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9. All the world’s a net; or not?
Guido Caldarelli and Michele Catanzaro
in Networks: A Very Short Introduction
‘All the world's a net; or not?’ summarizes the extent and limitations of network science. There are warnings about the generalizations drawn from network theory, but the results of network ...
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4. A combinatorial zoo
Robin Wilson
in Combinatorics: A Very Short Introduction
‘A combinatorial zoo’ presents a menagerie of combinatorial topics, ranging from the box (or pigeonhole) principle, the inclusion–exclusion principle, the derangement problem, and the Tower ...
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4. Connected and close
Guido Caldarelli and Michele Catanzaro
in Networks: A Very Short Introduction
‘Connected and close’ shows almost all the elements of networks take part in one large connected structure, called a giant connected component. Almost every node has a path of connections ...
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8. Designs and geometry
Robin Wilson
in Combinatorics: A Very Short Introduction
Block designs are used when designing experiments in which varieties of a commodity are compared. ‘Designs and geometry’ introduces various types of block design, and then relates them to ...
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7. Digging deeper into networks
Guido Caldarelli and Michele Catanzaro
in Networks: A Very Short Introduction
‘Digging deeper into networks’ shows that to capture the more subtle structure of networks one has to find measures that describe the surroundings of a node — degree distribution is not ...
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6. Emergence of networks
Guido Caldarelli and Michele Catanzaro
in Networks: A Very Short Introduction
‘Emergence of networks’ considers a range of network models that explain how local mechanisms, without global planning, can generate large-scale, complex, ordered, and efficient structures. ...
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2. Four types of problem
Robin Wilson
in Combinatorics: A Very Short Introduction
‘Four types of problem’ explains that combinatorics is concerned with four types of problem: existence problems (does x exist?); construction problems (if x exists, how can we construct ...
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2. A fruitful approach
Guido Caldarelli and Michele Catanzaro
in Networks: A Very Short Introduction
‘A fruitful approach’ shows how the vertices and edges of graphs identify relations between things and how Jacob Moreno became the first to apply this to the analysis of social networks ...
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6. Graphs
Robin Wilson
in Combinatorics: A Very Short Introduction
Graph theory is about collections of points that are joined in pairs, such as a road map with towns connected by roads or a molecule with atoms joined by chemical bonds. ‘Graphs’ revisits ...
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1. A network point of view on the world
Guido Caldarelli and Michele Catanzaro
in Networks: A Very Short Introduction
On a typical day, we check emails, update social network profiles, make mobile phone calls, use public transportation, take planes, and transfer money and goods. In all these cases — ...
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9. Partitions
Robin Wilson
in Combinatorics: A Very Short Introduction
How many ways can a number be split into two, three, or more pieces? ‘Partitions’ considers this interesting problem and the way in which Leonard Euler started to investigate them around ...
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8. Perfect storms in networks
Guido Caldarelli and Michele Catanzaro
in Networks: A Very Short Introduction
‘Perfect storms in networks’ shows that the range of possible dynamics in networks is enormous. Domino effects, co-extinctions, cascading failures, breakdown avalanches, systemic failure of ...
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3. Permutations and combinations
Robin Wilson
in Combinatorics: A Very Short Introduction
Permutations and combinations have been studied for thousands of years. ‘Permutations and combinations’ considers selecting objects from a collection, either in a particular order (such as ...
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7. Square arrays
Robin Wilson
in Combinatorics: A Very Short Introduction
‘Square arrays’ is concerned with magic squares and latin squares. An n × n magic square, or a magic square of order n, is a square array of numbers (usually the numbers from 1 to n
...
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5. Superconnectors
Guido Caldarelli and Michele Catanzaro
in Networks: A Very Short Introduction
‘Superconnectors’ compares homogeneous networks, where all nodes have more or less the same degree, and heterogeneous networks, where hubs, or superconnectors, are present. In homogeneous ...
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5. Tilings and polyhedra
Robin Wilson
in Combinatorics: A Very Short Introduction
A tiling of the plane (or tessellation) is a covering of the whole plane with tiles so that no tiles overlap and there are no gaps. Polyhedra are three-dimensional solids that are bounded ...
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1. What is combinatorics?
Robin Wilson
in Combinatorics: A Very Short Introduction
Combinatorics can loosely be described as the branch of mathematics concerned with selecting, arranging, constructing, classifying, and counting or listing things. ‘What is combinatorics?’ ...
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3. A world of networks
Guido Caldarelli and Michele Catanzaro
in Networks: A Very Short Introduction
‘A world of networks’ considers how the world's networks are interconnected. In cyberspace, power grids support the Internet, which hosts the WWW, which in turn enables email services, ...
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