Networks| Networks |
In General > s.a. Cellular
Automaton; complexity; graph.
* Idea: Networks are
finite non-empty set of objects called vertices, and a finite set of edges
associated each with
an unordered pair of vertices (its endpoints);
A generalization of the concept of graph (they can have multiple
edges).
* Embedded in a manifold: A
system of segments or edges which intersect only at their endpoints,
called vertices; For example, the edges of a tessellation, which are also an
embedded graph.
* Applications: Cells, networks of chemicals linked by chemical reactions,
the Internet or computer networks.
@ General references: Buchanan 02 [I]; Barabási; Newman PT(08)nov.
@ Complex networks: Albert & Barabási RMP(02)cm/01 [statistical
mechanics];
Evans CP(04)
[rev]; Boccaletti et al PRP(06)
[dynamics]; Bogacz et al PhyA(06)
[homogeneous, Monte Carlo]; Costa & Silva JSP(06)
[hierarchical model]; West et al PRP(08)
[new concepts, information exchange]; Radicchi et al PRL(08)
[renormalization]; Horak et al JSM(09)-a0811-in
[persistent homology].
@ Quantum localization models: Torma et al qp/01; Cardy CMP(05).
@ Quantum, other: Finkelstein et al qp/96 [of
quantum points, and standard
model]; Törmä PRL(98)qp [transitions];
Somma et al PRA(02)
[simulating physical phenomena]; Altman et al IJTP(04)qb.NC/03 =
IJTP(04)
[superpositional]; Chiribella et al a0904 [theoretical
framework]; Novotny et al JPA(09) [random unitary dynamics].
And Dynamical Systems > s.a. graphs
in physics.
@ Evolving: Krapivsky & Derrida PhyA(04)
[growing, properties]; Minnhagen et al PhyA(04)
[merging and creation]; Grönlund et al PS(05)
[correlations and preferential growth]; Shi
et al mp/05;
Shi et al PhyA(07)
[clustering coefficients]; Gu & Sun PLA(08)
[with node addition and deletion]; Hou et al a0808 [degree-distribution
stability of growing networks], a0901 [stable
degree distribution].
@ Transport, flows on networks: Jordan et al JMP(04)
[fluctuations]; Stinchcombe PhyA(05)
[regular and disordered
networks].
@ Critical phenomena: Goltsev et al PRE(03)cm/02 [phenomenological
theory]; Giuraniuc PRL(05)
+ pn(05)aug
[interactions vs network structure]; Dorogovtsev et al RMP(08)-a0705; > s.a.
renormalization, {scale-free below}.
@ Random: Dorogovtsev & Samukhin PRE(03)cm/02 [fluctuations],
et al NPB(03)cm/02 [statistical
mechanics],
cm/02, cm/02 [construction],
cm/02-in
[overview], NPB(03)cm/02 [path
lengths];
Derényi et al PhyA(04)
[topological phase transition]; Li et al
PhyA(04)
[transition
to
chaos];
Resendis-Antonio &
Collado-Vides PhyA(04)
[growth as diffusion]; Luque & Ballesteros PhyA(04)
[random
walk networks]; Kramer et al PRP(05)
[and 2D quantum phase transitions]; Franceschetti & Meester 07 [r JSP(09)];
Nowotny & Requardt JCA(07)cm/06 [emergent
properties]; Ben-Naim & Krapivsky JPA(07)
[addition-deletion]; Novotny et al a0904 [random
unitary quantum dynamics]; > s.a. non-extensive
statistical mechanics [entropy]; random
tilings.
@ Related topics: Balachandran & Ercolessi IJMPA(92)
[single-particle statistics];
Golubitsky & Stewart BAMS(06)
[grupoid formalism]; La Mura & Swiatczak qp/07 [Markovian
Entanglement Networks]; Passerini & Severini a0812 [entropy]; > s.a. entanglement,
spin models.
Neural Network
* Applications: Classification
of galaxies and other astronomical objects (> see astronomy).
@ References: Amit 89; Beale & Jackson 90; Biehl & Schwarze JPA(93);
Dotsenko 95; Altaisky qp/01 [quantum].
Scale-Free Networks
* Idea: The network
continually grows by the addition of new nodes; A new node connects to two
existing nodes in the network at time t + 1; This new node is much
more likely to connect to highly connected nodes (preferential attachment);
The function P(k) does not have a peak and decays as a power
law at large k, so most nodes have one or two links, but a few nodes
(hubs) have a large number of links, which guarantees that the system is fully
connected.
$ Def: A network in which
the probability that any given vertex is of degree k is Prob[d(v)
= k]
= kgamma, where often 
[1,3].
@ References: Dorogovtsev et al PRE(02)
[properties]; Barabási & Bonabeau SA(03)may; Dangalchev PhyA(04)
[stochastic models]; Chen & Shi PhyA(04)
[modeling]; Shiner & Davison CSF(04)
[connectivity]; Rodgers et al JPA(05)
[eigenvalue spectrum of adjacency matrix].
Other Concepts and Types > s.a. cell
complex; graph types; technology [internet]; tiling.
* Network connectivity: Can
be characterized by the probability P(k) that a node has k links.
* Random: (Erdös-Renyi)
Each pair of nodes is connected with probability p; The function P(k)
is highly peaked at some k, and decays exponentially at large k,
so most nodes have approximately the same number of links; (Uniform random
graph) Pick one uniformly at random.
* May-Wigner stability theorem:
Increasing the complexity of a network inevitably leads to its destabilization,
such that a small perturbation will
be able to disrupt the entire system.
@ Random cellular networks: Vincze et al JGP(04) [Aboav-Weaire law].
@ Small-world networks: Watts 99 [r PT(00)nov];
Araújo et al
PLA(03);
Sinha PhyA(05)
[complexity vs stability]; Cont & Tanimura AAP(08).
@ Related topics: Kim PRL(04)cm [coarse-graining];
Gelenbe PRS(08) [intro to stochastic networks].
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 24
sep
2009