# Random graph ensembles with many short loops

The analysis of real networks which contain many short loops requires novel methods, because they break the assumptions of tree-like models.

*ESAIM: Proceedings and surveys* 47, 97 (2014)

E. Roberts, A. Coolen

Networks observed in the real world often have many short loops. This violates the tree-like assumption that underpins the majority of random graph models and most of the methods used for their analysis. In this paper we sketch possible research routes to be explored in order to make progress on networks with many short loops, involving old and new random graph models and ideas for novel mathematical methods. We do not present conclusive solutions of problems, but aim to encourage and stimulate new activity and in what we believe to be an important but under-exposed area of research. We discuss in more detail the Strauss model, which can be seen as the ‘harmonic oscillator’ of ‘loopy’ random graphs, and a recent exactly solvable immunological model that involves random graphs with extensively many cliques and short loops.

#### More in Intelligence of graphs

### Transitions in loopy graphs

The generation of large graphs with a controllable number of short loops paves the way for building more realistic random networks.

### Bursting dynamic networks

A mathematical model captures the temporal and steady state behaviour of networks whose two sets of nodes either generate or destroy links.

### Exactly solvable random graphs

An explicit analytical solution reproduces the main features of random graph ensembles with many short cycles under strict degree constraints.

### Tailored random graph ensembles

New mathematical tools quantify the topological structure of large directed networks which describe how genes interact within a cell.

### Entropies of graph ensembles

Explicit formulae for the Shannon entropies of random graph ensembles provide measures to compare and reproduce their topological features.

### Unbiased randomization

Unbiased randomisation processes generate sophisticated synthetic networks for modelling and testing the properties of real-world networks.

### Spin systems on Bethe lattices

Exact equations for the thermodynamic quantities of lattices made of d-dimensional hypercubes are obtainable with the Bethe-Peierls approach.