On Graph Theory: On Structure (2010-2019)

"As it turns out, complex networks are everywhere. Or, to be more precise, it turns out that if we model real-world situations in terms of networks, we often discover new things. What is striking, is that many real-world net- works look alike: the structure of the Internet resembles the organization of our brain, but also the organization of online social communities. Where these similarities come from is still a mystery, just as it is often very difficult to understand how certain networks were actually structured." (Maarten van Steen, "Graph Theory and Complex Networks: An Introduction", 2010)

"First, what are the 'graphs' studied in graph theory? They are not graphs of functions as studied in calculus and analytic geometry. They are (usually finite) structures consisting of vertices and edges. As in geometry, we can think of vertices as points (but they are denoted by thick dots in diagrams) and of edges as arcs connecting pairs of distinct vertices. The positions of the vertices and the shapes of the edges are irrelevant: the graph is completely specified by saying which vertices are connected by edges. A common convention is that at most one edge connects a given pair of vertices, so a graph is essentially just a pair of sets: a set of objects." (John Stillwell, "Mathematics and Its History", 2010)

"[...] the contexts in which a social network is embedded will generally have significant effects on its structure. Each individual in a social network has a distinctive set of personal characteristics, and similarities and compatibilities between two people’s characteristics can strongly influence whether a link forms between them. Each individual also engages in a set of behaviors and activities that can shape the formation of links within the network. These considerations suggest what we mean by a network’s surrounding contexts: factors that exist outside the nodes and edges of a networks, but which nonetheless affect how the network’s structure evolves." (David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010)

"When people talk about the 'connectedness' of a complex system, in general they are really talking about two related issues. One is connectedness at the level of structure – who is linked to whom – and the other is connectedness at the level of behavior – the fact that each individual’s actions have implicit consequences for the outcomes of everyone in the system."(David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010)

"Cybernetics is the study of systems which can be mapped using loops (or more complicated looping structures) in the network defining the flow of information. Systems of automatic control will of necessity use at least one loop of information flow providing feedback." (Alan Scrivener, "A Curriculum for Cybernetics and Systems Theory", 2012)

"All living systems are networks of smaller components, and the web of life as a whole is a multilayered structure of living systems nesting within other living systems - networks within networks." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"The great strength of node–link layouts is that for sufficiently small networks they are extremely intuitive for supporting many ofthe abstract tasks that pertain to network data. They particularly shine for tasks that rely on understanding the topological structure of the network, such as path tracing and searching local topological neighborhoods a small number of hops from a target node, and can also be very effective for tasks such as general overview or finding similar substructures. The effectiveness of the general idiom varies considerably depending on the specific visual encoding idiom used [...]" (Tamara Munzner, "Visualization Analysis and Design", 2014)

"Although cascading failures may appear random and unpredictable, they follow reproducible laws that can be quantified and even predicted using the tools of network science. First, to avoid damaging cascades, we must understand the structure of the network on which the cascade propagates. Second, we must be able to model the dynamical processes taking place on these networks, like the flow of electricity. Finally, we need to uncover how the interplay between the network structure and dynamics affects the robustness of the whole system." (Albert-László Barabási, "Network Science", 2016)