Connectivity in Random Forests and Credit Networks

Ashish Goel ∗ Sanjeev Khanna† Sharath Raghvendra ‡ Hongyang Zhang §

AbstractRecent work has highlighted credit networks as an effectivemechanism for modeling trust in a network: agents issuetheir own currency and trust each other for a certainamount of each other’s currency, allowing two nodes totransact if there is a chain of sufficient residual trust betweenthem. Under a natural model of repeated transactions, theprobability that two agents can successfully transact in acredit network (i.e. the liquidity between these two agents)is the same as the probability that they are connected toeach other in a uniformly random forest of the network.

Motivated by this connection, we define the RF-connectivity between a pair of nodes in a graph G as theprobability that the two nodes belong to the same connectedcomponent in a uniformly random forest of G. Our first re-sult is that for an arbitrary subset S of nodes in G, theaverage RF-connectivity between pairs of nodes in S is atleast 1− 2/h(GS), where h(GS) is the edge expansion of thesubgraph GS induced by S. Informally, this implies that awell-connected “community” of nodes S in a credit networkwill have high liquidity among themselves, regardless of thestructure of the remaining network. We extend this resultto show that in fact every node in S has good average RF-connectivity to other nodes in S whenever S has good edgeexpansion. We also show that our results are nearly tight byproving an upper bound on the liquidity of regular graphs.

For our motivating application, it is important that werelate the average RF-connectivity in S to the expansioninside S and not merely to expansion of G since we wouldlike to assert that a well-connected community has highliquidity even if the graph as a whole is not well-connected.This naturally leads to a monotonicity conjecture: the RF-connectivity of two nodes can not decrease when a new edgeis added to G. We show that the monotonicity conjecture isequivalent to showing negative correlation between inclusionof any two edges in a random forest, a long-standing openproblem. Our result about the average RF-connectivity ofnodes in S may be viewed as establishing a weak version ofthe monotonicity conjecture.

Keywords. Uniformly random forests, Liquidity in credit

networks, Edge expansion, Markov chains.

∗Department of Management Science and Engineering, Stan-

ford University. Email: ashishg@stanford.edu. Supported in

part by the DARPA GRAPHS program, via grant FA9550-12-1-0411, and NSF grant 0904325.†Department of Computer and Information Science, Uni-

versity of Pennsylvania, Philadelphia, PA 19104. Email:sanjeev@cis.upenn.edu. Supported in part by National Science

Foundation grants CCF-1116961 and IIS-1447470.‡Supported in part by the DARPA GRAPHS program, via

grant FA9550-12-1-0411.§Department of Computer Science, Stanford University.

Email: hongyz@stanford.edu. Supported by an Enlight Foun-dation Engineering Fellowship.

1 Overview

Given a multi-graph G = (V,E), a subset of edgesis said to be a forest of G if it does not contain anycycles. In this paper, we will study uniformly randomforests of G. Unlike random spanning trees [23, 7, 1,30], several simple questions regarding random forestsremain unresolved: Does there exist a polynomial timealgorithm to approximately sample a random forest(or equivalently, approximately count the number offorests) of a given graph [9, 24, 2]? Does the presence ofan edge in a random forest make the presence of anotheredge less likely [19, 29]? And of interest to us in thispaper: can we characterize the component sizes of arandom forest [22]?

In addition to being fundamental graph-theoreticobjects (forests are independent sets of the graphic ma-troid, and correspond to an important class of Tuttepolynomials [32]), random forests have many applica-tions, most notably in machine learning [6, 3]. How-ever, the motivating application for our work is creditnetworks, which have emerged as an effective mecha-nism for modeling trust in transaction-oriented socialnetworks [11, 13, 15]. In a credit network, every node(i.e. user) acts as a bank and prints its own currency.Further, a weighted edge (u, v) in the network withweight w implies that u has extended a credit line ofw currency units to v, i.e., u is willing to trust up to wunits of v’s currency, at which point this credit line issaturated. However, this saturation (in fact, any trans-fer of currency from v to u) results in an edge formingin the reverse direction, from v to u, since u now hassome currency from v which it can return to v in ex-change for some service. Payments are routed alongfeasible paths; if there is no way to route a paymentfrom a payee to a payer, the transaction fails. A centralquestion in credit networks is liquidity: what fractionof desired transactions between nodes u and v actuallysucceed? Under a natural model of transaction rates,this question, surprisingly, is equivalent to the followingquestion [11, 20]: what is the probability that the twonodes u and v belong to the same component in a ran-dom forest of the underlying graph? This connection ledto a characterization of the liquidity of credit networksthat are lines, trees, and complete graphs [11]. However,realistic social networks are very far from each of these

idealized models. Instead, there is considerable evidencethat real-life social networks often contain (either over-lapping, or in a core-whisker structure) sub-networksor communities that have high expansion [5, 21]. Thisleads us to ask the following questions:

If G has high edge-expansion, what is theprobability that two nodes u and v are in thesame component in a random forest of G? Anddo the same bounds hold if u and v belongto a subgraph of G with high edge-expansion,even though G itself may not have high edge-expansion?

In the rest of the paper, we will use the term “RF-connectivity of u and v” to refer to the probability thatu and v are in the same connected component in a(uniformly) random forest.

1.1 Our results Our main result is that for anarbitrary subset S of nodes in G, the average RF-connectivity between pairs of nodes in S is at least1 − 2/h(GS), where h(GS) is the edge expansion ofthe subgraph GS induced by S. Informally, this im-plies that a well-connected “community” of nodes S ina credit network will have high liquidity among them-selves, regardless of the structure of the remaining net-work. Two points about this result are worth noting.First, in our motivating application, it is important thatwe relate the average RF-connectivity in S to the expan-sion inside S and not merely to expansion of G sincewe would like to assert that a well-connected commu-nity has high liquidity even if the graph as a whole isnot well-connected. Second, since many random graphmodels (such as Erdos-Renyi and preferential attach-ment [4]) have high edge expansion, our result automat-ically applies to these models. For example, above theconnectivity threshold, the edge expansion of an n-nodeErdos-Renyi graph is Ω(log n), giving an average pair-wise liquidity of 1 − O(1/ log n); this resolves an openproblem from earlier work on liquidity [11].

In order to prove our main result, we use a simpleMarkov Chain whose state space is all forests of G.In this Markov chain, at each step we pick an edgeuniformly at random. If this edge is part of the currentstate, the chain transitions to the forest obtained bydeleting this edge. If this edge is not part of the currentforest, and adding it does not cause a cycle, the edge isadded to the forest. The stationary distribution of thiswalk is uniform; thus, to analyze the RF-connectivityof two nodes, it suffices to estimate the probabilitythat they are in the same component in this Markovchain. This walk is simpler than the more traditionalone used to generate random forests [24] which also

allows edges to be swapped. It is plausible that themore involved walk mixes faster. However, since we arenot interested in designing a sampling algorithm in thispaper, we choose to use the simpler walk which is easierto analyze. We essentially show that under this walk,we are very likely to have a giant component wheneverthe graph G has good expansion. We then extend thisanalysis to the case where only S has good expansionby partitioning the set of forests into classes, with eachclass corresponding to the set of edges outside S that arein the forest; this extension is non-trivial, and suggestsa natural monotonicity conjecture, described later.

We then establish a nearly matching upper-bound:the average RF-connectivity in a d-regular graph is atmost 1 − 1/(d + 1). Since the existence of d-regulargraphs with Ω(d) edge-expansion is well known [27],this yields an upper bound of 1 − Ω(1/h(GS)) on theaverage RF-connectivity for this graph family, whereS = V . Thus, our result is essentially tight, up toconstant factors in the lower order term 1/h(GS). Wealso extend our main result to show that in fact everynode in S has good average RF-connectivity to othernodes in S whenever S has good edge expansion, thoughwith a slightly weaker bound of 1−O(log n)/h(GS).

As mentioned before, it is important that we relatethe average RF-connectivity in S to the expansioninside S and not merely to expansion of G. Thisnaturally leads to a monotonicity conjecture: the RF-connectivity of two nodes can not decrease when a newedge is added to G. Our result about the average RF-connectivity of nodes in S may be viewed as establishinga weak version of the monotonicity conjecture. Weshow that the monotonicity conjecture is equivalent toshowing negative correlation between inclusion of anytwo edges in a random forest, a long-standing openproblem [19, 25, 17]. This is in sharp contrast to the caseof random spanning trees, where negative correlation iswell understood [30]. We show this equivalence via anargument that directly counts the number of forests, asopposed to the Markov Chain approach outlined earlier.

While we were motivated to study RF-connectivitybecause of its connections to liquidity in credit networks,it is worth pointing out that this is a fundamental ques-tion in its own right. For example, distributions of com-ponent sizes of forests of complete graphs have beenshown to possess interesting phase transitions [22]. Ourwork reinforces the importance of several open prob-lems relating to random forests, in particular negativecorrelation and efficient approximate sampling (whichin our application will allow an easy estimate of the liq-uidity between any two nodes). Further, it would beinteresting to use the new analysis tools developed inthis paper to study the question of strategic formation

of credit networks: how do nodes decide how much trustto extend to each other [10, 12]?

Organization: The rest of this paper is organized asfollows. In Section 2 we formalize the notion of RF-connectivity and its connection to liquidity in creditnetworks. We present in Section 3 our results on therelationship between RF-connectivity and expansion ina (sub)graph. In Section 4, we establish the equivalencebetween negative correlation property and the mono-tonicity conjecture.

2 Preliminaries

2.1 Forests and RF-connectivity Let G = (V,E)be an undirected multi-graph with n vertices and mlabeled edges. If there are multiple labeled edgesbetween two vertices, each of them is associated witha unique label and is a distinct element in E.

A forest F is a subset of E that does not induceany cycles and contains no multiple edges. Let F (G),or simply F if the graph G is clear from the context,denote the set of all forests of G. For 1 ≤ k ≤ n − 1,let Fk denote the set of forests that contain exactly kedges. Let C (F ) denote the set of components in F ;we represent each component by the subset of verticesit contains, instead of edges. For each vertex u ∈ V ,let Cu(F ) denote the component that contains u in F .We use F to denote a uniformly random forest of G, wedefine:

• Φu,v(G) = Pr(u ∈ Cv(F)) as the RF-connectivitybetween any two vertices u and v in G.

• ΦS(G) =∑u∈S

∑v∈S:v 6=u

Φu,v(G)|S|(|S|−1) as the average

RF-connectivity between any two vertices of S forany S ⊆ V .

• Φu,S(G) =∑v∈S\u

Φu,v(G)|S|−1 as the average RF-

connectivity between any vertex u of S and therest of vertices of S.

2.2 RF-connectivity and Liquidity in CreditNetworks In this section, we describe the connectionbetween credit networks and random forests. A creditnetwork G = (V, E ; c) is a directed graph over agentsin V. Edges in the network represent pairwise creditlimits between agents. A state s in the network is simplythe vector of credit capacities along all the edges in thenetwork. An edge (u, v) ∈ E with capacity cuv(s) > 0represents a credit line extended from u to v worthcuv(s) units in v’s currency. Assume that capacitiesare integral. Successful transactions between nodes ofthe network result in a change in s. We denote by σ theset of all states that G can be in.

A transaction is specified by a tuple 〈u, v〉, wherenode u ∈ V is the payer (buyer), node v ∈ V is the payee(seller). Given a state s, the transaction can go throughprovided there is a feasible path for one unit of creditflow from v to u: that is, a path P from v to u suchthat each edge on the path has capacity at least one.It is assumed that all currencies are the same. If thetransaction goes through, then for each edge (ui, ui+1)on P , the credit capacity on each goes down by onewhile the credit capacity on the edge (ui+1, ui) goes upby one. 1

Consider a repeated transaction model where thetransaction rates are given by an n × n matrix Λ. Theentry corresponding to ith row and jth column in Λgives the probability of i initiating a transaction withj. In this paper, we work under the assumption thatthe transaction rates matrix is symmetric. Then, theserepeated transactions will induce a set of equivalenceclasses S over σ, and a Markov Chain whose steadystate distribution is uniform over these equivalenceclasses. 2

Definition 2.1. (Liquidity [11]) Let u and v be twonodes in G and let C be a uniformly random equiv-alence class of S . The steady-state transaction suc-cess probability from u to v is defined to be Ψu,v(G) =Pr(〈u, v〉 goes through in C),

Note that G induces an undirected multi-graph G =(V,E) in a natural way: the vertices V are equal to Vand there are cuv(s) + cvu(s) labeled edges between uand v in E, where s is any state in σ. The edges arewell-defined since the sum of credit limits of u and v toeach other remain the same in any state of the network.

Proposition 2.1. Φu,v(G) = Ψu,v(G) for any twonodes u and v in V.

The proof of Proposition 2.1 is left to the Ap-pendix A. Note that Proposition 2.1 also impliesΦS(G) = ΨS(G) for any subset of vertices S ⊆ V.

2.3 Uniform Sampling of a Random ForestConsider a simple random walk M on F : Let F0 beany forest.

Fi+1 =

Fi ∪ e if Fi ∪ e ∈ FFi\e if e ∈ FiFi otherwise

1Dandekar et al. [10] characterizes when there are multiple

path from v to u, the choice of P is without loss of generality,by a path-independence property: More details can be found in

Appendix A.2We defer a formal treatment of state space and how they

induce the equivalence classes to Appendix A.

where e is a uniformly random edge from E.

Proposition 2.2. The stationary distribution of M isuniform over F .

For some other random walks that sample a uni-formly random forest, we refer the reader to [9].

3 Connectivity in a Random Forest and Edge-Expansion

In this section, we establish a connection betweenaverage RF-connectivity in a community and the edgeexpansion of the community. Recall that the edgeexpansion of an undirected graph G is defined to be

h(G) = minS⊆V : 0≤|S|≤n2

∂(S)

|S|

where ∂(S) is the number of edges between S and V \Sin E, and the minimum is over all nonempty subsetsof V with at most n

2 vertices. Let S be any subset ofvertices in G and GS be the subgraph of G induced by S.Let h(GS) denote the edge expansion of the subgraphGS .

Our first main result is that for any subset S ofvertices in the graph, the average RF-connectivity insidethe subset S is at least 1 − 2

h(GS) . Thus vertices

in a community S that are well-connected to eachother (i.e. have high edge-expansion in the subgraphinduced by them) have high average connectivity amongthemselves, regardless of the structure of the remaininggraph. We also show that this bound is essentiallytight by establishing that average RF-connectivity inany d-regular graph is at most 1 − 1/(d + 1). Oursecond main result is that not only is the average RF-connectivity large in a well-connected community S, butevery vertex in any such community S also has highaverage connectivity to other nodes in S. Specifically,we will show that each vertex in a community S hasaverage RF-connectivity at least 1− lnn+2

h(GS)+1 inside S.

3.1 Average RF-connectivity in a Community

Theorem 3.1. Given a multi-graph G = (V,E) and asubset of vertices S ⊆ V , the average RF-connectivitybetween pairs of nodes in S is at least 1− 2

h(GS) , where

h(GS) is the edge expansion of its induced subgraph GS.

The proof proceeds in two stages. First, we parti-tion F into classes according to their set of edges out-side GS in Lemma 3.1. Secondly, within each class offorests, we show that the average RF-connectivity is atleast 1− 2

h(GS) in Lemma 3.3, obtaining the desired re-

sult.

We start by defining some additional notations. LetES be the set of edges in GS , and let ES = E\ES bethe set of edges outside GS . We partition F by itsrestriction to ES . We say that a set of edges E′ inES is feasible if there exists a forest F ∈ F such thatE′ = F ∩ ES . Then for any feasible set E′, we defineFE′ to be the set of forests such that F ∩ ES = E′ forany F ∈ FE′ .

Consider the average connectivity of forests in FE′ .This is captured by the RF-connectivity of a vertex-weighted multi-graph G(E′). To define this graph, notethat the connectivity created by edges of E′ naturallyinduces an equivalence relation ∼ on S. For two verticesu, v in S, u ∼ v if and only if u are connected to v byE′. Let V ∗ denote the equivalence classes of ∼ and letβ : V ′ → N 3 denote the number of elements in eachequivalence class. Secondly, let I = (v, v′, κ) denotethe labeled edges in GS . we define E∗ = ([v], [v′], κ),where [·] is the equivalence class of a vertex under ∼.Let G(E′) = (V ∗, E∗).

We define φ(F ) =∑

T∈C(F )

(β(T )

2

)/(β(V ∗)

2

)to cap-

ture pairwise connectivity of F in this weighted graphG(E′). Lemma 3.1 shows that ΦS(G) can be redefinedin terms of φ(F ).

Lemma 3.1. Let E be the set of feasible subsets of ES.Then ΦS(G) =

∑E′∈E

E [φ(F)]·Pr(F ′∩ES = E′), where

F is a uniformly random forest of G(E′) and F ′ is auniformly random forest of G.

Proof. Note that

ΦS(G) =∑E′∈E

Pr(F ′ ∩ ES = E′)×∑u∈V

∑v∈V :v 6=u

Pr(u ∈ Cv(F ′) | F ′ ∩ ES = E′)

|S|(|S| − 1)

.

It suffices to show that the above sum over u and vis equal to E [φ(F)]. We define a mapping Γ : FE′ →F (G(E′)) as Γ(F ) = ([v], [v′], κ) : ∀ (v, v′, κ) ∈ F\E′,and prove that the mapping Γ is bijective:

• By the definition of G(E′), it is clear that Γ(F ) isin F (G(E′)), hence the mapping is valid.

• Two different forests from FE′ map to differentforests in F (G(E′)), because they differ over theset of edges ES .

3Unless specified otherwise, from here on, the vertex weightfunction is only assumed to take positive integer values.

Note that the mapping preserves forest connectiv-ity. That is,

∑T∈C(F )

(|T∩S|2

)(|S|2

) = φ(Γ(F )),

since any two vertices v and v′ in S are connected inF if and only if [v] = [v′] or [v] are connected to [v′] inΓ(F ). Therefore,

E

∑T∈C(F′′)

(|T∩S|2

)(|S|2

) = E [φ(Γ(F ′′))]

where F ′′ is a uniform sample from FE′ .It is clear that∑

u∈S∑v∈S:v 6=u

Pr(u∈Cv(F ) |F∩ES=E′)|S|(|S|−1)

= E[∑

T∈C(F′′)

(|T∩S|2 )(|S|2 )

]= E [φ(Γ(F ′′))].

Finally, we observe that E [φ(Γ(F ′′))] = E [φ(F)] sinceΓ(·) is a one-to-one mapping; thus completing the proof.

We now concentrate on bounding the average RF-connectivity of G(E′) for each E′ in E . The notion ofedge expansion function can be naturally extended tographs with vertex weights β(·): we define it to be theminimum of the quantity ∂(S)/β(S), taken over all setsS for which β(S) is at most half of the total weight. Wewill use the following easy but important monotonicityproperty of expansion.

Lemma 3.2. The edge expansion function is monotoneunder vertex contraction, that is, h(G(E′)) ≥ h(GS) forany E′ ∈ E .

Proof. It suffices to consider G(E′) obtained by con-tracting a pair of vertices v and v′ in G to v′′. Let ∂′(·)denote the number of edges between any subset of ver-tices in G(E′) and its complement. Let T be any subset

of vertices of G(E′) such that β(T ) ≤ |S|2 . If T does notcontain v′′, then it’s clear that

∂′(T )

β(T )=∂(T )

β(T )≥ h(GS).

Otherwise, T contains v′′; let T ′ = T ∪ v, v′\v′′.Then

∂′(T )

β(T )=∂′(T ′)

β(T )≥ h(GS),

since β(u′′) = β(u) + β(u′) and ∂′(T ′) = ∂(T ). Henceedge expansion of G(E′) is at least h(GS).

We are now ready to prove the following key lemma.

Lemma 3.3. Let G∗ = (V ∗, E∗;β(·)) be a multi-graphwith vertex weight β(·) and edge expansion h(G∗).Then, for a uniformly random forest F of G∗, we have

E[φ(F)] ≥ 1− 2h(G∗) .

We will prove Lemma 3.3 by analyzing the randomwalk M introduced in 2.3. Let n be the number ofvertices and m be the number of edges in G∗. LetF ∗ denote the set of forests in G∗. We define α(F ) =max

T∈C(F )β(T ) as the maximum weight of any tree in

the forest F . Then

Lemma 3.4. For 1 ≤ k ≤ n− 2,∑F∈F∗k

min(β(V ∗)

2, β(V ∗)−α(F ))·h(G∗) ≤ |F ∗k+1|·(k+1)

Proof. Consider the overall probability that any forestF with k+ 1 edges moves to any forest F ′ with k edgesin the random walk: it happens when any edge in F isdeleted after a transition. Since there are k + 1 suchedges, the probability of this event happening at stateF is k+1

m . Therefore,∑F∈F∗k+1

∑F ′∈F∗k

PF,F ′

|F ∗|=

∑F∈F∗k+1

k + 1

m · |F ∗|

where PF,F ′ is the probability F moves to F ′ inM and1/|F ∗| is the stationary probability of a uniformlyrandom forest.

Next consider the overall probability that any forestF with k edges moves to any forest F ′ with k+ 1 edges:it happens when any edge between two trees of F isadded after a transition in the random walk. For thenumber of such edges, consider two cases:

• if α(F ) ≤ β(V ∗)2 , then for each component C in

C (F ), the number of edges from C to the othercomponents is at least h(G∗) times the number ofvertices in C. Therefore, the total amount of edgesbetween any two trees is at least h(G∗) · β(V ∗)/2;

• if α(F ) > β(V ∗)2 , then it will be at least (β(V ∗) −

α(F ))h(G∗).

In summary, the total number of edges between anytwo trees in F is at least x = min(β(V ∗)/2, β(V ∗) −α(F )) · h(G∗). Hence with probability at least x/m,two trees merge together after one move from state F :∑

F∈F∗k

∑F ′∈F∗k+1

PF,F ′

|F ∗|≥

∑F∈F∗k

min(β(V ∗)2 , β(V ∗)− α(F )) · h(G∗)

m · |F ∗|

Since the transition matrix is symmetric (PF,F ′ =PF ′,F ), the Lemma is proved.

To bound the RF-connectivity, we only consider themaximum weighted component of a forest. Such anapproximation is able to capture most of the connectedpairs, as we will see in the following analysis. This isbecause we expect that there will be a giant componentof size n − Ω(n/h(G∗)) in a uniformly random forest.Hence it does not lose too much to ignore the rest ofcomponents in the forest.

Proof of Lemma 3.3: Note that

φ(F ) ≥ (α(F ))2 − β(V ∗)

(β(V ∗))2 − β(V ∗)

=1− (β(V ∗)− α(F ))(β(V ∗) + α(F ))

(β(V ∗))2 − β(V ∗)

≥1− 2(β(V ∗)− α(F ))

β(V ∗)− 1

On the other hand, φ(F ) ≥ 0 ≥ 1− β(V ∗)β(V ∗)−1 . Combined

together,

φ(F ) ≥ 1−2 min(β(V ∗)

2 , β(V ∗)− α(F ))

β(V ∗)− 1.

With this inequality,∑F∈F′

k

φ(F )

|F ′|

≥∑F∈F′

k

1

|F ′| −∑F∈F′

k

2 min(β(V∗)

2, β(V ∗)− α(F ))

(β(V ∗)− 1)|F ′|

≥∑F∈F′

k

1

|F ′| −∑

F∈F′k+1

2(k + 1)

h(G∗)(β(V ∗)− 1)|F ′|

≥∑F∈F′

k

1

|F ′| −∑

F∈F′k+1

2

h(G∗)|F ′|

where we use Lemma 3.4 in the second inequality andβ(V ∗) ≥ k + 2 in the third inequality. By summing upthis inequality from k = 0 to n − 2, and taking intoaccount that the right hand term is zero for k = n− 1,

E[φ(F)] =

n−1∑k=0

∑F∈F′

k

φ(F )

|F ′|

≥n−1∑k=0

∑F∈F′

k

1

|F ′|−n−1∑k=1

∑F∈Fk

2

h(G∗) · |F ′|

≥1− 2

h(G∗)

Proof of Theorem 3.1: By Lemma 3.3, for each E∗ ∈ Eand a uniformly ranfom forest F ofG(E∗), we know that

E[φ(F)] ≥ 1 − 2/h(G(E∗)). Since Lemma 3.2 impliesh(G(E∗)) ≥ h(GS), therefore E[φ(F)] ≥ 1 − 2/h(GS).This inequality, combined with Lemma 3.1, implies thatΦS(G) ≥ 1− 2/h(GS).

We note that the bound shown in Lemma 3.3 isasymptotically tight in the following sense. Thereexist d-regular graphs with edge expansion Ω(d) (takea random d-regular graph, for instance). Lemma 3.3asserts that average connectivity in such graphs is1−Θ(1/d). On the other hand, the lemma below showsthat in a d-regular graph, the average connectivity isbounded by 1− 1/(d+ 1).

Lemma 3.5. The average RF-connectivity in any d-regular graph G = (V,E) is at most d/(d+ 1).

Proof. Let there be n vertices in the graph and eachvertex’s weight is one. We will prove a stronger result:the average RF-connectivity of any vertex u is at mostd/(d+ 1).

Group F by their restrictions to G\u. Morespecifically, for any forest F ∈ F (G\u), let ψ(F ) ⊆F denote the set of forests whose restriction to G\uis equal to F . We claim that for a uniform sample Ffrom ψ(F ), E [ |Cu(F)| ] ≤ (n− 1)(d/(d+ 1)).

Let du(T ) denote the number of edges between uand any subset T of V \u. Then

E [ |Cu(F)| ] = E [∑

T∈C (F)

|T | · (1− 1

du(T ) + 1)]

≤ E

∑T∈C (F)

|T | · (1− 1

d+ 1)

= (n− 1) · d

d+ 1.

Finally, by linearity of expectations, for a uniformlyrandom forest F ′ of G, E [ |Cu(F ′)| ] ≤ (n−1)·d/(d+ 1).Hence the avrage RF-connectivity of u is at mostd/(d+ 1).

As another remark, the idea of Lemma 3.4 alsoleads to some other interesting consequences regardinguniformly random forests. Without repeating similararguments, we simply list the results here.

Theorem 3.2. Given a graph G = (V,E) with nvertices and a uniformly random forest F of G, then

1. Pr(α(F) ≤ n2 ) ≤ 2

h(G) ;

2. E[α(F)] ≥ n(1− 2h(G) ), which also implies that the

expected number of components is at most 2nh(G) +1.

3.2 Average RF-connectivity of Any Vertex ina Community We continue this section by establish-ing that not only average RF-connectivity is large whena community has good expansion but in fact everynode in the community has high RF-connectivity (al-beit slightly weaker than Theorem 3.1). To this end wewill use more information from structures of a tree.

Theorem 3.3. Given a multi-graph G = (V,E), asubset of vertices S ⊆ V , and a vertex u ∈ S, theaverage RF-connectivity between u and the rest of nodesin S is at least 1 − lnn+2

h(GS)+1 , where h(GS) is the edge

expansion of its induced subgraph GS.

The proof again proceeds in two stages. We firstclassify the set of forests based on how edges in ES helpconnect pairs of vertices in S. Then we bound the RF-connectivity of u for each class of forests, by examiningthe random walk M and tree structures.

We will keep using notations introduced at thebeginning Section 3.1. We overload the function φ(·)and define φu(F ) = β(Cu(F ))/β(V ∗).

Lemma 3.6. Let E be the set of feasible subsets of ES.Then Φu,S(G) =

∑E′∈E

E[φu(F)] · |FE′ |/|F |, where Fbe a uniformly random forest of G(E′).

Proof. Follows from the proof of Lemma 3.1.

Lemma 3.7. Let G∗ = (V ∗, E∗;β) be a vertex-weightedmulti-graph with edge expansion h(G∗). Then, for auniformly random forest F of G∗, E[φu(F)] ≥ 1 −

lnn+2h(G∗)+1 .

The proof of Lemma 3.7 turns out to be moreintricate than Lemma 3.3. When an edge is removedfrom the component of u, it’s not impossible that morethan half of the component get disconnected with u.However, it is a simple fact of spanning trees thatthere are at most k edge whose removal will reduce thecomponent size of u to below k, for k = 1 up to the sizeof this tree. With this information, we first bound theprobability that the weight of Cu(F) is at most β(V ∗)/2;

Lemma 3.8. Pr(β(Cu(F)) ≤ β(V ′)2 ) ≤ lnn+1

h(G′) .

The following two facts will be useful in the proofof Lemma 3.8.

Fact 3.1. Let xj and yj be two sequences of realnumbers with length k. Then, subject to the set ofconstraints

i∑j=1

jxj ≤i∑

j=1

yj

for any i = 1, . . . , k, the sum of xj is at most∑kj=1 yj/j.

Proof. Multiply the k-th constraint by 1/k on both sidesand the i-th constraint by 1/i(i + 1) as well, for alli = 1, . . . , k − 1. Then, if we sum up the k inequalitiestogether, it will give us the desired inequality.

It remains to verify if the coefficient of each variableequals the coefficient of the desired inequality. For xkand yk, it’s true. For xj , where j < k, its coefficient is

given by jk +

∑k−1i=j

ji(i+1) = 1. For yj , its coefficient is

given by 1k +

∑k−1i=j

1i(i+1) = 1

j .

Fact 3.2. Let T = (V,E;β(·)) be a vertex weightedtree of n vertices and let u be any vertex of the tree.An edge e is called k-bad if β(Cu(T\e)) ≤ k, where1 ≤ k ≤ n− 1. Then there are at most k k-bad edges inT .

Proof. We introduce some additional notation first. Letλ(T, k) denote the number of k-bad edges in T . Assumethat T is a rooted tree at u, wlog. If v is a neighbor ofu, then all of v’s descendants plus the root node (andthe edges connecting these nodes) are called a branch ofT . It’s clear that a branch is still a rooted tree at u.

We will prove by an induction on k. The basecase is easily verified. Assume that the fact holdsfor any cases when j < k. If u has one child, thenλ(T, k) = 1+λ(T ′, k−1) ≤ k, where T ′ is the minor of Tobtained by contracting u and u’s neighbor. Otherwiseif the root has at least two children, then there exists away to divide T into two trees T ′ and T ′′ rooted at u,each including at least one branch of u. Let w′ denotethe total vertex-weights of T ′. Then the total weightsof T ′′ is w′′ = β(V )− w′ + β(u).

Consider the number of k-bad edges in T ′′: it’s notzero only if w′ ≤ k, in which case there are at mostλ(T ′′, k−w′+1) ≤ k−w′+1 (by induction hypothesis)k-bad edges. Similarly, the number of k-bad edges inT ′ is not zero only if w′′ ≤ k, and can bounded byλ(T ′, k − w′′ + 1) ≤ k − w′′ + 1 (again by inductionhypothesis). Summing up these two arguments, itbecomes clear that λ(T, k) ≤ max(0, k − w′ + 1, k −w′′ + 1, 2k − w′ − w′′ + 2) ≤ k, therefore the inductionhypothesis is verified for the case of k.

Proof of Lemma 3.8: Let n be the number of verticesand m be the number of edges in G∗. Let F ∗ denote theset of forests in G∗. The lemma follows from summingup the following inequality over k = 1, . . . , n− 2:

Pr(F ∈ F ∗k ∧β(Cu(F))

β(V ∗)≤ 1

2)

≤ lnn+ 1

h(G∗)· Pr(F ∈ F ∗k+1)(3.1)

Now we focus on proving inequality (3.1) for each

k. For j = 1, . . . , bβ(V ∗)2 c, we define H (j) ⊆ F ∗k

as the family of forests such that β(Cu(F )) = j foreach F ∈ H (j). We then use the notation H (≤ i)to represent the union of H (j), from j = 1 up to i.It is clear that the LHS of inequality (3.1) is equal to

Pr(F ∈ H (≤ bβ(V ∗)2 c)). We’ll find a set of constraints

that govern these sets. Let i be any integer between 1

and bβ(V ∗)2 c.

Consider the event in M when an edge in u’scomponent is deleted in a forest F with k + 1 edges,and then moves to F ′ such that β(Cu(F ′)) ≤ i. ByFact 3.2, there’re at most min(|Cu(F )|−1, i) such edges.By aggregating all such events, we get∑

F∈F∗k+1

∑F ′∈H (≤i)

Pr(F = F )PF,F ′

≤∑

F∈F∗k+1

Pr(F = F ) · min(|Cu(F )| − 1, i)

m=⇒

∑F∈F∗

k+1

∑F ′∈H (≤i)

PF,F ′

≤∑

F∈F∗k+1

min(|Cu(F )| − 1, i)

m(3.2)

since Pr(F = F ) is the same for any forest F in G∗.From the perspective of a forest F ′, which has k

edges and such that β(Cu(F ′)) ≤ i, there are at leastβ(Cu(F ′))h(G∗) number of ways to add an edge into F ′

in the next random walk. Again by aggregating all suchevents, we get ∑

F ′∈H (≤i)

β(Cu(F ′))h(G∗)

m

≤∑

F ′∈H (≤i)

∑F∈F∗

k+1

PF ′,F =⇒

∑F ′∈H (≤i)

β(Cu(F ′))h(G∗)

≤∑

F∈F∗k+1

min(|Cu(F )| − 1, i)(3.3)

by applying inequality 3.2 and then note that PF,F ′ =PF ′,F for any pair of forests F and F ′.

Now for j = 1, . . . , bβ(V ∗)2 c, we define K (j) as the

family of forests in F ∗k+1 such that |Cu(F )| > j for eachF ∈ K (j). Then note that

∑F∈H (≤i)

β(Cu(F ))

|F ′|=

i∑j=1

Pr(F ∈H (j)) · j

and ∑F∈Fk+1

min(|Cu(F )| − 1, i)

|F ′|=

i∑j=1

Pr(F ∈ K (j))

by the definition of H (·).Here we could use Fact 3.1. Let

xj = Pr(F ∈H (j)) · h(G∗) and yj = Pr(F ∈ K (j))

note that yj = 0 when j ≥ n. It’s clear that inequality(3.3) shows that the two sequences xj and yj satisfythe set of constraints

i∑j=1

jxj ≤i∑

j=1

yj ,

for i = 1, . . . , bβ(V ∗)2 c. Therefore

b β(V∗)

2 c∑j=1

Pr(F ∈H (j))h(G∗)

≤b β(V

∗)2 c∑j=1

Pr(F ∈ K (j))

j

=

n∑i=2

Pr(F ∈ Fk+1 ∧ |Cu(F)| = i)

·

min(i−1,b β(V∗)

2 c)∑j=1

1

j

≤

n∑i=2

Pr(F ∈ Fk+1 ∧ |Cu(F)| = i)(1 + lnn)

≤ Pr(F ∈ Fk+1)(1 + lnn)

Hence (3.1) is proved.

Proof of Lemma 3.7: Let ∂u(F ) denote the amount ofedges between Cu(F ) and the rest of nodes in G∗. Notethat the expected number of transitions that adds oneedge to F is equal to the number of transitions thatdeletes one edge from F . This is because the Markovchain M is symmetric. Therefore

E [|Cu(F)| − 1] = E [∂u(F)]

≥E [min(β(Cu(F)), β(V ∗)− β(Cu(F)))] · h(G∗).

After rearranging the inequality, we get

E [φu(F)] ≥ 1−Pr(β(Cu(F))

β(V ∗) ≤ 12 ) · h(G∗) + 1

h(G∗) + 1.

And the proof is complete after applying Lemma 3.8to the above equation.

Proof of Theorem 3.3: Let E′ ∈ E and F be a uniformlyrandom forest of G(E′). By combining Lemma 3.7and Lemma 3.2, E[φu(F)] ≥ 1−(lnn+ 2)/(h(GS) + 1).This inequality, combined with Lemma 3.6, implies thatΦu,S(G) ≥ 1− 2

h(GS) .

We conclude this section with a conjecture on RF-connectivity of a vertex in the Erdos-Renyi graph. LetG(n, p) denote a random graph over n nodes where everyedge is present with probability p, and let V be thevertex set of G. We denote Φu,V (G) by simply Φu(G)below.

Conjecture 3.1. Consider a random graph G(n, p)where p > 2 lnn

n . There exists a constant c and a func-tion ε(n), such that Pr(Φu(G) ≥ 1 − c

np , ∀u in G) ≥1− ε(n), where ε(n) goes to 0, as n goes to infinity.

We believe that the key to resolving this conjec-ture is to better understand the structure of a uniformlyrandom spanning tree of G(n, p). We know that whenp > 2lnn/n, the diameter of a uniformly random span-ning tree of G is O

(√n log n

)[8, 18], with probability

tending to 1 as n becomes large. However, not muchimprovement can be achieved by using this informationalone in the proof of Lemma 3.7. In particular, any fur-ther improvements seem to require understanding thedistribution of degree 1 and 2 nodes in a random span-ning tree as well as the expected sizes of subtrees whena random edge is deleted. We would like to note thatthese questions are of independent interest [26].

4 Negative Correlation and Monotonicity ofConnectivity

We will now define negative correlation and monotonic-ity of connectivity property of graphs and show (proofsdeferred to the appendix) that negative correlation im-plies that connectivity is monotone. We begin by defin-ing the negative correlation property. Let F be a forestchosen uniformly at random from F . The probabilitythat X ⊆ E is present in F is given by PX(G). If FX

be the largest subset of F with the property that X isincluded in every forest of FX , then,

PX(G) =|FX ||F |

.

Negative correlation property: Given a graphG = (V,E), we say that the edges (u, v), (u′, v′) ∈ E arenegatively correlated if

P(u,v),(u′,v′)(G) ≤ P(u,v)(G) · P(u′,v′)(G).

or,

(4.4) P(u′,v′)(G | (u, v)) ≤ P(u′,v′)(G).

Here P(u′,v′)(G | (u, v)) is the probability of (u′, v′)being present given that the forest is chosen uniformlyat random from all forests that have the edge (u, v). Wesay that a graph has negative correlation property if allpairs of edges are negatively correlated.

Monotonicity of connectivity property. Givena graph G = (V,E), let (u, v) 6∈ E. Let G′ = (V,E′)be such that E′ ← E ∪ (u, v). For any pair ofvertices u, v ∈ V , we say that their RF-Connectivity ismonotonically non-decreasing if ∀(u′, v′) 6∈ E and E′ =E ∪ (u′, v′) with G′ = (V,E′), Φu,v(G

′) ≥ Φu,v(G).We say that a graph has the monotonicity propertyif every pair (u, v) ∈ V × V has monotonically non-decreasing RF-Connectivity.

Theorem 4.1. The negative correlation property holdsfor all graphs if and only if connectivity is monotone.

Let F (u, v) denote the largest subset of F with theproperty that u and v belong to the same component ofevery forest of F (u, v), then

Φu,v(G) =|F (u, v)||F |

.

The following lemma proves some useful relations:

Lemma 4.1. Let G = (V,E) be any graph and letG′ = (V,E′) be a graph with E′ ← E ∪ (u, v) where(u, v) 6∈ E. Then,

P(u,v)(G′) =

1− Φu,v(G)

2− Φu,v(G)

Φu,v(G′) =

1

2− Φu,v(G).

Proof. We use F ′ to denote the set of forests of G′.Recollect that Φu,v(G) is the fraction of forests of Gthat have u and v in the same component. Therefore,1 − Φu,v(G) is the fraction of forests that do not haveu and v in the same component. For every forest whichhas u and v in different component, we construct a newforest by adding the edge (u, v) to it . It is easy to seethat the resulting set of new forests is precisely F ′(u,v).

Therefore, F ′ can be simply obtained taking the unionof F with F ′(u,v). The new probability of connectivity,

Φu,v(G′) is given by

(4.5) Φu,v(G′) =

|F (u, v)|+ |F ′(u,v)||F |+ |F ′(u,v)|

=1

2− Φu,v(G).

The above equality is obtained by dividing the numera-

tor and denominator by F and the fact that|F ′(u,v)||F | =

1−Φu,v(G). Similarly, we also get the probability thatthe edge (u, v) is present in a random forest of G′ as

(4.6) P(u,v)(G′) =

|F ′(u,v)||F |+ |F ′(u,v)|

=1− Φu,v(G)

2− Φu,v(G)

Proof of Theorem 4.1:First, assuming negative correlation property, we

show that the all pairs of vertices have non-decreasingmonotonicity.

We begin by showing that all pairs that are non-adjacent in G i.e., all pairs (u, v) 6∈ E, have themonotonicity property. Next, we use an easy extensionto prove monotonicity for all adjacent pairs as well.

We will now show that monotonicity holds for apair of non-adjacent vertices. Let u, v be any pair ofnon-adjacent vertices, and let (u′, v′) be the edge thatis added to G to obtain G′. We augment the graph G′

by adding the edge (u′, v′) to it; let the resulting graphbe G′′. There are two cases:

1. First, the pair (u′, v′) is the same as the pair (u, v).In this case, we need to show that adding the edge(u, v) to G increases the connectivity of u and v.From equation (4.5), for all 0 ≤ Φu,v(G) ≤ 1, weget

Φu,v(G′) ≥ Φu,v(G).

Next, we prove the claim for all non-adjacent pairs(u, v) where (u, v) and (u′, v′) are distinct. Fromthe negative correlation in G′′, we have

P(u,v)(G′′ | (u′, v′)) ≤ P(u,v)(G

′′).

We will now show that Φu,v(G) ≤ Φu,v(G′), i.e.,

adding the edge (u′, v′) to G will only increase theRF-Connectivity of u and v.

From (4.6), we have

(4.7) P(u,v)(G′′) =

1− Φu,v(G′)

2− Φu,v(G′).

Next, suppose we restrict ourselves to all the forestsof G′′ that contain the edge (u′, v′)4. An analysissimilar to the proof of equation (4.6), we get

(4.8)

P(u,v)(G′′ | (u′, v′)) =

1− Φu,v(G′ | (u′, v′))

2− Φu,v(G′ | (u′, v′)).

4We extend all the definitions to this set. We use the notation

G|(u′, v′) to indicate that the definition is restricted to the set offorests containing the edge (u′, v′)

Applying (4.7) and (4.8) to (4.4) we get,

Φu,v(G′) ≤ Φu,v(G

′ | (u′, v′))|F (u, v)|+ |F ′(u′,v′)(u, v)|

|F |+ |F ′(u′,v′)|≤|F ′(u′,v′)(u, v)||F ′(u′,v′)|

Here, F ′(u′,v′)(u, v) is the set of forests of G′ that

contain the edge (u′, v′) and have u and v in thesame component. The previous equation can bere-written as,

|F (u, v)||F |

≤|F (u, v)|+ |F ′(u′,v′)(u, v)|

|F |+ |F ′(u′,v′)|

=⇒ Φu,v(G) ≤ Φu,v(G′)

Now we describe the case where u and v areadjacent in graph G.

2. For the case where (u, v) is an adjacent pair. Let Gbe the graph obtained by removing the edge (u, v).Let G′ be the graph after adding edge (u′, v′) to G.Since u and v are non-adjacent, from the proof ofcase(ii), we know that Φu,v(G) ≤ Φu,v(G

′). Next,

we add the edge (u, v) to both G and G′ to obtainG and G′. From equation (4.5), we get

Φu,v(G) =1

2− Φu,v(G)

Φu,v(G′) =

1

2− Φu,v(G′).

From the above equations and the fact thatΦu,v(G) ≤ Φu,v(G

′), we get Φu,v(G) ≤ Φu,v(G′).

This concludes the proof in one direction, i.e., givennegative correlation holds, we show that connectivityis non-decreasing. Now, we prove the other directionthat if connectivity is non-decreasing then the negativecorrelation property holds.

We prove this for any two edges (u, v) and (u′, v′)in G′′. Given (u, v) and (u′, v′), let G,G′ and G′′

as defined before. From the monotonicity property,Φu,v(G) ≤ Φu,v(G

′), we have

Φu,v(G′) ≤ Φu,v(G

′ | (u′, v′))

Note that G′′ is obtained by adding the edge (u, v) toG′. From (4.7) and (4.8), we get

(4.9) P(u,v)(G′′) ≥ P(u,v)(G

′′ | (u′, v′)).

But, P(u,v)(G′′ | (u′, v′)) =

P(u,v),(u′,v′)(G′′)

P(u′,v′)(G′′) . Plugging

this to (4.9), we get

P(u′,v′)(G′′)P(u,v)(G

′′) ≥ P(u,v),(u′,v′)(G′′)

f

e

Figure 1: Consider the set of forests with six edges.It can be verified that Pr(ef) = 5

24 > Pr(e)Pr(f) =724 ×

1724 .

implying negative correlation of (u, v) and (u′, v′).

Remark: We know that the negative correlationproperty does not extend to the truncation of the set offorests by a fixed number of edges (a counter exampleis already implicitly mentioned in [14]. We draw it inFigure 1 for the completeness of our paper). We alsoknow that the property holds when G is a completegraph [29] or a series-parallel graph [31]. And theconjecture has been verified when G has eight or fewervertices [17]. However, not much is known beyond that.

Acknowledgements

We are grateful to Chandra Chekuri and Pranav Dan-dekar for many useful discussions in the early stages ofthis work.

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A Omitted Details from Section 2.2

Let s, s′ ∈ σ be two states in a credit network G =(V, E ; c(·)). Let du(s) denote the indegree of a node u in

state s and ~d(s) denote the indegree sequence associatedwith s. Two sates s and s′ are said to be equivalentif and only if they correspond to the same indegreesequence.

Dandekar et al. (Lemma 2.2 in [10]) characterizesthat two states s and s′ are equivalent if and only ifthe network can be transformed from state s to states′ by routing transactions along feasible cycles. Thisobservation implies that the set of transactions thatgoes through in s is equal to the set of transactionsthat goes through in s′. It also leads naturally toa path-independence property (Theorem 2.4 in [10]):a successful payment routing from u to v along anypath leads to the same equivalence class in the Markovchain. These two properties ensure that the repeatedtransaction model reduces to a Markov chain over theset of indegree sequences S . Under a symmetrictransaction regime, the stationary distribution of thisMarkov chain is uniform over S (Corollary 2.10 in[10]). Therefore, if Su,v is the set of indegree sequencesin which transactions from u to v go through, then

Ψu,v(G) = |Su,v|/|S |.It is well-known that the number of feasible indegree

sequences of G is equal to the number of forests of G[20, 16, 28]. Moreover, the proofs from [20] implicitlyimplies Proposition 2.1. We present a brief proof herefor the completeness of this section.

Proof of Proposition 2.1: Let F (u, v) denote the set offorests in which u and v are connected. Slightly abusingthe notation, for any edge e ∈ E, we let Fe denote theset of forests that contain the edge e, and let Fe denotethe set of forests that do not contain the edge e. Wewill prove that |F (u, v)| = |Su,v|. Since |F | = |S |,therefore Φu,v(G) = Ψu,v(G). Consider two cases.

• If there exists a labeled edge e between u and vin G: Let S ′ = ~d(s) | ∀s ∈ σ s.t. s(e) = 〈v, u〉,where s(e) = 〈v, u〉 means that e is directed fromv to u in state s. It is not hard to see that theamount of indegree sequences in S ′ is equal tothe amount of forests in G\e. Next, for any

state s′ such that ~d(s′) ∈ S \S ′, by definition eis oriented from u to v. And by Lemma 2.2 in[10], there is no path from v to u in s′. In otherwords, a transaction 〈v, u〉 cannot go through in s′.Hence Su,v is equal to the set S ′. Finally, note thesimple fact |F (u, v)| = |F |−|Fe| = |Fe|, therefore|F (u, v)| = |Su,v|.

• If there does not exist any labeled edge between uand v in G: Think of adding an edge e between thetwo vertices in G and a unit of credit limit betweenu and v in G. Let G′ denote the new graph and G′denote the new network. Let S ′′ denote the set ofindegree sequences in G′ such that the transaction〈u, v〉 does not go through. We would like to showthat |Su,v| = |S | − |S ′′| and |S ′′| = |Fe(G

′)|.Combined with |F (u, v)| = |F | − |Fe(G

′)|, wewould get |F (u, v)| = |Su,v|. The arguments isquite similar to the first case, so we omit the detailshere.