Propagation Kernels

We introduce propagation kernels, a general graph-kernel framework for efficiently measuring the similarity of structured data. Propagation kernels are based on monitoring how information spreads through a set of given graphs. They leverage early-stage distributions from propagation schemes such as random walks to capture structural information encoded in node labels, attributes, and edge information. This has two benefits. First, off-the-shelf propagation schemes can be used to naturally construct kernels for many graph types, including labeled, partially labeled, unlabeled, directed, and attributed graphs. Second, by leveraging existing efficient and informative propagation schemes, propagation kernels can be considerably faster than state-of-the-art approaches without sacrificing predictive performance. We will also show that if the graphs at hand have a regular structure, for instance when modeling image or video data, one can exploit this regularity to scale the kernel computation to large databases of graphs with thousands of nodes. We support our contributions by exhaustive experiments on a number of real-world graphs from a variety of application domains

Tight Bounds on Information Dissemination in Sparse Mobile Networks

Motivated by the growing interest in mobile systems, we study the dynamics of information dissemination between agents moving independently on a plane. Formally, we consider $k$ mobile agents performing independent random walks on an $n$-node grid. At time $0$, each agent is located at a random node of the grid and one agent has a rumor. The spread of the rumor is governed by a dynamic communication graph process ${G_t(r) | t \geq 0}$, where two agents are connected by an edge in $G_t(r)$ iff their distance at time $t$ is within their transmission radius $r$. Modeling the physical reality that the speed of radio transmission is much faster than the motion of the agents, we assume that the rumor can travel throughout a connected component of $G_t$ before the graph is altered by the motion. We study the broadcast time $T_B$ of the system, which is the time it takes for all agents to know the rumor. We focus on the sparse case (below the percolation point $r_c \approx \sqrt{n/k}$) where, with high probability, no connected component in $G_t$ has more than a logarithmic number of agents and the broadcast time is dominated by the time it takes for many independent random walks to meet each other. Quite surprisingly, we show that for a system below the percolation point the broadcast time does not depend on the relation between the mobility speed and the transmission radius. In fact, we prove that $T_B = \tilde{O}(n / \sqrt{k})$ for any $0 \leq r < r_c$, even when the transmission range is significantly larger than the mobility range in one step, giving a tight characterization up to logarithmic factors. Our result complements a recent result of Peres et al. (SODA 2011) who showed that above the percolation point the broadcast time is polylogarithmic in $k$.Comment: 19 pages; we rewrote Lemma 4, fixing a claim which was not fully justified in the first version of the draf

Coined quantum walks on percolation graphs

Quantum walks, both discrete (coined) and continuous time, form the basis of several quantum algorithms and have been used to model processes such as transport in spin chains and quantum chemistry. The enhanced spreading and mixing properties of quantum walks compared with their classical counterparts have been well-studied on regular structures and also shown to be sensitive to defects and imperfections in the lattice. As a simple example of a disordered system, we consider percolation lattices, in which edges or sites are randomly missing, interrupting the progress of the quantum walk. We use numerical simulation to study the properties of coined quantum walks on these percolation lattices in one and two dimensions. In one dimension (the line) we introduce a simple notion of quantum tunneling and determine how this affects the properties of the quantum walk as it spreads. On two-dimensional percolation lattices, we show how the spreading rate varies from linear in the number of steps down to zero, as the percolation probability decreases to the critical point. This provides an example of fractional scaling in quantum walk dynamics.Comment: 25 pages, 14 figures; v2 expanded and improved presentation after referee comments, added extra figur

Decoherence in quantum walks - a review

The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, led rapidly to several new quantum algorithms. These all follow unitary quantum evolution, apart from the final measurement. Since logical qubits in a quantum computer must be protected from decoherence by error correction, there is no need to consider decoherence at the level of algorithms. Nonetheless, enlarging the range of quantum dynamics to include non-unitary evolution provides a wider range of possibilities for tuning the properties of quantum walks. For example, small amounts of decoherence in a quantum walk on the line can produce more uniform spreading (a top-hat distribution), without losing the quantum speed up. This paper reviews the work on decoherence, and more generally on non-unitary evolution, in quantum walks and suggests what future questions might prove interesting to pursue in this area.Comment: 52 pages, invited review, v2 & v3 updates to include significant work since first posted and corrections from comments received; some non-trivial typos fixed. Comments now limited to changes that can be applied at proof stag