ISSN: 1083-589X in PROBABILITY
Rumor source detection for rumor spreading
on random increasing trees
Michael Fuchs
*Pei-Duo Yu
*Abstract
In a recent paper, Shah and Zaman proposed the rumor center as an effective rumor source estimator for rumor spreading on random graphs. They proved for a very general random tree model that the detection probability remains positive as the number of nodes to which the rumor has spread tends to infinity. Moreover, they derived explicit asymptotic formulas for the detection probability of randomd-regular trees and random geometric trees. In this paper, we derive asymptotic formulas for the detection probability of grown simple families of random increasing trees. These families of random trees contain important random tree models as special cases, e.g., binary search trees, recursive trees and plane-oriented recursive trees. Our results show that the detection probability varies from0to1across these families. Moreover, a brief discussion of the rumor center for unordered trees is given as well.
Keywords: Rumor spreading; rumor center; detection probability; random increasing trees. AMS MSC 2010: 05C80; 60C05.
Submitted to ECP on August 17, 2014, final version accepted on November 30, 2014.
1
Introduction and Results
Rumor spreading on random trees has a long history in the biology, computer science and probability literature and has been investigated from many different angles. In a recent paper, Shah and Zaman [10, 11] added a new angle by putting forth the rumor source detection problem which asks for the correct identification of the rumor source when only information about the underlying model and the infected nodes is known. In [10, 11], this problem was discussed for randomd-regular trees and random geometric trees. Then, in [12], the authors generalized their approach to obtain results for very general families of random trees. Their studies, even though all of them very recent, have attracted a lot of attention and have led to many follow-up works (e.g., according to a google scholar search from August 14, 2014, the number of citations of the paper [11] had already reached 80).
From now on, we assume that some random tree model is fixed. After some time has elapsed, the rumor has spread tonnodes which form a treeΓ. The main idea in [10, 11] was to assign a score to the nodes ofΓ. The so-called rumor center is then the node which receives the highest score (where ties are either ignored or broken uniformly at random). In [10, 11], the authors showed that the rumor source estimator obtained in this way is the maximum likelihood (ML) estimator if the underlying random tree model are randomd-regular trees. However, for most other random tree models, the rumor
*Department of Applied Mathematics, National Chiao Tung University, Taiwan. E-mail: [email protected]
source estimator is not the ML estimator. Nevertheless, it was shown in [12] that for very general families of random trees, the rumor source estimator is still effective in the sense that the detection probability tends to a positive value as the number of infected nodesntend to infinity.
Precise asymptotic values for detection probabilities have so far only been found in the special cases ofd-regular trees and geometric trees. It is the purpose of this work to derive detection probabilities for other classes of random trees, namely, all subclasses of simple families of random increasing trees whose random model arises from a (natural) tree evolution process. These subclasses will containd-regular trees and, e.g., the following important random tree models:
• Recursive Trees: they have been proposed as a simple model for the spread of epidemics (a situation very similar to rumor spreading); see Moon [8]. We will show that they constitute the limiting case ofd-regular trees asdtends to infinity. • Plane-oriented Recursive Trees: they are one of the most simplest models for real
complex networks; see the important paper of Barabási and Albert [1].
We will give a precise mathematical definition of simple families of random increasing trees below and describe some of their properties; for more information see Bergeron, Flajolet, and Salvy [2].
We now provide some more details in order to be able to state our results. We fix some notations. Recall thatΓdenotes the tree of the nodes to which the rumor has spread. We will denote byV (Γ)the nodes ofΓwith|Γ| = #V (Γ)and byE(Γ)the edges ofΓ. Ifv ∈ V (Γ),Γvwill denoteΓrooted atvwith an (arbitrary) embedding in the plane,
where we will drawΓin such a way thatvis at the top (and the subtrees are below). If
u ∈ V (Γ), thenΓvuwill denote the subtree at the fringe ofΓvrooted atu.
Rumor Center. In this paragraph, we will recall the definition of the rumor center from [10, 11]. Forv ∈ V (Γ), we define a score as follows
R(v, Γ) = n! Y u∈V (Γ) 1 |Γv u| .
This is the so-called shape functional; see for instance Fill [4]. In order to explain its meaning, we need to recall some further notation from graph theory. We call a rooted tree ordered if it comes with a fixed embedding into the plane (where in this paper, we always draw the root at the top); otherwise, the tree is called unordered. Moreover, a rooted increasing tree ofnnodes is a tree whose nodes are labeled with labels from the set{1, . . . , n}in such a way that every sequence of labels from to the root to a leaf forms an increasing sequence. Now, we can explain the meaning of the shape functional: it gives the number of rooted ordered increasing trees which are isomorphic toΓv.
We next recall the definition of rumor center from [10, 11].
Definition 1.1. LetΓbe a tree. A nodev ∈ V (Γ)which maximizesR(v, Γ)is called a rumor center ofΓ.
Thus, a rumor center v of Γ is a node such that the number of rooted ordered increasing trees which are isomorphic toΓv is maximal. Every such increasing tree
corresponds to a spreading order in which the rumor has spread from the sourcev. Consequently, if all spreading orders are equally likely (as is the case, e.g., ford-regular trees; see [10, 11] and below), then the rumor center is the most likely rumor source or in other words the rumor center is the ML estimator for the rumor source.
It was shown in [10, 11] that the rumor center has a surprisingly easy characterization. We will give two versions of this characterization. For the first, we need the following definition.
Definition 1.2. Let Γ be a tree. A node v ∈ V (Γ) is called a local rumor center if
R(v, Γ) ≥ R(u, Γ)for allu ∈ V (Γ)with{u, v} ∈ E(Γ).
Then, Shah and Zaman proved the following result in [10, 11].
Theorem 1.3 (Shah and Zaman; 2010 - Version 1). LetΓbe a tree. Then, every local rumor center is a rumor center.
The second version of Sha and Zaman’s result (which is in fact only a more precise version of the first one) characterizes a rumor center by graph-theoretical properties. Theorem 1.4 (Shah and Zaman; 2010 - Version 2). LetΓbe a tree withnnodes. Then,
v ∈ V (Γ)is a rumor center ofΓif and only if|Γv
u| ≤ n/2for allu ∈ V (Γ)with{u, v} ∈ E(Γ).
Moreover, if all inequalities are strict there is only one rumor center; otherwise, there are exactly two adjacent rumor centers.
The rumor source estimator is now defined as follows: if there is only one rumor center, then we choose this node; if there are two, we either ignore them or choose one of them uniformly at random.
The appropriateness of the rumor source estimator as defined above depends on the random model. In the definitions above, we considered ordered trees. This, however, might not be always appropriate, for instance if the underlying tree model has not a fixed but dynamic structure (e.g., if a node can spread the rumor to an arbitrary large number of neighbors; see the definition of recursive trees below). Then, considering unordered trees might be advantageous. For such trees, the above definition ofR(v, Γ)
has to be suitable modified. Unfortunately, the resulting characterization of nodesv
which maximize the score becomes messier; see Section 5 of this paper for details.
(Grown) Simple Families of Increasing Trees. In this paragraph, we are going to
explain the random tree models which will be used in this paper. First, consider the set of all rooted ordered increasing trees. A simple family of increasing trees consists of this set together with a sequence of weights(φi)i≥0withφ0 > 0andφi > 0for somei ≥ 2.
For every treeT, we define its weight as
w(T ) = Y
v∈V (t)
φd(v),
whered(v)is the out-degree ofv(=the number of edges ofvwhich point away from the root). Moreover, set
τn :=
X
V (T )=n
w(T ).
Then, a probability space on trees of sizenis defined as follows: a treeT of sizenhas probabilityw(T )/τn. The resulting family of random trees is called a simple family of
random increasing trees.
We give some prominent examples.
• d-ary trees:φi= di, 0 ≤ i ≤ dandφi= 0for alli > d(here,d ∈ {2, 3, . . .}).
• Recursive trees:φi= 1/i!for alli ≥ 0.
• Generalized plane-oriented recursive trees:φi= r+i−2i
for alli ≥ 0(here,r > 1is a real number).
These three families contain, e.g., random binary trees (d-ary trees withd = 2) which are equivalent to random binary search trees from computer science and plane-oriented recursive trees (PORTs for short; these are generalized PORTs withr = 2); see the introduction and [2] for more explanation concerning the relevance of these two random tree models.
The above three families of random increasing trees are very special; see Panholzer and Prodinger [9]. More precisely, it was shown in [9] that out of all families of random increasing trees they are the only ones for which the random model alternatively can also be obtained from a (natural) tree evolution process. Consequently, they have been nicknamed grown simple families of random increasing trees; see, e.g., Kuba and Panholzer [6].
We briefly describe the tree evolution process for the above three families.
• d-ary trees: the first node is the root andd empty leaves are attached; for the second node, one leaf is chosen uniformly at random and the node together withd
empty leaves is placed there; for the third node, again one of the leafs is chosen uniformly at random, etc.
• Recursive trees: assume that a tree withn − 1nodes was already constructed; for the next node, choose one of the nodes uniformly at random and add the next node as child. (Note that the tree here is unordered.)
• Generalized plane-oriented recursive trees: again assume that a tree withn − 1
nodes was already constructed; for the next node, choose an existing nodevwith probability proportional tod(v) + r − 1(d(v) is the out-degree ofv) and add the next node as child. (The tree is again unordered; however, forr = 2, this random model is equivalent to the uniform model on rooted ordered increasing trees.) From these descriptions, it is obvious that the random model ofd-ary trees is the uniform model on rooted ordered increasingd-ary trees and the random model for PORTs (generalized PORTs withr = 2) is the uniform model on rooted ordered increasing trees (as already mentioned above). Moreover, the random model for recursive trees is the uniform model on rooted unordered increasing trees. Thus, the rumor source estimator described in the previous paragraph is a ML estimator only for the former two families of random increasing trees but not for the latter (and also not for generalized PORTS withr 6= 2).
For later purpose, we need some more properties of the above three families of random increasing trees. Therefore, set
φ(z) =X i≥0 φizi, τ (z) = X n≥1 τn zn n!.
Then, it is straightforward to show that
τ0(z) = φ(τ (z)).
Solving this differential equation for the above families gives the following: • d-ary trees:τ (z) = −1 + (1 − (d − 1)z)−1/(d−1).
• Recursive trees:τ (z) = log(1/(1 − z)).
• Generalized plane-oriented recursive trees:τ (z) = 1 − (1 − rz)1/r. From thisτnis easy to derive by standard Taylor series expansion.
Results. In this paragraph, we explain our results. Consider a random increasing tree withnnodes (as random model, we choose one of the three random models from the previous paragraph). We denote byCnthe probability that the node obtained from the
rumor source estimator is indeed the rumor source, where we use here the strategy that ties are ignored (since ties anyway occur only with asymptotic probability zero; see below). Then, we have the following result for grown simple families of random increasing trees.
Theorem 1.5. (a) (d-ary Trees) We have,
lim
n→∞P (Cn) = kd-ary= 1 − d + d2
−1/(d−1)
withkd-aryincreasing indand
lim d→∞kd-ary= 1 − ln 2. Thus, ford ≥ 3, 0.12132 · · · =3 2 √ 2 − 2 ≤ kd-ary< 1 − ln 2.
(b) (Recursive Trees) We have,limn→∞P (Cn) = 1 − ln 2.
(c) (Generalized PORTs) We have,
lim
n→∞P (Cn) = kr= r − (r − 1)2 1/r
withkrdecreasing inrand
lim
r→1kr= 1 and r→∞lim kr= 1 − ln 2.
Thus, forr > 1,
1 − ln 2 < kr< 1.
Remark 1.6. Due to part (b), recursive trees can be seen as the limiting case ofd-ary trees asdtends to infinity. Moreover, note that the detection probability increases from
0(ford-ary trees withd = 2) all the way to1as one goes fromd-ary trees to recursive trees to generalized PORTs.
Part (a) of Theorem 1.5 will follow from a result on a more general family of random trees: the subtree of the root hasd1subtrees and all other subtrees haved2subtrees
(subtrees are possibly empty). The random model of this family of trees is as follows: the first node is the root andd1empty leafs are attached; for the next node, one leaf is
chosen uniformly at random and the next node together withd2 empty leafs is placed
there; for the third node, again one leaf is chosen uniformly at random, etc. Such random tree models where the root is treated different have appeared before in literature; see for instance [7].
For this more general random tree model, we have the following result. Theorem 1.7. We have, lim n→∞P (Cn) = 1 − d1+ d1I1/2 1 d2− 1 ,d1− 1 d2− 1 ,
whereIx(a, b)is the regularized incomplete beta function.
Note that ford1= d2= d, we obtain the above result ford-ary trees. Moreover, this
result also contains one of the main results from [11], namely,d1 = dandd2 = d − 1
which ared-regular trees.
Theorem 1.8 (d-regular Trees; see also [12]). Ford-regular trees, we have
lim n→∞P (Cn) = kd-reg= 1 − d 2 + dΓ(d−2d ) 2d−2d Γ(d−1 d−2) 2.
Remark 1.9. As observed in [12], Stirling’s formula implies that
lim
n→∞kd-reg= 1 − ln 2.
Hence, recursive trees are also the limiting case ofd-regular trees asdtends to infinity (this is of course not surprising).
We conclude the introduction with a brief sketch of the paper. In the next section, we prove Theorem 1.7. In contrast to [12] this will be done by using tools from Analytic Combinatorics (in [12] the authors used Pólya urn models and tools from the theory of stochastic processes). As a consequence, we will obtain part (a) of Theorem 1.5 and Theorem 1.8. In Section 3, we will prove part (b) of Theorem 1.5. In Section 4, we will prove part (c) of Theorem 1.7. Finally, in Section 5, we will give a brief discussion of the rumor center for rooted unordered trees.
2
Generalized
d
-ary Trees
In this section, we will prove Theorem 1.7.
We start by fixing some notation. First, recall the definition of the trees from Theorem 1.7 (see the paragraph preceding the theorem). The number of these trees withnnodes will be denoted byτ˜n. Moreover, we will denote byτn the number ofd2-ary trees withn
nodes. Then, observe that
˜ τn= X j1+···+jd1=n−1 n − 1 j1, . . . , jd1 τj1· · · τjd1,
wherej1, . . . , jd1 ≥ 0are the sizes of thed1subtrees of the root andτ0:= 1. Consequently,
˜
τ0(z) = (1 + τ (z))d1, (2.1) whereτ (z)is as in the introduction and
˜ τ (z) =X n≥1 ˜ τn zn n!. Recall that τ (z) = −1 + (1 − (d2− 1)z)−1/(d2−1). (2.2)
Now, we turn to the probability ofCn. By Theorem 1.4, we have
P (Cn) = 1 − d1P (size of leftist subtree≥ n/2). (2.3)
Denote byIthe size of the leftist subtree. Then,
P (I = j) = 1 ˜ τn X j+j2+···+jd1=n−1 n − 1 j, j2, . . . , jd1 τjτj2· · · τjd1 =(n − 1)!τj j!˜τn X j2+···+jd1=n−1−j τj2 j2! · · ·τjd1 jd1! =(n − 1)!τj j!˜τn [zn−1−j](1 + τ (z))d1−1 =(n − 1)!τj j!˜τn [zn−1−j](1 − (d2− 1)z)− d1−1 d2−1 =(n − 1)!τj j!˜τn (d2− 1)n−1−j[zn−1−j](1 − z) −d1−1 d2−1. (2.4)
In the sequel, we need the following standard lemma from analytic combinatorics. Theorem 2.1 (Theorem VI.1 in [5]). Forα ∈ C \ Z≤0 set
Then, asn → ∞, [zn]f (z) ∼ n α−1 Γ(α) 1 + ∞ X k=1 ek(α) nk ! ,
whereek(α)is a polynomial of degree2k.
Applying this result to (2.2) gives
τn = n![zn]τ (z) ∼ n!(d2− 1)n
nd2−11 −1
Γ( 1 d2−1)
.
Similarly, applying the result to (2.1) yields
˜ τn= (n − 1)![zn−1]˜τ0(z) ∼ (n − 1)!(d2− 1)n−1 nd2−1d1 −1 Γ( d1 d2−1) . By (2.3), we need to compute X n/2≤j≤n−1 P (I = j),
whereP (I = j)is given by (2.4). To accomplish this task, we again use Theorem 1 and the expansions forτn andτ˜nfrom above. This gives
X n/2≤j≤n−1 P (I = j) ∼ (n − 1)! ˜ τn X n/2≤j≤n−1 τj j!(d2− 1) n−1−j(n − 1 − j) d1−1 d2−1−1 Γ(d1−1 d2−1) ∼ Γ( d1 d2−1) Γ(d1 2−1)Γ( d1−1 d2−1) · 1 nd2−1d1 −1 X n/2≤j≤n−1 jd2−11 −1(n − 1 − j)d1−1d2−1−1 ∼ Γ( d1 d2−1) Γ(d1 2−1)Γ( d1−1 d2−1) · 1 n X n/2≤j≤n−1 j n d2−11 −1 n − 1 − j n d1−1d2−1−1 ∼ Γ( d1 d2−1) Γ(d1 2−1)Γ( d1−1 d2−1) Z 1 1/2 xd2−11 −1(1 − x)d1−1d2−1−1dx. (2.5) Observe that Z 1 1/2 xd2−11 −1(1 − x)d1−1d2−1−1dx = Γ( 1 d2−1)Γ( d1−1 d2−1) Γ(d1−1 d2−1) − B 1/2; 1 d2− 1 ,d1− 1 d2− 1 ,
whereB(x; a, b)denotes the incomplete beta function. Plugging this into (2.5) and (2.5) in turn into (2.3) yields Theorem 1.7.
Proof of Theorem 1.5, part (a). Setting d1 = d2 = d and evaluating the expression
obtained in Theorem 1.7 yields the claimed result for kd-ary. Moreover, the claims
concerning monotonicity and limit behavior ofkd-aryfollow by simple calculus.
Next, we consider the case ofd-regular trees, where we setd1 = dandd2= d − 1.
We need the following lemma. Lemma 2.2. Forα > 0, Z 1 1/2 xα−1(1 − x)αdx =1 2 B(α, α + 1) − 1 α22α ,
Proof. First, observe that B(α, α + 1) = Z 1 0 xα−1(1 − x)αdx = Z 1/2 0 xα−1(1 − x)αdx + Z 1 1/2 xα−1(1 − x)αdx.
Now, call the first and second integral on the right hand sideLandR, respectively. By integration by parts and substitution, we have
L = 1 αx α(1 − x)α 1/2 0 + R. Thus, R = 1 2 B(α, α + 1) − 1 α22α
which is the claimed result.
This lemma can be used to evaluate the integral in (2.5). Plugging the result then in turn into (2.3) yields Theorem 1.8.
3
Recursive Trees
We consider now recursive trees and will prove part (b) of Theorem 1.5. Recall that
τ (z) = ln 1
1 − z
and thusτn= (n − 1)!. Similar tod-ary trees, we have that
P (Cn) = 1 − P (one subtree of the root has size≥ n/2).
In order to find the latter probability observe that at most one subtree of the root has size at leastn/2. Consequently, since recursive trees are unordered trees, we can arrange the subtrees such that this subtree is the leftist one. Then,
P (one subtree of the root has size= j) = 1 τn X `≥1 1 (` − 1)! X j+j2+...j`=n−1 n − 1 j, j2, . . . , j` τjτj2. . . τj` =(n − 1)!τj j!τn [zn−1−j]X `≥1 τ (z)`−1 (` − 1)! =(n − 1)!τj j!τn [zn−1−j] 1 1 − z = 1 j,
wherej ≥ n/2. Plugging this into the expression above gives
P (Cn) = 1 −
X
n/2≤j≤n−1
1
j = 1 − Hn−1+ Hdn/2e−1,
whereHn denotes then-th harmonic number. We summarize this in a result.
Theorem 3.1 (Recursive Trees). We have,P (Cn) = 1 − Hn−1+ Hdn/2e−1.
From this part (b) of Theorem 1.5 follows from standard expansions for harmonic numbers.
4
Generalized Plane-oriented Recursive Trees
Finally, we consider generalized plane-oriented recursive trees and prove part (c) of Theorem 1.8. Recall that
τ (z) = 1 − (1 − rz)1/r. (4.1) Then, as in the last section
P (Cn) = 1 − P (one subtree of the root has size≥ n/2).
Since now the subtrees are ordered, we obtain
P (one subtree of the root has size= j) (4.2)
= 1 τn X `≥1 `φ` X j+j2+...j`=n−1 n − 1 j, j2, . . . , j` τjτj2. . . τj` = (n − 1)!τj j!τn [zn−1−j]X `≥1 `φ`τ (z)`−1 = (n − 1)!τj j!τn (r − 1)[zn−1−j](1 − τ (z))−r = (n − 1)!τj j!τn rn−1−j(r − 1), (4.3) wherej ≥ n/2.
We now turn to asymptotic expansions. First, applying Theorem 2.1 to (4.1) gives
τn ∼ −n!rn
n−1/r−1 Γ(−1/r).
Plugging this into (4.3) yields
P (one subtree of the root has size≥ n/2) ∼ r − 1 r · 1 n X n/2≤j≤n−1 j n −1/r−1 ∼ r − 1 r Z 1 1/2 x−1/r−1dx = (r − 1)(21/r− 1).
This proves the claimed limit result. The claimed properties of monotonicity and limit behavior ofkrfollow by simple calculus.
Remark 4.1. Alternatively to the above asymptotic derivation, one can also derive an exact expression (similar as in the last section). To give more details, note that from (4.1), one obtains that
τn = n!(−1)n+1rn 1/r n . Consequently, from (4.3),
P (one subtree of the root has size≥ n/2) = r − 1 r · (−1)n n 1/rn X n/2≤j≤n−1 (−1)j1/r j Note that X n/2≤j≤n−1 (−1)j1/r j = nr(−1)n+11/r n + dn/2er(−1)dn/2e 1/r dn/2e .
Theorem 4.2 (Generalized PORTs). We have, P (Cn) = r − (r − 1)(−1)n+dn/2e dn/2e 1/r dn/2e n 1/rn .
5
Rumor Center for Unordered Trees
In this final section, we will discuss the rumor center in unordered trees. We will use the same notation as in Section 1. Moreover, recall thatR(v, Γ)gives the number of rooted ordered increasing trees which are isomorphic toΓv.
If trees are now considered to be unordered instead of ordered, R(v, Γ)has to be replaced by the shape functional for unordered trees, i.e., by the number of rooted unordered increasing trees which are isomorphic toΓv. This shape functional has been
introduced and studied by Feng and Mahmoud in [3]. Following this paper, forv ∈ V (Γ), we define now the score by
S(v, Γ) = (n − 1)! Y u∈V (Γ) w(Γvu) |Γv u| , where w(Γvu) = r Y i=1 1 mi!
with(m1, . . . , mr)the multiplicities of the subtrees ofuinΓvu.
With a slight abuse of notation, we give the following definition.
Definition 5.1. Let Γ be a tree. A node v ∈ V (Γ) is called a local rumor center if
S(v, Γ) ≥ S(u, Γ)for allu ∈ V (Γ)with{u, v} ∈ E(Γ).
Then, as for ordered trees, we have the following theorem.
Theorem 5.2. LetΓbe a tree. Then, every local rumor center is a rumor center. Proof. Denote byvthe local rumor center and considerΓv. Letu ∈ V (Γ)with{u, v} ∈ E(Γ)be fixed. In the sequel, we will use the following notation: bymv
uwe denote the
multiplicity ofΓv
uamongst the subtrees ofvinΓv.
First consider S(v, Γ) S(u, Γ) = muv|Γu v| mv u|Γvu| = m u v(n − |Γ v u|) mv u|Γvu| ,
where we used the (trivial) fact that|Γu
v| + |Γvu| = n. Since, due to the assumptions, the
above ratio must be at least one, we have
|Γv u| ≤ muvn mu v+ mvu . (5.1)
We will fix now anu ∈ V (Γ)˜ withu 6= v˜ and{˜u, u} ∈ E(Γ); see the tree on the left in Figure 1. Observe that S(v, Γ) S(˜u, Γ) = S(v, Γ) S(u, Γ)· S(u, Γ) S(˜u, Γ) ≥ mu˜ u(n − |Γuu˜|) mu ˜ u|Γuu˜| (5.2) and we have to show that this is at least one. For this, we will consider two cases.
In the first case, we will assume that|Γv
u| ≤ n/2. Then,|Γuv| ≥ n/2and hencemuv = 1.
Similarly,|Γu˜
u| ≥ n/2andmuu˜= 1. Moreover, we have
v u ˜ u . . . u v ˜ u . . .
Figure 1: The two trees from the proof of Theorem 5.2
This implies that|Γu ˜
u| ≤ n/(2muu˜)which in turn implies that|Γuu˜| ≤ n/(1 + muu˜). Thus, (5.2)
is indeed at least one and henceS(v, Γ) ≥ S(˜u, Γ).
It should be clear that the above argument can be repeated. Consequently, if we choose a path fromvviau˜to a leaf, then theS-value of the nodes is non-increasing as required.
Next, we consider the second case, where we assume that|Γv
u| > n/2. Consider the
tree on the right in Figure 1 which is just the left one rooted atu(i.e.,Γu). Due to (5.1),
we have that |Γu v| ≥ n mu v+ mvu . (5.3) IfΓu ˜
uis isomorphic toΓuv, then theS-value ofu˜andvare the same and nothing has to be
proved. So, assume thatΓuu˜is not isomorphic toΓuv. Then, a simple counting argument shows that mu˜u|Γuu˜| + m u v|Γ u v| ≤ n. Using (5.3), we obtain |Γu ˜ u| ≤ n mu ˜ u(muv+ mvu) ≤ n 1 + mu ˜ u .
This in particular implies that |Γu ˜
u| ≤ n/2 and hencem˜uu = 1. Consequently, (5.2) is
again shown to be at least one. Finally, similar to the first case, this argument can be iterated such that again theS-value of nodes along paths fromv to a leaf (viau˜) are non-increasing. This concludes the proof.
Remark 5.3. From the proof, we find the following sufficient and necessary condition for
v ∈ V (Γ)to be a rumor center (compare with Theorem 1.4 from the introduction). Theorem 5.4. LetΓbe a tree withnnodes. Then,v ∈ V (Γ)is a rumor center ofΓif and only if the following holds for allu ∈ V (Γ)with{u, v} ∈ E(Γ):
|Γv u| ≤ mu vn mu v+ mvu .
Remark 5.5. In contrast to the ordered case, hereΓcan have more than two rumor centers; also, rumor centers are not necessary adjacent; see Figure 2 for examples. Remark 5.6. As mentioned in Section 1, the random model of recursive trees is the uniform model on rooted unordered increasing trees. Thus, the rumor center defined in this section is the ML estimator for the rumor source. It would be interesting to compute the detection probability for this estimator (which will be at least1 − ln 2as follows from Theorem 1.5). However, the more complicated characterization of the rumor center from Remark 5.3 makes this a seemingly complicated task.
v1 v2
Figure 2: Every node of the tree on the left is a rumor center; the nodesv1andv2of the
tree on the right are (non-adjacent) rumor centers.
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Acknowledgments. Parts of this research was done while the first author visited the Institut für Diskrete Mathematik und Geometrie, Technical University of Vienna. He thanks the department for hospitality and the NSC for financial support (NSC-102-2918-I-009-012). In addition, he was also partially supported by the Ministry of Science and Technology, Taiwan under the grant MOST-103-2115-M-009-007-MY2.
