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Heiko Rieger

Publications and source records attributed to Heiko Rieger.

6 recordsLinked to original sources
PubMed2004

Constrained spin-dynamics description of random walks on hierarchical scale-free networks.

We study a random walk problem on the hierarchical network which is a scale-free network grown deterministically. The random walk problem is mapped onto a dynamical Ising spin chain system in one dimension with a nonlocal spin update rule, which allows an analytic approach. We show analytically that the characteristic relaxation time scale grows algebraically with the total number of nodes N as T--N(z). From a scaling argument, we also show the power-law decay of the autocorrelation function C(sigma)(t)--t(-alpha), which is the probability to find the Ising spins in the initial state sigma after t time steps, with the state-dependent nonuniversal exponent alpha. It turns out that the power-law scaling behavior has its origin in a quasiultrametric structure of the configuration space.

Journal Article↗
PubMed2004

Random walks on complex networks.

We investigate random walks on complex networks and derive an exact expression for the mean first-passage time (MFPT) between two nodes. We introduce for each node the random walk centrality C, which is the ratio between its coordination number and a characteristic relaxation time, and show that it determines essentially the MFPT. The centrality of a node determines the relative speed by which a node can receive and spread information over the network in a random process. Numerical simulations of an ensemble of random walkers moving on paradigmatic network models confirm this analytical prediction.

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PubMed2003

Critical properties of loop percolation models with optimization constraints.

We study loop percolation models in two and in three space dimensions, in which configurations of occupied bonds are forced to form a closed loop. We show that the uncorrelated occupation of elementary plaquettes of the square and the simple cubic lattice by elementary loops leads to a percolation transition that is in the same universality class as the conventional percolation. In contrast to this, an optimization constraint for the loop configurations, which then have to minimize a particular generic energy function, leads to a percolation transition that constitutes a universality class for which we report the critical exponents. Implication for the physics of solid-on-solid and vortex glass models are discussed.

Journal Article↗
PubMed2002

Stability of shortest paths in complex networks with random edge weights.

We study shortest paths and spanning trees of complex networks with random edge weights. Edges which do not belong to the spanning tree are inactive in a transport process within the network. The introduction of quenched disorder modifies the spanning tree such that some edges are activated and the network diameter is increased. With analytic random-walk mappings and numerical analysis, we find that the spanning tree is unstable to the introduction of disorder and displays a phase-transitionlike behavior at zero disorder strength epsilon =0. In the infinite network-size limit (N--> infinity ), we obtain a continuous transition with the density of activated edges Phi growing like Phi approximately epsilon (1) and with the diameter-expansion coefficient Upsilon growing like Upsilon approximately epsilon (2) in the regular network, and first-order transitions with discontinuous jumps in Phi and Upsilon at epsilon=0 for the small-world (SW) network and the Barabási-Albert scale-free (SF) network. The asymptotic scaling behavior sets in when N>>N(c), where the crossover size scales as N(c) approximately epsilon (-2) for the regular network, N(c) approximately exp(alpha epsilon (-2)) for the SW network, and N(c) approximately exp(alpha|ln epsilon | epsilon (-2)) for the SF network. In a transient regime with N<<N(c), there is an infinite-order transition with Phi approximately Upsilon approximately exp[-alpha/(epsilon (2)ln N)] for the SW network and approximately exp[-alpha/(epsilon (2)ln N/ln ln N)] for the SF network. It shows that the transport pattern is practically most stable in the SF network.

Journal Article↗
PubMed2002

Numerical study of the disorder-driven roughening transition in an elastic manifold in a periodic potential.

We study the roughening transition of a (3+1)-dimensional elastic manifold, which is driven by the competition between a periodic pinning potential and a random impurity potential. The elastic manifold is modeled by a solid-on-solid-type interface model, and the universal properties of the transition from a flat phase (for strong periodic potential) to a rough phase (for strong random potential) are investigated at zero temperature using a combinatorial optimization algorithm technique. We find that the transition is a continuous one. Critical exponents are estimated numerically, and compared with analytic results and those for a periodic elastic medium.

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