Maximum matchings in scale-free networks with identical degree distribution

Abstract

The size and number of maximum matchings in a network have found a large variety of applications in many fields. As a ubiquitous property of diverse real systems, power-law degree distribution was shown to have a profound influence on size of maximum matchings in scale-free networks, where the size of maximum matchings is small and a perfect matching often does not exist. In this paper, we study analytically the maximum matchings in two scale-free networks with identical degree sequence, and show that the first network has no perfect matchings, while the second one has many. For the first network, we determine explicitly the size of maximum matchings, and provide an exact recursive solution for the number of maximum matchings. For the second one, we design an orientation and prove that it is Pfaffian, based on which we derive a closed-form expression for the number of perfect matchings. Moreover, we demonstrate that the entropy for perfect matchings is equal to that corresponding to the extended Sierpiński graph with the same average degree as both studied scale-free networks. Our results indicate that power-law degree distribution alone is not sufficient to characterize the size and number of maximum matchings in scale-free networks.

Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

“…In other words, even for small m, the upper and lower bounds in Eq. (22) converge to the quoted value Z(A n ) = 0.43017 . .…”

“…Due to the wide applications, it is of theoretical and practical significance to study matching number and domination number in networks, as well as the number of maximum matchings and MDSs. In the past years, concerted efforts have been devoted to developing algorithms for the problems related to maximum matchings [13,14,15,16,17,18,19,20,21,22] and minimum dominating sets [23,24,25,26,27,28] from the community of mathematics and theoretical computer science. However, solving these problems is a challenge and often computationally intractable.…”

Lower lower-critical spin-glass dimension from quenched mixed-spatial-dimensional spin glasses

“…see Refs. [24][25][26][27][28][29][30][31][32]. Thus, previous works have generally used a hierarchical lattice generated by a single graph and spatial dimensionality that is microscopically uniform throughout the system.…”

Phase transitions between different spin-glass phases and between different chaoses in quenched random chiral systems

“…Exact calculations on hierarchical lattices are also currently widely used in a variety of statistical mechanics [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], finance [37], and, most recently, DNA-binding [38] problems.…”