Chapter 5 of Dynamical Processes on Complex Networks includes a discussion of an Ising model on a small-world network. It considers a system of $N$ Ising spins $\sigma_i = \pm 1$, $i = 1, ..., N$ with Hamiltonian $$H(\sigma) = -J \sum_{i=1}^N \sigma_i \sum_{j=1}^m \sigma_{\mu (i, j)}$$
There are periodic boundary conditions (i.e. $\sigma_{N+j}= \sigma_j$).
$\mu (i, j)$ are independent and identically distributed numbers drawn from the following probability distribution: $$P(\mu (i, j)) = (1 - p) \delta_{\mu (i, j), i + j} + \frac{p}{N} \sum_{t=1}^N \delta_{\mu (i, j),l}$$
So, I understand that $N$ is the network size, $J$ is the amount that the spins are influencing each other, and $\sigma_i$ is the spin of node $i$. However, I don't understand exactly what $m$ is, and I'm very confused about what $\mu (i, j)$, $\delta_{\mu (i, j), l} $ and $\delta_{\mu (i, j), i + j}$ are.
Can someone explain what these terms are referring to?
I've no clue what $m$ is supposed to be, but $\delta_{j,k}$ is just the Kronecker $\delta$. That is, $\delta_{j,k}=1$ if $j=k$ and $0$ else. Thus, $\mu_{i,j}$ is equal to $i+j$ with probability $(1-p)+\frac{p}{N}$ and is equal to any other $j$ with probability $\frac{p}{N}$.
So I guess your model has spins interacting with a random set of $m$ neighbours - with a bias towards intearcting with the $m$ next spins. Studying an Ising model with non-symmetric interactions seems very exotic to me, but I guess that's besides the point.