The Laplacian operator is a linear operator that measures how much a function varies over a simplicial complex. The eigencochains of the Laplacian operator are functions that have a special relationship with the Laplacian operator.

Construction can be seen as follows - Paper 1 has 100 citations and has been published by A, B and C. Face with ABC as authors will have a value of 100. Paper two has a total of citations 50 and is published by A and B. We see the edge between A and B has a value of 150. We also observe the edge values >= face value. Since A and B have a total of 150 citations together we say that the value of the face corresponding to A and B will be 150. Face in this case will be an edge as the data is bounded by only two dimensions. The value of any vertices is the value of the total citations received by the author. A has a total of 160 citations with co-authorship with B, C, and D. Similarly we can see that the citations are an accumulation of citations received by each paper.

Persistent homology is a method for computing topological features of a space at different spatial resolutions. It is used to detect more persistent features over a wide range of spatial scales and identify them as more likely to be significant, as opposed to noise or a particular choice of parameters. To find the persistent homology of a space, the space must first be represented as a simplicial complex, and a distance function on the underlying space corresponds to a filtration of the simplicial complex. The essence of persistent homology is to track the birth and death of homological features, providing a robust and informative way to study qualitative features of data that persist across different spatial scales. It is a valuable tool in topological data analysis, as it is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input.