1,273 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
Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus, and Minor-Free Graphs
A (1 + eps)-approximate distance oracle for a graph is a data structure that
supports approximate point-to-point shortest-path-distance queries. The most
relevant measures for a distance-oracle construction are: space, query time,
and preprocessing time. There are strong distance-oracle constructions known
for planar graphs (Thorup, JACM'04) and, subsequently, minor-excluded graphs
(Abraham and Gavoille, PODC'06). However, these require Omega(eps^{-1} n lg n)
space for n-node graphs. We argue that a very low space requirement is
essential. Since modern computer architectures involve hierarchical memory
(caches, primary memory, secondary memory), a high memory requirement in effect
may greatly increase the actual running time. Moreover, we would like data
structures that can be deployed on small mobile devices, such as handhelds,
which have relatively small primary memory. In this paper, for planar graphs,
bounded-genus graphs, and minor-excluded graphs we give distance-oracle
constructions that require only O(n) space. The big O hides only a fixed
constant, independent of \epsilon and independent of genus or size of an
excluded minor. The preprocessing times for our distance oracle are also faster
than those for the previously known constructions. For planar graphs, the
preprocessing time is O(n lg^2 n). However, our constructions have slower query
times. For planar graphs, the query time is O(eps^{-2} lg^2 n). For our
linear-space results, we can in fact ensure, for any delta > 0, that the space
required is only 1 + delta times the space required just to represent the graph
itself
The Order Dimension of the Poset of Regions in a Hyperplane Arrangement
We show that the order dimension of the weak order on a Coxeter group of type
A, B or D is equal to the rank of the Coxeter group, and give bounds on the
order dimensions for the other finite types. This result arises from a unified
approach which, in particular, leads to a simpler treatment of the previously
known cases, types A and B. The result for weak orders follows from an upper
bound on the dimension of the poset of regions of an arbitrary hyperplane
arrangement. In some cases, including the weak orders, the upper bound is the
chromatic number of a certain graph. For the weak orders, this graph has the
positive roots as its vertex set, and the edges are related to the pairwise
inner products of the roots.Comment: Minor changes, including a correction and an added figure in the
proof of Proposition 2.2. 19 pages, 6 figure
Digital Switching in the Quantum Domain
In this paper, we present an architecture and implementation algorithm such
that digital data can be switched in the quantum domain. First we define the
connection digraph which can be used to describe the behavior of a switch at a
given time, then we show how a connection digraph can be implemented using
elementary quantum gates. The proposed mechanism supports unicasting as well as
multicasting, and is strict-sense non-blocking. It can be applied to perform
either circuit switching or packet switching. Compared with a traditional space
or time domain switch, the proposed switching mechanism is more scalable.
Assuming an n-by-n quantum switch, the space consumption grows linearly, i.e.
O(n), while the time complexity is O(1) for unicasting, and O(log n) for
multicasting. Based on these advantages, a high throughput switching device can
be built simply by increasing the number of I/O ports.Comment: 24 pages, 16 figures, LaTe
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
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