Introduction: Long and Winding Roads

Once upon a time, there was a cock and a mouse. One day the mouse said to the cock, “Friend cock, shall we go and eat some nuts on yonder tree?” “As you like.” So they both went under the tree, and the mouse climbed up at once and began to eat. The poor cock began to fly, and flew and flew, but could not come where the mouse was. When it saw that there was no hope of getting there, it said, “Friend mouse, do you know what I want you to do? Throw me a nut.” The mouse went and threw one and hit the cock on the head. The poor cock, with its head broken and all covered with blood, went away to an old woman. “Old aunt, give me some rags to cure my head.” “If you will give me two hairs, I will give you the rags.” The cock went away to a dog. “Dog, give me some hairs. The hairs I will give the old woman. The old woman will give me rags to cure my head.” “If you will give me a little bread,” said the dog, “I will give you the hairs.” The cock went away to a baker. “Baker, give me bread. I will give the bread to the dog. The dog will give hairs. The hairs I will carry to the old woman. The old woman will give me rags to cure my head.…

Amazing Scientific Discoveries: Aspirin, Cattle, Business Communication and Others

Besides the philosophical arguments, one could think of a series of possibilities where all these observations about paths could be used. Have you ever wondered, for example, what happens in your body after you swallow an Aspirin? As incredible as it sounds, this question was answered only 74 years after the development of the medicine. It was 1897 when the young German chemist Dr. Felix Hoffmann managed to stabilize the agent of Aspirin. After patenting in Germany in 1899 and in the US in 1900, Aspirin started its great triumph and became the most popular painkiller worldwide. Even Neil Armstrong took an Aspirin pill in his medical-kit when going to the Moon on the Apollo 11. In the early 1970s, more and more researchers asked the question: How and where does Aspirin work in the body? Pharmacologist Sir John Vane was the first to demonstrate the classical effect profile of Aspirin, for which he received the Nobel Prize in 1982

PATHS : Why is life filled with so many detours?

This open access book explores the amazing similarity between paths taken by people and many other things in life, and its impact on the way we live, teach and learn. Offering insights into the new scientific field of paths as part of the science of networks, it entertainingly describes the universal nature of paths in large networked structures. It also shows the amazing similarity in the ways humans and other – even nonliving – things navigate in a complex environment, to allow readers to easily grasp how paths emerge in many walks of life, and how they are navigated. Paths is based on the authors recent research in the area of paths on networks, which points to the possible birth of the new science of “paths” as a natural consequence ‘and extension) of the science of “networks.” The approach is essentially story-based, supported by scientific findings, interdisciplinary approaches, and at times, even philosophical points of view. It also includes short illustrative anecdotes showing the amazing similarities between real-world paths and discusses their applications in science and everyday life. Paths will appeal to network scientists and to anyone interested in popular science. By helping readers to step away from the “networked” view of many recent popular scientific books and start to think of longer paths instead of individual links, it sheds light on these problems from a genuinely new perspective. --------------------------------------------------------------------------------- The path is the goal. The essence behind this short sentence is known to many people around the world, expressed through the interpretations of some of the greatest thinkers like Lao-Tze and Gandhi. It means that it is the journey that counts, not the destination. When speaking about such subjective and intangible things, philosophy and religion are some of the only approaches that are addressed. In this book, the authors address this conventional wisdom from the perspective of natural science. They explore a sequence of steps that leads the reader closer to the nature of paths and accompany him on the search for “the path to paths”

The Skeleton of Hyperbolic Graphs for Greedy Navigation

Random geometric (hyperbolic) graphs are impor-tant modeling tools in analyzing real-world complex networks.Greedy navigation (routing) is one of the most promising infor-mation forwarding mechanisms in complex networks. This paperis dealing with greedy navigability of complex graphs generatedby using a metric (hyperbolic) space. Greedy navigability meansthat every source-destination pairs in the graph can communicatein such a way that every node passes the information towards thatneighboring node which is ”closest” to the destination in terms ofnode coordinates in the metric space. A set of compulsory linksin greedy navigable graphs called Greedy Skeleton is identified.Because the two-dimensional hyperbolic plane (H2, also knownas the two dimensional Bolyai-Lobachevsky Space [2]) turnedout to be extremely useful in modelling and generating real-like networks, we deal with the statistical properties of theGreedy Skeleton when the metric space isH2. Some examples ofnumerical studies and simulation results supporting the analyticalformulae are also performed. The significance of the results liesin that every (either artificial or natural) network formationprocess aiming at greedy navigability must contain this GreedySkeleton. Furthermore, this could be an important step towardsthe formal argumentation of the very high greedy navigabilityof some models observed only experimentally for the time being,and also to analyze equilibrium of greedy network navigationgames onH2

Paths

This open access book explores the amazing similarity between paths taken by people and many other things in life, and its impact on the way we live, teach and learn. Offering insights into the new scientific field of paths as part of the science of networks, it entertainingly describes the universal nature of paths in large networked structures. It also shows the amazing similarity in the ways humans and other – even nonliving – things navigate in a complex environment, to allow readers to easily grasp how paths emerge in many walks of life, and how they are navigated. Paths is based on the authors recent research in the area of paths on networks, which points to the possible birth of the new science of “paths” as a natural consequence ‘and extension) of the science of “networks.” The approach is essentially story-based, supported by scientific findings, interdisciplinary approaches, and at times, even philosophical points of view. It also includes short illustrative anecdotes showing the amazing similarities between real-world paths and discusses their applications in science and everyday life. Paths will appeal to network scientists and to anyone interested in popular science. By helping readers to step away from the “networked” view of many recent popular scientific books and start to think of longer paths instead of individual links, it sheds light on these problems from a genuinely new perspective. --------------------------------------------------------------------------------- The path is the goal. The essence behind this short sentence is known to many people around the world, expressed through the interpretations of some of the greatest thinkers like Lao-Tze and Gandhi. It means that it is the journey that counts, not the destination. When speaking about such subjective and intangible things, philosophy and religion are some of the only approaches that are addressed. In this book, the authors address this conventional wisdom from the perspective of natural science. They explore a sequence of steps that leads the reader closer to the nature of paths and accompany him on the search for “the path to paths”

Greedy Navigational Cores in the Human Brain

Greedy navigation/routing plays an important role in geometric routing of networks because of its locality and simplicity. This can operate in geometrically embedded networks in a distributed manner, distances are calculated based on coordinates of network nodes for choosing the next hop in the routing. Based only on node coordinates in any metric space, the Greedy Navigational Core (GNC) can be identified as the minimum set of links between these nodes which provides 100% greedy navigability. In this paper we perform results on structural greedy navigability as the level of presence of Greedy Navigational Cores in structural networks of the Human Brain

The Path is the Goal!

There is something compelling about shortest paths. They are so simple and reasonable. They seem to be the most efficient paths for traveling between nodes in a network. They may take the lowest amount of distance, time or energy. For grasping the idea of shortest paths, let’s consider the network in the first figure of this chapter. In this network, the shortest path between nodes D and H is the path (D → C → E → G → H) marked with red arrows. Its length is the number of edges crossed which is 4 and this is the only shortest path between D and H. Green arrows mark the shortest paths from node C to node F. There are two shortest paths (C → B → A → F) and (C → E → G → F) and both have a length of 3. Shortest paths are also pretty straightforward to compute by a few lines of code e.g., by using Edsger W. Dijkstra’s (Numer Math 1:269–271, 1959) method

On the Memory Requirement of Hop-by-hop Routing: Tight Bounds and Optimal Address Spaces

Routing in large-scale computer networks today is built on hop-by-hop routing: packet headers specify the destination address and routers use internal forwarding tables to map addresses to next-hop ports. In this paper we take a new look at the scalability of this paradigm. We define a new model that reduces forwarding tables to sequential strings, which then lend themselves readily to an information-theoretical analysis. Contrary to previous work, our analysis is not of worst-case nature, but gives verifiable and realizable memory requirement characterizations even when subjected to concrete topologies and routing policies. We formulate the optimal address space design problem as the task to set node addresses in order to minimize certain network-wide entropy-related measures. We derive tight space bounds for many well-known graph families and we propose a simple heuristic to find optimal address spaces for general graphs. Our evaluations suggest that in structured graphs, including most practically important network topologies, significant memory savings can be attained by forwarding table compression over our optimized address spaces. According to our knowledge, our work is the first to bridge the gap between computer network scalability and information-theory