Phase transition in the bipartite z-matching

Till Kahlke,Martin Fränzle,Alexander K Hartmann

doi:10.1140/epjb/s10051-021-00252-2

Till Kahlke, Martin Fränzle + Show 1 more

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https://doi.org/10.1140/epjb/s10051-021-00252-2

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Abstract

We study numerically the maximum z-matching problems on ensembles of bipartite random graphs. The z-matching problems describes the matching between two types of nodes, users and servers, where each server may serve up to z users at the same time. Using a mapping to standard maximum-cardinality matching, and because for the latter there exists a polynomial-time exact algorithm, we can study large system sizes of up to 10^6 nodes. We measure the capacity and the energy of the resulting optimum matchings. First, we confirm previous analytical results for bipartite regular graphs. Next, we study the finite-size behaviour of the matching capacity and find the same scaling behaviour as before for standard matching, which indicates the universality of the problem. Finally, we investigate for bipartite Erdős–Rényi random graphs the saturability as a function of the average degree, i.e. whether the network allows as many customers as possible to be served, i.e. exploiting the servers in an optimal way. We find phase transitions between unsaturable and saturable phases. These coincide with a strong change of the running time of the exact matching algorithm, as well with the point where a minimum-degree heuristic algorithm starts to fail.Graphical

Highlights

We study a phase transition of the satisfiable–unsatisfiable type for the z-matching problem, which is a generalisation of the standard matching
In the main part of our work, for the case of the Poissonian random graphs with average user degree k, we investigate the model with respect to the phase transition between saturability and unsaturability for some typical parameter combinations of z and the ratio N/S
We have studied the saturable–unsaturable phase transition for the z-matching problem on bipartite Erdos– Renyi random graphs

Summary

Introduction

Ensembles of polynomially solvable problems such as shortest paths, maximum flows or graph matching may be of interest and show corresponding phase transitions In physics, such algorithms are used to investigate models like random magnets [21,22]. Our study is motivated by a previous work of Kreacıc and Bianconi [41], who have studied, to our knowledge for the first time in statistical mechanics, the z-matching problem analytically with the approximate cavity approach and numerically with a message-passing algorithm. They have obtained the capacity of the system for two ensembles, namely for fixed degree and Poissonian bipartite graphs. They showed that for both cases, parameter combinations exists, where the capacity converges to its maximum possible value, i.e. a saturable phase, when increasing the average node degree

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