4.1. Network Formation
Networks, often called graphs, are abstract representations of interrelated elements in mathematics and computer science. They are excellent models for understanding and analyzing complex systems. These systems encompass social interactions, information flow in systems [
34,
35], biological interactions in ecological systems [
36,
37], and molecular structures in chemistry [
38,
39,
40]. The core elements of a network are nodes (or vertices) and edges (or links). Nodes symbolize the entities in the system, such as individuals, cities, genes, or abstract notions, while edges define the links or relationships between these entities. A network can be described mathematically as
, where
V represents the nodes and
E represents the edges. Every edge in
E links two nodes from
V, representing a type of interaction, reliance, or connection between them. This mathematical framework enables the formalization and examination of intricate systems, offering insights into their structure, behavior, and emergent characteristics.
Moreover, in weighted networks, weights are numerical values assigned to the edges, introducing an additional dimension. The weights indicate a level of importance, ability, or expense related to moving across the edge. A weighted network can be represented mathematically as
, where
is a function that assigns weights to edges [
41,
42].
Examining systems using weighted network topology is a fundamental aspect of contemporary network science, providing deep understanding of the structural and dynamic characteristics of intricate networks. This technique extends beyond the basic binary perspective of connections in traditional network analysis by recognizing that not all interactions or links hold the same level of importance. Weighted networks assign a weight to edges to represent the intensity, capacity, or frequency of interactions between nodes. This multifaceted perspective allows for a more precise and complete comprehension of the network’s fundamental structure and operation.
The weighted degree
of node
i in a weighted network is defined as the total of the weights of all edges connected to
i, represented mathematically as
where
is the set of neighbors of node
i (or the set of adjacent nodes) and
is the weight of the edge between nodes
i and
j. This formulation offers a more detailed assessment of node centrality by considering the intensity of connections [
43,
44].
Understanding the concept of shortest paths in weighted networks is essential for grasping the efficiency of information or resource transfer. The weighted shortest path length between two nodes,
i and
j, is the minimum total weight of the edges traveled, represented as
. If
denotes all feasible paths from
i to
j, then
where
p represents a path in
made up of edges
with weights
. Dijkstra’s and the Bellman–Ford algorithm are frequently employed to determine the shortest paths in weighted graphs [
45].
A clique or a topological triangle in weighted networks can be defined by including edge weights [
46,
47,
48]. A weighted clique with threshold
t is a subset of nodes
C from the set
V of all network nodes, where for every pair of different nodes
in
C, there is an edge
with weight
. This term encapsulates the intuitive notion of closely interconnected groupings with strong internal bonds.
The Minimum Spanning Tree (MST) is a fundamental concept in graph theory, specifically in the examination and improvement of networks where links are assigned weights that represent distances or costs [
49,
50]. In a graph, a MST is a set of edges that forms a tree and includes all the vertices. The total weight of the edges in the MST is minimized, making it a direct representation of hierarchical clustering in a weighted network. The tree structure is crucial in situations where the objective is to minimize the cost or distance of linking different vertices. The lowest
values are crucial in the construction of a MST, especially through algorithms such as Prim’s and Kruskal’s [
51]. These algorithms aim to include edges that help achieve the lowest
between any two vertices in the tree. This technique guarantees that the tree covers all vertices in the graph while minimizing the total weight, effectively utilizing the concept that shorter distances generally indicate stronger or more desired connections. When building an MST, the rule is to choose edges with smaller weights, as these reflect stronger relationships or more desirable connections [
52,
53]. The preference for minimizing edge weights guarantees that the resulting tree will cover all vertices while incurring the lowest possible total cost. The MST possesses mathematical elegance by efficiently representing the most optimal route through a network. It showcases a crucial characteristic: out of all the potential spanning trees, the MST ensures the lowest total of distances or expenses, thereby enhancing connection.
Assigning weights to network edges based on distances is a crucial method in network analysis, providing an advanced way to measure the connections or interactions between nodes in a network. The weight of an edge often denotes the distance between nodes, using different distance metrics based on the network’s characteristics and the data it conveys. This method allows for the quantification of the strength, similarity, or dissimilarity of nodes using numerical values, which helps in gaining a more profound insight into the structure and behavior of the network.
The selection of a distance metric significantly impacts the success of this strategy. Euclidean distance is often used in spatial networks, whereas Hamming or cosine distances are better suited for networks that involve text or category data [
54,
55]. Integrating these weighted edges alters the network analysis, enabling algorithms to take into account not only the existence of a connection but also its qualitative intensity or importance. This sophisticated method of network analysis presents difficulties. Choosing a suitable distance measure and properly scaling distances can greatly influence the results of the investigation. In highly dynamic networks, it is essential to have strong mechanisms in place to update weights in response to changes over time. However, the advantages of edge weighting using distances, such as increased analytical accuracy, greater model realism, and the capacity to provide profound insights into network activity, establish it as a fundamental technique in the realm of network science.
Euclidean distance in descriptor space is particularly useful in molecular networks where nodes represent molecules, and edges signify the similarity or dissimilarity between their chemical properties. For molecules represented by numerical descriptors, the Euclidean distance between two molecules,
A and
B with descriptor vectors
and
can be calculated as
where
n is the number of descriptors, and
and
are the values of the
ith descriptor for molecules
A and
B, respectively.
This work considers not only the general descriptors of molecules, such as physicochemical properties, but also geometric and topological descriptors, which delve into the spatial arrangement and connectivity of atoms within a molecule.
In the context of clustering BACE-1 inhibitors using Euclidean distance, several molecular descriptor vectors are employed to capture the diverse properties of these inhibitors. The first descriptor considered is the molecular mass, reflecting the overall size and weight of the molecule. This descriptor is fundamental as it influences many physical and chemical properties, including the molecule’s behavior in biological systems.
The Merck Molecular Force Field (MMFF) Energy is a computational estimate of a molecule’s potential energy, reflecting its stability and reactivity. Lower energy indicates a more stable structure, which is crucial for understanding the energetics of BACE-1 inhibitors. The MMFF energy is computed using several components:
In equation
is the bond force constant,
r is the bond length,
is the equilibrium bond length. This term accounts for the energy associated with the stretching of bonds between atoms. Angle bending energy is defined by
where
is the angle force constant,
is the bond angle,
is the equilibrium bond angle. This term represents the energy required to bend bond angles from their equilibrium positions. Torsional (dihedral) energy is also defined by
where
is the torsional barrier height,
n is the periodicity,
is the dihedral angle,
is the phase angle. This term describes the energy associated with the rotation around bonds. Out-of-plane bending energy is defined by
where
is the out-of-plane bending force constant,
is the out-of-plane angle. This term accounts for the energy needed to distort atoms out of the plane defined by their neighboring atoms. Van der Waals energy is defined by
where
and
are van der Waals parameters for the interaction between atoms
i and
j,
is the distance between atoms
i and
j. This term describes the van der Waals forces, including repulsive and attractive interactions. Finally, electrostatic energy is defined by
where
and
are the partial charges on atoms
i and
j,
is the dielectric constant. This term accounts for the electrostatic interactions between charged atoms.
The MMFF energy descriptor provides a comprehensive estimate of the potential energy of a molecule, encompassing various intra- and intermolecular interactions such as bond stretching, angle bending, torsional strain, van der Waals forces, and electrostatic interactions. These energy components are fundamental to understanding the overall stability and reactivity of the inhibitors. Even though BACE-1 inhibitors are not isomers, the MMFF energy offers insights into how stable and reactive these molecules are under standard conditions which are crucial for their biochemical efficacy. To address the challenge of comparing MMFF energies across non-isomeric compounds, we normalize the energy values. This involves calculating the relative MMFF energy by subtracting the energy of a reference structure within the same class of compounds from the MMFF energy of each inhibitor. This normalization mitigates the differences in the number and types of atoms, allowing for a more meaningful comparison that focuses on the relative stability and reactivity of the molecules.
Crippen ClogP, the calculated logarithm of the partition coefficient, measures the lipophilicity of a compound, influencing its solubility, permeability, and interaction with biological membranes. Different inhibitors can exhibit varying logP values, affecting their bioavailability and distribution within biological systems. Crippen MR, or molar refractivity, measures the volume occupied by a molecule’s electrons, relating to molecular size and polarizability, which impacts interactions with the environment.
Eccentricity is a topological descriptor that measures how much a molecule’s shape deviates from being circular or spherical, affecting how it fits into binding sites and interacts with other molecules. This provides insight into the spatial configuration of inhibitors, which is important for understanding their functional roles. The Kappa indices (, , and ) are topological descriptors that measure molecular shape based on the number and arrangement of atoms and bonds. provides information on the complexity and compactness of a molecule’s structure, while and offer additional details on molecular shape and branching patterns, essential for differentiating inhibitors based on their structural intricacies and potential interactions.
The Synthetic Accessibility Score estimates the ease with which a molecule can be synthesized based on its structural complexity, indicating how readily a molecule can be produced for experimental or therapeutic use. Inhibitors with lower synthetic accessibility might be more challenging to work with but could offer unique properties. The Topological Polar Surface Area (TPSA) measures the surface area occupied by polar atoms, affecting a molecule’s ability to interact with polar environments, influencing properties like solubility, permeability, and transport. Inhibitors with different TPSA values have varied interactions with biological systems, impacting their pharmacokinetic and pharmacodynamic profiles.
Two-dimensional autocorrelation refers to the analysis of the spatial distribution of molecular properties (e.g., atomic charges, polarizabilities) on a two-dimensional representation of the molecule, such as its connectivity graph. It involves calculating the correlation of a property at one atom with the same property at another atom, separated by a certain topological distance (number of bonds). This descriptor can include the autocorrelation of atomic charges, masses, electronegativities, or other properties over the molecular graph. Three-dimensional autocorrelation extends this analysis to the three-dimensional structure of the molecule, taking into account the spatial arrangement of atoms in three-dimensional space. It involves calculating the correlation of a property at one atom with the same property at another atom, separated by a certain Euclidean distance in 3D space. This descriptor can include the autocorrelation of properties such as van der Waals volumes, solvent-accessible surface areas, or electrostatic potentials.
When using 2D and 3D autocorrelations as descriptors, their entropy is considered because they are obtained vectorially. Entropy, in the context of autocorrelation, measures the randomness or complexity of the distribution of a particular property across the molecule. High entropy indicates a more random and complex distribution, while low entropy indicates a more ordered distribution. In this study, the Shannon entropy function is used for the entropies of 2D and 3D autocorrelation vectors.
Each of these descriptors is linked to the molecular energies of BACE-1 inhibitors in different ways. For instance, MMFF Energy directly measures the potential energy, while descriptors like Crippen ClogP and TPSA relate to how the structure of the molecule influences its interaction with the environment, indirectly affecting its energy state. Eccentricity and Kappa indices describe shape and branching, which can affect molecular stability and reactivity, thereby influencing the energy profile. The Synthetic Accessibility Score, although not directly related to energy, provides a practical aspect of how molecular properties translate to synthetic feasibility, which is crucial for the experimental validation of theoretical models. Entropy is a fundamental thermodynamic property contributing to the molecule’s free energy. In this context, higher entropy can lower the Gibbs free energy, influencing molecular stability and reaction favorability. Therefore, the entropy of 2D and 3D autocorrelation provides valuable information about molecular complexity and distribution, crucial for understanding molecular stability, reactivity, and overall energy in chemical and biological systems.
We use various molecular descriptors, including MMFF energy, cLogP, and TPSA, to capture the biochemical properties of BACE-1 inhibitors comprehensively. Each of these descriptors provides critical insights into different aspects of the molecular behavior and potential efficacy of the inhibitors. MMFF Energy is crucial for understanding the stability and binding affinity of BACE-1 inhibitors. The MMFF energy calculation provides an estimate of the potential energy of the molecule, taking into account various intra- and intermolecular interactions such as bond stretching, angle bending, torsional strain, van der Waals forces, and electrostatic interactions. A lower MMFF energy value indicates a more stable molecular structure, which is often associated with higher binding affinity to the target enzyme, making it a key predictor of inhibitory activity. cLogP measures the lipophilicity of the inhibitors. Lipophilicity is a critical factor that influences a molecule’s ability to permeate cellular membranes, which is essential for its bioavailability and distribution within biological systems. High cLogP values suggest that a molecule is more hydrophobic, enhancing its potential to traverse lipid-rich environments such as cell membranes. However, excessive lipophilicity can also lead to issues such as poor solubility in aqueous environments and increased risk of off-target effects. Therefore, an optimal cLogP balance is necessary for effective BACE-1 inhibition. TPSA measures the surface area of a molecule occupied by polar atoms, typically oxygen and nitrogen, and their attached hydrogen atoms. TPSA is a significant descriptor as it affects a molecule’s ability to form hydrogen bonds, which in turn influences its solubility, permeability, and overall interaction with biological environments. Molecules with an appropriate TPSA are more likely to exhibit favorable pharmacokinetic properties, such as adequate absorption and permeation, while maintaining sufficient solubility. This balance is particularly important for achieving effective inhibition of BACE-1, as it ensures that the inhibitors can reach and interact with their target site efficiently. By integrating these descriptors into our network analysis, we can map out the structural and functional landscape of BACE-1 inhibitors, highlighting how specific molecular features contribute to their biochemical performance. This approach enables us to identify key structural attributes that enhance inhibitory potency and provides a robust framework for optimizing these molecules for therapeutic applications.
The Tanimoto molecule distance function offers another approach, specifically tailored for comparing molecular structures [
56]. It is used to calculate the similarity between two molecules based on their chemical fingerprints, which are binary vectors representing the presence or absence of particular structural elements. The Tanimoto coefficient (also known as the Jaccard index) is widely used for this purpose and is defined for two molecules with fingerprint vectors
and
as
where
is the dot product of the fingerprints and
and
are the squares of the norms of the fingerprints (essentially the count of bits set to 1). The Tanimoto coefficient ranges from 0 (completely different) to 1 (identical), and a distance measure can be derived as
The Tanimoto coefficient is a ratio that measures the overlap between the fingerprints of two molecules, making it an ideal distance measure for weighting edges in molecular networks, particularly when assessing structural similarity.
Hamming distance for fingerprint differences is relevant for networks where the focus is on the binary representation of molecular features. The Hamming distance counts the number of positions at which the corresponding elements (bits) of two fingerprints differ [
57]. For two molecular fingerprints
and
, the Hamming distance is given by
where
n is the length of the fingerprints and
and
are the bits at position
i in fingerprints
and
, respectively. This measure is particularly useful for quickly assessing the similarity between molecules in large datasets, providing a basis for weighting network edges that reflect minor structural differences or mutations [
58,
59].
Edit distance for structural comparisons is crucial for analyzing networks of molecules. The edit distance quantifies the minimum number of operations (such as the addition, removal, or substitution of a bond or atom) required to transform one molecular structure into another through string-based representations [
60]. In scenarios where molecules are compared based on their structural representations, Simplified Molecular Input Line Entry System (SMILES) strings, the Levenshtein distance [
61,
62] can quantify how many edits (insertions, deletions, substitutions) are required to convert one molecular representation into another. This measure is more qualitative and depends heavily on the representation scheme. Since it reveals the structural similarities and differences among molecules with the same chemical formula but different spatial configuration, this distance metric is very helpful for building weighted networks of molecules.
The network formed by utilizing the molecular distances between molecules is mathematically represented by a graph that consists of nodes equal in number to the total number of molecules. Among the molecules, the ones that are nearest to each other in terms of distance are connected by edges that have low values, which depend on how the distance functions are defined. These distances are non-zero in practice, except for identical molecules. Therefore, it is anticipated that there will be distinctions established for each node in the graph model that is constructed. The mathematical term for these types of graph structures is n-complete graphs. To provide consistent network analysis between molecules, it is imperative to exclude high-weighted edges that signify a weak edge.
The Triangulated Maximally Filtered Graph (TMFG) is a network analysis technique that simplifies intricate networks while retaining crucial information [
63]. It is also valuable for examining entire graphs that depict intricate systems, such as molecule networks, where the number of connections increases exponentially with the number of components, resulting in a significant degree of intricacy.
A complete graph represents a network in which each node represents a molecule, and every pair of nodes is connected by edge . The dissimilarity of the molecules is quantified by a distance measure (such as Euclidean, Tanimoto, Hamming, Levenshtein), and each edge is weighted accordingly. The TMFG chooses a subset of edges E from the complete graph to create a minimum spanning tree. This is accomplished by assigning priority to edges based on their weight, which indicates the intensity of the connection (inversely proportional to the distance) between nodes, and choosing those that contribute to a network that is as planar as possible.
A graph is considered planar if it can be represented on a two-dimensional plane without any edges intersecting. The TMFG guarantees planarity by following Euler’s formula for planar graphs, which establishes a relationship between the number of vertices V, edges E, and faces F in the graph: . The TMFG is a triangulated graph, which means that every face of the graph, except for the outer face, is a 3-clique. This is accomplished by augmenting the initial spanning tree with more edges until the requirement of maximal planarity is achieved without any violations.
The reason for using TMFG filtration for molecule networks is its ability to extract the core of the network, reducing complexity while preserving important information about chemical similarities and differences. The decision to utilize this methodology is justified by its capacity to highlight important molecular connections through a simplified yet informative network structure, improving analytical precision and enabling a more targeted investigation of molecular similarities. Moreover, the TMFG’s ability to effectively process extensive datasets, along with its resilience to interference and slight deviations, establishes it as an indispensable tool in the investigation of molecular networks. It skillfully emphasizes the significant relationships between molecules that are crucial in chemistry and biology, allowing for insightful interpretations that are both computationally possible and scientifically substantial. Therefore, the TMFG filtration not only simplifies but also strategically enhances the analysis, allowing for a more profound comprehension of the intricate connections within molecule networks.
The network representations obtained by filtering the structures of molecules provide a limited set of visual data in this study. To enhance comprehension and offer a more thorough analysis, this study includes quantitative measures that illustrate the interactions between molecules, along with graphical representations. This methodology guarantees a more profound understanding of the intricate dynamics and connections among the molecules, surpassing a simple visual study to encompass a thorough analysis of the interactions between these molecules.
Edge betweenness centrality of a node is a metric employed in network analysis to evaluate the significance of an edge in enabling communication or connectivity within the network. The term refers to the count of the most direct routes connecting pairs of points that traverse a specific link. In essence, it quantifies the frequency with which an edge serves as a bridge on the shortest route between two nodes [
64,
65]. Mathematically, for a given edge
e in a network, the edge betweenness centrality can be calculated using the following formula:
where
is the total number of shortest paths from node
s to node
t and
is the number of those shortest paths that passes through node
v.
A high betweenness edge in a molecule network signifies a crucial connection or pathway that is traversed by numerous shortest paths. These findings indicate that the two molecules connected by this edge have important functions in connecting other molecules in the network. They may act as central points or connectors that aid in the communication or transitions between different sections of the network. In the context of molecule networks, a high betweenness edge can indicate a pair of molecules that play a crucial role as intermediates in the transformation between two other molecules, or it can emphasize a significant similarity that is shared by many other molecules. Edges with high betweenness play a crucial role in maintaining the integrity and connection of the network from a structural standpoint. Eliminating these edges can greatly disturb the network, potentially causing it to be split into separate and isolated parts. Within the realm of molecules, this implies that the elimination or alteration of specific crucial molecules or transformations could significantly impact the range of potential isomermolecule transformations or resemblances.
In the field of network analysis, particularly when examining networks among molecules, the constraint score is a metric that measures the extent to which a node relies on other nodes in the network for its connections. The evaluation primarily assesses the distribution of a node’s connections among its neighbors, specifically examining whether the node’s network relationships are concentrated on a small number of nodes or spread out among a larger number. A higher constraint score signifies a higher degree of reliance on a smaller number of nodes, implying that the node has fewer but more concentrated connections. Conversely, a lower score implies that a node is more uniformly connected across multiple nodes, indicating a lesser amount of dependency.
The constraint score for node
i in a network can be calculated using the following formula:
where
is the proportion of node
i’s direct ties that is allocated to node
j and
represents the indirect ties from
i to
j through a third node,
k [
66].
Within a network of molecules, a high constraint score for a certain molecule signifies its strong reliance on a limited number of other molecules. This reliance can be due to direct transformation pathways, structural similarities, or other types of interactions. This indicates that the molecule holds a specific position or specialized function within the network, potentially serving as a vital mediator in particular reactions or transformations. An analysis of the constraint scores throughout the network can provide insights into the general organization and behavior of the network. A network containing numerous nodes with high constraints tends to exhibit a tightly ordered structure centered around a limited number of crucial pathways or links. On the other hand, a network with lower overall constraint scores tends to be more flexible, featuring multiple pathways and linkages between molecules. Molecules that have high constraint ratings may be promising candidates for chemical synthesis or modification endeavors. Modifying these molecules or their connections, given their crucial location and interdependencies in the network, could have substantial consequences on the network’s behavior. This could potentially lead to the emergence of new routes or changes in the distribution of molecules.
The scores of the constraints can also offer valuable information concerning the network’s resiliency. Networks that consist of numerous nodes with high constraint scores may exhibit greater susceptibility to disruptions. The removal of a crucial molecule or connection could have a substantial influence on the network. On the other hand, networks with nodes that have lower constraint scores are probably more resilient, as molecules rely less on specific connections.
Eigenvector centrality is a metric that quantifies the impact of a node in a network, taking into account both the quantity and quality of its connections [
67]. The idea underlying its operation is that connections to nodes that are highly linked or influential have a greater impact on the centrality of a node compared to connections of similar strength to less influential nodes. Eigenvector centrality for node
i, denoted as
, is defined as
where
W is the weighted adjacency matrix,
is the largest eigenvalue of
W.
Within a molecule network, a node with a high eigenvector centrality is linked not only to several other nodes but also specifically to nodes that are themselves highly connected or important. This implies that the molecule has a substantial impact on the network, maybe serving as a crucial intermediary in reaction pathways or as a pivotal hub in a network of structural resemblances. A molecule with high eigenvector centrality suggests that it is a pivotal component in a sequence of reactions, capable of producing numerous other molecules either directly or through densely interconnected intermediary compounds. It can identify molecules that play a crucial role in the synthesis of a diverse range of other chemicals. Eigenvector centrality can be used to find molecules that are not only structurally related to many others but also central in a cluster of very similar molecules, when edges signify structural similarity. This can be beneficial for discerning characteristic structures or comprehending the arrangement of chemical space surrounding fundamental patterns.
4.2. Network Communities
Network communities in the context of molecule networks can be mathematically defined using notions of graph theory. In such a network, nodes represent molecules, and edges represent the similarity or dissimilarity between these molecules, often quantified by measures such as Euclidean distance in descriptor space, Tanimoto molecule distance, Hamming distance for fingerprint differences, or edit distance for structural comparisons. Identifying communities involves partitioning the network into subgraphs (communities) where nodes (molecules) are densely connected internally but sparsely connected to nodes outside the community. This clustering based on structural similarities or differences is crucial for understanding the collective behaviors of molecules in chemical reactions or biological systems. Mathematically, if we represent our network as a filtered weighted graph
where
V is the set of nodes and
is the set of filtered edges (distances between molecules), community detection seeks to maximize modularity
Q of a partition of
V defined as
where
represents the edge weight between nodes
i and
j,
and
are the sum of the weights of the edges attached to nodes
i and
j, respectively,
m is the sum of all of the edge weights in the network,
and
are the communities containing nodes
i and
j, and
is the Kronecker delta function, which is 1 if
and 0 otherwise. We can better understand and control the features and interactions of molecules by classifying them into communities that reflect their significant chemical or structural similarities.
The Leading Eigenvector algorithm [
68] utilizes the eigenspectrum of the modularity matrix
B of a network, where
. Here,
is the adjacency matrix,
and
are the weighted degrees of nodes
i and
j, and
m is the total number of edges. The network is partitioned by the sign of the components of the leading eigenvector of
B, corresponding to its largest eigenvalue, to identify communities.
The Leiden algorithm [
69] enhances this by iteratively refining communities through a three-phase process: (1) local moving of nodes to improve modularity, (2) aggregation of nodes into super nodes based on the refined communities, and (3) the creation of a new network of these super nodes. This process, which ensures connectivity within identified communities, repeats until modularity
Q can no longer be increased, leading to more accurate and finely grained community detection.
Within the scope of molecule networks, the Leading Eigenvector and Leiden algorithms employ mathematical models to group molecules together based on their similarity in structure or properties. The Leading Eigenvector technique includes dividing the network based on the positive and negative components of the principal eigenvector of the modularity matrix. This grouping process forms communities that maximize the density of edge weights inside each community. The Leiden algorithm utilizes iterative refinement to enhance modularity by locally moving nodes and subsequently aggregating communities into super nodes for further partitioning. This approach not only groups molecules based on their similarities, but also guarantees that these groups have significance in terms of chemical behavior or physical properties. This results in more refined classifications that can indicate shared or distinct characteristics among the molecules, such as patterns of reactivity or solubility.
The clustering process is improved when these approaches are applied to distance-based molecule networks because it allows for the utilization of the particular advantages that are associated with each distance measure. For instance, the Euclidean Distance in Descriptor Space makes it possible to cluster based on subtle physicochemical features, which allows for the capture of minor changes in the molecular composition of molecules. By comparing molecular fingerprints, Tanimoto distance makes it possible to categorize molecules according to their more general structural characteristics. In the meantime, the Hamming distance places an emphasis on minute, precise changes in the structure of molecules, and the edit distance offers insight into the degree of structural modification that is required between molecules. An improved comprehension of molecular variety can be achieved by the utilization of this multi-faceted method, which provides a comprehensive and thorough panorama of molecule similarities and variances.
This study examines two distinct network communities within molecule networks and uses a range of network metrics to assess the performance differences between these communities.
The Variation of Information (VI) is a metric derived from information theory that precisely measures the quantity of information that is both lost and gained when comparing two community architectures [
70,
71]. Given two partitions of the same dataset,
and
, the Variation of Information is defined as
where
and
are the entropies of communities
and
, respectively, and
is the mutual information between
and
. The entropy of a partition measures the uncertainty or randomness in the partition, and the mutual information quantifies the amount of shared information. A lower
indicates a higher similarity between two community structures, with a
of zero denoting identical community assignments.
Normalized Mutual Information (NMI) is a metric based on information theory that measures the similarity between two sets of community structures [
72]. For two communities
and
, NMI is defined as
NMI values range from 0 to 1, where 1 indicates perfect agreement between two community structures and 0 denotes no mutual information.
The Split–Join distance measures the similarity between two clusterings by counting the minimum number of split and join operations required to transform one clustering into another. It effectively counts the number of elements that need to be moved to reconcile the differences between two partitions [
73]. Given two communities, the Split-Join distance (SJ) is defined as
where
is the number of splits required to transform
into
, and
is the number of joins required to transform
into
. A SJ of zero means that the two community structures are identical.
The Rand Index (RI) is a measure of the similarity between two data clusterings. It is calculated by considering all pairs of elements in the dataset and counting the pairs that are either assigned to the same or different clusters in both clusterings [
74]. The RI between two communities is defined as
where
is the number of pairs of elements that are in the same subset in
and in the same subset in
,
is the number of pairs of elements that are in different subsets in
and different subsets in
, and the denominator is the total number of possible pairs. RI is a measure that ranges from 0 to 1. A value of 1 indicates that the clusterings are completely identical, while a value of 0 indicates that there is no agreement between the clusterings beyond what would be expected by chance.
The Adjusted Rand Index (ARI) is a corrected-for-chance version of the Rand Index that adjusts for the fact that the RI tends to increase simply with the number of clusters, even if the clusterings are randomly generated, and it is defined as
where
is the expected value of the RI under random classifications and
is the maximum possible value of the RI.