Traffic congestion in interconnected complex networks

Traffic congestion in isolated complex networks has been investigated extensively over the last decade. Coupled network models have recently been developed to facilitate further understanding of real complex systems. Analysis of traffic congestion in coupled complex networks, however, is still relatively unexplored. In this paper, we try to explore the effect of interconnections on traffic congestion in interconnected BA scale-free networks. We find that assortative coupling can alleviate traffic congestion more readily than disassortative and random coupling when the node processing capacity is allocated based on node usage probability. Furthermore, the optimal coupling probability can be found for assortative coupling. However, three types of coupling preferences achieve similar traffic performance if all nodes share the same processing capacity. We analyze interconnected Internet AS-level graphs of South Korea and Japan and obtain similar results. Some practical suggestions are presented to optimize such real-world interconnected networks accordingly.

💡 Research Summary

This paper investigates how interconnections between two Barabási‑Albert (BA) scale‑free networks influence traffic congestion, focusing on three coupling preferences—assortative (high‑load nodes linked together), disassortative (high‑load nodes linked to low‑load nodes), and random—and on the coupling probability P, defined as the ratio of inter‑network links to the total number of nodes. Each node may have at most one inter‑network link, so P ranges from 0 to 1.

Two traffic‑resource allocation schemes are considered. In the uniform (UNI) scheme every node receives a processing capacity of one packet per time step. In the node‑usage‑probability (NUP) scheme a node’s capacity is proportional to its algorithmic betweenness (i.e., its expected traffic load) in its original isolated network. Two routing protocols are examined: the classic shortest‑path (SP) routing and the efficient routing (ER) protocol that deliberately avoids hub nodes to spread load more evenly.

Traffic dynamics follow a standard packet‑generation and packet‑processing model. At each time step R new packets with random source‑destination pairs are generated. Each node processes up to its capacity Ck packets; excess packets join a FIFO queue of unlimited length. The macroscopic order parameter η(R) measures the growth rate of the total number of packets in the system. When η = 0 the system is in free‑flow; when η > 0 it is jammed. The critical packet‑generation rate Rc, at which η first becomes positive, quantifies the network’s traffic capacity. Theoretical analysis shows that Rc is proportional to the total number of nodes divided by the maximum ratio Bk/Ck, where Bk is algorithmic betweenness and Ck is processing capacity. Under NUP, Bk and Ck scale together, reducing the maximum ratio and thus increasing Rc.

Simulations are performed on two identical BA networks with N = 600 nodes and average degree ⟨k⟩ = 4. The main findings are:

1. Assortative coupling exhibits a non‑monotonic dependence of Rc on P. As P increases from zero, Rc rises sharply because the few inter‑network links connect high‑capacity nodes, efficiently sharing the extra load. Beyond a certain P, however, Rc declines slightly because additional links start to overload the same high‑capacity nodes, reducing the benefit of load balancing. Consequently, an optimal coupling probability exists that maximizes traffic capacity.

2. Disassortative and random coupling both show a monotonic increase of Rc with P. Linking high‑load nodes to low‑load nodes (or randomly) gradually spreads traffic across the two networks, improving capacity without the oversaturation effect seen in the assortative case.

3. Interaction with routing and allocation:

   • With shortest‑path routing, the NUP allocation yields the highest Rc for all three coupling types. Since SP concentrates traffic on hubs, assigning larger capacities to those hubs (NUP) is essential to avoid congestion.
   • With efficient routing, the UNI allocation outperforms NUP. ER already redistributes traffic away from hubs, so giving every node the same modest capacity is sufficient and even advantageous.

4. Real‑world validation: The authors analyze inter‑AS (Autonomous System) graphs of South Korea and Japan, which are not perfectly symmetric but share similar scale‑free characteristics. The same qualitative trends appear: assortative inter‑AS links give the greatest capacity, and an optimal inter‑AS link density can be identified.

The paper’s contributions are threefold: (i) it systematically quantifies how coupling preference and coupling density affect congestion in interconnected networks; (ii) it demonstrates that allocating node capacities proportional to usage probability (NUP) combined with assortative coupling maximizes traffic capacity when shortest‑path routing is used; and (iii) it translates these insights into practical design guidelines for real communication infrastructures, such as encouraging high‑capacity backbone nodes to interconnect directly and tuning the number of inter‑domain links to the identified optimal range.

Overall, the study provides a comprehensive framework for evaluating and optimizing traffic flow in coupled complex networks, bridging theoretical modeling with empirical Internet data, and offering actionable recommendations for network operators seeking to mitigate congestion in multi‑domain environments.