*Sergey N. Dorogovtsev*

- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199548927
- eISBN:
- 9780191720574
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199548927.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces the basic notions of graph theory and discusses the starting point of network science, namely the Konigsberg bridge problem. A few examples of different graphs, lattices, and ...
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This chapter introduces the basic notions of graph theory and discusses the starting point of network science, namely the Konigsberg bridge problem. A few examples of different graphs, lattices, and fractals are considered. These are used to explain the notions of a node degree, the shortest path length, clustering and so on. Milgram's experiment is also considered, and the notion of a random network is explained. A fundamental difference between small worlds and lattices and fractals, is discussed.Less

This chapter introduces the basic notions of graph theory and discusses the starting point of network science, namely the Konigsberg bridge problem. A few examples of different graphs, lattices, and fractals are considered. These are used to explain the notions of a node degree, the shortest path length, clustering and so on. Milgram's experiment is also considered, and the notion of a random network is explained. A fundamental difference between small worlds and lattices and fractals, is discussed.

*Stefan Thurner, Rudolf Hanel, and Peter Klimekl*

- Published in print:
- 2018
- Published Online:
- November 2018
- ISBN:
- 9780198821939
- eISBN:
- 9780191861062
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198821939.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Understanding the interactions between the components of a system is key to understanding it. In complex systems, interactions are usually not uniform, not isotropic and not homogeneous: each ...
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Understanding the interactions between the components of a system is key to understanding it. In complex systems, interactions are usually not uniform, not isotropic and not homogeneous: each interaction can be specific between elements.Networks are a tool for keeping track of who is interacting with whom, at what strength, when, and in what way. Networks are essential for understanding of the co-evolution and phase diagrams of complex systems. Here we provide a self-contained introduction to the field of network science. We introduce ways of representing and handle networks mathematically and introduce the basic vocabulary and definitions. The notions of random- and complex networks are reviewed as well as the notions of small world networks, simple preferentially grown networks, community detection, and generalized multilayer networks.Less

Understanding the interactions between the components of a system is key to understanding it. In complex systems, interactions are usually not uniform, not isotropic and not homogeneous: each interaction can be specific between elements.Networks are a tool for keeping track of who is interacting with whom, at what strength, when, and in what way. Networks are essential for understanding of the co-evolution and phase diagrams of complex systems. Here we provide a self-contained introduction to the field of network science. We introduce ways of representing and handle networks mathematically and introduce the basic vocabulary and definitions. The notions of random- and complex networks are reviewed as well as the notions of small world networks, simple preferentially grown networks, community detection, and generalized multilayer networks.

*Ernesto Estrada*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199591756
- eISBN:
- 9780191774959
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199591756.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter describes random models frequently used for studying complex networks. These include the Erdös-Rényi, Barabási-Albert and its variations, small-world models of Watts-Strogatz and ...
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This chapter describes random models frequently used for studying complex networks. These include the Erdös-Rényi, Barabási-Albert and its variations, small-world models of Watts-Strogatz and Kleinberg, and random geometric, range-dependent, lock-and-key, and stickiness models. It also discusses the topological properties of the networks generated with these modes, including clustering, metric, and spectral properties.Less

This chapter describes random models frequently used for studying complex networks. These include the Erdös-Rényi, Barabási-Albert and its variations, small-world models of Watts-Strogatz and Kleinberg, and random geometric, range-dependent, lock-and-key, and stickiness models. It also discusses the topological properties of the networks generated with these modes, including clustering, metric, and spectral properties.

*David D. Nolte*

- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198844624
- eISBN:
- 9780191880216
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198844624.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

A language of nodes and links, degree and moments, and adjacency matrix and distance matrix, among others, is defined and used to capture the wide range of different types and properties of network ...
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A language of nodes and links, degree and moments, and adjacency matrix and distance matrix, among others, is defined and used to capture the wide range of different types and properties of network topologies. Regular graphs and random graphs have fundamentally different connectivities that play a role in dynamic processes such as diffusion and synchronization on a network. Three common random graphs are the Erdös–Rényi (ER) graph, the small-world (SW) graph, and the scale-free (SF) graph. Random graphs give rise to critical phenomena based on static connectivity properties, such as the percolation threshold, but also exhibit dynamical thresholds for the diffusion of states across networks and the synchronization of oscillators. The vaccination threshold for diseases propagating on networks and the global synchronization transition in the Kuramoto model are examples of dynamical processes that can be used to probe network topologies.Less

A language of nodes and links, degree and moments, and adjacency matrix and distance matrix, among others, is defined and used to capture the wide range of different types and properties of network topologies. Regular graphs and random graphs have fundamentally different connectivities that play a role in dynamic processes such as diffusion and synchronization on a network. Three common random graphs are the Erdös–Rényi (ER) graph, the small-world (SW) graph, and the scale-free (SF) graph. Random graphs give rise to critical phenomena based on static connectivity properties, such as the percolation threshold, but also exhibit dynamical thresholds for the diffusion of states across networks and the synchronization of oscillators. The vaccination threshold for diseases propagating on networks and the global synchronization transition in the Kuramoto model are examples of dynamical processes that can be used to probe network topologies.