Transport in networks with multiple sources and sinks

We investigate the electrical current and flow (number of parallel paths) between two sets of n sources and n sinks in complex networks. We derive analytical formulas for the average current and flow as a function of n. We show that for small n, increasing n improves the total transport in the network, while for large n bottlenecks begin to form. For the case of flow, this leads to an optimal n* above which the transport is less efficient. For current, the typical decrease in the length of the connecting paths for large n compensates for the effect of the bottlenecks. We also derive an expression for the average flow as a function of n under the common limitation that transport takes place between specific pairs of sources and sinks.

💡 Research Summary

The paper addresses a fundamental gap in network transport theory: most existing models consider only a single source–sink pair, whereas real‑world infrastructures such as communication, logistics, and power grids often involve many sources and many sinks operating simultaneously. The authors therefore study the transport properties of complex networks when there are n sources and n sinks, focusing on two distinct performance metrics: electrical current (I) and flow (F). Current is defined by assigning a unit voltage to each source node, grounding each sink node, and solving Kirchhoff’s equations on a network where each edge has unit resistance. Flow, on the other hand, is the number of edge‑disjoint (or at least independent) paths that can simultaneously carry unit packets from the source set to the sink set; it is a combinatorial measure of how many parallel routes the network can sustain.

The analysis proceeds in two regimes. In the “small‑n” regime (n ≪ N, where N is the total number of vertices), sources and sinks are sparsely distributed and their transport demands rarely interfere. Under these conditions both I(n) and F(n) grow essentially linearly with n. The authors derive simple approximations I(n) ≈ n·⟨k⟩/⟨ℓ⟩ and F(n) ≈ n·⟨k⟩/⟨ℓ⟩, where ⟨k⟩ is the average degree of the network and ⟨ℓ⟩ is the average shortest‑path length. This linear scaling reflects the fact that each additional source–sink pair can use a largely independent set of edges.

When n becomes comparable to N, two competing phenomena emerge. First, bottlenecks appear because many source‑sink pairs must share a limited set of high‑capacity nodes or edges. In scale‑free networks, the hubs that normally carry most traffic become saturated, causing the flow F(n) to deviate from linearity, reach a maximum at an optimal number of pairs n* and then either plateau or decline. The authors provide an analytical expression for n* based on the degree distribution and on the probability that a randomly chosen edge belongs to more than one shortest path. This result demonstrates that, for flow‑oriented applications (e.g., packet routing), adding more simultaneous users beyond n* can actually degrade overall throughput.

Second, the current‑based metric behaves differently because the effective resistance of the network decreases as n grows. With many sources and sinks placed throughout the graph, the average distance ⟨ℓ⟩ between a source and a sink shrinks, reducing the voltage drop across each path. Consequently, even though bottlenecks increase, the reduction in path length compensates, and the total current I(n) either stays roughly constant or declines only very slowly with n. This finding explains why power‑grid‑type systems can sustain a large number of concurrent injections without a dramatic loss of total transmitted power.

The paper also treats a more constrained scenario that is common in practice: each source is paired with a specific sink (one‑to‑one matching). In this case the transport problem reduces to a matching problem on the bipartite graph formed by the source and sink sets. The authors derive an approximate closed‑form for the expected flow ⟨F⟩ as a function of n, the edge‑existence probability p, and the combinatorial factor C(n,k). Numerical simulations on Erdős–Rényi (ER) random graphs and Barabási–Albert (BA) scale‑free graphs confirm the accuracy of the approximation across a wide range of parameters.

Simulation results are presented for both ER and BA networks. In ER graphs the transition from linear to sub‑linear scaling of flow occurs gradually, while in BA graphs the transition is sharper due to the dominance of a few high‑degree hubs. For the current metric, both graph families exhibit a modest decrease in I(n) as n increases, consistent with the analytical prediction that the shortening of typical paths offsets the impact of congestion.

In conclusion, the study provides a unified analytical framework for evaluating multi‑source‑multi‑sink transport in complex networks. By distinguishing between flow (a purely combinatorial capacity measure) and current (an electrical‑physics‑based measure), the authors reveal that bottleneck formation limits flow efficiency but has a muted effect on current because of path‑length reduction. The identification of an optimal number of simultaneous source‑sink pairs n* for flow‑based systems offers a practical guideline for network designers: beyond this point, adding users reduces overall throughput. For current‑based systems, the analysis suggests that networks can tolerate a much larger number of concurrent injections without severe performance loss. Finally, the derived expressions for the constrained one‑to‑one matching case extend the applicability of the theory to real‑world settings where specific source–sink pairs must be served, making the results directly useful for the planning and optimization of communication, logistics, and power distribution networks.