Let $X$ be the topological realization of a finite simplicial complex, $G$ a finite group and $p: P \to X$ a principal $G$-bundle.
Let's recall the standard fact that more generally for any numerable principal G-bundle $p: P \to X$ over paracompact base space $X$ after having fixed a trivializing cover $(U_i)_{i \in I}$ of $p$ the bundle can be encoded (up to isomorphism) by cocycle datum $(g_{ij}: U_i \cap U_j \to G)_{j,i \in I}$ satisfying cocycle condition. Now from practical reasons, it might be in hard to find such a trivializing cover of $p$.
I would like to discuss here if there is a reasonable notion of a certain version of "discrete $1$-skeleton cocycle datum" more determined by combinatorical that topological structure and how much information about the principal $G$-bundle $p$ we consider it retains.
The leading question is how much "information" about the
bundle is concentrated in $1$-skeleton if our $X$ is
"simple enough". For example, does it suffice (sometimes?) to reconstruct the complete bundle up to isomorphism? If not always, when it's possible? For which class of base spaces? For all simplicial complexes like our $X$?
How does the situation change if we allow a more general class of spaces, eg the Delta sets? The crucial difference is that Delta sets allow more flexible glueing, eg the vertices of $n$-simples do not necessarily pairwise differ anymore as required for simplicial complexes.
I propose following construction for such "discrete $1$-skeleton cocycle datum" associated to $p$ (because it appears rather natural and nothing more sophisticated came into my mind) but up to now I have nowhere seen before; it might be garbage, but then I would like at least to know where it flaws and if it's improvable:
Let $X^0$ the set of vertices and $X^1$ the set of edges or $1$-simplices of $X$. Fix for any vertex $v \in X_0$ any open subset $U_v \subset X$ over which $p:P \to X$ trivialize, $P_{\vert{U_v}} \cong U_v \times G$ and such that $v$ is the only vertex which $U_v$ contains. Note that we do not require that the $(U_v)_{v \in X_0}$ cover $X$, just the set of vertices.
Next let $e_{vw} \in X^1$ be an edge or $1$-simplex with vertices
$v,w \in X^0$. Note that as we consider $X$ to be a simplicial complex and in contrast to a delta set the two vertices of a $1$-simplex of a simplicial complex
are different, so $1$-simplices cannot become loops in $X$.
Therefore the $1$-simplex $e_{vw}$ is a line
isomorphic to closed interval $[0,1]$ and therefore contractible.
Thus we can extend the trivializing neighborhood $U_v$ of $v$
to contain $[0,3/4)$ and $U_w$ of $w$
to contain $(1/4,1]$ such that the intersection $U_v \cap U_w$
contain the $(1/4,3/4)$ middle piece of $e_{vw}$.
Let $U_{e_{vw}} $ a connected component containing this
$(1/4,3/4)$ piece of $e_{vw}$ but not the middle pieces of over
$1$-simplices $e'_{vw}$ with possibly same vertices.
We consider the restricted map $ U_{e_{vw}} \to G $ induced
by usual cocycle associated to trivializations
$P_{\vert{U_v}} \cong U_v \times G$. This map is constant and
therefore we obtain a $g_{e_{vw}} \in G$ is its image.
We obtain a datum
$$ ( g_{e_{vw}})_{e_{vw} \in X^1} $$
I would like to call it "discrete $1$-skeleton cocycle datum" and this raises the question if this datum suffices to reconstruct up to isom our principal $G$-bundle $p: P \to X$ as long as our space $X$ is "simple enough" to be a simplicial complex. My guess is that should be the case, because:
Next, we use the Delta set description of a model of classifying space $BG$ -the "bar construction" -which can be found e.g. in A. Hatcher's Algebraic Topology on p 89. We observe that its $1$-simplices can be identified with $G$.
Let's do the most naive thing we can do now: we restrict us to $1$-skeleta, and map every vertex $v$ of $X$ to unique vertex of $BG$ and every $1$-simplex $e_{vw}$ to $g_{e_{vw}}$.
This gives a map $X^1 \to BG^1$ on a level of $1$-skeleta. This raises the natural question can this map be extended to $f_p: X \to BG$ canonically? If yes then in the next step, we temper to compare the original $p:P \to X$ with a pullback of universal bundle $U: EG \to BG$ along $f_p$.
Firstly, I would be happy about any feedback about this proposed construction. Does it work at all, are there serious flaws, are there more "natural" ways to construct such a "discrete" version of cocycle datum, does it contain enough information to reconstruct the original $G$-bundle (up to Isom)?
The most serious problem I see at this stage is if this map defined a priori on $1$-skeleton indeed extends to $X \to BG$. Extension to $2$-skeleton should be by construction of $BG$ equivalent to that the condition that the $g_{e_j} $ associated to three boundaries $e_1,e_2,e_3$ of a $2$-simplex satisfy cocycle condition, right?
What about the extension to higher skeleta?
It seems that if we can extend our $X^1 \to BG^1$ to $X \to BG$ "naturally", then we have good chances that this "discrete $1$-skeleton cocycle datum" $ ( g_{e_{vw}})_{e_{vw} \in X^1} $ knows enough to reconstruct $p$, at least as long as our base $X$ is a simplicial complex.
Now a sidenote about what might change if we consider $X$ to be delta set. The construction from above seemingly one level of $1$-skeleton goes through except that there might appear $1$-simplices with two vertices beeing identified. A proposal how I would try to fix it:
If $e_{vw} \in X^1$ with $v=w$, then $e_{vw}$ is a cirle in our now delta set $X$. We next pullback $p: P \to X$ along inclusion $e_{vw} \subset X$ and obtain obtain induced $G$-bundle over $e_{vw} \cong S^1$. This can be encoded by a single $g_{e_{vw}} \in G$. Let's take it and together with $1$-simplices whose vertices differ which we handle as for simplicial complex we obtain a datum
$$ ( g_{e_{vw}})_{e_{vw} \in X^1} $$
We obtain as before a map on $1$-skeleta $X^1 \to BG^1$, but I think here the lifting problem becomes more subtle...
I would like also to remark that I posed a nearly identical question few days ago in MO but realized that maybe it should be reasonable to discuss it on more elementary level as my approach uses at only all elementary techniques.