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In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e. all of its edges are bidirectional), the adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory.
The network generation process does not exclude the development of a self-loop or a multi-link. If we designed the process where self-loops and multi-edges are not allowed, the matching of the stubs would not follow a uniform distribution. The expected total number of multi-links in a configuration model network would be:
Maybe the example graph can contain a self loop, to show how it can be represented into the adjacency matrix. That's a great idea. Deco 01:39, 21 Mar 2005 (UTC) Most software packages show a binary adjacency matrix, even on the diagonal. But loops are always counted twice, and some books show an adjacency matrix like this one, with 2 on the ...
The proof is bijective: a matrix A is an adjacency matrix of a DAG if and only if A + I is a (0,1) matrix with all eigenvalues positive, where I denotes the identity matrix. Because a DAG cannot have self-loops, its adjacency matrix must have a zero diagonal, so adding I preserves the property that all matrix coefficients are 0 or 1. [13]
where is the matrix of node representations , is the matrix of node features , () is an activation function (e.g., ReLU), ~ is the graph adjacency matrix with the addition of self-loops, ~ is the graph degree matrix with the addition of self-loops, and is a matrix of trainable parameters.
Given a network with nodes, where each node has a node degree , the configuration model cuts each edge into two halves, and then each half edge, called a stub, is rewired randomly with any other stub in the network, even allowing self-loops (which occur when a stub is rewired to another stub from the same node) and multiple-edges between the ...
In particular, the resulting graph contains no self-loops. ... When the graph is represented using adjacency lists or an adjacency matrix, ...
A multigraph is a graph that allows multiple adjacencies (and, often, self-loops); a graph that is not required to be simple. multiple adjacency A multiple adjacency or multiple edge is a set of more than one edge that all have the same endpoints (in the same direction, in the case of directed graphs).