BioNet.qmd

---
title: "Analyzing Biological Networks"
subtitle: "Protein-Protein Interaction Network"
title-block-banner: true
title-block-banner-color: "#6c9685"

author: "Alexandra Mpekou"
date: "08-08-2024"

toc: true
toc-title: "Contents"
toc-location: left
toc-depth: 4
number-sections: true

format:
  html:
    embed-resources: true
    code-fold: true
    code-tools:
      source: https://github.com/darwinsorchid/Biological-Networks
    theme: minty
    css: styles.css
    html-math-method: katex
  
editor: visual
bibliography: references.bib
---

## Introduction

Most phenotypes, from organism to cell, emerge from complex interactions of biological elements. Reductionist approaches to the elucidation of biological processes have gained immense popularity due to the relative simplicity and clear nature of such research workflows, focusing on a single molecule or a few of them at a time. However, in order to understand life from its simplest to its most complex forms, we need to delve into the world of networks.

Network theory has long been studied because of the many fields it can be applied to. The majority of complex systems can be modeled or explained by networks with, more or less, the same architectural features, highlighting the universality of such systems and the laws they are governed by. Despite disparities in the details that characterize each network, the abstract nature of network theory and analyses allow for the extraction of significant information that can help with comprehending critical constituents of a network's organization, function and robustness, and can have major impacts on the fields of interest and their respective advancements [@barabási2004].

The following protein network analysis is performed on protein-protein interaction (PPI) data that were derived from two-hybrid experiments. These data constitute part of a differential epistasis mapping paper published by Bandyopadhyay et. al. in 2010 [@bandyopadhyay2010]. Pertaining to comparison of genetic networks across conditions, this study reveals differences in interactions of DNA damage response genes with and without cellular exposure to methane methylsulfonate or MMS, a DNA-damaging agent.

Two-hybrid experiments are a set of molecular biology techniques that aim to study protein-protein interactions and allow researchers to determine if two proteins can physically interact with each other in a living cell. The most common type of two-hybrid system is the yeast two-hybrid (Y2H) system, although variations exist for use in other organisms.

The process consists of a bait protein of interest that is fused to a DNA-binding domain (DBD) and a prey, which is another protein of interest, fused to an activation domain (AD). The system's function relies on the reconstitution of a transcription factor when the two proteins of interest interact inside the living cell. If such interaction does occur after the cellular introduction of the bait and prey proteins, the functional transcription factor will bind to the promoter of a reporter gene. Through this process, protein-protein interactions can be detected in vivo and lead to advancements in the study of such complex networks that underlie cellular processes [@suter2008].

This brief report is part of a personal project exploring the world of complex systems, network biology and the statistic measures that describe their properties and behavior. The aim of this project is to better understand biological interactions and network theory, while getting familiar with using Quarto publishing system and the appropriate R libraries for graph visualization, as well as the calculation of a few selected network metrics, including node degree and degree distribution, betweenness centrality, transitivity, modularity and community detection.

## Results

### Graph Visualization

A network graph is created using only the unique nodes of the protein network and the appropriate edges between them. In this case, the generated graph is undirected, meaning its edges do not point to a defined direction, from or to a specific node, in each path of the network.

```{r setup}
knitr::opts_chunk$set(message = FALSE, warning = FALSE)
options(warn = -1)
#install.packages("ggraph")
#install.packages("tidygraph")
#install.packages("ggplot2")
#install.packages("igraph",  repos = "https://cran.r-project.org")
#install.packages("gt")

suppressPackageStartupMessages({
  library(igraph)
  library(ggraph)
  library(tidygraph)
  library(gt)
})
```

```{r}
#| label: fig-graph
#| fig-cap: "Graph of protein-protein interaction network with node size being proportional to node degrees."

# Read data from txt file
data <- read.table("Bandyopadhyay2010.txt", header = T)

# Isolate only unique proteins of dataset
unique_nodes <- unique(c(data$BAIT_OFFICIAL_SYMBOL, data$PREY_OFFICIAL_SYMBOL))

# Create edges by binding together columns of initial dataset
edges <- cbind(data$BAIT_OFFICIAL_SYMBOL, data$PREY_OFFICIAL_SYMBOL)

# Create graph object
graph <- graph_from_data_frame(edges, directed = F)

# Add label attribute to graph vertices for protein names
V(graph)$label <- unique_nodes

# Plot graph with node size according to node degrees
ggraph(graph, layout = "fr") + 
  geom_edge_link(alpha = 0.25) + 
  geom_node_point(aes(size = degree(graph)), shape = 21, fill = "#7ea695", color ="black") + 
  theme_void() +
  ggtitle("Protein-Protein Interaction Network") +
  theme(plot.title = element_text(hjust = 0.5, margin = margin(t =20)),
        plot.title.position = "plot",
        plot.margin = margin(t = 30, r = 10, b = 10, l = 10))+
  labs(size = "Node Degree")
```

Plotting the graph as it is leads to an output that has too much information to be interpreted and, thus, further analysis is needed in order to get to meaningful conclusions. It does, however, convey a few propensities characteristic of biological networks, such as the high degree of clustering, with the obvious existence of closely linked groups in the system around central nodes, so-called "hubs". Also, the graph indicates the degree of different nodes, with a few nodes having a great number of links and forming big groups and most of the vertices being connected to little to no other nodes, hinting to the scale-free nature of the network. This is a tendency especially prominent in PPI and protein domain networks that denotes the intertwined modularity of such systems; proteins mostly work in functional and structural groups that cooperate to give rise to cellular phenotypes [@barabási2004].

### Node Degree (k)

In the context of network theory and graph analysis, node degree refers to the number of edges or direct connections to other vertices that are incident to a specific node in the graph [@opsahl2010]. High-degree nodes, often referred to as "hubs", are typically crucial for the network's structure and function and they often represent key proteins or genes in biological networks. On the contrary, low-degree nodes might pertain to more specialized or peripheral components in the system, since they have less interactions with the rest of the vertices. Understanding the information that is denoted by node degree significantly helps in analyzing the importance and influence of specific nodes in the network, elucidating the overall network topology and even identifying critical points for potential intervention [@przulj2004].

#### Degree distribution (P(k))

```{r}
#| label: fig-degree-distribution
#| fig-cap: "Barplot of degree distribution of nodes in PPI network."

# Calculate node degrees of graph
node_degrees <- degree(graph)

# Create frequency table of node degrees
degree_count <- table(node_degrees)

# Calculate degree distribution of graph nodes
deg_dist <- degree_count / vcount(graph)

# Turn into dataframe
deg_df <- data.frame(Degree = as.numeric(names(deg_dist)), Frequency = as.numeric(deg_dist))

# Degree distribution barplot
ggplot(deg_df, aes(x = Degree, y = Frequency)) + 
  geom_bar(stat = "identity", fill = "#7ea695",color = "black") +
  labs(title = "Degree Distribution (P(k))", x = "Degree (k)", y = "Frequency") +
  theme_minimal() +
  theme(plot.title = element_text(hjust = 0.5, margin = margin(t=20)))
```

Degree distribution (P(k)) is a network property that gives the probability for a randomly chosen node to have k links or, in other words, a degree equal to k. It is defined as follows:

$$
P(k) = \frac{\left|\{i \mid \deg(i) = k\}\right|}{n}
$$

where the numerator represents the number of nodes in the graph that have degree equal to ${\displaystyle k}$ and the denominator ${\displaystyle n}$ is the total number of nodes in the network.

Degree distribution conveys information about how the system is structured and allows for distinguishing between different network classes. Degree distribution of random networks tends to follow the Poisson distribution, with most of the nodes having the same amount of edges or links, a value that is closer to the average ${\displaystyle <k>}$. On the contrary, real complex systems, including PPI networks, showcase a highly asymmetric degree distribution that follows Power law and has a right skewness, which is obvious in @fig-degree-distribution. It clearly denotes that the vast majority of the system's nodes have a low degree, close or equal to 1, while a few vertices constitute hubs with extraordinarily high values of degrees. This is a characteristic attribute of scale-free networks, which mostly describe complex systems of the real world. Briefly, this property indicates the absence of a typical element or node in the network that could describe all other nodes, which could exist in random networks, where most node degrees approximate the average degree, but not in real networks [@barabási2004].

Node degree and degree distribution have significant consequences for the network's function and robustness. Generally, robustness is a property that pertains to the system's capacity to react to changes in its environment or internal structure, while continuing to behave in a relatively typical manner. As its immunology counterpart of homeostasis, robustness is critical to the viability of the network, especially since most complex systems in the real world interact with highly variable and dynamic environments [@albert2000].

There is a strong relationship between the hub status of a node and its role in maintaining the network's connectivity and viability, especially when it comes to topological robustness. Generally, scale-free networks with long tail degree distributions are resilient to random removal of nodes, a phenomenon known as "failure", but they are vulnerable to the removal of hub vertices, which is known as "attack". Thus, this type of networks are robust against accidental failures, since most of their components are low degree nodes, they do, however, rely on hubs for their structure and flow of information. As a result, the attack vulnerability of such networks can potentially affect both dynamical and functional robustness, besides the obviously afflicted topological one [@liu2017].

### Clustering and Communities

#### Community Detection

For better visualization of protein-protein interactions, the vertices that pertain to unique proteins of the given network can be grouped into clusters that constitute protein communities; that is, groups of proteins that seem to interact with each other in a greater degree than they do with other proteins of the network. Here, community detection was performed using the Louvain clustering algorithm, a popular modularity-based algorithm for the identification of communities in large networks. This method is widely preferred due to its agglomerative nature, since it is based on a multi-scale modularity optimization process. However, it must be pointed out that the Louvain algorithm does have a resolution limit that stems from its size-dependence; the larger the system, the higher the probability for detection of larger clusters/communities [@blondel2008].

```{r}
#| label: fig-louvain-communities
#| fig-cap: "Graph of PPI network after applying the Louvain clustering algorithm for community detection. Communities are discerned by color."

# Apply Louvain clustering algorithm to graph for community detection
net_communities <- cluster_louvain(graph)

# Extract community membership vector from community detection result object
# Indicates which community each node in the network belongs to
membership <- membership(net_communities)

# Convert graph object into tbl_graph object for better visualization of graph
tbl_graph <- as_tbl_graph(graph)

# Add community attribute to tbl_graph vertices to match each node to its community
V(tbl_graph)$community <- membership

# Plot graph with node colors according to community membership of each node
# using the Fruchterman-Reingold layout 
ggraph(tbl_graph, layout = "fr") +
  geom_edge_link(alpha = 0.25) + 
  geom_node_point(aes(fill = as.factor(community)), size = 2,
                  shape = 21, stroke = 0.08, border = "grey") +
  theme_void() +
  labs(title = "PPI Network Communities",
       subtitle = "Louvain Clustering",
       fill = "Community") +
  theme(plot.title = element_text(hjust = 0.5),
        plot.subtitle = element_text(hjust = 0.5, vjust = -1),
        legend.key.height = unit(0.1, "line"))+
  guides(fill = guide_legend(ncol = 2))
```

```{r}
#| label: fig-community-sizes
#| fig-cap: "Barplot of community sizes in PPI network."

# Assign membership vector to variable com_member_net
com_member_net <- membership

# Identify all unique communities in the network
unique_com_net <- unique(membership)

# Create placeholder list variable
community_sizes_net <- list()

# Iterate through unique communities of network to count each community's nodes
for (i in 1:length(unique_com_net)) {
  nodes_in_com <- which(com_member_net == unique_com_net[i])
  community_sizes_net[[i]] <- length(nodes_in_com)
}

# Create dataframe of community sizes
com_size_net_df <- data.frame(Community_ID = as.factor(unique_com_net),
                              Size = as.numeric(unlist(community_sizes_net)))

# Barplot of community sizes
ggplot(com_size_net_df, aes(x= Community_ID, y= Size, fill= Size)) +
  geom_bar(stat = "identity")+
  labs(title = "Community Sizes", x = "Community", y = "Number Of Nodes")+
  theme_minimal() +
  theme(plot.title.position= "plot",
        plot.title = element_text(hjust = 0.5))
```

After applying the Louvain clustering algorithm to the network for community detection, it is clear that there are 21 communities, each one indicated by a different color in @fig-louvain-communities. @fig-community-sizes shows the sizes of the communities detected in the network, with over half of the communities having more than 20 nodes as members, and 4 of them having 50 or more members.

#### Modularity

Modularity measures the extend to which the network is organized into communities. The modularity of a given set of communities within a network is defined as follows:

$$
Q = \frac{|E_{in}| - \langle E_{in} \rangle}{|E|}
$$

where $|Ein|$ is the number of edges within a community that do not cross boundaries between communities, $\langle Ein \rangle$ is the expected number of within-community edges if the topology were completely random and $|E|$ is the total number of edges. The value of modularity for a specific network ranges from -1 to 1, with positive values denoting that the number of edges within communities are more than the expected number.

```{r modularity}
# Modularity calculates the modularity of a graph with respect to the given membership vector
modularity_score <- modularity(graph, membership, resolution = 1)
rounded_modularity <- round(modularity_score, 2)
cat("Q =", rounded_modularity)
```

The network in question has a modularity score of 0.75, which indicates a relatively strong community structure, with well-clustered nodes, many edges within communities and few edges between discrete communities. In biological terms, modularity refers to groups of interlinked molecules, such as proteins, that cooperate to achieve a specific function. As expected, most complex biological systems, like PPI networks, usually brim with modular functionality and/or structure, a property that is correlated with high network robustness [@newman2006].

#### Betweenness Centrality

Betweenness centrality denotes the number of times a node acts as a bridge along the shortest path between two other nodes. The betweenness centrality of a node ${\displaystyle v}$ is defined as follows:

$$
{\displaystyle g(v)=\sum _{s\neq v\neq t}{\frac {\sigma _{st}(v)}{\sigma _{st}}}}
$$

where ${\displaystyle \sigma _{st}}$ is the number of shortest paths from node ${\displaystyle s}$ to node ${\displaystyle t}$ and ${\displaystyle \sigma _{st}(v)}$ is the number of shortest paths from node ${\displaystyle s}$ to node ${\displaystyle t}$ that go through node ${\displaystyle v}$ (without node ${\displaystyle v}$ being an ending or starting point).

```{r}
#| label: fig-betweenness-centrality
#| fig-cap: "Barplot of betweenness centrality values of nodes in PPI network."

# Calculate betweenness centrality of each node in graph
betweenness_values <- betweenness(graph, normalized = T)

# Add attribute Betweenness to tbl_graph vertices
V(tbl_graph)$Betweenness <- betweenness_values

# Create vector of only non zero betweenness centrality values
non_zero_betweeness <- betweenness_values[betweenness_values > 0]

# Create vector of only nodes with non-zero betweenness centrality values
non_zero_names <- V(tbl_graph)$name[(V(tbl_graph)$Betweenness) > 0]

# Create dataframe of nodes and non-zero betweenness centrality values
betweenness_df <- data.frame(Proteins = as.factor(non_zero_names),
                             Betweenness = as.numeric(non_zero_betweeness))

# Betweenness centrality barplot
ggplot(betweenness_df, aes(x =Proteins, y = Betweenness))+
  geom_bar(stat = "identity")+
  theme_minimal()+
  labs(title = "Betweenness Centrality of Proteins",
       x = "Nodes",
       y = "Betweenness Centrality") +
  theme(plot.title = element_text(hjust=0.5),
        axis.text.x = element_blank())
```

```{r}
#| label: fig-high-betweenness-nodes
#| fig-cap: "Barplot of nodes with betweenness centrality values of >0.1."

# Create vector of only relatively high betweenness centrality values
# Betweenness centrality > 0.1
high_betweenness <- betweenness_values[betweenness_values > 0.1]

# Create vector of nodes with betweenness centrality > 0.1
high_betweenness_nodes <- V(tbl_graph)$name[(V(tbl_graph)$Betweenness) > 0.1]

# Create dataframe of nodes and betweenness centrality values [> 0.1]
high_betweenness_df <- data.frame(Proteins = as.factor(high_betweenness_nodes),
                             Betweenness = as.numeric(high_betweenness))

# Barplot of nodes with betweenness centrality > 0.1
# Central nodes for information flow in network
ggplot(high_betweenness_df, aes(x = Proteins, y = Betweenness))+
  geom_bar(stat = "identity", fill = "#7ea695", color = "black", width= 0.8)+
  theme_minimal()+
  labs(title = "Proteins with Betweenness Centrality > 0.1",
       x = "Nodes",
       y = "Betweenness Centrality") +
  theme(plot.title = element_text(hjust=0.5),
        axis.text.x = element_text(angle=90))
```

As shown in @fig-betweenness-centrality, most of the network's nodes have low betweenness centrality values, with only a few exceeding 0.1. The proteins with relatively high betweenness centrality values (\>0.1) are plotted in @fig-high-betweenness-nodes, with APC or adenomatous polyposis coli, a Wnt signalling pathway regulator, reaching over 0.3. High betweenness centrality indicates that a node significantly influences the flow of information within a network and is crucial for maintaining the network's functionality. Usually, such proteins are evolutionarily conserved and constitute some of the oldest vertices in the PPI system, as is APC, which exhibits an extreme degree of conservation across taxa with most mutations having adverse effects for cellular processes, especially in the epithelium [@abbott2023].

#### Transitivity

Transitivity, also known as global clustering coefficient, quantifies the propensity of a group of nodes to cluster together in a network. It is defined as the ratio of the number of closed triplets or triangles, to the number of all triplets, closed and open, of the network. Global clustering coefficient provides an average measure of the clustering in a network and is defined as follows:

$$
C_T = \frac{\left| \{ (i, j, k) \mid d(i, j) = d(i, k) = d(j, k) = 1 \} \right|}{\left| \{ (i, j, k) \mid d(i, j) = d(i, k) = 1 \} \right|}
$$

where the denominator is the total number of possible node pairs within node ${\displaystyle i}$ 's neighborhood and the numerator is the number of actually connected node pairs among them.

```{r}
#| label: fig-transitivity
#| fig-cap: "Barplot of transitivity values of nodes in PPI network."

# Calculate transitivity values of nodes in network
transitivity_values <- transitivity(graph, type = "local")

# Add transitivity values as attribute of tbl_graph vertices
V(tbl_graph)$Transitivity <- transitivity_values

# Create vector of only non-zero transitivity values
non_zero_trans <- transitivity_values[transitivity_values > 0]

# Create vector of nodes with non-zero transitivity values
non_zero_trans_nodes <- V(tbl_graph)$name[(V(tbl_graph)$Transitivity) > 0]

# Create dataframe of nodes and non-zero transitivity values
transitivity_df <- data.frame(Proteins = as.factor(non_zero_trans_nodes),
                             Transitivity = as.numeric(non_zero_trans))

# Barplot of non-zero transitivity values
ggplot(transitivity_df, aes(x =Proteins, y = Transitivity))+
  geom_bar(stat = "identity", fill = "#495c51", width = 0.8)+
  theme_minimal()+
  labs(title = "Transitivity Values of Proteins",
       x = "Nodes",
       y = "Transitivity") +
  theme(plot.title = element_text(hjust=0.5),
        axis.text.x = element_text(angle = 90, vjust = 0.5, hjust = 1, size = 7))
```

Transitivity values range from 0 to 1, with ${\displaystyle C_T}$ values close to 0 implying no closed triad for a given node, whereas ${\displaystyle C_T}$ values close or equal to 1 denote vertices that are part of completely closed triads. @fig-transitivity shows that a small group of proteins in the network have low transitivity values, with most of the nodes having ${\displaystyle C_T}$ values above 0.5.

```{r}
#| label: tbl-APC-transitivity
#| tbl-cap: "Transitivity value of APC."

# Create table of APC transitivity value
APC_trans <- subset(transitivity_df, Proteins == "APC")
APC_trans |>
  select(Proteins, Transitivity) |>
  gt()
```

@fig-transitivity shows that APC, the protein that has the highest betweenness centrality value of all other vertices in the network, exhibits the lowest possible value of transitivity. More specifically, APC has a transitivity value of about 0.00095, as shown in @tbl-APC-transitivity. This possibly constitutes a counter-intuitive combination for a node; having both high betweenness centrality and substantially low transitivity values. However, a closer look into the community's structure and how the nodes are organized around APC will elucidate the two statistical measures and their meanings.

### The Importance Of Hubs

Focusing on a community can be helpful when it comes to looking into possible important constituents of a network. The close examination of the interactions of proteins in a particular community can lead to the identification of protein hubs, proteins that interact with many others within the cell and often play a crucial role in the organization and regulation of PPI networks, being essential for the stability and functionality of cellular processes. Many hub proteins are evolutionarily conserved, indicating their essential role across different species [@vallabhajosyula2009a].

```{r}
#| label: fig-APC-community
#| fig-cap: "Graph of nodes belonging to the same community as APC. Node sizes are proportionate to node degrees."

# Assign the community APC is a part of to a variable 
target_community <- V(tbl_graph)$community[V(tbl_graph)$name == "APC"]

# Create vector with nodes that belong to the community APC belongs to
nodes_in_community <- which(membership == target_community)

# Create subgraph object of APC community nodes and turn it into tbl_graph object
subgraph1 <- subgraph(graph, nodes_in_community)
tbl_subgraph1 <- as_tbl_graph(subgraph1)

# Add degree attribute for vertices in tbl_subgraph1
V(tbl_subgraph1)$Degree <- degree(subgraph1)

# APC Community plot with node size according to node degree
ggraph(tbl_subgraph1, layout = "fr") + 
  geom_edge_link(alpha = 0.8, color = "darkgrey") + 
  geom_node_point(aes(size = V(tbl_subgraph1)$Degree), shape = 21, border="black",
                  fill="#174e50", stroke = 0.1) +
  geom_node_text(aes(label=name), vjust = 1.5, hjust = 1, size = 2.3) +
  theme_void() +
  labs(title = "APC Community") +
  theme(plot.title = element_text(hjust = 0.5),
        plot.margin = margin(t = 40, r = 40, b = 40, l = 40),
        legend.position = "none")
```

```{r}
#| label: fig-APC-neighbors
#| fig-cap: "Graph of APC neighbors."

# Find neighbors of APC and assign them to vector
APC_neighbors <- neighbors(subgraph1, V(subgraph1)$name == "APC")

# Assign APC neighbors' names to vector
APC_net <- c("APC", V(subgraph1)$name[APC_neighbors])

# Create subgraph of APC neighbors and turn into tbl_graph object
APC_net_graph <- subgraph(subgraph1, APC_net)
tbl_APC <- as_tbl_graph(APC_net_graph)

# APC neighbors plot
ggraph(tbl_APC, layout = "fr") + 
  geom_edge_link(alpha = 0.8, color = "darkgrey") + 
  geom_node_point(aes(size = 4), shape = 21, border="black",
                  fill="#174e50", stroke = 0.1) +
  geom_node_text(aes(label=name), vjust = 2, hjust = 0.5, size = 2.3) +
  theme_void() +
  labs(title = "APC Neighbors") +
  theme(plot.title = element_text(hjust = 0.5),
        plot.margin = margin(t = 40, r = 40, b = 40, l = 40),
        legend.position = "none")
```

@fig-APC-community depicts the community of APC and its neighboring nodes. It is clear that APC is a central node and has numerous connections to the rest of the vertices in the community, which hints to the high degree and betweenness centrality of the protein in question. In order to clearly display what a low transitivity value can be translated to one needs to look into the neighboring nodes of APC, that is, nodes that are connected to the protein with a direct link. As shown in @fig-APC-neighbors, the neighbors of APC are hardly connected to each other, which indicates low degree of triad formation or local clustering in the community.

It is important to highlight the fact that the central position, high connectivity, node degree and betweenness centrality of APC constitute characteristic properties of hub nodes. In other words, APC, being one of the few central vertices of the system that has lots of connections, contributes to the increase of the network's topological robustness and flow of information. This example demonstrates the fact that some highly connected hubs can have low transitivity values, linking otherwise not communicating nodes.

```{r}
#| label: fig-ATF7IP-community
#| fig-cap: "Graph of nodes belonging to the same community as ATF7IP. Node sizes are proportionate to node degrees."

# Assign the community ATF7IP is a part of to a variable 
target_com <- V(tbl_graph)$community[V(tbl_graph)$name == "ATF7IP"]

# Create vector with nodes that belong to the community ATF7IP belongs to
nodes_in_com <- which(membership == target_com)

# Create subgraph object of ATF7IP community nodes and turn it into tbl_graph object
subgraph2 <- subgraph(graph, nodes_in_com)
tbl_subgraph2 <- as_tbl_graph(subgraph2)

# Add degree attribute for vertices in tbl_subgraph2
V(tbl_subgraph2)$Degree <- degree(subgraph2)

# ATF7IP Community plot with node size according to node degree
ggraph(tbl_subgraph2, layout = "fr") + 
  geom_edge_link(alpha = 0.8, color = "darkgrey") + 
  geom_node_point(aes(size = V(tbl_subgraph2)$Degree), shape = 21, border="black",
                  fill="#174e50", stroke = 0.1) +
  geom_node_text(aes(label=name), vjust = 1.8, hjust = 0.5, size = 2.3) +
  theme_void() +
  labs(title = "ATF7IP Community") +
  theme(plot.title = element_text(hjust = 0.5),
        plot.margin = margin(t = 40, r = 40, b = 40, l = 40),
        legend.position = "none")
```

```{r}
#| label: fig-ATF7IP-neighbors
#| fig-cap: "Graph of ATF7IP neighbors."

# Find neighbors of ATF7IP and assign them to vector
ATF7IP_neighbors <- neighbors(subgraph2, V(subgraph2)$name == "ATF7IP")

# Assign ATF7IP neighbors' names to vector
ATF7IP_net <- c("ATF7IP", V(subgraph2)$name[ATF7IP_neighbors])

# Create subgraph of ATF7IP neighbors and turn into tbl_graph object
ATF7IP_net_graph <- subgraph(subgraph2, ATF7IP_net)
tbl_ATF <- as_tbl_graph(ATF7IP_net_graph)

# ATF7IP neighbors plot
ggraph(tbl_ATF, layout = "fr") + 
  geom_edge_link(alpha = 0.8, color = "darkgrey") + 
  geom_node_point(aes(size = 4), shape = 21, border="black",
                  fill="#174e50", stroke = 0.1) +
  geom_node_text(aes(label=name), vjust = 1.8, hjust = 0.5, size = 2.3) +
  theme_void() +
  labs(title = "ATF7IP Neighbors") +
  theme(plot.title = element_text(hjust = 0.5),
        plot.margin = margin(t = 40, r = 40, b = 40, l = 40),
        legend.position = "none")
```

For comparison purposes, the same analysis was done for one of the proteins with a transitivity value equal to 1, ATF7IP. @fig-ATF7IP-community displays the community that ATF7IP is a member of, showcasing the relatively low node degree of the protein, as well as the vastly different structure of the community, conveying more complex relationships between nodes, when compared to @fig-APC-community. The high transitivity value of ATF7IP is shown in @fig-ATF7IP-neighbors, where it is clear that all the possible triads that could be formed between ATF7IP's neighboring vertices already exist in the small group.

## Conclusion

Network Biology is an interdisciplinary field of research that can be of immense importance for the deeper understanding of biological systems, including protein-protein interaction networks. Network theory provides the tools for the study of such complex systems, using a variety of statistical measures, including node degree, betweenness centrality, transitivity and modularity to distinguish between different types of networks and their distinct behavior and evolution. These types of analyses can be a starting point for more detailed experimental studies to explore these interactions and their functional implications.