## 05C81 Random walks on graphs

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The aim of this thesis is to deepen our understand of how
IDA* heuristics influence the number of nodes expanded during
search. To this end, we develop Korf's formula for the number
of expanded nodes into a heuristic quality η which
expresses the quality of a heuristic function as a constant factor
on the number of expanded nodes, independent of a particular problem
instance.
We proceed to show how to compute η for some common kinds of
heuristics and how to estimate η by means of a random sample for
arbitrary heuristics. Using the value of η for some concrete
examples, we then inspect for which parts of the search space the
values of h(v) are particularly critical to the performance of the
heuristic, allowing us to build better heuristics for future problems.
This report originally appeared as a master thesis at Humboldt
University of Berlin.

We present a comprehensive theory for analysis and understanding of transition events between an initial set A and a target set B for general ergodic finite-state space Markov chains or jump processes, including random walks on networks as they occur, e.g., in Markov State Modelling in molecular dynamics. The theory allows us to decompose the probability flow generated by transition events between the sets A and B into the productive part that directly flows from A to B through reaction pathways and the unproductive part that runs in loops and is supported on cycles of the underlying network. It applies to random walks on directed networks and nonreversible Markov processes and can be seen as an extension of Transition Path Theory. Information on reaction pathways and unproductive cycles results from the stochastic cycle decomposition of the underlying network which also allows to compute their corresponding weight, thus characterizing completely which structure is used how often in transition events. The new theory is illustrated by an application to a Markov State Model resulting from weakly damped Langevin dynamics where the unproductive cycles are associated with periodic orbits of the underlying Hamiltonian dynamics.

The problem of decomposing networks into modules (or clusters) has gained much attention in recent years, as it can account for a coarsegrained description of complex systems, often revealing functional subunits of these systems. A variety of module detection algorithms have been proposed, mostly oriented towards finding hard partitionings of undirected networks. Despite the increasing number of fuzzy clustering methods for directed networks, many of these approaches tend to neglect important directional information. In this paper, we present a novel random walk based approach for finding fuzzy partitions of directed, weighted networks, where edge directions play a crucial role in defining how well nodes in a module are interconnected. We will show that cycle decomposition of a random walk process connects the notion of network modules and information transport in a network, leading to a new, symmetric measure of node communication. Finally, we will use this measure to introduce a communication graph, for which we will show that although being undirected it inherits all necessary information about modular structures from the original network.