# Heatmap Centrality

#### Definition

A new centrality measure, termed the heatmap centrality, utilizes both local and global network information by comparing the farness of each node with the average sum of the farness of its neighbor nodes. In particular, the heatmap centrality for node $v_i$, denoted as $CHM_{(v_i)}$, is defined formally as:

$$C_{HM}(v_i)= {\underset{j=1}{\overset{N}{\sum}}} s(v_i, v_j) - {{\sum}_{j=1}^N a_{i,j} . {\sum}_{k=1}^N s(v_i, v_j)\over {\sum}_{j=1}^N a_{i,j}}$$

The heatmap measure identifies the “hot spot” node within its neighborhood, as it considers a node with a smaller farness than that of the average of its neighbors to be an influential node within the network. Intuitively, a node with the smaller farness among that of its neighbors is more likely to have information pass specifically through it, rather than through any of the adjacent nodes. When the sign of $CHM_{(v_i)}$ transitions from negative to positive, then the average sum of the farness of the neighbors of $v_i$ becomes smaller than that of node $v_i$, decreasing the likelihood of information passing specifically through node $v_i$. Therefore, using this intuition, the heatmap centrality can be considered a shortest path based measure and utilized in the identification of super-spreader nodes that control the flow of information within a scale-free network.

$$C_{HM}(v_i)= {\underset{j=1}{\overset{N}{\sum}}} s(v_i, v_j) - {{\sum}_{j=1}^N a_{i,j} . {\sum}_{k=1}^N s(v_i, v_j)\over {\sum}_{j=1}^N a_{i,j}}$$

The heatmap measure identifies the “hot spot” node within its neighborhood, as it considers a node with a smaller farness than that of the average of its neighbors to be an influential node within the network. Intuitively, a node with the smaller farness among that of its neighbors is more likely to have information pass specifically through it, rather than through any of the adjacent nodes. When the sign of $CHM_{(v_i)}$ transitions from negative to positive, then the average sum of the farness of the neighbors of $v_i$ becomes smaller than that of node $v_i$, decreasing the likelihood of information passing specifically through node $v_i$. Therefore, using this intuition, the heatmap centrality can be considered a shortest path based measure and utilized in the identification of super-spreader nodes that control the flow of information within a scale-free network.

#### References

- Duron C., 2020. Heatmap centrality: A new measure to identify super-spreader nodes in scale-free networks. PLoS ONE, 15(7 July). DOI: 10.1371/journal.pone.0235690