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A Markov Random Field Approach to Multicast-Based Network Inference Problems Jian Ni and Sekhar Tatikonda Department of Electrical Engineering, Yale University, New Haven, CT 06520, USA Email:{jian.ni, sekhar.tatikonda}@yale.edu Abstract— In this paper, we provide a new unified approach to analyze and solve multicast-based network inference problems. We show that the outcome variables induced by the transmission of a multicast packet form a Markov random field on the multicast tree. We present an algorithm that recovers the multicast tree topology based on the values of an additive tree metric on pairs of the terminal nodes. We prove the correctness of the algorithm. We also give several examples of an additive tree metric for which the values on pairs of the terminal nodes can be estimated from traffic measurements taken at the receivers. In addition, we propose an algorithm to recover the link performance parameters from the joint distribution of the outcome variables at the terminal nodes.

I. I NTRODUCTION Network inference, also known as network tomography, involves estimating the network topology and performance parameters based on traffic measurements taken at a limited subset of the nodes in a network [3], [6]. As today’s communication networks (e.g., the Internet) grow in size and diversity, accurate and scalable network inference techniques will become increasingly important for many network control and management tasks, including traffic engineering, service provision, network failure and anomaly detection, etc. Three classes of network inference problems have been investigated recently: 1) link performance parameter estimation based on end-to-end traffic measurements [2], [5], [10], [13]; 2) network topology inference [7], [8]; 3) traffic matrix estimation based on link traffic measurements [9], [12]. In this paper we focus on the first two classes of network inference problems. Link performance parameters (e.g., link loss rates, delay distributions, bandwidth) can be estimated from direct measurements if all nodes in a network can cooperate. More realistically we need to consider and design inference techniques only requiring cooperation among a limited subset of the nodes in the network, for example, when traffic measurements can only be conducted by the end users. For link performance parameter estimation, one normally assumes that the network topology is known. Such knowledge, however, may not be readily available in practice. When the network topology is unknown, tools like traceroute could be used to perform topology inference. But these tools require close cooperation from network internal devices (e.g., routers), which may not be feasible in practice. Therefore, one may also need to infer the network topology based on traffic measurements taken at the boundary of the network. For both classes of network inference problems, there are two probing techniques one can apply: unicast or multicast.

Due to its effectiveness and efficiency, the use of end-toend multicast traffic measurements has been proposed to infer network link performance parameters such as packet loss rates, delay distributions, as well as the topology of the logical multicast tree [2], [8], [10]. Since multicast is not universally deployed, unicast-based network inference techniques have also been explored. However, unicast measurements are much more difficult to work with because unlike multicast measurements, the outcomes of the unicast packets at different receivers are normally not highly correlated. A clever idea to handle this is to use backto-back unicast packet pair (or packet sequence) to mimic the transmission of a multicast packet [5], [7], [13]. In this paper, we propose a Markov random field (MRF) approach to analyze and solve network inference problems based on multicast traffic measurements. The same methodology developed here can be applied with unicast packet-pair/sequence measurements. The rest of the paper is organized as follows. In Section II, we introduce the model and the network inference problems. In Sections III and IV, we discuss multicast tree topology inference and link performance parameter inference respectively. The paper is concluded in Section V. II. T HE M ODEL AND T HE I NFERENCE P ROBLEMS The physical multicast tree from a multicast source to a group of multicast receivers consists of actual network nodes (hosts, routers, etc.) and the physical communication links that join them. We can construct a logical multicast tree including the source, the receivers, and the branching nodes (nodes with two or more outgoing links) of the physical multicast tree [2], [8]. Therefore, each internal node on the logical multicast tree has two or more children (i.e., the degree of an internal node is at least three). A logical link comprises one or more physical links. An example is shown in Fig. 1. Let T = (V, E) denote a logical multicast tree, with node set V and link set E. Let r be the root (multicast source) and W be the set of leaves (multicast receivers). Let U = W ∪ r be the set of terminal nodes, which are nodes with degree one. Each node k ∈ V has a parent f (k) ∈ V such that (f (k), k) ∈ E, and a set of children c(k) = {j ∈ V : f (j) = k}, except that the root has no parent and the leaves have no children. For notational simplification, sometimes we use lk to denote link (f (k), k). Each link lk is associated with a performance parameter θk (either a scaler or a vector) that we want to estimate. For example, for link loss inference, θk could be the

Source

Theorem 1. Under the spatial independence assumption that the link states Sk ’s are independent from link to link, ∆ XV = (Xk : k ∈ V ) forms a Markov random field on T . Specifically, for each k ∈ V , the conditional distribution of Xk given other random variables (Xj : j 6= k) on T is the same as the conditional distribution of Xk given just its neighboring random variables (Xj : j ∈ f (k) ∪ c(k)) on T . Proof: For notational simplification, let p(xA ) denote P(Xk = xk : k ∈ A) for any subset A ⊂ V . First we prove by induction

Root

X

r

Router

1

X Router

X

Router

Receiver

2

3

X Receiver

4

Router

5

Receivers W={3,4,5} Receiver

(a) A physical multicast tree.

Fig. 1.

(b) The logical multicast tree.

success rate of lk (hence 1 − θk is the loss rate of lk ); for link delay distribution inference, the delay distribution of lk could be parameterized by θk . The multicast source will send a sequence of multicast packets (probes) to all the receivers. Our goal is to estimate the topology of the logical multicast tree and the performance parameters θk of all links based on traffic measurements taken at the receivers. Now we introduce a new unified framework to analyze and solve these inference problems. For a multicast packet sent by the source, we define a set of link state variables Sk for all lk ∈ E. Sk takes value in a state set S. The distribution of Sk is parameterized by θk , e.g., P(Sk = s) = θk (s) for s ∈ S. The transmission of the multicast packet will induce a set of outcome variables on T as follows. For each node k ∈ V , we associate it with a random variable Xk denoting the outcome of the packet at node k. Xk takes value in a outcome set X . When an internal node (router) receives the multicast packet, it will send a copy to all its children. We assume that the outcome of the packet at node k (i.e., Xk ) is determined by the outcome of the packet at f (k) (i.e., Xf (k) ) and the link state of lk (i.e., Sk ): (1)

Example 1: Link Loss Inference. In this case, the link state variable Sk is modelled as a Bernoulli random variable: Sk takes value 1 with probability θk if link lk is in ‘good’ state and the packet can go through it; Sk takes value 0 if link lk is in ‘bad’ state and the packet will be lost on it. The outcome variable Xk is also a Bernoulli random variable, which takes value 1 if the packet successfully reaches node k. It is clear that for link loss inference Xk = Xf (k) · Sk .

(2)

Example 2: Link Delay Inference. In this case, the link state variable Sk is a random variable denoting the (random) delay of link lk . For link delay variance inference, θk = var(Sk ); for link delay distribution inference, θk (s) = P(Sk = s), s ∈ S. The outcome variable Xk denotes the cumulative delay of the packet from the source to node k. For link delay inference Xk = Xf (k) + Sk .

(3)

p(xk |xf (k) ).

(4)

k∈V \r

A physical multicast tree and its associated logical multicast tree.

Xk = g(Xf (k) , Sk ).

Y

p(xV ) = p(xr )

(4) is clearly true for T with |V | = 1 or |V | = 2. Assume (4) is true for T with |V | ≤ n. Now consider T with |V | = n+1. Let u be a leaf of T , then by (1) and the spatial independence assumption we have p(xV ) =p(xu |xV \u )p(xV \u ) = p(g(xf (u) , su )|xV \u )p(xV \u )

(5)

=p(xu |xf (u) )p(xV \u ).

XV \u is defined on T 0 = (V \ u, E \ (f (u), u)), a tree with Q n nodes. By induction assumption, p(xV \u ) = p(xr ) k∈V \u\r p(xk |xf (k) ). Substituting it into (5) we have that (4) holds for T with |V | = n + 1. Then by induction (4) is true for any T . Now for any k ∈ V , from (4) we have

Y

p(xV ) =p(xk |xf (k) ) p(xV \k ) =

X

p(xj |xk ) · q(xV \k ),

j∈c(k)

Y

p(xk |xf (k) )

xk



p(xj |xk ) · q(xV \k ),

j∈c(k)

where q(xV \k ) is a function that does not depend on xk . Then p(xk |xV \k ) =

p(xV ) = p(xV \k )

=P

p(xk |xf (k) )

P  xk

xk

Q

Q

p(xf (k) )p(xk |xf (k) )

=p(xk |xf (k) S c(k) ).

j∈c(k)

p(xk |xf (k) )

p(xf (k) )p(xk |xf (k) )



Q

j∈c(k)

Q

p(xj |xk )

j∈c(k)

p(xj |xk )

j∈c(k)



p(xj |xk )



p(xj |xk )

Therefore XV is a Markov random field on T . 2 This completes the description of the probabilistic model for the transmission of a single multicast packet, with the outcome random field XV . In actual network inference problems, the source will multicast a sequence of n packets, and there are (i) (i) totally n outcomes xV = (xk : k ∈ V ), i = 1, 2, ..., n, one for each multicast packet. Under the temporal independence assumption that the packets sent by the source at different time (1) (n) epochs are independent, xV , ..., xV are independent. (i) (i) For the i-th multicast packet, only the outcome xW = (xk : k ∈ W ) at the receivers can be measured and observed. The network inference problem involves using the observations (1) (n) xW , ..., xW at the receivers to estimate (1) the topology of the logical multicast tree if the network topology is unknown; (2) the link performance parameters θk of all links.

r

W = W \ {u∗ , v ∗ }. V = V ∪ {u∗ , v ∗ }, E = E ∪ {(u∗ v ∗ , u∗ ), (u∗ v ∗ , v ∗ )}. 3.2 Find all w ∈ W such that d(r, u∗ w) = d(r, u∗ v ∗ ): W = W \ w, V = V ∪ w, E = E ∪ (u∗ v ∗ , w). 3.3 For each w ∈ W : d(w, u∗ v ∗ ) = d(w, u∗ ) − d(u∗ v ∗ , u∗ ). Compute d(r, wu∗ v ∗ ), d(wu∗ v ∗ , w), d(wu∗ v ∗ , u∗ v ∗ ) by (6). W = W ∪ u∗ v ∗ . 4. If |W | = 1, for the w ∈ W : V = V ∪ w, E = E ∪ (r, w). Otherwise, repeat Step 3. Output: The tree T = (V, E), and d(e) for all e ∈ E.

uv

u

Fig. 2.

v

The topology of a canonical tree with three terminal nodes.

III. M ULTICAST T REE T OPOLOGY I NFERENCE This multicast-based network inference model is similar to the evolutionary tree inference model studied in [1], [4]. A common assumption for both logical multicast trees and evolutionary trees is that the degree of a nonterminal node is at least three (which is a prerequisite for identifiability). We will assume this and call such a tree canonical.

d(r, uv) + d(uv, u) = d(r, u), d(r, uv) + d(uv, v) = d(r, v), d(uv, u) + d(uv, v) = d(u, v). (7)

A. Canonical Tree Topology Recovery Let T = (V, E) be a canonical tree. For any two nodes k, j on T , let path(k, j) denote the unique path consisting of a sequence of links in E that connects k and j. We say that d is an additive tree metric on T if (a) 0 < d(e) < ∞, all e ∈ E; (b) d(k, j) =

X

d(e), all k, j ∈ V.

e∈path(k,j)

It is clear that an additive tree metric d is uniquely determined by the link lengths d(e) of all links e ∈ E. Let U be the set of terminal nodes. The following theorem, proved in [1], is a key result for canonical tree topology recovery. Theorem 2. The values of an additive tree metric on pairs of the terminal nodes specifies a unique tree. That is, if d is an additive tree metric on T , then the topology of T is uniquely determined by the values of d(u, v) for all u, v in U . Let d be an additive tree metric on T . We now present an algorithm that recovers the topology of the tree as well as the link lengths based on the values of d on pairs of nodes in U . To avoid trivial cases, we assume |U | ≥ 3. Algorithm 1 Input: Terminal node set U and d(u, v) for all u, v ∈ U . 1. Select a node r ∈ U as the root. Let W = U \ r, V = {r}, E = ∅. 2. For every pair of nodes u, v ∈ W , let uv denote their nearest common ancestor (i.e., the ancestor of both u and v that is farthest from r). Compute: d(r, uv)

=

d(uv, u)

=

d(uv, v)

=

d(r, u) + d(r, v) − d(u, v) 2 d(u, v) + d(r, u) − d(r, v) 2 d(u, v) + d(r, v) − d(r, u) 2

Theorem 3. Algorithm 1 correctly recovers the tree topology and the link lengths d(e) of all e ∈ E. Proof: We prove this theorem by induction on the cardinality of U . (1) |U | = 3, i.e., if there are three terminal nodes, then the only possible topology of a canonical tree is shown in Fig. 2. It is easy to verify that Algorithm 1 returns the correct topology. In addition, since d is an additive tree metric, we have

(6)

3.1 Find u∗ , v ∗ ∈ W with the largest d(r, uv) (break the tie arbitrarily).

Solving (7) we obtain the correct link lengths as computed in (6). Hence Algorithm 1 is correct when |U | = 3. (2) Assume that Algorithm 1 is correct for a tree with |U | ≤ n. Now consider a tree with |U | = n + 1. As in Algorithm 1, for any u, v ∈ W , let uv be their nearest common ancestor. Claim 1: u∗ , v ∗ found in Step 3.1 are siblings. If u∗ , v ∗ are not siblings, then either f (u∗ ) or f (v ∗ ) (or both) is descended from u∗ v ∗ . Without loss of generality, assume that f (u∗ ) is descended from u∗ v ∗ . Since the tree is canonical, u∗ must have at least one sibling u0 which is a child of f (u∗ ). If u0 ∈ W , d(r, u∗ u0 ) = d(r, f (u∗ )) > d(r, u∗ v ∗ ) since f (u∗ ) is descended from u∗ v ∗ and the link lengths are positive, a contradiction to the maximality of d(r, u∗ v ∗ ). If u0 ∈ / W , then u0 must have at least two descendants 0 w, w ∈ W such that d(r, ww0 ) > d(r, u∗ u0 ) > d(r, u∗ v ∗ ), again a contradiction. Claim 2: w found in Step 3.2 is a sibling of u∗ and v ∗ . If w is not a sibling of u∗ , then either 1) f (u∗ ) is descended from u∗ w, which implies d(r, u∗ w) < d(r, f (u∗ )) = d(r, u∗ v ∗ ), a contradiction; or 2) f (w) is descended from u∗ w, let w0 be the sibling of w: if w0 ∈ W , then d(r, ww0 ) = d(r, f (w)) > d(r, u∗ w) = d(r, u∗ v ∗ ), a contradiction; otherwise w0 must have at least two descendants z, z 0 ∈ W such that d(r, zz 0 ) > d(r, ww0 ) > d(r, u∗ v ∗ ), again a contradiction. From Claims 1 and 2, we know that after the first run of Step 3, Algorithm 1 correctly finds out a pair of siblings u∗ , v ∗ , and all their other siblings (if any). In addition, based on (7), the algorithm returns the correct link lengths as computed in (6). Then |W | is decreased at least by 1, this implies that the cardinality of the terminal node set of the remaining part of the tree (which is also a tree) is at most n. Hence by induction assumption, Algorithm 1 will correctly recover the remaining part of the tree and the corresponding link lengths. This completes our proof of Theorem 3. 2

For an additive tree metric d on T , let dU = (d(u, v) : u, v ∈ U ) denote the values of d on pairs of the terminal nodes. Algorithm 1 and Theorem 3 establish a one-to-one mapping between (T , d) and (U, dU ), hence we provide an alternative algorithmic proof to Theorem 2. B. Multicast Tree Topology Inference For (logical) multicast tree topology recovery, the natural choice of the root r in Algorithm 1 is the multicast source. The key thing is to construct an additive tree metric. Let XV = (Xk : k ∈ V ) be an outcome Markov random field on T = (V, E). For each link (f (k), k) ∈ E we can define an M × M (assume |X | = M ) forward link transition matrix Pf (k)k and an M ×M backward link transition matrix Pkf (k) with entries Pij (x, y)

=

P(Xj = y|Xi = x), x, y ∈ X .

(8) ∆

If the link transition matrices are invertible so that |Pij | = | det(Pij )| > 0, not equal to a permutation matrix (a matrix with exactly one entry in each row and column being 1) so that |Pij | < 1, and there exists a node k ∈ V with positive marginal distribution, then we can verify that d0 defined as follows is an additive tree metric [4]: d0 (lk ) = − log |Pf (k)k | − log |Pkf (k) |, lk = (f (k), k) ∈ E.

For any pair of nodes u, v ∈ U , d0U can be calculated by d0U (u, v) = − log |Puv | − log |Pvu |, u, v ∈ U.

(9)

There are other choices of an additive tree metric for the specific network inference problem. For Example 1 (link loss inference) in Section II, if 0 < θk < 1 for all links, then we can construct an additive tree metric d1 by defining the link lengths as follows: d1 (lk ) = − log θk ,

lk ∈ E.

(10)

From (2) and let Xr = 1 we have d1 (r, u) = d1 (r, uv) + d1 (uv, u) = − log P(Xu = 1), d1 (r, v) = d1 (r, uv) + d1 (uv, v) = − log P(Xv = 1), d1 (r, uv) + d1 (uv, u) + d1 (uv, v) = − log P(Xu = 1, Xv = 1).

Hence d1U on pairs of the terminal nodes can be calculated by d1U (u, v) = log

P(Xu = 1)P(Xv = 1) , u, v ∈ U. P2 (Xu = 1, Xv = 1)

(11)

For Example 2 (link delay inference) in Section II, if 0 < var(Sk ) < ∞ for all links, then we can construct an additive tree metric d2 by defining the link lengths as follows: d2 (lk ) = var(Sk ),

lk ∈ E.

(12)

From (3) and let Xr = 0 we have d2 (r, u) = var(Xu ), d2 (r, v) = var(Xv ), d2 (r, uv) = cov(Xu , Xv ).

Hence d2U on pairs of the terminal nodes can be calculated by d2U (u, v) = var(Xu ) + var(Xv ) − 2cov(Xu , Xv ), u, v ∈ U. (13)

From (9), (11), (13), if we know the pairwise joint distributions of the outcome variables at the terminal nodes, then we know d0U , d1U , d2U . And we can apply Algorithm 1 to recover the topology of the multicast tree correctly. In actual network inference problems, we are not given such (1) (n) distributions, what we have are some data, i.e., xW , ..., xW , the measured outcomes at the receivers. We can use the data to estimate the distributions (e.g., using empirical distributions), calculate the estimated additive tree metric on the terminal nodes, and then apply Algorithm 1 to infer the topology of the logical multicast tree. Algorithm 1 is similar to the grouping algorithms presented in [8], [11]. Here is a short summary of the differences and advantages of Algorithm 1: 1. Algorithm 1 is general and provides a guideline for multicast tree topology recovery: what we need is to construct an additive tree metric with the values on pairs of the terminal nodes can be measured or estimated. 2. Algorithm 1 is more computationally efficient because a) it only needs pairwise joint distributions of the outcome variables at the terminal nodes in order to construct an additive tree metric, as in (9), (11), (13); b) for an additive tree metric, the link lengths can be computed explicitly and simply as in (6). For a multicast tree with N receivers, the computational complexity of Algorithm 1 is O(N 2 ). On the contrary, the grouping algorithms in [8], [11] have an O(N 2 ) complexity only for binary trees. For general trees one needs to search among all subsets of the terminal nodes (# of searches is on the order of 2N ), and numerical root finding procedure is needed when the number of leaves is greater than five [8]. IV. L INK P ERFORMANCE PARAMETER I NFERENCE If there is a one-to-one mapping between the link performance parameters and the link lengths under a certain additive tree metric, then since Algorithm 1 also returns the link lengths, we can use the link lengths to recover the link performance parameters. This is true for link loss inference (see (10)) and link delay variance inference (see (12)). More generally if the link performance parameters are vectors, then they may not be uniquely determined by the link lengths under any additive tree metric. Nevertheless, we can recover the link performance parameters from the forward link transition matrices, since the parameters are just entries of the matrices. For example, for link delay distribution inference, Pf (k)k (x, y) = P(Xk = y|Xf (k) = x) = P(Sk = y−x) = θk (y−x)

for x, y ∈ X , y − x ∈ S. Under mild conditions, for a Markov random field defined on a canonical tree, the link transition matrices are uniquely determined by the joint distributions of the variables at the terminal nodes [4]. Note that we only need the forward link transition matrices Pf (k)k of all links in order to recover the link performance parameters. We design the following algorithm to recover the forward link transition matrices. Assume that the topology of the logical multicast tree T is known. If not, then we can apply Algorithm 1 to recover the

topology first. Let r be the root of T , and W = U \ r be the set of leaves. To avoid trivial cases, we assume |W | ≥ 2 (i.e., |U | ≥ 3). Let p(xu , xv |xr ) denote P(Xu = xu , Xv = xv |Xr = xr ). Algorithm 2 Input: T = (V, E) and p(xu , xv |xr ) for all u, v ∈ W . 1. Pick a pair of siblings u, v ∈ W . Let m = f (u). Without loss of generality (see [4]), we can assume that Pmv has one column Pmv (:, x ¯v ) with distinct entries. ∆ x ¯v Let Pru (xr , xu ) = p(¯ xv , xu |xr ). Compute Pmu from the −1 x ¯v eigenvalue-eigenvector decomposition of Pru Pru : −1 −1 x ¯v Pmu diag(Pmv (:, x ¯v ))Pmu = Pru Pru .

(14)

−1 −1 2. Prm = Pru Pmu , Pmv = Prm Prv . W = W \ {u, v}. −1 3. For all w ∈ c(m), w 6= u, v: Pmw = Prm Prw . W = W \ w. 4. If |W | = 0, Stop. Else, for all w ∈ W : ∆ r let Qxwm (xw , xm ) = p(xw , xm |xr ), compute −1 r r Qxwm = Qxwu Pmu , for all xr ∈ X .

(15)

W = W ∪ m, goto Step 1. Output: Pf (k)k of all (f (k), k) ∈ E.

Theorem 4. If the forward link transition matrices are invertible, not equal to a permutation matrix, reconstructible from rows (i.e., the matrix can be uniquely determined by its unordered set of rows); and if there is a node with positive marginal distribution, then Algorithm 2 correctly recovers the forward link transition matrices. The proof of Theorem 4 follows by a similar argument as in [4], and is omitted here due to space constraint. Note that all operations in Algorithm 2 are purely algebraic (matrix operations), so the algorithm is computationally efficient and can be applied to large-scale problems. In addition, the recursive nature of Algorithm 2 makes it amenable to distributed implementation. In order to recover the forward link transition matrices, we only need to know the conditional distributions p(xu , xv |xr ) for u, v ∈ W . In actual network inference problems, those distributions are not given. We can estimate them from traffic measurements collected at the receivers. Once the forward link transition matrices are recovered/estimated, we can use them to derive the link performance parameters of all links. Example 2 (Continued): For link delay distribution inference, as in [10], let X = S = {0, t, 2t, ..., M t, ∞} denote the set of possible link delays and cumulative delays, where ∞ means “packet lost” or “delay greater than M t.” For link lk , θk (i) = P(Sk = it), i = 0, 1, ..., M, ∞. The forward link transition matrix Pf (k)k is an (M + 2) × (M + 2) matrix with entries:

8 > > > < Pf (k)k (i, j) = > > > :

0, θk (j − i),

PM

c=M −i+1

1,

θk (c) + θk (∞),

j < i; j ≥ i, j 6= ∞; j > i, j = ∞; i = ∞, j = ∞.

We can verify that if θk is a positive vector (i.e., θk (i) > 0 for all i) for all links, then the forward link transition matrices satisfy the assumptions of Theorem 4 (invertible, not equal to a permutation matrix, reconstructible from rows). In addition, for any lk the matrix Pf (k)k has one column Pf (k)k (:, M ) with distinct entries (and these entries uniquely determine the link performance parameter θk ). Hence we can apply Algorithm 2 to recover the link delay distributions efficiently. V. C ONCLUSION In this paper, we proposed a Markov random field approach to analyze and solve network inference problems based on multicast traffic measurements. We presented an algorithm that recovers the multicast tree topology based on the values of an additive tree metric on pairs of the terminal nodes. We proved the correctness of the algorithm. We also gave several examples of an additive tree metric for which the values on pairs of the terminal nodes can be estimated from traffic measurements taken at the receivers. In addition, we proposed an algorithm to recover the link performance parameters when the parameters cannot be uniquely determined by the link lengths under any additive tree metric. In the future, we will extend the methodology developed in this paper to network inference problems with unicast packetpair/sequence measurements. We will also consider distributed implementation of the proposed network inference algorithms. R EFERENCES [1] P. Buneman, “The Recovery of Trees from Measures of Dissimilarity,” Mathematics in the Archaeological and Historical Sciences, Edinburgh University Press, pp. 387-395, 1971. [2] R. Caceres, N. G. Duffield, J. Horowitz, D. Towsley, “Multicast-Based Inference of Network-Internal Loss Characteristics,” IEEE Transactions on Information Theory, vol. 45, no. 7, pp. 2462-2480, Nov. 1999. [3] R. Castro, M. Coates, G. Liang, R. Nowak, B. Yu, “Network Tomography: Recent Developments,” Statistical Science, vol. 19, no. 3, pp. 499-517, 2004. [4] J. T. Chang, “Full Reconstruction of Markov Models on Evolutionary Trees: Identifiability and Consistency,” Mathematical Biosciences, vol. 137, pp. 51-73, 1996. [5] M. Coates and R. Nowak, “Network Loss Inference using Unicast Endto-End Measurement,” Proceedings of ITC Conference on IP Traffic, Modelling and Management, Monterey, CA, Sep. 2000. [6] M. Coates, A. O. Hero III, R. Nowak, B. Yu, “Internet Tomography,” IEEE Signal Processing Magazine, vol. 19, no. 3, pp. 47-65, May 2002. [7] M. Coates, R. Castro, M. Gadhiok, R. King, Y. Tsang, R. Nowak, “Maximum Likelihood Network Topology Identification from EdgeBased Unicast Measurements,” Proc. ACM Sigmetrics 2002, Jun. 2002. [8] N. G. Duffield, J. Horowitz, F. Lo Presti, D. Towsley, “Multicast Topology Inference From Measured End-to-End Loss,” IEEE Transactions on Information Theory, vol. 48, no. 1, pp. 26-45, Jan. 2002. [9] A. Medina, N. Taft, K. Salamatian, S. Bhattacharyya, C. Diot, “Traffic Matrix Estimation: Existing Techniques and New Directions,” Proc. ACM SIGCOMM 2002, Aug. 2002. [10] F. L. Presti, N. G. Duffield, J. Horowitz, D. Towsley, “MulticastBased Inference of Network-Internal Delay Distributions,” IEEE/ACM Transactions on Networking, vol. 10, no. 6, pp. 761-775, Dec. 2002. [11] S. Ratnasamy and S. McCanne, “Inference of Multicast Routing Trees and Bottleneck Bandwidths using End-to-end Measurements,” Proc. IEEE INFOCOM 1999, Mar. 1999. [12] Y. Vardi, “Network Tomography: Estimating Source-Destination Traffic Intensities from Link Data,” Journal of the American Statistical Association, vol. 91, no. 433, pp. 365-377, 1996. [13] Y. Tsang, M. Coates, R. Nowak, “Network Delay Tomography,” IEEE Transactions on Signal Processing, Special Issue on Signal Processing in Networking, vol. 51, no. 8, pp. 2125-2136, Aug. 2003.