ODES: Overlapping DEnse Sub-graphs

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News: A paper on ODES has been accepted, Bioinformatics, Vol. 26, No. 21. (1 November 2010), pp. 2788-2789; doi: 10.1093/bioinformatics/btq514

ODES is a pthreads parallelized exact algorithm to enumerate all maximal sub-graphs of a graph that exceed a specified cutoff density of at least 1/2, even if they overlap. The following is proved in our paper:

A connected graph G, with density den(G) >= 0.5, and number of verticies n >= 3, has at least one non-cut vertex w where degree d(w) <= the average degree of vertices in G. Removal of w from G does not decrease the density of G.

The theorem says that vertices can iteratively be removed from a sufficiently dense graph, without decreasing its density or cutting it into disconnected pieces, until all that remains is a single pair of connected vertices. The algorithm goes the other way: Starting with the set S of all connected pairs of vertices (single-edge sub-graphs), an iteration consists of looking for adjacent vertices that can be added to each member of S consistent with the theorem. A member m of S to which a vertex can be added is placed with the added vertex into a new set S' for the next iteration, one for each new vertex that can be added to m. The brute-force search space of a non-clique graph G is thus confined to the actual dense sub-graphs of G, and since each iteration is independent, it can be handled by its own thread. During any iteration, a dense subgraph is saved for output if it is of sufficient size, and no more vertices can be added.

Features yet to implement (see paper):

  • run algorithm within a series of density bins that span the interval 0.5 to 1.0
  • replace binary search with a fast hash function
  • output some indication of overlaps

  • Due to its high complexity, ODES does not scale well to very large dense sub-graphs. This can be ameliorated, however, by using ODES in conjunction with a heuristic method. If heuristically determined edges from large dense sub-graphs H are excluded from the initial single-edge sub-graph list S for ODES, but retained in the subsequent search space, ODES will find all other overlapping dense sub-graphs outside of H, along with those dense sub-graphs that overlap H containing at least one edge E outside of H that can be the last one chosen according to our theorem. Dense sub-graphs overlapping H that have no such edge E outside of H will not be found. See the README file for details to turn this feature on.

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    James Long
    International Arctic Research Center
    University of Alaska Fairbanks
    PO Box 757340
    Fairbanks, AK 99775-7340
    Voice: (907) 474-2440

    International Arctic Research Center

    Chris Hartman
    Department of Computer Science
    University of Alaska Fairbanks
    PO Box 756670
    Fairbanks, AK 99775-6670