Detecting and Characterizing Small Dense Bipartite-like Subgraphs by the Bipartiteness Ratio Measure

We study the problem of finding and characterizing subgraphs with small \textit{bipartiteness ratio}. We give a bicriteria approximation algorithm \verb|SwpDB| such that if there exists a subset $S$ of volume at most $k$ and bipartiteness ratio $\theta$, then for any $0<\epsilon<1/2$, it finds a set $S'$ of volume at most $2k^{1+\epsilon}$ and bipartiteness ratio at most $4\sqrt{\theta/\epsilon}$. By combining a truncation operation, we give a local algorithm \verb|LocDB|, which has asymptotically the same approximation guarantee as the algorithm \verb|SwpDB| on both the volume and bipartiteness ratio of the output set, and runs in time $O(\epsilon^2\theta^{-2}k^{1+\epsilon}\ln^3k)$, independent of the size of the graph. Finally, we give a spectral characterization of the small dense bipartite-like subgraphs by using the $k$th \textit{largest} eigenvalue of the Laplacian of the graph.Comment: 17 pages; ISAAC 201

Managing community membership information in a small-world grid

As the Grid matures the problem of resource discovery across communities, where resources now include computational services, is becoming more critical. The number of resources available on a world-wide grid is set to grow exponentially in much the same way as the number of static web pages on the WWW. We observe that the world-wide resource discovery problem can be modelled as a slowly evolving very-large sparse-matrix where individual matrix elements represent nodes’ knowledge of one another. Blocks in the matrix arise where nodes offer more than one service. Blocking effects also arise in the identification of sub-communities in the Grid. The linear algebra community has long been aware of suitable representations of large, sparse matrices. However, matrices the size of the world-wide grid potentially number in the billions, making dense solutions completely intractable. Distributed nodes will not necessarily have the storage capacity to store the addresses of any significant percentage of the available resources. We discuss ways of modelling this problem in the regime of a slowly changing service base including phenomena such as percolating networks and small-world network effects

Comparing bird communities within shrubby transmission line rights-of-way managed by mowing or by selective herbicide application in Maine and New Hampshire

In the northeastern U.S., thousands of miles of shrub-dominated transmission line rights-of-way (ROW) extend across the landscape and provide some of the largest and most stable shrubland habitats in the region. These ROW are used as nesting and post-fledging habitat by the region’s entire community of shrubland-dependent songbirds, but evidence for how ROW are used by songbirds that require other habitats for nesting is lacking. Mist-netting surveys conducted in regenerating clearcuts indicate that adult and fledgling mature-forest songbirds comprise a large proportion of the bird community in clearcuts during the post-fledging portion of the breeding season, a time when juvenile birds and molting adults require dense cover to avoid predators and abundant food resources to prepare for migration. In 2017, we began the first comprehensive mist-netting survey ever conducted in shrubby ROW in southern Maine and New Hampshire to inventory the entire community of songbirds using ROW during the nesting and post-fledging periods. In this preliminary year of our study, we investigated whether differences in the height, density, and species composition of plants between three ROW maintained by mowing and three ROW maintained with selective herbicide treatment resulted in differences in the community of shrubland-dependent or other-habitat-dependent songbirds. We conducted six mist net surveys in each ROW from late May-late August and captured 1,153 individual birds of 44 unique species. There was no difference in the richness or diversity of “Shrubland Species,” “Other Species,” or the entire songbird community between the different ROW types

Characterizing the community structure of complex networks

Community structure is one of the key properties of complex networks and plays a crucial role in their topology and function. While an impressive amount of work has been done on the issue of community detection, very little attention has been so far devoted to the investigation of communities in real networks. We present a systematic empirical analysis of the statistical properties of communities in large information, communication, technological, biological, and social networks. We find that the mesoscopic organization of networks of the same category is remarkably similar. This is reflected in several characteristics of community structure, which can be used as ``fingerprints'' of specific network categories. While community size distributions are always broad, certain categories of networks consist mainly of tree-like communities, while others have denser modules. Average path lengths within communities initially grow logarithmically with community size, but the growth saturates or slows down for communities larger than a characteristic size. This behaviour is related to the presence of hubs within communities, whose roles differ across categories. Also the community embeddedness of nodes, measured in terms of the fraction of links within their communities, has a characteristic distribution for each category. Our findings are verified by the use of two fundamentally different community detection methods.Comment: 15 pages, 20 figures, 4 table