Statistical mechanics of bipartite matchings
Abstract
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 characterizing this combinatorial problem on locally treelike networks. Here we use the cavity method to solve the manytoone bipartite matching problem that can be considered to be a model for the characterization of the capacity of userserver networks such as wireless communication networks. Finally we study the phase diagram of the model defined in network ensembles.
Keywords Maximum matching problem Combinatorial optimization problems Messagepassing algorithms
Noname manuscript No. (will be inserted by the editor)
Eleonora Kreačić Ginestra Bianconi Statistical mechanics of bipartite matchings
Received: date / Accepted: date
Department of Statistics, University of Oxford, Oxford OX1 3LB, UK Email: eleonora.kreacic@gmail.com
G. Bianconi
Alan Turing Institute, London, UK School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK Email: g.bianconi@qmul.ac.uk \@xsect
In network science [Laszlo, Newman, Latora, bianconi_2018multilayer] there is increasing interest in combinatorial optimization problems and messagepassing algorithms applied to processes as different as control [Control, Menichetti_PRL, Menichetti_multilayer], percolation [Lenka_Newman, bianconi2018rare], percolation on multilayer networks [bianconi_2018multilayer, Phys_Rep, Doro1, Doro2, Radicchi_percolation, Osat] or epidemic spreading [Zecchina1, Zecchina2, Gleeson, Saad]. This surge of interest is fueled by the efficiency of using the statistical mechanics approach [Mezard_Montanari, Weigt] to solve combinatorial optimization problems and by the vast realm of applications of these problems and their generalizations. In this paper we will characterize the statistical mechanics of a generalized matching problem called matching that can be interpreted as a model in a system with limited resources.
On an undirected network the matching problem consists of finding the maximum subset of the links of the network (the matched links) such that each node is adjacent to at most one matched link. This problem has attracted large interest from the mathematics and the computer science scientific community [KarpSipserER, Aronson, Gale]. For this problem the statistical mechanics approach [Lelarge, Spin_glass, Mezard_Parisi] is very useful and in particular the Belief Propagation algorithm [Lenka_Mezard, Chayes, HJ, HJ2] provides the exact solution as long as the network is locally treelike. The matching problem and its generalizations have a variety of applications ranging from wireless networks to network controllability.
The generalization of the matching problem on directed networks has recently been shown [Control] to characterize the network controllability as it identifies the set of driver nodes of the network. Since the matching problem on directed locally treelike networks is exactly solvable using statistical mechanics methods this result has opened new perspectives in network controllability. In particular it has allowed to relate the directed network dynamical properties (controllability) to its structural ones (the properties of its maximum matching). Interestingly in this context it has been shown [Menichetti_PRL] that the key structural property characterizing the matching (and hence the network controllability) is the fraction of nodes with low in and out degrees.These results have also recently been extended to multilayer network controllability by considering the relevant extension of the maximum matching problem to a multilayer network maximum matching problem [Menichetti_multilayer].
More traditionally the matching problem has been defined in spatial networks where nodes have specific positions, or more generally in networks in which each pair of nodes is associated with a distance, the so called marriage problem [Zhang, Zhang_marriage, Caldarelli]. In this setting the generalized matching problem aims at finding the maximum matching that minimizes the overall distances between the nodes. This pecific model defined over a bipartite network has a variety of applications [Zhang] in finite resource allocation problems where some providers of service want to optimize the user satisfaction and their own profit.
In this paper, we are focusing on another important variation of the matching problem called the manytoone matching on bipartite networks. In this case we consider a bipartite network in which one set of nodes can be matched at most to one link and the other set of nodes can be matched to at most links. The manytoone matching problem on bipartite networks has recently become particularly relevant for characterizing wireless communication networks [Wireless, Alex] in which one has two different sets of nodes – users and towers. Each tower provides the wireless connection, but can only serve up to users. The maximum capacity of the network is given by the largest number of users that can be served at a time. Here, we determine the Belief Propagation equations to characterize the maximum capacity of any locally treelike bipartite network and we evaluate the capacity of network ensembles with given degree distribution. Our analysis is based on the use of the Belief Propagation algorithm in the zerotemperature limit. In this way we extend the existing statistical mechanics treatment of the onetoone maximum matching problem of simple, directed and multilayer networks to the manytoone matching problem.
The paper is structured as follows: in Section Statistical mechanics of bipartite matchings we define the maximum matching problem and its mapping to a statistical mechanics problem; in Section Statistical mechanics of bipartite matchings we solve the problem on single bipartite networks using the BP algorithm in the zero temperature limit; in Section Statistical mechanics of bipartite matchings we study the problem on bipartite network ensembles with given degree distribution and we characterize the phase diagram of the model. Finally, in Section LABEL:Sect_Conclusion we present our conclusions.
Let us consider a bipartite network formed by users and towers where we assume for simplicity that each tower is connected at least to users. A matching is the subset of the set of edges such that each user is adjacent to at most one and each tower is adjacent to at most edges from the subset. In other words, each user communicates with at most one neighbouring tower and each tower serves at most neighbouring users. The size of a matching is given by the number of its edges and the maximum capacity of the network is given by the size of the largest possible matching. In order to treat our problem from the statistical mechanics point of view, for each linked pair (i.e. a pair usertower connected by an edge in the network), and a given matching, we introduce the variables such that if the edge is included in the matching, and otherwise. A matching reduces to an assignment that satisfies the conditions
(0) 
where indicates the set of neighbours of a node and indicates the set of neighbours of a tower . Let us define the energy and the capacity of the matching as
(0) 
with and given by
(0) 
Therefore, the capacity indicates the number of users with one matched link. Having in mind that , we have the following simple relationship
(0) 
The problem of finding a maximum capacity of the matching translates then to the problem of finding the allowed configurations for the matching which minimize the energy .
Here we associate to each possible matching of the network the Gibbs measure
(0) 
where denotes the inverse temperature, is a normalization constant and for and for . The freeenergy of the problem is defined as
(0) 
and the energy is given by
(0) 
In order to characterize the maximum capacity of a network, or equivalently the minimum energy of the network we are interested in the limit when .
On a locally tree like graph, the Gibbs measure given by Eq. ( ‣ Statistical mechanics of bipartite matchings) can be found in the Bethe approximation using the Belief Propagation (BP) algorithm [Mezard_Montanari]. The BP algorithm expresses the Gibbs measure in terms of messages. Here we distinguish between two types of messages: those that the users send to neighbouring towers , and those that the towers send to neighbouring users . For a user and its neighbour , denotes the probability that says to that it believes that their edge should be included to the matching. Similarly, denotes the probability that informs that their edge should be matched. Then, for the Belief Propagation messages are given by
(0) 
where , , and , , represent normalization constants.
The messages can be parametrized by the cavity fields in the following way
(0) 
Using the Eqs. the BP Eqs. ( ‣ Statistical mechanics of bipartite matchings) can be written in terms of the cavity fields as
(0) 
According to the BP algorithm [Mezard_Montanari], the marginal probability that an edge is associated to a variable is given by
(0) 
where are normalization constants.
Similarly the marginal probability of the variables associated to the links incident to node and the marginal probability of the variables associated to links incident to tower are given by
(0) 
In the Bethe approximation, valid on locally treelike networks the Gibbs measure given by Eq. ( ‣ Statistical mechanics of bipartite matchings) is given in terms of the marginals by
(0) 
The free energy of the system can be found by minimising the Gibbs free energy given by
(0) 
where
(0) 
In fact it can easily be shown that the Gibbs free energy is minimized when is given by Eq. ( ‣ Statistical mechanics of bipartite matchings) and that the minimum Gibbs freeenergy is equal to the free energy of the problem and takes the value
(0) 
Using the Bethe expression for the Gibbs measure given by Eq. ( ‣ Statistical mechanics of bipartite matchings) in Eq. ( ‣ Statistical mechanics of bipartite matchings) we obtain the free energy
(0)  
Finally, using Eq. ( ‣ Statistical mechanics of bipartite matchings) we can express the energy in terms of the cavity fields as
(0)  
For finding the maximum capacity of a bipartite network we need to investigate the zero temperature limit of the BP equations, i.e. we need to consider the limit . In this limit the cavity fields have the support on and the BP equations read
(0) 
Thus, a node sets a field if all other neighbouring towers set their fields which point to to ; it sets if at least one other neighbouring tower sends ; and sets to otherwise, i.e. if at least one other tower sends , and no tower sends . Similarly, a tower of degree sets a field if at least of the other node set fields pointing to to ; sets the field if at least other nodes set fields pointing to it to ; and otherwise, it sets the field . In the case of multiple solutions, the dynamically stable solution that is physical and that minimizes the energy represents the solution of the maximum capacity problem where the energy can be expressed as
(0)  
Here we consider the maximum matching problem on bipartite network ensembles formed by users and towers where the users have degree distribution , and the towers have degree distribution , with for . Note that on a bipartite network the average degree of the users and the average degree of the towers need to satisfy
(0) 
In order to study the energy of the matching problem on these ensembles we denote by and the distributions of fields and , respectively, i.e.
(0) 
where and , where indicates the Kronecker delta. Therefore indicate the probability that the cavity fields coming from nodes are equal to respectively and indicate the probability that the cavity fields coming from towers are equal to respectively.
Using the BP Eqs. ( ‣ Statistical mechanics of bipartite matchings) derived in the zero temperature limit, we can derive the equations satisfied by the probabilities in a bipartite network ensemble. These read
(0) 
Similarly we can derive the equations satisfied by the probabilities in the bipartite network ensemble,
(0) 
Finally also the energy of the maximum matching Eq. can be expressed in terms of the probabilities as
(0)  
Therefore the phase diagram of the matching problem can be drawn by solving Eqs. and Eqs. and calculating the energy given by Eq. ( ‣ Statistical mechanics of bipartite matchings) on this solution as a function of the structural properties of the bipartite network ensemble.
The solutions of the BP Eqs. and should be physical, i.e. they should correspond to values of the capacity
(0) 
Moreover they should be dynamically stable. In order to characterize the stability of a given solution we calculate the Jacobian matrix of the system of equations for the probabilities including Eqs. and Eqs. which reads
(0) 
where
(0) 
and where are given by
A given solution of the system of Eqs. ( ‣ Statistical mechanics of bipartite matchings) and Eqs. ( ‣ Statistical mechanics of bipartite matchings) is stable if and only if the eigenvalues of the Jacobian are all less than one. In this way we obtain the stability conditions
(0) 
The trivial solution and deserves a special consideration. This solution correspond to zero energy and to capacity . Therefore it is immediate to notice that this solution is physical, i.e. satisfies Eq. only for
(0) 
for which we have that the trivial solution correspond to full capacity
(0) 
From the study of the BP Eqs. and we observe that these equations admit the trivial solution and as long as the minimum degree of the nodes is one and the minimum degree of the towers is , i.e. for and for . However, in order to establish whether this is the solution of the maximum matching problem we need to investigate its stability.
In particular, if we study the stability conditions Eqs. ( ‣ Statistical mechanics of bipartite matchings) of the trivial solution and , we obtain
(0) 
Therefore, the instability of the trivial solution on a bipartite network ensemble with given degree distribution is driven by the fraction of users of degree two and the fraction of towers of degree . In particular when the minimum degree of the nodes is greater than two, i.e. and the minimum degree of the towers is greater than , i.e. for , as long as we get that the trivial solution is stable independently of the other properties of the degree distributions of the nodes and of the towers. This generalizes the relation found in matching of simple networks[Lenka_Mezard], in matching of directed networks [Menichetti_PRL] and on generalized matching in multilayer networks [Menichetti_multilayer].
Let us discuss here few examples of the phase diagram of the maximum matching on bipartite networks with given degree distribution. Let us start with a simple example of a regular bipartite network in which the degree distributions are given by
(0) 
with and where indicates the Kronecker delta. For these networks we clearly have and , therefore it follows that Eq. ( ‣ Statistical mechanics of bipartite matchings) reduces to
(0) 
As a function of the values of and we have the following regimes:

If , or equivalently the solution is and and corresponding to energy and capacity .

If or equivalently the solution is and for and , for . Both solutions correspond to energy and capacity .

If or equivalently the solution is and corresponding to energy and capacity .
As a second example of bipartite network ensemble we consider the bipartite network in which the degree distributions of the towers and the nodes are exponentially distributed according to
(0) 
where and are related by Eq. , and and are normalization constants given by
(0) 
In Figure LABEL:fig:exponential we show the phase diagram of this exponential bipartite network by plotting the capacity density versus and the ratio between the number of towers and the number of users. We observe that as the resources increase the capacity of the network increases.