Functional Distance: A Novel Measure of Network Connectivity
Abstract
Network connectivity is a critical factor in the analysis of network structure and dynamics. Traditional measures of connectivity, such as edge betweenness, are limited in their ability to capture the impact of node position on the overall network structure. In this paper, we present a novel measure of network connectivity, referred to as functional distance, which takes into account the impact of node position on the overall network structure. We demonstrate the effectiveness of this measure by applying it to a variety of real-world and simulated networks. Our results show that functional distance is a powerful tool for measuring network connectivity and can be used to identify important nodes, trends, and patterns in complex networks.
Keywords: Network Connectivity, Functional Distance, Network Structure, Network Dynamics
Introduction
Network connectivity is an important factor in the analysis of network structure and dynamics. Traditional measures of connectivity, such as edge betweenness, are limited in their ability to capture the impact of node position on the overall network structure (Borgatti & Everett, 2006). This is due to the fact that these measures are based solely on the number of edges connecting a node to others, without taking into account the relative position of the node within the network (Onnela & Saramaki, 2007). In order to more effectively measure the impact of node position on the overall network structure, we propose a novel measure of network connectivity, referred to as functional distance.
Functional distance is based on the principle that nodes that are close together, both in terms of physical distance and in terms of their position in the network, are more likely to interact than those that are further apart. By taking into account both the physical and network distances between nodes, this measure is able to capture the impact of node position on the overall network structure more effectively than traditional measures.
Methods
To calculate functional distance, we first define a “functional neighborhood” for each node in the network. This neighborhood is defined as the set of nodes that are within a given distance from the focal node, both in terms of physical distance and network distance. The physical distance is measured as the Euclidean distance between the nodes, while the network distance is measured as the length of the shortest path connecting the nodes.
Once the functional neighborhood for each node is defined, we can then calculate the functional distance between two nodes as the average of the physical and network distances between them. In other words, the functional distance between nodes i and j is calculated as:
Functional Distance(i,j) = (Physical Distance(i,j) + Network Distance(i,j))/2
Results
To demonstrate the effectiveness of functional distance, we applied it to a variety of real-world and simulated networks. Our results show that functional distance is a powerful tool for measuring network connectivity and can be used to identify important nodes, trends, and patterns in complex networks.
In our first set of experiments, we applied functional distance to a simulated social network with 1,000 nodes and 10,000 edges. We then compared the results to a baseline measure of connectivity, edge betweenness (Freeman, 1977). The results showed that functional distance was able to more accurately identify important nodes and trends in the network than edge betweenness.
In our second set of experiments, we applied functional distance to a real-world network of airlines. We then compared the results to a baseline measure of connectivity, degree centrality (Degree Centrality, 2019). Again, the results showed that functional distance was able to more accurately identify important nodes and trends in the network than degree centrality.
Finally, we applied functional distance to a simulated biological network with 1,000 nodes and 10,000 edges. We then compared the results to a baseline measure of connectivity, closeness centrality (Freeman, 1978). The results showed that functional distance was able to more accurately identify important nodes and trends in the network than closeness centrality.
Conclusion
In summary, functional distance is a novel measure of network connectivity that takes into account the impact of node position on the overall network structure. Our results demonstrate that functional distance is a powerful tool for measuring network connectivity and can be used to identify important nodes, trends, and patterns in complex networks.
References
Borgatti, S. P., & Everett, M. G. (2006). A graph-theoretic perspective on centrality. Social Networks, 28(4), 466-484.
Degree centrality. (2019). In Wikipedia. Retrieved from http://en.wikipedia.org/wiki/Degree_centrality
Freeman, L. C. (1977). A set of measures of centrality based on betweenness. Sociometry, 40(1), 35–41.
Freeman, L. C. (1978). Centrality in social networks conceptual clarification. Social Networks, 1(3), 215-239.
Onnela, J.-P., & Saramäki, J. (2007). Structure and tie strengths in mobile communication networks. Proceedings of the National Academy of Sciences, 104(18), 7332–7336.