A weighted network is a network where the ties among nodes have weights assigned to them. A network is a system whose elements are somehow connected. [1] The elements of a system are represented as nodes (also known as actors or vertices) and the connections among interacting elements are known as ties, edges, arcs, or links. The nodes might be neurons, individuals, groups, organisations, airports, or even countries, whereas ties can take the form of friendship, communication, collaboration, alliance, flow, or trade, to name a few.
In a number of real-world networks, not all ties in a network have the same capacity. In fact, ties are often associated with weights that differentiate them in terms of their strength, intensity, or capacity [2] [3] On the one hand, Mark Granovetter (1973) [4] argued that the strength of social relationships in social networks is a function of their duration, emotional intensity, intimacy, and exchange of services. On the other, for non-social networks, weights often refer to the function performed by ties, e.g., the carbon flow (mg/m2/day) between species in food webs, [5] the number of synapses and gap junctions in neural networks, [6] or the amount of traffic flowing along connections in transportation networks. [7]
By recording the strength of ties, [8] a weighted network can be created (also known as a valued network).
Weighted networks are also widely used in genomic and systems biologic applications. [3] For example, weighted gene co-expression network analysis (WGCNA) is often used for constructing a weighted network among genes (or gene products) based on gene expression (e.g. microarray) data. [9] More generally, weighted correlation networks can be defined by soft-thresholding the pairwise correlations among variables (e.g. gene measurements). [10]
Although weighted networks are more difficult to analyse than if ties were simply present or absent, a number of network measures has been proposed for weighted networks:
A theoretical advantage of weighted networks is that they allow one to derive relationships among different network measures (also known as network concepts, statistics or indices). [3] For example, Dong and Horvath (2007) [15] show that simple relationships among network measures can be derived in clusters of nodes (modules) in weighted networks. For weighted correlation networks, one can use the angular interpretation of correlations to provide a geometric interpretation of network theoretic concepts and to derive unexpected relationships among them Horvath and Dong (2008) [16]
In network theory, intrinsically dense weighted networks represent a distinctive class of complex structures characterized by a near-completeness of links and associated weights, transcending the conventional constraints of sparser network configurations. Unlike sparse networks where the absence of links typically indicate lack of interaction, intrinsically dense networks exhibit a comprehensive interconnection among nodes, where each node is intricately linked to all others. Such systems do not have obvious natural limits for a node to have connection with any or all of the other nodes.
The term "intrinsically dense" emphasizes that edges within these networks may not solely represent positive relationships but can encompass randomness or even negative associations based on their respective weights. For instance, in scenarios where edge weights denote similarity between nodes, lower weights don't just signify a lack of similarity but may connote dissimilarity or negative underlying links. The study by Gursoy & Badur (2021) [17] introduced methods to extract meaningful and sparse signed backbones from these networks, showcasing their significance in preserving the intricate structures inherent in intrinsically dense weighted networks across various domains including certain migration, voting, human contact, and species cohabitation networks. This distinctive network paradigm expands the understanding of complex systems observed in natural, social, and technological domains, offering insights into nuanced interactions and relationships within these densely interconnected networks.
There are a number of software packages that can analyse weighted networks; see social network analysis software. Among these are the proprietary software UCINET and the open-source package tnet. [18]
The WGCNA R package implements functions for constructing and analyzing weighted networks in particular weighted correlation networks. [10]
Social network analysis (SNA) is the process of investigating social structures through the use of networks and graph theory. It characterizes networked structures in terms of nodes and the ties, edges, or links that connect them. Examples of social structures commonly visualized through social network analysis include social media networks, meme proliferation, information circulation, friendship and acquaintance networks, business networks, knowledge networks, difficult working relationships, collaboration graphs, kinship, disease transmission, and sexual relationships. These networks are often visualized through sociograms in which nodes are represented as points and ties are represented as lines. These visualizations provide a means of qualitatively assessing networks by varying the visual representation of their nodes and edges to reflect attributes of interest.
A generegulatory network (GRN) is a collection of molecular regulators that interact with each other and with other substances in the cell to govern the gene expression levels of mRNA and proteins which, in turn, determine the function of the cell. GRN also play a central role in morphogenesis, the creation of body structures, which in turn is central to evolutionary developmental biology (evo-devo).
In mathematics, computer science and network science, network theory is a part of graph theory. It defines networks as graphs where the vertices or edges possess attributes. Network theory analyses these networks over the symmetric relations or asymmetric relations between their (discrete) components.
In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes.
In graph theory and network analysis, indicators of centrality assign numbers or rankings to nodes within a graph corresponding to their network position. Applications include identifying the most influential person(s) in a social network, key infrastructure nodes in the Internet or urban networks, super-spreaders of disease, and brain networks. Centrality concepts were first developed in social network analysis, and many of the terms used to measure centrality reflect their sociological origin.
Computational phylogenetics, phylogeny inference, or phylogenetic inference focuses on computational and optimization algorithms, heuristics, and approaches involved in phylogenetic analyses. The goal is to find a phylogenetic tree representing optimal evolutionary ancestry between a set of genes, species, or taxa. Maximum likelihood, parsimony, Bayesian, and minimum evolution are typical optimality criteria used to assess how well a phylogenetic tree topology describes the sequence data. Nearest Neighbour Interchange (NNI), Subtree Prune and Regraft (SPR), and Tree Bisection and Reconnection (TBR), known as tree rearrangements, are deterministic algorithms to search for optimal or the best phylogenetic tree. The space and the landscape of searching for the optimal phylogenetic tree is known as phylogeny search space.
Biological network inference is the process of making inferences and predictions about biological networks. By using these networks to analyze patterns in biological systems, such as food-webs, we can visualize the nature and strength of these interactions between species, DNA, proteins, and more.
In a connected graph, closeness centrality of a node is a measure of centrality in a network, calculated as the reciprocal of the sum of the length of the shortest paths between the node and all other nodes in the graph. Thus, the more central a node is, the closer it is to all other nodes.
A biological network is a method of representing systems as complex sets of binary interactions or relations between various biological entities. In general, networks or graphs are used to capture relationships between entities or objects. A typical graphing representation consists of a set of nodes connected by edges.
The clique percolation method is a popular approach for analyzing the overlapping community structure of networks. The term network community has no widely accepted unique definition and it is usually defined as a group of nodes that are more densely connected to each other than to other nodes in the network. There are numerous alternative methods for detecting communities in networks, for example, the Girvan–Newman algorithm, hierarchical clustering and modularity maximization.
In graph theory, betweenness centrality is a measure of centrality in a graph based on shortest paths. For every pair of vertices in a connected graph, there exists at least one shortest path between the vertices, that is, there exists at least one path such that either the number of edges that the path passes through or the sum of the weights of the edges is minimized. The betweenness centrality for each vertex is the number of these shortest paths that pass through the vertex.
A social network is a social structure consisting of a set of social actors, sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures. The study of these structures uses social network analysis to identify local and global patterns, locate influential entities, and examine network dynamics. For instance, social network analysis has been used in studying the spread of misinformation on social media platforms or analyzing the influence of key figures in social networks.
Weighted correlation network analysis, also known as weighted gene co-expression network analysis (WGCNA), is a widely used data mining method especially for studying biological networks based on pairwise correlations between variables. While it can be applied to most high-dimensional data sets, it has been most widely used in genomic applications. It allows one to define modules (clusters), intramodular hubs, and network nodes with regard to module membership, to study the relationships between co-expression modules, and to compare the network topology of different networks. WGCNA can be used as a data reduction technique, as a clustering method, as a feature selection method, as a framework for integrating complementary (genomic) data, and as a data exploratory technique. Although WGCNA incorporates traditional data exploratory techniques, its intuitive network language and analysis framework transcend any standard analysis technique. Since it uses network methodology and is well suited for integrating complementary genomic data sets, it can be interpreted as systems biologic or systems genetic data analysis method. By selecting intramodular hubs in consensus modules, WGCNA also gives rise to network based meta analysis techniques.
In graph theory and network analysis, node influence metrics are measures that rank or quantify the influence of every node within a graph. They are related to centrality indices. Applications include measuring the influence of each person in a social network, understanding the role of infrastructure nodes in transportation networks, the Internet, or urban networks, and the participation of a given node in disease dynamics.
A gene co-expression network (GCN) is an undirected graph, where each node corresponds to a gene, and a pair of nodes is connected with an edge if there is a significant co-expression relationship between them. Having gene expression profiles of a number of genes for several samples or experimental conditions, a gene co-expression network can be constructed by looking for pairs of genes which show a similar expression pattern across samples, since the transcript levels of two co-expressed genes rise and fall together across samples. Gene co-expression networks are of biological interest since co-expressed genes are controlled by the same transcriptional regulatory program, functionally related, or members of the same pathway or protein complex.
The rich-club coefficient is a metric on graphs and networks, designed to measure the extent to which well-connected nodes also connect to each other. Networks which have a relatively high rich-club coefficient are said to demonstrate the rich-club effect and will have many connections between nodes of high degree. The rich-club coefficient was first introduced in 2004 in a paper studying Internet topology.
In mathematical modeling of social networks, link-centric preferential attachment is a node's propensity to re-establish links to nodes it has previously been in contact with in time-varying networks. This preferential attachment model relies on nodes keeping memory of previous neighbors up to the current time.
Network medicine is the application of network science towards identifying, preventing, and treating diseases. This field focuses on using network topology and network dynamics towards identifying diseases and developing medical drugs. Biological networks, such as protein-protein interactions and metabolic pathways, are utilized by network medicine. Disease networks, which map relationships between diseases and biological factors, also play an important role in the field. Epidemiology is extensively studied using network science as well; social networks and transportation networks are used to model the spreading of disease across populations. Network medicine is a medically focused area of systems biology.
Steve Horvath is a German–American aging researcher, geneticist, and biostatistician. He is a professor at the University of California, Los Angeles known for developing the Horvath aging clock, which is a highly accurate molecular biomarker of aging, and for developing weighted correlation network analysis. His work on the genomic biomarkers of aging, the aging process, and many age related diseases/conditions has earned him several research awards. Horvath is a principal investigator at the anti-aging startup Altos Labs and co-founder of nonprofit Clock Foundation.
In a social network analysis, a positive or a negative 'friendship' can be established between two nodes in a network; this results in a signed network. As social interaction between people can be positive or negative, so can be links between the nodes.