16,433 research outputs found
Detecting 2-joins faster
2-joins are edge cutsets that naturally appear in the decomposition of
several classes of graphs closed under taking induced subgraphs, such as
balanced bipartite graphs, even-hole-free graphs, perfect graphs and claw-free
graphs. Their detection is needed in several algorithms, and is the slowest
step for some of them. The classical method to detect a 2-join takes
time where is the number of vertices of the input graph and the number
of its edges. To detect \emph{non-path} 2-joins (special kinds of 2-joins that
are needed in all of the known algorithms that use 2-joins), the fastest known
method takes time . Here, we give an -time algorithm for both
of these problems. A consequence is a speed up of several known algorithms
Gregarious Behaviour of Evasive Prey
Gregarious behavior of potential prey was explained by Hamilton (1971) on the basis of risk-sharing: The probability of being picked up by a predator is small when one makes part of a large aggregate of prey. This argument holds only if the predator chooses its victims at random. It is not the case for herds of evasive prey in the open, where prey’s gregarious behavior, favorable for the fast group members, makes it easier for the predator to home in on the slowest ones. We show conditions under which, gregarious behavior of the relatively fast prey individuals leaves slowest prey with no other choice but to join the group.Failing to do so would signal their vulnerability, making them a preferred target for the predator. Analysis of an n + 1 player game of a predator and n unequal prey individuals clarifies conditions for fully gregarious, partially gregarious, or solitary behavior of the prey.
Finding community structure in very large networks
The discovery and analysis of community structure in networks is a topic of
considerable recent interest within the physics community, but most methods
proposed so far are unsuitable for very large networks because of their
computational cost. Here we present a hierarchical agglomeration algorithm for
detecting community structure which is faster than many competing algorithms:
its running time on a network with n vertices and m edges is O(m d log n) where
d is the depth of the dendrogram describing the community structure. Many
real-world networks are sparse and hierarchical, with m ~ n and d ~ log n, in
which case our algorithm runs in essentially linear time, O(n log^2 n). As an
example of the application of this algorithm we use it to analyze a network of
items for sale on the web-site of a large online retailer, items in the network
being linked if they are frequently purchased by the same buyer. The network
has more than 400,000 vertices and 2 million edges. We show that our algorithm
can extract meaningful communities from this network, revealing large-scale
patterns present in the purchasing habits of customers
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