mathematics

Null model

Null model is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Null model rather than just read about it. In short: In mathematics, for example in the study of statistical properties of graphs, a null model is a type of random object that matches one specific object in some of its features, or more generally satisfies a collection of constraints, but which is otherwise taken to be an unbiasedly random structure. The null model is used as a term of comparison, to verify whether the object in question displays some non-trivial feat…

Key takeaways

Null model belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Null model to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Null model from memory before moving on to harder problems.

Reference excerpt

In mathematics, for example in the study of statistical properties of graphs, a null model is a type of random object that matches one specific object in some of its features, or more generally satisfies a collection of constraints, but which is otherwise taken to be an unbiasedly random structure. The null model is used as a term of comparison, to verify whether the object in question displays some non-trivial features (properties that wouldn't be expected on the basis of chance alone or as a consequence of the constraints), such as community structure in graphs. An appropriate null model behaves in accordance with a reasonable null hypothesis for the behavior of the system under investigation. One null model of utility in the study of complex networks is that proposed by Newman and Girvan, consisting of a randomized version of an original graph G {\displaystyle G} , produced through edges being rewired at random, under the constraint that the expected degree of each vertex matches the degree of the vertex in the original graph. The null model is the basic concept behind the definition of modularity, a function which evaluates the goodness of partitions of a graph into clusters. In particular, given a graph G {\displaystyle G} and a specific community partition σ : V ( G ) → { 1 , . . . , b } {\displaystyle \sigma :V(G)\rightarrow \{1,...,b\}} (an assignment of a community-index σ ( v ) {\displaystyle \sigma (v)} (here taken as an integer from 1 {\displaystyle 1} to b {\displaystyle b} ) to each vertex v ∈ V ( G ) {\displaystyle v\in V(G)} in the graph), the modularity measures the difference between the number of links from/to each pair of communities, from that expected in a graph that is completely random in all respects other than the set of degrees of each of the vertices (the degree sequence). In other words, the modularity contrasts the exhibited community structure in G {\displaystyle G} with that of a null model, which in this case is the configuration model (the maximally random graph subject to a constraint on the degree of each vertex).

Worked examples

Example 1 — a first encounter with Null model

Start with the simplest possible case. Write down what Null model claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Null model before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Null model ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Null model

In research: Null model appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Null model in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Null model is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph theory, Graph theory stubs, Statistical methods, so understanding it makes those chapters shorter.
In everyday life: Look for Null model outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Null model in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Null model means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Null model out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Null model in simple terms?

Why does Null model matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Null model?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Null model.