68,464 research outputs found
A sufficient condition for the existence of an anti-directed 2-factor in a directed graph
Let D be a directed graph with vertex set V and order n. An anti-directed
hamiltonian cycle H in D is a hamiltonian cycle in the graph underlying D such
that no pair of consecutive arcs in H form a directed path in D. An
anti-directed 2-factor in D is a vertex-disjoint collection of anti-directed
cycles in D that span V. It was proved in [3] that if the indegree and the
outdegree of each vertex of D is greater than (9/16)n then D contains an
anti-directed hamilton cycle. In this paper we prove that given a directed
graph D, the problem of determining whether D has an anti-directed 2-factor is
NP-complete, and we use a proof technique similar to the one used in [3] to
prove that if the indegree and the outdegree of each vertex of D is greater
than (24/46)n then D contains an anti-directed 2-factor
Reciprocity in Social Networks with Capacity Constraints
Directed links -- representing asymmetric social ties or interactions (e.g.,
"follower-followee") -- arise naturally in many social networks and other
complex networks, giving rise to directed graphs (or digraphs) as basic
topological models for these networks. Reciprocity, defined for a digraph as
the percentage of edges with a reciprocal edge, is a key metric that has been
used in the literature to compare different directed networks and provide
"hints" about their structural properties: for example, are reciprocal edges
generated randomly by chance or are there other processes driving their
generation? In this paper we study the problem of maximizing achievable
reciprocity for an ensemble of digraphs with the same prescribed in- and
out-degree sequences. We show that the maximum reciprocity hinges crucially on
the in- and out-degree sequences, which may be intuitively interpreted as
constraints on some "social capacities" of nodes and impose fundamental limits
on achievable reciprocity. We show that it is NP-complete to decide the
achievability of a simple upper bound on maximum reciprocity, and provide
conditions for achieving it. We demonstrate that many real networks exhibit
reciprocities surprisingly close to the upper bound, which implies that users
in these social networks are in a sense more "social" than suggested by the
empirical reciprocity alone in that they are more willing to reciprocate,
subject to their "social capacity" constraints. We find some surprising linear
relationships between empirical reciprocity and the bound. We also show that a
particular type of small network motifs that we call 3-paths are the major
source of loss in reciprocity for real networks
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