Poceedings of the IEEE Conf. on Decision and Contol, Olando, FL, 011 Reaanging tees fo obust consensus Geoge Foest Young, Luca Scadovi and Naomi Ehich Leonad Abstact In this pape, we use the H nom associated with a communication gaph to chaacteize the obustness of consensus to noise. In paticula, we estict ou attention to tees, and by systematic attention to the effect of local changes in topology, we deive a patial odeing fo undiected tees accoding to the H nom. Ou appoach fo undiected tees povides a constuctive method fo deiving an odeing fo diected tees. Futhe, ou appoach suggests a decentalized manne in which tees can be eaanged in ode to impove thei obustness. I. INTRODUCTION The study of linea consensus poblems has gained much attention in ecent yeas [1] [4]. This attention has aisen, in pat, fom the wide ange of applications of linea consensus, including collective decision-making [3], fomation contol [5], senso fusion [6], distibuted computing [7] and analysis of biological goups [8]. In most of these applications, infomation passed between agents can be coupted by noise. It is theefoe necessay to undestand the obustness of consensus when noise is pesent, with the goal of designing systems that can efficiently filte noise and emain close to consensus. Fo a linea system with additive white noise, a natual measue of obustness is the H nom [9]. Fo the study of consensus, most of the impotant details ae descibed by the communication gaph. In fact, the popeties of the Laplacian matix of the gaph ae deeply elated to the pefomance of the consensus potocol. In this way, the study of consensus often educes to studying the undelying gaph, and elating gaph popeties to the pefomance of the oiginal system [1], [3], [10], [11]. Communication in a multi-agent system is likely to be of a diected natue, simply because each agent may teat the infomation it eceives diffeently than its neighbos. Additionally, diected communication can aise when infomation is tansfeed though sensing o when agents have limited capabilities and may choose to only eceive infomation fom a subset of possible neighbos. In many eal systems, the gaph between agents may change ove time depending on the behavio and decisions of individual agents [1]. Theefoe, when seaching fo ways in which to impose effective gaphs on multi-agent systems, it is highly advantageous to conside whethe such gaphs could be fomed in a decentalized manne. This eseach was suppoted in pat by AFOSR gant FA9550-07-1-0- 058 and ONR gant N00014-09-1-1074. G. F. Young and N. E. Leonad ae with the Depatment of Mechanical and Aeospace Engineeing, Pinceton Univesity, Pinceton, NJ 08544, USA. L. Scadovi is with the Depatment of Electical Engineeing and Infomation Technology, Technical Univesity of Munich, 80333 Munich, Gemany. gfyoung@pinceton.edu, scadovi@tum.de, naomi@pinceton.edu In this pape, we study the obustness of a paticula family of gaphs, namely tees, accoding to thei H noms. We develop a patial odeing among tees that allows us to find a tee with minimal H nom, given cetain constaints. Although most of this patial odeing has aleady been developed in the liteatue on Wiene indices [1] [15], ou methods of poof ae new. In paticula, we ely only on local changes in which one o moe leaf nodes ae moved fom a single location in the tee to a new location. This appoach povides insight into ways in which tees can be eaanged in a decentalized manne in ode to impove thei obustness. Additionally, ou methods can be used to deive a simila odeing fo diected tees that could not be done using the Wiene index liteatue. This pape is oganized as follows. In Section II we summaize notation. In Section III we discuss the H nom in moe detail. In Sections IV and V we discuss the elationship between the H nom and othe gaph indices. In Section VI we intoduce a system of teminology to descibe tee gaphs, and in Section VII we deive ou patial odeing. Finally, in Section VIII, we discuss the potential fo a decentalized algoithm to impove the H nom of a tee. II. PRELIMINARIES AND NOTATION The state of the system is x = [x 1, x,..., x N ] R N, whee x i is the state of agent i. We call the state consensus when x = γ1 N, whee 1 N = [1, 1,..., 1] T R N and γ R. Fo each agent i we define the set of neighbos, N i, to be the set of agents that supply infomation to agent i. We associate to the system a communication gaph G = (V, E, A), whee V = {1,,..., N} is the set of nodes, E V V is the set of edges and A R N N is a weighted adjacency matix with nonnegative enties a i,j coesponding to the weight on edge (i, j). Evey node in the gaph coesponds to an agent, while the gaph contains edge (i, j) when j N i. Then a i,j is the weight given by agent i to the infomation fom agent j. An edge (i, j) E is said to be undiected if (j, i) is also in E and a i,j = a j,i. A gaph is undiected if evey edge is undiected, that is, if A is symmetic. Two nodes ae said to be adjacent when thee is an edge between them. The out-degee of node k is defined as d out k = N j=1 a k,j. G has an associated Laplacian matix L, defined by L = D A, whee D = diag (d out 1, d out,..., d out ) is the diagonal N matix of out-degees. The ow sums of the Laplacian matix ae zeo, that is L1 N = 0. Thus 0 is always an eigenvalue of L with coesponding eigenvecto 1 N. Futhemoe, all eigenvalues of L have non-negative eal pat (by Gešgoin s
Theoem). Fo an undiected gaph, L is symmetic, so in addition 1 T N L = 0 and all the eigenvalues of L ae eal. A path in G is a (finite) sequence of nodes containing no epetitions and such that each node is a neighbo of the pevious one. The length of a path is given by the sum of the weights on all edges tavesed by the path. The gaph G is connected if it contains a globally eachable node k; i.e. thee is a path in G fom i to k fo evey node i. It can be shown that 0 will be a simple eigenvalue of L if and only if G is connected [16]. If an undiected gaph is connected, thee will be a path in G between evey pai of nodes. We use λ i to efe to the i th eigenvalue of the Laplacian matix, when aanged in ascending ode by eal pat. Thus λ 1 = 0 fo any Laplacian matix, and Re {λ } > 0 if and only if G is connected. The distance, d i,j, between nodes i and j in a gaph is the shotest length of any path fom i to j. If no such path exists, d i,j is infinite. The diamete, d, of a gaph is the maximum distance between all pais of nodes in the gaph. A tee on N nodes is a connected undiected gaph in which evey pai of nodes is connected by a unique path. This implies that a tee contains exactly N 1 undiected edges and that it contains no cycles (paths with positive length connecting a node to itself). A ooted tee is a tee in which one paticula node has been identified as the oot (note that othe than being called the oot, thee is nothing special about this node). A diected tee is a connected gaph containing exactly N 1 diected edges. In a diected tee, the globally eachable node is identified as the oot. III. ROBUST NOISY CONSENSUS AND THE H NORM In this pape we assume that evey agent is independently affected by white noise of the same intensity. The esulting consensus dynamics ae (as in [5], [9], [17]) ẋ(t) = Lx(t) + ξ(t) (1) with x R N and whee ξ(t) R N is a zeo-mean mutually independent white stochastic pocess. Since (1) is only maginally stable in the noise-fee case (coesponding to the fact that thee is no pefeed o coect value fo the agents to agee upon), we only conside the dynamics on the subspace of R N othogonal to the subspace spanned by 1 N. We let Q R (N 1) N be a matix with ows that fom an othonomal basis of this subspace. This is equivalent to equiing that Q1 N = 0, QQ T = I N 1 and Q T Q = I N 1 N 1 () N 1 T N. Next, we define y := Qx. Then y = 0 if and only if x = γ1 N, γ R. A measue of the distance fom consensus is the dispesion of the system y(t). Diffeentiating y(t), we obtain ẏ(t) = Ly(t) + Qξ(t) (3) whee L = QLQ T is the educed Laplacian matix. L has the same eigenvalues as L except the zeo eigenvalue, which implies that L is Huwitz pecisely when the gaph is connected [9]. Thus, fo a connected gaph in the absence of noise, system (3) will convege exponentially to zeo. In the pesence of noise, (3) will no longe convege, but will emain in motion about zeo. We define the obustness of consensus to noise as the expected dispesion of the system in steady state. Note that this definition is analogous to the steady-state mean-squae deviation used in [7]. Ou measue of obustness coesponds to the H nom of system (3), with output equation z(t) = I N 1 y(t). H noms of consensus systems have also been studied in [5], [17]. In [9] we poved that fo a system with an undiected communication gaph 1, the H nom is given by ( N ) 1 1 H =. (4) λ i= i In geneal, the H nom of a diected gaph can be computed as H = [t(σ)] 1, whee Σ is the solution to the Lyapunov equation [9] LΣ + Σ LT = I. (5) Since this H nom can be computed using only the communication gaph, in the est of this pape we associate the H nom with the gaph. Thus when we efe to the H nom of a gaph, we mean the H nom of system (3) (with output z = y) with L computed fom the given gaph. IV. EFFECTIVE RESISTANCE AS A MEASURE OF THE H NORM Although equation (4) allows us to compute the H nom fo any undiected gaph, it does not eadily allow us to infe elationships between stuctual featues of the gaph and the H nom. Howeve, the concept of the effective esistance, o Kichhoff index, of a gaph can help us in this espect. The effective esistance esults fom consideing a given gaph as an electical netwok, whee evey edge coesponds to a esisto with esistance given by the invese of the edge weight. The esistance between two nodes in the gaph is given by the esistance between those two points in the electical netwok, and the effective esistance of the gaph is the sum of the esistances between all pais of nodes[18]. The effective esistance of a gaph is elated to the eigenvalues of the gaph Laplacian [18] by the fomula 1 K f = N, leading to the elationship λ i i= ( ) 1 Kf H =. (6) N We theefoe see that fo gaphs with equal numbes of nodes, any odeing induced by the effective esistance is the same as that induced by the H nom. Although computing the effective esistance can be difficult fo most gaphs, it is vey staightfowad fo tees. In a tee with unit weights on evey edge, the esistance between two nodes is given by the distance between them [19]. Hence, the effective esistance of a tee with unit edge weights is K f = i<j i,j = i<j d i,j. (7) 1 In fact, the esult in [9] is moe geneal and extends to diected gaphs with a nomality condition on thei Laplacian matix.
Although the concept of effective esistance does not apply to diected gaphs, we can define an extension so that equation (6) applies to all gaphs. The esistance between two nodes of an undiected gaph can be computed as [18] i,j = (L ) i,i + (L ) j,j (L ) i,j (8) whee L is the Mooe-Penose pseudoinvese of L. Fo an undiected gaph, we can explicitly wite L = Q T ΣQ, whee Σ is the solution to the Lyapunov equation (5). Thus, if we let X = Q T ΣQ, we can compute fo any gaph i,j = (X) i,i + (X) j,j (X) i,j. (9) Using equation (9), we can compute diected esistances (and hence Kichhoff indices) fo diected gaphs. Though this constuction, we can show that equation (6) will hold fo diected gaphs as well. Then, in a diected tee (as in an undiected tee) the esistance between two nodes only depends on the paths between them. The poofs and esults fo diected tees will appea in a futue publication. In the following sections we detemine a patial odeing of undiected tees with unit edge weights. The same odeing will apply to the set of tees with a given constant edge weight, as all esistances will be popotional to those in the coesponding tee with unit weights. V. THE H NORM AND OTHER GRAPH INDICES In addition to the Kichhoff index, many othe topological indices of gaphs have aisen out of the mathematical chemisty liteatue [0]. One of the ealiest to aise was the Wiene index, W [0]. The Wiene index fo any (undiected) gaph is defined as W = d i,j. (10) i<j Thus, fo tees with unit edge weights, the Kichhoff and Wiene indices ae identical. Howeve, the two indices diffe fo any othe gaph. Hence, while the esults in Section VII apply equally to Wiene indices, we choose to intepet them only in tems of the Kichhoff index and H nom. Much wok has aleady been done on compaing tees based on thei Wiene indices. It is aleady well-known that the Wiene index of a tee will fall between that of the sta and that of the path [1], [1]. Fo tees with a fixed numbe of nodes, the 15 tees with smallest Wiene index and the 17 tees with lagest Wiene index have been identified [13], [14]. Futhe, fo tees with a fixed numbe of nodes and a fixed diamete, the tee with smallest Wiene index has been found [15]. Theefoe, most of the main esults in Section VII have aleady been deived. Ou contibution includes new methods of poof that ely on local changes of topology and povide constuctive means to ode diected tees and deive decentalized stategies fo impoving obustness. A diffeent gaph index, developed in the mathematical liteatue, is the maximum eigenvalue of the adjacency matix A []. Simić and Zhou developed a patial odeing of tees with fixed diamete accoding to this index in []. Thei wok, in paticula the families of tees they consideed and the ode in which they poved thei esults, has motivated the appoach taken in this pape. VI. A SYSTEM OF TERMINOLOGY FOR TREES We fist intoduce a system of teminology elating to tees. Much of ou teminology coesponds to that in [] and ealie papes. T N,d is the set of all tees containing N nodes and with diamete d. Fo N 3, a tee must have d, and T N, contains only one tee. This tee is called a sta, and is denoted K 1,N 1. Fo all positive N, the maximum diamete of a tee is N 1, and T N,N 1 contains only one tee. This tee is called a path, and is denoted P N. A leaf (o pendant) is a node with degee 1. A bouquet is a non-empty set of leaf nodes, all adjacent to the same node. A node which is not a leaf is called an intenal node. A catepilla is a tee fo which the emoval of all leaf nodes would leave a path. The set of all catepillas with N nodes and diamete d is denoted by C N,d (see Figue 1). Any catepilla in C N,d contains a path of length d, with all othe nodes adjacent to intenal nodes of this path. In paticula, we efe to the catepilla that contains a single bouquet attached to the i th intenal node along this path P N,d,i (see Figue ). To avoid ambiguity, we equie 1 i d. The tee fomed fom P N 1,d, d by attaching an additional node to one of the leaves in the cental bouquet is denoted by N N,d (see Figue 3). Fig. 1. Geneal fom of a catepilla in C N,d, with n j 0 additional leaf nodes attached to each intenal node j in the path of length d. Fig.. The catepilla P N,d,i, a path of length d with a bouquet containing N d 1 leaf nodes attached to the i th intenal node on the path. Fig. 3. The tee N N,d, fomed fom P N 1,d, d by attaching an additional node to one of the leaves in the cental bouquet. Note that N d 3 could be 0. The double palm tee (also efeed to as a dumbbell in [1]) is a catepilla with two bouquets, one at each end of the path (see Figue 4). We use D N,p,q to denote the double palm tee on N nodes, with bouquets of sizes p and q. If we take a ooted tee T (with oot ) and attach two sepaate paths containing l and k nodes to the oot, we call the esulting tee a vine and denote it by Tl,k (see Figue 5). VII. MANIPULATIONS TO REDUCE THE EFFECTIVE RESISTANCE OF TREES We can now stat to descibe a patial odeing on tees based on thei H noms. Evey tee is assumed to have a
Now, in D N,p,q, the path length between node 1 and any of the emaining p 1 nodes in the bouquet of size p is. Similaly, the path length between node 1 and any node in the bouquet of size q is N p q + 1. Finally, the path lengths between node 1 and the intenal nodes take on each intege value fom 1 to N p q. Convesely, in D N,p 1,q+1, the path length between node 1 and any of the nodes in the bouquet of size p 1 is N p q + 1. The path length between node 1 and any of the emaining q nodes in the bouquet of size q + 1 is. Again, the path lengths between node 1 and the intenal nodes take on all intege values fom 1 to N p q. Thus, D N,p 1,q+1 (compaed to D N,p,q ) has moe nodes at a distance fom node 1 and fewe nodes at a distance N p q + 1, while the sum of distances to all intenal nodes emains the same. Theefoe K f (D N,p,q ) > K f (D N,p 1,q+1 ). Hence, by equation (6), the esult holds. Fig. 4. Double palm tee D N,p,q, with bouquets of sizes p and q at each end of a path. Fig. 5. The vine Tl,k, fomed fom a ooted tee T by sepaately connecting paths containing l and k nodes to the oot. unit weight on evey edge. Fist, we detemine the effect of moving a leaf fom one end of a double palm tee to the othe, and use this to deive a complete odeing of all tees in T N,3 (Theoem 1). Second, we conside moving a leaf fom one end of a vine to the othe, and use this to pove that the path has the lagest H nom of any tee with N nodes (Theoem ), and to deive a complete odeing of T N,N (Theoem 3). Finally, by moving all (o almost all) nodes in a bouquet to an adjacent node, we show that P N,d, d has the smallest H nom of any tee with diamete d (Theoem 4) and that fo any tee that is not a sta, we can find a tee of smalle diamete with a smalle H nom (Theoem 5). Fom Theoem 5 we also conclude that the sta has the smallest H nom of any tee with N nodes. A. Double Palm Tees We begin ou patial odeing by showing that the H nom of a double palm tee is educed when we move a single node fom the smalle bouquet to the lage one. Lemma 1: Let 1 < p q and p + q N. Then H (D N,p,q ) > H (D N,p 1,q+1 ). Poof: In D N,p,q, let one of the nodes in the bouquet of size p be node 1. The emaining nodes ae labelled though N. To fom D N,p 1,q+1, we take node 1 and move it to the othe bouquet. Since all othe nodes emain unchanged, we can use equation (7) to wite K f (D N,p,q ) K f (D N,p 1,q+1 ) = j= d 1,j DN,p,q j= d 1,j. DN,p 1,q+1 Although Lemma 1 applies to double palm tees with any diamete, we can apply it to tees with d = 3 in ode to pove ou fist main esult. Theoem 1: Fo N 4, we have a complete odeing of T N,3, namely H (D N,1,N 3 ) ) < H (D N,,N 4 ) <... < H (D N, N, N. Poof: Any tee with d = 3 must have a longest path of length 3. Any additional nodes in the tee must be connected though some path to one of the two intenal nodes on this longest path. In addition, any node adjacent to one of the intenal nodes of the longest path foms a path of length 3 with the node at the fa end of the path. Hence all such nodes must be leaves and so evey tee with d = 3 is a double palm tee. The odeing follows fom Lemma 1. B. Vines Ou next task is to find an odeing of tees with the lagest possible diamete. Lemma applies to tees of any diamete, but again we can specialize it to give the esults we need. Lemma : Let T be a tee containing moe than one node and with a oot, and let k ) be any positive ( integes ) such that 1 l k. Then H (Tl,k < H Tl 1,k+1. Poof: Let the total numbe of nodes in Tl,k be N (so N k l > 1), and let the leaf at the end of the path containing l nodes be node 1. Let the emaining nodes in the two paths be nodes though l + k, and let the oot of T be node l + k + 1. The emaining nodes in T ae labelled l + k + though N. To fom Tl 1,k+1, we take node 1 and move it to the end of the othe path. Since all othe nodes emain unchanged, we can use equation (7) to wite ( ) ( ) K f T l 1,k+1 Kf T l,k =. (11) j= d 1,j T l 1,k+1 j= d 1,j T l,k Now, in both Tl,k and T l 1,k+1, the path lengths between node 1 and all nodes along the paths (including the oot of T ) take on each intege value between 1 and l + k. Hence the sum of these path lengths does not change between the two tees. Futhemoe, since the oot of T lies on evey path between node 1 and any othe node in T, we can wite d 1,j = d 1,l+k+1 + d l+k+1,j, j l + k +. Theefoe, fo Tl,k, the sum of the distances fom node 1 to all the nodes in T is (N l k + 1)l plus the sum of the distances fom node to each node in T. Howeve, in Tl 1,k+1, the sum of the distances fom node 1 to all the nodes in T is (N l k + 1)(k + 1) plus the sum of the distances ) fom node) to each node in T. Thus K f (Tl 1,k+1 > K f (Tl,k and so by equation (6), the esult holds The fist consequence of Lemma is that the tee with lagest d (i.e. d = N 1) also has the lagest H nom.
Theoem : The path P N has the lagest H nom of any tee with N nodes. Poof: Any tee T 1 which is not a path will contain a node with degee geate than. We can locate one such node that has two paths (each with fewe than N nodes) attached. Let T be the tee fomed by emoving these two paths fom T 1, and let this node be the oot of T. Then T 1 = Tl,k, and by Lemma we can find a tee with lage H nom. We can also use Lemma to deive an odeing of those tees with d one less than its maximum value (i.e. d = N ). Theoem 3: Fo N 4, we have a complete ) odeing of T N,N, namely H (P N,N, N < ) H (P N,N, N 1 <... < H (P N,N,1 ). Poof: Evey tee in T N,N must contain a path of length N (which contains N 1 nodes), and one additional node. This node must be adjacent to an intenal node of the path, since othewise we would have a path of length N 1. Thus evey tee in T N,N is of the fom P N,N,i, fo some 1 i N. Now, P N,N,i = Ti,N i, with T a path containing nodes (and one identified as the oot). Suppose that i < N. Then i ( < N i ), and so by Lemma, H (P N,N,i ) = H T ( ) i,n i > H T i+1,n 3 i = H (P N,N,i+1 ). Each tee in T N,N consists of a path of length N with one leaf attached to an intenal node. Theoem 3 ensues that the H nom is smallest when this intenal node is at the cente of the path. C. Catepillas We now have complete odeings fo T N, (tivial, since T N, contains only the sta), T N,3 (by Theoem 1), T N,N (by Theoem 3) and T N,N 1 (tivial, since T N,N 1 contains only the path). We next conside the emaining families of tees with 4 d N 3 (and hence, N 7). Rathe than deiving complete odeings, the main goal of the next two lemmas is to find the tee in T N,d with lowest H nom. Howeve, we use two steps to attain ou esult as this povides geate insight into the odeing amongst the emaining tees. Lemma 5 then allows us to combine the esults to pove (Theoem 4) that among tees of diamete d, the one with lowest H nom is P N,d, d. Theoem 5 povides a compaison of tees with diffeent diamete. Lemma 3: Suppose N 7 and 4) d N 3. If T C N,d, then H (T ) H, with equality if and only if T = P N,d, d. Poof: Since d N 3 and T C N,d, a longest path in T contains N d 1 leaves attached to intenal nodes (othe than the two leaves in the longest path). Suppose that P T is a longest path in T. Fo the est of this poof, when we efe to leaf nodes and bouquets, we mean leaves not pat of P T, and bouquets made up of these leaves. Suppose T contains a single bouquet. Thus T = P N,d,i fo some 1 i d. If i d, then by Lemma, H (P N,d,i ) > H (P N,d,i+1 ). Suppose T contains multiple bouquets. Locate a bouquet futhest fom the cente of P T, and move evey leaf in this bouquet one node futhe fom the closest end of P T. Call this new tee T, and label the nodes that wee moved 1 though n. Then between T and T, the path lengths between each of these leaves and any othe leaf decease by 1. The path lengths between each of these leaves and d+1 nodes on P T incease by 1, and the path lengths between each of these leaves and d+1 nodes on P T decease by 1. Thus the sum of the path lengths in T is less than the sum in T, and so by equations (7) and (6), H (T ) < H (T ). Thus, if T is not P N,d, d, thee is a tee in C N,d with stictly smalle H nom. Lemma 4: Suppose that N 7 and 4 d N 3. Let T be a tee in T N,d \ C N,d. Then H (T ) H (N N,d ), with equality if and only if T = N N,d. Poof: Let P T be a longest path in T (of length d), and let m be the numbe of nodes with distances to P T geate than 1 (the distance between a node and P T is the shotest distance between that node and any node on the path). If m > 1, locate a bouquet with the geatest distance fom P T, label the leaves in this bouquet 1 though n, and label the adjacent node n+1. Suppose that eithe the distance between this bouquet and P T is geate than, o the distance is and anothe bouquet exists the same distance fom P T. Let T be the tee fomed by moving all leaves in this bouquet one node close to P T. By ou assumptions, T T N,d \ C N,d. Then d i,n+1 inceases by 1 fo i = 1,..., n. Convesely, d i,j deceases by 1 fo i = 1,..., n and j > n + 1. Since thee must be at least d + 6 of these othe nodes (with labels above n + 1), the sum of all distances in T is smalle than the sum of all distances in T. Thus H (T ) < H (T ). If the bouquet we found has a distance of to P T, and is the only such bouquet, fom T by moving leaves 1 though n 1 one node close to P T. Then T T N,d \C N,d. Now, d i,n and d i,n+1 both incease by 1 fo i = 1,..., n 1. Howeve, d i,j deceases by 1 fo i = 1,..., n 1 and j one of the emaining d + 1 5 nodes. Thus H (T ) < H (T ). If m = 1, then T contains a single node at a distance fom P T, and all othe nodes in T ae eithe on P T o adjacent to nodes on P T. Locate a node on P T with additional nodes attached that is futhest fom the cente of the path. Label all nodes attached to this node 1 though n (including the node at distance fom P T if it is connected to P T though this node), and label this node n + 1. If n + 1 is not the d th intenal node on the path (i.e. if T N N,d ), then fom T by moving all nodes not on P T that ae adjacent to n+1 (including the node at distance, if pesent) one node futhe fom the closest end of P T. Then T T N,d \ C N,d. Futhemoe, fo i = 1,..., n, d i,j deceases by 1 fo any j not on P T, inceases by 1 fo d nodes on P T and deceases by 1 fo d + 1 nodes on P T. Thus the sum of the distances in T is less than the sum in T, and so H (T ) < H (T ). Hence fo evey tee in T N,d \C N,d othe than N N,d, thee exists anothe tee in T N,d \ C N,d with smalle H nom. Lemma 5: Suppose that N 7 and 4 d N 3. Then H ) < H (N N,d ).
Poof: Label the node in N N,d that is a distance fom the longest path as node 1, and label the node it is adjacent to as node. Then we can fom P N,d, d fom N N,d by moving node 1 one node close to the longest path. Then d 1,j deceases by 1 fo j = 3,..., N, and d 1, inceases by 1. Since N 7, the sum of all path lengths in P N,d, d is less than in N N,d. Thus, by equations (7) and ) (6), H < H (N N,d ). Now, we have enough to detemine the tee in T N,d with smallest H nom. Theoem 4: Let N 4 and d N. The tee in T N,d with smallest H nom is P N,d, d. Poof: Fo d =, T N,d only contains K 1,N 1, which is the same as P N,,1. Fo d = 3, the esult follows fom Theoem 1 since D N,1,N 3 = P N,3,1. Fo 4 d N 3, this is a simple consequence of Lemmas 3, 4 and 5. Fo d = N, the esult follows fom Theoem 3. Finally, we can combine seveal of ou ealie esults to obtain a basic compaison between tees of diffeent diametes. Theoem 5: Let 3 d N 1. Fo any tee in T N,d, thee is a tee in T N,d 1 with a smalle H nom. Hence, the sta K 1,N 1 has the smallest H nom of any tee with N nodes. Poof: By Lemma, H (P N,N,i ) < H (P N ) (fo any 1 i N ). Let 4 d N. Suppose T T N,d. Then by Theoem 4, H (T ) H. But by Lemma, H ) ) > H (P N,d 1, d 1 ). Thus H (P N,d 1, d 1 < H (T ). Let T T N,3. Then by Theoem 1, H (T ) H (D N,1,N 3 ). But by Lemma, H (K 1,N 1 ) < H (D N,1,N 3 ). Thus H (K 1,N 1 ) < H (T ). ) VIII. DISCUSSION We wee able to deive the esults in Section VII by detemining the effect of moving leaves on effective esistances. With ou well-defined definition of diected esistance fo diected gaphs, the same calculations can be made fo diected tees as well. Thus ou appoach in this pape povides a constuctive method fo deiving a patial odeing of diected tees accoding to thei H noms. Futhe details will appea in a futue publication. Pevious known esults on the Wiene index of tees do not povide the same oppotunity fo the examination of diected tees. Additionally, the manipulations we used to pove ou esults suggest how tees can be eaanged to impove thei H noms in a decentalized fashion. In paticula, we showed that fo a non-sta tee, the H nom can always be educed eithe by moving a single node to somewhee else in the tee, o by moving a bouquet of nodes to an adjacent node. These manipulations ae local in the sense that nodes ae moved only fom a single location in the tee at a time, and the est of the nodes in the tee ae not equied to take any additional action. Thus, we can popose the following decentalized method to impove the obustness of a tee. Conside a lage tee connecting many nodes, each of which has only local infomation. Fo example, suppose each node knows the gaph between it and a fixed numbe of othe nodes, as well as the degees of each of these nodes. 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