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A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as
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.
A small-world network is a graph characterized by a high clustering coefficient and low distances. On an example of social network, high clustering implies the high probability that two friends of one person are friends themselves. The low distances, on the other hand, mean that there is a short chain of social connections between any two people. Specifically, a small-world network is defined to be a network where the typical distance L between two randomly chosen nodes grows proportionally to the logarithm of the number of nodes N in the network, that is:
A transport network, or transportation network, is a network or graph in geographic space, describing an infrastructure that permits and constrains movement or flow. Examples include but are not limited to road networks, railways, air routes, pipelines, aqueducts, and power lines. The digital representation of these networks, and the methods for their analysis, is a core part of spatial analysis, geographic information systems, public utilities, and transport engineering. Network analysis is an application of the theories and algorithms of graph theory and is a form of proximity analysis.
In the context of network theory, a complex network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in networks representing real systems. The study of complex networks is a young and active area of scientific research inspired largely by empirical findings of real-world networks such as computer networks, biological networks, technological networks, brain networks, climate networks and social networks.
A spatial network is a graph in which the vertices or edges are spatial elements associated with geometric objects, i.e., the nodes are located in a space equipped with a certain metric. The simplest mathematical realization of spatial network is a lattice or a random geometric graph, where nodes are distributed uniformly at random over a two-dimensional plane; a pair of nodes are connected if the Euclidean distance is smaller than a given neighborhood radius. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks and biological neural networks are all examples where the underlying space is relevant and where the graph's topology alone does not contain all the information. Characterizing and understanding the structure, resilience and the evolution of spatial networks is crucial for many different fields ranging from urbanism to epidemiology.
In the study of complex networks, a network is said to have community structure if the nodes of the network can be easily grouped into sets of nodes such that each set of nodes is densely connected internally. In the particular case of non-overlapping community finding, this implies that the network divides naturally into groups of nodes with dense connections internally and sparser connections between groups. But overlapping communities are also allowed. The more general definition is based on the principle that pairs of nodes are more likely to be connected if they are both members of the same community(ies), and less likely to be connected if they do not share communities. A related but different problem is community search, where the goal is to find a community that a certain vertex belongs to.
In applied probability theory, the Simon model is a class of stochastic models that results in a power-law distribution function. It was proposed by Herbert A. Simon to account for the wide range of empirical distributions following a power-law. It models the dynamics of a system of elements with associated counters. In this model the dynamics of the system is based on constant growth via addition of new elements as well as incrementing the counters at a rate proportional to their current values.
In network science, a gradient network is a directed subnetwork of an undirected "substrate" network where each node has an associated scalar potential and one out-link that points to the node with the smallest potential in its neighborhood, defined as the union of itself and its neighbors on the substrate network.
An ecological network is a representation of the biotic interactions in an ecosystem, in which species (nodes) are connected by pairwise interactions (links). These interactions can be trophic or symbiotic. Ecological networks are used to describe and compare the structures of real ecosystems, while network models are used to investigate the effects of network structure on properties such as ecosystem stability.
Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Modularity is often used in optimization methods for detecting community structure in networks. Biological networks, including animal brains, exhibit a high degree of modularity. However, modularity maximization is not statistically consistent, and finds communities in its own null model, i.e. fully random graphs, and therefore it cannot be used to find statistically significant community structures in empirical networks. Furthermore, it has been shown that modularity suffers a resolution limit and, therefore, it is unable to detect small communities.
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.
The global shipping network is the worldwide network of maritime traffic. From a network science perspective ports represent nodes and routes represent lines. Transportation networks have a crucial role in today's economy, more precisely, maritime traffic is one of the most important drivers of global trade.
Luís A. N. Amaral is a Portuguese physicist recognized for his research in complex systems and complex networks. His specific research interests include the emergence, evolution, and stability of complex social and biological systems. He is best known for his work in network classification and cartographic methods for uncovering the organization of complex networks. He is currently professor at McCormick School of Engineering and Feinberg School of Medicine, Northwestern University.
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.
Individual human mobility is the study that describes how individual humans move within a network or system. The concept has been studied in a number of fields originating in the study of demographics. Understanding human mobility has many applications in diverse areas, including spread of diseases, mobile viruses, city planning, traffic engineering, financial market forecasting, and nowcasting of economic well-being.
Targeted immunization strategies are approaches designed to increase the immunization level of populations and decrease the chances of epidemic outbreaks. Though often in regards to use in healthcare practices and the administration of vaccines to prevent biological epidemic outbreaks, these strategies refer in general to immunization schemes in complex networks, biological, social or artificial in nature. Identification of at-risk groups and individuals with higher odds of spreading the disease often plays an important role in these strategies, since targeted immunization in high-risk groups is necessary for effective eradication efforts and has a higher return on investment than immunizing larger but lower-risk groups.
Robustness, the ability to withstand failures and perturbations, is a critical attribute of many complex systems including complex networks.
Trophic coherence is a property of directed graphs. It is based on the concept of trophic levels used mainly in ecology, but which can be defined for directed networks in general and provides a measure of hierarchical structure among nodes. Trophic coherence is the tendency of nodes to fall into well-defined trophic levels. It has been related to several structural and dynamical properties of directed networks, including the prevalence of cycles and network motifs, ecological stability, intervality, and spreading processes like epidemics and neuronal avalanches.
Marta Sales-Pardo is a Spanish statistical physicist, complex systems scientist, and network scientist. She is an associate professor at Rovira i Virgili University, in the Department of Chemical Engineering. She is notable for her contributions to complex networks, where she has studied both social and biochemical systems, especially the contribution of each node or individual to the overall network. Her research has earned several mentions in the Fulbright Program and the ICREA Program, among others.
References
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- ↑ Barrat, A.; Barthelemy, M.; Pastor-Satorras, R.; Vespignani, A. (2004). "The architecture of complex weighted networks". Proc. Natl. Acad. Sci. 101 (11): 3747–3752. arXiv: cond-mat/0311416 . Bibcode:2004PNAS..101.3747B. doi: 10.1073/pnas.0400087101 . PMC 374315 . PMID 15007165.
- ↑ X. Hu and E. Di Paolo "A Genetic Algorithm Based on Complex Networks Theory for the Management of Airline Route Networks", Studies in Computational Intelligence (SCI) 129, 495–505 (2008) (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.434.8417&rep=rep1&type=pdf#page=495).
- ↑ Wang, Junwei; Zhou, Yaoming; Huang, George Q. (2019-06-01). "Alternative pair in the airport network". Transportation Research Part A: Policy and Practice. 124: 408–418. doi:10.1016/j.tra.2019.04.010. ISSN 0965-8564. S2CID 159312236.