Your question fits into the area of random graphs, rather than extremal graph theory; and also expander graphs are relevant.

As mentioned previously, Erdos-Renyi graphs are a good and simple model for random graphs. For example $G_{n,p}$ has $n$ vertices and each edge is independently randomly determined to exist with probability $p$.

If you're talking about sparse graphs, you have to quantify how sparse. Say, for example, $p = \frac{\log n}{n}$? Above a certain point (Alon and Spencer, "The Probabilistic Method", will have many details) there is essentially a single "giant component" to the graph. Below that, there is a transition (which they also understand in detail) and then everything should be a tree below that.

Expander graphs (there are many constructions) are typically sparse graphs which however are sufficiently connected that a random walk mixes rapidly. With expanders there should be a result about the typical cycle size and distribution of cycles by length, compared to the second eigenvalue of the Laplacian of the graph, which governs its expansion.

It appears you're looking at LDPC codes, whose vertices have (if I recall correctly from undergrad days) edges independently chosen at random, with each vertex choosing a number $d$ as its total number of edges, where $d$ comes from some distribution chosen to maximize efficiency as a code. Mitzenmacher, Luby, and others were involved in their creation and have analyzed the efficiency extensively. "Digital Fountain" is/was a company doing this.

LDPC codes offer a bit of independence if they are as described, but locally the edge probabilities will be correlated because of the distribution of $d$.

It might be possible to use Janson's inequality (Ch8 of Alon and Spencer) to analyze this, as long as you're in the situation where there are no "negatively correlated" pairs of probabilities in your sum. It only uses the second (and first) probability moments.

LDPC codes are probably good expanders, so you could use bounds from expander graph literature if true.

Off the top of my head, that's where this problem fits ... maybe I'll be able to fill in more details for some of this later.