Branching Processes: How Cascades Propagate

This essay is adapted from my MSc thesis (University of Copenhagen & VU University Amsterdam, 2007). The mathematical framework described here — originally developed to understand activity propagation in neuronal networks — applies to any system where events can trigger further events: epidemic spread, error propagation, incident escalation, information cascades. The formalism has not changed; what has expanded is our understanding of where it applies. For the wider framework, see Introduction to Self-Organised Criticality; for the evidence in the brain, see The Brain as a Critical System.

If citing this material: S.-S. Poil, Temporal correlations and criticality in models of neuronal networks, MSc Thesis, University of Copenhagen & VU University Amsterdam, 2007.

What is a branching process?#

Whenever activity in one element of a system can trigger activity in connected elements, the resulting propagation can be described as a branching process. The concept has a surprisingly old pedigree: it was first formalised by Francis Galton and Henry William Watson in the late 19th century, in a study on the extinction of surnames in the British aristocracy.

The question they asked was simple: if each generation of a family produces some number of sons, will the family name persist or die out? The answer depends on a single parameter — the average number of descendants per individual. This parameter, generalised beyond families, turns out to govern the behaviour of cascading processes across an extraordinary range of systems.

In neuronal networks, the “elements” are neurons or groups of neurons, and “activity” is the propagation of electrical signals. In an epidemic, the elements are people and the activity is infection. In an error cascade, the elements are components and the activity is failure. The mathematics is the same.

The branching parameter#

The key quantity in any branching process is the branching parameter σ: the average number of new activations triggered by each active element. This single number determines the qualitative behaviour of the entire system.

Sub-critical (σ < 1). Each active element triggers, on average, fewer than one successor. Activity dies out. Cascades are short and small. In a neuronal network, this corresponds to low excitability — signals are damped. In an epidemic, it means each infected person infects fewer than one other on average: the outbreak fizzles.

Critical (σ = 1). Each active element triggers, on average, exactly one successor. The process is balanced — activity neither grows nor decays on average, but fluctuates. In this regime, cascades of all sizes can occur: most are small, but occasionally a cascade spans the entire system. The size distribution follows a power law. This is the regime that produces the scale-free dynamics characteristic of critical systems.

Super-critical (σ > 1). Each active element triggers more than one successor on average. Activity grows exponentially. In a neuronal network, this corresponds to a seizure — runaway excitation. In an epidemic, it is exponential spread.

The transition between these regimes is sharp. A small change in σ around the critical value of 1 produces a qualitative change in system behaviour — from rapid extinction to sustained, variable cascading to explosive growth.

Why the critical regime is special#

At σ = 1, the branching process produces cascades — also called “avalanches” — whose sizes and durations follow power-law distributions. This means there is no characteristic or “typical” cascade size. Small cascades are common, large ones are rare, but cascades of all sizes occur according to a precise mathematical relationship.

Specifically, for a critical branching process, the probability of an avalanche of size s scales as:

This exponent of −3/2 is a universal result for the simplest class of branching processes (Harris, 1963; Zapperi et al., 1995). It does not depend on the details of the system — only on the branching parameter being at its critical value. This universality is one of the reasons the branching process framework has been so influential: very different systems can produce the same statistical signature when their propagation dynamics are near-critical.

Away from criticality, the power-law distribution is truncated. In the sub-critical regime, an exponential cut-off suppresses large cascades. In the super-critical regime, a finite fraction of cascades become system-spanning.

Branching processes in the brain#

In the context of brain dynamics, the branching process framework provides a way to connect local neural activity — one neuron firing and triggering others — with the large-scale patterns observed in EEG and MEG recordings.

Beggs and Plenz (2003) showed that spontaneous activity in cortical networks propagates as cascades whose size distributions follow a power law with an exponent near −3/2, consistent with a critical branching process. They coined the term “neuronal avalanches” for these cascading events.

In my own work (Poil et al., 2008), I compared the temporal structure of ongoing oscillations recorded with MEG during rest with activity propagation in model networks at different branching ratios. Models with a branching ratio near the critical value of 1 reproduced the power-law scaling in lifetime distributions observed in the human data. This supported the idea that neuronal networks generating resting-state oscillations operate near a critical state.

Later, in Poil et al. (2012), we showed that critical-state dynamics of avalanches on short timescales and long-range temporal correlations on longer timescales jointly emerge in a neuronal network model when excitation and inhibition are balanced — connecting the branching process framework to the broader picture of multi-level criticality.

Finite systems and real-world complications#

The classical theory of branching processes assumes an infinite system with no spatial correlations between elements. Real systems violate both assumptions.

In finite systems, the power-law distribution of cascade sizes is truncated at large sizes — the system simply isn’t big enough to support arbitrarily large cascades. In my thesis work, I explored how finite-size effects alter the observed avalanche distributions, a subject not well covered by the classical theory. This matters practically because all empirical systems are finite, and the apparent branching parameter estimated from data can be biased by the system size.

More importantly, real systems have correlated branching processes: the probability of activating neighbours depends on the history of past activity, not just the current state. In neuronal networks, this correlation arises through synaptic plasticity and the local balance of excitation and inhibition. In other systems, it arises through whatever mechanism creates memory of past events. These correlations can change the universality class of the process and produce temporal structure on timescales much longer than any individual cascade.

Beyond neuroscience#

The branching process framework applies wherever events can trigger further events through a connected network:

Epidemics. The basic reproduction number R₀ is precisely the branching parameter. The critical threshold R₀ = 1 determines whether an outbreak grows or dies. Public health interventions aim to push σ below 1.

Forest fires. In the Drossel-Schwabl model (1992), fire spreads from tree to tree with a local propagation rule. The density of trees controls the effective branching parameter, and the fire-size distribution follows a power law at critical density.

Error propagation. In a system of interdependent components, the failure of one component can trigger failures in connected components. The effective branching parameter — how many downstream failures each failure causes on average — determines whether errors remain localised or cascade through the system.

Financial contagion. Default by one institution can trigger defaults in connected institutions. The network structure and the strength of dependencies determine the effective branching ratio.

The value of the branching process framework is not that it provides detailed predictions for each of these systems. It is that it identifies the single most important parameter — the average propagation ratio — and shows that the qualitative behaviour of the system changes abruptly at a critical value. Whether you are studying neurons, epidemics, or error propagation, the first question to ask is the same: what is σ, and is the system near 1?

References#

Beggs JM, Plenz D (2003) Neuronal avalanches in neocortical circuits. J Neurosci 23:11167.

Drossel B, Schwabl F (1992) Self-organized critical forest-fire model. Phys Rev Lett 69:1629.

Harris TE (1963) The Theory of Branching Processes. Springer.

Poil SS, van Ooyen A, Linkenkaer-Hansen K (2008) Avalanche dynamics of human brain oscillations: relation to critical branching processes and temporal correlations. Hum Brain Mapp 29:770–777.

Poil SS, Hardstone R, Mansvelder HD, Linkenkaer-Hansen K (2012) Critical-state dynamics of avalanches and oscillations jointly emerge from balanced excitation/inhibition in neuronal networks. J Neurosci 32:9817–9823.

Zapperi S, Lauritsen KB, Stanley HE (1995) Self-organized branching processes: mean-field theory for avalanches. Phys Rev Lett 75:4071.