= Marcie P. Driscoll (2004, 2nd edition) defines rhizome as: a tangle of tubers with no apparent beginning or end. It constantly changes shape, and every point in it appears to be connected with every other point (p. 389). 
"Rhizome takes it name from plants such as bamboo, aspen, or ginger that spread via a connected underground root system. As metaphor, Gilles Deleuze and Felix Guattari used rhizome to refer to a non-hierarchal form of organization. I have extended this metaphor, refering to rhizome as an alternative mode of human organization consisting of a network of minimally self-sufficient nodes that leverage non-hierarchal coordination of economic activity. The two keys concepts in my formulation of rhizome are 1) minimal self-sufficiency, which eliminates the dependencies that accrete hierarchy, and 2) loose and dynamic networking that uses the "small worlds" theory of network information processing to allow rhizome to overcome information processing burdens that normally overburden hierarchies." (http://www.jeffvail.net/2007/01/what-is-rhizome.html)
1. From the Wikipedia:
"In philosophy, the term rhizome has been used by Carl Jung as a metaphor, and by Gilles Deleuze as a concept, and refers to the botanical rhizome.
Gilles Deleuze and Félix Guattari used the term "rhizome" to describe theory and research that allows for multiple, non-hierarchical entry and exit points in data representation and interpretation. In A Thousand Plateaus, they opposed it to an arborescent conception of knowledge, which worked with dualist categories and binary choices. A rhizome works with horizontal and trans-species connections, while an arborescent model works with vertical and linear connections. Their use of the "orchid and the wasp" was taken from the biological concept of mutualism, in which two different species interact together to form a multiplicity (i.e. a unity that is multiple in itself). Horizontal gene transfer would also be a good illustration.
Rhizome theory is also gaining currency in the educational field, as a means of framing knowledge creation and validation in the online era. In 'Innovate - Journal of Online Education, Vol. 4, Issue 5', Dave Cormier critiques the limitations of the expert-centered pedagogical planning and publishing cycle, and posits instead a rhizomatic model of learning. In this rhizomatic model, knowledge is negotiated, and the learning experience is a social as well as a personal knowledge creation process with mutable goals and constantly negotiated premises. The rhizome metaphor, which represents a critical leap in coping with the loss of a canon against which to compare, judge, and value knowledge, may be particularly apt as a model for disciplines on the bleeding edge where the canon is fluid and knowledge is a moving target." (http://en.wikipedia.org/wiki/Rhizome_(philosophy))
2. Giorgio Bertini:
"“The ‘arborescenť model of thought designates the epistemplogy that informs all of Western thought, from botany to information sciences to theology”. Arbolic thought is a model to describe a system that is hierarchical, centered around a core belief, reductivistic, increasingly specialized, non-cyclical, linear, and ripe with segmentation and striation. Similar to a tree-like description of biological evolution or genealogy, arborescent systems start from a central origin and continue to evolve by branching into successively specialized generations. Vertical in nature, the arbolic is ordered, structured and “scientific”: it has a distinct train of thought, a clear inheritance, an order.
In contrast, the rhizome is brought forward as a matted web of interlinked concepts. Inspired by the wandering, non-centered root systems of grasses and plants, the rhizome appears non-linear, horizontal, nomadic, deterritorialized and heterogeneous. The rhizome cuts across and between the order of vertical space, connecting multiple points simultaneously in a network of nodes. Connected to each other at arbitrary points, the rhizomatic system is more concerned with the multiplicitous interlinking of concept, action and being.
Although it lacks a central dogma of a trunk/brain, it is a horizontal, bottom-up system that produces an emergent system of metabehavior that is strong, robust, and intelligent… in the non-standard sense of the word. Within nature, rhizomatic systems like ants or grassy weeds eventually win: “True, the weed produced no lilies, no battleships, no Sermons on the Mount… Eventually the weed gets the upper hand… The lily is beautiful, the cabbage is provender, the poppy is maddening – but the weed is rank growth… it points a moral.”
If intelligence could exist without a central brain, the rhizome would be it." (http://gfbertini.wordpress.com/2013/09/05/rhizomatic-systems-and-the-emergence-of-intelligence/)
Principles of the rhizome:
"Deleuze and Guattari introduce A Thousand Plateaus by outlining the concept of the rhizome (quoted from A Thousand Plateaus):
1 and 2: Principles of connection and heterogeneity: any point of a rhizome can be connected to anything other, and must be
3. Principle of multiplicity: only when the multiple is effectively treated as a substantive, "multiplicity" that it ceases to have any relation to the One
4. Principle of asignifying rupture: a rhizome may be broken, but it will start up again on one of its old lines, or on new lines
5 and 6: Principles of cartography and decalcomania: a rhizome is not amenable to any structural or generative model; it is a "map and not a tracing"
Add-ons, proposed by Ellen E. Berry and Carol Siegel is taken from their article, "Rhizomes, Newness, and the Condition of Our Postmodernity 
 exists to suggest ways out of this all-too-common paralysis of our critical imaginations by providing sites for the emergence of new thinking, the not-yet-conceived. We see speculative impulses and experimental strategies as vital components of the political agenda of contemporary cultural studies: Today more than ever we require acts of radical imagination and psychic mobility as preludes to the invention of historically new modes of relationship.
 Although we cannot (and would not wish to) predict the nature of the strange attractions that might migrate to Rhizomes, we are particularly interested in soliciting the following:
 creative and critical practices that generate alternative thinking by deliberately pursuing those alternatives embedded in any idea or system, particularly what a system omits or deems unworthy of serious scrutiny. Such thinking prevents any system from promoting itself as definitive and leaves it open to other ways of knowing and being.
 creative and critical practices that encourage us to unite ideas that seem most disparate or incompatible, thereby deliberately dislocating us from the known.
 creative and critical practices that train us actively to desire multiple differences rather than simply tolerating them or projecting them as objects of analysis. Such practices would be unpredictable, performative, and incomplete. By "hailing" us in ways that permit entry into relation with the other even as we forego full comprehension of him/her, they thereby will also extend our empathetic and ethical capacities." (http://integral-options.blogspot.ca/2013/02/trevor-malkinson-rhizomatic-for-people.html)
John R. Dreyer:
"Deleuze and Guattari discuss the rhizome, a subterranean stem that is much different from a root19. Rhizomes are all inclusive, the best and worst, animal or plant; it matters little. The most important part for our purpose is that rhizomes can be connected to anything other, there is not fixed point like a root has. Instead the rhizome, as the authors state, establish connections between semiotic chains, organizations of power and builds relations between the arts, social struggles and the sciences. The rhizome leads us to the idea of the multiplicity, which is a concept that contains neither a subject nor an object but instead is composed of determinations, magnitudes and dimensions. Instead of a single guide directing the multiplicity, there are connections to what the author’s term as “multiplicity of nerve fibers”. These connections are indeed the primary girders that hold the rhizome together in the absence of points or positions.
The rhizome is made up of connections, yet is not merely a tracing but a map that constructs the unconscious20. This map is open and it is continuously being modified, re‐mapped, and re‐engineered to represent different and new connections between the disparate points of the rhizome. Perhaps another conception of this is of an acentered system; what is termed finite networks in which communications run from any neighbors to any others. The channels which allow this are not pre‐existing but instead are constructed as the communications occur. Further the neighbors and individuals within the systems are interchangeable. These individuals are defined by the state they are in and it is through this that local operations can expand worldwide without a central coordinating agency. Here a central point is not a central point. Instead the rhizome connects any point to any other point. The traits of one point in no way need join to a like point. Instead the traits of various points are mixed and unmatched. There is no beginning, no middle and no end to a rhizome; it keeps growing, keeps connecting disparate points to one another with little thought to organization. The rhizome operates by variation, expansion, conquest, capture and offshoots. The map of the rhizome is constantly being re‐ drawn, modified, detached, connected, and reversed while always showing new entrances and exits. Importantly the rhizome is non‐hierarchical and non‐signifying without a leader and without an organizing memory or central automaton. The rhizome is, as Deleuze and Guattari state, a plateau as plateaus are always in the middle, neither a beginning nor an end. A plateau is a continuous, self‐ vibrating region whose development avoids any orientation toward a cumulation point or an external end.
The rhizome as radical democracy is based on a logic of collective action by elements that challenge the relation of subordination23. It is the relation of subordination is where the agent is given to the authority of another, such as an employer. The relation of subordination gives rise to the relation of oppression, where the relationship is no longer mere authority given down the chain of command but now that relationship if fraught with antagonism between the parties involved." (https://dl.dropboxusercontent.com/u/36447565/Dreyer%20-%20Technology%20%26%20the%20Death%20of%20Hegemony.pdf)
See also: Rhizomatic Learning
Several posts by Jeff Vail that elaborate on the application of rhizome to human systems include:
1. Problem of Growth. A capstone formulation of why our societal structure is unsustainable, how rhizome presents a solution, and how to implement it.
2. Envisioning a Hamlet Economy. Big-picture concpetion of how a rhizome economy will function.
3. Creating Resiliency and Stability in Horticulture. A more detailed analysis of how to implement a hybrid-horticultural scheme at the level of the rhizome node.
4. Rhizome & Central Place Theory. In response to a comment, a more detailed discussion of how rhizome can grow amidst existing hierarchal structures.
5. Rhizome Network Defense. A review of a Cambridge team's analysis of potential tacticts to defend rhizome structures against hierarchy.