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26 lines
1.5 KiB
TeX
26 lines
1.5 KiB
TeX
\section{Harmonic centrality}
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The algorithm shown in this paper is very versatile. We have tested it with two different graphs and obtained excellent results. But there could be more.
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\s \nd It can be adapted very easily to compute other centralities, as the harmonic one \cite{2000}. Given a graph $G = (V,E)$ and a node $v \in V$, it's defined as
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\begin{equation}
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h(v) = \sum_{w \neq v} \frac{1}{d(v,w)}
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\end{equation}
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\nd The main difference here is that we don't have a farness. Then we won't need a lower bound either. Since the biggest the number is the higher is the centrality, we have to adapt the algorithm. Instead of a lower bound, we need an upper bound $U_B$ such that
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\begin{equation}
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h(v) \leq U(v) \leq h(w)
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\end{equation}
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\nd A possibile upper bound can be taken considering the worst case that could happen at each state
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\begin{equation}
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U(v) = \sigma_{d-1} + \frac{n_d}{d} + \frac{n - r - n_d}{d+1}
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\end{equation}
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\nd When we are at the level $d$ of our exploration, we already know the partial sum $\sigma_{d-1}$. The worst case in this level happens when the node $v$ is connected to all the other nodes. To consider this possibility we add the factors $\frac{n_d}{d} + \frac{n - r - n_d}{d+1}$.
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\s \nd This method has been tested and works with excellent results. What needs to be adjusted is a formal normalization for the harmonic centrality and for the upper bound. In the Github repository, the script already gives the possibility to compute the top-k harmonic centrality of both graphs.
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