Networks interacting with matter

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

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

3. Traditional approach Dynamics of matter field and network: characteristic time-scales tMatter and tNetwork tM >> tN - network regarded as static object - dynamics of matter - results averaged over ensemble of networks tM << tN - dynamics of network (connections) - governed by distribution of matter (node fitness etc.) - results averaged over ensemble of matter Examples: Ising model, opinion formation models on various networks (PRL 94, 178701) Examples: hidden variable models (PRE 68, 036112), fitness models (PRE 70, 056126) tM = tN non-trivial interaction between matter & network Examples: coevolution of networks and opinions (PRE 74, 056108), self-organization of neural networks (PRL 84, 6114)

4. 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 qi neighbors m(i)-1 m(i) u(i) m(j)+1 i i j qi = ”degree” = number of n.n. u(m) – ”jumping rate” (can be arbitrary) The move depends only on occupation m(i) – Zero Range Process

5. ZRP on fixed graphs 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 qi 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 qmax q1=8, q2=...=qN=4 (almost k-regular graph) q1=N-1, q2=...=qN=1 (star graph)

6. Matter-geometry interaction Introduce coupling between balls and geometry 1) to each node ”i” ascribe the weight w(mi) 2) move balls with prob. 1-r, or with prob. r rewire links (Metropolis algorithm)   The characteristic time-scales: tN  1/r, tM 1/(1-r) • 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).

7. 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 large r phase transition? rewiring << transport of balls rewiring >> transport of balls enough time for condensation condensate destroyed before it forms

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

9. 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