21,180 research outputs found
Point patterns occurring on complex structures in space and space-time: An alternative network approach
This paper presents an alternative approach of analyzing possibly multitype
point patterns in space and space-time that occur on network structures, and
introduces several different graph-related intensity measures. The proposed
formalism allows to control for processes on undirected, directional as well as
partially directed network structures and is not restricted to linearity or
circularity
The Complexity of Routing with Few Collisions
We study the computational complexity of routing multiple objects through a
network in such a way that only few collisions occur: Given a graph with
two distinct terminal vertices and two positive integers and , the
question is whether one can connect the terminals by at least routes (e.g.
paths) such that at most edges are time-wise shared among them. We study
three types of routes: traverse each vertex at most once (paths), each edge at
most once (trails), or no such restrictions (walks). We prove that for paths
and trails the problem is NP-complete on undirected and directed graphs even if
is constant or the maximum vertex degree in the input graph is constant.
For walks, however, it is solvable in polynomial time on undirected graphs for
arbitrary and on directed graphs if is constant. We additionally study
for all route types a variant of the problem where the maximum length of a
route is restricted by some given upper bound. We prove that this
length-restricted variant has the same complexity classification with respect
to paths and trails, but for walks it becomes NP-complete on undirected graphs
On Temporal Graph Exploration
A temporal graph is a graph in which the edge set can change from step to
step. The temporal graph exploration problem TEXP is the problem of computing a
foremost exploration schedule for a temporal graph, i.e., a temporal walk that
starts at a given start node, visits all nodes of the graph, and has the
smallest arrival time. In the first part of the paper, we consider only
temporal graphs that are connected at each step. For such temporal graphs with
nodes, we show that it is NP-hard to approximate TEXP with ratio
for any . We also provide an explicit
construction of temporal graphs that require steps to be
explored. We then consider TEXP under the assumption that the underlying graph
(i.e. the graph that contains all edges that are present in the temporal graph
in at least one step) belongs to a specific class of graphs. Among other
results, we show that temporal graphs can be explored in steps if the underlying graph has treewidth and in
steps if the underlying graph is a grid. In the second part of the
paper, we replace the connectedness assumption by a weaker assumption and show
that -edge temporal graphs with regularly present edges and with random
edges can always be explored in steps and steps with high
probability, respectively. We finally show that the latter result can be used
to obtain a distributed algorithm for the gossiping problem.Comment: This is an extended version of an ICALP 2015 pape
Discriminative Distance-Based Network Indices with Application to Link Prediction
In large networks, using the length of shortest paths as the distance measure
has shortcomings. A well-studied shortcoming is that extending it to
disconnected graphs and directed graphs is controversial. The second
shortcoming is that a huge number of vertices may have exactly the same score.
The third shortcoming is that in many applications, the distance between two
vertices not only depends on the length of shortest paths, but also on the
number of shortest paths. In this paper, first we develop a new distance
measure between vertices of a graph that yields discriminative distance-based
centrality indices. This measure is proportional to the length of shortest
paths and inversely proportional to the number of shortest paths. We present
algorithms for exact computation of the proposed discriminative indices.
Second, we develop randomized algorithms that precisely estimate average
discriminative path length and average discriminative eccentricity and show
that they give -approximations of these indices. Third, we
perform extensive experiments over several real-world networks from different
domains. In our experiments, we first show that compared to the traditional
indices, discriminative indices have usually much more discriminability. Then,
we show that our randomized algorithms can very precisely estimate average
discriminative path length and average discriminative eccentricity, using only
few samples. Then, we show that real-world networks have usually a tiny average
discriminative path length, bounded by a constant (e.g., 2). Fourth, in order
to better motivate the usefulness of our proposed distance measure, we present
a novel link prediction method, that uses discriminative distance to decide
which vertices are more likely to form a link in future, and show its superior
performance compared to the well-known existing measures
- …