Flow graphs: Interweaving dynamics and structure

The behavior of complex systems is determined not only by the topological organization of their interconnections but also by the dynamical processes taking place among their constituents. A faithful modeling of the dynamics is essential because different dynamical processes may be affected very differently by network topology. A full characterization of such systems thus requires a formalization that encompasses both aspects simultaneously, rather than relying only on the topological adjacency matrix. To achieve this, we introduce the concept of flow graphs, namely weighted networks where dynamical flows are embedded into the link weights. Flow graphs provide an integrated representation of the structure and dynamics of the system, which can then be analyzed with standard tools from network theory. Conversely, a structural network feature of our choice can also be used as the basis for the construction of a flow graph that will then encompass a dynamics biased by such a feature. We illustrate the ideas by focusing on the mathematical properties of generic linear processes on complex networks that can be represented as biased random walks and their dual consensus dynamics, and show how our framework improves our understanding of these processes.

Figure 1

Visual representation of (a) an unweighted graph $\mathcal{G}$ and of its flow graphs ${\mathcal{G}}^{{}^{\prime }}$ defined for BRWs with attributes (b) ${\alpha }_i=s_i$ and (c) ${\alpha }_i=s_i^{-2}$, and (d) a continuous-time random walk with $t=2,r_i=s_i$. The width of the links is proportional to their weight, and the surface of the nodes to their strength. The strength leader, i.e., the node with the highest strength, is darkened if it exists. In order to make the graphs comparable, we renormalized the weights in order to ensure that $W=W^{{}^{\prime }}$. These examples clearly show that different dynamics lead to different patterns and that important nodes for one dynamics might be less important for other dynamics.