(343g) The Role of Community Structures in Network Control: A Case for the Evolution of Modular Networks in Biology

Community structures [1] in a network refer to such subnetworks that the interconnections within these subnetworks are much denser than the interconnections between them. Networks with community structures, called modular networks and often quantified by a high value of modularity value, are widely noticed in biological as well as social networks. Developing algorithm to detect the community structures from a network has also become a major area of study in network science. However, the role of community structures, as a common network topological feature, in the control of the dynamics associated with networks has not yet been revealed.

In this work, we consider the proportional feedback control of a linear consensus dynamics on networks, and explore the tradeoff between control performance and feedback sparsity under different cost of feedback. For the solution of the corresponding optimal control problem, we employ the distributed optimization algorithm proposed in [2]. We show that as the feedback channel cost increases, the sparsity of the feedback gains increases. When the feedback channel cost is less than or comparable to the control performance cost, the control performance can be retained close to that of centralized control by sacrificing inter-community channels while preserving the intra-community ones. As inter-community channels are shown to have little contribution to control performance, the feedback sparsity can be promoted faster in modular networks than in non-modular networks, resulting in a lower total control cost. When the feedback channel cost becomes so high that the intra-community channels cannot be preserved, the control performance also significantly deteriorates.

By restricting the feedback channels to exist only inside the communities, we obtain a decentralized control architecture which we call community-decentralized control. The corresponding structured optimal control problem is solved by the algorithm in [3]. We document that for modular networks, the community-decentralized control retains a good performance close to that of the centralized control. Given any decentralized control architecture, the corresponding control performance is shown to be strongly correlated to its structural closeness to the community-decentralized control architecture. Based on the above observations for consensus control on networks, we posit that the community structures in the network act as the “core” of feedback control playing a central role in achieving high performance at a low cost.

The above analysis suggests that modular networks possess higher robustness to exogenous disturbances than non-modular networks based on their inherent controllability. Motivated by this, we hypothesize that the enhanced controllability of modular networks plays a fundamental role in the evolution of biological networks into highly modular ones. This complements previous arguments from biologists which have focused on the adaptability of modular network functionality to environmental changes [4].

References:

[1] Fortunato, S. (2010). Community detection in graphs. Physics Reports, 486(3), 75-174.

[2] Lin, F., Fardad, M., & Jovanović, M. R. (2013). Design of optimal sparse feedback gains via the alternating direction method of multipliers. IEEE Transactions on Automatic Control, 58(9), 2426-2431.

[3] Lin, F., Fardad, M., & Jovanovic, M. R. (2011). Augmented Lagrangian approach to design of structured optimal state feedback gains. IEEE Transactions on Automatic Control, 56(12), 2923-2929.

[4] Clune, J., Mouret, J.-B., & Lipson, H. (2013). The evolutionary origins of modularity. Proceedings of the Royal Society, 280(1755), 20122863.