Effects of inspections in small world social networks with different contagion rules

•Inspections are used to prevent tax evasion or any other unlawful behavior.•The effect of inspections depends on the network topology and the contagion rule.•The network is modeled as a Watts–Strogatz Small World that is tuned from regular to random.•Two contagion rules are applied: continuous and discontinuous.•The equilibrium populations of payers and evaders are obtained in terms of these system parameters.

We study the way the structure of social links determines the effects of random inspections on a population formed by two types of individuals, e.g. tax-payers and tax-evaders (free riders). It is assumed that inspections occur in a larger scale than the population relaxation time and, therefore, a unique initial inspection is performed on a population that is completely formed by tax-evaders. Besides, the inspected tax-evaders become tax-payers forever. The social network is modeled as a Watts–Strogatz Small World whose topology can be tuned in terms of a parameter p∈[0,1]p∈[0,1] from regular (p=0p=0) to random (p=1p=1). Two local contagion rules are considered: (i) a continuous one that takes the proportion of neighbors to determine the next status of an individual (node) and (ii) a discontinuous (threshold rule) that assumes a minimum number of neighbors to modify the current state. In the former case, irrespective of the inspection intensity νν, the equilibrium population is always formed by tax-payers. In the mean field approach, we obtain the characteristic time of convergence as a function of νν and pp. For the threshold contagion rule, we show that the response of the population to the intensity of inspections νν is a function of the structure of the social network pp and the willingness of the individuals to change their state, rr. It is shown that sharp transitions occur at critical values of νν that depends on pp and rr. We discuss these results within the context of tax evasion and fraud where the strategies of inspection could be of major relevance.