Electrical networks

Mathematically an eletrical network is just a non negative weighted graph. We call this edge weights conductances, their reciprocals resistences. The reason for this terminology is that physics can aid our intuition.

Def A Network is a tuple $(V,E,c)$ where $c:E\to [0,\infty ]$ is a non negative weight function on the edges of the graph (possibly with loops).

In a natural way we can define transition probabilities between state $i,j\in V$:

$$p(i,j):=\frac{c(ij)}{{\sum }_{v\in \Gamma (i)}c(iv)}$$

Claim The transition matrix $P$ such defined is reversibile: Proof We show that $c(i):={\sum }_{j\in \Gamma (i)c(ij)}$ is an invariant measure, since it satisfies detailed balance:

$$c(i)p(i,j)=c(ij)=c(j)p(j,i),\square$$

The converse is also true: evert reversible markov chain can be turned into a network (we can find the right conductances) by using the function $\lambda$ that satisfies D.B:

$$c(ij):=P(i,j)\lambda (i)$$

We now show that this works. Note that reversibility is necessary, since $c(ij)=c(ji)$. Also we have the desired transition probabilities, since

$$c(i)=\sum _{j\in \Gamma (i)}c(ij)=\sum _{j}P(i,j)\lambda (i)=\lambda (i)$$

so that the transition probabilities are

$$\frac{c(ij)}{c(i)}=\frac{P(i,j)}{\lambda (i)}\lambda (i)=P(i,j)\square$$

Def (Voltage) Given two disjoint subsets $A$ and $Z$ of vertices of a network, a voltage is a function on the vertices of the network such that is harmonic on all $x\notin A\cup Z$. Usually the voltage is specified to be $1$ on $A$ and $0$ on $Z$.

Def (Current) Given a voltage on a network the current is a function $i:V\times V\to \mathbb{R}$ defined as:

$$i(x,y):=c(x,y)[v(x)-v(y)]$$

Remarks From the definition we see that $i$ is antisymmetric

$$i(x,y)=-i(y,x)$$

Also if $v$ is harmonic at $x$ (that is $x\notin A\cup Z$) we can se that

$$\sum _{y\sim x}i(x,y)=v(x)\sum _{y\sim x}c(x,y)-\sum _{y\sim x}v(y)c(x,y)=0$$

A function that satisfies these two properties is called a flow from source $A$ to sink $Z$.

This definition and remarks can be stated as the familiar Ohm’s and Kirchhoff’s laws:

• Ohm’s law

$$v(x)-v(y)=i(x,y)r(x,y)$$

• Kirchhoff’s node law The current is a flow from $A$ to $Z$.

• Kirchhoff’s cycle law If $x_1{x}_2\dots x_n=x_1$ is a cycle, then

$$\sum _{i=1}^{n}r(x_i,x_{i+1})i(x_i,x_{i+1})=0$$

The cycle law can be proven by summing Ohm’s law.

A more general statement can be proven:

Claim Given an antisymmetric function $j$ on the edges of a connected network that satiesfies Kirchhoff node and cycle law, show that we can construct a voltage function up to an additive function such that $j$ is its associeted current. Proof Choose a vertex $\overline{x}\in V$, and fix the value $v(\overline{x})=K$. Define the voltage $v$ for every other vertex $x\in V$ as:

$$v(x):=\sum _{\mathcal{P}}i(x_i,x_{i+1})r(x_i,x_{i+1})+K$$

where $\mathcal{P}$ is any path from $x$ to $\overline{x}$. This function is well defined, since at least one such paths exists (the network is connected), and the value of $v(x)$ doesn’t depend on the path choosen: suppose $\mathcal{P}$ and ${\mathcal{P}}^{\prime }$ are two different path from $x$ to $\overline{x}$, joining them (one in reverse order) we get a cycle, using Kirchhoff cycle law their difference is zero:

$$\sum _{\mathcal{P}}j(x_i,x_{i+1})r(x_i,x_{i+1})-\sum _{{\mathcal{P}}^{\prime }}j(x_i,x_{i+1})r(x_i,x_{i+1})=$$ $$\sum _{\mathcal{P}}j(x_i,x_{i+1})r(x_i,x_{i+1})+\sum _{{\mathcal{P}}^{\prime }}j(x_{i+1},x_i)r(x_{i+1},x_i)=0$$

where we have used the antisymmetri of the flow and the symmetry of the resistences.

It remains to show that $v$ is harmonic. Let $y$ be a neighbour of $x$, then

$$v(x)=r(x,y)j(x,y)+v(y)$$

rearrenging this identity and summing over all neighbours $y$ we get

$$v(x)\sum _{y\sim x}c(x,y)=\sum _{y\sim x}j(x,y)+\sum _{y\sim x}v(y)c(x,y)$$

the first term on the right is zero since $j$ is a flow. We are left with the definition of an harmonic function at $x$. $\square$