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In neuronal systems, inhibition contributes to stabilizing dynamics andregulating pattern formation. Through developing mean field theories ofneuronal models, using complete graph networks, inhibition is commonly viewedas one ``control parameter'' of the system, promoting an absorbing phasetransition. Here, we show that for low connectivity sparse networks, inhibitionweight is not a control parameter of the transition. We present analytical andsimulation results using generic stochastic integrate-and-fire neurons that,under specific restrictions, become other simpler stochastic neuron modelscommon in literature, which allow us to show that our results are valid forthose models as well. We also give a simple explanation about why theinhibition role depends on topology, even when the topology has adimensionality greater than the critical one. The absorbing transitionindependence of the inhibitory weight may be an important feature of a sparsenetwork, as it will allow the network to maintain a near-critical regime,self-tuning average excitation, but at the same time, have the freedom toadjust inhibitory weights for computation, learning, and memory, exploiting thebenefits of criticality.

Less is different: why sparse networks with inhibition differ from complete graphs