R. Lambiotte
Department of Mathematics University of Namur
Dynamics on networks: beyond Markov
In models defined on networks, e.g. synchronization, diffusion, etc., transitions tend to depend only on the current state of the system. A network representation hides the effect of memory.
Implicit Markov assumption
No memory on where the system comes
from: a random walker follows a link at
random wherever it comes from
No memory on when the system last
evolved: the walker moves either at
discrete times or following a Poisson process
Implicit Markov assumption
No memory on where the system comes
from: a random walker follows a link at
random wherever it comes from
No memory on when the system last
evolved: the walker moves either at
discrete times or following a Poisson process
How realistic are is this assumption when compared to real-world data?
If unrealistic, is it possible to add some memory into the modelling?
In models defined on networks, e.g. synchronization, diffusion, etc., transitions tend to depend only on the current state of the system. A network representation hides the effect of memory.
Implicit Markov assumption
No memory on where the system comes
from: a random walker follows a link at
random wherever it comes from
No memory on when the system last
evolved: the walker moves either at
discrete times or following a Poisson process
In models defined on networks, e.g. synchronization, diffusion, etc., transitions tend to depend only on the current state of the system. A network representation hides the effect of memory.
1. Stochastic dynamics on stochastic temporal networks: random walks
When arriving on a node i, a walker waits until the first link leaving i appears and it follows it.
The time at which link ij appears is determined by the waiting time distribution on link ij.
Links exist for an infinitely small amount of time.
The probability to have two links at the same time is zero.
Generalized Master Equations for Non-Poissonian Dynamics on Networks, Till Hoffmann, Mason Porter and R.L., Physical Review E, in press
In systems where all edges have the same waiting time
distribution, one can find a simple, closed expression for the stationary distribution
Broader waiting time distributions tend to decrease the density of walkers on hubs
Large-scale analysis
2. Decentralized routing on stochastic temporal networks
Decentralized Routing on Spatial Networks with Stochastic Edge Weights, Till Hoffmann, R.L. and Mason Porter, submitted
Instead of letting a walker randomly explore the network, we would like to guide him to find paths at a minimal cost
Ideally, the algorithm should be decentralized: finding good paths does not require the knowledge of the whole system.
Routing on deterministic networks
Routing on stochastic networks
The weights are now stochastic variables, distributed with a given probability.
The probability for a path to have a certain weight is:
What is the best path?
There is no longer a unique concept of optimality Frank defines a path to be
optimal if its cdf surpasses a threshold within the
shortest time
Fan suggests maximizing the cdf for a given time budget
Y. Y. Fan, R. E. Kalaba, and J. E. Moore, Journal of Optimization Theory and Applications 127, 497(2005).
H. Frank, Operations Research 17, pp. 583(1969).
What is the best path?
There is no longer a unique concept of optimality Frank defines a path to be
optimal if its cdf surpasses a threshold within the
shortest time
Fan suggests maximizing the cdf for a given time budget
Y. Y. Fan, R. E. Kalaba, and J. E. Moore, Journal of Optimization Theory and Applications 127, 497(2005).
H. Frank, Operations Research 17, pp. 583(1969).
Short time budget BAD
GOOD
What is the best path?
There is no longer a unique concept of optimality Frank defines a path to be
optimal if its cdf surpasses a threshold within the
shortest time
Fan suggests maximizing the cdf for a given time budget
Y. Y. Fan, R. E. Kalaba, and J. E. Moore, Journal of Optimization Theory and Applications 127, 497(2005).
H. Frank, Operations Research 17, pp. 583(1969).
BAD GOOD
Long time budget
What is the best path?
There is no longer a unique concept of optimality Frank defines a path to be
optimal if its cdf surpasses a threshold within the
shortest time
Fan suggests maximizing the cdf for a given time budget
Y. Y. Fan, R. E. Kalaba, and J. E. Moore, Journal of Optimization Theory and Applications 127, 497(2005).
H. Frank, Operations Research 17, pp. 583(1969).
Joint criterion: Frank
criterion if it finds a path, otherwise we use Fan
Finding the best path: A centralized algorithm
Routing table to reach a target r from all other nodes, solution of
Y. Y. Fan, R. E. Kalaba, and J. E. Moore, Journal of Optimization Theory and Applications 127, 497(2005).
ui(t) is the probability to arrive at node r starting from node i with a total time no longer than t
When on i, the node to chose to optimize the path is:
Finding the best path: A centralized algorithm
Routing table to reach a target r from all other nodes, solution of
Y. Y. Fan, R. E. Kalaba, and J. E. Moore, Journal of Optimization Theory and Applications 127, 497(2005).
ui(t) is the probability to arrive at node r starting from node i with a total time no longer than t
When on i, the node to chose to optimize the path is:
... but requires global knowledge of the system because of the recurrence
Finding the best path: A decentralized algorithm
We build an estimation function f(i,j,t) that gauges the arrival probability between i and j.
1. The network is embedded in a metric space. Let dij the physical distance between two nodes
(proximity will help us to guide travellers)
Navigation in small world networks, Kleinberg, Nature 2000
Finding the best path: A decentralized algorithm
We build an estimation function f(i,j,t) that gauges the arrival probability between i and j.
1. The network is embedded in a metric space. Let dij the physical distance between two nodes
2. Let gij be the physical distance of the shortest path between two nodes, and assume that
Finding the best path: A decentralized algorithm
We build an estimation function f(i,j,t) that gauges the arrival probability between i and j.
1. The network is embedded in a metric space. Let dij the physical distance between two nodes
2. Let gij be the physical distance of the shortest path between two nodes, and assume that
3. One estimates the number of steps between i and j by
The weight on each step is chosen uniformly at random from the known weights associated with edges
Finding the best path: A decentralized algorithm
Under these assumptions, we build the estimation function
estimating the probability for a path from i to j to have a weight smaller than t
Advantage: the estimation function is calculated by using properties of the spatial embedding alone, and does not involve the knowledge of the graph as a whole.
Finding the best path: A decentralized algorithm
Under these assumptions, we build the estimation function
and use it to estimate the exact cdf for nodes where we have no information
known
unknown
Numerical tests
Simulations on two-dimensional lattices with short-cuts (a la Kleinberg)
Numerical tests
Simulations on two-dimensional lattices with short-cuts (a la Kleinberg)
When budget
increases, the joint
criterion outperforms Fan criterion, as
expected
Mean travel time of successful routing
Numerical tests
Simulations on two-dimensional lattices with short-cuts (a la Kleinberg)
When budget
increases, the joint
criterion outperforms Fan criterion, as
expected
Numerical tests
Simulations on two-dimensional lattices with short-cuts (a la Kleinberg)
When budget
increases, the joint
criterion outperforms Fan criterion, as
expected
Numerical tests
Chicago road network (with 542 junctions)
When budget
increases, the joint
criterion outperforms Fan criterion, as
expected
Effect of the time evolution of the network on diffusion: possibility for a mathematical analysis, but stationary process (no circadian rhythm, etc.) and the bursty nature of the process is only modeled by inter-event statistics (no higher order).
Conclusion
Models going beyond standard network theory and accounting for memory in the dynamics
Dynamics on stochastically evolving networks, where the presence (or weight) of a link is a stochastic process
Random walk: generalized master equation. The stationary solution (pagerank) is
defined by an effective transition matrix accounting for the ordering of the appearance of edges. Time to reach stationarity?
Routing: centralized and decentralized algorithms
