Abstract networks as in the bridges of Kaliningrad, graph theory and so on. the prevalence of online social networks in modern life leads us to naturally think of those, but networks provide a natural substrate for a bunch of processes, and passing backwards and forwards between linear-algebraic formalisms and graph formalisms can illuminate both.
specific graph counting based on generating functions (which is a “spectral” method but not normally what one means by spectral methods.)
- esp. use of approximating Fourier inversion/Cauchy path integrals. (the “asymptotic enumeration method”, (McKay and Wormald 1990; Odlyzko 1995) I just saw Mikhael Isaev present on work with Brendan McKay and Catherine Greenhill (Greenhill et al. 2016; Isaev and McKay 2016)) on an interesting construction here based on concentration and complex martingales, which was surprisingly elementary, and constructed some complex martingales based on novel concentration equality based on complex generalisation of McDiarmid and Catoni type concentration inequalities.
What graph theory gets us in understanding factor graphs and other graphical models.
Spectral methods, which is to say least “graph Laplacian” ones. (For that see also matrix factorisation.)
“Pagerank” et al. See Iterative reputation methods
graphons and graphexes
“Persistent Homology” – what is that?
To not mention:
- “Modularity” without domain-specific notion of modularity.
- Small world/scale-free/Erdős–Rényi, which are covered to the point of suffocation.
- Calculate eigenvalues of your connectivity matrix and learn some stuff..
- Or take the Fourier transform and also learn some stuff.
- Kirchhoff’s Theorem gives us, roughly, that the number of spanning trees in a graph is equal to the determinant of any cofactor of the graph Laplacian matrix (where a cofactor is a matrix with row and column \(i\) deleted). Wow.
e.g. Volodymyr Miz, Kirell Benzi, Benjamin Ricaud, and Pierre Vandergheynst, Wikipedia graph mining: dynamic structure of collective memory
As a topology for other processes
It’s not just nodes and edges and possibly a probability distribution over the occurrence of each. Networks are presumably interesting because they provide a topology upon which other processes occur. And the interaction between this theory and pure driven topology is much more complex and rich. Such models include circuit diagrams, probabilistic graphical models, neural networks, contagion processes reaction networks and others.
Scientists and engineers use diagrams of networks in many different ways. The Azimuth Project is investigating these, using the tools of modern mathematics. You can read articles about our research here:
…You can watch 4 lectures, an overview of network theory, here:
For now, I’m interested in conductance in electrical networks, random walks on graphs and the connection betwixt them. Where can I find out more about that? And how about the connection from those to harmonic functions?
Stochastic Petri Nets
What are these?
If you add in probability, are they the same as stochastic Petri nets?
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