19,944 research outputs found
Clustering in Complex Directed Networks
Many empirical networks display an inherent tendency to cluster, i.e. to form
circles of connected nodes. This feature is typically measured by the
clustering coefficient (CC). The CC, originally introduced for binary,
undirected graphs, has been recently generalized to weighted, undirected
networks. Here we extend the CC to the case of (binary and weighted) directed
networks and we compute its expected value for random graphs. We distinguish
between CCs that count all directed triangles in the graph (independently of
the direction of their edges) and CCs that only consider particular types of
directed triangles (e.g., cycles). The main concepts are illustrated by
employing empirical data on world-trade flows
Using Triangles to Improve Community Detection in Directed Networks
In a graph, a community may be loosely defined as a group of nodes that are
more closely connected to one another than to the rest of the graph. While
there are a variety of metrics that can be used to specify the quality of a
given community, one common theme is that flows tend to stay within
communities. Hence, we expect cycles to play an important role in community
detection. For undirected graphs, the importance of triangles -- an undirected
3-cycle -- has been known for a long time and can be used to improve community
detection. In directed graphs, the situation is more nuanced. The smallest
cycle is simply two nodes with a reciprocal connection, and using information
about reciprocation has proven to improve community detection. Our new idea is
based on the four types of directed triangles that contain cycles. To identify
communities in directed networks, then, we propose an undirected edge-weighting
scheme based on the type of the directed triangles in which edges are involved.
We also propose a new metric on quality of the communities that is based on the
number of 3-cycles that are split across communities. To demonstrate the impact
of our new weighting, we use the standard METIS graph partitioning tool to
determine communities and show experimentally that the resulting communities
result in fewer 3-cycles being cut. The magnitude of the effect varies between
a 10 and 50% reduction, and we also find evidence that this weighting scheme
improves a task where plausible ground-truth communities are known.Comment: 10 pages, 3 figure
A strong geometric hyperbolicity property for directed graphs and monoids
We introduce and study a strong "thin triangle"' condition for directed
graphs, which generalises the usual notion of hyperbolicity for a metric space.
We prove that finitely generated left cancellative monoids whose right Cayley
graphs satisfy this condition must be finitely presented with polynomial Dehn
functions, and hence word problems in NP. Under the additional assumption of
right cancellativity (or in some cases the weaker condition of bounded
indegree), they also admit algorithms for more fundamentally
semigroup-theoretic decision problems such as Green's relations L, R, J, D and
the corresponding pre-orders.
In contrast, we exhibit a right cancellative (but not left cancellative)
finitely generated monoid (in fact, an infinite class of them) whose Cayley
graph is a essentially a tree (hence hyperbolic in our sense and probably any
reasonable sense), but which is not even recursively presentable. This seems to
be strong evidence that no geometric notion of hyperbolicity will be strong
enough to yield much information about finitely generated monoids in absolute
generality.Comment: Exposition improved. Results unchange
- …