# HellRank

#### Definition

HellRank is based on the Hellinger distance between two nodes on the same side of a bipartite network. HellRank is a similarity-based centrality measure. Since the similarity measure is usually inverse of the distance metrics, in proposed method it first choose a suitable distance measure, namely Hellinger distance.

Hellinger distance is one type of the f-divergence metric that is a function $D_f (P||Q)$ that measures the difference between two probability distributions $P$ and $Q$. In another word, a type of f-divergence measure, that indicates structural similarity of each node to other network nodes. Hellinger distance defines as fallows:

$$D_H (P||Q)={1\over \sqrt 2} || \sqrt P -\sqrt Q||_2$$

were $P$ and $Q$ are two discrete probability distributions.

$$HellRank^{*}(x)=HellRank(x).{\underset{z\in v_1}{min}}(HellRank(z))$$

$HellRank^{*}(x)$ is the normalized $HellRank$ of node $x$, $ ". "$ denotes the multiplication dot, and ${\underset{z\in v_1}{min}}(HellRank(z))$ is the minimum possible HellRank for each node. The HellRank is a Hellinger-based centrality measure and defined as fallows:

$$HellRank(x)= {n_1\over {\sum}_{z\in V_1}d(x,z)} $$

were $d(x,z)$ is Hellinger distance between two nodes of the weighted bipartite network. $z$ is any other nodes in the same side of the bipartite network and $n$ denotes nodes in each side of a bipartite network

The nodes with high HellRank centrality are more behavioral representative nodes in bipartite social networks. The computation of the HellRank centrality measure can be distributed, by letting each node uses local information only on its immediate neighbors. Consequently, one does not need a central entity that has full knowledge of the network topological structure

Hellinger distance is one type of the f-divergence metric that is a function $D_f (P||Q)$ that measures the difference between two probability distributions $P$ and $Q$. In another word, a type of f-divergence measure, that indicates structural similarity of each node to other network nodes. Hellinger distance defines as fallows:

$$D_H (P||Q)={1\over \sqrt 2} || \sqrt P -\sqrt Q||_2$$

were $P$ and $Q$ are two discrete probability distributions.

$$HellRank^{*}(x)=HellRank(x).{\underset{z\in v_1}{min}}(HellRank(z))$$

$HellRank^{*}(x)$ is the normalized $HellRank$ of node $x$, $ ". "$ denotes the multiplication dot, and ${\underset{z\in v_1}{min}}(HellRank(z))$ is the minimum possible HellRank for each node. The HellRank is a Hellinger-based centrality measure and defined as fallows:

$$HellRank(x)= {n_1\over {\sum}_{z\in V_1}d(x,z)} $$

were $d(x,z)$ is Hellinger distance between two nodes of the weighted bipartite network. $z$ is any other nodes in the same side of the bipartite network and $n$ denotes nodes in each side of a bipartite network

The nodes with high HellRank centrality are more behavioral representative nodes in bipartite social networks. The computation of the HellRank centrality measure can be distributed, by letting each node uses local information only on its immediate neighbors. Consequently, one does not need a central entity that has full knowledge of the network topological structure

#### References

- Taheri S.M., Mahyar H., Firouzi M., Ghalebi E., Grosu R., Movaghar A., 2017. HellRank: a Hellinger-based centrality measure for bipartite social networks. Social Network Analysis and Mining, 7(1). DOI: 10.1007/s13278-017-0440-7