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 $O(n^3m)$ time where $n$ is the number of vertices of the input graph and $m$ 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 $O(n^4m)$. Here, we give an $O(n^2m)$-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