  In network science, the non-homogeneity of node degrees has been a concerned
issue for study. Yet, with the modern web technologies today, the traditional
social communication topologies have evolved from node-central structures to
online cycle-based communities, urgently requiring new network theories and
tools. Switching the focus from node degrees to network cycles, it could reveal
many interesting properties from the perspective of totally homogeneous
networks, or sub-networks in a complex network, especially basic simplexes
(cliques) such as links and triangles. Clearly, comparing to node degrees it is
much more challenging to deal with network cycles. For studying the latter, a
new clique vector space framework is introduced in this paper, where the vector
space with a basis consisting of links has the dimension equal to the number of
links, that with a basis consisting of triangles has the dimension equal to the
number of triangles, and so on. These two vector spaces are related through a
boundary operator, e.g., mapping the boundary of a triangle in one space to the
sun of three links in the other space. Under the new framework, some important
concepts and methodologies from algebraic topology, such as characteristic
number, homology group and Betti number, will have a play in network science
leading to foreseeable new research directions. As immediate applications, the
paper illustrates some important characteristics affecting the collective
behaviors of complex networks, some new cycle-dependent importance indexes of
nodes, and implications for network synchronization and brain network analysis.
