PLoS One. 2012; 7(12): e51883
Veremyev A, Boginski V, Krokhmal PA, Jeffcoat DE
Recent reports suggest that evolving large-scale networks exhibit "explosive percolation": a large fraction of nodes suddenly becomes connected when sufficiently many links have formed in a network. This phase transition has been shown to be continuous (second-order) for most random network formation processes, including classical mean-field random networks and their modifications. We study a related yet different phenomenon referred to as dense percolation, which occurs when a network is already connected, but a large group of nodes must be dense enough, i.e., have at least a certain minimum required percentage of possible links, to form a "highly connected" cluster. Such clusters have been considered in various contexts, including the recently introduced network modularity principle in biological networks. We prove that, contrary to the traditionally defined percolation transition, dense percolation transition is discontinuous (first-order) under the classical mean-field network formation process (with no modifications); therefore, there is not only quantitative, but also qualitative difference between regular and dense percolation transitions. Moreover, the size of the largest dense (highly connected) cluster in a mean-field random network is explicitly characterized by rigorously proven tight asymptotic bounds, which turn out to naturally extend the previously derived formula for the size of the largest clique (a cluster with all possible links) in such a network. We also briefly discuss possible implications of the obtained mathematical results on studying first-order phase transitions in real-world linked systems.
With regards to the 
is a rational number, its 
, which is uniquely determined by the following four properties, analogous to the properties uniquely specifying 
, where 
is the groupoid with one object and one morphism.
(
), then 
.
.
is an 
-sheeted covering map, then 
.
is a morphism in 
is an object in 
, then there exists a unique morphism 
in 
. This axiomatizes the path lifting property satisfied by a covering map of topological spaces. A covering map is 
, where 
is a finite group (since we are assuming essential finiteness). Associated to any such groupoid is a canonical 
-sheeted cover
is the action groupoid for the action of 
between any pair of objects). This covering map sends the morphism 
of 
by the cardinality axiom, and the covering axiom then implies that 
. In conclusion, we find that if 
.
be essentially finite groupoids. Then their product 
is also essentially finite, and 
.
satisfied all of the axioms above. 
be a finite set and 
is 
.
, a point which trips up many beginners in combinatorics. 
(thinking of 
, which consists of pairs 
where 
, and where there is a unique morphism 
for every 
. Since 
and the corresponding map
.
and 
. We can now apply the covering axiom, and the conclusion follows. 
given by the condition that a given isomorphism class 
occurs with probability
.
is the groupoid of finite sets and bijections, then
. In other words, “size of a random finite set” is Poisson with parameter 
. It is unclear to me what the significance of this observation is, if any. 
be a finite set and consider the groupoid of 
equipped with a map 
(assigning to each element of 
compatible with colors. The cardinality of this groupoid may be computed in two ways. On the one hand, there are 
isomorphism types of objects where 
, and the groupoid consisting these isomorphism types is equivalent to the action groupoid of 
acting on the set of all functions from an 
.
copies of the groupoid of finite sets; the equivalence is given by sending an 
.
, and along the way to seeing this we have shown that two ways of defining 
give the same answer (and also implicitly given a combinatorial proof that 
). 
.
and such that all higher homotopy groups are trivial; in other words, it is the 
. Such spaces are also known as 
is the circle 
. 
if and only if its universal cover is contractible; in particular any 
is 
. 
ought to be 
is the number of 
-cells of 
, and thus we ought to have
.
with a map 
will be denoted 
(rather than the more typical 
). This will make it easier to translate between diagrams and non-diagrams. All diagrams were drawn in 
is represented by a box labeled 
with one input string labeled 
and one output string labeled 
. Composition of linear maps 
is given by pairing input and output wires with matching labels.
is a map 
represented graphically by stacking boxes vertically. Note that 
has two input wires and two output wires.
-dimensional vector space 
. Thus a vector in a vector space is a morphism 
, which is just a box with no input wire and one output wire, and a linear functional or covector is a morphism 
, which is just a box with no output wire and one input wire. 
is represented by a wire with no attached box, to reflect the fact that it is an identity for composition, and the identity map 
is not represented by anything at all. 
and 
output wires labeled 
.
is a bilinear map on a vector space 
, the following is the statement that 
.
(in the sense that the two ways of evaluating it using tensor products and compositions produce the same result): 
.
-dimensional version of associativity. It allows us to “drop generalized parentheses” in the sense that we do not have to specify what order we tensor and compose a family of maps as above. 
of vector spaces there is a distinguished symmetry map
.
. 
.
, Reidemeister II and III are precisely the relations in a well-known 
– to begin with. Then an observer can be specified by choosing a particular point 
in spacetime, as well as a unit future-directed timelike vector 
. This vector is a tangent to the observer’s worldline at 
. The observer space is therefore a bundle over 
, the “future unit tangent bundle”. However, using the notion of a “Cartan geometry”, one can give a general definition of observer space which makes sense even when there is no underlying 
of all symmetries on an underlying space. Any given point 
will be fixed by some symmetries, and not others, so one also has a subgroup 
. This is the “stabilizer subgroup”, consisting of all symmetries which fix 
, the subgroups 
and 
are conjugate (by a symmetry taking 
). Then the homogeneous space, or Klein geometry, associated to 
is, up to isomorphism, just the same as the quotient space 
of the obvious action of 
on 
-dimensional Euclidean space. the Euclidean group 
is precisely the group of transformations that leave the data of Euclidean geometry, lengths and angles, invariant. It acts transitively on 
. Any point will be fixed by the group of rotations centred at that point, which is a subgroup of 
isomorphic to 
. Klein’s insight is to reverse this: we may define Euclidean space by 
.
. The Euclidean group has been replaced by the Poincaré group, and rotations by the Lorentz group (of rotations and boosts), but otherwise the situation is essentially the same.
. That is, the stabilizer of any point is the Lorentz group – so things look locally rather similar to Minkowski space around any given point. But the global symmetries of de Sitter space are different. Even more, it looks like Minkowski space locally in the sense that the Lie algebras give representations 
and 
are identical, seen as representations of 
. It’s natural to identify them with the tangent space at a point. de Sitter space as a whole is easiest to visualize as a 4D hyperboloid in 
. This is supposed to be seen as a local model of spacetime in a theory in which there is a cosmological constant that gives empty space a constant negative curvature.
– in fact, this whole theory goes through for any of the last three examples: Minkowski; de Sitter; and anti-de Sitter, each of which acts as a “local model” for spacetime in General Relativity with the cosmological constant, respectively: zero; positive; and negative.
and a vierbein (coframe field) called 
, with dynamics encoded in the Palatini action, which is the integral over ![R[\omega] \wedge e \wedge e](http://s0.wp.com/latex.php?latex=R%5B%5Comega%5D+%5Cwedge+e+%5Cwedge+e&bg=ffffff&fg=29303b&s=0 "R[\omega] \wedge e \wedge e")
, where 
is the curvature 2-form for 
.
. This “breaks the full symmetry” at each point. Then one has a fairly natural action on the 
-connection:
is the 
part of the curvature of the big connection. The splitting of the connection means that 
, and the action above is rewritten, up to a normalization, as the Palatini action for General Relativity (plus a topological term, which has no effect on the equations of motion we get from the action). So General Relativity can be written as the theory of a Cartan geometry modelled on de Sitter space.
, and one can really speak of tangent “vectors”. The tangent homogeneous space is not linear. In these first cases, the fibres are not vector spaces, precisely because the large group of symmetries doesn’t contain a group of translations, but they are Klein geometries constructed in just the same way as Minkowski space. Thus, the local description of the connection in terms of 
-valued forms can be treated in the same way, regardless of which Klein geometry 
to 
similarly reflects the symmetry breaking that occurs when one chooses a specific time direction (a future-directed unit timelike vector). These are the tangent vectors to the worldline of an observer at the chosen point, so 
the model Klein geometry, is the space of such possible observers. The stabilizer subgroup for a point in this space consists of just the rotations of space around the corresponding observer – the boosts in 
. This is just the future unit tangent bundle of de Sitter space, which we think of as a homogeneous model for the “space of observers”
, however, is just a Cartan geometry modelled on 
, the contact form, which is such that the top-dimensional form 
is nowhere zero. This is the odd-dimensional analog of a symplectic manifold. Contact manifolds are, intuitively, configuration spaces of systems which involve “rolling without slipping” – for instance, a sphere rolling on a plane. In this case, it’s better to think of the local space of observers which “rolls without slipping” on a spacetime manifold 
. But a general observer space 
. Every Cartan geometry 
) on the observer space 
-valued form. This decomposes into four Lie-algebra valued forms, interpreted as infinitesimal transformations of the model observer by: (1) spatial rotations; (2) boosts; (3) spatial translations; (4) time translation. The four-fold division is based on two distinctions: first, between the base event at which the observer lives, and the choice of observer (i.e. the reduction of 
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