli and Mendes. “Small-world of communities: communication andcorrelation of the meta-network”. J. Stat. Mech., L08004, (2009).

• Allard, Noel, Dube, and Pourbohloul. “Heterogeneous bond percolation onmultitype networks with an application to epidemic dynamics”. Phys. Rev.E, 79:036113, (2009).

Connectivity in interacting networks – Modular Erdos-Renyi

• Divide nodes initially into two groups (A and B):

A B

• Add internal a-a edges with rate λ.

• Add internal b-b edges with rate λ/r1, with r1 > 1.

• Add intra-group a-b edges with rate λ/r2, with r2 > 1, r2 6= r1.

What happens? (Anything different?)

Percolation on interacting networks,using random graph models

(E. Leicht and R. D’Souza, arXiv:0907.0894)

a

b

edges to aedges to b

System of two networks Connectivity for an individual node

• Probability distribution nodes in network a: pakakb• For the the system: {pakakb, p

bkakb}

• Build generating function formalism for interacting networks.(i.e., properties of the ensemble of networks with {pakakb, p

bkakb})

Wiring which respects group structures percolates earlier!A B

0.0

0.2

0.4

0.6

0.8 1.0 1.2 1.4 1.6

Frac

tion

of n

odes

–k

SL1

SL2

SER

“Cooperative enhancement”

aa rate λ

bb rate λ/r1, w/ r1 = 2

ab rate λ/r2, w/ r2 = 6

Generating functions, topology, and percolation cascadesin coupled infrastructure networks

• Sergey Buldyrev, Roni Parshani, Gerald Paul, H. Eugene Stanley, ShlomoHavlin, “Catastrophic cascade of failures in interdependent networks”Nature 464, 1025-1028

• Though power law degree dist make a single network more robust torandom failure, makes a coupled network more susceptible.

• Modeling a blackout in Italy

continue this process until no further splitting and link removal canoccur (Fig. 2d). We find that this process leads to a percolation phasetransition for the two interdependent networks at a critical threshold,p5 pc, which is significantly larger than the equivalent threshold for asingle network. As in classical network theory21–25, we define the giantmutually connected component to be themutually connected clusterspanning the entire network. Below pc there is no giant mutuallyconnected component, whereas above pc a giant mutually connectedcluster exists.

Our insight based on percolation theory is that when the networkis fragmented, the nodes belonging to the giant component connect-ing a finite fraction of the network are still functional, whereas thenodes that are part of the remaining small clusters become non-functional. Therefore, for interdependent networks only the giant

mutually connected cluster is of interest. The probability that twoneighbouring A-nodes are connected by A«B links to two neigh-bouring B-nodes scales as 1/N (Supplementary Information). Hence,at the end of the cascade process of failures, above pc only very smallmutually connected clusters and one giant mutually connected clus-ter exist, in contrast to traditional percolation, wherein the clustersize distribution obeys a power law. When the giant componentexists, the interdependent networks preserve their functionality; ifit does not exist, the networks split into small fragments that cannotfunction on their own.

We apply our model first to the case of two Erdo0 s–Renyi net-works21–23 with average degrees ÆkAæ and ÆkBæ. We remove a randomfraction, 12 p, of the nodes in network A and follow the iterativeprocess of forming a1-, b2-, a3-, …, b2k- and a2k11-clusters as

a11

a12

a13

a11

a12

a13

a31

a32

a33

a34

b21

b22

b23

b24

b21

b22

b23

b24

Attack

Stage 1 Stage 2 Stage 3A B

a b c d

Figure 2 | Modelling an iterative process of a cascade of failures. Eachnode in network A depends on one and only one node in network B, and viceversa. Links between the networks are shown as horizontal straight lines, andA-links and B-links are shown as arcs. a, One node from network A isremoved (‘attack’). b, Stage 1: a dependent node in network B is alsoeliminated and network A breaks into three a1-clusters, namely a11, a12 anda13. c, Stage 2: B-links that link sets of B-nodes connected to separate a1-clusters are eliminated and network B breaks into four b2-clusters, namely

b21, b22, b23 and b24. d, Stage 3: A-links that link sets of A-nodes connected toseparate b2-clusters are eliminated and network A breaks into four a3-clusters, namely a31, a32, a33 and a34. These coincidewith the clusters b21, b22,b23 and b24, and no further link elimination and network breaking occurs.Therefore, each connected b2-cluster/a3-cluster pair is a mutually connectedcluster and the clusters b24 and a34, which are the largest among them,constitute the giant mutually connected component.

a b c

Figure 1 | Modelling a blackout in Italy. Illustration of an iterative process ofa cascade of failures using real-world data from a power network (located onthe map of Italy) and an Internet network (shifted above the map) that wereimplicated in an electrical blackout that occurred in Italy in September200320. The networks are drawn using the real geographical locations andevery Internet server is connected to the geographically nearest powerstation. a, One power station is removed (red node on map) from the powernetwork and as a result the Internet nodes depending on it are removed fromthe Internet network (red nodes above the map). The nodes that will bedisconnected from the giant cluster (a cluster that spans the entire network)

at the next step are marked in green. b, Additional nodes that weredisconnected from the Internet communication network giant componentare removed (red nodes above map). As a result the power stationsdepending on them are removed from the power network (red nodes onmap). Again, the nodes that will be disconnected from the giant cluster at thenext step are marked in green. c, Additional nodes that were disconnectedfrom the giant component of the power network are removed (red nodes onmap) as well as the nodes in the Internet network that depend on them (rednodes above map).

LETTERS NATURE |Vol 464 | 15 April 2010

1026Macmillan Publishers Limited. All rights reserved©2010

Other interesting power grid models (Tune in Mar 14)

Topology based (approximate flow by structural features):• Albert, Albert, Nakarado, “Structural vulnerability of the North American

power grid”, Phys Rev E 69, 2004.

• Kinney, Crucitti, Albert, Latora, “Modeling cascading failures in the NorthAmerican power grid”, Eur, Phys. J. B 46 (2005).

• Kong, Yeh, “Resilience to degree-dependent and cascading node failure inrandom geometric graphs”, IEEE Trans Info Theory 56 (11) 2010.

Complete graph but better flow models• Dobson, Carrera, Newman ... various works

Comparing topology estimators and real flows• P. Hines, E. Cotilla-Sanchez, S. Blumsack. “Do topological models provide

good information about vulnerability in electric power networks?” Chaos 20(3) 2010.

“Dynamic Networks” – Questions / Challenges / Issues

• Timescales– Establishing them: dynamics of the network and dynamics on.– Separation of timescales – (what do we do when can’t separate?)– Coupling across systems

• Flows are the network ... (need good heuristic models – topologicalindicators – characteristic states)

• Warning indicators of phase transitions in network structure or function :Can fluctuations in time-series data / fluctuation-response data be used aspredictive?

• Co-evolution, linkages and interdependencies (land-use and populationgrowth / gene regulatory networks and protein interactions / human-ecoevolution / critical infrastructure....)

• Random graphs (incld. generating functions, esp dynamic PGFs):percolation, motifs, null models, phase transitions...

• Spectral statistics : timescales, dynamic communities?

• Visualization (e.g. http://vis.cs.ucdavis.edu/∼ogawa/codeswarm/