---- > [!definition]+ Definition. ([[adjacency matrix]]) > For an [[weighted network|unweighted]] [[graph]] with nodes indexed by $1,2,\dots$, the **adjacency [[matrix]]** $A$ of the [[network]] is defined with elements $a_{ij}=k$, where $k = \text{rank}(\text{number of edges from node $j$ to node $i$)}.$ ^definition > [!basicproperties]+ > - The **adjacency matrix** of an [[network|undirected network]] is [[symmetric matrix|symmetric.]] > - The **adjacency matrix** of a [[simple graph]] is binary, with $0$ along the [[diagonal]]. > - Be careful with [[simple graph|self-edges]] in [[network|undirected graphs]]! These will have the value *2*, since technically the node is entered twice. > - The $i^{th}$ row of an **adjacency matrix** tell you which nodes have edges incoming to node $i$. So, a row of zeros implies that node $i$ has no incoming edges. Likewise, a column $j$ of zeros implies node $j$ has no outgoing edges. ^basic-properties > [!definition]+ Definition. (Adjacency Matrix of Weighted Graph) > For a [[weighted network|weighted graph]] that is [[simple graph|simple]], we instead let $k=(\text{weight of the edge from node $i$ to node $j$}).$ ^definition > [!basicexample]+ > ![[CleanShot 2023-09-11 at 08.37.01.jpg]] > The **adjacency matrix** of this [[network]] is $\begin{bmatrix} 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \end{bmatrix}.$ This [[matrix]] is [[rank]]-deficient because node $5$ has no incoming connections. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```