The inspiration for this neologism -- which mashes up rhizome + ohm -- are the writings of Gilles Deleuze and Felix Guattari, in particular the introduction to their book, A thousand plateaus , in which they articulate a vision and explication of rhizomatic systems. A rhizome, in botany, is a horizontal stem of a plant that sends out roots and shoots from its nodes.
Deleuze and Guattari use the rhizome as a metaphor to think about non-hierarchical, "anexact" systems, which are applicable far beyond botany. They provide examples of animal rhizomes in their pack form such as rats or burrow and my favorite example being ants: "You can never get rid of ants because they form an animal rhizome that can rebound time and again after most of it has been destroyed." (p.9)
(Which makes me think of Rudy Rucker's The Hacker and the Ants  but that's another whole post.)
But the most interesting application of rhizomatic structures applies to human systems. They outline four characteristics of rhizomatic systems:
- Principle of Connection: "Any point of a rhizome can be connected to anything other and must be. This is very different from the tree or root, which plots a point, fixes an order." (p.7)
- Principle of Heterogeneity: "There is no ideal speaker-listener, nay more than there is a homogenous linguistic community. Language is, in Weinreich's words, 'an essentially heterogeneous reality.'" (p.7)
- Principle of Multiplicity: "Unity always operates in an empty dimension supplementary to that of the system considered (overcoding). The point is that a rhizome or multiplicity never allows itself to be overcoded, never has available a supplementary dimension over and above its number of lines..." (p.8-9)
- Principle of Asignifying Rupture: "Every rhizome contains lines of segmentarity according to which it is stratified, territorialized, organized, signified, attributed, etc., as well as lines of deterritorialization down which it constantly flees." (p.) This last point is very important, in that it complicates the very simple binary or dialectic that starts to appear and is tempting to reduce their thinking to.
Delueze and Guattari go on to explicate how rhizomatic systems work, especially in contrast to arboreal systems. They provide many other great examples, such as the human brain itself: "Many people have a tree growing in their heads, but the brain itself is much more a grass than a tree." (p.15). Ultimately, I can't do justice to their explication and I would highly recommend reading the introduction on rhizomes -- and the whole book for that matter.
Where things get interesting is in applying some of their theories to computer science, the internet and the Web 2.0 phenomenon, which is the crux of my neologism, rhizohm, where the organic meets the digital, where we witness the marriage of hierarchy (XML, file systems, object hierarchies) with rhizomes (multiple personas, wikis, trackbacks). I see rhizohms as the manifestation of rhizomes online. Consider the following quote:
"To these centered systems, the authors contrast acentered systems, finite networks of automata in which communication runs from any neighbor to any other, the stems or channels do not preexist, and all individuals are interchangeable, defined only by their state at a given moment - such that the local operations are coordinated and the final, global result synchronized without a central agency." (p.17)
To me, this single statement can inform and provide models for much of what happens online. The obvious application is to the structure of the web itself. Already, there has been some good work on this front: check out Rhizome Navigation and some of their work:
But I want to extend my thinking about rhizohms beyond navigation and structure to the very concepts of identity and authorship on the web. A single blog entry becomes rhizomatic when on one page exists comments from other "authors", embedded images and videos from other sources, and, of course, hyperlinks. We each create rhizohms of the web, our own sprawls of content and presence on wikis, forums, social networks, etc. And, perhaps most interesting, is this dialectic that unfolds when rhizohms and hierarchies clash. (Wikipedia, anyone?)
Much more to be written on this topic, but that at least gets it going...
 Deleuze, Gilles and Guattari, Felix. A thousand plateaus; translation and forward by Brian Massumi. University of Minnesota Press, 1987.
 Rucker, Rudy. The Hacker and the Ants. Four Walls Eight Windows; Version 2.0 edition, 2002.