7 research outputs found
Estimating Infection Sources in Networks Using Partial Timestamps
We study the problem of identifying infection sources in a network based on
the network topology, and a subset of infection timestamps. In the case of a
single infection source in a tree network, we derive the maximum likelihood
estimator of the source and the unknown diffusion parameters. We then introduce
a new heuristic involving an optimization over a parametrized family of Gromov
matrices to develop a single source estimation algorithm for general graphs.
Compared with the breadth-first search tree heuristic commonly adopted in the
literature, simulations demonstrate that our approach achieves better
estimation accuracy than several other benchmark algorithms, even though these
require more information like the diffusion parameters. We next develop a
multiple sources estimation algorithm for general graphs, which first
partitions the graph into source candidate clusters, and then applies our
single source estimation algorithm to each cluster. We show that if the graph
is a tree, then each source candidate cluster contains at least one source.
Simulations using synthetic and real networks, and experiments using real-world
data suggest that our proposed algorithms are able to estimate the true
infection source(s) to within a small number of hops with a small portion of
the infection timestamps being observed.Comment: 15 pages, 15 figures, accepted by IEEE Transactions on Information
Forensics and Securit
On the robustness of the metric dimension of grid graphs to adding a single edge
The metric dimension (MD) of a graph is a combinatorial notion capturing the
minimum number of landmark nodes needed to distinguish every pair of nodes in
the graph based on graph distance. We study how much the MD can increase if we
add a single edge to the graph. The extra edge can either be selected
adversarially, in which case we are interested in the largest possible value
that the MD can take, or uniformly at random, in which case we are interested
in the distribution of the MD. The adversarial setting has already been studied
by [Eroh et. al., 2015] for general graphs, who found an example where the MD
doubles on adding a single edge. By constructing a different example, we show
that this increase can be as large as exponential. However, we believe that
such a large increase can occur only in specially constructed graphs, and that
in most interesting graph families, the MD at most doubles on adding a single
edge. We prove this for -dimensional grid graphs, by showing that
appropriately chosen corners and the endpoints of the extra edge can
distinguish every pair of nodes, no matter where the edge is added. For the
special case of , we show that it suffices to choose the four corners as
landmarks. Finally, when the extra edge is sampled uniformly at random, we
conjecture that the MD of 2-dimensional grids converges in probability to
, and we give an almost complete proof