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Degree-preserving edge rewiring replaces two edges $(u,v)$ and $(x,y)$ by $(u,y)$ and $(x,v)$, preserving every endpoint's degree of a vertex. On simple graphs, reject proposed self-loops and duplicate edges. Symmetric random proposals with rejected moves retained preserve a uniform distribution on the reachable realizations; adequate mixing remains a separate requirement. Connectivity can change unless explicitly constrained.

A network null model is an ensemble representing a specified baseline against which to assess an observed biological network. It should preserve the features that would otherwise confound the test: at least the number of vertices and edges, and often the degree sequence, direction, edge type, or relevant spatial constraints. It should randomize the feature whose biological organization is under investigation. An ensemble that preserves that very feature cannot test whether the feature is unusual.

For example, test whether a measured protein-protein interaction network contains more closed three-protein interaction groups than can be explained by the interaction counts of individual proteins. Define vertices as the measured proteins and an undirected edge as a reproducible detected interaction, using a fixed assay threshold and treatment of duplicates. Preselect the triangle count as the test statistic. Generate many simple random graphs by degree-preserving edge rewiring, rejecting self-loops and duplicate edges, with enough accepted swaps and separation to obtain a well-mixed reference sample. This retains the observed degree sequence while disrupting higher-order association.

Compute the same triangle count for every reference graph. Compare the observed count with this distribution; an empirical upper-tail p-value can be reported as $(1+\#\{T_{\mathrm{null}}\ge T_{\mathrm{obs}}\})/(M+1)$ for $M$ suitably sampled reference graphs. Report the size of the excess as well as its tail frequency, and check sensitivity to the assay threshold and any required connectivity constraint. A significant excess is evidence beyond the chosen degree baseline, while assay bias or spatial organization omitted from that baseline can still explain it.

A generative network model specifies a mechanism for producing networks, such as growth with preferential attachment, and asks what structures that mechanism creates. A network null model is defined by its role as a comparison baseline. These roles can overlap: the same random mechanism can be a proposed biological explanation in one analysis and a deliberately simplified null in another.

The supplied adjacency matrix describes a four-vertex cycle graph, with every vertex degree equal to two. Degree-preserving edge rewiring is random, so it does not determine one unique labelled output adjacency matrix. For example, replace edges $AB$ and $CD$ by $AC$ and $BD$, retaining $AD$ and $BC$. This valid switch gives, in order $A,B,C,D$,Each row still sums to two. Under the simple graph convention, all allowed outputs have the same unlabelled four-cycle topology: there are three possible labelled four-cycles. A rejected switch or later return can also leave the original adjacency matrix.

It is not a scale-free network. Its degree distribution is concentrated entirely at degree two, with no power-law tail; degree-preserving edge rewiring cannot change that.

Use degree-preserving edge rewiring, often called a double-edge swap. Select two edges $(u,v)$ and $(x,y)$ with four distinct endpoints, then replace them by $(u,y)$ and $(x,v)$, choosing the reconnection symmetrically. Reject a proposal that would create a self-loop or an already present edge if the target is a simple graph. Each endpoint loses one edge and gains one, so every degree of a vertex, as well as the numbers of vertices and edges, is preserved exactly.

Repeat many such random proposals to generate reference networks. With symmetric proposals, retaining rejected steps as self-transitions, the uniform distribution on the reachable degree-preserving realizations is stationary. Adequate mixing must be checked; a few successful swaps do not establish an independent random sample. Degree preservation alone does not preserve connectedness, so impose and check that extra restriction if the graph distance comparison requires connected networks.