arXiv:1308.4301v1 [nlin.AO] 16 Aug 2013 vra xeddpro.Ec acd a h ffc of effect the has cascade repeatedly Each ex- occur be cascades period. may which extended behaviour in an similar over system that a show by to breakdown. hibited aim complete our bundles to close is in in and It events [5] tension snapping under discovered small of fibres was of bursts of behaviour of that of distribution to type the law This power different large distri- cascades. of a size distribution follows the cascade cascades when in occurs crossover Crossover law butions. power of nomenon diverges. size toward cascade criti- rapidly expected this increasingly the moves approaches and infinity, cut–off system is the the distribution point, the As cal which power-law. at pure instability ability exists a of inherent there level its systems, may critical or infinite cut–off uni- a For system the as the cascade. of the known of origin propagate size phenomenon physical physical a the The – be often 9]. details but [2, system, fine cut–off versality the the its of and nature on the exponent, not on law depend power 9]. will [2, the point cut–off” of “exponential values an The as to referred power probabil- density, the decaying ity exponentially of a an end over by the marked big, appears with region law magnitude very often of the scaling orders mathemati- to of Such number small the scaling. very and power-law the events, is links such distri- which of the property of sizes cal feature the common [2], of A events fires bution [6–8]. Such landslides forest ropes and snapping [5] [1], [4], reactions catastrophe. chain failures nuclear [3], a network avalanches in small electrical a end include with begin and which disturbance phenomena of occurring, rally ∗ [email protected] nti oki soramt netgt h phe- the investigate to aim our is it work this In natu- and man–made both examples, many are There ASnmes 56.b 46.d 25.r 89.90.+n 02.50.-r, 64.60.fd, a 05.65.+b, before numbers: exponents PACS res t the a over determine as performed analytically that is to average show able We an are We when region. distribution s subcritical cascade cascade the the mean to the p and systems pure power, models, large propagation cascading of many level with critical common a In determi occur. they properties Casca when distributional occurs. with cascade process driftin a branching upward time each an discontinuously follows drops susceptibi power and mean propagation the the or that repres connectivity, assume could internal number its a system, such a Physically, propagate. cascades hc scpbeo xeinigcsaigfiue norm the our number, In single a failure. by cascading described is experiencing cascades of capable is which epooeamdlwihepan o oe–a rsoe be crossover power–law how explains which model a propose We .INTRODUCTION I. eateto ahmtc,Uiest fPrsot,P13 PO1 Portsmouth, of University Mathematics, of Department rsoe eaiu nDie Cascades Driven In Behaviour Crossover Dtd etme 2 2018) 12, September (Dated: ae Burridge James rpgto power propagation edtrie ytopoess First, processes. two by determined be ecie yBona os 1]s hti h absence the in that so cascades: [13] of noise unpredictability, Brownian of element by an described with but time, over crease where tnadWee rcs 1] and [13], process Wiener standard a eedo h bouevlm ftesse ti an is of it properties – system dynamical the The of volume property. absolute intensive not will the eas- it how on system of of depend parts measure Because between a propagate failure. as cascades to power ily propagation system the a view of prox- we average components the the or system, of den- a imity the in nodes, material unstable unstable by of of sity network determined a be of might connectivity quantity the this terms, physical In ecie yasnl number, be single will a which cascade, by a described propagate to system the of ability phe- cascading real of features key nomena. and some tractable, preserves mathematically is that that explanation) cascades, (and driven is model of approach possible simplest Our the explanation arise. produce might to generic behaviour a crossover forward how physical puts of particular it one rather to but tied system, not is [12]. evidence model distributions some size our However, landslide is in there behaviour fact, crossover terrain in In of effect similar landslides. a to see to susceptible expect vaccinated) transmission also might (or preventing We firewalls recovered [14]. population as of a act presence through who the individuals, disease by of reduced spread exam- is the Another reduced where [11]. been is areas has limited ple where fires of fires burning large forest planned of by to frequency prone and regions extent is in the effect example, stabilizing for a subsequent Such seen, that freely. so less system propagate the cascades of stability the increasing h udmna uniyo neett swl ethe be will us to interest of quantity fundamental The e ytevleo h rpgto power propagation the of value the by ned iyo t opnn at ofiue We failure. to parts component its of lity ∗ > µ z iegs hsdvrec constrains divergence This diverges. ize dltessetblt ftesse to system the of susceptibility the odel n h est fusal aeilin material unstable of density the ent e r ecie yacniuu state continuous a by described are des datrtecrossover. the after nd rwinmto ewe cascades, between motion Brownian g edsrbto fpoaainpower. propagation of distribution he hc esrstees ihwhich with ease the measures which , l,cosvrbhvorapasin appears behaviour crossover ult, stema aeo nraeof increase of rate mean the is 0 wrlwbhvori xiie at exhibited is behaviour law ower aiu a rs nasystem a in arise can haviour dp F ntdKingdom. United HF, ( t = ) µdt + p σdW h rpgto power. propagation the , ( t > σ (1) ) p iltn oin- to tend will otosthe controls 0 p , W p ( t will is ) 2 magnitude of the noise. The physical origin of such a divergence of the mean cascade size as the critical point is process in, for example, a forest fire model might be the approached from below will create a self–stabilizing effect drying out of vegetation due to unpredictable weather, or which pushes the system away from pc. The combina- in a landslide model, the natural variability of pore water tion of this automatic stabilization and the upward drift pressure. We assume that the magnitude of the noise is of the driving process means that the propagation power independent of system size, and therefore it represents will fluctuate about a mean value lying in the subcritical an external driving process. region. We will find that the magnitude of fluctuations The second influence on p arises from the cascades is controlled largely by σ for large systems. themselves, which act to stabilize the system. To cap- Of central interest to us is the long term cumulative ture this stabilizing effect, we suppose that if the size of record of cascade sizes in the system, which we describe the kth cascade since an arbitrary reference time is given with a probability density function ψ¯(z). This will be an by the continuous random variable Jk, then in response average of ψp(z) over the values of p at which cascades p changes by ǫJk. The parameter ǫ measures the sen- take place. We define f(p,t) to be the probability density sitivity of p to− cascades, and will depend on the size of function for the value of the propagation power at time t. the system, and the stabilizing effect on those parts of The “steady state” density function for p is then f(p) := it that are involved in the cascade. For convenience, we limt→∞ f(p,t). We then have that will refer to ǫ as the “inverse system size”. Note that the distribution of Jk will itself depend on p. We define J(t) ψ¯(z)= f(p)ψp(z)dp. (5) to be the sum of all cascade sizes that have occurred since Z t = 0 and let N count the number that have occurred. t This expression may also be thought of as the probability We then have that: density function for the size of the next cascade observed
Nt after some arbitrary (but large) time. One of our main J(t)= Jk. (2) results is to show analytically that fluctuations in p about k=1 its typical subcritical value, causing it to approach tem- X porarily closer to criticality, are what generates crossover Together with the driving dynamics (1) we have a com- behaviour in the “averaged” cascade distribution ψ¯(z). plete stochastic differential equation for p(t): From the above arguments it is clear that typical value of p, corresponding to the peak of f(p), will be increased dp(t)= µdt + σdW (t)+ ǫ dJ(t). (3) by increasing the driving rate µ. In the limit of infinite system size we will find that the stabilizing effect of di- We will assume that cascades begin when a small part verging mean cascade size near pc means that it is always of the system spontaneously fails. Since a larger system the case that p
X0−z is the Laplace transform of G. Given this definition, X p−kz (z X ) kz−1e p n ∗n 0 0 Φ(s) is the Laplace transform of G where n ψp(z)= − (11) zΓ(kz) 0, 1, 2, 3 ... . By relaxing the constraint that n be an∈ { } X −κz integer, we may define the Laplace transform of X1, con- k −kX + 0 e X e 0 p as z (12) ditional on X , to be E(e−sX1 X )=Φ(s)X0 , and then 0 3/2 0 0 ∼ "r2π # z → ∞ extend this rule to later generations:| where: E(e−sXn+1 X )=Φ(s)Xn−1 . (7) | n 1 kp κ = k ln(kp)+ − . (13) This recursive equation defines the relationships between p the distributions of successive generations, giving a com- In the context of equation (12) the symbol means plete characterisation of the process once G is defined. “tends asymptotically to”. To the author’s knowledge,∼ We will take G to be the Gamma distribution [1, 19, 20] equation (11) is a new result in the theory of continu- Γ(k,θ) which has density function ous state branching processes [25]. Considering equation k−1 − x (12), when the mean of the offspring distribution is one x e θ g(x)= , (8) (kp = 1), then κ = 0, so the distribution is asymptoti- Γ(k)θk 3 cally a pure 2 power law with infinite moments. The − 1 defined for x 0, where the variables k > 0 and θ > 0 value of propagation power pc = k at which this be- are referred to≥ as the shape and scale parameters. Our haviour occurs is referred to as the critical point. When −1 choice of G is motivated by the requirement that it have p
1.0 and W−1 is one of the two real branches of the Lambert W function [21], the other being W0. We will cap cascades at a maximum size 0.8
p 0.6
Z = (18) L max t H
ǫ p 0.4 so that they cannot cause the propagation power to take negative values. This cap scales in proportion with the 0.2 system size. Assuming that cascades will stop abruptly once Z>Z , then the tail of ψ (z) will replaced with a max p 0.0 0 10 20 30 40 50 delta function ω(Zmax)δ(z Zmax) where ω(Zmax) is the t total probability weight in− the tail, plus the probability of an infinite cascade: FIG. 1. Simulation of p(t) using parameter values are ǫ = −5 ∞ 10 , µ = 10 and σ = 0.5 and k = 1 giving a critical p value χ(k,p)X0 ω(Z )= ψ (z)dz + (1 e ). (19) of pc = 1.0. Values of p(t) were recorded when the jump max p − ZZmax which preceded them exceeded 1% of the system size. The capped cascade distribution is therefore: power is y. Note that W (r) is independent of n. The ψ (z; Z ) := 1 (z)ψ (z)+ω(Z )δ(z Z ), y p max [X0,Zmax] p max max master equation governing the evolution of f (p) is then: − (20) n and the nth cascade moment in a finite system is: f (p)= W (r)f (p r)dr. (22) n+1 p−r n − Zmax Z E(Zn) := znψ (z)dz + ω(Z )Zn , (21) p max max Our aim is to find the steady state distribution: ZX0 limn→∞ fn(p). where the second term arises from integration over the Before we continue our analysis of the master equa- delta function in the capped cascade density. For sim- tion we will make clear the link between the distribu- plicity, for the remainder of the paper we will begin all tions of discrete time process, and the underlying con- cascades with a failure of volume X0 = 1. tinuous time process (3). In section I, we defined f(p) to be the steady state probability density function for the continuous process, p(t), which may intuitively be III. THE DISTRIBUTION OF PROPAGATION thought of as the density function for the value of p POWER observed at an arbitrary time T , large enough so that the influence of initial conditions is insignificant. Due For arbitrary ǫ, the steady state distribution of the to the properties of exponential waiting times [22], the propagation power equation (3) cannot be found analyt- time interval, ∆T , between T and the last cascade be- ically, but may be determined by simulation. However, fore T will be exponentially distributed. Letting p∗ be for small ǫ, jumps which are not small with respect to the the value of p immediately after this last cascade, we see system size become increasingly rare, and the dynamics that p(T )= p∗ +µ∆T +σW (∆T ). Now let T ′ be another of p may be approximated by diffusion with drift, giv- large time, but restricted to the set of cascade times so ing rise to a pure diffusion equation which is analytically that we are making an observation of the discrete time tractable. In the limit ǫ 0, this approximation be- process. Again using the properties of exponential wait- → comes exact. To illustrate the rarity of large jumps, Fig- ing times, the time ∆T ′ since the previous cascade will ure 1 shows a simulation of p(t) when ǫ = 10−5, µ = 10, have the same distribution as ∆T (intuitively this arises σ =0.5 and k =1.0. In this caseout of 0.5 107 cascades because T is likely to lie in a larger than average waiting × only six exceeded 10% of the system size, and the system interval). Because the value of p just after the previous reached criticality only once out of all recorded times. cascade will be drawn from the same distribution as p∗, To derive our diffusion approximation we view p as we have that p(T ) =d p(T ′) where =d denotes equality evolving in discrete steps, with its value being recorded in distribution. As a result, observations of the discrete immediately before each cascade. Each change in p is time and the continuous time process have the same dis- therefore comprised of a cascade, followed by a period tribution at large times: of diffusion until the next cascade occurs. This defines a discrete time stochastic process p ,p ,p ... where p lim fn(p)= f(p). (23) 1 2 3 n n→∞ is the propagation power immediately before the the nth cascade. We let fn(p) be the probability density func- If the times between cascades were not exponentially dis- tion for pn. We then define Wy(r) to be the probability tributed then the probability distribution of the state of density function for the size, r, of the next single step of the system immediately preceding a cascade would not, the process given that the current value of propagation in general, be the same as the distribution at a randomly 5 selected time. In that case equation (5) would be incor- In equation (29) we have retained the ǫ2 term because the rect because ψp(z) must be averaged over the distribution moments of the cascade distribution become very large of p immediately preceding a cascade. as p pc. For any finite system we expect our ap- To approximate the steady state solution to (22) we proximation→ to break down near this critical point, and derive the corresponding Kramers–Moyal equation [23] to breakdown globally if there is sufficient probability by expanding the integrand of (22) in powers of r: weight in the supercritical region where infinite cascades have nonzero probability. We will explore this breakdown
Wp−rfn(p r) Wp(r)fn(p) r∂p Wp(r)fn(p) using simulations. The first moment of the cascade dis- − ≈ − { } tribution has asymptotic behaviour: r2 + ∂pp Wp(r)fn(p) + ... 1 2 { } E(Z) as ǫ 0 (30) ∼ 1 kp → − Substituting this approximation back into the master 1 equation we find that: which diverges near p = pc = k , creating an infinite neg- ative drift. The probability weight at the critical point therefore declines to zero as ǫ 0 and the divergence → fn+1(p)= fn(p) ∂p fn(p) rWp(r)dr in B(p) will never be realised. We may therefore drop − 2 Z the ǫ term for infinite systems. Making use of A(p) and 1 the simplified B(p) we see from equation (27) that the + ∂ f (p) r2W (r)dr + ... 2 pp n p limiting form of f(p) satisfies: Z 1 σ2 where we have made use of the normalisation of the step µ f(p)= f ′(p). (31) − 1 kp 2 size distribution to simplify the first term. We now define − the first two moments of step size to be: Although the cascade distribution ψp(z) is not defined for p < 0, in the interests of tractability, we will take A(p) := rW (r)dr (24) equation (31) as valid over the interval [ , 0], yielding p −∞ Z the following expression for f(p): 2 2 B(p) := r Wp(r)dr. (25) 2µ kσ2 kµ kσ2 2 2µ Z kσ2 f(p)= 2 (1 kp) exp 2 (1 kp) Γ 2 − −kσ − As ǫ 0, both the time between cascades and their kσ (32) effect→ on p decline so the step distribution becomes in- For all parameter values of interest to us, the probability creasingly sharply peaked about r = 0. We therefore weight in the invalid region p [ , 0] is less than 10−12. ignore moments of higher order than two and obtain a Note that this solution is independent∈ −∞ of ǫ, but we expect discrete time analogue of the Fokker–Planck equation for it to become an increasingly good approximation to the pure diffusion: true solution as ǫ 0 for values of p
We have extended the lower limit of integration to 10-8
−∞ L
for tractability, since the integrand will be negligible z H when p< 0. The integral (35) is not tractable. However, Ψ 10-12 as z , the weight of the integrand becomes concen- → ∞ trated in a shrinking neighborhood of the critical point. -16 We may therefore approximate the integral asymptoti- 10 cally by replacing the first exponent in the integrand with 10-20 1 4 6 8 its Taylor expansion to quadratic order about pc = k : 1 100 10 10 10 z 1 1 κz k 1+(1 kp) (z 2)(1 kp)2 . (37) p − ≈ − − 2 − − FIG. 2. Log–log plot of the mean cascade distributions (dashed lines) ψ¯(z) in the limit ǫ → 0 when k = 2, σ = 0.5 Making the change of variables s =1 kp, our approxi- − and µ takes the values 10 and 60. The larger µ value produces mation becomes: a crossover point at a higher value of z. Also shown are the ∞ asymptotic power–law predictions (42) (solid lines) and the 2 k 3 k 3 ¯ − 2 α −(αµ−k)s− 2 (z−2)s − ψ(z) z s e ds. (38) function z 2 as a dotted line. ∼C 2 r Z0
We now make a second change of variables: 1 (z 2)s2 = t2, (39) − 0.001 which gives: 10-6 1+α
2 L 3 k − 2 z 10-9 ¯ 2 H
ψ(z) z Ψ ∼Cr 2 k(z 2) − 1 10-12 ∞ 2 2 2 tαe−t exp (αµ k)t dt. (40) × − k(z 2) − 10-15 Z0 " − # 10-18 We may extract the asymptotic behaviour of this integral 1 100 104 106 108 by noting that, for finite t, as z the z dependent ex- z ponential term tends to one. We→ now ∞ note that provided α> 1: FIG. 3. Log–log plot of the mean cascade distributions ∞ (dashed lines) ψ¯(z) in the limit ǫ → 0 when k = 2, µ = 40 and 2 1 1+ α tαe−t dt = Γ . (41) σ takes the values 0.5 and 0.8. The larger σ value produces a 2 2 Z0 shallower tail in the cascade distribution. Also shown are the asymptotic power–law predictions (42) (solid lines) and the The asymptotic behaviour of the mean cascade distribu- − 3 tion is therefore a pure power law: function z 2 as a dotted line.
−2− 1 ψ¯(z) z kσ2 (42) ∼ A V. SIMULATION RESULTS where α 1 2 2 1+ α Using simulations we will now test the validity of our = Γ . (43) A 2 k 2 C large system crossover predictions, explore the influence of ǫ and the effect of the model parameters on the aver- So, in the limit of large system size, the tail of the mean aged cascade distribution, ψ¯(z). cascade distribution, rather than being exponential, fol- lows a power law with an exponent which is an increasing function of the variance of the destabilisation process. For smaller z, when most of the probability weight in A. Techniques of simulation f(p) corresponds to cut-offs at larger z values, we see − 3 z 2 behaviour preserved. Together with our asymptotic We make use of two simulation methods; a “naive” predictions this gives rise to a power–law crossover. The technique where every cascade is simulated, and an “ac- theoretical distribution ψ¯(z), in the limit ǫ 0, is illus- celerated” technique which uses a diffusion approxima- trated in Figures 2 and 3 together with the→ asymptotic tion when the probability of jumps of significant size in result (42). comparison to ǫ−1 is sufficiently small. 7
1 0.01 0.001
10-6 10-6
-10 L L 10
z -9 10 z H H Ψ Ψ
-14 10-12 10
10-15 10-18
10-18 -22 4 5 6 10 1 10 100 1000 10 10 10 1 100 104 106 z z
FIG. 4. The circles show the simulated cascade distribution FIG. 5. The circles show the simulated cascade distribution −6 − ψ¯(z) when ǫ = 10 , µ = 10, σ = 1.5 and k = 0.1. The results ψ¯(z) when ǫ = 10 9, µ = 10, σ = 1.5 and k = 0.1. The results were obtained by simulating the propagation power process 9 were obtained by simulating the propagation power process over 10 cascades. The black line shows the theoretical cas- using the accelerated technique over 109 time steps. The black cade distribution in the limit ǫ → 0. line shows the theoretical cascade distribution in the limit ǫ → 0. The dashed lines are pure power laws. The shallow − 3 gradient line has exponent 2 , whereas the steep gradient 1. Naive simulation line is our asymptotic prediction (42). The dotted line shows ψp(z) when p is equal to its mean value. Note the presence The simplest method to determine the averaged cas- of the exponential cut–off. cade distribution is to simulate the stochastic process de- scribed by equation (22). The results (shown in Figure 4) were obtained by simulating 109 cascades, and recording region where the quality of the diffusion approximation their sizes in bins of increasing width. It is clear from the is uncertain. We refer to the sum total of long (δt) and figure that to fully examine the tail behaviour of the dis- short (mean length ǫ) time steps to be the number of tribution would require us to simulate significantly more steps in the simulation. Provided δt >> ǫ, then, because cascades. This would be prohibitively time consuming so the system spends most of its time in the region p
1 size of the region over which the crossover occurs.
0.001
10-6 VI. CONCLUSIONS L
z 10-9 H Ψ We have presented a driven cascade model which, 10-12 in the limit of large system size, exhibits power–law crossover behaviour in its cascade size distribution. For -15 10 smaller systems, the crossover partially develops, but the distribution moves back toward the initial 3 power law 10-18 2 1 10 100 1000 104 105 106 at larger cascade sizes, because the system is− able to reach z and exceed the critical value of propagation power. The mechanism which generates the crossover is a com- FIG. 6. The open circles show the simulated cascade distribu- − tion ψ¯(z) when ǫ = 10 5, µ = 15, σ = 0.35 and k = 1.5. The petition between the driving process, which increases the filled circles show the simulated cascade distribution when instability of the system, and the cascade process, which − ǫ = 10 7 and all other parameters are the same. The results reduces it. As the propagation power nears the critical were obtained by simulating the propagation power process point, pc, the mean cascade size diverges, so that in the 9 using the accelerated technique over 10 time steps. The solid limit of large system size, pc is not accessible for finite µ. line shows the theoretical cascade distribution in the limit The upper tail of the distribution of propagation power, → ǫ 0. f(p), therefore decays to zero at pc. The asymptotic behaviour of the averaged cascade distribution, ψ¯(z), is determined by averaging the asymptotic behaviour of the the exponent begins to increase again. This occurs be- cascade distribution for fixed propagation power, ψp(z), cause the distribution of propagation power does not de- over the upper tail of f(p). The result is a power–law cay to zero at the critical point, so the averaged cascade with exponent lower than 3 , producing a crossover. distribution includes significant contributions from val- − 2 We suggest that the presence of a crossover in a cas- ues of p for which ψ (z) is approximately a pure power p cade size distribution may indicate that the system is able law for z<ǫ−5. The breakdown of the diffusion approx- to self stabilize through frequent but non–catastrophic imation occurs in part because the frequency of cascades cascades, and that the mechanism which drives the in- is insufficient to realise their divergent mean size on short stability of the system to cascading failure is inherently time scales. In the larger system where ǫ = 10−7 we see noisy. The stabilizing cascades act to prevent the system that the crossover develops more fully. Because cascades from reaching a fully critical state where very large cas- occur with greater frequency, fluctuations in the short cades can occur. Because our system will be near critical term average of the cascade size are reduced. for a wide range of parameter values (large enough µ), it may be considered to exhibit “Self Organised Critical- ity”. However, the crossover is more obvious if the typical C. The influence of µ and σ value of propagation power is not too close to criticality, and it has been our focus to explore crossover. It should In Figures 2 and 3, we illustrate the role of the parame- also be noted that an infinite system will only become ters µ and σ in determining the behaviour of the cascade fully critical in the limit µ , whereas a finite system distribution ψ¯(z) in the limit of large system size. Fig- may become critical or supercritical→ ∞ through fluctuations. ure 2 shows that for fixed σ, the driving rate µ, which The location of the crossover indicates how close to crit- determines the location of the maximum of f(p), fixes icality the system will typically be found, because it is the location of the crossover. At larger driving rates, determined by the peak of f(p). It remains to adapt the the crossover point shifts to larger cascade sizes. This is ideas contained in our simple model to investigate real because, for larger µ, the peak of f(p) is nearer to the physical systems such as forest fires and landslides. critical point. Therefore the typical exponential cut–off 3 scale is larger and the 2 scaling region is extended. In Figure 3 we see that− a noisier driving process reduces 3 Appendix A: Cascade distribution for Gamma the magnitude of the tail exponent, but the size of the 2 region remains unaffected. Increased variability in− the branching process driving process means that although the typical distance from the critical point is unchanged, the system spends The purpose of this appendix is to derive the proba- more time in close proximity to it and therefore large bility density function for the total cascade size in the cut–offs are more heavily weighted. continuous state branching process defined by relation- In both Figures the asymptotic power–law predictions ship (10), and also to determine the probability of an (42) for the cascade distribution give an indication of the infinite size cascade. 9
a. Mapping to a first passage problem We begin by noting that the negative binomial distri- bution, which has probability mass function: We begin by showing that the our problem may be Γ(n + r) interpreted as a first passage time problem. Consider b(n, r, q)= (1 q)rqn (A6) n!Γ(r) − the branching process X0,X1,X2,... defined by the re- lationship Xn Γ(kXn−1,θ) with X0 given. In order to provides an arbitrarily close discrete approximation to ∼ ∞ calculate the distribution of Z = k=0 Xk, we show that the gamma distribution for appropriate choice of the pa- Z may be viewed as the first passage time of a stochastic rameters r and q. The approximation is set up in the process through the origin. WeP first give the definition following way. We divide [0, ] into a discrete lattice ∞ of the “Gamma process” [25], which takes place in con- of constant spacing δ, and let Xδ be a discrete random tinuous time on [0, ]. If St is a Gamma process with variable which approximates X Γ(k,θ). Let X have ∞ δ parameters k and θ then: the probability mass function: ∼ P 1. S0 = 0. (Xδ = nδ)= b(n, r, q). (A7) E δqr 2. It has independent increments, in the sense that for The mean and variance of Xδ are then (Xδ)= 1−q and 2 any 0 t0 < t1 <... 3. Ss+t Ss Γ(kt,θ). kθ − ∼ r∗ = (A8) θ δ We now define a new stochastic process Qt = X0 +St t, − − ∗ δ where X0 is the size of the first generation of our branch- q =1 . (A9) n − θ ing process. By defining Zn := k=0 Xk to be the cumu- lative cascade size up to the nth step, we may show that With these choices of r and q, in the limit δ 0 the P → the processes Qt and Xn have the following relationship: discrete distribution converges to Γ(k,θ) in the following sense: d Xn+1 = QZn (A1) k−1 − x 1 x e θ lim b [ x/δ , r∗, q∗]= , (A10) provided that the cascade has not ended for some k c. First passage time of the discrete process limit. We find expressions for the moments of the cascade size using a similar method. Using a martingale method Having shown how to construct the discrete state ran- we determine the probability that the cascade is of finite dom walk, we now solve the first passage problem using size in the supercritical case. generating functions [4] and the Lagrange inversion for- We obtain the continuum cascade density function, mula [26]. which we will call ψ(z), by setting n = z/δ and m = X0/δ P Let m = X /δ and Z (m) be the first passage time and then taking the limit δ 0 of Zδ(m)= n : ⌊ 0 ⌋ δ → { } of the walk (A13), starting from position m. Considering 1 the first step, which will have size A 1, we have that: ψ(z) = lim P Zδ(m)= n (A25) − δ→0 δ { } X −z −kz kz−1 0 Zδ(m)=1+ Zδ(m + A 1). (A14) X θ (z X ) e θ − = 0 − 0 . (A26) Since the time for the walk to get from position m to the zΓ(kz) origin is equal to the time it takes to get to position 1 The asymptotic properties of ψ(z) may be determined plus the time to get from position 1 to the origin, then d by making use of Stirling’s approximation: Γ(z + 1) Zδ(m) = Zδ(m 1) + Zδ(1). The quantity Zδ(m) is z ∼ − √2πz z . The result is: therefore the sum of m independent copies of Zδ(1). We e have from (A14) that: X −κz k −kX + 0 e ψ(z) X e 0 θ as z (A27) Z (1)= 1+ Z (A). (A15) 0 3/2 δ δ ∼ "r2π # z → ∞ If H(s) and F (s) are the probability generating functions where for Zδ(1) and A, then from equation (A15) we have 1 kθ E 1+Zδ (A) κ = k ln(kθ)+ − . (A28) H(s)= (s ) (A16) θ = sE[E(sZδ (A) A)] (A17) | Provided kθ < 1, the distribution ψ(z) is normalised = sE[(H(s))A] (A18) and its moments are defined. It is useful to have explicit = sF (H(s)). (A19) expressions for the first two moments of ψ(z) in this case. We may compute the moments of the (discrete) distribu- From the negative binomial mass function we have that: tion of Zδ(1) by differentiating the generating function re- ′ ∗ ∞ δr lationship: H(s)= sF (H(s)), and then solving for H (s) ∗ ′′ n ∗ ∗ 1 q and H (s). Using the expression for F (s), together with F (s)= s b(n,δr , q )= − . (A20) 1 q∗s the fact that when kθ < 1, H(1) = F (1) = 1, we find n=0 X − that: We are interested in the probability generating function m n δ of Zδ(m), which is just H (s). The coefficient of s E(Z (1)) = (A29) δ 1 kθ in this function may be determined using the Lagrange − inversion formula [26]: δkθ2 Var(Z (1)) = . (A30) δ (1 kθ)3 1 d − [sn]Hm(s)= [Hn−1] Hm F n(H) (A21) n dH The total cascade size, Z, has the same distribution −1 E m n−m n as the sum of X0δ copies of Zδ(1), so (Z) = = [H ]F (H) (A22) −1E 2 −1 n X0δ (Zδ(1)) and Var(Z )= X0δ Var(Zδ(1)), yield- ∗ m Γ(n(1 + δr ) m) ∗ ing the first two moments of the cascade distribution in = − (1 q∗)δr n(q∗)n−m n Γ(nδr∗)Γ(n m + 1) − exact form: − (A23) X E(Z)= 0 (A31) = P Z (m)= n (A24) 1 kp { δ } − X2(1 kθ)+ X kθ2 where the notation [xn]f(x) stands for the coefficient E 2 0 0 (Z )= − 3 . (A32) n (1 kθ) of x in the Taylor series of f(x). We now have the − probability mass function for the cascade size in the dis- Numerical integration of the exact distribution (A26) re- crete branching process which approximates the contin- veals that it is not normalized when kθ > 1. This is the uum process that we are interested in. “supercritical” regime. In general the total probability weight is equal to P Z < , which is less than one in the supercritical case{ because∞} there is a non–zero proba- d. Continuum limit of the discrete process bility of seeing an infinite cascade. We may deduce this probability by considering the stochastic process: Now that we have the solution to the discrete problem −qQt we solve the continuous problem by taking the continuum Mt = e . (A33) 11 Taking the expectation value of this process, conditional the origin, then we have that on its value at t = 0 we find that: ∗ E(M )= P(Z < )= e−q X0 . (A36) Z ∞ E(M )= e−qX0+t(q−k ln(1+qθ)). (A34) t This gives the result presented in equation (21). If we let q be the solution to the equation q k ln(1+qθ)= − 0 then this expectation will be independent of time. The ACKNOWLEDGMENTS value of q which solves this equation is: The author would like to acknowledge Murad Banaji, ∗ 1 1 − 1 q = kW e kθ 0. 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