SHIFT-COUPLING OF RANDOM ROOTED GRAPHS AND NETWORKS

Shift-Coupling of Random Rooted Graphs and Networks

Ali Khezeli School of Mathematics
Institute for Research in Fundamental Sciences (IPM)
P.O. Box 19395-5746, Tehran, Iran
alikhezeli@ipm.ir
August 17, 2019
Abstract

In this paper, we present a result similar to the shift-coupling result of Thorisson (1996) in the context of random graphs and networks. The result is that a given random rooted network can be obtained by changing the root of another given one if and only if the distributions of the two agree on the invariant sigma-field. Several applications of the result are presented for the case of unimodular networks. In particular, it is shown that the distribution of a unimodular network is uniquely determined by its restriction to the invariant sigma-filed. Also, the theorem is applied to the existence of an invariant transport kernel that balances between two given (discrete) measures on the vertices. An application is the existence of a so called extra head scheme for the Bernoulli process on an infinite unimodular graph. Moreover, a construction is presented for balancing transport kernels that is a generalization of the Gale-Shapley stable matching algorithm in bipartite graphs. Another application is on a general method that covers the situations where some vertices and edges are added to a unimodular network and then, to make it unimodular, the probability measure is biased and then a new root is selected. It is proved that this method provides all possible unimodularizations in these situations. Finally, analogous existing results for stationary point processes and unimodular networks are discussed in detail.

Shift-coupling, random graphs, random networks, unimodularity, invariant transports, stable transports, network extension, unimodularization.
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equationsection \theoremstyletheorem \theoremstyledefinition \theoremstyleremark \theoremstyletheorem \numberwithinequationsection

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[2010]Primary 60C05; Secondary: 60K99; 05C80

1 Introduction

This paper deals with random rooted graphs, which are possibly infinite, but finite-degree connected graphs with a distinguished vertex called the root. Roughly speaking, each vertex and edge of a graph can be equipped with marks to form a network. Unimodular random rooted networks have been of great interest in the last two decades. They satisfy a formulation of the heuristic property that all vertices are equally likely to be the root, although there may be infinitely many vertices. The formulation, called the mass transport principle, will be recalled in Section 2. This concept is introduced in [4] and developed further in [1] to generalize some properties of Cayley graphs, which are highly homogeneous, to more general classes of graphs and random graphs. It also arises in the study of limits of sequences of finite graphs, which is the novel work of [2], and also in stationary point processes. Many concepts and results in stationary point processes have analogues in the context of unimodular networks. This analogy will be addressed many times in this paper.

To introduce the idea of this work, we get help from the following general construction method. Let be a given (non-unimodular) random rooted network, where stands for the root. In some examples in the literature, a unimodular network is constructed from by the following two steps: Bias the probability measure by an appropriate function and then choose a new random root with an appropriate distribution on the vertices of . Denote the resulting random rooted network by . Explicit examples of such constructions in the literature will be recalled in Section 5. One may intuitively accept that is equivalent to if we disregard the root ([1]), or that and have the same non-rooted networks. However, to state this in a mathematically precise way, one should answer the following question.

Question 1

When do two given (not necessarily unimodular) random rooted networks have the same non-rooted networks?

Note that the question is not limited to the setting of the above example. In general, no special relation is assumed between the two random rooted networks and they might be given only by two probability distributions. The answer to this question is not straightforward since the space of non-rooted networks is non-standard. Several definitions of unrooted-equivalence are provided in Section 3 as answers to this question, where some of the definitions are shown to be equivalent. It will be shown that in the above example, and are weakly unrooted-equivalent, to be defined later. The strong sense in our definition is that (the distribution of) the second one can be obtained from the first by a root-change (note that there is a biasing in the above definition of before changing the root). Another definition is that the two random rooted networks agree on the invariant sigma-field; i.e. any event that does not depend on the root occurs with equal probabilities. Some other definitions will also be given (Definition 7).

The main theorem (Theorem 1) in this work is that the last two definitions mentioned above are equivalent; namely, if two random rooted networks agree on the invariant sigma-field, then they can be obtained from each other by a root-change. This theorem, in its spirit, is similar to a well known result by Thorisson [19] that studies shift-coupling of random elements in a space equipped with a group action.

In Section 4, we discuss applications of the main theorem in the unimodular case. Theorem 3 says that the distribution of a unimodular network is uniquely determined by the distribution of its non-rooted network, or equivalently, by its restriction to the invariant sigma-field. Theorem 4 deals with invariant balancing transport kernels, which are transport kernels that transport a given measure to another given one. In the context of stationary random measures and point processes, this concept has been studied by many authors recently. In this context, under suitable assumptions, the existence of a (random) balancing transport kernel that is invariant under translations is implied by the result of [19] (proved in the general case in [14]). Based on this abstract result, several constructions have been provided, starting from [15] and [10], where the latter provides a transport kernel balancing between (a multiple of) the Lebesgue measure and the counting measure of the Poisson point process. Here, in the context of unimodular networks, we consider two discrete measures on the vertices of the random network. In Theorem 4, it will be proved that, roughly speaking, a balancing transport kernel between them exists if and only if the measures have equal sample intensities; i.e. have the same expectation conditioned on the invariant sigma-field. A construction of such transport kernels is discussed in Section 6 (Theorem 7) based on the construction of stable transports in [8], which is by itself based on [10]. It is a generalization of the Gale-Shapley stable matching algorithm in bipartite graphs [7].

In Section 5, we describe a general method for constructing unimodular networks. In some of the examples in the literature, such a network is constructed by the following steps: Adding some vertices and edges to another unimodular network (called a network extension here), then biasing the probability measure and finally applying a root-change. These examples are unified in the method presented in Theorem 5. It is also proved in Theorem 6 that this method gives all possible ways to unimodularize the extension (to be defined more precisely later).

Many of the definitions and results in this paper have analogues in the context of point processes and random measures, which are discussed in Section 8.

This paper is structured as follows. The definition and basic properties of unimodular random networks are given in Section 2. The definition of unrooted-equivalence and the main shift-coupling theorem are presented in Section 3. The applications of the theorem to the unimodular case are studied in Section 4. Section 5 deals with extensions of unimodular networks. Section 6 presents a construction of balancing transport kernels using stable transports. The proofs of some results are moved to Section 7 to help to focus on the main thread of the paper. Finally, Section 8 reviews the analogous results in the context of point processes.

2 Random Rooted Graphs and Networks

In this section, we recall the concepts of random networks and unimodularity mainly from [1]. A network is a (multi-) graph equipped with a complete separable metric space , called the mark space and with two maps from and to , where the symbol is used for adjacency of vertices or edges. The image of (resp. ) in is called its mark. The degree of a vertex is denoted by and the graph-distance of vertices and is denoted by . The symbol is used for the closed ball centered at with radius ; i.e. the set of vertices with distance at most to .

In this paper, all networks are assumed to be locally finite; that is, the degrees of every vertex is assumed to be finite. Moreover, a network is assumed to be connected except when explicitly mentioned. An isomorphism between two networks is a graph isomorphism that also preserves the marks. A rooted network is a pair in which is a network and is a distinguished vertex of called the root. An isomorphism of rooted networks is a network isomorphism that takes the root of one to that of the other. Let denote the set of isomorphism classes of connected networks and the set of isomorphism classes of connected rooted networks. The set is defined similarly for doubly-rooted networks; i.e. those with a pair of distinguished vertices. The isomorphism class of a network (resp. or ) is denoted by (resp. or ).

The sets and can be equipped with natural metrics that make them a complete separable metric space and equip them with the corresponding Borel sigma-fields. The distance of two rooted networks is defined based on the similarity of finite neighborhoods of their roots. See [1] for the precise definition. There are two natural projections obtained by forgetting the second and the first root respectively. These projections are continuous and measurable. In contrast, there is no useful metric on . However, as will be defined in Definition 4, the natural projection of forgetting the root induces a sigma-field on . This sigma-field is extensively used in this paper although it does not make a standard space.

Definition 1

A random rooted network is a random element in and is represented in either of the following ways.

  • A probability measure on .

  • A measurable function from some probability space to that is denoted by bold symbols . Here, and represent the network and the root respectively.

Note that the whole symbol represents one random object, which is a random equivalence class of rooted networks. Therefore, any formula using and should be well defined for equivalence classes of rooted networks; i.e. should be invariant under rooted isomorphisms. Moreover, bold symbols are used only in the random case.

The relation between the two representations is expressed by the equation for events ; i.e. is the distribution of the random object. These representations are mostly treated equally in this paper. Therefore, all definitions and results expressed for random rooted networks also make sense for probability measures on .

For a measurable function , a network and , let

where brackets are used as a short form of . Also, for , let

Definition 2

A random rooted network is unimodular if for all measurable functions ,

(1)

where the expectations may be finite or infinite. The term unimodular network is used as an abbreviation for unimodular random rooted network. A probability measure on is called unimodular when, by considering it as a random rooted network, one gets a unimodular network.

Remark 1

For a function as above, can be regarded as a function on (or a transport kernel on ) defined for all networks . One can interpret as the amount of mass that is transported from to . Using this intuition, (resp. ) can be seen as the amount of mass that goes out of (resp. comes into) and \eqrefeq:unimodular expresses some conservation of mass in expectation. It is referred to as the mass transport principle in the literature. With this analogy, a measurable function is also called an invariant transport kernel in this paper.

The invariant sigma-field on is the family of events in that are invariant under changing the root; i.e., events such that for every rooted network and every , if , then . Events in are also called invariant events here. A unimodular network is called extremal if any invariant event has probability 0 or 1.

A measurable function on is -measurable if and only if it doesn’t depend on the root; i.e. for every rooted network and every , one has . Also, if is a random rooted network, we say doesn’t depend on the root almost surely if almost surely, for all , one has .

The following definition is borrowed from [3].

Definition 3

A covariant subset (of the vertices) is a function which associates to each network a set such that is a well-defined and measurable function on . By an abuse of notation, we use the same symbols for the subnetwork induced by (i.e. the restriction of to ) for all networks . This is called a covariant subnetwork.

Note that in this definition, should be covariant under network isomorphisms, that is, for all isomorphisms , one should have . Moreover, For any event , is a covariant subset. This easily implies that covariant subsets are in one-to-one correspondence with measurable subsets of .

The following lemma is straightforward and its proof is skipped. See [3] or [1].

Lemma 1

Let be a unimodular network and be a covariant subset of the vertices. Then if and only if . Equivalently, a.s. if and only if a.s.

3 Shift-Coupling of Random Rooted Networks

In this section, different formulations of unrooted-equivalence are defined and the main theorem of this paper is presented, which studies the implications between these formulations. The proofs of most of the results are moved to Section 7 to help to focus on the main thread. The reader can either see the proofs first or proceed to the next results with no problem.

3.1 Definitions

The following definitions are needed for stating the main definition (Definition 7).

Definition 4

The projection defined by induces a sigma-field, namely , on as follows.

A random non-rooted network is a random element in, or a probability measure on , although it does not form a standard probability space (Proposition 1 below). If is a random rooted network with distribution , the symbol is used for its corresponding random non-rooted network whose distribution is . It can also be seen as a natural coupling of and .

Non-standardness of is stated in the following proposition. It is essentially an easy result in theory of smooth Borel equivalence relations. See the notes in Subsection 3.4.

Proposition 1

The measurable space is not a standard Borel space. More precisely, there is no metric on that makes it a Polish space whose Borel sigma-field is .

Due to non-standardness, several classical tools of probability theory may fail for random non-rooted networks; e.g. conditional expectation. However, it poses no problem for the arguments in this paper; e.g. equality of distributions, pull-back and push-forward of distributions, etc.

Note also that the map corresponds bijectively to the invariant sigma-field on . Therefore, probability measures on are in one-to-one correspondence with probability measures on .

Definition 5

Let be a probability measure on and be a measurable function. Assume . By biasing by we mean the following measure on .

The choice of the denominator ensures that the result is a probability measure. It is the unique probability measure on whose Radon-Nikodym derivative w.r.t. is proportional to . Biasing a probability measure on is defined similarly.

It can be seen that biasing by is equal to if and only if is essentially constant (w.r.t. ); i.e. for some constant one has , -a.s. Note that is not assumed to be equal to one. As an example, for an event , conditioning on is just biasing by the indicator function .

Lemma 2

By biasing the distribution of a random rooted network by a function , the distribution of becomes biased by , where the latter, which is -measurable, is considered as a function of with a slight abuse of notation (see Section 2).

This lemma is straightforward and we skip its proof.

Definition 6

Let be a (not necessarily unimodular) random rooted network and be a measurable function. Assume a.s. Conditioned on , choose a new root in with distribution ; i.e. consider the following probability measure on .

(2)

Any random rooted network with this distribution is called the root-change of by kernel .

Lemma 3

If is a root-change of , then is also a root-change of .

We are now ready to present the main definition.

Definition 7

Let and be (not necessarily unimodular) random rooted networks. The following conditions are different definitions for and to be unrooted-equivalent (or to have the same non-rooted networks).

(B)

The distribution of each one is obtained from the other by a biasing and then a root-change.

(R)

The distribution of is obtained from by a root-change.

(C)

There is a coupling of them (i.e. a probability measure on whose marginals are identical with the distributions of ’s) which is concentrated on the set of pairs of rooted networks with the same non-rooted networks; i.e. .

(D)

There is a random doubly-rooted network such that and have the same distributions as and respectively.

(F)

By forgetting the roots, the random non-rooted networks and have the same distribution on . Equivalently, the distributions of and agree on the invariant sigma-field .

As mentioned in the introduction, the definition with Condition 7 is used heuristically in some examples in the literature, some of which will be mentioned in Section 5.

3.2 Main Theorems

Here, we study the implications between the conditions in Definition 7. At first sight, Condition 7 may seem weaker than the other ones, because the other conditions assume the existence of a third object. But this is not the case as shown below.

Theorem 1 (Shift-Coupling)

Let and be (not necessarily unimodular) random rooted networks. Then, can be obtained from by a root-change if and only if their distributions agree on the invariant sigma-field. In other words, conditions 7 and 7 are equivalent.

Most results of this paper are based on the above Theorem. Also, the chosen name shift-coupling is justified in the notes in Subsection 3.4. This result is the main part in the following implications.

Theorem 2

Conditions 7, 7, 7 and 7 are equivalent and imply Condition 7.

It should be noted that Condition 7 does not imply the other conditions (see Remark 2 below). Theorem 2 allows us to define the following.

Definition 8

Under the assumptions of Definition 7, and are weakly unrooted-equivalent if Condition 7 holds and strongly unrooted-equivalent if the other equivalent conditions hold.

Remark 2

According to Condition 7, when two random rooted networks are weakly unrooted-equivalent, the distributions of the corresponding non-rooted networks may be different (but are always mutually absolutely continuous by Lemma 2). This difference can be seen clearly in Example 6.

3.3 Some Applications

The following propositions are presented here as corollaries of Theorem 1. More important applications of the theorem will be presented in the next sections.

Proposition 2

Let be a (not necessarily unimodular) random rooted network and be a covariant subset (Definition 3) such that . Denote by the random rooted network obtained by conditioning on . Then, the following are equivalent.

  1. can be obtained from by a root-change.

  2. is essentially constant.

{proof}

The distribution of is obtained from that of by biasing by the function . Lemma 2 implies that the distribution of is obtained from that of by biasing by (considered as a function on ).

First, assume the bias function is essentially constant. It follows that and are identically distributed; i.e. the distributions of and agree on the invariant sigma-field. Thus, Theorem 1 implies that can be obtained from by a root-change.

Conversely, assume can be obtained from by a root-change. Theorem 1 implies that their distributions agree on the invariant sigma-field. In other words, and have the same distribution. Since the former is obtained by biasing the latter by , it follows that the bias function is essentially constant and the claim is proved.

Proposition 3 (Extra Head Scheme)

Let be a unimodular graph. Add i.i.d. marks in to the vertices with Bernoulli distribution with parameter . If is infinite a.s. then there exists a root-change that when applied to , the result is the same (in distribution) as except that the mark of the root is forced to be 1.

The condition of being infinite is necessary in this proposition as explained in Remark 3. See also [5] for the precise definition of adding i.i.d. marks to the vertices. The name extra head scheme is borrowed from an analogous definition in [12] as will be explained in Section 8.

{proof}

[Proof of Proposition 3] Note that the desired random rooted network can be obtained by conditioning on , where denotes the marks of the vertices. Therefore, by Proposition 2, it is enough to prove that is essentially constant. Let be an invariant event. By Lemma 4 below, is -valued and does not depend on the root a.s. Therefore, conditioned on , is independent of any random variable including . Thus,

This equation for all implies that a.s. So a.s. and the claim is proved.

The following lemma is used in the proof of Proposition 3 and is interesting in its own. It is similar to the ergodicity of the Bernoulli point process on or the Poisson point process in (see Section 8).

Lemma 4

Let be a unimodular graph and be a random network obtained by adding i.i.d. marks to the vertices of . If is extremal and almost surely infinite, then so is . More generally, if is infinite a.s., then for any invariant event ,

(3)

and the left hand side does not depend on the root a.s.

Note that in the statement of the lemma, the natural coupling of and is considered to enable us to condition on . The proof is presented in Section 7.

Remark 3

The claims of Lemma 4 and Proposition 3 are false for any finite unimodular network. Note that in this case, conditioned on , with positive probability the marks of all vertices are 0. This contradicts \eqrefeq:ergodic:1. Also, the same property holds in any root-change of the network, contradicting the claim of Proposition 3.

3.4 Notes

The name shift-coupling for Theorem 1 is borrowed from the analogous result of [19]. This result studies when two random elements in a space equipped with some group action have a coupling such that the second one is obtained from the first by a shift corresponding to a random element of the group, called a shift-coupling in the literature. Here, instead of a group action, we have root-changes as in Condition 7 of Definition 7, which don’t form a group. In Section 7, a proof of Theorem 1 is presented by mimicking that of [19]. A second proof is also presented using the result of [19]. With this proof, one can generalize Theorem 1 to the context of Borel equivalence relations as follows.

The following definitions are borrowed from [6]. An equivalence relation on a Polish space is a countable Borel equivalence relation if when considered as a subset of , it is a Borel subset and each equivalence class is countable. The -invariant sigma-field on consists of Borel subsets of which are formed by unions of -equivalence classes. In the following result, a Borel automorphism is called -stabilizing if for each .

Theorem 1’

Let be a countable Borel equivalence relation on and and be random elements in . Then there exists a random -stabilizing Borel automorphism such that has the same distribution as if and only if the distributions of and agree on the -invariant sigma-field.

In fact, in the converse, can be chosen to be supported on countably many automorphisms. As mentioned above, the proof of this theorem is similar to one of the proofs given for Theorem 1 and is skipped here.

A Borel equivalence relation is smooth if the quotient space with the induced Borel structure is a standard Borel space. Therefore, Proposition 1 just claims that the equivalence relation on induced by (see the proof of Theorem 1) is not smooth, which is implied by Corollary 1.3 of [9]. A direct proof is also presented in Section 7.

4 The Unimodular Case and Balancing Transport Kernels

In this section, some applications of Theorem 1 are presented for the case of unimodular networks. The main results are theorems 3 and 4 whose proofs are postponed to the end of the section after presenting some minor results.

Theorem 3 (Uniqueness)

The distribution of a unimodular network is uniquely determined by its restriction to the invariant sigma-field (or equivalently, by the distribution of the non-rooted network ). In other words, if two unimodular networks are strongly unrooted-equivalent, then they are identically distributed.

Theorem 3 is a precise formulation of a comment in [1] saying that ‘intuitively, the distribution of the root is forced given the distribution of the unrooted network’. Note also that if we replace ‘strongly’ with ‘weakly’ in this theorem, the claim no longer holds. This case will be considered in Lemma 7 and Proposition 4 below.

Theorem 4 (Balancing Transport Kernel)

Let be a unimodular network and be measurable functions for . Assume a.s. Then, the following are equivalent.

  1. There is an invariant transport kernel that almost surely balances between the functions and on the vertices; i.e. a measurable function such that almost surely, and for all .

  2. One has

    (4)

Theorem 4 is analogous to similar results for stationary point processes and random measures ([12] and [14]). This analogy will be explained in Section 8.

Remark 4

A result similar to Proposition 2 holds with the assumptions of Theorem 4. For , consider biasing the distribution of a (not necessarily unimodular) random rooted network by a function . Then, the resulting random rooted networks are always weakly unrooted-equivalent, but this holds strongly if and only if the ratio is essentially constant. However, the existence of a balancing transport kernel as in Theorem 4 is only proved for the unimodular case.

Before proving the above theorems, we present some other minor results in the unimodular case.

Lemma 5

Let be a unimodular network and be an arbitrary random rooted network.

  1. If is a root-change of by kernel , then it can also be obtained by biasing by the function .

  2. If is weakly unrooted-equivalent to , then the distribution of is obtained from that of by only a biasing (i.e. is absolutely continuous w.r.t. the distribution of ).

{proof}\eqref

lemma:unimodularBias:1. Given an event , define By \eqrefeq:rootChange and unimodularity of , one gets

By letting , one gets . Therefore, the above equation means that the distribution of is obtained from that of by the desired biasing (see Definition 5).

\eqref

lemma:unimodularBias:2. By part \eqreflemma:unimodularBias:1 and Definition 8, is obtained by biasing by a composition of two biasings, say by functions and . It is easy to show that the result is just biasing by and the claim is proved.

Lemma 6

Let be a unimodular network and be a measurable function. Then, biasing by gives a unimodular probability measure if and only if doesn’t depend on the root a.s.

{proof}

Let be a random rooted network whose distribution is obtained by biasing that of by . Let . For a measurable function , one has

where . By unimodularity of , one obtains

On the other hand,

Therefore, is unimodular if and only if

(5)

First, suppose that almost surely, for all . This implies that \eqrefeq:lemma:unimodularBiasUni holds and thus, is unimodular. Conversely, assume is unimodular. By substitute with the positive and negative parts of respectively, \eqrefeq:lemma:unimodularBiasUni gives that almost surely, for all . So, the claim is proved.

Lemma 7

Let and be random rooted networks that are weakly unrooted-equivalent. If both are unimodular, then the distribution of can be obtained by biasing that of by a function that doesn’t depend on the root and is almost surely positive.

{proof}

Since both are unimodular, by lemmas 5 and 6, is obtained from by biasing by a function that doesn’t depend on the root. As a result, the distribution of is absolutely continuous w.r.t. that of . The same holds by swapping the roles of and . Therefore, the Radon-Nikodym derivative, which is proportional to , is positive almost surely. This proves the claim.

Proposition 4

Let and be random rooted networks which are weakly unrooted-equivalent. If at least one of them is an extremal unimodular network, then they are also strongly unrooted-equivalent.

{proof}

Assume is an extremal unimodular network. Lemma 5 implies that can be obtained from by biasing by a measurable function . Lemma 2 implies that the distribution of is obtained from that of by biasing by . On the other hand, since is extremal, the -measurable function is essentially constant. It follows that and have the same distribution, which shows that is strongly unrooted-equivalent to .

We are now ready to prove the main theorems of this section.

{proof}

[Proof of Theorem 3]

Let and be unimodular networks such that their distributions agree on the invariant sigma-field. Therefore, they are strongly unrooted-equivalent (Condition 7). The same holds weakly by Theorem 2. Thus, Lemma 7 implies that is obtained by biasing the distribution of by a measurable function that doesn’t depend on the root. It is enough to show that is essentially constant.

By Lemma 2, the distribution of is obtained from that of by biasing by . Since the latter distributions are equal by assumption, it follows that is essentially constant. On the other hand, since doesn’t depend on the root, it is -measurable and thus, a.s. It follows that the bias function is essentially constant. Therefore, and are identically distributed.

{proof}

[Proof of Theorem 4] \eqrefthm:balancing:1 \eqrefthm:balancing:2. Let be an invariant event. Define By the assumption, one gets that almost surely, and . By unimodularity, one gets

By considering this for all , one obtains \eqrefeq:thm:balancing0.

\eqref

thm:balancing:2 \eqrefthm:balancing:1. For , let be a random rooted network obtained by biasing by . Assumption \eqrefeq:thm:balancing0 and Lemma 2 imply that has the same distribution as . In other words, and are strongly unrooted-equivalent (Condition 7). By Theorem 1, can be obtained from by a root-change; i.e. there is a measurable function such that a.s. and

for any measurable function . Fix arbitrarily. By the definition of , one obtains

(6)

where the equation is used (which holds by \eqrefeq:thm:balancing0) to cancel out the denominators. Define an invariant transport kernel by By unimodularity, one has

So, \eqrefeq:thm:balancing1 implies that . Since this holds for any , it follows that a.s. On the other hand, by a.s., one gets that a.s. Therefore, Lemma 1 implies that the same holds for all vertices; i.e. almost surely, for all , one has and . So the theorem is proved.

5 Network Extension and Unimodularization

In this section, the method of network extension is introduced and the shift-coupling theorem is applied to it. This method unifies some of the examples in the literature to construct unimodular networks. First, in Subsection 5.1 we study unimodularizations of a random non-rooted network in general. Then, network extension is studied in Subsection 5.2.

5.1 Unimodularizations of a Non-Rooted Network

Definition 9

Let be a probability measure on (or similarly, on ). We say that a random rooted network is unrooted-equivalent to ,

  • strongly if the distribution of is identical to .

  • weakly if the distribution of and are mutually absolutely continuous.

If in addition is unimodular, we say it is a (weak or strong) unimodularization of and can be unimodularized.

Heuristically, unimodularization means to choose a random root for a given random non-rooted network to obtain a unimodular network.

To see why the weak sense is ever defined here, it will turn out that some well known examples in the literature are weak unimodularizations (see examples 7 and 8 of Subsection 5.2). Moreover, the notions of weak and strong here are analogous to the previous notions as described in the following lemma.

Lemma 8

In the case is the distribution of , where is a random rooted network, Definition 9 is reduced to Definition 8.

This lemma is straightforward and we skip its proof.

Proposition 5

Under the assumptions of Definition 9, if can be unimodularized (either weakly or strongly), then there is a unique strong unimodularization of .

{proof}

Suppose is a weak unimodularization of . Let be the Radon-Nikodym derivative of w.r.t. the distribution of . Let be the random rooted network obtained by biasing by . Lemma 6 implies that is unimodular. Lemma 2 implies that the distribution of is equal to , which means that is a strong unimodularization of . Now, Theorem 3 implies that this is the unique strong unimodularization of .

Note that some probability measures on (i.e. some random non-rooted networks) cannot be unimodularized; e.g. a deterministic semi-infinite path.

Example 1 (Planar Dual I)

Let be a unimodular plane graph (see Example 9.6 of [1] for how to regard a plane graph as a network and define its dual). With no need to select a vertex of the dual graph as a root, makes sense as a random non-rooted network. In [1], a unimodular network is constructed based on the dual graph, which in our language, is a weak unimodularization of . This construction will be discussed in Example 8.

Example 2 (Subnetwork)

Let be a unimodular network and be a covariant subnetwork (Definition 3). Assume is nonempty and connected a.s. Therefore, is a random non-rooted network; i.e. a random element in (note that doesn’t need to contain and no root is chosen for ). Condition on the event ; i.e. consider the probability measure on . Considering as a random rooted network, we claim that it is a weak unimodularization of (the distribution of) . Let be the distribution of the non-rooted network under . By taking conditional expectation w.r.t. , one obtains for any measurable function that Therefore,

In other words, is just biasing the distribution of by , where the latter is considered as a function on . Similar to Lemma 1, one can deduce from a.s. that a.s., thus, and the distribution of are mutually absolutely continuous. On the other hand, it is easy to use \eqrefeq:unimodular directly to see that is unimodular. Thus, is a weak unimodularization of (the distribution of) .

To obtain a strong unimodularization of , one can bias the distribution of by and then consider the subnetwork rooted at (see Lemma 2). Here, the denominator can be regarded as the sample intensity of , which is a random variable and a function of .

5.2 Unimodularizations of a Network Extension

In some examples in the literature, given a unimodular network , another (not necessarily unimodular) random rooted network is obtained by adding some vertices and edges to the original network, called an extension here (Definition 10). Then, by biasing the probability measure and changing the root, another unimodular network is constructed. In this subsection, first a general method is presented that covers such examples and helps to construct new unimodular networks. Then, using the previous theorems, it is shown that this method gives all unimodularizations of in the sense given in Subsection 5.1. A number of basic examples are provided as applications of the definitions and results, although the examples are not new.

The method presented here needs that the original network can be reconstructed from the extension, as explained in the following definition. In applications, to ensure the reconstruction is possible, one may add extra marks to the newly added vertices and edges (e.g. see Example 5). Nevertheless, after a new unimodular network is successfully constructed using the method, one may forget the extra marks and unimodularity will be preserved.

Definition 10

Let be a unimodular network. An extension of is a pair , where is a (not necessarily unimodular) random rooted network and is a covariant subnetwork with the conditions that a.s., is connected a.s. and has the same distribution as . It is called a proper extension if

(7)

where runs over all measurable functions . Here, is allowed to have a larger mark space than .

Note that by Lemma 1, is non-unimodular except when a.s. Note also that is not necessarily a function of ; i.e. the newly added vertices and edges might be random. Moreover, \eqrefeq:mtpOnS is stronger than unimodularity of (compare it with \eqrefeq:unimodular for ). We are interested in proper extensions only, since the results in this section only hold in the proper case. See the following simple examples for more clarification of \eqrefeq:mtpOnS.

Example 3

Let be the usual deterministic graph of . Let be the graph obtained by adding a new vertex for any even number and connecting it to the vertex . For the networks that have a unique bi-infinite path, let be the subnetwork that represents that path. Now, has the same distribution as , but \eqrefeq:mtpOnS does not hold (e.g. let be zero except when and ). So, is an improper extension of .

However, if one chooses uniformly at random in , then it can be seen that \eqrefeq:mtpOnS holds and is a proper extension of . Moreover, by choosing uniformly at random in , is unimodular and is a strong unimodularization of . (Definition 9).

Example 4

Let be a unimodular network and be a covariant subnetwork such that is nonempty and connected a.s. Let be the random rooted network obtained by conditioning on (see Example 2). It can be seen that is a proper extension of and by Example 2, is a weak unimodularization of . This holds strongly if and only if the sample intensity of in is essentially constant.

We are now ready to state the results of this section. All proofs are postponed to the end of the subsection.

Lemma 9

Let be a proper extension of a unimodular network. If can be unimodularized, then there is a unique strong unimodularization of and it can be obtained by applying a root-change to .

Theorem 5 provides a general method to construct unimodularizations of a given proper extension. Moreover, part \eqrefthm:extensionT:1 of the theorem gives a criteria for verifying existence or non-existence of a unimodularization.

Theorem 5 (Unimodularization of an Extension)

Let be a proper extension of a unimodular network. Assume is a measurable function such that is almost surely a Markovian transport kernel from to ; i.e. almost surely, for all , and on . Let . Then,

  1. can be unimodularized if and only if a.s.

  2. If , then the following probability measure gives a weak unimodularization of .

  3. If a.s., then the following probability measure gives the unique strong unimodularization of .

Remark 5

The probability measure (resp. ) in Theorem 5 can be described as biasing the distribution of by (resp. ) and then changing the root to a random vertex with distribution .

As an example, it can be seen that the unimodularization of Example 3 can be obtained by the constructions in Theorem 5. More examples are provided at the end of this section.

Corollary 1

Under the assumptions of Theorem 5, if for all invariant events , then and are equal and extremal. Moreover, has a unique weak unimodularization.

In Theorem 5, the function is assumed to be given. In the following theorem, the existence of is studied using Theorem 1.

Theorem 6 (Existence of )

Let be a proper extension of a unimodular network. If can be unimodularized, then

  1. There exists a function satisfying the assumptions in Theorem 5 such that