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Combinatorial probability is an area of probability that studies random geometric objects using techniques from combinatorics, ranging from enumeration to graph-theoretic methods.

Counting and understanding how many structures exist with a particular collection of properties is one of the most creative areas of mathematics, and has many applications, especially to computer science. Combinatorial probability includes the study of random graphs or networks, random trees, and P贸lya urns.

From theoretical tool to real world implications

Initially, random graphs were introduced as a tool to investigate deterministic properties of graphs; this tool, pioneered by Erdos, is known as the probabilistic method. However, it soon became clear that random graphs were fascinating objects in themselves, often with global structures arising from purely local random rules, and with broad applications to all sorts of real-world networks and structures.

Indeed, random graphs provide a flexible way to model any network whose connections form unpredictably. We might think of social networks, the spread of disease or misinformation, or telecommunication networks. Depending on the underlying rules鈥攈ow likely an edge is, how vertices are arranged, or what constraints are imposed鈥攖he resulting graphs can exhibit a wide variety of behaviours.

Probabilistic methods help reveal when large connected regions form, how local neighbourhoods behave, and how global structure arises from purely random choices. We can study connectivity thresholds, component size distributions, or scaling limits. We can also investigate how graphs might change over time: how their properties might be affected by sudden changes, or how unusual behaviours can occasionally appear.

The study of random trees

Random trees are an important special class of random graphs. They appear naturally themselves in many applications, particularly computer algorithms, but also frequently arise as local approximations to more complicated random structures鈥攎any large sparse graphs, when examined locally, look 鈥渢ree-like,鈥� making the study of random trees a crucial tool.

More broadly, combinatorial probability provides a toolkit for analysing these structures, often blending ideas from probability theory, combinatorial enumeration, and asymptotic analysis. The study of random graphs has implications across mathematics and computer science, from network theory to algorithms, and from statistical physics鈥搃nspired models to the behaviour of large data structures.

Members of ProbLaB with research in these areas include:

• , Reader; an expert on P贸lya urns and other models involving reinforcement. This can include recursive trees, preferential attachment networks, and reinforcement learning.
• , Reader; interested in dynamically growing networks such as preferential attachment models and their emerging properties. Studying interacting particle systems defined on top of random graph models, modelling e.g. the spread of diseases or the formation of voting patterns.
• , Lecturer; an expert in graph theory, has utilised techniques from this area to study percolation and self-avoiding walks on various graphs with different geometric properties. Interested in models arising from statistical physics on complete graphs and trees.
• , Professor; has written an important monograph on random geometric graphs, and continues to make contributions in this field, for example, in recent work on the asymptotic probability that these graphs are connected, and on the size of the giant component when they are not quite connected.
• , Professor; has looked at the probability that a critical random graph has an unusually large component, and applied these bounds to study dynamical random graphs. He has also looked at the asymptotic behaviour of the random binary search tree, which appears in the quicksort algorithm.