This course from Stanford is great: EE263: Introduction to Linear Dynamical SystemsThe course includes videos and lecture notes. Prof Boyd tried to present the information in a straight way, demystifying the difficult points as much as possible.
It is a pity that the following course: EE363 does not include video lectures.
The isoperimetric number or Cheeger constant is an indicator of the “bottleneckedness” of a network.
This measure is an NP measure and can only be calculated partially. It gives an estimation of the existence of a big enough group of nodes with too few connections to the rest of the network, meaning that this group of nodes will act as a bottleneck.
There is a curious relationship between the Isoperimetric number of a network A and the capacity to synchronize. This relationship is connected with a very interesting number that is associated with any network which is the algebraic connectivity which is the is the second-smallest eigenvalue of the Laplacian matrix of A. If this value is zero implies that the network is disconnected. The bigger it is this value the better will synchronize the network. Then, the interesting relationship is that ; where is the second smallest eigenvalue of the net Laplacian. Therefore a network that synchronize well will have a big meaning that will not have bottlenecks.
That seems to imply that a network with bottlenecks synchronize worst that one without them.
Our brain dedicates a big part of its volume to connect different parts of the cortex (white matter) and other different regions. The white matter made of axons seems to try to give a good connection opportunity to the neurons, reducing bottlenecks.
I recommend this book. It is simply fantastic.
Great introduction to geometry from a computer science viewpoint.
There is as well an additional web page from the author: Gallier, in Penn University:
¿Does anyone know how to obtain these video lectures from Spivak at Keio University?
It seems should be avalilable at this url, but it is not, and it seems very very interesting.
SYNC book from Steven Strogatz is great. For all interested in the misteries of synchronization.
A review of all aspects related to sync: synchronization in biology, physics, sociology …. how it looks so normal and how difficult it is really to understand.
At the end gives an overview of the small-world networks, which owed much of its importance to his own work. He tells how ubiquitous are these networks in natural processes and how they improve in general synchronization.
He terminado este libro y creo que es un buen libro para pasar el rato, está entre la literatura de divulgación de las matemáticas con algunos aspectos mas literarios ya que va haciendo un recorridio a su trayectoria personal utilizando la correspondencia que intercambio con un antiguo profesor de matemáticas del instituto.
Lo interesante del libro es ver el interés y amor por las matemáticas del profesor y del alumno, ya que sus cartas van fundamentalmente de problemas de matemáticas que se intercambian a modo de acertijos y juegos.
Los problemas van sobre Calculo y a distintos niveles de dificultad.
Una buena lectura para estudiantes de mates y fisica
Look to this page http://www.facebook.com/note.php?note_id=469716398919
It is a very interesting entry of a person in Facebook who has made a diagram on the friendship network in the world (using facebook data). I have it right now as my screen saver. It is great.
I am not sure but looking to the diagram I think you can appreciate it is small-world and scale free.
- Small world: there are clear big connections but between the big connected points there are also clear small connections to many other points. So, the typical patterns of small world is maintained, a big number of links to your neighborhood plus always a very small number to other remote locations.
- Scale free: It is difficult to say it without looking to the distribution of the number of links coming out of any single point. If this distribution follows a power law , then, we can say that the network is scale free. For the Barabassi & Adler scale free networks the constant c is between 2 and 3. Well, this is difficult to say looking to the diagram, but another curious property of these networks is that the number of links per node looks like a fractal, meaning that if you erase the nodes with highest number of links the remaining nodes maintain the link connectivity architecture (the overall aspect of the network). A consequence of that: the distribution of links coming out of a node (what is called the degree of the node) is a heavy tail distribution, that is, it is quite possible to have nodes with very, very small number of links and with very big, big numbers as well. In contrast, with the light tail distributions where it is impossible to have so small and big values (for example, the human heights distribution is light tail as it is impossible to find people 10 m tall or 10 cm tall, but, curiously 🙂 the richness distribution is heavy tail, I don’t need to give examples….)