5 research outputs found
Randomized Shortest Paths with Net Flows and Capacity Constraints
This work extends the randomized shortest paths (RSP) model by investigating
the net flow RSP and adding capacity constraints on edge flows. The standard
RSP is a model of movement, or spread, through a network interpolating between
a random-walk and a shortest-path behavior [30, 42, 49]. The framework assumes
a unit flow injected into a source node and collected from a target node with
flows minimizing the expected transportation cost, together with a relative
entropy regularization term. In this context, the present work first develops
the net flow RSP model considering that edge flows in opposite directions
neutralize each other (as in electric networks), and proposes an algorithm for
computing the expected routing costs between all pairs of nodes. This quantity
is called the net flow RSP dissimilarity measure between nodes. Experimental
comparisons on node clustering tasks indicate that the net flow RSP
dissimilarity is competitive with other state-of-the-art dissimilarities. In
the second part of the paper, it is shown how to introduce capacity constraints
on edge flows, and a procedure is developed to solve this constrained problem
by exploiting Lagrangian duality. These two extensions should improve
significantly the scope of applications of the RSP framework
Randomized Optimal Transport on a Graph: framework and new distance measures
The recently developed bag-of-paths (BoP) framework consists in setting a
Gibbs-Boltzmann distribution on all feasible paths of a graph. This probability
distribution favors short paths over long ones, with a free parameter (the
temperature ) controlling the entropic level of the distribution. This
formalism enables the computation of new distances or dissimilarities,
interpolating between the shortest-path and the resistance distance, which have
been shown to perform well in clustering and classification tasks. In this
work, the bag-of-paths formalism is extended by adding two independent equality
constraints fixing starting and ending nodes distributions of paths (margins).
When the temperature is low, this formalism is shown to be equivalent to a
relaxation of the optimal transport problem on a network where paths carry a
flow between two discrete distributions on nodes. The randomization is achieved
by considering free energy minimization instead of traditional cost
minimization. Algorithms computing the optimal free energy solution are
developed for two types of paths: hitting (or absorbing) paths and non-hitting,
regular, paths, and require the inversion of an matrix with
being the number of nodes. Interestingly, for regular paths on an undirected
graph, the resulting optimal policy interpolates between the deterministic
optimal transport policy () and the solution to the
corresponding electrical circuit (). Two distance
measures between nodes and a dissimilarity between groups of nodes, both
integrating weights on nodes, are derived from this framework.Comment: Preprint paper to appear in Network Science journal, Cambridge
University Pres
Sparse Randomized Shortest Paths Routing with Tsallis Divergence Regularization
This work elaborates on the important problem of (1) designing optimal
randomized routing policies for reaching a target node t from a source note s
on a weighted directed graph G and (2) defining distance measures between nodes
interpolating between the least cost (based on optimal movements) and the
commute-cost (based on a random walk on G), depending on a temperature
parameter T. To this end, the randomized shortest path formalism (RSP,
[2,99,124]) is rephrased in terms of Tsallis divergence regularization, instead
of Kullback-Leibler divergence. The main consequence of this change is that the
resulting routing policy (local transition probabilities) becomes sparser when
T decreases, therefore inducing a sparse random walk on G converging to the
least-cost directed acyclic graph when T tends to 0. Experimental comparisons
on node clustering and semi-supervised classification tasks show that the
derived dissimilarity measures based on expected routing costs provide
state-of-the-art results. The sparse RSP is therefore a promising model of
movements on a graph, balancing sparse exploitation and exploration in an
optimal way
New Resistance Distances with Global Information on Large Graphs.
Appearing in Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS) 2016, Cadiz, Spain. JMLR: W&CP volume 51.We consider the problem that on large random geometric graphs, random walk-based distances between nodes do not carry global information such as cluster structure. Instead, as the graphs become larger, the distances contain mainly the obsolete information of local density of the nodes. Many distances or similarity measures between nodes on a graph have been proposed but none are both proved to overcome this problem or computationally feasible even for small graphs. We propose new distance functions between nodes for this problem. The idea is to use electrical flows with different energy functions. Our proposed distances are proved analytically to be metrics in L^p spaces, to keep global information, avoiding the problem, and can be computed efficiently for large graphs. Our experiments with synthetic and real data confirmed the theoretical properties and practical performances of our proposed distances
New Resistance Distances with Global Information on Large Graphs
We consider the problem that on large random geometric graphs, random walk-based distances between nodes do not carry global information such as cluster structure. Instead, as the graphs become larger, the distances contain mainly the obsolete information of local density of the nodes. Many distances or similarity measures between nodes on a graph have been proposed but none are both proved to overcome this problem or computationally feasible even for small graphs. We propose new distance functions between nodes for this problem. The idea is to use electrical flows with different energy functions. Our proposed distances are proved analytically to be metrics in spaces, to keep global information, avoiding the problem, and can be computed efficiently for large graphs. Our experiments with synthetic and real data confirmed the theoretical properties and practical performances of our proposed distances.Peer reviewe