# Object-Based Dynamics: Applying Forman–Ricci Flow on a Multigraph to Assess the Impact of an Object on The Network Structure

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

- 1.
- The upper boundary window size (WS) defines the maximum number of words between ${v}_{i}\mathrm{and}{v}_{j}$. The sublist p is therefore taken into account when the number of words between ${v}_{i}$ and ${v}_{j}$ is less than or equal to the number WS.
- 2.
- MS is a number defining the lower boundary, which is the minimum number of p’s containing ${v}_{i}$ and ${v}_{j}$ in that order, and with, at most, WS words between them.

## 3. Results

#### 3.1. Forman–Ricci Flow

#### 3.1.1. Forman–Ricci Curvature

- 1.
- The smaller the weight ${w}_{e}$ of the edge $e$, the more positive the curvature of $e$.
- 2.
- The greater the weighted in/out-degree of ${v}_{1}$ and ${v}_{1}$, the more positive the curvature of $e$. The weighted in-degree of vertex ${v}_{1}$ takes into account the weights ${e}_{1j}$, where ${e}_{1j}=\left({v}_{1,}{v}_{j}\right),j\ne 2$, while the weighted out-degree of vertex ${v}_{2}$ takes into account the weights ${e}_{j2},$ where ${e}_{j2}=\left({v}_{j,}{v}_{2}\right),j\ne 1$.
- 3.
- The smaller the $\mathrm{deg}({v}_{1})$ and/or $\mathrm{deg}({v}_{2})$, the more positive the curvature of $e$.

#### 3.1.2. Forman–Ricci Flow

- 1.
- Apply Dijkstra’s shortest path algorithm for any edge based on ${w}_{ij}^{t}$ to calculate ${d}_{ij}^{t}$;
- 2.
- Compute the Forman–Ricci curvature ${\kappa}_{ij}^{t}$ for any edge based on ${d}_{ij}^{t}$;
- 3.
- Update the edge weight by Formula (4);
- 4.
- Repeat steps 1–3 for N iterations.

#### 3.1.3. Analysis

#### 3.2. Object-Based Dynamic Measure

- 1.
- Define index $I$ for graph $\widehat{G}$ such that $I=\left\{{\left.\widehat{G}\right|}_{s},{\left.\widehat{G}\right|}_{\neg s}\right\},$ so that $s$ is represented by two graphs.
- 2.
- For each element in $I$, define the weights for ${\left.\widehat{G}\right|}_{s}$ as ${w}_{ij}\in {\left.\widehat{G}\right|}_{s}:={w}_{ij}|{s}_{i-k}$, for ${\left.\widehat{G}\right|}_{\neg s}$ as ${w}_{ij}\in ,{\left.\widehat{G}\right|}_{\neg s}:={w}_{ij}|\neg {s}_{i-k}$, for $k=1\cup 2$ and $j\le i+WS$.
- 3.
- Compute steps 1–4 of the Forman–Ricci flow algorithm for each of the graphs $\left\{{\left.\widehat{G}\right|}_{s},{\left.\widehat{G}\right|}_{\neg s}\right\}$.
- 4.
- For any ${w}_{ij}$ and ${w}_{ij}\in {\left.\widehat{G}\right|}_{s}$ and ${w}_{ij}\in {\left.\widehat{G}\right|}_{\neg s}$.

#### 3.2.1. Study Case

#### 3.2.2. Analysis

#### 3.3. Exploratory Comparison Analysis

#### 3.3.1. Centrality Measures

- The number of triangles: This measure calculates the number of undirected three-cliques for each vertex in the graph. The number of triangles is used to detect vertices that belong to numerous cliques. It is worth noting that this measure is closely related to the clustering coefficient.
- Closeness: This measure is the inverse of farness, which is defined as the mean of the shortest paths to all other vertices [30,31]. Closeness can be interpreted as the expected time of arriving at a word through the graph’s shortest paths. The gist of this metric is to assign more importance to the vertices that are closest. The definition is as follows:$$C\left(v\right)=\frac{N-1}{{{\displaystyle \sum}}_{u}\delta \left(v,u\right)}$$
- PageRank: The basic idea of the PageRank algorithm, first introduced in a Google paper [32], is that a central vertex is determined not only by the number of incoming edges (in-degree) but also by the level of importance of the incoming vertices. T represents the set of vertices, ${N}_{u}$ represents the number of vertices to which vertex v points, and ${\mathrm{S}}_{V}$ represents the set of vertices pointing to vertex v. Finally, $\alpha $ is the damping factor of the probability of jumping from a given vertex to another random vertex in the graph. PageRank is computed as follows:$$\mathrm{PR}\left(\mathrm{v}\right)={\displaystyle \sum}_{u\in {S}_{V}}\frac{PR\left(u\right)}{{N}_{u}}+\frac{1-\alpha}{T}$$
- Betweenness: This measure was introduced by Freeman [33] in order to quantify the extent to which a vertex tends to be on the shortest paths between other vertices—in other words, to serve as an intermediary. Betweenness for a vertex v is defined as follows:$$\mathsf{\gamma}\left(v\right)={\displaystyle \sum}_{i\ne v\ne j\in V}\frac{{\sigma}_{ij}\left(v\right)}{{\sigma}_{ij}}$$
- LDC: In Cohen et al. [34], we recently proposed local detour centrality (LDC), a novel centrality measure for weighted and directed graphs that quantifies the tendency of a given vertex to lie on the shortest possible path between other vertices. In other words, LDC indicates whether the possible path between ${v}_{1}$ and ${v}_{2}$ is significantly closer when $v$ lies between them, in contrast to cases in which $v$ does not lie between them. Let $G=\left(V,E\right)$ be an edge-weighted and directed graph in which $w\left({v}_{\mathrm{i}},{v}_{\mathrm{j}}\right)$ represents the weights from ${v}_{\mathrm{i}}$ to ${v}_{\mathrm{j}}$ and let $\delta \left({v}_{\mathrm{i}},{v}_{\mathrm{j}}\right)$ denote the shortest path from ${v}_{\mathrm{i}}$ to ${v}_{\mathrm{j}}$ based on Dijkstra’s shortest path algorithm. For any vertex $v$,
- Let $L=\left\{{v}_{1},{v}_{2}\dots {v}_{n}\right\}$ such that any ${v}_{\mathrm{i}}\in L$ if $\delta \left(v,{v}_{\mathrm{i}}\right)\le r$ or $\delta \left({v}_{\mathrm{i}},v\right)\le r$. The number $r=\frac{1}{\left|\mathrm{V}\left(G\right)\right|}{\displaystyle \sum}_{{v}_{i,}{v}_{j}\in V}\delta \left({v}_{i},{v}_{j}\right)$ is called the threshold.
- Let ${G}_{v}\subset G$ be a complete graph with $V\left({G}_{v}\right)=L$ where, in this case, $w\left({v}_{i},{v}_{j}\right)=\delta \left({v}_{\mathrm{i}},{v}_{\mathrm{j}}\right)$.

#### 3.3.2. Scalar Curvature

- 1.
- Haantjes curvature: This curvature [36] is a path-based measure and can be generalized to a path of length $n$ by replacing path ab ∪ bc with a path of length $n$. For a general discrete notion of Haantjes curvature, let $\pi =\left({v}_{0},\dots ,{v}_{n}\right)$ be a directed path between the vertices ${v}_{0}$ and ${v}_{n}$. The simplified Haantjes-Ricci curvature of the path $\pi $ is then defined as follows [37]:$${\kappa}_{H}^{2}\left(\pi \right)=\frac{l\left(\pi \right)-d\left({v}_{0},{v}_{n}\right)}{d\left({v}_{0},{v}_{n}\right)}$$Next, the Haantjes-scalar curvature of $v$ in $G$ is defined as follows:$$\kappa {s}_{H}^{2}\left(v\right)={\displaystyle \sum}_{\pi ~v}{\kappa}_{H}^{2}\left(\pi \right)$$
- 2.
- Menger–Ricci curvature: In 1930 Menger introduced a discrete definition of Ricci curvature K(T) for any three vertices in the network. Let (M, d) be a metric space and T = T(a, b, c) be a triangle with sides lengths a, b, c, and denote $p=\frac{a+b+c}{2}$. Then the Menger curvature of T is given by$${K}_{m}\left(T\right)=\frac{\sqrt{p\left(p-a\right)\left(p-b\right)\left(p-c\right)}}{abc}$$The Menger–Ricci curvature of a vertex e in a network can be defined as$${K}_{m}\left(v\right)={\displaystyle \sum}_{{T}_{e}~v}{K}_{m}\left(e\right)$$
_{e}∼ e denotes the triangles adjacent to the vertex v. Intuitively, if an edge is part of several triangles in the network, that edge will have a high positive Menger-Ricci curvature.Note that the definition of Menger-Ricci in (12) relies on the geometry of the Euclidian plane. This geometric assumption is incorrect in many cases, however, including hyperbolic spaces [14]. By contrast with Menger-Ricci curvature, Haantjes and Forman–Ricci curvatures do not assume any background geometry. - 3.
- Forman–Ricci curvature: There are different ways to calculate Forman–Ricci curvature of a directed and weighted network. Formula (1) described 1-dimensional Forman–Ricci curvature, and here we present an extension by Saucan, Sreejith, Vivek-Ananth, Jost, and Samal [16], the 2-dimensional Augmented Forman–Ricci curvature (AFR). FR1 considers only the pairwise correlation between vertices; while computing FR1 for the directed edge $e=\left({v}_{1},{v}_{2}\right)$, we consider only the incoming edges to ${v}_{1}$ and the outcoming edges from ${v}_{2}$. By contrast, AFR considers 2-dimensional faces—cases in which three vertices form a triangle—and potentially higher-order faces as well. Although there are four different types of directed triangles, we consider only two types (see Figure 6). The directed triangular face t formed by vertices (${v}_{1},{v}_{2},{v}_{3})$ and edges {(${v}_{1},{v}_{2}),\left({v}_{2},{v}_{3}\right),\left({v}_{3},{v}_{1}\right)\}$ makes a positive contribution. In this case, since the information from ${v}_{3}$ “flows” back to ${v}_{1}$, the triangle represents a spherical structure that is consistent with the interpretation of a positive Forman–Ricci curvature. The directed triangular face t formed by edges {(${v}_{1},{v}_{2}),\left({v}_{1},{v}_{3}\right),\left({v}_{3},{v}_{2}\right)\}$ makes a negative contribution. This triangle represents a tree-like structure since the information “flows” out from vertex ${v}_{1}$ to vertex ${v}_{2}$ and so also from vertex ${v}_{1}$ to ${v}_{3}$ and from there to ${v}_{2}$.

_{t}denotes the weight of face t, and σ < τ means that σ is a face of τ. The augmented Forma–Ricci curvature of a vertex v in a network can be defined as

#### 3.3.3. Frequency and Word Location

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Descriptive Analysis

**Figure A1.**(

**a**) This graph presents the percentage of words belonging to the following frequency ranges: (1) 0–5; (2) 6–15; (3) 16–30; (4) 31–60; (5) 61–100; (6) 101–500; (7) >500. (

**b**) This pie chart presents the empirical cumulative distribution of word frequency.

**Figure A2.**(

**a**) Each cell in this heatmap represents the number of vertices that receive at least one weight given by the “distance function” for the pair of parameters MS (minimum subjects per edge) and WS (window size). (

**b**) This heatmap is for the number of edges for the pair of parameters MS and WS. The x-axis denotes the range of MS values, and the y-axis denotes the range of WS values. The color highlights the number of vertices/edges. Bluer cells indicate a lower number of vertices/edges; redder cells indicate a higher number of vertices/edges.

**Figure A3.**The number of object-based dynamic scores that each word received in 90 networks where the free parameters of the object-based dynamic were WS, MS, and the threshold.

_{1}= 61, n

_{2}= 69, p < 0.001, two-tailed). The greater the number of scores, the higher the probability that the word will not be significant.

**Figure A4.**(

**a**) This panel presents the correlation between the number of scores and the mean object-based dynamic score. (

**b**) This panel presents the correlation after removing dots that are 1SD higher or lower than the mean. (

**c**) This panel presents histograms of the number of scores for sets of significant and non-significant words.

#### Appendix A.2. Internal Validity Scores

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**Figure 1.**The distribution of (

**a**) weights, (

**b**) Forman–Ricci curvature, and (

**c**) weights after 20 iterations of Forman–Ricci flow.

**Figure 2.**The evolution of edge weights as a function of Forman–Ricci curvature (represented by the different colors) over 20 iterations of Forman–Ricci flow on a semantic network with 183 vertices and 6018 edges. Each line represents the evolution of an edge weight along iterations of Forman–Ricci flow.

**Figure 3.**The clustering performance of the Forman–Ricci flow metric compared to graph weights for four different cluster algorithms and two internal validity scores.

**Figure 4.**The upper panel represents the set of non-significant words, and the lower panel the set of significant words. Each bar represents the range between the upper and lower boundaries given by the bootstrap test. The numbers represent the number of values out of 90 networks that each word received.

**Figure 5.**Spearman correlations between object-based dynamic and centrality measures. “Dynamic” denotes the directed approach, and “Dynamic abs” denotes the undirected approach.

**Figure 6.**Four archetypical triangles in a directed network. For a directed edge from ${v}_{1}$ to ${v}_{2}$, colored in black, the incoming edges are colored in blue and the outgoing edges are colored in red. Type (

**a**) represents a backward loop that contributes positively to the curvature. Type (

**b**) represents a forward loop that contributes negatively to the curvature. Since Types (

**c**,

**d**) have no defined direction, we do not take them into account in our calculations.

**Figure 7.**Spearman correlations between object-based dynamic and scalar curvature measures. “Dynamic” denotes the directed approach, and “Dynamic abs” denotes the undirected approach.

**Figure 8.**This graph presents the Spearman correlations between the object-based dynamic and log-frequency, average location, degree-in, and degree-out.

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## Share and Cite

**MDPI and ACS Style**

Cohen, H.; Nachshon, Y.; Maril, A.; Naim, P.M.; Jost, J.; Saucan, E.
Object-Based Dynamics: Applying Forman–Ricci Flow on a Multigraph to Assess the Impact of an Object on The Network Structure. *Axioms* **2022**, *11*, 486.
https://doi.org/10.3390/axioms11090486

**AMA Style**

Cohen H, Nachshon Y, Maril A, Naim PM, Jost J, Saucan E.
Object-Based Dynamics: Applying Forman–Ricci Flow on a Multigraph to Assess the Impact of an Object on The Network Structure. *Axioms*. 2022; 11(9):486.
https://doi.org/10.3390/axioms11090486

**Chicago/Turabian Style**

Cohen, Haim, Yinon Nachshon, Anat Maril, Paz M. Naim, Jürgen Jost, and Emil Saucan.
2022. "Object-Based Dynamics: Applying Forman–Ricci Flow on a Multigraph to Assess the Impact of an Object on The Network Structure" *Axioms* 11, no. 9: 486.
https://doi.org/10.3390/axioms11090486