On the Difference of Expected Lengths of Minimum Spanning Trees

An exact formula for the expected length of the minimum spanning tree of a connected graph, with independent and identical edge distribution, is given, which generalizes Steele's formula in the uniform case. For a complete graph, the difference of expected lengths between exponential distribution, with rate one, and uniform distribution on the interval (0, 1) is shown to be positive and of rate ζ(3)/n. For wheel graphs, precise values of expected lengths are given via calculations of the associated Tutte polynomials.

ELIMINATION OF MULTIPLE ARROWS AND SELF-CONNECTIONS IN COUPLED CELL NETWORKS

A coupled cell network is a finite directed graph in which nodes and edges are classified into equivalence classes. Such networks arise in a formal theory of coupled systems of differential equations, as a schematic indication of the topology of the coupling, but they can be studied independently as combinatorial objects. The edges of a coupled cell network are "identical" if they are all equivalent, and the network is "homogeneous" if all nodes have isomorphic sets of input edges. Golubitsky et al. [2005] proved that every homogeneous identical-edge coupled cell network is a quotient of a network that has no multiple edges and no self-connections. We generalize this theorem to any coupled cell network by removing the conditions of homogeneity and identical edges. The problem is a purely combinatorial assertion about labeled directed graphs, and we give two combinatorial proofs. Both proofs eliminate self-connections inductively. The first proof also eliminates multiple edges inductively, the main feature being the specification of the inductive step in terms of a complexity measure. The second proof obtains a more efficient result by eliminating all multiple edges in a single construction.

Edge Assignment in Cells of Monkey Area V2

One of the processes of visual perception is to organise 2-D images into figure and ground, assigning the borders to the figure. We have studied the neural basis of this phenomenon. We recorded from orientation-selective cells of areas V1 and V2 in the awake, fixating monkey. A square (typically 4 deg) of uniform colour or gray was displayed in a uniform surround field (11 deg) of different colour or gray. The square was much larger than the response fields of the cells studied. Its orientation and colour were optimised for each cell. In interleaved tests, we centred two opposite edges of the square in the RF, and also reversed the colours of square and surround, resulting in four different display combinations. Flipping edges and colours produced pairs of displays with an identical edge in the response field, but the figure on opposite sides. The display was static for each period of fixation, and mean spike numbers per second were measured. Many cells were selective for the sign of local contrast. In V2 we found cells that were highly discriminative for the direction of the figure, eg responding 10 times more to the left edge of a gray square with white surround than to the right edge of a white square with gray surround. In some cells, this discrimination was nearly independent of the figure size. The response could either be independent of local contrast (general edge assignment), or conditional on figure colour (joint assignment of edge and colour). We have observed direction-of-figure preference also in V1, but with smaller discrimination ratios. We conclude that figural edge assignment is part of early cortical processing.