# Graph-Theoretical Analysis of Biological Networks: A Survey

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## Abstract

**:**

## 1. Introduction

- Topological Analysis: This analysis is based on the topological properties of the network, providing information to be used in further analysis, as described in the following sections.
- Clustering: This is the process of discovering dense regions of a biological network that may indicate important activity for the survival of the organism, or sometimes, disease states.
- Network Motifs: These are frequently repeating subgraph patterns in biological networks that may indicate some specific function performed by them.
- Network Alignment: The alignment of two networks shows the similarity between them, which may be used to deduce hereditary relationships. This affinity may help to discover conserved regions in organisms to aid with the understanding of the evolutionary process.

## 2. Biological Networks

#### 2.1. Networks in the Cell

- Protein Networks: Proteins are the workhorses of the cell, performing the vital functions of organisms. A protein is basically a sequence of amino acids constructed by the code in a gene, which is part of the DNA. The 3-D structure of a protein plays an important role in its function, so that various drug treatment methods use this property to disable the functioning of a disease-causing virus such as the HIV. A protein interacts with various other proteins through biochemical reactions forming a protein–protein-interaction (PPI) network. Nodes with high degrees in a PPI network has fundamental functions in the cell [5]. The PPI network of T. pallidum is depicted in Figure 1, where proteins involved in DNA metabolism are shown as enlarged red circles.
- Gene Regulation Networks: The main function of a gene in DNA is to provide the code to be used through transcription and translation processes to produce a protein. This process is called gene expression, and the mechanism of specific gene expression is controlled and affected by proteins that are coded by other genes, denoted regularity interactions. For example, gene X regulates gene Y if a change in the expression of gene X results in a change in the expression of gene Y. A gene regulation network (GRN) is made of genes, proteins, and various other molecules, and it may be modeled using a directed graph, with nodes representing these entities and the edges showing their biochemical interactions leading to regulations, as shown in Figure 2. Typically, a GRN is a sparse graph with small-world and power-law properties, which means there are only a few nodes that have very high out-degrees that regulate the expression of other genes. Moreover, the distance between any two nodes in a GRN network is small compared to the size of the network as being consistent with small-world properties.
**Figure 1.**The PPI network of T. pallidum, taken from [6]. - Metabolic Pathways: The main ingredients of the cell, such as sugars, amino acids, and lipids, are produced by the basic chemical system called metabolism that works on ingredients called metabolites. The biochemical reactions in the cell that result in metabolisms can be modeled by directed or undirected graphs, with nodes representing metabolites and edges showing biochemical reactions that transform one metabolite to another one [7,8,9]. An edge in such a graph may also represent an enzyme that catalyzes a biochemical reaction. An undirected edge in the graph model denotes a reversible reaction where a directed edge means an irreversible one. A metabolic pathway is a sequence of biochemical reactions to perform a specific metabolic function. An example of a metabolic function is glycolysis, in which a glucose molecule is divided into two sugars that generate adenosine triphosphates (ATPs) to produce energy. Graphs representing metabolic pathways have the small-world and scale-free properties. A study of metabolic pathways may provide insight into pathogens causing infections in search of cures for diseases [10].

#### 2.2. Networks outside the Cell

- Brain Networks: We can analyze brain networks at the cell (neuron) level, or at a coarser functional level. A neuron in the brain fires when the sum of its input signal strengths exceeds a threshold. A neural network made of neurons performs various cognitive tasks such as problem solving, reasoning and, image processing. The artificial neural networks function similar to biological neural networks and have been used widely to implement various tasks in deep learning, which is a component of machine learning to be used for artificial intelligence tasks. At a coarser level, we can investigate the functions performed by the brain, using brain structural networks (BSNs) or brain functional networks (BFNs). A BSN basically reflects the structures of neural connections, whereas a BFN models the connnectedness of the functional regions of the brain. Studies of BFNs have shown that these networks are also small-world and scale-free networks, like most of the biological networks [11].
- Phylogenetic Networks: A phylogenetic tree shows evolutionary relationships among organisms, with leaves representing living organisms and the intermediate nodes, their common ancestors. A phylogenetic network is the general form of a phylogenetic tree where a node may have more than one parent.
- The Food Chain: Living organisms rely on food for survival. The food chain directed graph shows the relationships between the predators and the prey, where the direction of an edge is from the predator to the prey.

## 3. Large Graph Analysis

#### 3.1. Degree Distribution

**Definition**

**1**

#### 3.2. Density

**Definition**

**2**

#### 3.3. Clustering Coefficient

**Definition**

**3**

#### 3.4. Matching Index

**Definition**

**4**

#### 3.5. Centrality

#### 3.5.1. Closeness Centrality

#### 3.5.2. Vertex Betweenness Centrality

#### 3.5.3. Edge Betweenness Centrality

#### 3.6. Topological Index

- The first Zagrep index$${M}_{1}\left(G\right)=\sum _{v\in G}d{\left(v\right)}^{2}$$
- The second Zagrep index$${M}_{1}\left(G\right)=\sum _{v\in G}d\left(u\right)d\left(v\right)$$
- The Wiener index$${M}_{1}\left(G\right)=\sum _{v\in G}d(u,v)$$

#### 3.7. Network Perturbation Analysis

## 4. Large Network Models

- Random networks: These types of networks, proposed by Erdos and Renyi, assume that an edge $(u,v)$ between the vertices u and v is formed with the probability $p=2m/\left(n\right(n-1\left)\right)$. The degree distribution in random networks is binomial, following a Poisson distribution. A random network has a short average path length and it has a clustering coefficient that is inversely proportional to the size of the network.
- Small-world networks: These types of networks are characterized by low average path lengths and short diameters. Biological networks such as PPI networks, GRNs, and metabolic pathways; and other complex networks such as social networks and the Internet exhibit this property. The diameter of a small-world network is proportional to $logn$, where n is the number of nodes in the network.
- Scale-free networks: Most biological networks have few high-degree nodes, with many low-degree ones. The PPI network of T. pallidum in Figure 1 exhibits small-world and scale-free network properties, as can be seen. These networks, along with various other complex networks, obey the power-law degree distribution shown by the following equation,$$P\left(k\right)\approx k-\gamma ,\gamma >1$$
- Growth: A new node is added to the network at each discrete time t.
- Preferential Attachment: A new node u is attached to any node v in the network with a probability proportional to the degree of v, which means that higher degree nodes tend to have more neighbors at each attachment.

- Hierarchical Networks: A study of biological networks shows that the clustering coefficients of nodes are inversely proportional to their degrees. This unexpected result means that lower degree nodes in these networks have higher clustering coefficients than the hubs. A hierarchical network model of a biological network captures all of the observed properties, such as small-world and scale-free, with an additional property that is exhibited by dense clusters of low-degree nodes, connected by high-degree hubs. That is, the neighbors of low-degree nodes in such networks are highly connected but the nodes around the high-degree nodes are sparsely connected.

## 5. Cluster Discovery in Biological Networks

#### 5.1. Hierarchical Clustering

- Single Link: The distance between two closest points, with one of them in ${C}_{i}$ and other in ${C}_{j}$, is considered.
- Complete Link: The distance between the two points in two clusters that are farthest is used.
- Average Link: The average distance between every pair of points in ${C}_{i}$ and ${C}_{j}$ is considered.

#### 5.2. Density-Based Clustering

#### 5.3. Flow-Based Clustering

#### 5.4. Spectral Clustering

#### 5.5. Fuzzy Clustering

## 6. Network Motifs

#### 6.1. Motif Discovery

- The detection of ${m}_{k}$ in G may be performed via exact counting, which involves the enumeration of all subgraphs of order k. This method evidently has a high time complexity; alternatively, sampling-based methods that work in a representative sample of the graph may provide approximate solutions.
- Isomorphic classes of the discovered motifs should be determined, since various motifs may be isomorphic to each other.
- Statistical significance of the discovered motifs in G should be determined. Commonly, a similar structured set $\mathcal{R}$ of random graphs are generated, and motifs are searched in these graphs. If the motifs found in G are statistically higher in number than the ones found in the graphs of set $\mathcal{R}$, we can conclude that they do represent some biological function in the network represented by G.

#### 6.2. Background

- P-value: This parameter is calculated by finding the number of elements of the randomly generated set $\mathcal{R}={R}_{1},\dots ,{R}_{n}$ that have a greater frequency of motif m than in the target graph G. A motif m is considered a significant motif if P-value of m, $P\left(m\right)$, given below, is less than 0.01.$$P\left(m\right)=\frac{1}{n}\sum _{i=1}^{n}{\sigma}_{{R}_{i}}\left(m\right)$$
- Z-score: The Z-score of a motif m, $Z\left(m\right)$, in a graph G, is evaluated using the following formula:$$Z\left(m\right)=\frac{{F}_{m}-\overline{{F}_{r}}}{\sqrt{{\sigma}_{r}^{2}}}$$
- Motif significance profile: The motif significance profile vector SP is structured with elements as Z-scores of motifs ${m}_{1},{m}_{2},\dots ,{m}_{k}$, and normalized to unity as below. Various graphs may then be compared for any common motifs contained in them.$$SP\left({m}_{i}\right)=\frac{Z\left({m}_{i}\right)}{{\sum}_{i=1}^{n}Z{\left({m}_{i}\right)}^{2}}$$

#### 6.3. Review of Motif Searching Algorithms

#### 6.3.1. Network Centric Search Algorithms

#### 6.3.2. Motif Centric Search Algorithms

#### 6.3.3. Parallel Motif Search Algorithms

## 7. Network Alignment

#### 7.1. Background

- Form the similarity matrix R with entry ${r}_{ij}$ showing the similarity score of the nodes $i\in {V}_{1}$ and $j\in {V}_{2}$ in input networks ${N}_{1}$ and ${N}_{2}$, respectively.
- Implement a weighted matching algorithm to assess the similarity of the networks ${N}_{1}$ and ${N}_{2}$.

#### 7.2. Alignment Quality

#### 7.3. Review of Network Alignment Algorithms

## 8. Discussion

## Funding

## Informed Consent Statement

## Conflicts of Interest

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**Figure 3.**Degree distribution of a sample graph with vertices a–g. (

**a**) The graph. (

**b**) Its degree distribution.

**Figure 4.**Classification of biological network clustering algorithms, adapted from [4].

**Figure 5.**Commonly found biological network motifs with vertices a–h. (

**a**) Feed-Forward-Loop, (

**b**) Bifan, (

**c**) Multi-input motifs.

**Figure 6.**Network motif search algorithms, adapted from [4].

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Erciyes, K.
Graph-Theoretical Analysis of Biological Networks: A Survey. *Computation* **2023**, *11*, 188.
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Erciyes K.
Graph-Theoretical Analysis of Biological Networks: A Survey. *Computation*. 2023; 11(10):188.
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