# A Note on Matricial Ways to Compute Burt’s Structural Holes

## Abstract

**:**

## 1. Introduction

## 2. Notation

## 3. Effective Size and Redundancy for Undirected Binary Networks

#### An Example

## 4. Local Constraint (a.k.a. Dyadic Constraint)

- $\mathbf{x}=(A+{A}^{T})\mathbf{1}$;
- $\mathbf{y}=\mathbf{1}\oslash \mathbf{x}$;
- $P=Diag\left(\mathbf{y}\right)\phantom{\rule{4pt}{0ex}}(A+{A}^{T})$;
- $L=\left[P+(A\odot P)P\right]\odot \left[P+(A\odot P)P\right]$.

## 5. Constraint

## 6. Improved Structural Holes

- The edge weight between two connected nodes i and j is ${w}_{ij}:={d}_{i}+{d}_{j}$. In matricial form,$$W=A\odot \left(\mathbf{d}{\mathbf{1}}^{T}+\mathbf{1}{\mathbf{d}}^{T}\right)=A\odot \left(A\mathbf{1}{\mathbf{1}}^{T}+\mathbf{1}{\mathbf{1}}^{T}{A}^{T}\right).$$
- The node weight is ${w}_{i}:={\sum}_{j\in N\left(i\right)}{w}_{ij}$, so that $\mathbf{w}=W\mathbf{1}\in {\mathbb{R}}^{n}$.
- The relative importance is ${q}_{ij}:=\frac{{w}_{ij}}{{w}_{i}}$, with ${\sum}_{j}{q}_{ij}=1$, and can be represented as$$Q=W\oslash \left(W\mathbf{1}{\mathbf{1}}^{T}\right).$$

## 7. Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Adapted from Burt’s and Borgatti’s works. Let us compute the effective size for node A. Consider one of its neighbors, say G. They have 3 “common neighbors”: B, E, and F. Divide this number by A’s degree: $3/4$. Take another of A’s neighbors, say E. A and E have just 1 common neighbor, which is G. Therefore, in this case, dividing by A’s degree gives $1/4$. To determine A’s redundancy, repeat this process for all 4 neighbors of A (respectively, B, E, F, and G) and sum up all the numbers obtained: $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{3}{4}=\frac{6}{4}=1.5$. Lastly, A’s effective size is its degree minus its redundancy: $4-1.5=2.5$. In the example of this section, the matricial computation is also done for the remaining nodes.

**Figure 2.**As an example, we perform a basic comparison of computational speed between this paper’s algorithms (orange) and NetworkX’s algorithms (blue) for effective size (left) and constraint (right), as the number of nodes increases from 1000 to 10,000. The networks are Barabasi-Albert graphs with parameter $m=5$ (top row) and Erdos-Renyi random graphs with parameter $p=0.01$ (bottom row). In particular, in orange, effective size and constraint are, respectively, computed using the formulas in Equations (4) and (17). All of the code to implement it is available here: https://github.com/alessiomuscillo/matricial_ways_for_Burt_structural_holes (accessed on 27 October 2022). However, a more precise comparison in a controlled environment should show that more efficient methods could scale better and be better suited for large networks. (All links accessed on 27 October 2022).

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**MDPI and ACS Style**

Muscillo, A.
A Note on Matricial Ways to Compute Burt’s Structural Holes. *Symmetry* **2023**, *15*, 211.
https://doi.org/10.3390/sym15010211

**AMA Style**

Muscillo A.
A Note on Matricial Ways to Compute Burt’s Structural Holes. *Symmetry*. 2023; 15(1):211.
https://doi.org/10.3390/sym15010211

**Chicago/Turabian Style**

Muscillo, Alessio.
2023. "A Note on Matricial Ways to Compute Burt’s Structural Holes" *Symmetry* 15, no. 1: 211.
https://doi.org/10.3390/sym15010211