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Concepts: connectivity, components, and cliques

If it’s possible to build a path between any pair of nodes in a network, then this has implications for navigation and how processes work. Sometimes networks are connected; sometimes they are composed of disconnected components; and sometimes somewhere in between.

(This is a chapter from Complex networks, complex processes.)

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The mathematics of ER networks

Because Erdős-Rényi networks have a very simple and regular generation process, it’s reasonable to ask whether we can do better than numerical studies when trying to find its degree distribution, mean degree, and the like. And indeed we can.

(This is a chapter from Complex networks, complex processes.)

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Erdős-Rényi networks

Let’s leave epidemic spreading for a little while to return to the static structure of networks, and look at perhaps the most common class of network, the Erdős-Rényi or ER network. These networks are incredibly simple to describe, incredibly simple to build and simulate – and incredibly useful as at least first-attempt network models for a range of processes. We’ll also run up against the limitations of animation for presenting networks.

(This is a chapter from Complex networks, complex processes.)

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Compartmented models of disease

When modelling diseases computationally we need to step back from the biological complexity somewhat, and this leads to a set of approaches called compartmented models. A disease progresses through a set of stages, modelled as compartments: at any point, each individual sits in a particular compartment and the disease evolves by people moving between the compartments as their disease progresses. Classically these models are studied using differential equations, and it’s useful to understand this before seeing how networks provide a more reliable setting for the same models.

(This is a chapter from Complex networks, complex processes.)

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Epidemic disease processes

There are lots of different spreading processes, but perhaps the most engaging — as well as the most socially important — is the way epidemics spread. Before we think about how epidemics spread over networks, we should first explore how epidemics are typically encountered and understood in medicine.

(This is a chapter from Complex networks, complex processes.)

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Concepts: processes

So far we’ve talked about networks as static things with static properties. But the real interest of networks comes from how they influence processes, the dynamics that runs over a network: the “traffic” in the “streets”, as it were. Making this happen requires defining local rules that aggregate together as global phenomena.

(This is a chapter from Complex networks, complex processes.)

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Concepts: neighbourhoods and paths

If we think of networks as the streets and junctions of towns, then paths are the ways we can navigate around those towns. They give us a notion of locality and distance, and also a notion of the “width” or “diameter” of a network, even when there’s no inherent geometry.

(This is a chapter from Complex networks, complex processes.)

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Concepts: degree

The degree of a node is a local property that, when integrated over an entire network, leads to the idea of degree distribution that is one of the most important features of complex networks..

(This is a chapter from Complex networks, complex processes.)

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Concepts: simple networks

The idea of a “simple” complex network, without directed edges, parallel edges, or self-loops.

(This is a chapter from Complex networks, complex processes.)

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Concepts: networks and geometry

Drawing a network, and how this might (or might not) aid intuition.

(This is a chapter from Complex networks, complex processes.)

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