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I was reading chapter 5 of Dynamical Processes on Complex Networks, which discusses the Ising model, where I encountered the following equation for the average magnetization of the class of nodes with degree k: $$\langle \sigma \rangle_k = \frac{1}{N_k}\sum_{i/k_i = k} \langle \sigma_i \rangle$$

I'm particularly confused by the notation $$\sum_{i/k_i = k}$$

Can anyone explain what this means?

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$$\sum\limits_{i\mid k_i=k} a_i,$$ also written $$\sum\limits_{i: k_i=k} a_i,$$ means to sum $a_i$ over indices $i$ such that $k_i=k$.

To make it clearer that the formula is an average, I probably would have first defined $N_k = \{i \in N: k_i = k\}$ as the set of nodes with degree $k$ and then used the expression $$\langle \sigma \rangle_k = \frac{1}{|N_k|}\sum_{i\in N_k} \langle \sigma_i \rangle$$

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  • $\begingroup$ / and | are equivalent notations in this case, then? $\endgroup$ – EJoshuaS - Reinstate Monica Apr 13 at 2:18
  • $\begingroup$ Yes, but $\mid$ and $:$ are more common. $\endgroup$ – RobPratt Apr 13 at 2:19
  • $\begingroup$ Thanks, that makes a lot of sense in this context. I'll accept the answer as soon as the system allows me to. $\endgroup$ – EJoshuaS - Reinstate Monica Apr 13 at 2:20
  • $\begingroup$ If $i/k_i=k$ for integers, then it seems that $k\mid i$, rather than $i\mid k$. Does dynamical systems really standardly reverse things/misuse $/$ in this way? $\endgroup$ – Mark S. Apr 13 at 10:47
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    $\begingroup$ @MarkS. there is no division here. The $/$ or $\mid$ indicates such that here. Better notation would have been $i\in N: k_i=k$. $\endgroup$ – RobPratt Apr 13 at 12:48

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