2,761 research outputs found
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
Switching Reconstruction of Digraphs
Switching about a vertex in a digraph means to reverse the direction of every
edge incident with that vertex. Bondy and Mercier introduced the problem of
whether a digraph can be reconstructed up to isomorphism from the multiset of
isomorphism types of digraphs obtained by switching about each vertex. Since
the largest known non-reconstructible oriented graphs have 8 vertices, it is
natural to ask whether there are any larger non-reconstructible graphs. In this
paper we continue the investigation of this question. We find that there are
exactly 44 non-reconstructible oriented graphs whose underlying undirected
graphs have maximum degree at most 2. We also determine the full set of
switching-stable oriented graphs, which are those graphs for which all
switchings return a digraph isomorphic to the original
Small feedback vertex sets in planar digraphs
Let be a directed planar graph on vertices, with no directed cycle of
length less than . We prove that contains a set of vertices
such that has no directed cycle, and if ,
if , and if . This
improves recent results of Golowich and Rolnick.Comment: 5 pages, 1 figure - v3 final versio
Packing tight Hamilton cycles in 3-uniform hypergraphs
Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C
\subset H is a collection of N edges for which there is an ordering of the
vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i,
v_{i+1}, v_{i+2}} is an edge of C (indices are considered modulo N). We develop
new techniques which enable us to prove that under certain natural
pseudo-random conditions, almost all edges of H can be covered by edge-disjoint
tight Hamilton cycles, for N divisible by 4. Consequently, we derive the
corollary that random 3-uniform hypergraphs can be almost completely packed
with tight Hamilton cycles w.h.p., for N divisible by 4 and P not too small.
Along the way, we develop a similar result for packing Hamilton cycles in
pseudo-random digraphs with even numbers of vertices.Comment: 31 pages, 1 figur
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