1. Discover a Network by Walking on it! Andrea Asztalos & Zoltán Toroczkai Department of Physics University of Notre Dame Department of Physics University of Notre Dame NetSci07

2. MotivationMotivation  What is the structure of a graph? Nodes, Edges, Loops Knowing the network, and some features of the walk  What will be the network seen by the walker? - example: gradient network  How can one optimize an exploration?

3. Exploring the Network Connected graph - transition probabilities define the walk on the graph EXPLORATION – recording the set of nodes and edges which have been visited RANDOM WALK Jumping only to adjacent sites The underlying, “unexplored” network NetSci07 Graph exploration algorithm:

4. Mathematical Approach NetSci07 - probability of being at site s on the n th step, given that the walk started from site s 0 - probability of being at site s for the first time on the n th step, given that the walk started from site s 0 Evolution law: - number of visited distinct edges up to n steps, averaged over many realizations of the walk - number of visited distinct sites up to n steps, averaged over many realizations of the walk Generating Function Formalism Generating function of A n asymptotic behavior of A n

5. Discovering the Nodes NetSci07 Valid for arbitrary graph *B.D. Hughes, Random Walks and Random Environments, Vol. 1, Oxford (1995) Introduce an indicator function I n Using the first-passage probability distribution for the nodes: Discovering the Edges - probability of arriving at edge e for the first time on the n th step, given that the walk started at site s 0 First - passage probabilities for the EDGES? ?? Generating function for

6. Discovering the Edges … z – an auxiliary node placed on the edge ( s,s’) EXTENDED GRAPH NetSci07 Generating function for Valid for arbitrary graph Notation:

7. Complete Graph  Homogeneous walk :  No directional bias : Estimate of steps needed to explore the majority of nodes edges NetSci07

8. 1d lattice and Trees  Homogeneous, without any directional bias  Translationally invariant walk Step distribution: 1D infinite lattice1D finite lattice For trees and 1d lattices : NetSci07 *B.D. Hughes, Random Walks and Random Environments, Vol. 1, Oxford (1995)

9. d>1 Infinite Lattices  Homogeneous, without any directional bias  Translationally invariant walk Step distribution: d=2 d>2 Generating function: *B.D. Hughes, Random Walks and Random Environments, Vol. 1, Oxford (1995) NetSci07

10. Simulation results for three different graphs of the same size N=2 10 In the case of ER graph with high degree the walker makes fewer steps in which it does not discover new nodes. In the case of SF graph there are many small nodes, the probability to go backwards increases, thus the discovering process is slow. ER Random Graphs & Scale-free Graphs

11. Simulation results for three different graphs of the same size N=2 10 S n =1 X n =0 S n =2 X n =1 S n =1 X n =0 S n =3 X n =2 S n =2 X n =1 S n =1 X n =0 S n =4 X n =3 S n =3 X n =2 S n =2 X n =1 S n =1 X n =0 S n =5 X n =4 S n =4 X n =3 S n =3 X n =2 S n =2 X n =1 S n =1 X n =0 S n =5 X n =4 S n =4 X n =3 S n =5 X n =5 S n =2 X n =1 S n =1 X n =0 Ln(( +1)/ ) - a measure of discovering loops in the graph This quantity only grows when a new edge is discovered, which means a new loop in the graph. In a SF graph, this quantity grows faster than in an ER graph.

12. SummarySummary Exploring graphs via a general class of random walk Increase of the set of revealed nodes as a function of time Counting edges by introducing an auxiliary node, thus extending the original graph Deriving expressions for particular cases: complete graph, infinite and finite hypercubic lattices, analyze random and scale-free graphs NetSci07