Cascade models

a.k.a. cluster distributions, Galton-Watson models

October 11, 2019 β€” August 4, 2021

Figure 1

\(\newcommand{\rv}[1]{\mathsf{#1}}\)

Models for, loosely, the total population size arising from all generations the offspring of some progenitor.

Let us suppose that each individual \(i\) who catches a certain strain of influenza will go on to infect a further \(\rv{n}_i\sim F\) others. Assume the population is infinite, that no one catches influenza twice and that the number of transmission of the disease is distributed the same for everyone who catches it. How many people, will ultimately catch the influenza, starting from one infected person?

The Galton-Watson version of this model considers this byu generation; We write $(k)=_{i k} _i $ for the number of people infected in the \(k\)th generation. Writing \(F^{*k}\) for the \(k\)-fold convolution of \(F\), we have \[\rv{x}(k) \sim F^{\ast \rv{x}(k-1)}\] The sum over all these \[\sum_k \rv{x}(k)\] is the cascade size.

A type of count model for a Markov stochastic pure-birth branching process.

I say it is a count model, but it turns out there are continuous-state generalisations. See, e.g. (Burridge 2013a, 2013b).

The distribution of subcritical processes are sometimes tedious to calculate, although we can get a nice form for the generating function of a geometric offspring distribution cascade process.

Set \(\frac{1}{\lambda+1}=p\) and \(q=1-p\). We write \(G^{n}\equiv G\cdot G\cdot \dots \cdot G\cdot G\) for the \(n\)-fold composition of \(G\). Then the (non-critical) geometric offspring distribution branching process obeys the identity

\[ 1-G^n(s;\lambda) = \frac{\lambda^n(\lambda-1)(1-s)}{\lambda(\lambda^n-1)(1-s)+\lambda-1} \]

This can get us a formula for the first two factorial moments, and hence the vanilla moments and thus mean and variance etc

More generally the machinery of Lagrangian distributions is all we need to analyse these.

Maybe I should use (Dwass 1969) to get the moments? Dominic Yeo explains beautifully as always.

πŸ— πŸ— πŸ—

1 Lagrangian distributions

A clade of count distributions, which I would call β€œcascade size distribution”. For now, let’s get to the interesting new ones contained in this definition. They are unified to my mind by modelling cascade size of cluster processes. Specifically, if I have a given initial population and a given offspring distribution for some population of… things… a Lagrangian distribution gives me a model for the size of the total population. There are other interpretations of course (queueing is very popular), but this one is extremely useful for me. See (P. C. Consul and Shoukri 1988; P. C. Consul and Famoye 2006 Ch 6.2) for a deep dive on this. They introduce various exponential_families via the pgf, which is powerful and general, although it does obscure a lot of simplicity and basic workaday mathematics where the forms of the mass functions do in fact turn out to be easy.

Terminology: the total cascade size of a subcritical branching process has a β€œdelta Lagrangian” or β€œgeneral Lagrangian” distribution, depending on whether the cluster has, respectively, a deterministic or random starting population. We define the offspring distribution of such a branching process as \(G\sim G_Y(\eta, \alpha)\). Usually we also assume \(EG:=\eta< 1\), because otherwise the cascade size is infinite.

1.1 Borel-Tanner distribution

A delta Lagrangian distribution, the Borel distribution is the distribution of a cascade size starting from a population size of \(k=1\). We can generalize it to \(k>\), in which case it is the Borel-Tanner distribution.

Spelled
\(\operatorname{Borel-Tanner}(k,\eta)\)
Pmf
$$ (X=x;k,)={}{}}
Mean
( $$
Variance
\(\frac{k\eta}{(1-\eta)^3}\)

Note to self: Wikipedia mentions an intriguing-sounding correspondence with random walks, which I should follow up Dwass (1969).

The only R implementation I could find for this is in VGAM, although it is not so complicated.

1.2 Poisson-Poisson Lagrangian

See (P. C. Consul and Famoye 2006 Ch 9.3). Also known as the Generalised Poisson, although there are many things called that.

Spelled
\(\operatorname{GPD}(\mu,\eta)\)
Pmf
\(\mathbb{P}(X=x;\mu,\eta)=\frac{\mu(\mu+ \eta x)^{x-1}}{x!e^{\mu+x\eta}}\)
Mean
\(\frac{\mu}{1-\eta}\)
Variance
\(\frac{\mu}{(1-\eta)^3}\)

Returning to the cascade interpretation: Suppose we have

  • an initial population is distributed \(\operatorname{Poisson}(\mu\))
  • and everyone in the population has a number of offspring distributed \(\operatorname{Poisson}(\eta\)).

Then the total population is distributed as \(\operatorname{GPD}(\mu, \eta)\).

Notice that this can produce long tails, in the sense that it can have a large variance with finite mean, but not heavy tails, in the sense of the variance becoming infinite while retaining a finite mean; both variance and expectation go to infinity together.

Here, I implemented the GPD for you in python. There are versions for R, presumably. A quick search turned up RMKDiscrete and LaplacesDemon.

1.3 General Lagrangian distribution

A larger family of Lagrangian distributions (the largest?) family is summarised in (P. Consul and Shenton 1972), in an unintuitive (for me) way.

One parameter: a differentiable (infinitely differentiable?) function, not necessarily a pgf, \(g: [0,1]\rightarrow \mathbb{R}\) such that \(g(0)\neq 0\text{ and } g(1)=1\). Now we define a pgf \(\psi(s)\) implicitly as the smallest root of the Lagrange transformation \(z=sg(z)\). The paradigmatic example of such a function is \(g:z\mapsto 1βˆ’p+pz\); let’s check how this fella out.

πŸ—

Spelled
?
Pmf
?
Mean
?
Variance
?

2 References

Bacry, and Muzy. 2016. β€œFirst- and Second-Order Statistics Characterization of Hawkes Processes and Non-Parametric Estimation.” IEEE Transactions on Information Theory.
Baldwin, and Freeman. 2022. β€œRisks and Global Supply Chains: What We Know and What We Need to Know.” Annual Review of Economics.
Bowman, and Shenton. 1989. β€œThe Distribution of a Moment Estimator for a Parameter of the Generalized Poision Distribution.” Communications in Partial Differential Equations.
Burridge. 2013a. β€œCascade Sizes in a Branching Process with Gamma Distributed Generations.” arXiv:1304.3741 [Math].
β€”β€”β€”. 2013b. β€œCrossover Behavior in Driven Cascades.” Physical Review E.
Consul, P. C. 2014. β€œLagrange and Related Probability Distributions.” In Wiley StatsRef: Statistics Reference Online.
Consul, P. C., and Famoye. 1992. β€œGeneralized Poisson Regression Model.” Communications in Statistics - Theory and Methods.
β€”β€”β€”. 2006. Lagrangian Probability Distributions.
Consul, P. C., and Felix. 1989. β€œMinimum Variance Unbiased Estimation for the Lagrange Power Series Distributions.” Statistics.
Consul, P., and Shenton. 1972. β€œUse of Lagrange Expansion for Generating Discrete Generalized Probability Distributions.” SIAM Journal on Applied Mathematics.
Consul, P. C., and Shenton. 1973. β€œSome Interesting Properties of Lagrangian Distributions.” Communications in Statistics.
Consul, P.C., and Shoukri. 1984. β€œMaximum Likelihood Estimation for the Generalized Poisson Distribution.” Communications in Statistics - Theory and Methods.
Consul, P.C., and Shoukri. 1988. β€œSome Chance Mechanisms Related to a Generalized Poisson Probability Model.” American Journal of Mathematical and Management Sciences.
Dwass. 1969. β€œThe Total Progeny in a Branching Process and a Related Random Walk.” Journal of Applied Probability.
Elliott, and Golub. 2022. β€œNetworks and Economic Fragility.” Annual Review of Economics.
Haight, and Breuer. 1960. β€œThe Borel-Tanner Distribution.” Biometrika.
HoudrΓ©. 2002. β€œRemarks on Deviation Inequalities for Functions of Infinitely Divisible Random Vectors.” The Annals of Probability.
Imoto. 2016. β€œProperties of Lagrangian Distributions.” Communications in Statistics - Theory and Methods.
JΓ‘nossy, and Messel. 1950. β€œFluctuations of the Electron-Photon Cascade - Moments of the Distribution.” Proceedings of the Physical Society. Section A.
β€”β€”β€”. 1951. β€œInvestigation into the Higher Moments of a Nucleon Cascade.” Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences.
Li, Famoye, and Lee. 2010. β€œOn the Generalized Lagrangian Probability Distributions.” Journal of Probability and Statistical Science.
Messel. 1952. β€œThe Solution of the Fluctuation Problem in Nucleon Cascade Theory: Homogeneous Nuclear Matter.” Proceedings of the Physical Society. Section A.
Messel, and Potts. 1952. β€œNote on the Fluctuation Problem in Cascade Theory.” Proceedings of the Physical Society. Section A.
Mishra, Rizoiu, and Xie. 2016. β€œFeature Driven and Point Process Approaches for Popularity Prediction.” In Proceedings of the 25th ACM International Conference on Information and Knowledge Management. CIKM ’16.
Mutafchiev. 1995. β€œLocal Limit Approximations for Lagrangian Distributions.” Aequationes Mathematicae.
Neyman. 1965. β€œCertain Chance Mechanisms Involving Discrete Distributions.” Sankhyā: The Indian Journal of Statistics, Series A (1961-2002).
Otter. 1948. β€œThe Number of Trees.” Annals of Mathematics.
β€”β€”β€”. 1949. β€œThe Multiplicative Process.” The Annals of Mathematical Statistics.
Pardoux, and Samegni-Kepgnou. 2017. β€œLarge Deviation Principle for Epidemic Models.” Journal of Applied Probability.
Pazsit. 1987. β€œNote on the Calculation of the Variance in Linear Collision Cascades.” Journal of Physics D: Applied Physics.
Pitman. 1998. β€œEnumerations of Trees and Forests Related to Branching Processes and Random Walks.” In Microsurveys in Discrete Probability. DIMACS Series in Discrete Mathematics and Theoretical Computer Science.
Ramakrishnan, and Srinivasan. 1956. β€œA New Approach to the Cascade Theory.” In Proceedings of the Indian Academy of Sciences-Section A.
Rizoiu, Xie, Sanner, et al. 2017. β€œExpecting to Be HIP: Hawkes Intensity Processes for Social Media Popularity.” In World Wide Web 2017, International Conference on. WWW ’17.
Shoukri, and Consul. 1987. β€œSome Chance Mechanisms Generating the Generalized Poisson Probability Models.” In Biostatistics.
Sibuya, Miyawaki, and Sumita. 1994. β€œAspects of Lagrangian Probability Distributions.” Journal of Applied Probability.
Tanner. 1961. β€œA Derivation of the Borel Distribution.” Biometrika.