Flow Subgraphs and Flow Network Design under End-to-End Power Dissipation Constraints

TL;DR

A heuristic algorithm, \quotes{Resistor Gap Pruning} (RGP), is proposed, which provides sparse graphs closely approximating the demand effective resistance and which shows stable performance across different demand scenarios.

Abstract

We investigate how the underlying graph of a network supports a flow between a source node and a destination node and propose to compute the expected number of nodes and links that contribute to transferring items in random graphs. Since the transportation is associated with a \quotes{cost} or \quotes{power dissipation}, we further address how to construct a graph given predetermined end-to-end power dissipation, which can be reduced to the \quotes{inverse effective resistance problem} that asks for a weighted graph in which the effective resistance matrix equals a predetermined demand matrix. We propose a heuristic algorithm, \quotes{Resistor Gap Pruning} (RGP), which provides sparse graphs closely approximating the demand effective resistance and which shows stable performance across different demand scenarios.

Figures (14)

• Figure 1: Illustration of the excess degree $d_i'$. When node $i$ is reached by following a randomly chosen link $l$, the remaining number of neighbors of node $i$ is $d_i' = d_i - 1$.
• Figure 2: Examples illustrating the role of link-weight degeneracy in the flow subgraph. The nodes and links of the flow subgraph are highlighted in red. In (a), link weights are i.i.d. continuous random variables, which yields distinct nodal potentials almost surely. Thus, Property \ref{['property1']} is sufficient. In (b), all link weights are identical and structural symmetries create equipotential nodes $u$ and $v$. Consequently, node $w$ does not belong to the flow subgraph, although it satisfies Property \ref{['property1']}.
• Figure 3: Backbone–branch decomposition: Backbone nodes (each node has $\ge 2$ neighbors within the backbone) are highlighted with green outlines; other nodes are branches (finite subgraph) attached to the backbone.
• Figure 4: Examples for different source--destination pairs. Red nodes and links indicate the components of the flow subgraph. Backbone nodes are highlighted with green node outlines.
• Figure 5: Intersections of $y=p$ (black) and $y=1-e^{-E[D] p}$ (colored) for different expected degrees $E[D]$. When the expected degree $E[D] \le 1$, only the trivial solution $p=0$ exists. For $E[D]>1$, a non-trivial fixed point appears and increases with $E[D]$.

Theorems & Definitions (5)

• proof
• Lemma 1
• proof
• Lemma 2
• proof