  Influence competition finds its significance in many applications, such as
marketing, politics and public events like COVID-19. Existing work tends to
believe that the stronger influence will always win and dominate nearly the
whole network, i.e., "winner takes all". However, this finding somewhat
contradicts with our common sense that many competing products are actually
coexistent, e.g., Android vs. iOS. This contradiction naturally raises the
question: will the winner take all?
  To answer this question, we make a comprehensive study into influence
competition by identifying two factors frequently overlooked by prior art: (1)
the incomplete observation of real diffusion networks; (2) the existence of
information overload and its impact on user behaviors. To this end, we attempt
to recover possible diffusion links based on user similarities, which are
extracted by embedding users into a latent space. Following this, we further
derive the condition under which users will be overloaded, and formulate the
competing processes where users' behaviors differ before and after information
overload. By establishing the explicit expressions of competing dynamics, we
disclose that information overload acts as the critical "boundary line", before
which the "winner takes all" phenomenon will definitively occur, whereas after
information overload the share of influences gradually stabilizes and is
jointly affected by their initial spreading conditions, influence powers and
the advent of overload. Numerous experiments are conducted to validate our
theoretical results where favorable agreement is found. Our work sheds light on
the intrinsic driving forces behind real-world dynamics, thus providing useful
insights into effective information engineering.
