Adaptive Geographical Search in Networks Andrea Wiggins EECS 549, Winter 2007

The Problem Geographical search in networks can be very inefficient Need good strategies for finding the shortest (geodesic) paths Network characteristics vary widely, and search strategies accordingly What works for A doesn’t work for B

Geographical search in networks can be very inefficient

Need good strategies for finding the shortest (geodesic) paths

Network characteristics vary widely, and search strategies accordingly

What works for A doesn’t work for B

The Example Example Algorithm (from Lada Adamic, SI 614, Winter 2006) current node = start node while (current node is not the target), mark current node as visited if one or more of the neighbors of the current node has not been visited, pick the unvisited neighbor with the smallest distance to the target otherwise, pick a visited neighbor at random set the current node to the neighbor selected In each network there are 4,000 nodes placed randomly on a two dimensional square area. Each node is connected to its two closest neighbors (note that it may be the closest neighbor from another node's point of view, so it may gain more than two edges from this requirement). Each node additionally adds one edge to another random node with probability 1/ d r , where d is the Euclidean distance (sqrt( x 2 + y 2 )) between the two nodes, and r is the parameter that varies between the networks and takes on values 1,2, and 4.

Example Algorithm (from Lada Adamic, SI 614, Winter 2006)

current node = start node

while (current node is not the target), mark current node as visited

if one or more of the neighbors of the current node has not been visited, pick the unvisited neighbor with the smallest distance to the target

otherwise, pick a visited neighbor at random

set the current node to the neighbor selected

In each network there are 4,000 nodes placed randomly on a two dimensional square area.

Each node is connected to its two closest neighbors (note that it may be the closest neighbor from another node's point of view, so it may gain more than two edges from this requirement).

Each node additionally adds one edge to another random node with probability 1/ d r , where d is the Euclidean distance (sqrt( x 2 + y 2 )) between the two nodes, and r is the parameter that varies between the networks and takes on values 1,2, and 4.

The Results 12.4 2.2 6.2 28.6 22.9 28.0 Standard Deviation 9.7 0.9 5.1 49.4 27.2 36.6 Mean Revisits r = 4 Revisits r = 2 Revisits r = 1 Steps r = 4 Steps r = 2 Steps r = 1 10 trials

The Proposal Test many diverse search algorithms in parallel on a broad spectrum of network topologies with varied parameters Adaptive agents created from elements of known successful algorithms are the search strategies being tested Agents weight their own genes and recombine for new search algorithms

Test many diverse search algorithms in parallel on a broad spectrum of network topologies with varied parameters

Adaptive agents created from elements of known successful algorithms are the search strategies being tested

Agents weight their own genes and recombine for new search algorithms

The Simulation Environments are graphs New but statistically similar graph for each turn prevents local optimization Agents’ task is to find a goal node from a starting node in the fewest possible steps Agents are recombined according to the relative length of their traversals (fitness)

Environments are graphs

New but statistically similar graph for each turn prevents local optimization

Agents’ task is to find a goal node from a starting node in the fewest possible steps

Agents are recombined according to the relative length of their traversals (fitness)

The Environments Use stochastically generated graphs, on a lattice, with similar network properties Start with Erdös-Rényi random graphs as a control - well studied standard random graphs Study other well-known models (small worlds, etc.) Use network growth models from the literature to create more experiments

Use stochastically generated graphs, on a lattice, with similar network properties

Start with Erdös-Rényi random graphs as a control - well studied standard random graphs

Study other well-known models (small worlds, etc.)

Use network growth models from the literature to create more experiments

The Agents Agents are made up of weighted combinations of graph traversal rules Genetic structure determines movement Agents know the relative direction of the goal node (in 2D space) Must have memory of traversed nodes to allow backtracking & prevent loops Usually achieved by coloring nodes

Agents are made up of weighted combinations of graph traversal rules

Genetic structure determines movement

Agents know the relative direction of the goal node (in 2D space)

Must have memory of traversed nodes to allow backtracking & prevent loops

Usually achieved by coloring nodes

The Interactions Condition of limited information: each node knows and can report whether it is the goal node, if it has been visited, its degree, and vector direction of its edges Agents can ask nodes for this info, but only this info Agents traverse the graph from a start node along the graph edges to find the goal node

Condition of limited information: each node knows and can report whether it is the goal node, if it has been visited, its degree, and vector direction of its edges

Agents can ask nodes for this info, but only this info

Agents traverse the graph from a start node along the graph edges to find the goal node

The Rules At each turn, agents traverse the graph according to their genetic instructions At the end of the traversal, each agent adjusts its weights to credit useful strategies After adjusting weights, agents recombine with probability based on traversal length relative to other agents

At each turn, agents traverse the graph according to their genetic instructions

At the end of the traversal, each agent adjusts its weights to credit useful strategies

After adjusting weights, agents recombine with probability based on traversal length relative to other agents

The Outcomes Agent populations expected to converge to a few good algorithms for each graph Rules and weights for successful algorithms will vary across graph types Current algorithms will be discovered and surpassed Future work can explore which search strategies work for graph characteristics

Agent populations expected to converge to a few good algorithms for each graph

Rules and weights for successful algorithms will vary across graph types

Current algorithms will be discovered and surpassed

Future work can explore which search strategies work for graph characteristics

Thank you! Questions?

Questions?