  We investigate, at the fundamental level, the questions of `why', `when' and
`how' one could or should reach out to poor and vulnerable people to support
them in the absence of governmental institutions. We provide a simple and new
approach that is rooted in linear algebra and basic graph theory to capture the
dynamics of income circulation among economic agents. A new linear algebraic
model for income circulation is introduced, based on which we are able to
categorize societies as fragmented or cohesive. We show that, in the case of
fragmented societies, convincing wealthy agents at the top of the social
hierarchy to support the poor and vulnerable will be very difficult. We also
highlight how linear-algebraic and simple graph-theoretic methods help explain,
from a fundamental point of view, some of the mechanics of class struggle in
fragmented societies. Then, we explain intuitively and prove mathematically
why, in cohesive societies, wealthy agents at the top of the social hierarchy
tend to benefit by supporting the vulnerable in their society. A number of new
concepts emerge naturally from our mathematical analysis to describe the level
of cohesiveness of the society, the number of degrees of separation in business
(as opposed to social) networks, and the level of generosity of the overall
economy, which all tend to affect the rate at which the top wealthy class
recovers its support money back. In the discussion on future perspectives, the
connections between the proposed matrix model and statistical physics concepts
are highlighted.
