Controllability and maximum matchings of complex networks

Zhao, Jin Hua; Zhou, Haijun

doi:10.1103/physreve.99.012317

Cited by 12 publications

(9 citation statements)

References 68 publications

(105 reference statements)

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“…It immediately follows that this solution is physical, i.e., it satisfies eq. ( 27) only for N = zM, (30) in which case it corresponds to full capacity…”

Section: Stability Conditionmentioning

confidence: 99%

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Statistical mechanics of bipartite z-matchings

Kreacic¹,

Bianconi²

2019

EPL

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The matching problem has a large variety of applications including the allocation of competitive resources and network controllability. The statistical mechanics approach based on the cavity method has shown to be exact in characterising this combinatorial problem on locally tree-like networks. Here we use the cavity method to solve the many-to-one bipartite z-matching problem that can be considered to be a model for the characterisation of the capacity of user-server networks such as wireless communication networks. Finally we study the phase diagram of the model defined in network ensembles.

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“…It immediately follows that this solution is physical, i.e., it satisfies eq. ( 27) only for N = zM, (30) in which case it corresponds to full capacity…”

Section: Stability Conditionmentioning

confidence: 99%

“…This problem has attracted large interest from combinatorics, probability and the computer science communities [21][22][23]. For this problem the statistical mechanics approach [24][25][26] is very useful and in particular the Belief Propagation algorithm [27][28][29][30] provides the exact solution as long as the network is locally tree-like.…”

mentioning

confidence: 99%

Statistical mechanics of bipartite z-matchings

Kreacic¹,

Bianconi²

2019

EPL

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“…A link only connects a node in + set and a node in − set and there is no link within each set of nodes. To identify the minimum driver nodes necessary and sufficient to control the networked system G, one can find a maximum matching in the bipartite network [2,50], where a node can at most match another node through one link. If node j − is unmatched in a maximum matching configuration, node j in G is a driver node for control in this configuration.…”

Section: Problem Descriptionmentioning

confidence: 99%

The network asymmetry caused by the degree correlation and its effect on the bimodality in control

Yu,

Liang,

Wang

et al. 2021

Preprint

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Our ability to control a whole network can be achieved via a small set of driver nodes. While the minimum number of driver nodes needed for control is fixed in a given network, there are multiple choices for the driver node set. A quantity used to investigate this multiplicity is the fraction of redundant nodes in the network, referring to nodes that do not need any external control. Previous work has discovered a bimodality feature characterized by a bifurcation diagram: networks with the same statistical property would stay with equal probability to have a large or small fraction of redundant nodes.Here we find that this feature is rooted in the symmetry of the directed network, where both the degree distribution and the degree correlation can play a role. The in-in and out-out degree correlation will suppress the bifurcation, as networks with such degree correlations are asymmetric under network transpose. The out-in and in-out degree correlation do not change the network symmetry, hence the bimodality feature is preserved. However, the out-in degree correlation will change the critical average degree needed for the bifurcation. Hence by fixing the average degree of networks and tuning out-in degree correlation alone, we can observe a similar bifurcation diagram. We conduct analytical analyses that adequately explain the emergence of bimodality caused by out-in degree correlation. We also propose a quantity, taking both degree distribution and degree correlation into consideration, to predict if a network would be at the upper or lower branch of the bifurcation. As is well known that most real networks are not neutral, our results extend our understandings of the controllability of complex networks.

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“…Further, the MM problem on a directed graph can be mapped onto its bipartite graph representation [18]. There are results from the cavity method at zero-temperature limit [19], yet a core percolation analysis [20] gives a corrected estimation of matching sizes in percolated regime. The GLR procedure is also applied on the minimum vertex cover problem [12,17], and reproduces in a simple way the results of energy densities from a calculation with replica trick on Erdös-Rényi (ER) random graphs [21].…”

Section: Introductionmentioning

confidence: 99%

A local algorithm and its percolation analysis of bipartite z-matching problem

Zhao¹

2023

J. Stat. Mech.

Self Cite

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A z-matching on a bipartite graph is a set of edges, among which each vertex of two types of the graph is adjacent to at most 1 and at most z ( ⩾ 1 ) edges, respectively. The z-matching problem concerns finding z-matchings with the maximum size. Our approach to this combinatorial optimization problem is twofold. From an algorithmic perspective, we adopt a local algorithm as a linear approximate solver to find z-matchings on any graph instance, whose basic component is a generalized greedy leaf removal procedure in graph theory. From a theoretical perspective, on uncorrelated random bipartite graphs, we develop a mean-field theory for the percolation phenomenon underlying the local algorithm, leading to an analytical estimation of z-matching sizes on random graphs. Our analytical theory corrects the prediction by belief propagation algorithm at zero-temperature limit in (Kreačić and Bianconi 2019 Europhys. Lett. 126 028001). Besides, our theoretical framework extends a core percolation analysis of k-XORSAT problems to a general context of uncorrelated random hypergraphs with arbitrary degree distributions of factor and variable nodes.

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Controllability and maximum matchings of complex networks

Cited by 12 publications

References 68 publications

Statistical mechanics of bipartite z-matchings

Statistical mechanics of bipartite z-matchings

The network asymmetry caused by the degree correlation and its effect on the bimodality in control

A local algorithm and its percolation analysis of bipartite z-matching problem

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