# NISQ-Ready Community Detection Based on Separation-Node Identification

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## Abstract

**:**

## 1. Introduction

**C**ommunity

**D**etection based on

**S**eparation-

**N**ode identification (CDSN). This approach is specialized for (quantum heuristic) QUBO solving that uses a smaller search space than the state-of-the-art quantum modularity maximization approach [13]. This objective led to the sociologically inspired approach of defining a community by its extreme ends, similar to, e.g., differentiating political parties by their position on the left–right spectrum. For graphs, we translate this idea to the existence of what we later define as a bijective set of separation nodes. The removal of the nodes contained in this set then yields connected components, which represent the “cores” of the communities. We subsequently conduct experiments that indicate that this essentially solves the computationally hard part of the community detection problem, as the community assignment for the separation nodes can typically be obtained using a greedy optimizer.

## 2. Background

- Randomly split the given graph $G=\left(\right)open="("\; close=")">V,E$ into two equally sized partitions $A\dot{\cup}B=V$ and delete all edges inside the partitions to yield a bipartite graph.
- Find subsets $X\subseteq A$ and $Y\subseteq B$ such that $X=\left(\right)open="\{"\; close="\}">{v}_{i}\in A\mid {s}_{i}=1$ and$Y=\left(\right)open="\{"\; close="\}">{v}_{j}\in B\mid {s}_{j}=1$ where $s=\left(\right)open="("\; close=")">{s}_{1},\dots ,{s}_{\left|V\right|}$ is the solution to the quadratic program given by$$\underset{s\in {\left(\right)}^{0}\left|V\right|}{arg\; min}{s}_{i}{s}_{j}.$$
- Identify $C:=X\cup Y$ to be a community and repeat Steps 1 and 2 for the subgraph induced on G by $V\setminus C:=\left(\right)open="\{"\; close="\}">v\in V\mid v\notin C$.

## 3. Proposed Model

#### 3.1. Separation-Node Sets

- (1)
- Identifying a set of nodes separating communities and thus revealing the fundamental community structure (see Section 3.4 and Section 3.5).
- (2)
- Classifying the community of each separation node to finalize community detection (see Section 3.6).

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

#### 3.2. Proving Theorem 1

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Proof.**

- $P\left(\tilde{x}\right)=0$ and the separation-node set $\tilde{S}$ is smaller than S;
- $P\left(\tilde{x}\right)>0$ and the separation-node set $\tilde{S}$ is much smaller than S.

- $\left(\right)open="|"\; close="|">{S}^{*}$;
- $\left(\right)open="|"\; close="|">{S}^{*}$.

#### 3.3. Constructing Penalty Terms for the In- and Surjectivity Constraints

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

#### 3.4. Modularity-Based Separation Edge Estimation

- ${m}_{ij}>0$, if less connectivity between ${v}_{i}$ and ${v}_{j}$ was to be expected, indicating that ${v}_{i}$ and ${v}_{j}$ likely belong to the same community;
- ${m}_{ij}<0$, if more connectivity between ${v}_{i}$ and ${v}_{j}$ was to be expected, indicating that ${v}_{i}$ and ${v}_{j}$ likely belong to different communities.

#### 3.5. Separation Edge Estimation Based on Edge Neighborhood Connectivity

- (1)
- Consider connections between r-neighborhoods with radius $r\ge 0$;
- (2)
- Consider paths of length 2.

#### 3.6. Assigning the Separation Nodes to Communities

- (1)
- Count the number of edges to every identified community for each separation node.
- (2)
- Assign the node with the most edges to a single community to that community. In case of a tie, the community that reached the highest number of edges first during the iteration over all adjacent nodes is selected.
- (3)
- Update the counts for every neighboring separation node.
- (4)
- Repeat steps two and three until every separation node is properly assigned to a community.

## 4. Evaluation

- (1)
- The assignment of separation nodes to their communities is computationally easy given a good enough estimator, i.e., it can be executed accurately in linear runtime with respect to the number of communities for each separation node.
- (2)
- Neighborhood connectivity constitutes a suitable estimator for separation edges, i.e., it can be employed to identify an adequate separation-node set in the here-proposed approach to conduct community detection in practice.

- Modularity. For the comparability between different datasets, we use the approximation ratio based on the best known solution. This yields values between 0 (bad) and 1 (good).
- NMI score. The NMI score is used to compare the community assignments with known ground truth. It yields values between 0 (bad) and 1 (good).
- ${R}^{2}$ score. This score is used to estimate predictive performance of the separation edge classification. It yields values between 0 (bad) and 1 (good).

#### 4.1. Evidence That Separation-Node Assignment Is Computationally Cheap

#### 4.2. Neighborhood Connectivity Constitutes a Suitable Estimator for Separation Edges

#### 4.3. Evaluating the Performance of Edge Neighborhood Connectivity

**Figure 5.**This box plot displays the fraction of the achieved modularity score by the best known solution for selected standard benchmark datasets: (1) the social network of a karate club [46], (2) the social interactions between dolphins [47], (3) the collectively appearing characters in the book “Les Miserables” [48], (4) protein–protein interactions [49] and (5) jointly bought political books [50]. Each graph was analyzed 10 times using simulated annealing. Our approach clearly does not work well for the karate club network. Closer inspections yield the result signifying that the connected components resulting from the found separation-node sets often only consist of single nodes, indicating suboptimality in using neighborhood connectivity for this relatively small dataset.

#### 4.4. Evaluating the Separation-Node Assignment

**Figure 7.**${R}^{2}$ score of the edge-neighborhood-connectivity-based separation edge estimator. In practice, an ${R}^{2}$ score of 30% implies that merely 30% of the variability of the ground truth has been accounted for. A strict trend towards worse results for harder datasets is clearly visible. This shows that the performance of the estimator decreases for harder problem instances as to be expected while still yielding somewhat accurate results.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Outline of the workflow for the proposed approach of community detection via separation-node identification. The computationally expensive tasks of identifying a set of separation nodes (

**b**) and classifying the communities for these nodes (

**d**) are performed using quantum computing, while the computationally cheap tasks of removing the classified separation-nodes and identifying the resulting connected components (

**c**) are performed classically.

**Figure 2.**Counterexample proving no-free-lunch when using Theorem 1 to find surjective separation-node sets.

**Figure 3.**Counterexample indicating no-free-lunch when using Theorem 1 to find injective separation-node sets.

**Figure 4.**This figure shows the Normalized Mutual Information (NMI) score of the presented approach for 50 different graphs each based on ground truth and a perfect separation edge estimator coupled with the greedy separation-node assignment. The NMI score as defined in [43,44] was used, as it resembles a well-proven measure for the accuracy of a community given the ground truth [45]. The different probabilities for intra-community edges in the chosen SBM model resemble different difficulties according to the phase transition known for this model. The lower the stated probability, the harder the problem. The probabilities were chosen such that the hardest graphs barely differed from a null model inheriting no measurable structure up to the hardest that still allowed perfect NMI scores. For this dataset, the phase transition can be calculated to be at a probability of 0.2865 for the intra-community edges. As modularity maximization has been shown to perform very well up until the sharp phase transition (which is not reached here), the constantly good results for the SA based approach appear to be reasonable. Additional tests show a sharp performance drop off to NMI values at around 0.5 for smaller intra-probabilities such as 0.23.

**Figure 6.**The y-axis depicts the deviation factor from the best-known separation-node set in size. Notably, the absolute sizes of the identified separation-node sets are typically similar over the different difficulties, while they rise slightly for larger graphs.

**Figure 8.**This figure depicts the normalized mutual information score of the selected SBM benchmark graphs using the greedy assignment of separation nodes to communities. A substantial drop off in performance can be observed for the harder datasets. Meanwhile, as all problem instances are significantly above the phase transition for modularity maximization in these datasets (an intra-prob of 0.2865), our classical baseline easily identifies close to optimal solutions. Notably, however, it is promisingly slightly outperformed by our approach in the case of the easiest dataset.

**Figure 9.**This figure depicts the normalized mutual information score of the selected SBM benchmark graph using a simulated annealing-based approach of assigning the separation nodes to communities. The worse performance for the easy dataset clearly indicates that the chosen simulated annealing approach based on the QUBO as described in Section 3.6 is suboptimal in general.

**Table 1.**Summary of all employed datasets. Note that five different SBM graphs were utilized, each with the same number of nodes and communities, but with varying probabilites of edges being inside communities. For details on all datasets, see the following sections.

No. of Nodes | No. of Communities | Intra Prob | |
---|---|---|---|

SBM graphs | 250 | 7 | $[0.75,0.625,0.5,0.4,0.3]$ |

Karate Club | 24 | 2 | cannot be specified |

Dolphins | 62 | 4 | cannot be specified |

Miserables | 77 | 5 | cannot be specified |

Protein | 83 | 9 | cannot be specified |

Books | 105 | 3 | cannot be specified |

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## Share and Cite

**MDPI and ACS Style**

Stein, J.; Ott, D.; Nüßlein, J.; Bucher, D.; Schönfeld, M.; Feld, S.
NISQ-Ready Community Detection Based on Separation-Node Identification. *Mathematics* **2023**, *11*, 3323.
https://doi.org/10.3390/math11153323

**AMA Style**

Stein J, Ott D, Nüßlein J, Bucher D, Schönfeld M, Feld S.
NISQ-Ready Community Detection Based on Separation-Node Identification. *Mathematics*. 2023; 11(15):3323.
https://doi.org/10.3390/math11153323

**Chicago/Turabian Style**

Stein, Jonas, Dominik Ott, Jonas Nüßlein, David Bucher, Mirco Schönfeld, and Sebastian Feld.
2023. "NISQ-Ready Community Detection Based on Separation-Node Identification" *Mathematics* 11, no. 15: 3323.
https://doi.org/10.3390/math11153323