scholarly journals Phase transition in the bipartite z-matching

2021 ◽  
Vol 94 (12) ◽  
Author(s):  
Till Kahlke ◽  
Martin Fränzle ◽  
Alexander K. Hartmann

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 $$10^6$$ 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 Abstract

Author(s):  
Yingping Huang ◽  
Xihui Zhang ◽  
Paulette S. Alexander

Business matchmaking is a service dedicated to providing one-on-one appointments for small businesses (or sellers) to meet with government agencies and large corporations (or buyers) for contracting opportunities. Business matchmaking scheduling seeks to maximize the total number of appointments with the maximum objective that weighs the preferences of both buyers and sellers. In this paper, the authors transformed the business matchmaking scheduling problem into a 3-dimensional planar assignment problem and solved it heuristically using a series of bipartite maximum weighted maximum cardinality matching problems. Simulation experiments and real data showed that this algorithm outperforms human experts and prior algorithm in terms of number of appointments, the objective that weighs buyer and seller’s preferences, and the execution time.


2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Nicole Balashov ◽  
Reuven Cohen ◽  
Avieli Haber ◽  
Michael Krivelevich ◽  
Simi Haber

Abstract We consider optimal attacks or immunization schemes on different models of random graphs. We derive bounds for the minimum number of nodes needed to be removed from a network such that all remaining components are fragments of negligible size.We obtain bounds for different regimes of random regular graphs, Erdős-Rényi random graphs, and scale free networks, some of which are tight. We show that the performance of attacks by degree is bounded away from optimality.Finally we present a polynomial time attack algorithm and prove its optimal performance in certain cases.


2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.


2008 ◽  
Vol 19 (12) ◽  
pp. 1777-1785 ◽  
Author(s):  
F. P. FERNANDES ◽  
F. W. S. LIMA

The zero-temperature Glauber dynamics is used to investigate the persistence probability P(t) in the Potts model with Q = 3, 4, 5, 7, 9, 12, 24, 64, 128, 256, 512, 1024, 4096, 16 384, …, 230 states on directed and undirected Barabási–Albert networks and Erdös–Rényi (ER) random graphs. In this model, it is found that P(t) decays exponentially to zero in short times for directed and undirected ER random graphs. For directed and undirected BA networks, in contrast it decays exponentially to a constant value for long times, i.e., P(∞) is different from zero for all Q values (here studied) from Q = 3, 4, 5, …, 230; this shows "blocking" for all these Q values. Except that for Q = 230 in the undirected case P(t) tends exponentially to zero; this could be just a finite-size effect since in the other "blocking" cases you may have only a few unchanged spins.


2013 ◽  
Vol 88 (18) ◽  
Author(s):  
U. Ferrari ◽  
C. Lucibello ◽  
F. Morone ◽  
G. Parisi ◽  
F. Ricci-Tersenghi ◽  
...  

2020 ◽  
Vol 117 (27) ◽  
pp. 15394-15396
Author(s):  
Timothy W. Sirk

The chordless cycle sizes of spatially embedded networks are demonstrated to follow an exponential growth law similar to random graphs if the number of nodesNxis below a critical valueN*. For covalent polymer networks, increasing the network size, as measured by the number of cross-link nodes, beyondN*results in a crossover to a new regime in which the characteristic size of the chordless cyclesh*no longer increases. From this result, the onset and intensity of finite-size effects can be predicted from measurement ofh*in large networks. Although such information is largely inaccessible with experiments, the agreement of simulation results from molecular dynamics, Metropolis Monte Carlo, and kinetic Monte Carlo suggests the crossover is a fundamental physical feature which is insensitive to the details of the network generation. These results show random graphs as a promising model to capture structural differences in confined physical networks.


2009 ◽  
Vol 388 (17) ◽  
pp. 3413-3425 ◽  
Author(s):  
Julien Barré ◽  
Antonia Ciani ◽  
Duccio Fanelli ◽  
Franco Bagnoli ◽  
Stefano Ruffo

10.37236/4205 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
M. Abreu ◽  
G. Araujo-Pardo ◽  
C. Balbuena ◽  
D. Labbate ◽  
J. Salas

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of $(q+1,8)$-cages, for $q$ a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain $(q+1)$-regular graphs of girth 7 and order $2q^3+q^2+2q$ for each even prime power $q \ge 4$, and of order $2q^3+2q^2-q+1$ for each odd prime power $q\ge 5$. A corrigendum was added to this paper on 21 June 2016.


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