Is the effect of mass density on the expansion of the universe linear? The average expansion rate for large scales parsec and above measured is $\approx 73\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$. This expansion is calculated for average densities of space volume across space. However this expansion must vary significantly between voids and large mass concentrations. 
My question in regards to this is if the effect on the expansion increases linearly with a linear decrease of mass density or it does not. Meaning for if there are other effects for this expansion even if minimal that are not accounted by mass. 
 A: The Standard cosmology model, the $\Lambda$CDM model, rearranged from the Friedmann equations, looks like,  
$H = H_0  {\sqrt{L_m  a^{-3} + L_r  a^{-4} + L_{de}}}$
This assumes zero curvature space, pretty well measured now (note, not zero spacetime curvature, just the spatial slices). 
H is the Hubble parameter as function of time, from the Big Bang. $H_0$ is H now. 
H = $\dot a/a$    where  a is the universe scale parameter, which has been growing since the Big Bang. H is the expansion velocity per unit distance, often expressed in terms of kms/sec/Mparsec. Today it is about 67.74 km/sec/Mparsec. This and the other numbers in this post are from the Planck satellite Cosmic Microwave Background measurements and collaboration.
$L_m$ is the matter density, including normal and dark matter. 
$L_r$ is the radiation density, things like photons. 
$L_{de}$ is the dark energy density.
All the L's are in percent of the total matter-energy density of the universe and add up to 1.{Note: in most papers and textbooks the Greek letter $\Omega$ is used instead of L}
Clearly the $L_r$ term dominates when a is very small, then as it increases the $L_m$ dominates, and as a gets larger the $L_{de}$ term is the dominant one.    
$L_m$ nowadays is about 31% of the total energy density of the universe
$L_r$ is too small to count now. For small a, early on during the Big Bang expansion, it dominated. Its effect has gone down as a increased, and 
$L_m$ dominated after the radiation era. For a few billion years it was the matter dominated era. 
$L_{de}$ nowadays is about 69% of the total energy of the universe, and dominates the calculation for the Hubble parameter H today. 
So, you see, matter indeed was important for billions of years, not linearly but it dominated the effect on H, i.e. the expansion. Nowadays it is the dark energy.
Now, before you worry too much about dark energy, it does seem, more or less, to have been constant during the universe's history, per unit volume. But as the universe expands and its volume increases it increases, that is, same density, more volume, more energy. Seems stable for the next few billion years. But you can see also why the interest in discovering exactly what it is. We don't know yet, though there are theories. 
Dark matter, which is most of the matter (it's about 25-26% of the total energy, normal matter is about 5%) is in a sense less mysterious. It is believed, but not known yet, to be some exotic matter that survived the Big Bang, but interacts very weakly with normal matter or radiation. It is expected to be a new type of particle that froze out of the Big Bang. There is some theories there also. 
The critical piece of information is that the parameter for the cosmological model as very well known now, and are expected to be measured even better over time, to try to distinguish between different possible models for dark matter and dark energy, and also for the evolution to form galaxies and sstars. The cosmological model is not a piece of mythological belief, there's observations and measurements for it all, with some unknowns like dark matter and energy remaining (though measured how much, not known what). 
The Wikipedia articles on these are pretty good. Also look for the Lambda-CDM model. Lambda is the term used in the Einstein equations for the cosmological constant, representing the dark energy. CDM is cold dark matter, as opposed to the now less relevant radiation. The derived equations and graphs are pretty easy. 
Hope this helps
A: Bob Bee's answer already covers a lot of extra detail, so I just want to give the very concise answer to your specific question. One form of one of the Friedmann equations is:
$$H(t) = \sqrt{\frac{8\pi G}{3}\rho(t) - \frac{kc^2}{(a(t))^2}}$$
In a universe with zero global spatial curvature ($k=0$), like ours is thought to be, then the expansion rate $H(t)$ scales as $\sqrt{\rho(t)}$. Not linearly, but to the $\frac{1}{2}$ power. Note that $\rho(t)$ refers to any kind of (energy) density, which includes mass, but also radiation (photons, relativistic neutrinos) and dark energy. For a long time the total energy density was dominated by mass density, but now we are thought to be entering the epoch of dark energy domination. In the very early Universe, radiation was the most important contribution.
