Random Graphs and Complex Network Dynamics

Random graphs provide a mathematical framework for modelling networks in which connections between nodes occur with prescribed probabilities. Classical models such as the Erdős–Rényi graph establish the emergence of a giant component and a sharp phase transition in connectivity as the average degree crosses unity. Extensions including the configuration model and inhomogeneous random graphs account for arbitrary degree distributions or vertex weights, yielding heavy-tailed topologies and community structure. Critical phenomena in these settings are characterised by the scaling behaviour of component sizes and distances, often captured by coalescent processes or continuum random trees in the limit of large system size. Beyond static topology, complex network dynamics explores processes unfolding on graphs, ranging from percolation and epidemic spreading to cascading failures and synchronisation. Epidemic models such as the susceptible-infected-recovered (SIR) framework on multiplex networks illustrate how layered contact patterns and targeted interventions shape outbreak size and resilience. The study of universality classes reveals that diverse network models share common scaling laws near criticality, while metric-space convergence techniques formalise continuum limits of large components as measured metric spaces. Together, these developments underpin applications in epidemiology, communication systems, infrastructure robustness and social dynamics.

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Random Graphs and Complex Network Dynamics publication trend

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Technical terms

Random graph model: A mathematical construction in which nodes are connected by edges according to specified probability rules.

Erdős–Rényi graph: A fundamental random graph in which each pair of vertices is joined independently with equal probability.

Configuration model: A random graph ensemble that realises a given degree sequence by uniformly pairing half-edges.

Inhomogeneous random graph: A generalisation in which vertices carry weights that bias the probability of edge formation.

Giant component: A connected subgraph whose size grows proportionally with the total number of vertices.

Phase transition: An abrupt change in global connectivity or dynamical behaviour as a control parameter (e.g., average degree) crosses a threshold.

Percolation: A process in which edges or vertices are randomly retained or removed to study robustness and connectivity.

Multiplex network: A multilayer network in which the same set of nodes is connected by different types of edges representing varied interaction contexts.

SIR model: An epidemiological compartmental model dividing individuals into susceptible, infected and recovered classes to describe disease transmission dynamics.

References

1. Limits of multiplicative inhomogeneous random graphs and Lévy trees: limit theorems. Probability Theory and Related Fields (2021).
2. Universality for critical heavy-tailed network models: Metric structure of maximal components. Electronic Journal of Probability (2020).
3. Modeling partial lockdowns in multiplex networks using partition strategies. Applied Network Science (2021).