{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Graphs and Networks in Linear Algebra\n",
"\n",
"This notebook is based on section 10.1 of Strang's *Linear Algebra* textbook.\n",
"\n",
"One interesting source of large matrices in linear algebra is a [graph](https://en.wikipedia.org/wiki/Graph_(discrete_mathematics), a collection of *nodes* (vertices) and *edges* (arrows from one vertex to another). Graphs are used in many applications to represent *relationships* and *connectivity*, such as:\n",
"\n",
"* For computer networks, nodes could represent web pages, and edges could represent links.\n",
"* For circuits, edges could represent wires (or resistors) and nodes junctions.\n",
"* For transportation, nodes could represent cities and edges roads.\n",
"* In bioinformatics, graphs can represent gene regulatory networks.\n",
"* In sociology, nodes could represent people and edges relationships.\n",
"* ... and many, many other applications ...\n",
"\n",
"In this notebook, we explain how a graph can be represented by a *matrix*, and how linear algebra can tell us properties of the graph and can help us do computations on graph-based problems. There is a particularly beautiful connection to Kirchhoff's laws of circuit theory."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Packages for this notebook\n",
"\n",
"To run the code in this notebook, you'll need to install a few Julia packages used below. To do so, uncomment the following line and run it:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"# Pkg.add.([\"LightGraphs\", \"MetaGraphs\", \"GraphPlot\", \"NamedColors\", \"RowEchelon\", Interact\", \"SymPy\"])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"...and then run this cell to import the packages:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
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"source": [
"using Interact, RowEchelon, LightGraphs, MetaGraphs, GraphPlot, NamedColors, LinearAlgebra\n",
"import SymPy\n",
"using SymPy: Sym"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Julia Graph-visualization code\n",
"\n",
"There are several Julia packages for manipulating graphs, e.g. [LightGraphs](https://github.com/JuliaGraphs/LightGraphs.jl), along with several packages for visualizing graphs, e.g. [GraphViz](https://github.com/Keno/GraphViz.jl). LightGraphs is oriented towards fast and sophisticated graph computations, however, and here I just want to do some simple and pretty visualizations with simple algorithms based on those in Strang's 18.06 textbook.\n",
"\n",
"So, here I define a simple `MyGraph` wrapper around LightGraphs directed graphs, with metadata attached via the MetaGraphs package, for basic plotting via the GraphPlots package."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"labels (generic function with 1 method)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"struct MyGraph\n",
" g::MetaDiGraph\n",
"end\n",
"Base.copy(mg::MyGraph) = MyGraph(copy(mg.g))\n",
"function MyGraph(edges::Pair{<:Integer,<:Integer}...)\n",
" g = SimpleDiGraphFromIterator(Edge(e) for e in edges)\n",
" MyGraph(MetaDiGraph(g))\n",
"end\n",
"\n",
"using Random\n",
"function deterministic_spring_layout(g::AbstractGraph; seed::Integer=0, kws...)\n",
" rng = MersenneTwister(seed)\n",
" spring_layout(g, 2 .* rand(rng, nv(g)) .- 1.0, 2 .* rand(rng,nv(g)) .- 1.0; kws...)\n",
"end\n",
"function Base.show(io::IO, m::MIME\"image/svg+xml\", mg::MyGraph)\n",
" show(io, m, \n",
" gplot(mg.g, layout=deterministic_spring_layout,\n",
" nodelabel=map(v -> get(MetaGraphs.props(mg.g, v), :label, v), vertices(mg.g)),\n",
" nodefillc=map(v -> get(MetaGraphs.props(mg.g, v), :color, \"gray\"), vertices(mg.g)),\n",
" edgelabel=map(ie -> get(MetaGraphs.props(mg.g, ie[2]), :label, ie[1]), enumerate(edges(mg.g))),\n",
" edgestrokec=map(e -> get(MetaGraphs.props(mg.g, e), :color, \"lightgray\"), edges(mg.g)),\n",
" ))\n",
"end\n",
"\n",
"function nodecolors!(g::MyGraph, nodes::AbstractVector{<:Integer}, color::String=\"red\")\n",
" for n in nodes\n",
" set_prop!(g.g, n, :color, color)\n",
" end\n",
" g\n",
"end\n",
"nodecolors(g, nodes, color) = nodecolors!(copy(g), nodes, color)\n",
"edgearr(g, e) = e\n",
"edgearr(g, e::AbstractVector{<:Integer}) = collect(edges(g))[e]\n",
"edgearr(g, e::AbstractVector{<:Pair}) = Edge.(e)\n",
"function edgecolors!(g::MyGraph, edges::AbstractVector, color::String=\"red\")\n",
" for e in edgearr(g.g, edges)\n",
" set_prop!(g.g, e, :color, color)\n",
" end\n",
" g\n",
"end\n",
"edgecolors(g::MyGraph, edges::AbstractVector, color::String=\"red\") = edgecolors!(copy(g), edges, color)\n",
"\n",
"# A little code so that we can label graph nodes/edges with SymPy expressions.\n",
"# convert strings like \"v_2 - v_0\" from SymPy to nicer Unicode strings like \"v₂ - v₀\"\n",
"subchar(d::Integer) = Char(UInt32('₀')+d)\n",
"subchar(c::Char) = subchar(UInt32(c)-UInt32('0'))\n",
"subchar(s::String) = replace(s, r\"_[0-9]\" => s -> subchar(s[2]))\n",
"labelstring(s::SymPy.Sym) = subchar(repr(\"text/plain\", s))\n",
"labelstring(x) = x\n",
"\n",
"function labels!(g::MyGraph; edges=nothing, nodes=nothing)\n",
" if edges !== nothing\n",
" for (e,E) in zip(MetaGraphs.edges(g.g), edges)\n",
" set_prop!(g.g, e, :label, labelstring(E))\n",
" end\n",
" end\n",
" if nodes !== nothing\n",
" for (n,N) in zip(vertices(g.g), nodes)\n",
" set_prop!(g.g, n, :label, labelstring(N))\n",
" end\n",
" end\n",
" g\n",
"end\n",
"labels(g::MyGraph; kws...) = labels!(copy(g); kws...)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
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"text/plain": [
"randgraph (generic function with 1 method)"
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"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
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"source": [
"# generate a random graph with a given average #edges per node\n",
"function randgraph(numnodes::Integer, edgespernode::Real)\n",
" p = edgespernode/numnodes # probability of each edge\n",
" e = Vector{Pair{Int,Int}}()\n",
" for i = 1:numnodes, j = 1:numnodes\n",
" if i != j && rand() < p\n",
" push!(e, i=>j)\n",
" end\n",
" end\n",
" return MyGraph(e...)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
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"data": {
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"incidence (generic function with 1 method)"
]
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"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
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"source": [
"# returns the incidence matrix for g\n",
"function incidence(g::MyGraph)\n",
" A = zeros(Int, ne(g.g), nv(g.g))\n",
" for (i,e) in enumerate(edges(g.g))\n",
" A[i,e.src] = -1\n",
" A[i,e.dst] = +1\n",
" end\n",
" return A\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
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"data": {
"text/plain": [
"leftnullspace (generic function with 1 method)"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"# Find the loops in g by the simplest \"textbook\" manner:\n",
"# get a basis for the left nullspace incidence matrix.\n",
"# We do this via the rref form, rather than nullspace(A'), because\n",
"# we want a \"nice\" basis of ±1 and 0 entries.\n",
"function leftnullspace(g::MyGraph)\n",
" A = incidence(g)\n",
" R = rref(Matrix(A'))\n",
" m, n = size(R)\n",
" pivots = Int[]\n",
" for i = 1:m\n",
" j = findfirst(!iszero, R[i,:])\n",
" j !== nothing && push!(pivots, j)\n",
" end\n",
" r = length(pivots) # rank\n",
" free = Int[j for j=1:n if j ∉ pivots]\n",
" N = zeros(Int, n, n-r)\n",
" k = 0\n",
" for (k,j) in enumerate(free)\n",
" N[pivots, k] = -R[1:r, j]\n",
" N[j, k] = 1\n",
" end\n",
" return N\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
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"data": {
"text/plain": [
"tree (generic function with 1 method)"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"# color the edges of a spanning tree of g, by the textbook\n",
"# method of finding the pivot rows of the incidence matrix\n",
"function pivotrows(g::MyGraph)\n",
" A = incidence(g)\n",
" R = rref(Matrix(A'))\n",
" m, n = size(R)\n",
" pivots = Int[]\n",
" for i = 1:m\n",
" j = findfirst(!iszero, R[i,:])\n",
" j !== nothing && push!(pivots, j)\n",
" end\n",
" return pivots\n",
"end\n",
"colortree(g::MyGraph, color::String=\"red\") = edgecolors(g, pivotrows(g), color)\n",
"tree(g::MyGraph) = MyGraph(MetaDiGraph(SimpleDiGraphFromIterator(edgearr(g.g, pivotrows(g)))))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Graphs and incidence matrices\n",
"\n",
"Let's start by looking at an example graph with 6 nodes 8 edges. Computers are pretty good at drawing graphs for us:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
],
"text/plain": [
"MyGraph({6, 8} directed Int64 metagraph with Float64 weights defined by :weight (default weight 1.0))"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g = MyGraph(1=>4, 4=>5, 5=>6, 6=>3, 3=>2, 2=>1, 2=>6, 4=>6)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A key way to represent a graph in linear algebra is the [incidence matrix](https://en.wikipedia.org/wiki/Incidence_matrix). As defined in Strang's textbook, this is a matrix where the **rows correspond to edges** and the **columns correspond to nodes**. (Some authors use the transpose of this instead.)\n",
"\n",
"In particular, in the row for each edge going **from node N to node M**, there is a **-1 in column N** and a **+1 in column N**.\n",
"\n",
"For example, the incidence matrix of the graph above is:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"8×6 Array{Int64,2}:\n",
" -1 0 0 1 0 0\n",
" 1 -1 0 0 0 0\n",
" 0 -1 0 0 0 1\n",
" 0 1 -1 0 0 0\n",
" 0 0 0 -1 1 0\n",
" 0 0 0 -1 0 1\n",
" 0 0 0 0 -1 1\n",
" 0 0 1 0 0 -1"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = incidence(g)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There is an interesting structure if you think about *loops* in the graph. For example, in the graph above there is a loop among nodes 6, 3, 2, via edges 4,3,8. Let's look at the rows of $A$ corresponding to those edges:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3×6 Array{Int64,2}:\n",
" 0 1 -1 0 0 0\n",
" 0 -1 0 0 0 1\n",
" 0 0 1 0 0 -1"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A[[4,3,8],:]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If we **add these rows** we get **zero**:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1×6 Adjoint{Int64,Array{Int64,1}}:\n",
" 0 0 0 0 0 0"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A[4,:]' + A[3,:]' + A[8,:]'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In general, it is easy to see that **any loop in the graph** corresponds to **dependent rows**: if we sum the rows going around the loop (with a minus sign for arrows in the wrong direction), we get zero.\n",
"\n",
"The reason is simple: we get a -1 in a column when we *leave* a node, and a +1 in the column when we *enter* a node. When we go around the loop, we leave and enter each node, so the sum is zero.\n",
"\n",
"But dependent rows correspond to **elements of the left nullspace**:"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1×6 Array{Int64,2}:\n",
" 0 0 0 0 0 0"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"[0 0 1 1 0 0 0 1] * A"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"That means that the number of \"independent\" (primitive) loops in a graph is related to the **rank** of the incidence matrix, and the **independent rows of A have no loops**.\n",
"\n",
"Let's look at the row-reduced echelon (rref) form of $A^T$:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6×8 Array{Int64,2}:\n",
" 1 0 0 0 0 -1 -1 0\n",
" 0 1 0 0 0 -1 -1 0\n",
" 0 0 1 0 0 1 1 -1\n",
" 0 0 0 1 0 0 0 -1\n",
" 0 0 0 0 1 0 -1 0\n",
" 0 0 0 0 0 0 0 0"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Matrix{Int}(rref(Matrix(A')))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can see that the rank of $A$ is 5:"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rank(A)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This means that there are **five loop-free (independent) edges**, and there are **three** (8 - 5) primitive loops. Using the rref form of $A^T$, we can read off a basis for the left nullspace from the free columns (6,7,8):"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"8×3 Array{Int64,2}:\n",
" 1 1 0\n",
" 1 1 0\n",
" -1 -1 1\n",
" 0 0 1\n",
" 0 1 0\n",
" 1 0 0\n",
" 0 1 0\n",
" 0 0 1"
]
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"execution_count": 15,
"metadata": {},
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],
"source": [
"N = leftnullspace(g)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
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{
"data": {
"text/plain": [
"6×3 Array{Int64,2}:\n",
" 0 0 0\n",
" 0 0 0\n",
" 0 0 0\n",
" 0 0 0\n",
" 0 0 0\n",
" 0 0 0"
]
},
"execution_count": 16,
"metadata": {},
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],
"source": [
"A' * N"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's visualize these loops by plotting the edges in a different color (red) one by one,\n",
"with help from the Interact package to give us an interactive widget:"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"animloops (generic function with 1 method)"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"colorloop(g::MyGraph, n::Vector) = edgecolors!(edgecolors(g, findall(n .> 0), \"red\"), findall(n .< 0), \"blue\")\n",
"function animloops(g::MyGraph)\n",
" L = leftnullspace(g)\n",
" @manipulate for loop in 1:size(L,2)\n",
" colorloop(g, L[:,loop])\n",
" end\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 18,
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"It is fun to do the same thing for bigger graphs, chosen at random:"
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"animloops(gbig)"
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"cell_type": "markdown",
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"source": [
"Conversely, the *independent* rows of $A$ (corresponding to the **pivot columns** of the rref form of $A^T$) form a **maximal set of edges with no loops**. A graph with no loops is called a [tree](https://en.wikipedia.org/wiki/Tree_(graph_theory)), and this particular tree is called a [spanning tree](https://en.wikipedia.org/wiki/Spanning_tree) because it touches all of (\"spans\") the nodes (assuming the graph is connected).\n",
"\n",
"Let's color the spanning tree (loop-free edges) of our example graph red:"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
],
"text/plain": [
"MyGraph({6, 8} directed Int64 metagraph with Float64 weights defined by :weight (default weight 1.0))"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"colortree(g)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can also discard all of the edges that are *not* in the spanning tree, and we are left with a more boring graph of *just* the spanning tree:"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
],
"text/plain": [
"MyGraph({6, 5} directed Int64 metagraph with Float64 weights defined by :weight (default weight 1.0))"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"tree(g)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can do the same thing for our bigger random example:"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
],
"text/plain": [
"MyGraph({15, 23} directed Int64 metagraph with Float64 weights defined by :weight (default weight 1.0))"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"colortree(gbig)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
],
"text/plain": [
"MyGraph({15, 14} directed Int64 metagraph with Float64 weights defined by :weight (default weight 1.0))"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"tree(gbig)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And we can make trees from even larger graphs, for fun:"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
],
"text/plain": [
"MyGraph({50, 49} directed Int64 metagraph with Float64 weights defined by :weight (default weight 1.0))"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"tree(randgraph(50, 8))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Graphs and Kirchhoff's circuit laws\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"An elegant application of the incidence matrix and its subspaces arises if we think of the graph as representing an **electrical circuit**:\n",
"\n",
"* Each edge represents a wire/resistor, with an unknown current $i$. The *direction* of the edge indicates the *sign convention* ($i>0$ indicates current flowing in the direction of the arrow).\n",
"* Each node represents a junction, with an unknown voltage $v$.\n",
"\n",
"Let's visualize this by re-labeling our graph from above. We'll use the [SymPy](https://github.com/JuliaPy/SymPy.jl) package to allow us to do *symbolic* (not numeric) calculations with the incidence matrix."
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
],
"text/plain": [
"MyGraph({6, 8} directed Int64 metagraph with Float64 weights defined by :weight (default weight 1.0))"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"labels(g, edges=[Sym(\"i_$i\") for i = 1:size(A,1)], nodes=[Sym(\"v_$i\") for i = 1:size(A,2)])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Kirchhoff's voltage law (KVL)\n",
"\n",
"Let's start doing some linear algebra. What happens if we *multiply* our incidence matrix $A$ by a vector of voltages, one per node?"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[ \\left[ \\begin{array}{r}v_{1}\\\\v_{2}\\\\v_{3}\\\\v_{4}\\\\v_{5}\\\\v_{6}\\end{array} \\right] \\]"
],
"text/plain": [
"6-element Array{Sym,1}:\n",
" v_1\n",
" v_2\n",
" v_3\n",
" v_4\n",
" v_5\n",
" v_6"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v = [Sym(\"v_$i\") for i = 1:size(A,2)]"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"8×6 Array{Int64,2}:\n",
" -1 0 0 1 0 0\n",
" 1 -1 0 0 0 0\n",
" 0 -1 0 0 0 1\n",
" 0 1 -1 0 0 0\n",
" 0 0 0 -1 1 0\n",
" 0 0 0 -1 0 1\n",
" 0 0 0 0 -1 1\n",
" 0 0 1 0 0 -1"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[ \\left[ \\begin{array}{r}- v_{1} + v_{4}\\\\v_{1} - v_{2}\\\\- v_{2} + v_{6}\\\\v_{2} - v_{3}\\\\- v_{4} + v_{5}\\\\- v_{4} + v_{6}\\\\- v_{5} + v_{6}\\\\v_{3} - v_{6}\\end{array} \\right] \\]"
],
"text/plain": [
"8-element Array{Sym,1}:\n",
" -v_1 + v_4\n",
" v_1 - v_2\n",
" -v_2 + v_6\n",
" v_2 - v_3\n",
" -v_4 + v_5\n",
" -v_4 + v_6\n",
" -v_5 + v_6\n",
" v_3 - v_6"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A * v"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"What we get are the **voltage difference** (and in particular, the **voltage rise**) across each edge. It is easier to see this if we use the elements of $Av$ to directly label the edges of our graph:"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
],
"text/plain": [
"MyGraph({6, 8} directed Int64 metagraph with Float64 weights defined by :weight (default weight 1.0))"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"labels(g, edges=A*v, nodes=v)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now, let's ask the inverse question: **what voltage differences $d=Av$** can possibly arise? i.e. what $d$ are in $C(A)$? \n",
"\n",
"Remember, $A$ is **not full rank**: its rank is 5, but there are 8 rows (8 edges). So, $C(A)$ is 5-dimensional (\"missing\" three dimensions). Equivalently $C(A)$ is **orthogonal to the left nullspace**, which has three rows. What does this mean?\n",
"\n",
"Let's visualize the differences d:"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
],
"text/plain": [
"MyGraph({6, 8} directed Int64 metagraph with Float64 weights defined by :weight (default weight 1.0))"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d = [Sym(\"d_$j\") for j = 1:size(A,1)]\n",
"labels(g, edges=d, nodes=v)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If $N$ is a basis for the left nullspace, we must have $N^T d = 0$, or:"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[ \\left[ \\begin{array}{r}d_{1} + d_{2} - d_{3} + d_{6}\\\\d_{1} + d_{2} - d_{3} + d_{5} + d_{7}\\\\d_{3} + d_{4} + d_{8}\\end{array} \\right] \\]"
],
"text/plain": [
"3-element Array{Sym,1}:\n",
" d_1 + d_2 - d_3 + d_6\n",
" d_1 + d_2 - d_3 + d_5 + d_7\n",
" d_3 + d_4 + d_8"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"N' * d"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"But what is this? Remember, each element of the left nullspace corresponded to a **loop in the graph**. Saying $N^T d = 0$, or $d \\perp N(A^T)$, is equivalent to saying that the **sum of the voltage rises around each loop = 0**.\n",
"\n",
"But this is precisely [Kirchhoff's voltage law](https://en.wikipedia.org/wiki/Kirchhoff's_circuit_laws) from circuit theory!"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Kirchhoff's current law (KCL)\n",
"\n",
"To actually solve circuit problems, we need three additional ingredients:\n",
"\n",
"* The voltage difference $d$ must be divided by a resistance $R$ to get the *current* $i$ through that edge: $i = -d/R = -Yd$ (where $Y=1/R$ is the \"admittance\"), by [Ohm's law](https://en.wikipedia.org/wiki/Ohm's_law). Note that we need a minus sign to get the current in the direction of the arrow, since $d$ was the the voltage *rise* across the edge.\n",
"\n",
"* The sum of the currents $i$ entering each node must be zero, by Kirchhoff's current law (KCL).\n",
"\n",
"* To get a nontrivial solution, we need some kind of *source*: a battery or current source, to start currents flowing.\n",
"\n",
"How do we represent each one of these steps by linear-algebra operations?"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Ohm's law\n",
"\n",
"To represent Ohm's law, we need to multiply the voltage differences $d=Av$ by a *diagonal matrix* of admittances:"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\left[ \\begin{array}{rrrrrrrr}Y_{1}&0&0&0&0&0&0&0\\\\0&Y_{2}&0&0&0&0&0&0\\\\0&0&Y_{3}&0&0&0&0&0\\\\0&0&0&Y_{4}&0&0&0&0\\\\0&0&0&0&Y_{5}&0&0&0\\\\0&0&0&0&0&Y_{6}&0&0\\\\0&0&0&0&0&0&Y_{7}&0\\\\0&0&0&0&0&0&0&Y_{8}\\end{array}\\right]\\]"
],
"text/plain": [
"8×8 Array{Sym,2}:\n",
" Y_1 0 0 0 0 0 0 0\n",
" 0 Y_2 0 0 0 0 0 0\n",
" 0 0 Y_3 0 0 0 0 0\n",
" 0 0 0 Y_4 0 0 0 0\n",
" 0 0 0 0 Y_5 0 0 0\n",
" 0 0 0 0 0 Y_6 0 0\n",
" 0 0 0 0 0 0 Y_7 0\n",
" 0 0 0 0 0 0 0 Y_8"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Y = diagm(0=>[Sym(\"Y_$i\") for i = 1:size(A,1)])"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[ \\left[ \\begin{array}{r}Y_{1} d_{1}\\\\Y_{2} d_{2}\\\\Y_{3} d_{3}\\\\Y_{4} d_{4}\\\\Y_{5} d_{5}\\\\Y_{6} d_{6}\\\\Y_{7} d_{7}\\\\Y_{8} d_{8}\\end{array} \\right] \\]"
],
"text/plain": [
"8-element Array{Sym,1}:\n",
" Y_1*d_1\n",
" Y_2*d_2\n",
" Y_3*d_3\n",
" Y_4*d_4\n",
" Y_5*d_5\n",
" Y_6*d_6\n",
" Y_7*d_7\n",
" Y_8*d_8"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Y*d"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[ \\left[ \\begin{array}{r}- Y_{1} v_{1} + Y_{1} v_{4}\\\\Y_{2} v_{1} - Y_{2} v_{2}\\\\- Y_{3} v_{2} + Y_{3} v_{6}\\\\Y_{4} v_{2} - Y_{4} v_{3}\\\\- Y_{5} v_{4} + Y_{5} v_{5}\\\\- Y_{6} v_{4} + Y_{6} v_{6}\\\\- Y_{7} v_{5} + Y_{7} v_{6}\\\\Y_{8} v_{3} - Y_{8} v_{6}\\end{array} \\right] \\]"
],
"text/plain": [
"8-element Array{Sym,1}:\n",
" -Y_1*v_1 + Y_1*v_4\n",
" Y_2*v_1 - Y_2*v_2\n",
" -Y_3*v_2 + Y_3*v_6\n",
" Y_4*v_2 - Y_4*v_3\n",
" -Y_5*v_4 + Y_5*v_5\n",
" -Y_6*v_4 + Y_6*v_6\n",
" -Y_7*v_5 + Y_7*v_6\n",
" Y_8*v_3 - Y_8*v_6"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Y*A*v"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Net current into each node\n",
"\n",
"Given the currents $i$, a little thought shows that the net current flowing into each node is precisely $A^T i$:"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[ \\left[ \\begin{array}{r}- i_{1} + i_{2}\\\\- i_{2} - i_{3} + i_{4}\\\\- i_{4} + i_{8}\\\\i_{1} - i_{5} - i_{6}\\\\i_{5} - i_{7}\\\\i_{3} + i_{6} + i_{7} - i_{8}\\end{array} \\right] \\]"
],
"text/plain": [
"6-element Array{Sym,1}:\n",
" -i_1 + i_2\n",
" -i_2 - i_3 + i_4\n",
" -i_4 + i_8\n",
" i_1 - i_5 - i_6\n",
" i_5 - i_7\n",
" i_3 + i_6 + i_7 - i_8"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"i = [Sym(\"i_$j\") for j=1:size(A,1)]\n",
"A'*i"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
],
"text/plain": [
"MyGraph({6, 8} directed Int64 metagraph with Float64 weights defined by :weight (default weight 1.0))"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"labels(g, edges=i, nodes=A'*i)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Why is this? The reason is that each row $A^T$ corresponds to a node, and has $\\pm 1$ for each edge going into or out of the node, exactly the right sign to sum the net currents flowing in:"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6×8 Adjoint{Int64,Array{Int64,2}}:\n",
" -1 1 0 0 0 0 0 0\n",
" 0 -1 -1 1 0 0 0 0\n",
" 0 0 0 -1 0 0 0 1\n",
" 1 0 0 0 -1 -1 0 0\n",
" 0 0 0 0 1 0 -1 0\n",
" 0 0 1 0 0 1 1 -1"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Putting it together, given voltages $v$, the net current flowing **out of** each node is\n",
"\n",
"$$\n",
"A^T Y A v\n",
"$$\n",
"\n",
"The matrix $A^T Y A$ is a very special and important kind of matrix. It is obviously **symmetric**, and later on in the course we will see that any matrix of this form is necessarily **positive semidefinite** (all pivots are ≥ 0). Many important matrices in science, engineering, statistics, and other fields take on this special form.\n",
"\n",
"If we multiply $A^T Y A$ together, not all of its specialness is apparent. It is often better to leave it in \"factored\" form:"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\left[ \\begin{array}{rrrrrr}Y_{1} + Y_{2}&- Y_{2}&0&- Y_{1}&0&0\\\\- Y_{2}&Y_{2} + Y_{3} + Y_{4}&- Y_{4}&0&0&- Y_{3}\\\\0&- Y_{4}&Y_{4} + Y_{8}&0&0&- Y_{8}\\\\- Y_{1}&0&0&Y_{1} + Y_{5} + Y_{6}&- Y_{5}&- Y_{6}\\\\0&0&0&- Y_{5}&Y_{5} + Y_{7}&- Y_{7}\\\\0&- Y_{3}&- Y_{8}&- Y_{6}&- Y_{7}&Y_{3} + Y_{6} + Y_{7} + Y_{8}\\end{array}\\right]\\]"
],
"text/plain": [
"6×6 Array{Sym,2}:\n",
" Y_1 + Y_2 -Y_2 0 … 0 0\n",
" -Y_2 Y_2 + Y_3 + Y_4 -Y_4 0 -Y_3\n",
" 0 -Y_4 Y_4 + Y_8 0 -Y_8\n",
" -Y_1 0 0 -Y_5 -Y_6\n",
" 0 0 0 Y_5 + Y_7 -Y_7\n",
" 0 -Y_3 -Y_8 … -Y_7 Y_3 + Y_6 + Y_7 + Y_8"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A' * Y * A"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Null space\n",
"\n",
"If we just say that the net current flowing out of each node is zero, we get the equation:\n",
"$$\n",
"A^T Y A v = 0\n",
"$$\n",
"or $v \\in N(A^T Y A) = N(A)$.\n",
"\n",
"It is an amazing and important fact that $N(A^T Y A) = N(A)$!! (You saw a version of this in homework.) Why is this? Clearly, if $Ax = 0$ then $A^T Y Ax=0$. But what about the converse? Here is a trick: if $A^T Y Ax =0$, then $x^T A^T Y A x=0 = (Ax)^T Y (Ax)$. Let $y=Ax$. It is easy to see that $y^T Y y = \\sum_i Y_i y_i^2 = 0$ only if $y=0$, since all of the admittances $Y_i$ are positive. (We will later say that $Y$ is a \"positive-definite matrix\".) This means that $A^T Y Ax =0$ implies that $y=Ax=0$, which implies that $x \\in N(A)$.\n",
"\n",
"What is $N(A)$? The rank of $A$ is 5, so $N(A)$ must be **1-dimensional**. A basis for it is:"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6×1 Array{Float64,2}:\n",
" -0.4082482904638629 \n",
" -0.408248290463863 \n",
" -0.4082482904638628 \n",
" -0.40824829046386313\n",
" -0.4082482904638631 \n",
" -0.4082482904638629 "
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"nullspace(A)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"But this is, of course, just the space of vectors where **all voltages are equal**. In hindsight, this should be obvious: if all the voltages are equal, then their difference are zero, and the currents are zero, and KCL is satisfied."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Current sources\n",
"\n",
"Of course, it is much more interesting to think about circuits when the currents are nonzero!\n",
"\n",
"To do this, we must consider a **source term** in the equations, and in particular we could try to solve\n",
"\n",
"$$\n",
"A^T Y A v = s\n",
"$$\n",
"\n",
"for some $s\\ne 0$. What does $s$ represent? It is precisely an **external source of current** flowing **out of each node**.\n",
"\n",
"For this to have a solution, however, we must have $s \\in C(A^T Y A) = N((A^T Y A)^T)^\\perp = N(A^T Y A)^\\perp = N(A)^\\perp$ (since $A^T Y A$ is symmetric, the left and right nullspaces are equal). We know a basis for $N(A)$ from above, so this boils down to:\n",
"\n",
"$$\n",
"\\begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 \\end{pmatrix} s = 0 = \\sum_{i=1}^6 s_i\n",
"$$\n",
"\n",
"That is, to have a solution, **all current that flows in must flow out**, so that the net current flowing into the circuit is zero. This makes a lot of physical sense!"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Just for fun, let's solve this circuit problem when the current is flowing **into node 2** and **out through node 1**, with slider controls for the 8 admittances, and label the edges with the currents."
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"twodigits (generic function with 1 method)"
]
},
"execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"twodigits(x) = round(x, digits=2)"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"application/vnd.webio.node+json": {
"children": [
{
"children": [
{
"children": [
{
"children": [
{
"children": [
{
"children": [
"Y₁"
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"source": [
"@manipulate for Y₁=0.1:0.1:10,\n",
" Y₂=0.1:0.1:10,\n",
" Y₃=0.1:0.1:10,\n",
" Y₄=0.1:0.1:10,\n",
" Y₅=0.1:0.1:10,\n",
" Y₆=0.1:0.1:10,\n",
" Y₇=0.1:0.1:10,\n",
" Y₈=0.1:0.1:10\n",
" s = [1,-1,0,0,0,0]\n",
" Y = diagm(0=>[Y₁,Y₂,Y₃,Y₄,Y₅,Y₆,Y₇,Y₈])\n",
" nodecolors!(labels(g, edges=[subchar(\"i_$j = $i\") for (j,i) in enumerate(twodigits.(Y*A*(pinv(A'*Y*A) * s)))]),\n",
" [1,2])\n",
"end"
]
},
{
"cell_type": "markdown",
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"source": [
"Notice that if we make the admittance $Y_2$ really large compared to all of the other admittances, then nearly all of the current should flow just over that one edge. Conversely, if we make $Y_2$ really *small*, it is almost like \"cutting\" that wire: almost all of the current should flow through the *other* edges.\n",
"\n",
"Hooray, math (and physics) works!"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Sparsity\n",
"\n",
"The case of matrices arising from graphs illustrates another point that I've made many times: **really large matrices are often sparse (mostly 0)** in practice.\n",
"\n",
"For example, imagine a circuit with a million nodes. For the most part, there will only be *wires between nearby nodes*. Or imagine a graph where the nodes are websites and the edges are links: there are billions of sites, but *each site only links to a few other sites* (a few hundred at most, usually). In such cases, the **incidence matrix is mostly zero**, and similarly for $A^T Y A$ etcetetera.\n",
"\n",
"This is hugely important, because solving $Ax=b$ and most other matrix equations scale as $\\sim n^3$ for $n \\times n$ matrices. $1000 \\times 1000$ matrices are easy (< 1 second), but $n=10^6$ would require supercomputers, and $n=10^9$ would be impossibly hard. What saves us is that there are **much faster algorithms for sparse matrices**. We won't learn much about such algorithms in 18.06, but the key point is to know that they exist.\n",
"\n",
"If you encounter a large sparse matrix problem in the future, go read about sparse matrix algorithms!"
]
}
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