Networks interacting with matter Bartlomiej Waclaw Jagellonian University, Poland and Universität Leipzig, Germany

Motivation simplicial quantum gravity neural networkstransportation network: space triangulation matter: e.g. Ising spins network: neurons+axons matter: neuron’s state network: bus stops+roads matter: buses, people

Traditional approach Dynamics of matter field and network: characteristic time-scales t Matter and t Network t M >> t N - network regarded as static object - dynamics of matter - results averaged over ensemble of networks t M << t N - dynamics of network (connections) - governed by distribution of matter (node fitness etc.) - results averaged over ensemble of matter t M = t N non-trivial interaction between matter & network Examples: coevolution of networks and opinions (PRE 74, ), self-organization of neural networks (PRL 84, 6114) Examples: Ising model, opinion formation models on various networks (PRL 94, ) Examples: hidden variable models (PRE 68, ), fitness models (PRE 70, )

Starting point: dynamics on fixed network The model: network – a graph with N nodes and L links initial state: M identical balls (particles) distributed randomly on nodes; m(i) = number of balls at i-th node Evolution: at each time step pick up one node ”i” at random with probability u(m(i)) move a ball from ”i” to one of its q i neighbors i m(i) j u(i) i m(i)-1 m(j)+1 u(m) – ”jumping rate” (can be arbitrary) The move depends only on occupation m(i) – Zero Range Process q i = ”degree” = number of n.n.

What we measure: the distribution of balls  (m) at i-th node: the probability that we find m balls at site i What we find: - the inhomogeneity due to various q i dominates the dependence on u(m) (from now on we assume u(m)=1) - above a certain critical density : condensation on the node with maximum degree q max ZRP on fixed graphs q 1 =8, q 2 =...=q N =4 (almost k-regular graph) q 1 =N-1, q 2 =...=q N =1 (star graph)

Matter-geometry interaction Introduce coupling between balls and geometry 1) to each node ”i” ascribe the weight w(m i ) 2) move balls with prob. 1-r, or with prob. r rewire links (Metropolis algorithm) Two limits: for r=1 only rewirings, balls do not move (forget about them) uniform distribution of balls  random graph for r=0 pure ZRP discussed earlier We know how to make calculations What is in between? This depends strongly on w(m).  The characteristic time-scales: t N  1/r, t M  1/(1-r)

Different choices of w(m) w(m)=1: balls perform a random walk on evolving network. The network rewires independently of balls, but balls feel it w(m) grows with m: links tend to connect sites with many balls  more balls flow to these sites  more links ect...  condensation of balls and links w(m) falls with m: links avoid sites with multitude of balls. But even a small inhomogeneity triggers the condensation. Which effect is stronger? This depends on r. small r enough time for condensation large r rewiring << transport of ballsrewiring >> transport of balls condensate destroyed before it forms phase transition ?

Some examples below r critical condensation above r critical no condensation for different r jumping rate u(m)=1, rewiring weight w(m)=1/m

Summary in pure ZRP process on fixed network, the condensation takes place always on the node with highest degree when rewirings are possible, the condensate may be destroyed before it forms there is a critical ratio of the two time-scales (dynamics of balls/dynamics of links) which separates phases with/without condensate Questions: is it possible to produce heavy tails in pi(m) in this model by changing w(m),u(m)? is it possible to find an analytical solution to that model? can one apply a similar model to ”physical” problems like changes in transportation networks or transport chemicals by cytoskeleton in living cells? Thanks: W. Janke, Z. Burda