Detecting Community Structures in Networks Using a Linear-Programming Based Approach: a Review
We give an account of an approach to community-structure detection in networks using linear programming: given a finite simple graph G, we assign penalties for manipulating this graph by either deleting or adding edges, and then consider the problem of turning G, by performing these two operations, at minimal total cost into a graph that represents a community structure, i.e., that is a disjoint union of complete subgraphs. We show that this minimization problem can be reformulated (and solved!) in terms of a one-parameter family of linear-programming problems relative to which some kind of a “second-order phase transition” can be observed, and we demonstrate by example that this interpretation provides a viable alternative for dealing with the much studied task of detecting community structures in networks. And by reporting our discussions with a leading ecologist, we demonstrate how our approach can be used to analyse food webs and to support the elucidation of their “global” implications.
KeywordsNetworks Graph Graph manipulation Community structure Zachary’s karate club Food web Modularity GN algorithm Linear programming (LP) Integer linear programming (ILP) One-parameter families of LP-problems Second-order phase transitions for parametrized LP-problems Perturbation (of community data)
The authors are grateful to R.Ulanowicz and C. Bondavalli for their expert comments on our results and many helpful suggestions, and to M. Briesemann for his help on the data packing.
This work was financially supported by the Max Planck Society of Germany, the 973 Project on Mathematical Mechanization, the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China.
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