Apollonian network

Apollonian network
David Eppstein · CC0 · source
Name	Apollonian network

Apollonian network is a class of planar graphs generated by recursive circle or triangle packing procedures related to Apollonius of Perga, combining geometric, combinatorial, and algorithmic aspects. These graphs arise in studies linking Apollonius of Perga, Descartes' circle theorem, Soddy circle, Francis Guthrie, and constructions in Euclidean geometry and Projective geometry. They serve as canonical examples connecting Graph theory, Computational geometry, Complex networks, and Statistical physics.

Definition and construction

An Apollonian network is produced by iteratively subdividing a planar triangular region: start from a triangle with vertices associated to three base nodes often tied to historical figures like Apollonius of Perga and René Descartes through the circle-packing interpretation. At each step, insert a new node inside any existing triangular face and connect it to the three vertices of that face, echoing operations studied under Circle packing theorem, Delaunay triangulation, Voronoi diagram, Sierpiński triangle, and Kuratowski's theorem. The combinatorial dual of this process relates to iterative constructions in Plantri-type enumerations used by researchers at institutions such as Universität Bayreuth and University of Waterloo.

Mathematical properties

Apollonian networks are maximal planar graphs and hence are 3-vertex-connected by results similar to Tutte's theorem and Whitney's theorem. They are chordal in a restricted sense connected to Perfect graph theorem contexts and exhibit scale-free degree distributions reminiscent of models by Barabási–Albert model though derived deterministically like constructions studied by Paul Erdős and Alfréd Rényi. Their spectral properties have been linked to investigations by groups around Chung Fan Chung and László Lovász concerning eigenvalues and expansion, while metric properties such as small-world behavior relate to analyses in the tradition of Duncan Watts and Steve Strogatz. Planarity constraints engage classical results like Euler's formula and modern work connected to Hadwiger conjecture-style questions.

Variants and generalizations

Variants include higher-dimensional analogues analogous to simplicial complexes used in Hatcher's algebraic topology courses or generalizations to nonplanar triangulations inspired by constructions of Coxeter groups and Sierpiński gasket families. Weighted Apollonian networks incorporate edge or node weights studied in contexts like Erdős–Rényi model modifications and network science research from groups such as Santa Fe Institute and Institute for Advanced Study. One can also consider stochastic insertion orders yielding ensembles related to Markov chains explored by teams at Princeton University and Massachusetts Institute of Technology.

Algorithms and generation

Generation algorithms employ depth-first or breadth-first traversals and are implemented in libraries influenced by tools from Boost C++ Libraries and software like Mathematica and NetworkX. Efficient enumeration uses combinatorial generation techniques akin to those developed by Brendan McKay (author of nauty) and planar embedding routines that reference work at DIMACS-centered collaborations. Circle-packing realizations leverage numerical approaches related to algorithms from Ken Stephenson and implementations in projects tied to Wolfram Research and computational efforts from Carnegie Mellon University.

Applications and examples

Apollonian networks model hierarchical fragmentation patterns observed in geoscience studies related to Albert Einstein-era granular theories and in urban research connected to studies at MIT Media Lab. They have been applied in designing communication topologies influenced by work at Bell Labs and in peer-to-peer overlays echoing protocols researched at University of California, Berkeley. Examples appear in art and architecture referencing Islamic geometric patterns and historical constructions examined by scholars at British Museum and Louvre Museum. In physics, they are used as substrates for percolation and spin models in traditions stemming from Ludwig Boltzmann and Ising model analyses.

Open problems and research directions

Open questions include rigorous classification of spectral gaps tied to conjectures in the spirit of Alon–Boppana bound, precise mixing-time bounds for random walks reminiscent of problems studied by Persi Diaconis, and scaling limits connecting to conformal invariance themes pursued by researchers like Oded Schramm and Stanislav Smirnov. Further directions explore embeddings in non-Euclidean geometries inspired by Poincaré models, optimization of routing metrics relevant to work at Google and IBM Research, and enumeration complexity questions linked to counting triangulations analogous to problems addressed by Richard Stanley and Miklós Bóna.

Category:Planar graphs Category:Network science Category:Graph theory