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It is noted that human behaviour changes can have a significant impact on energy consumption, however, qualitative study on such an impact is still very limited, and it is necessary to develop the corresponding mathematical models to describe how much energy savings can be achieved through human engagement. In this paper a mathematical model of human behavioural dynamic interactions on a social network is derived to calculate energy savings. This model consists of a weighted directed network with time evolving information on each node. Energy savings from the whole network is expressed as mathematical expectation from probability theory. This expected energy savings model includes both direct and indirect energy savings of individuals in the network. The savings model is obtained by network weights and modified by the decay of information. Expected energy savings are calculated for cases where individuals in the social network are treated as a single information source or multiple sources. This model is tested on a social network consisting of 40 people. The results show that the strength of relations between individuals is more important to information diffusion than the number of connections individuals have. The expected energy savings of optimally chosen node can be 25.32% more than randomly chosen nodes at the end of the second month for the case of single information source in the network, and 16.96% more than random nodes for the case of multiple information sources. This illustrates that the model presented in this paper can be used to determine which individuals will have the most influence on the social network, which in turn provides a useful guide to identify targeted customers in energy efficiency technology rollout programmes.