"The first attempts to consider the behavior of so-called 'random neural nets' in a systematic way have led to a series of problems concerned with relations between the 'structure' and the 'function' of such nets. The 'structure' of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of 'this' neuron synapsing on 'that' one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport, "Cycle distributions in random nets", The Bulletin of Mathematical Biophysics 10(3), 1948)
"Networks are not en route from a random to an ordered state. Neither are they at the edge of randomness and chaos. Rather, the scale-free topology is evidence of organizing principles acting at each stage of the network formation process."
"In a random network the loss of a small number of nodes can cause the overall network to become incoherent - that is, to break into disconnected subnetworks. In a scale-free network, such an event usually won’t disrupt the overall network because most nodes don’t have many links. But there’s a big caveat to this general principle: if a scale-free network loses a hub, it can be disastrous, because many other nodes depend on that hub."
"If a network is solely composed of neighborhood connections, information must traverse a large number of connections to get from place to place. In a small-world network, however, information can be transmitted between any two nodes using, typically, only a small number of connections. In fact, just a small percentage of random, long-distance connections is required to induce such connectivity. This type of network behavior allows the generation of 'six degrees of separation' type results, whereby any agent can connect to any other agent in the system via a path consisting of only a few intermediate nodes." (John H Miller & Scott E Page, "Complex Adaptive Systems", 2007)
"A graph enables us to visualize a relation over a set, which makes the characteristics of relations such as transitivity and symmetry easier to understand. […] Notions such as paths and cycles are key to understanding the more complex and powerful concepts of graph theory. There are many degrees of connectedness that apply to a graph; understanding these types of connectedness enables the engineer to understand the basic properties that can be defined for the graph representing some aspect of his or her system. The concepts of adjacency and reachability are the first steps to understanding the ability of an allocated architecture of a system to execute properly." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)
"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)
"Some connected graphs are 'more connected' than others. That is, a connected graph’s vulnerability to disconnection by edge- or vertex-deletion varies. Two numerical parameters, vertex-connectivity and edge-connectivity, are useful in measuring a graph’s connectedness. Intuitively, a network’s vulnerability should be closely related also to the number of alternative paths between each pair of nodes. There is a rich body of mathematical results concerning this relationship, many of which are variations of a classical result of Menger, and some of these extend well beyond graph theory." (Jonathan L Gross et al, "Topics in Graph Theory", 2023)
"Spanning trees capture the connectedness of a graph in the most efficient way, and they provide a foundation for a systematic analysis of the cycle structure of a graph. Mathematicians regard the algebraic structures underlying the collection of cycles and edge-cuts of agraph as beautiful in their own right. Establishing connections between linear algebra and graph theory provides some powerful analytical tools for understanding a graph’s structure." (Jonathan L Gross et al, "Topics in Graph Theory", 2023)
No comments:
Post a Comment