1 Consensus and Coherence in Fractal Networks Stacy Patterson, Member, IEEE and Bassam Bamieh, Fellow, IEEE

Abstract—We consider first and second order consensus algo- determined by the spectrum of the Laplacian, though in a rithms in networks with stochastic disturbances. We quantify the slightly more complex way [5]. deviation from consensus using the notion of network coherence, Several recent works have studied the relationship between which can be expressed as an H2 norm of the stochastic system. We use the setting of fractal networks to investigate the question network coherence and topology in first-order consensus sys- of whether a purely topological measure, such as the fractal tems. Young et al. [4] derived analytical expressions for dimension, can capture the asymptotics of coherence in the large coherence in rings, path graphs, and star graphs, and Zelazo system size limit. Our analysis for first-order systems is facilitated and Mesbahi [11] presented an analysis of network coherence by connections between first-order stochastic consensus and the in terms of the number of cycles in the graph. Our earlier global mean first passage time of random walks. We then show how to apply similar techniques to analyze second-order work [5] presented an asymptotic analysis of network coher- stochastic consensus systems. Our analysis reveals that two ence for both first and second order consensus algorithms in networks with the same fractal dimension can exhibit different torus and lattice networks in terms of the number of nodes asymptotic scalings for network coherence. Thus, this topological and the network dimension. These results show that there characterization of the network does not uniquely determine is a marked difference in coherence between first-order and coherence behavior. The question of whether the performance of stochastic consensus algorithms in large networks can be second-order systems and also between networks of different captured by purely topological measures, such as the spatial spatial dimensions. For example, in a one-dimensional ring dimension, remains open. network with first-order dynamics, the per-node variance of Index Terms—distributed averaging, autonomous formation the deviation from consensus grows linearly with the number control, networked dynamic systems nodes, while in a two-dimensional torus, the per-node variance grows logarithmically with the number of nodes. Even more importantly, this work shows that these coherence scalings are I. INTRODUCTION the best achievable by any local, linear consensus algorithm. Thus, in lattice and torus graphs, the network dimension im- ISTRIBUTED CONSENSUS is a fundamental problem poses a fundamental limitation on the scalability of consensus in the context of multi-agent systems and distributed algorithms. formationD control [1], [2]. In these settings, agents must reach A natural question is whether the same dimension- agreement on values like direction, rate of travel, and inter- dependent limitations exist in networks with different struc- agent spacing using only local communication, A critical ques- tures, namely graphs that do not have an integer dimension. tion is how robust these systems are to external disturbances, As a first step towards answering this question, we analyze and in particular, how this robustness depends on the network the coherence of first-order and second-order consensus algo- topology. In the presence of stochastic disturbances, a network rithms in self-similar, tree-like fractal graphs. For first-order never reaches consensus, and the best that can be hoped for is systems, we draw directly from literature on random walks that certain measures of deviation from consensus are small. on fractal networks [12], [13] to show that, in a network with In this work, we investigate the performance of systems 1/df N nodes, the network coherence scales as N where df with first-order and second-order consensus dynamics in the is the fractal dimension (also the Hausdorff dimension, in our presence of additive stochastic disturbances. In particular, we case) of the network. We then show how the techniques used study algorithm performance for two classes of self-similar for the analysis of random walks can be extended to analyze networks with non-integer topological dimension. We use an the coherence of second-order consensus systems, and we H2 norm as a measure of deviation from consensus, which present asymptotic results for the per-node variance in terms thus quantifies a notion of network coherence. For systems of the network size and fractal dimension. An interesting and with first-order dynamics, it has been shown that this H2 norm perhaps unexpected result of our analysis is that the fractal di- can be characterized by the trace of the pseudo-inverse of the mension does not uniquely determine the asymptotic behavior Laplacian matrix [3]–[6]. This value has important meaning of network coherence. We show that two self-similar graphs not just in consensus systems, but in electrical networks [7], with the same fractal dimension exhibit different coherence [8], random walks [9], and molecular connectivity [10]. For scalings. We note that a preliminary version of this work, systems with second-order dynamics, this H2 norm is also without mathematical derivations, appeared in [14]. The remainder of this paper is organized as follows. In This work was supported in part by NSF INSPIRE award 1344069 and NSF ECCS award 1408442. Section II, we present the models for first-order and second- S. Patterson is with the Department of Computer Science, Rensselaer order noisy consensus systems and give a formal definition of Polytechnic Institute, Troy, NY, 12180, USA, [email protected]. network coherence for each setting. We also present several B. Bamieh is with the Department of Mechanical Engineering, Univer- sity of California, Santa Barbara, Santa Barbara, California 93106, USA properties from other domains that are mathematically similar [email protected]. to network coherence. In Section III, we describe the fractal 2

1 graph models, and in Section IV we present analytical results where J is the projection operator J := I 11⇤, with 1 the N on the coherence scalings for these fractal graphs. Section V N-vector of all ones. It is well known that HFO is given by provides a discussion of the relationship between graph dimen- the H2 norm of the system defined in (1) and (2), sion and coherence, followed by our conclusions in Section VI. 1 1 L⇤t Lt HFO = tr e Je dt . N 0 II. NETWORK COHERENCE ✓Z ◆ It has been shown that HFO is s completely determined by the We consider local, linear first-order and second-order con- spectrum of L [3]–[5]. Let the eigenvalues of L be denoted sensus algorithms over a network modeled by an undirected, 0= < ... . The first-order network coherence 1 2   N connected graph G with N nodes and M edges. We denote is then equal to, the adjacency matrix of G by A, and D is a diagonal matrix N where the diagonal entry D is the degree of node i. The 1 1 ii H = . (3) Laplacian matrix of the graph G is denoted by L and is defined FO 2N i=2 i as L := D A. X Our objective is to study the robustness of consensus algo- rithms when the nodes are subject to external perturbations and B. Coherence in Networks with Second-Order Dynamics to analytically quantify the relationship between the system In the second-order consensus problem, each node j has robustness and the graph topology. We capture this robustness two state variables x1,j(t) and x2,j(t). The state of the entire using a quantity that we call network coherence. We now system is thus captured in two N-vectors, x1(t) and x2(t). formally define the system dynamics and the notion of network Nodes update their states based on local feedback, i.e., the coherence (Sections II-A and II-B). We then present some states of their neighbors in the graph, and they are also subject mathematically related properties that we can leverage in our to random external disturbances that enter through the x2(t) robustness analysis (Section II-C). terms. The dynamics of the system are, x˙ (t) 0 I x (t) 0 1 = 1 + w(t), (4) A. Coherence in Networks with First-Order Dynamics x˙ (t) L L x (t) I  2   2  In the first-order consensus problem, each node j has a where w(t) is a 2N disturbance vector with zero-mean, unit single state xj(t). The state of the entire system at time t is variance, and uncorrelated second-order processes. given by the vector x(t) RN . Each node state is subject to 2 These system dynamics arise in the problem of autonomous stochastic disturbances, and the objective is for the nodes to vehicle formation control (e.g., see [5]). Here, x1(t) contains maintain consensus at the average of their current states. the vehicles’ positions and x2(t) contains the vehicles’ veloc- The dynamics of this system are given by, ities. The vehicles attempt to maintain a specified formation traveling at a fixed velocity while subject to stochastic external x˙(t)= Lx(t)+w(t), (1) perturbations. The non-zero entries of L specify the communi- cation links of the formation, i.e., if L = 1, then node i can where is the gain on the communication links, and w(t) ij is a size N disturbance vector with zero-mean, unit variance, observe the position and velocity of node j, and vice versa. and uncorrelated second-order processes. As is well known for It has been shown that similar system dynamics also arise in linear systems driven by second-order processes [15]–[17], one problems in phase synchronization in power networks [18]. can work directly with signal correlations without explicitly The network coherence of the second-order system (4) writing stochastic differential equations, and we interpret the is defined in terms of x1(t) only, and as with first-order above equation in this sense. coherence, it captures the deviation from the average of x1(t). In the absence of the disturbance processes, the system Definition 2.2: The second-order network coherence is the converges asymptotically to consensus at the average of the mean (over all nodes), steady-state variance of the deviation initial states [2]. With the additive noise term, the nodes do from the average of x1(t), not necessarily converge to consensus, but instead, node values 1 N 1 N fluctuate around the average of the current node states. HSO := lim var x1,j(t) x1,k(t) . N t N The concept of network coherence captures the variance of j=1 !1 ( ) X kX=1 these fluctuations in the first-order consensus system. In the vehicle formation problem, this quantity captures the Definition 2.1: The first-order network coherence is defined deviation of the vehicle positions from a rigid formation as the mean (over all nodes), steady-state variance of the traveling at the average position. deviation from the average of the current node states, We define the output for the system (4) as N N 1 1 x1(t) HFO := lim var xj(t) xk(t) . y(t)= J 0 , (5) t N ( N ) x2(t) !1 j=1 k=1  X X ⇥ ⇤ We define the output of the system (1) to be where J is again the projection operator. The second-order network coherence is given by the H2 norm of the system de- y(t)=Jx(t), (2) fined by (4) and (5). This value is also completely determined 3 by the eigenvalues of the Laplacian matrix [5], specifically, III. SELF-SIMILAR GRAPHS 1 N 1 Ideally, one would like to find an analytical expression for H = . (6) SO 22N 2 network coherence that depends on the graph topology. While i=2 (i) X it is a difficult problem to characterize the spectrum of the Laplacian matrix for a general graph, for graphs with special C. Related Concepts structure, it is sometimes possible to find a closed form for The eigenvalues of the Laplacian are linked to the topology the either the eigenvalues themselves or for the sum of their of the network, and therefore, it is not surprising that these inverses. For example, for d-dimensional torus and lattice net- eigenvalues play a role in many graph problems. In fact, the works, the Laplacian is a circulant operator. Its eigenvalues can sum, be determined analytically using a Discrete Fourier Transform, N 1 and analytical expressions that relate coherence to the network S := (7) size and dimension have been derived [5], [22], [23]. i=2 i X To extend this type of analysis to other graph topologies, that appears in the expression for first-order coherence in (3) we require that such graphs have both a dimension and a is an important quantity, not just in the study of consensus prescribed method of increasing the graph size that preserves algorithms, but in several other fields. We can leverage work this dimension. A class of graphs that exhibits these properties in these fields to develop analytic expressions for network is the class of self-similar graphs. Informally, a self-similar coherence for different graph topologies. We briefly review graph is one which exhibits the same structure at every scale. these related properties below. For a more formal definition, we refer the reader to [24]. 1) Effective resistance in an electrical network: Let the One notion of dimension of a self-similar graph is the fractal graph represent an electrical network where each edge is a dimension, which is defined as follows [25]. unit resistor. The resistance distance rij between two nodes Definition 3.1: Let G =(V,E) be an infinite, connected, i and j is the potential distance between them when a one undirected graph where each vertex has finite degree. Let v 2 ampere current source is connected from node j to node i. V be an arbitrary vertex, and define B(r) to be the of radius The total effective resistance of the network, also called the r, centered at v, i.e., Kirchoff index [7], [8], is the sum of the resistance distances over all pairs of nodes in the graph. It has been shown [19] B(r)= u V : d(u, v) r , { 2  } that the total effective resistance depends on the spectrum of the Laplacian matrix as R =2NS. where d(u, v) denotes the length (number of edges) in the 2) Global mean first passage time of a random walk: In a shortest path between u and v in G. The fractal dimension of simple random walk on a undirected graph, the probability of G is 1 ln( B(r) ) moving from a node i to a neighboring node j is d where df := lim sup | | . i r ln(r) di is the degree of node i. The first passage time fij is the !1 average number of steps it takes for a random walk starting Other notions of dimensions include the Hausdorff dimen- at node i to reach node j for the first time. The global mean sion and the box counting dimension. For self-similar graphs first passage time is the average first passage time over all of the type we study in this work, the values of these three pairs of nodes. It has been shown that, for a connected graph, dimensions are equivalent [24]. the mean first passage time between nodes i and j and the Torus and lattice graphs are both self-similar graphs, and resistance distance are related as fij + fji =2Mrij, where their fractal dimensions are equivalent to the natural dimension M is the number of edges in the graph [9]. The global mean definition, e.g., a 2-dimensional torus has df =2. In this work, first passage time is therefore related to the effective resistance we give coherence analysis for two classes of self-similar as graphs that have fractional dimensions, tree-like fractals and M 2M F = R = S. (8) Vicsek fractals. We now describe the construction of these N(N 1) N 1 graphs. If the graph is a tree, then M = N 1, and the global mean Tree-Like Fractals: Each family of tree-like fractal graphs first passage time is simply F =2S. is parameterized by a positive integer m. The graphs are 3) Quasi-Wiener index: The Wiener index is a measure constructed in an iterative manner, and each iteration yields of molecular connectivity [20]. Here, the graph represents a new graph generation. This generation 1 graph consists of a molecule where nodes are atoms and edges are chemical two vertices, or nodes, connected by a single edge. Given a bonds, and the distance between two atoms is the length graph of generation g, denoted Gg, the graph of generation (number of edges) of the shortest path between them. The g +1 is formed by replacing each edge with a path of length Wiener index is the sum of the distances between all pairs 2. In other words, each edge (i, j) in Gg, is replaced by two of non-hydrogen atoms. If the molecular graph is acyclic, edges (i, k) and (k, j) where k is a new node (not existing in this value is exactly S in (7) [10]. If the graph contains graph Gg). Then, m additional new nodes are added to every cycles, the Wiener index is no longer equal to the sum of new node k. The generation g graph thus has (m + 2)g +1 lengths of the shortest paths. However, this quantity is still nodes. The process is illustrated in Fig. 1 for m =2. This utilized in mathematical chemistry and is called the quasi- tree-like fractal model encompasses several well-known fractal Wiener index [21]. graphs including the T-graph (m =1) [26] and the Peano basin 4

A. Coherence in Tree-Like Fractals In their recent work, Lin et al. analyze the global mean first g = 1! passage time in tree-like fractals [12]. Their approach is based g = 2! on first deriving a recursion for the characteristic polynomial of the Laplacian matrix. They then show that the sum S in (7) can be obtained by solving for several coefficients of this `" characteristic polynomial. We first review the construction of this polynomial and then show how it can be used to derive expressions for network coherence. g = 3! Let Gg be the graph of generation g with Ng nodes, and let Lg denote its Laplacian. The characteristic polynomial for Fig. 1. First three generations of the tree-like fractal for m =2. Lg is P (x)=det(L xI ). (9) g g g Here I is the N N identity matrix. As we are only g g ⇥ g g = 1! interested in the non-zero eigenvalues of Lg, we instead consider a modified version of the characteristic polynomial, 1 P (x)= P (x). (10) g x g

The modified characteristic polynomial P g can be written as, N 1 N 1 g g (i) i (N 1) P (x)= p x = p g (x ¯ ), (11) g g g g,i i=0 i=1 X Y (i) i where pg denotes the coefficient of the term x and ¯g,i, g = 2! i =1...(Ng 1), are the roots of P g(x). g = 3! In the remainder of Section IV-A, we show how to deter- mine network coherence from coefficients of (11). Fig. 2. First three generations of the Vicsek fractal for v =4. 1) First-order network coherence: For systems first-order dynamics, network coherence depends on the sum of the roots fractal (m =2) [27]. The tree-like fractal graph has a fractal of (10), Ng Ng 1 dimension of d = log(m + 2)/ log(2) [12]. 1 1 f S = = , Vicsek Fractals: The family of Vicsek fractal graphs is g ¯ i=2 g,i i=1 g,i also parameterized by a positive integer v, and each family is X X constructed in an iterative manner [28], [29]. The generation where g,i, i =1...Ng are the roots of Pg(x). Lin et al. [12] 1 graph is a star graph with v +1 nodes. The graph for show that the sum Sg can be expressed in terms of the zeroth generation g +1 is generated from the generation g graph by order and first order coefficients of (11), (1) making v copies of Gg and arranging them in a star around pg Sg = . (12) Gg. These copies are connected to Gg by adding edges from (0) pg the v corners of the Gg, each one linking to a corner of a g Therefore, to find this sum, one only needs to determine these copy. Thus, the generation g graph has Ng =(v + 1) nodes. We illustrate the first three generations of the Vicsek fractal two coefficients. The coefficients can be found by first deriving a recursion for v =4in Fig. 2. The Vicsek fractal has a fractal dimension for P g(x), as follows. Let Qg be the characteristic polyno- of df = log(v + 1)/ log(3). mial of the (N 1) (N 1) submatrix of L formed g ⇥ g g IV. COHERENCE ANALYSIS by removing a single column and row corresponding to an outermost node. Let R be the characteristic polynomial of In this section, we present our analysis of network coherence g the (N 2) (N 2) submatrix of L formed by removing for the families of fractal graphs described in the previ- g ⇥ g g columns and rows corresponding to two outermost nodes. ous section. Several recent works have analyzed the global The following equations capture the relationship between the mean first passage time of a random walk for these graph modified characteristic polynomials for L and L , families [12], [13], [29]. Using the relationship described in g g+1 m+1 m+2 Section II-C, we can extend this analysis to derive asymptotic P g+1(x)=(m + 2)[Qg(x)] P g(x)+(m + 1)[Qg(x)] scalings for network coherence in systems with first-order (13) m+2 m+1 noisy consensus dynamics. Below, we briefly summarize the Qg+1(x)=[Qg(x)] +(m + 1)xRg(x)[Qg(x)] analytical techniques used in previous works and formally m +(m + 1)xRg(x)[Qg(x)] P g(x)) (14) state the asymptotic scalings for first-order network coherence m+1 2 m in fractal networks. We then derive expressions for network Rg+1(x)=2Rg(x)[Qg(x)] +(m + 1)x[Rg(x)] [Qg(x)] 2 m 1 coherence in fractal networks with second-order dynamics. + mx[Rg(x)] [Qg(x)] P g(x). (15) 5

(0) (0) (0) To find pg , first let qg and rg be the constant terms for P g and Qg in (13) and (14), respectively, Q (x) and R (x) respectively (when written in the form like g g (1) (0) m+1 (1) (0) (0) m (1) (11)). One can then write recursions on these constant terms rg+1 = 2[qg ] rg + 2(m + 1)rg [qg ] qg (0) 2 (0) m (0) 2 (0) m 1 (0) as, +(m + 1)[rg ] [qg ] + m[rg ] [qg ] pg (18) p(2) =(m + 2)[q(0)]m+1p(2) p(0) =(m + 2)[q(0)]m+1p(1) +(m + 1)[q(0)]m+2 g+1 g g g+1 g g g (0) m (2) (0) (0) (0) m+2 +(m + 1)(m + 2)[qg ] qg pg qg+1 =[qg ] + m(m+1)(m+2) [q(0)]m 1[q(1)]2p(0) (0) (0) (0) m+1 2 g g g rg+1 =2rg [qg ] . (0) m (1) (1) +(m + 1)(m + 2)[qg ] qg pg (0) m+1 (2) Given the initial values for the generation 1 graph, p(0) = +(m + 1)(m + 2)[qg ] qg 1 2 (0) (0) (0) (0) (m+1) (m+2) (0) m (1) 2 m 3, q1 =1, and r1 =2, one can solve for pg , qg , + 2 [qg ] [qg ] . (19) (0) and rg . Using a similar approach, one can also solve for (2) (0) m+1 (2) (m+1)(m+2) (0) m (1) 2 (1) qg+1 =(m + 2)[qg ] qg + 2 [qg ] [qg ] pg and, therefore, for the entire expression Sg (see [12] for (1) (0) m+1 2 (0) (0) m (1) details). +(m + 1)rg [qg ] +(m + 1) rg [qg ] qg (0) m (0) (1) (0) (0) (0) m 1 (1) With these results, Lin et al. arrive at the following expres- +(m + 1)[qg ] pg rg + m(m + 1)rg pg [qg ] qg sion for the asymptotic order of the global mean first passage (0) (0) m (1) +(m + 1)rg [qg ] pg . (20) time in a tree-like fractal of generation g with Ng nodes, We then solve for these coefficients. The full derivation is

Ng given in Appendix A. 1 1+log(2)/ log(m+2) F =2 N . As we are interested in the asymptotic behavior of HSO, g ⇠ g i=2 g,i we consider only the highest order terms of the coefficients in X (17), which are given by, By using the relationship between F and S defined in g g p(0) (m + 2)g Section II-C , we can easily obtain an analytical expression g ⇠ for the coherence of first-order consensus algorithms in tree- p(1) 2g(m + 2)2g g ⇠ like fractals. (2) 2g 3g pg 2 (m + 2) . Theorem 4.1: For a tree-like fractal with N nodes, param- ⇠ eterized by the integer m, the first-order network coherence Thus, the order of the sum in (16) is, of the system with dynamics defined in (1) is given by Ng 1 1 2g 2g 2+2 log(2)/ log(m+2) 2 2 (m + 2) Ng . 1 1 ¯ ⇠ ⇠ H N log(2)/ log(m+2) = N 1/df , i=1 g,i FO ⇠ X Here, the last expression follows from the fact that the gener- g ation g tree-like fractal graph has Ng =(m + 2) +1nodes. where df is the fractal dimension of the graph. We then substitute the expression for this sum into HSO in 2) Second-order network coherence: We now show how (6) to arrive at the following theorem. we extend the work above to derive an expression for second- Theorem 4.2: For a tree-like fractal with N nodes, param- order network coherence in tree-like fractals. In this case, for eterized by the integer m, the second-order coherence of the g the generation graph, we are interested in a sum of the form, system with dynamics as defined in (4) is given by

N 1 g 1 1+2 log(2)/ log(m+2) 1 1+(2/df ) 1 HSO N = N , . (16) ⇠ 2 2 ¯ 2 i=1 g,i X where df is the fractal dimension of the graph. Using equation (11) and Vieta’s formulae, we can express this sum in terms of coefficients of P (x). For a tree-like fractal of B. Generalized Vicsek Fractals generation g with Ng nodes, the sum is, To analyze the network coherence in Vicsek fractals, we exploit a different technique that was used in the analysis of 2 Ng 1 (1) (2) 1 pg pg the global mean first passage time. As with tree-like fractals, it = 2 , (17) ¯ 2 (0) (0) is not straightforward to find a closed form for the individual i=1 g,i pg ! pg X eigenvalues of the Laplacian matrix of a Vicsek fractal. In their recent work [13], Zhang et al. determined a closed-form (0) (1) The values for pg and pg were derived by Lin et al. [12] expression for the sum S in (7). With this sum, we can easily in their analysis of the global mean first passage time. What obtain the first-order network coherence. We first summarize (2) remains is to solve for pg . To do this, we first find the the results for first-order coherence, and we then show how we recursion equations for the coefficients corresponding to the can use a similar approach to obtain a closed-form expression first-order term in Rg in (15) and the second-order terms in for second-order coherence. 6

1) First-order network coherence: This analysis makes This expression can then be combined with the multiplicity of use of several previously derived properties relating to the the degenerate eigenvalues defined in Property 4.1 to find the eigenvalues of Lg for Vicsek fractals [29]–[31], which we sum of the inverses of the degenerate eigenvalues of Lg, restate here convenience. g 1 1 Property 4.1: Let Lg be the Laplacian matrix for a gen- D = g ii eration g Vicsek fractal. The eigenvalues of Lg satisfy the D g,i g,i ⌦g i=0 following: X2 X 1 2(v + 2) 3g(v + 1)g 1 = (v 2)(v + 1)g+1(3g 1) + . 1) The non-degenerate eigenvalues of L1 are 0 and v +1. 2 v +1 3v +2 L1 has one degenerate eigenvalue with value one and (26) multiplicity v 1. It is then straightforward to add (23) and (26) to obtain Sg, 2) Every eigenvalue of Lg is also an eigenvalue of Lg+1. g 1 g g g As a result, eigenvalues preserve their degeneracy in (v 2)(v + 1) (3 1) v +23 (v + 1) 1 Sg = + . subsequent generations. 2 v +1 3v +2 3) For Lg, the multiplicity of the one eigenvalue is, Using the facts that the number of nodes in generation g graph g g log(3)/ log(v+1) g 1 is Ng =(v + 1) and that 3 = Ng , the sum Sg := (v 2)(v + 1) +1. (21) g can be shown to be of the following order, A degenerate eigenvalue that appears for the first time 1+log(3)/ log(v+1) 1+1/df Sg Ng = Ng . at generation j has multiplicity g j. ⇠ The expression for network coherence in Vicsek fractals 4) Each non-zero eigenvalue g,i in Lg produces three new with first-order consensus dynamics immediately follows from eigenvalues in Lg+1 according the relation, Sg and is formally stated in the following theorem. g+1,i(g+1,i 3)(g+1,i v 1) = g,i. (22) Theorem 4.3: For a generalized Vicsek fractal graph with N nodes, parameterized by the positive integer v, the first- To find the sum Sg, Zhang et al. [13] consider the sums order coherence of the system with the dynamics defined in for degenerate and non-degenerate eigenvalues separately. Let (1) is given by ND ⌦g be the set of non-degenerate eigenvalues of Lg, exclud- 1 1 D log(3)/ log(v+1) 1/df ing 0, and let ⌦ be the set of degenerate eigenvalues. Thus HFO N = N , g ⇠ Sg is equivalent to, where df is the fractal dimension of the graph. 1 1 2) Second-order network coherence: To find the second- Sg = + . ND D order network coherence, we also determine a recursion over ⌦g ⌦g 2X X2 the eigenvalues of Lg, in this case, a recursion over the sum In the generation 1 graph, there is a single non-degenerate, of the squares of the inverses of the eigenvalues. non-zero eigenvalue, v +1. This eigenavalue produces three As the first step in this analysis, we consider the relationship non-degenerate eigenvalues in generation 2, according to (22). between the eigenvalues of Lg and Lg+1. Let g,i be a non- These eigenvalues are the first-generation descendants of v+1. zero eigenvalue of Lg. This eigenvalue produces three new The first generation descendants yield 32 second-generation eigenvalues in Lg+1, according to (22), which we denote by ND g+1,i , g+1,i , and g+1,i . We want to define the sum of descendants (also non-degenerate), and so on. Let i be 1 2 3 the sum of the reciprocals of the ith generation descendants of the squared inverses of these three eigenvalues in terms of the v +1. By employing the recursion on the eigenvalues defined parent eigenvalue. by (22) in Property 4.1, Zhang et al. obtain the following We first expand the expression for the sum of squared ND inverses, as follows, closed-form expression for i , 1 1 1 2 + 2 + 2 = ND 1 i i 1 (g+1,i1 ) (g+1,i2 ) (g+1,i3 ) i = =3(v + 1) . (23) 2 ND ND 1 1 1 (⌦i ⌦i 1 ) + + 2 X g+1,i1 g+1,i2 g+1,i3

⇣ g+1,i1 +g+1,i2 +⌘g+1,i3 They then sum over the generations of descendants, i = 2 . (27) g+1,i1 · g+1,i2 · g+1,i3 0,...,g 1, to find the sum of the inverses of the non- ⇣ ⌘ Then, leveraging the analysis in Zhang et al. [13] for Sg, we degenerate eigenvalues of Lg, obtain a recursive expression for the first term on the right- g 1 g g hand size of (27), 1 ND 1 3 (v + 1) 1 = i = . (24) 2 2 v +1 3v +2 1 1 1 (3(v+1)) ⌦ND i=0 + + = ( )2 . (28) 2Xg X g+1,i,1 g+1,i,2 g+1,i,3 g,i ⇣ ⌘ A similar approach can be used to find a closed-form For the second term, we use the following equivalences, expression for the sum of reciprocals of the ith generation obtained by application of Vieta’s formulae to (22), descendants of a single degenerate eigenvalue 1, = (29) g+1,i1 · g+1,i2 · g+1,i3 i,g D i i g+1,i + g+1,i + g+1,i =(v + 4). (30) i =3(v + 1) . (25) 1 2 3 7

With these equivalences, we can rewrite the second term on Finally, we combine the results in (34) and (35) to find the the right-hand size of (27) as, sum of the squared inverses of the non-zero eigenvalues of Lg, g+1,i1 + g+1,i2 + g+1,i3 2(v + 4) 2 = . (31) Ng g+1,i1 g+1,i2 g+1,i3 i,g 1 ✓ · · ◆ 32g(v + 1)2g = N 2+2 log(3)/ log(v+1). 2 ⇠ g Combining (28) and (31), we obtain to following relation- i=2 (g,i) X ship between g,i and the sum of the squared inverses of its From this result and the definition of second-order network children, coherence in (6), we arrive at the following theorem. 1 1 1 (3(v+1))2 2(v+4) Theorem 4.4: For a generalized Vicsek fractal graph with ( )2 + ( )2 + ( )2 = ( )2 . (32) g+1,i1 g+1,i2 g+1,i3 g,i g,i N nodes, parameterized by the integer v, the second-order We now use the expression (32) to find a recursion over coherence of the system with the dynamics defined in (4) is the sum of the squared inverses of the eigenvalues of Lg. 1 1 H N 1+2 log(3)/ log(v+1) = N 1+(2/df ), As in [13], we consider the non-degenerate and degenerate SO ⇠ 2 2 eigenvalues separately. where d is the fractal dimension of the graph. Recall that v +1 is the single, non-zero, non-degenerate f ND eigenvalue of L1. Let ⇥i be the sum of the squared inverses of the ith generation descendants of v +1, i.e., V. C OHERENCE AND GRAPH DIMENSION The coherence scalings for several families of self-similar ND 1 ⇥i = 2 . graphs are presented in Table I. We note that for the fractal ND ND ⌦i ⌦i 1 graphs studied in this paper, the coherence expressions for 2 X both first-order and second-order systems exhibit the same Using (32), we obtain a recursion over this sum, scalings a one-dimensional torus or lattice, i.e., coherence 1/d 1+2/d (3(v+1))2 2(v+4) scales as N f for first-order systems and N f for ⇥ND = i 2 second order systems. This is true even for the fractals with ND ND ⌦i ⌦i 1 2 X ⇣ ⌘ fractal dimension greater than or equal to two. These results 2 ND ND = (3(v + 1)) ⇥i 1 2(v + 4)i 1 . demonstrate that the fractal dimension does not uniquely ND determine the scaling behavior of network coherence. The We then substitute in the expression for i 1 in (23) and ND Peano Basin Fractal and the two-dimensional lattice both obtain a closed-form expression for ⇥i , have fractal dimension equal to two but exhibit strikingly ND 7v 12 2i 2i 2(v+4) i 1 i 2 different coherence scalings. Therefore, it seems that there ⇥ = 2 3 (v + 1) + 3 (v + 1) . i 9(v+2)(v+1) 3v+2 other characteristics of the graph that affect the asymptotic (33) scalings of network coherence. The full derivation of this expression is given in Appendix B. One possible characterization is the spectral dimension, By summing over the generations i =0,...,g 1, we can which is defined as follows [32]. find the sum of the squared inverses of the non-degenerate Definition 5.1: Let ⇢(x) be the eigenvalue counting func- eigenvalues of L . As we are interested in the asymptotic g tion of L, i.e., ⇢(x) is the number of eigenvalues of L that have behavior for second-order coherence, we only consider the magnitude less than or equal to x. The spectral dimension of highest order term, which is as follows, the graph is g 1 log(⇢(x)) 1 ND 2g 2g ds := 2 lim . = ⇥ 3 (v + 1) . (34) x log(x) 2 i ⇠ !1 ⌦ND i=0 2Xg X For torus and lattice graphs, the fractal dimension is equal to We now consider the descendants of a degenerate eigenvalue spectral dimension, and therefore all of the previous coherence results are the same for the fractal and spectral dimensions. of Lg with value 1. Using the recursion (32), we can obtain a similar expression for the ith generation descendants of this For the families of fractal graphs studied in this paper, the eigenvalue, spectral dimension and the fractal dimension are related as ds =(2df )/(df + 1) (see [33]). With this relationship, we D 2(v+4) 2i 2i can derive expressions for network coherence in terms of ⇥i = 1 34(v+1)3(v+2) 3 (v + 1) the spectral dimension. For first order-systems the network 2(v+4) i 3 ⇣ ⌘ 2/ds 1 3v+2 (3(v + 1)) . coherence is HFO N , and for second order systems, 4/d⇠ 1 it is H N s . We note that the fractals considered The full derivation of this expression is given in Appendix B. SO ⇠ in this paper all have spectral dimension less than two. Then, incorporating the multiplicity of the degenerate eigen- Therefore, by considering the spectral dimension, we eliminate values as stated in Property 4.1, we can find the sum of the the conflicting coherence results we obtained for graphs with squared inverses of the degenerate eigenvalues, fractal dimension equal to two. g 1 In general, there is not a straightforward relationship be- 1 D 2g 2g = g i⇥i 3 (v + 1) . (35) tween the fractal and spectral dimensions, and while the fractal 2 ⇠ ⌦D i=0 dimension is defined in terms of the graph topology, the X2 g X 8

TABLE I EXAMPLES OF SELF-SIMILAR GRAPHS WITH N NODES AND THEIR DIMENSIONS AND COHERENCE SCALINGS.

Network Fractal Dimension HFO HSO 1 1 3 1-dimensional torus 1 N 2 N Generalized Vicsek Fractal with v =4 log(5) 1.46 1 N log(3)/ log(5) 1 N 1+2(log(3)/ log(5)) log(3) ⇡ 2 T Fractal (tree-like fractal with m =1) log(3) 1.58 1 N log(2)/ log(3) 1 N 1+2(log(2)/ log(3)) log(2) ⇡ 2 1 p 1 2 Peano Basin Fractal (tree-like fractal with m =2) 2 N 2 N 1 1 2-dimensional torus 2 log N 2 N relationship between the topology and the spectral dimension Note that this is the modified characteristic polynomial defined is not generally understood. In addition, it is not yet known in (10). The coefficient corresponding to the second order whether the spectral dimension plays the same role in deter- terms (in x) is mining network coherence for families of graphs beyond those p(2) = (m + 1) (m+3)(m+1)m . (36) studied in this work. The question of how to generalize our 1 2 results to other graph structures is an open problem and a For Q1, the characteristic polynomial is subject for future work. Q (x)=(1+( m 3)x + x2)(1 x)m. 1 VI. CONCLUSION The coefficient for the second order term (in x) is We have investigated the relationship between the topologi- (2) 3 2 5 cal dimension of a graph and the per node variance of the devi- q1 = m + m +1. (37) ation from consensus in systems with noisy consensus dynam- 2 2 ics. In first-order systems, the coherence measure is closely For R1, the characteristic polynomial is related to concepts in electrical networks, random walks, and 2 m 1 R1(x)=(2+( m 3)x + x )(1 x) . molecular connectivity. Drawing directly from literature on random walks in fractal graphs, we have derived asymptotic The coefficients for the first order term (in x) is expressions for first-order coherence in terms of the network r(1) = 3m 1. (38) size and fractal dimension. We have then extended this line 1 (1) of analysis to derive asymptotic expressions for second-order We now solve the recursion equation for rg . Substituting (0) (0) (0) (1) coherence for several families of fractal graphs. Our analysis the expressions for pg , qg , rg , and qg from [12] into shows that the fractal dimension does not uniquely determine (18), we obtain, the asymptotic behavior of network coherence, and in fact, two r(1) =2r(1) self-similar graphs with the same fractal dimension exhibit g+1 g g+1 g 1 g g different coherence scalings. We conclude that the question 2 (m + 1)(m + 2) [1 2 +2 (m + 2)] of whether performance of stochastic consensus algorithms +(m + 1)22g + m22g( (m + 2)g 1) in large networks can be captured by purely topological 2 =2r(1) + 2m 5m 2 22g(m + 2)g measures, such as the network’s fractal dimension, remains g (m+2) 2(m+1) g g 2g open. (m+2) 2 (m + 2) +2 . The algorithms analyzed in this paper are standard diffusive- (1) type consensus algorithms common in the literature. An im- Then, using the value for r1 in (38), we obtain a closed-form portant question for further research is whether the asymptotic expression for the above recursion, scalings we obtained for these algorithms can be improved g 1 2 by more sophisticated consensus algorithms. Earlier work in (1) g 1 (1) ( 2m 5m 2) g i rg =2 r1 + (m+2) 2 (2(m + 2)) lattice-type networks [5] has shown corresponding asymptotic i=1 X bounds to be tight for a large class of local interactions. g 1 g 1 Whether similar conclusions can be made for the present case 2(m+1) 2g (m + 2)i +2g 2i (m+2) of self-similar networks remains to be investigated. i=1 i=1 X X 2m2+5m+2 2g g 1 = 2( 3m 1) 2 (m + 2) APPENDIX A 2m+3 4m2+10m+4 g g+1 g 1 2g DERIVATIONS FOR SECOND-ORDER NETWORK + 2 2 (m + 2) +2 . 2m+3 COHERENCE IN TREE-LIKE FRACTALS The highest order term of r(1) is thus, First we find the characteristic polynomials for the matrices g (1) 2g g for the generation 1 graph, P 1, Q1 and R1. We then find the r 2 (m + 2) . g ⇠ coefficients that we need to solve the recursions (18) - (20). (2) For P 1, we have, Next, we solve the recursion for qg , which is given by, P (x)=( (m + 3) + x)(1 x)m+1. q(2) =(m + 2)q(2) + D , (39) 1 g+1 g g 9 where Recall that v +1 is single, non-degenerate, non-zero eigen- ND 1 (m+1)(m+2) (1) 2 value of L1. Therefore, ⇥0 = (v+1)2 . Substituting this and D = [qg ] g 2 the value for ND given in (23), we simplify (41) as follows, 2 g g (1) j ((m +2m + 2)(m + 2) + 1)2 qg ND 2i 1 g (1) g (1) ⇥i = (3(v + 1)) (v+1)2 +2 (m + 1)pg (m + 1)(m + 2) rg . i 1 (1) (1) 2(v+4) 2(i j 1) j Note that the closed-form expressions for pg and qg are (3(v + 1)) (3(v + 1)) ) v+1 given in [12]. Solving the recursion (39), using the initial value j=0 (2) 2i X2i 2 of q1 in (37), we find the following closed-form expression, =3 (v + 1) (2) i 1 g 1 3 2 5 qg (m + 2) 2 m + 2 m +1 2(i 1) 2i 3 j ⇠ 2(v + 4)3 (v + 1) (3(v + 1)) g 1 g i j=0 +(m + 2) (4(m + 2)) . 2i 2i 2 X i=1 =3 (v + 1) X 2(v+4) 2i 1) 2i 2 i 1 i 2 (2) 3 (v + 1) 3 (v + 1) The highest order term of qg is thus, 3v+2 7v 12 2i 2i 2(v+4) i 1 i 2 (2) 2g 2g = 9(v+2)(v+1)2 3 (v + 1) + 3v+2 3 (v + 1) . qg 2 (m + 2) ⇠ For the degenerate eigenvalue of 1 of L the closed-form (2) 1 Finally, we consider the recursion for pg , expression for the recursion over its descendants is the same (2) (2) as for the non-degenerate eigenvalue (v + 1), pg+1 =(m + 2)pg + Eg, (40) ⇥D = (3(v + 1))2i ⇥D where i 0 i 1 g (2) 2(i j 1) D E =(m + 1)(m + 2)( (m + 2) 1)q 2(v + 4) (3(v + 1)) . (42) g g j m(m+1)(m+2) g (1) 2 j=0 + 2 ( (m + 2) 1)[qg ] X D D (1) (1) We note that ⇥0 =1, and the closed-form expression for j +(m + 1)(m + 2)pg qg 2 is given in (42). By substituting these values into (42), we +(m + 1)(m + 2)q(2) + (m+1) (m+2) [q(1)]2. D g 2 g obtain the following closed form expression for ⇥i , (2) D 2i Using the initial value of p1 in (36), we obtain a closed-form ⇥i = (3(v + 1)) (2) expression for pg , i 1 2(i j 1) j 2(v + 4) (3(v + 1)) (3(v + 1)) ) (2) g 1 (2) g 1 g i pg =(m + 2) p1 + i=1 (m + 2) Ei. j=0 X (2) = (3(v + 1))2i The highest order for of pg is thus,P 2(v+4) 2i 3 i 3 (2) 2g 3g 3v+2 (3(v + 1)) (3(v + 1)) pg 2 (m + 2) . ⇠ 2(v+4) 2i 2i = 1 4 3 3 (v + 1) We can now find the asymptotic order of the desired sum 3 (v+1) (v+2) ⇣ 2(v+4) ⌘ i 3 in (17), (3(v + 1)) . 3v+2 Ng 2 1 p(1) p(2) g g 2g 2g ACKNOWLEDGMENT 2 = (0) 2 (0) 2 (m + 2) . ( ) pg pg ⇠ i=2 i X ✓ ◆ The authors would like to thank Stephen Boyd for a useful insight and discussion regarding networks with fractional APPENDIX B Hausdorff dimension. DERIVATIONS FOR SECOND-ORDER NETWORK COHERENCE IN VICSEK FRACTALS REFERENCES ND Here we show the details of the derivation of ⇥i in (33). [1] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile ND The recursive expression for ⇥i is, autonomous agents using nearest neighbor rules,” IEEE Trans. Autom. 2 Control, vol. 48, pp. 988–1001, June 2003. ND (3(v+1)) 2(v+4) [2] R. Olfati-Saber and R. Murray, “Consensus problems in networks of ⇥i = 2 ND ND agents with switching topology and time-delays,” IEEE Trans. Autom. ⌦i ⌦i 1 2 X ⇣ ⌘ Control, vol. 49, pp. 1520–1533, September 2004. 2 ND ND [3] L. Xiao, S. Boyd, and S.-J. Kim, “Distributed average consensus with = (3(v + 1)) ⇥i 1 2(v + 4)i 1 . least-mean-square deviation,” J. Parallel Dist. Comput., vol. 67, no. 1, pp. 33–46, 2007. We solve the above recursion to obtain, [4] G. F. Young, L. Scardovi, and N. E. Leonard, “Robustness of noisy ND 2i ND consensus dynamics with directed communication,” in Proc. American ⇥i = (3(v + 1)) ⇥0 Control Conference, 2010, pp. 6312–6317. i 1 [5] B. Bamieh, M. Jovanovic,´ P. Mitra, and S. Patterson, “Coherence in 2(i j 1) ND large-scale networks: Dimension dependent limitations of local feed- 2(v + 4) (3(v + 1)) . (41) j back,” IEEE Trans. Autom. Control, vol. 57, no. 9, pp. 2235–2249, j=0 X September 2012. 10

[6] E. Lovisari, F. Garin, S. Zampieri et al., “A resistance-based approach [29] A. Blumen, C. von Ferber, A. Jurjiu, and T. Koslowski, “Generalized to performance analysis of the consensus algorithm,” in Proc. 19th Int. vicsek fractals: Regular hyperbranched polymers,” Macromolecules, Sym. Mathematical Theory of Networks and Systems, 2010. vol. 37, no. 2, pp. 638–650, 2004. [7] D. J. Klein and M. Randic,´ “Resistance distance,” J. Math. Chem., [30] C. Jayanthi, S. Wu, and J. Cocks, “Real space green’s function approach vol. 12, pp. 81–95, 1993. to vibrational dynamics of a vicsek fractal,” Phys. Rev. B, vol. 69, no. 13, [8] D. Bonchev, A. T. Balaban, X. Liu, and D. J. Klein, “Molecular cyclicity pp. 1955–1958, 1992. and centricity of polycyclic graphs. I. cyclicity based on resistance [31] C. Jayanthi and S. Wu, “Dynamics of a vicsek fractal: The boundary distances or reciprocal distances,” Internat. J. 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Autom. Control, vol. 56, no. 3, pp. 544–555, 2011. [12] Y. Lin, B. Wu, and Z. Zhang, “Determining mean first-passage time Stacy Patterson is the Clare Boothe Luce Assistant on a class of treelike regular fractals,” Phys. Rev. E, vol. 82, no. 3, p. Professor in the Department of Computer Science 031140, September 2010. at Rensselaer Polytechnic University. She received [13] Z. Zhang, B. Wu, H. Zhang, S. Zhou, J. Guan, and Z. Wang, “Determin- the BS in mathematics and computer science from ing global mean-first-passage time of random walks on Vicsek fractals Rutgers University in 1998 and the MS and PhD in using eigenvalues of Laplacian matrices,” Phys. Rev. E, vol. 81, no. 3, computer science from the University of California, p. 031118, March 2010. Santa Barbara in 2003 and 2009, respectively. From [14] S. Patterson and B. Bamieh, “Network coherence in fractal graphs,” in 2009-2011, she was a postdoctoral scholar at the Proc. 50th IEEE Conf. Decision and Control and European Control Center for Control, Dynamical Systems and Compu- Conf., 2011, pp. 6445–6450. tation at the University of California, Santa Barbara. [15] E. Wong and B. Hajek, Stochastic processes in engineering systems. From 20011-2013, she was a postdoctoral fellow Springer-Verlag New York, 1985. at the Department of Electrical Engineering at the Technion. Dr. Patterson [16] K. J. Astr˚ om,¨ Introduction to stochastic control theory. Courier Dover is the recipient of a Vitterbi postdoctoral fellowship and the IEEE CSS Publications, 2012. Axelby Outstanding Paper Award. Her research interests are in distributed [17] H. Kwakernaak and R. Sivan, Linear optimal control systems. Wiley- and networked systems, cooperative control, and sensor networks. Interscience New York, 1972, vol. 1. [18] B. Bamieh and D. F. Gayme, “The price of synchrony: Resistive losses due to phase synchronization in power networks,” in Proc. American Bassam Bamieh is Professor of Mechanical Engi- Control Conference, 2013, pp. 5815–5820. neering and Associate Director of the Center for [19] D. Aldous and J. Fill, Reversible Markov Chains and Control, Dynamical Systems and Computation at the Random Walks on Graphs, In preparation. [Online]. Available: University of California at Santa Barbara (UCSB). http://www.stat.berkeley.edu/ aldous/RWG/book.html He received his B.Sc. degree in Electrical Engineer- [20] H. Wiener, “Structural determination of paraffin boiling points,” J. Am. ing and Physics from Valparaiso University (Val- Chem. Soc., vol. 69, no. 1, pp. 17–20, 1947. paraiso, IN) in 1983, and his M.Sc. and PhD degrees [21] S. Markovic,´ I. Gutman, and Z.ˇ Banceviˇ c,´ “Correlation between wiener in Electrical and Computer Engineering from Rice and quasi-wiener indices in benzenoid hydrocarbons,” J. Serb. Chem. University (Houston, TX) in 1986 and 1992 respec- Soc., vol. 60, pp. 633–636, 1995. tively. Prior to joining UCSB in 1998, he was an [22] E. Montroll, “Random walks on lattices. III. calculation of first-passage Assistant Professor in the Department of Electrical times with application to exciton trapping on photosynthetic units,” J. and Computer Engineering and the Coordinated Science Laboratory at the Math. Phys., vol. 10, pp. 753–765, 1969. University of Illinois at Urbana-Champaign (1991-98). Professor Bamieh’s [23] P. Barooah and J. Hespanha, “Estimation from relative measurements: research interests are in the fundamentals of Control and Dynamical Systems, Electrical analogy and large graphs,” IEEE Trans. Signal Processing, as well as the applications of systems and feedback techniques in several vol. 56, no. 6, pp. 2181–2193, June 2008. physical and engineering systems. These areas include Robust and Optimal [24] B. Kron,¨ “Growth of self-similar graphs,” J. Graph Theory, vol. 45, pp. Control, distributed and networked control and dynamical systems, shear flow 224–239, Mar 2004. transition and turbulence, and the use of feedback in thermoacoustic energy [25] A. Telcs, “Spectra of graphs and fractal dimensions i,” Probab. Th. Rel. conversion devices. Fields, vol. 85, pp. 489–497, 1990. Professor Bamieh has co-authored over 100 refereed publications in Sys- [26] E. Agliari, “Exact mean first-passage time on the t-graph,” Phys. Rev. tems and Controls and allied fields. He has received several awards and E, vol. 77, no. 1, p. 011128, Jan 2008. honors for his research, including the IEEE Control Systems Society G. [27] S. De Bartolo, F. Dell’Accio, and M. Veltri, “Approximations on the S. Axelby Outstanding Paper Award, the AACC Hugo Schuck Best Paper peano river network: Application of the horton-strahler hierarchy to the Award, and a National Science Foundation CAREER award. He was elected a case of low connections,” Phys. Rev. E, vol. 79, no. 2, p. 026108, Feb Distinguished Lecturer of the IEEE Control Systems Society (2005), a Fellow 2009. of the International Federation of Automatic Control (IFAC) , and a Fellow [28] T. Vicsek, “Fractal models for diffusion controlled aggregation,” J. Phys., of the IEEE with the citation ”For contributions to robust, sampled-data and vol. 16, no. 17, pp. L647–L652, 1983. distributed control”.