35
Explosive percolation: What we’ve learned in the past two years Raissa D’Souza University of California, Davis External Professor, Santa Fe Institute
Explosive percolation:What we’ve learned in the past two years
Raissa D’SouzaUniversity of California, Davis
External Professor, Santa Fe Institute
Modeling networks as random graphs
• Erdos and Renyi random graphs (1959, 1960).Phase transition.
• Configuration models (Bollobas 1980, Molloy and Reed RSA1995). Enumerating over all networks with specified {pi}.
. . .
• Preferential attachment (Barbasi-Albert 1999, etc.)
• Growth by copying (Kumar, Raghavan, Rajagopalan, Sivakumar,Tomkins, Upfal FOCS 2000), including duplication/mutation(Vazquez, Flammini, Maritan, Vespignani, ComPlexUs 2003)
• Many more . . .
Building a random instance of a network, G(N, p)
• 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.
• Consider each possible edge, andadd it with probability p.
What does the resulting graph look like?(Typical member of the ensemble)
G(N=300,p)
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p = 1/400 = 0.0025 p = 1/200 = 0.005
Erdos-Renyi, Emergence of unique “giant component”
• t < 1, Cmax ∼ O(lnn)
• t = 1, Cmax = n2/3
• t > 1, Cmax ∼ An, with A > 1
• The critical windowBollobas, Trans. Amer. Math. Soc., 286 (1984).Luczak, Random Structures and Algorithms, 1 (1990).
t = 1 + λn−1/3 (where t = 2e/n)
• Mean field critical exponentse.g., Grimmett, Percolation. 2nd Edition. Springer-Verlag. 1999.
χ ∼ (tc − t)−γ, with γ = 1.
where χ is the expected size of the component to which an arbitrarilychosen vertex belongs.
Connectivity – good or bad?
• Communications, Transportation, Synchronization, ...
versus
• Spread of human or computer viruses
Percolation, onset of: large scale connectivity, epidemic threshold, globalcascades...
Can any limited perturbation change the phase transition?[Bohman, Frieze, RSA 19, 2001]
[Achlioptas, D’Souza, Spencer, Science 323, 2009]
• Possible to Enhance or Delay the onset?
• The “Product Rule”– Choose two edges at random each step.– Add only the desirable edge and discard the other.
(Enhance) (Delay)
• The Power of Two Choices in randomized algorithms.Azar; 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 –Extremely abrupt
The scaling window, ∆ from n1/2 to 0.5n
• Let e0 denote the last edge added for which Cmax < n1/2.(Recall 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.
PR From n1/2 to 0.5n in number of edges that is sublinear in n.
In terms of edge density or “time”, tc, where t = e/n
(Note, for ER, tc = 1/2)
• For t < tc, Cmax < n1/2.
• For t > tc, Cmax > 0.5n.
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”
(Compelling evidence that the transition is discontinuous)
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
Similar “Explosive percolation” ...• R. Ziff, Phys. Rev. Lett. 103, 045701 (2009).
“Explosive Growth in Biased Dynamic Percolation on 2-D Regular 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 the Achlioptas Process”(Chung-Lu weighted node power law growth model)• pc > 0 for γ > 2.3 or 2.4 and discontinuous.
• F. Radicchi, S. Fortunato Phys. Rev. Lett. 103, 168701 (2009).“Explosive percolation in scale-free networks”(Configuration model power law)• pc > 0 for γ > 2.2, discontinuous for γ > 3.
• E. J. Friedman, A. S. Landsberg Phys. Rev. Let. 103, 255701 (2009).“Construction and Analysis of Random Networks with Explosive Percolation”
• Y.S. Cho, B. Kahng, D. Kim; Phys. Rev. E 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”
• Araujo, Herrmann, Phys. Rev. Lett. 105, 035701 (2010).“Explosive percolation via control of the largest cluster”
More “Explosive percolation” ...• D’Souza, Mitzenmacher, Phys. Rev. Lett. 104, 195702 (2010).
“Local cluster aggregation models of explosive percolation”.
• Kumar Pan, Kivela, Jari Saramaki, Kaski, Kertesz, Phys. Rev. E 83, 046112 (2011).“Using explosive percolation in analysis of real-world networks”
• Araujo, Andrade Jr, Ziff, Herrmann, Phys. Rev. Lett. 106, 095703 (2011).“Tricritical point in explosive percolation”
• Chen, R.D., Phys. Rev. Lett. 106, 115701 (2011)“Explosive percolation with multiple giant components”
• Hooyberghs, Van Schaeybroeck, Phys. Rev. E 83, 032101 (2011)“Criterion for explosive percolation transitions on complex networks”
• Gomez-Gardenes, Gomez, Arenas, Moreno, Phys. Rev. Lett. in press. ∗
“Explosive Synchronization Transitions in Scale-free Networks”
• Cho, Khang“Explosive percolation transitions in diffusion-limited cluster aggregation model”
• Bastas, Kosmidis, Argyrakis“Explosive site percolation and finite size hysteresis”
• . . .(∗Percolations talk session)
Beyond “Product Rule”: Models with fixed choice
• Nothing special about two edges; need a fixed number greater than one.(An “Achlioptas process”: examine fixed number of edges, add the onethat optimizes a pre-set criteria.)
• “Sum rule”, Adjacent edge, Triangle rule, k-clique rule, etc., all also work.
• These rules keep largest components similar in size in subcritical regime:– “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
Puzzle 1: Hybrid Transitions!
(Discontinuous change, but scaling behavior)
• Component density ni ∼ i−τ
• “Susceptibility”, W ∼ |t− tc|−α
• Second largest, C2 ∼ |t− tc|−µ
PR AE TRtc 0.888 0.796 0.848τ 2.1∗ 2.1 2.1α 1.17 1.13 1.13µ 1.17 1.13 1.13
∗ Also seen in Radicchi, Fortunato, PRE (2010); Cho, Kahng, Kim, PRE, (2010),Manna, Chatterjee Physica A (2011).
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
Puzzle 2: “Weakly” discontinuousRather than ∆ (the number of edges in the scaling window),measure: the maximum impact from adding one single edge.
• C1 := the fraction of nodes in the largest component.
• ∆C1 := largest change in C1 due to addition of a single edge.
Author's personal copy
180 S.S. Manna, A. Chatterjee / Physica A 390 (2011) 177–182
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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, !)with link density r for! = 1/2, 1/4, 0, !0.1, !1/4, !1/2, !1, !2, !3, !4 and!5,! values decreasingfrom left to right, for a random graph of N = 4096. (b) The asymptotic values of the gap "(!), the percolation threshold rc(!) ! 1/2 and the largest jumpg(!) of the order parameter plotted with !.
For our problem the asymptotic jump g(!) = limN"# #Cm(!,N) for different ! are plotted in Fig. 3(b). For ordinarybond percolation g(! = 0,N) " 0 as Ndf /2!1, with df = 91/48, the fractal dimension of the incipient infinite percolationcluster [3]. However for ! < 0 the g(!) jumps to 0.16 at ! = !0.05 and then gradually increases to 1/2 as ! " !#. Thepercolation thresholds pc(N) are estimated by the average values of rm and t1 giving approximately the same asymptoticvalue for pc(!). The pc(0) = limN"# pc(0,N) " 1/2 limit is approached as N!1/2$ , $ = 4/3 [3] being the correlationlength exponent for the ordinary percolation. On the other hand for ! < 0, pc(!) increased continuously with ! (Fig. 3(b))and tends to unity as ! " !#.
In Fig. 5(a) we show C(r, !) vs. r for a random graph. The last five curves for ! $ !1 almost coincide with the verticalline at r = 1 implying first order transition with rc = 1 in this range. Curves are smoother for !1 < ! < 0. In Fig. 5(b)we show that rc(!) tends to 1/2 rapidly as ! " 0. The jump g(!) is almost zero for !1/4 $ ! $ 0 but then it rapidlyincreases and tends to 1/2 as ! " !#. The gap "(!) is almost zero for ! $ !1/2 and then slowly increases to % 0.193for a random graph. This result indicates that for random graph !c = !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 ! = !1. The individualdistributions have strong dependence on #p = pc ! p but not significantly on L. The best scaling form is P(s, p)#p!% &G(s#p& ) with % = & = 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 ! < !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.
∆C1 ∼ n−0.065
• Nagler, Levina, Timme, Nature Phys. 7, 265 (2011). (Nagler poster)
• Manna, Chatterjee Physica A 390 (2011).
(From Manna, Chatterjee)
Product Rule
The truth: In limit n→∞, the Product Rule is continuous!
• da Costa, Dorogovtsev, Goltsev, Mendes, Phys. Rev. Lett. 105, (2010).∗
Assume component density scales at critical point, ni ∼ i−δ, and then canshow ultimately continuous but with extremely slow decay.
For n = 1018 still see ∆C1 = 0.1 from a single edge (10% of system size!).(PR at tc, ∆C = 0.1 for 109, 0.07 for 1011.)
• Riordan, Warnke, arXiv:1102.5306:Rigorous proof: Any fixed choice process ultimately continuous!∆ will ultimately crossover to linear in n, but no estimate of crossover.
• Grassberger, Christensen, Bizhani, Son, Paczuski, arXiv:1103.3728∗
Unusual finite size behavior
• Lee, Kim, Park, arXiv:1103.4439Finite size scaling data collapse.
• Are any real social or technologial networks of size n ∼ 1018?(100 billion = 1011)
(*See Percolations talks)
The search for a truly discontinuous percolation transitionFriedman, Landsberg PRL (2009); Rozenfeld, et. al. EPJB (2010);
Nagler, Levina, Timme, Nature Phys. (2011)
A deterministic model
(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 emerges
(Giant emerges when only one component remains)
Re-visiting the Bohman Frieze Wormald model (BFW)(Random Structures & Algorithms, 25(4):432-449, (2004))
• A stochastic model, which exams a single-edge at a time.(Not a model with choice or edge competition).
• Like deterministic, start with n isolated vertices, and stage k = 2.
• Sample edges uniformly at random fromthe complete graph on n nodes.
• Can simply reject edges but in phase kmust accept at leastg(k) = 1/2 + (2k)−1/2
fraction of sampled edges.
• 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].– (tc < 1: an infinite number of components exist at transition.)
g(k)→ 1/2
The BFW model in words
• Start with n isolated vertices, and cap on maximum component set to k = 2.
• Sample an edge uniformly at random from the complete graph on n nodes,and examine the result of adding the edge:
1. If the resulting component size ≤ k, accept the edge.
2. Otherwise reject that edge if possible (meaning the fraction of acceptededges remains ≥ g(k)).
3. Else augment k → k+1, and repeat steps (1) and (2), with (3) if necessary.(Recall augmenting k decreases g(k)).
The BFW model stated formally
• Initially n isolated nodes with cap on maximum size set to k = 2.• Let u denote the total number of edges sampled• A the set of accepted edges (initially A = ∅)• t = |A| the number of accepted edges.
At each step u, select edge eu uniformly at random from complete graph, andapply the following loop:
Set l = maximum size component in A ∪ {eu}if (l ≤ k) {
A← A ∪ {eu}u← u+ 1 }
else if (t/u < g(k)) { k ← k + 1 }else { u← u+ 1 }
• If the edge eu is troubling and t/u < g(k), augment k repeatedly until either:(i) k increases sufficiently that eu is accepted or(ii) g(k) decreases sufficiently that eu is rejected.
Simultaneous emergence of multiple stable giantsin a strongly discontinuous transition(Wei Chen and R.D. Phys. Rev. Lett. 83 (2011).)
• Two stable giants!(C1 = 0.570, C2 = 0.405.)
– Fraction of internal cluster edges > 1/2.
– (If restrict to sampling only edges thatspan clusters, only one giant ultimately.)
“Strongly” discontinuous(gap independent of n)
∆Cmax ≈ 0.165
Tuning the number of stable giants(Wei Chen and R.D. Phys. Rev. Lett. 83 (2011).)
• Now let g(k) = α+ (2k)−1/2. Smaller α more edges can be rejected.
α determines number of stable giants!
• Multiple stable giants, not anticipated.(“uniqueness of the giant component” / gravitational coalescence)
• Applications for multiple giants? (Communications, epidemiology, buildingblocks for modular networks, polymerization (Krapivsky, Ben-Naim)...)
More generally, Discrete jump:multiple giants coexist in critical window
• Note, like Nagler, Manna, da Costa, etc, we define as the critical pointtc, the single edge who’s addition causes the biggest change, ∆Cmax.
(Recall C1 is the fraction of nodes in the largest component.)
• If ∆C1 > 0 (i.e. if we see a discrete jump) then there necessarily existedanother macroscopic component. e.g. If ∆Cmax = 0.1 (as in da Costa modelwith n = 1018), that means C1 merged with a component of size |Cj| = 0.1n.
• So, tc is just beyond the “post-critical”regime for Erdos-Renyi.
Deriving the underlying mechanism:Slow decay of g(k) leads to growth by overtaking
(Wei Chen and R.D, arxiv Fri/Mon,http://mae.ucdavis.edu/dsouza/Pubs/bfw.pdf)
• Instead of g(k) = 1/2 + (2k)−1/2 now let g(k) = 1/2 + (2k)−β
• Proceedure: analyze by how much k must grow before g(k) would decreasesufficiently to reject troubling edge.
• For β ∈ (0.5, 1], an increase in k ∼ nβ is always sufficient to reject atroubling edge. Slow increase in k means:– Growth by overtaking∗: two smaller components merge becoming new C1.– Multiple components of size O(n) before the largest jump.
• For β > 1, once stage k = n1/β, an increase in k = C1 would be needed,allowing C1 to even double (the max increase possible). Thus once k =n1/β troubling edges must be accepted at times, leading to large directgrowth of C1, and a weakly discontinuous transition.
∗ Consistent with Nagler, et. al., Nature Phys (2011), for direct growth forbidden.
S(n), Average size of direct growth in C1
For β > 1, once k = n1/β direct growth merging with size O(n).
β = 0.5
Evolution of component density for BFWβ = 0.5 β = 2.0
• For β = 0.5 no scaling. Separates into components of size O(n) and < log(n).
• For β = 0.5 and β = 2.0 no finite size effects in the location of the “hump” (inset), unlikefor PR where location depends on n. (c.f. Lee, Kim, Park: data collapse)
• No scaling, no “early warning signs” (Scheffer, et. al. Nature (2009).
A rigorous proof of a discontinuous transition fora variant on Erdos-Renyi
Panagiotou, Spohel, Steger, Thomas, arxiv:1104.1309
• An Erdos-Renyi-like evolution, but at every step connect twovertices, one chosen randomly from all vertices, and onechosen randomly from a restricted set of vertices.
Does EP apply to any real physical processor provide useful applications?
• Growth of wikipedia in 5 different languages:disconnected topical clusters merging together over a few days.(G. Bounova)
• Evolution of protein homology network: evolution of spanning clusterRozenfeld, Gallos, Makse; Eur. Phys. J. B, 75, 305-310, 2010.
• Mapping sub-critical clusters to community structure in real social networksKumar Pan, Kivela, Jari Saramaki, Kaski, Kertesz,Phys. Rev. E 83, 046112 (2011).
• Applications to network discovery (with Z. Toroczkai)
• Multiple giants...
From multiple giants to interacting networks
Networks:
TransportationNetworks/Power grid(distribution/collection networks)
Biological networks- protein interaction- genetic regulation- drug design
Computernetworks
Social networks- Immunology- Information- Commerce
Sandpile cascades on interacting networksC. Brummitt, R. D’Souza, E. A. Leicht (arxiv on Friday)
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Balancing the benefits and detriments of interconnectivity.
Explosive percolation in random graphs – Conclusions
• Models with fixed choice are “weakly” discontinuous (gap decays slowlywith system size)... Is there a better name?
• In the true n → ∞ limit, they are in fact continuous, but the crossover maybe greater than 1018. In what regime to real-world networks exist?
• Truly discontinuous percolation transitions do exist(BFW, restricted Erdos-Renyi).
• Multiple giant components necessarily exist before the discrete jump(and, c.f. BFW, can co-exist in supercritical region).
Thanks to
• Collaborators: D. Achlioptas, J. Spencer, M. Mitzenmacher, W. Chen
• Funding Agencies:– Army Research Laboratory under Cooperative Agreement Number W911NF-09-2-0053.– Defense Threat Reduction Agency, Basic Research Award No. HDTRA1-10-1-0088.– National Academies Keck Futures Initiative.
• To you for your attention.
Potential postdoctoral opportunities: [email protected]
