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Time-Delay Origins of Fundamental Tradeoffs Between Risk of Large Fluctuations and Network Connectivity

For the class of noisy time-delay linear consensus networks, we obtain explicit formulas for risk of large fluctuations of a scalar observable as a function of Laplacian spectrum and its eigenvectors. It is shown that there is an intrinsic tradeoff between risk and effective resistance of the underlying coupling graph of the network. The main implication is that increasing network connectivity, increases the risk of large fluctuations. For vector-valued observables, we obtain computationally tractable lower and upper bounds for joint risk measures. Then, we study behavior of risk measures for networks with specific graph topologies and show how risk scales with network size.


Introduction
The notion of systemic risk describes fragility in interconnected networks that can result in global or cascading failures due to either relatively small disturbances at the subsystem level or larger and more malicious types of disruptions affecting the whole network. The challenges of dealing with risk spread through many areas of engineering and economy [1,2,3,6,11,31]. Modern networked control systems bear innate complexities in their structure and evolving dynamics. There are numerous networks, which are playing major roles in fabrics of our society, that are susceptible to exogenous stochastic uncertainties and disturbances. Examples include platoon of autonomous vehicles in highways, synchronous power networks with integrated renewable sources, water supply networks, interconnected networks of financial institutions, and air traffic networks. The recent crisis show specific vulnerabilities of modern networks due to weaknesses in their structures, e.g., air traffic congestion, power outages, the 2008 financial crisis, and other major disruptions. The costly and severe in magnitude outcomes of these network-related crisis motivate risk-oriented analysis and synthesis of networked control systems.
There are some recent lines of research that are close in spirit to the subject of this paper. In [6], the authors introduce novel concepts of resilience for dynamic flow networks, where they study robustness of such network with respect to perturbations that reduce the flow functions on the links of the network. They also discuss a situation in which imposing finite density constraints on the links may result in cascaded failures. Other related works [4,21,36,37] utilize performance measures based on H 2 -norm to quantify quality of noise propagation in noisy linear consensus networks. The focus of these works are on linear consensus networks with no time delay. In [38,35], the authors introduce a class of spectral systemic performance measures (which includes H p -norms as its special case) to synthesize networks through growing and sparsification of links in noisy linear consensus networks in absence of time delay. In all these papers, performance measures assess holistic and macroscopic features of the networks, such as total deviation from average or uncertainty volume of the output. In this paper, we show that exogenous uncertainties along with time delay creates unorthodox behaviors in microscopic level that cannot be captured by the existing performance measures in literature.
The class of time-delay linear consensus model (6) has been previously studied within different disciplines [16,17,20,23,13,14]. In [16], the authors obtain expressions for the average size of fluctuations using modal behavior of the network. They also show how extreme fluctuations scale with network size and argue that distribution of those nodes with highest fluctuation takes specific forms. The focus of [17] is on synchronizability threshold and the scaling behavior of the width of the synchronization for linear consensus systems with uniform and multiple time delays, where their results and analysis are heavily relied on numerical simulations. Reference [23] proposes a method to study systems with multiple delays (scalar and multidimensional) in the presence of additive noise. Using Laplace transform, they obtain an integral formula for statistics of the state variables without providing closed form solutions for the integrals. Their results are applied to first and second order consensus networks with multiple delays and complete (all-to-all) graph topology. Extensive numerical solutions to draw conclusions are evident in all these works. In this paper, we adopt a rigorous terminology and develop tools based on notions of risk to quantify and measure the tendency of noisy time-delay linear consensus networks towards exhibiting undesirable systemic events (e.g., large fluctuations and violating critical constraints). Our analysis is based on the fact that a time-delay linear consensus network can be decomposed into many one-dimensional systems and then analyzed using tools developed in [20,13,14].
Systemic risk as a tool for decision making under uncertainty has been initially employed to study uncertainties in financial markets [2,3,11,31,32]. More recently, cascading failures in interconnected financial institutes have been studied [1,11]. In [41,40], the notion of value-at-risk has been applied to calculate probability of large fluctuations due to external disturbances in linear consensus networks with no time delay. The present work is an outgrowth of [40] and [41] in a number of ways. At first, we introduce and analyze two types of risk measures to assess possibility of large fluctuations in networks: one gauge risk in probability and the other one in expectation w.r.t. a utility (cost) function. By exploiting functional properties of risk measures, such as (quasi-) convexity and monotonicity, we calculate explicit formulas for them w.r.t. scalar network observables and show that these individual risk measures are spectral functions of Laplacian eigenvalues and eigenvectors. Furthermore, it is shown that the risk measures are not monotone functions of Laplacian eigenvalues, i.e., increasing connectivity (through adding new feedback interconnections or increasing feedback gains) may increase risk of large fluctuations and sparsification (eliminating existing feedback loops) may decrease risk of large fluctuations. We do not observe this peculiar behavior when there is no time delay. In addition to the fundamental limits reported to [40], we characterize several Heisenberg-like inequalities and reveal intrinsic tradeoff between the risk measure and network connectivity. Our results assert that in presence of time delay more connectivity results in higher risk of large fluctuations that in turn may lead to violation of critical constraints and network fragility. Finally, we extend our risk analysis to multiple network observables. In this case, the risk measures take vector values and one needs to calculate joint risk measures. In general, obtaining explicit closed form formulas for the joint risk measures is nearly impossible. However, one can employ numerical algorithms to compute values of such joint risk measures [12]. To tackle this challenge, we obtain tight lower and upper bounds for the joint risk measures and show that joint risk measures depend on the vector of individual risk measures, where we know how to calculate them explicitly. The big advantage of our bounds is that they are computationally tractable. Finally, we investigate behavior of risk for networks with particular graph topologies and show how the risk measures scale with network size. Several numerical examples support our theoretical findings. The proofs of all technical results, in this work, are placed in the Appendix.

Preliminaries
Throughout the paper, plain and bold lowercase letters are reserved for scalar-valued and vector-valued variables, respectively. The n-dimensional Euclidean space with elements z = ( z 1 , . . . , z n ) T is denoted by R n . The set of standard Euclidean basis for R n is {e 1 , . . . , e n }. The nonnegative orthant cone is defined by R n + = z ∈ R n z i ≥ 0 for all i = 1, . . . , n .
For x, z ∈ R n , relation x z holds iff z − x ∈ R n + , and x ≺ z iff all elements of z − x are strictly positive. The Euclidean vector and induced matrix norms is represented by · , while absolute value operation for vectors is defined element-wise, i.e., |z| = [ |z 1 |, . . . , |z n | ] T . The n × 1 vector of all ones is shown by 1 n and the n × n centering matrix is defined as M n := I n − 1 n 1 n 1 T n , where m i is the i'th column of M n . Representation of a vector in R n with respect to linearly independent columns of matrix Q = [q 1 | . . . | q n ] is shown by z Q = [ z Q 1 , . . . , z Q n ] T , where z Q i = q T i z. Algebraic Graph Theory: A weighted graph is the triple G = (V, E, w), where V is the set of nodes, E is the set of links (edges), and w : E → R + is the weight function of the graph that assigns a real number to every link. Assumption 1. For every pair of nodes i, j ∈ V, we assume: (i) nonnegativity of link weights, i.e., w(i, j) ≥ 0 (ii) indirectness of links, i.e., w(i, j) = w(j, i), and (iii) simpleness, i.e., w(j, j) = 0. Moreover, this paper only considers connected graphs.
Let us denote k ij = w(i, j) for every i, j ∈ V. The Laplacian matrix of graph G is an n × n matrix L = [l ij ] whose elements are defined as Laplacian matrix of a graph is symmetric and positive semi-definite. Our connectivity assumption implies that only the smallest Laplacian eigenvalue is equal to zero and all other ones are strictly positive, i.e., 0 = λ 1 < λ 2 ≤ . . . ≤ λ n . The eigenvector of L corresponding to λ k is denoted by q k . By letting Q = [ q 1 | . . . | q n ], it follows that L = QΛQ T with Λ = diag[ 0, λ 2 , . . . , λ n ]. We normalize the Laplacian eigenvectors such that Q becomes an orthogonal matrix, i.e., Q T Q = QQ T = I n with q 1 = 1 √ n 1 n .

Remark 1.
Suppose that E n is the n × n diagonal matrix that is obtained by setting element of (1, 1) of I n to zero. For an n × n unitary matrix Q, standard graph theory results are: M n = QE n Q T and B T n B n = nM n = nQE n Q T , where B n is the n(n−1) 2 × n complete incidence matrix.
The notion of effective resistance is a widely used metric to measure how strongly a graph is connected in the context of graph theory [19,25,37]. The effective resistance between every pair of nodes i and j of a graph can be interpreted as the voltage difference between terminals i and j if we replace every link with an electrical resistance (whose value is equal to that link's weight) and let unit current source to be applied between the two nodes; we refer to [8,19] for more information. The total effective resistance is the sum of all effective resistances over all distinct pair of nodes in G and is equal to [19] where λ 2 , . . . , λ n are the non-zero Laplacian eigenvalues. Probability Theory: A complete filtered probability space is the quadruple (Ω, F, {F t } t≥0 , P), where {F t } t≥0 is the filtration of a vector-valued standard Brownian motion {w t } t≥0 , with w t = w 1 (t), . . . , w n (t) T constructed out of n independent scalar Brownian motions. The space of all F-measurable, R q -valued random variables with finite second moment is denoted by L 2 (R q ). The stochastic process y = {y t } t∈R is {F t }-adapted if y t is F t -measurable for t ≥ 0. The multi-variable normal distribution is denoted by N (µ, Σ), where µ ∈ R q is the expected value and Σ is the q × q positive semi-definite covariance matrix. A random variable y ∼ N (µ, Σ) attains a density function if and only if Σ is positive definite. The error function is denoted as For fixed ε ∈ (0, 1), the following quantity is introduced for the sake of convenience: and S ε (0) = inf δ ∈ R + | erf(δ) ≥ 1 − ε .

Problem Statement
We consider the class of time-delay linear consensus networks that has found wide applications in engineering (e.g., clock synchronization in sensor networks, rendezvous in a space or time, and heading alignment in swarm robotics) and social sciences (e.g., agreement in opinion networks); we refer to [30,28] for more details. As a motivational application, we discuss a cooperative rendezvous problem where the control objective of a team of agents (e.g., ground vehicles or quadcopters) is to meet simultaneously at a pre-specified location known to all agents 1 . In this rendezvous problem agents do not have a priori knowledge of the meeting time as it may have to be adjusted in response to unexpected emergencies or exogenous uncertainties (see [30] for a detailed discussion). Thus, all agents should agree on a rendezvous time by achieving consensus. This is usually done by each agent creating a state variable, say x i ∈ R, that represents its estimation of the rendezvous time. Each agent's initial estimate is set to a preferred time at which it would be able to rendezvous with other agents. The rendezvous (more generally consensus) dynamics for each agent evolves in time according to the following stochastic differential equation for all i = 1, . . . , n. Each agent's control input is u i ∈ R. The source of uncertainty is diffused in the network as additive stochastic noise, the magnitude of which is uniformly scaled by the diffusion b ∈ R. The impact of uncertain environments on dynamics of agents are modeled by independent Brownian motions w 1 , . . . , w n . In real-world situations, each agent experiences a time delay in accessing, computing, or sharing its own state information with itself and other neighboring agents [30]. It is assumed 2 that all agents experience an identical time delay that is equal to a nonnegative number τ . The control inputs are determined via a negotiation process by forming a linear consensus network over a communication graph using the following feedback law where k ij are non-negative feedback gains. Let us denote the state vector by x t = x 1 (t), . . . , x n (t) T and the vector of exogenous disturbance by w t = w 1 (t), . . . , w n (t) T . The dynamics of the resulting closed-loop network can be cast as a linear consensus network that is governed by the stochastic delay differential equation for all t ≥ 0, where the initial function x t = φ(t) is statistically independent of w 0 for all t ∈ [−τ, 0], and B = b I n . The underlying coupling structure of the consensus network (6) is a graph G = (V, e, w) that satisfies Assumption 1 and whose Laplacian matrix is L. The underlying communication graph is considered to be time-invariant with a time-independent Laplacian matrix as the network of agents aim to reach consensus on a rendezvous time before each agent can perform motion planning to get to the meeting location. Upon reaching consensus, a properly designed internal feedback control mechanism steers each agent towards the rendezvous location.

Assumption 2.
The time delay satisfies τ < π 2λn . When there is no noise, i.e., b = 0, it is already known [29] that under Assumptions 1 and 2, states of all agents converge to average of all initial states 1 n 1 T n φ(0); whereas in presence of input noise, state variables fluctuate around the network average 1 n 1 T n x(t). In order to quantify the quality of consensus (e.g., rendezvous) and its fragility features, we consider the vector of observables for network (6) in which C is a generic q×n output matrix and y t = (y 1 (t), . . . , y q (t) ) T . Some relevant examples of network observables are discussed in Section 6. Assumption 1 implies that one of the modes of network (6) is marginally stable with eigenvector 1 n , while the rest are exponentially stable. The marginally stable mode, which corresponds to the zero eigenvalue of L, must be unobservable from the output (7) so that in the long run, y t stays bounded. This can be guaranteed by imposing that 1 n belongs to the null space of C, i.e.
1 Rendezvous in space is very similar to rendezvous in time by switching the role of time and location in the above explanation. 2 This assumption has been widely used by other researchers as it allows analytical derivations of formulas. For rendezvous in time, it is a common practice in robotics labs to use identical communication modules for all agents, which results in a uniform communication time delay. Moreover, in other related applications such as heading alignments, rendezvous in space, and velocity control of agents using a Motion Capture (MoCap) system to observe their spatial locations in indoor labs [22,24,39], all agents experience an identical time delay to access data through MoCap system. According to (6), when there is no input noise, the aforementioned condition implies y t → 0 as t tends to infinity. Consequently, the presence of exogenous noise input makes the network observables to fluctuate around zero. This implies that agents will not be able to agree upon an exact consensus state (e.g., rendezvous time). A practical resolution is to allow a tolerance interval for agents to concur.

Definition 1.
For observable (7) and some δ ∈ R q + , dynamical network (6) can tolerate some degrees of disagreement and reach δ-consensus if the following stochastic event lim t→∞ |y t | δ (8) holds with high probability 3 .
The notion of δ-consensus means that all agents have agreement on all points in x ∈ R n |Cx| δ . We interpret this definition using our rendezvous example, where C = M n . Suppose that event (8) holds with δ = δ1 n . Then, the network of agents will achieve a δ-consensus over the rendezvous time in the following sense: In steady state, the i'th agent is assured that by x i (t) + 2δ units of time, all other agents will arrive and meet each other in that time interval with high probability. Some alternative undesirable situations may also happen that we refer to as systemic events.

Definition 2.
For a given vector δ ∈ R q + , network (6) with observable (7) will be prone to a systemic event if the probability of happening the following stochastic event in steady-state, is nonzero, where ∨ is the disjunction operator.
In the rendezvous example, when the network of agents undergo systemic event (9) with some probability greater than ε > 0, it would be possible that the agents fail to reach δ-consensus and meet simultaneously during the rendezvous time interval. The objective of this paper is to identify how time-delay makes dynamical network (6) vulnerable to systemic events (9) and quantify probability of such undesired stochastic events that may lead to network fragility, i.e., not being able to reach δ-consensus.

The State and Output Dynamics
The dynamical network (6) satisfies all the well-posedness conditions of the classical theory of stochastic differential equations and it generates well-defined stochastic processes {x t } t≥−τ and {y t } t≥−τ ; we refer to [9,26] for more details. In this section, we discuss statistics of these stochastic processes and derive results that will be useful in derivations of our results.

Internal Stability
The solution of the unperturbed system, i.e., b = 0, can be described using the transition matrix of the system, which is the matrix solution of differential equatioṅ with initial conditions Φ L (t) = 0 if t ∈ [−τ, 0) and Φ L (0) = I n . According to [29], we have the following limit behavior as t → +∞ 1−sin(x) over its domain (0, π 2 ) that guarantees internal stability of network (6).
for t ≥ −τ . Namely, Φ(·) is the principal solution ofẋ = −Λx(t − τ ) and Λ the diagonal matrix of Laplacian eigenvalues. We remark that the diagonal elements of Φ(t) do not come in closed form for τ > 0, with the exception of ϕ 1 (t) ≡ 1. Nevertheless, under Assumption 2, it can be shown that is a function defined in (0, π 2 ) that takes values in R + . The plot of f is provided in Figure 1. The rigorous derivation of formula (10) is held in Appendix 12.

The Statistics of the Solution
The solution of the perturbed network (6) is an {F t } t≥0 -measurable stochastic process {x t } t≥−τ given by Moreover, it follows that x t ∼ N (µ t , Σ t ) with vector-valued mean µ t = E v t and covariance matrix

The Statistics of the Output
Consequently, the statistics of the output can be quantified as The following result characterizes output matrices for which y t ∈ L 2 (R q ), t ≥ 0.

Lemma 1. 1 n ∈ ker(C) if and only if
The result of Lemma 1 implies that if Cq 1 = 0, then one gets y = lim t→∞ y t ∈ L 2 (R q ) as both Σ = lim t→∞ CΣ t C T and µ = lim t→∞ Cµ t are well-defined. Moreover, y ∼ N (0, Σ) if and only if 1 n ∈ ker(C).

Systemic Risk Measures
There are several known quantifiers in the context of finance and economy to measure risk of a systemic event.
In this paper, we utilize two notions of risk measures, which are relevant to the context of dynamical systems, to quantify safety margins of network (6) before it experiences systemic event (9) and reaches a δ-consensus based on Definitions 1 and 2.

Risk of Large Fluctuations in Probability
When network observables admit a well defined probability measure, we may be able to calculate risk of a systemic event that is expressed in terms of violation of a set of critical constraints; for example, stochastic event (9) in the rendezvous example. The value-at-risk measure, also referred to as risk in probability, determines maximum allowable fluctuation, which we also refer to it as safety margin, before network experiences a systemic event with a pre-specified probability. In other words, this measure quantifies the smallest lower bound δ for systemic event z(ω) ∈ L 2 (R) before probability of event "z(ω) surpassing δ" exceeds a given threshold.

Definition 3.
For a given probability threshold ε ∈ [0, 1), the value-at-risk or risk in probability is an operator If the right hand-side has no solution, we set R ε (z) = +∞.
The value-at-risk measure enjoys several functional properties that will help us later on to derive fundamental limits on the best achievable risk value and tradeoffs between risk and network connectivity. Proposition 1. The measure R ε : L 2 (R) → R in Definition 3 satisfies the properties: Property (i) implies that if a z includes no uncertainty, the risk measure remains invariant. Property (ii) means that R ε is not affected when a fixed constant is added to the random variable. Property (iii) explains that if a random variable z 1 takes smaller values than z 2 almost surely, then the risk of the former is smaller than the risk of the latter. Next, (iv) is a positive homogeneity property and implies that if we scale a random variable (i.e., scale the uncertainty with it) then the risk will scale accordingly. Finally, property (v) is a weak convexity property of the risk measure and indicates that the risk of a convex combination of two random variables cannot get worse than the maximum of the risk of these random variables.

Risk of Large Fluctuations in Expectation
The second type of risk measures that are relevant to the context of dynamical systems is quantification of risk through statistical expectation of a utility or cost function. In several applications in economics, game theory, and decision and control theory, we evaluate violation of critical constraints against some meaningful utility or cost functions. These utility functions reflect perception of a decision maker towards a potential outcome when there is uncertainty [10]. In this setting, utility function can be interpreted as a distance function to measure how far a random variable is from a predefined threshold. This interpretation is particularly useful in the context of dynamical systems as one can employ energy or entropy based utility functions to quantify risk.

Definition 4.
Suppose that utility function v : R → R is convex and monotonically increasing. For a given threshold ε, the risk in expectation is an operator T ε : If the right hand-side has no solution, we set T ε (z) = +∞.
The risk in expectation quantifies possibility of large fluctuations by evaluating expected value of a predefined utility or cost function. A trivial choice for utility function is v(z) = z, where the value of risk is the smallest number for which inequality E[z(ω) − δ] ≤ ε holds. For more general utility functions, we can still relate to this simple inequality by applying Jensen's inequality and monotonicity property of the utility function. It is straightforward to show that T ε is positive if and only if E v(z) > v(ε) (we refer to [10] for more details). This helps us to define the following acceptance set that contains all random variables in L 2 (R) whose expected cost E[v(z)] is less than cost threshold v(ε). In the following, we discuss two important and relevant utility functions for risk analysis in dynamical systems.
Quadratic Utility: For scalar random variables, the quadratic utility function is v(z) = z 2 over nonnegative real numbers. The value of risk in this case represents the smallest value δ for which random variable z stays in a ball with center δ and radius v(ε). Depending on the application, quadratic form of the output (7) relates to the potential or kinetic energies of the system [37].
Exponential Utility: For scalar random variables, the exponential utility function is v(z) = e βz for some parameter β > 0. This function is monotonically increasing over all real numbers and close in spirit to the entropic risk measure [10].

Proposition 2.
The risk in expectation T ε : L 2 (R) → R as in Definition 4 satisfies the following properties: (iv) For all nonnegative z ∈ A, it follows that for all α ≥ 1, and for all nonpositive z ∈ A, we have Property (i) is the equivalent of Property (ii) in Proposition 1. It explains that adding fixed numbers (with no uncertainty) does not affect the value of risk in expectation. The convexity property of Property (ii) holds within the acceptance set (13) and can be generalized to any acceptance set that is convex [10]. Property (iii) means that risk in expectation is monotonically increasing. According to property (iv), the risk measure enjoys a weak positive homogeneity property that holds over a subset of A.

Risk Assessment of a Single Event
Suppose that the network observable is scalar with c ∈ R n . According to (12), it follows that where are the corresponding vectors c and φ(t) represented in terms of the basis spanned by the columns of Q = [q 1 | . . . | q n ], for q k is the k'th Laplacian eigenvector, and {ϕ k } k∈V the diagonal elements of principal matrix Φ.

Lemma 2.
For every time instant t, functions µ t = µ t (Q) and σ t = σ 2 (Q), as they are defined by (15) and (16), are eigenspace invariant, i.e., for all orthonormal matrices Q 1 and Q 2 for which is the solution of (6) and the network observable is scalar as in (14). Then, the risk of large fluctuations in probability is given by where S ε (·) is defined by (3), and the risk of output fluctuations violating a utility threshold can be calculated as follows: (ii) when u(z) = e βz for some β > 0, in which and µ t and σ t are defined by (15)- (16).
In case of no time delay, i.e., τ = 0, the eigensolutions of the nominal system take simple forms ϕ k (t) = e −λ k t for k = 2, . . . , n. This allows us to derive a more explicit form for the variance: When τ > 0 and t < ∞, the risk expressions in Theorem 1 do not admit more tractable representations in general. However, towards the end of this section, we show that in steady state (as t → ∞) one can obtain more explicit formulas for systemic risk measures; we also refer to [41] for more details.
The steady state behavior of systemic risk measures can be investigated by letting t tend to infinity. According to Lemma 1, if output matrix C is orthogonal to the vector of all ones, then y = lim t→∞ y t will belong to L 2 (R), i.e. with finite variance, denoted as σ. If c T 1 n = 0, then µ t vanishes exponentially fast. In steady state, the transient effect of initial conditions disappears and other origins of large volatility and fluctuation in network observables reveal themselves, e.g., communication topology, time delay, and the diffusion coefficient b. The steady-state risk analysis uncovers how exogenous noise propagates throughout the network, is amplified by various factors, and results in rapid and unpredictable variations in network observables.

Theorem 2.
Suppose that stochastic process x = {x t } t≥−τ is the solution of (6) and the network observable is scalar as in (14) and the output matrix satisfies c T 1 n = 0. Then, the risk of large fluctuations in probability in steady-state is given by in which where λ k is the k'th non-zero Laplacian eigenvalue. The risk of output fluctuations violating a utility threshold in steady-state w.r.t.: (ii) exponential utility v(x) = e βx for some β > 0 can be expressed by The expression for variance (21) can be equivalently written in the following compact form 4 where L o = cc T . In the following, we discuss some of the important and useful examples of scalar network observables.
(a) Deviation of a single state from the average: Suppose that we are interested in observing state of agent i. The deviation from average of x i can be obtained by choosing the i'th column of the centering matrix M n as the output matrix, i.e., c = m i . A direction calculation reveals that (b) Pairwise deviation: When one is interested in characterizing volatility of the disagreement between two specific fixed agents, say agents with labels i and j, we choose output matrix c = e i − e j . In this case, (c) Deviation from average of immediate neighbors: Comparing state of each agent with average of its own adjacent neighbors is another meaningful observable for network (6). In this case, the components of the output matrix are defined as where {j | j ∼ i} is equal to set {j | {i, j} ∈ E} and n i is its cardinality. In all these scenarios, the value of systemic risk measure T ε (|y t |) and R ε (|y t |) can be calculated using Theorem 1 with statistical parameters defined in (15) and (16). Moreover, Theorem 2 shows that value of the systemic risk measure depend on Laplacian spectrum.

Remark 2.
When the output matrix is c = 1 n 1 n , the network observable becomes the average of all states. It follows that c Q 1 = 1 √ n and c Q k = 0 for every k > 1. In this case, the mean and variance of the scalar observable reduces to Using these formulas, one can compute the value of transient systemic risk measures T ε (|y t |) and R ε (|y t |) according to Theorem 1. Since network (6) has a marginally stable mode corresponding to λ 1 = 0, the variance σ 2 t diverges as t grows (see also Lemma 1). This implies that all states fluctuate around their average and eventually diverge. This phenomenon is so-called "flocking to default" in finance and economy literature; we refer to [11,Ch. 17] for a discussion. Furthermore, one also observes that the stastistical parameters in (25) are independent of the network topology and time delay.
We conclude this section by investigating effects of time delay on systemic risk measures.  (6) and sparsification may improve it. Figure 1 depicts behavior of each spectral term in expression of variance (21). The unorthodox behavior of risk and its interplay with connectivity and performance in time-delay networks is highly counterintuitive and requires careful investigations. In Section 9, we characterize an intrinsic tradeoff that explains this behavior.

Joint Risk Assessment of Multiple Events
The extension of risk for network (6) with vector valued output observables is a problem with different approaches, each of which highlights the problem from a different perspective. In this section we introduce the most important multi-value risk measures for the steady state distribution of |y t | = |y 1 |, . . . , |y q | T with y k = c T k x t . We recall from Section 4 that in steady-state if and only if 1 n is in the kernel of output matrix C.
When different types of observations are used for risk analysis in a dynamical network, systemic risk measure becomes a vector, where every element of that vector corresponds to the value of risk of an event that has a similar nature to the corresponding observation. For example, suppose that our goal is to evaluate risk using vector of observables y t = [ y 1 (t), y 2 (t), y 3 (t) ] T , where stochastic variable y 1 measures deviation of a single state from the average of all states, y 2 shows pairwise deviation between two given agents, and y 3 gives deviation of a single state from average of its immediate neighbors.

Joint Risk Assessment of Multiple Events
The extension of risk for network (6) with vector valued output observables is a problem with different approaches, each of which highlights the problem from a different perspective. In this section we introduce the most important multi-value risk measures for the steady state distribution of |y t | = |y 1 |, . . . , |y q | T with y k = c T k x t . We recall from Section 4 that in steady-state if and only if 1 n is in the kernel of output matrix C.
When different types of observations are used for risk analysis in a dynamical network, systemic risk measure becomes a vector, where every element of that vector corresponds to the value of risk of an event that has a similar nature to the corresponding observation. For example, suppose that our goal is to evaluate risk using vector of observables y t = [ y 1 (t), y 2 (t), y 3 (t) ] T , where stochastic variable y 1 measures deviation of a single state from the average of all states, y 2 shows pairwise deviation between two given agents, and y 3 gives deviation of a single state from average of its immediate neighbors.

Risk of Large Fluctuations in Probability
The joint risk in probability measure is a vector-valued operator R ε : in which infimum operates on elements of δ. Obtaining a closed-form expression for the joint risk in probability is particularly difficult. To see this, we observe that (28) is equivalent to a joint chance constraint optimization problem [5].The solution set is not a singleton. In fact, the set of all optimal vectors δ constitute a Pareto set. Moreover, problems of this kind involve the quantification of joint dependent events are typically difficult to solve. One way to compute solution of such multi-objective optimization problems is via Monte Carlo sampling [34,27]. Robust Optimization techniques propose analytic approximations to this problem by decomposing the joint constraint problem into a problem with individual chance constraints [5]. By applying the latter approach to (28), we obtain the next result. Let us define the vector of individual risk measures by R ε (|y|) = R ε (|y 1 |), . . . , R ε (|y q |) T .

Theorem 4.
For dynamical network (6) with vector of observables (27), the joint vector-valued systemic risk measure (28) satisfies the following inclusion where Figure 2 illustrates geometric shape of W Rε(|y|) for q = 2. The set of solutions that constitute R ε (|y|) lies within a bounded set that is characterized by R ε (|y|).
When all output observables show evolution of similar stochastic events, a single safety margin can be employed to assess systemic risk throughout the network. For instance, all observables are similar or homogenous if every y k , for k = 1, . . . , q, measures either deviation of a single state from the average of all states, pairwise difference between two agents, or deviation of a single state from average of its immediate neighbors. In this case, systemic risk measure becomes scalar-valued according to The next result is a direct consequence of Theorem 4.

Risk of Large Fluctuations in Expectation
Suppose that function v : R q → R is convex and monotonically increasing on the cone of positive orthant, i.e., if x y, then v(x) ≤ u(y). The joint risk in expectation is an operator T ε : L 2 (R q ) → R q that is defined by where δ = [ δ 1 , . . . , δ q ] T is the vector of safety margins.

Theorem 5.
Let us assume that 1 n ∈ ker(C) for dynamical network (6) with observables (27). The constraint Moreover, the joint risk in expectation assumes a set value that is given by where σ = [ σ 1 , . . . , σ q ] T . Theorem 6. Suppose that assumptions of Theorem 5 hold. For all safety margins ε 1 , . . . , ε q > 0 that satisfy When all observed stochastic events are homogeneous, the joint risk in expectation (33) reduces to a scalar as follows The risk measure T ε (|y|) is the smallest number δ that can be added to every element of vector |y| such that the cost of the resulting vector stays below the predetermined safety threshold.

Corollary 2.
Let us assume that 1 n ∈ ker(C) for dynamical network (6) with observable (27). Let ε 2 > j σ 2 j − 2 π j σ j 2 , for σ j the standard deviation of y j . Then, the risk in expectation as in (34) with quadratic utility function satisfies For vector-valued risk measure in expectation w.r.t. exponential utility function, we may use utility function v(z) = e β(z1+...+zq) .
The definition of risk measure in this case is given by Following a simple continuity argument, it can be shown that the vector of risk T ε (|y|) = δ = (δ 1 , . . . , δ q ) satisfies The calculation of the right hand side of (36) becomes significantly harder as it involves calculation of functions of joint random variables, which is beyond the scope of this paper.

Remark 4.
The vector-valued risk metrics, enjoy similar functional properties with the scalar risk metrics of §5. Indeed, Propositions 1 and 2 can be appropriately modified and extended to vector-valued risk, along the lines of [18]. The details are omitted due to space limitation.

Fundamental Limits and Tradeoffs
The presence of time-delay induces fundamental limits and tradeoffs on the least achievable risk values for the class of dynamical network (6) in steady-state. We characterize some Heisenberg-like inequalities and reveal inherent interplay between the systemic risk measure and a measure of network connectivity. The first limit occurs due to stability condition of Assumption 2 that can be quantified as the following lower bound on the total effective resistance of the coupling graph of the network This implies that in time-delay network (6) the feedback structure (represented by the weighted graph G) may not have an overall strong couplings beyond the above limit, e.g., feedback loops may not have very large gains and/or dense topology especially when all gains are equal (which corresponds to unweighted G).  (14), i.e., q = 1. Then, there is a hard limit on the least achievable value for risk of large fluctuations as follows where and z + is the positive solution of equation cos(z) = z. Furthermore, the least achievable value for risk of high volatility cannot be improved beyond some lower bound that is characterized by: (a) for quadratic utility function: for exponential utility function: The lower limit can be achieved by graphs with Laplacian spectrum that satisfy λ i τ = z + for all i > 1, i.e., a complete graph over n nodes with coupling weights k ij ≈ 0.7391 nτ . Inequalities (37) and (38) represent hard limits as their lower bounds are independent of the coupling structure of the network; by fixing ε they only depend on the size of the network and time-delay. When there is no time-delay, i.e., τ = 0, hard limits disappear.  (10). For a detailed analysis on the derivation of the formulas we refer to Appendix C.

Risk of Large Fluctuations in Probability for Scalar Events
Because in no time delay case, one may simultaneously improve risk and connectivity by arbitrarily increasing the feedback gains. This shows that time-delay is the main reason for emergence of these fundamental limits on risk and connectivity.

Theorem 8. Let observable of dynamical network (6) be scalar, i.e., q = 1. There is an inherent fundamental tradeoff between least achievable values for risk of large fluctuations and network interconnectivity that can be quantified as follows
where Ξ G is the total effective resistance of the coupling graph of the network and ϑ * = c b S ε (0) p † with p † = min{p * 2 , . . . , p * n } in which positive numbers p * k is the minimum of over x ∈ (0, π/2) for all k = 2, . . . , n. Moreover, fundamental tradeoffs emerge between the least achievable values for volatility risk and network connectivity in the following sense for the quadratic utility function and ∆(τ ) = β 2 σ * * + erf(βσ * /2) β n(n − 1)τ π for the exponential utility function, where * = c |b| Inequality (40) implies that network (6) will be prone to higher levels of risk if the connectivity enhances, e.g., by adding new coupling links and/or increasing the feedback gains. The origin of this intrinsic tradeoff is time-delay; inequality (40) would be trivial if τ = 0. We observe that by increasing τ , while preserving stability, interplay between systemic risk and network connectivity becomes more apparent. The key observation is that by fixing the network size, time delay and ε inequalities (38) and (40) remain true for all dynamical networks (6) with arbitrary graph topologies and output matrices c; in other words, these fundamental limits and tradeoffs are universal. In Figure 3, we verify tightness of our bounds using extensive simulation results. The blue dots are numerically calculated based on the following procedure: (i) randomly generate a connected graph that satisfies stability condition in Assumption 2 , (ii) randomly generate an output vector c, and (iii) compute the scaled pair ( √ Ξ G , R ε (ȳ))/( √ 2S ε (0)). The orange color curve in Figure 3 illustrates a particular family of complete graphs with parametrized identical coupling strengths that is specifically constructed to highlight sharpness of our bounds in (38) and (40).

Theorem 9.
Let us consider dynamical network (6) with vector of observables (27). There is an intrinsic fundamental tradeoff between the least achievable values for the joint vector-valued risk measure on R q + and network interconnectivity that can be characterized by inequality

Proof of this theorem is based on results of Theorems 4 and 8 and is omitted.
Corollary 3. Suppose that 1 n ∈ ker(C) for dynamical network (6) with output matrix C = c 1 , . . . , c q T . The

vector-valued joint risk in expectation (33) cannot be improved beyond some hard limits that are characterized as lower bounds in the following inequalities:
(i) For quadratic utility function v(z) = z T z, it follows that and z + is the positive solution of equation cos(z) = z.

Exploiting Graph Topology
We show that for some graph topologies, the value of systemic risk measure can be calculated explicitly. Our focus will be only on risk in probability with respect to deviation of a single state from the average, where similar results can be obtained for other types of observations as well as risk in expectation. In order to exploit graph structure and determine how risk scales with network size, we consider the class of dynamical network (6) with unweighted coupling graphs. The results of this section are summarized in Table 1.
Complete graph. Due to the perfect symmetry of a complete graph K n , all elements of the vector of joint fluctuation risk measure are identical and the risk measure reduces to (31). The network is marginally stable if τ < π 2n , where this region shrinks as n grows. For two given linear consensus networks (6) with coupling graphs K n1 and K n2 and n 1 < n 2 , there exists a critical time delay τ * < π 2n2 such that R ε (|y|) for all τ ≤ τ * , and R (1) ε (|y|) < R (2) ε (|y|) for all τ * < τ < π 2n2 . The phenomenon is graphically illustrated on the left plot of Figure 4. In a more simple language, networks with larger complete graphs are safer than smaller ones, Wheel graph. Let us label the central node of the wheel graph W n+1 by 1 and the remaining nodes on the circumference by 2, . . . , n + 1. The largest Laplacian eigenvalue is λ n+1 = n + 1, which implies that network is marginally stable if τ < π 2(n+1) . The symmetric structure of W n+1 implies that R ε (|y 2 |) = . . . = R ε (|y n+1 |). According to explicit expressions for risk measures in Table 1, we can verify that 5 R ε (|y| 1 ) < R ε (|y| i ) for all i = 2, . . . , n+1. Thus, the central node in a dynamical network with wheel graph is riskier than the surrounding nodes.
Complete Bipartite graph. Let us denote each side of the complete bipartite graph K n1,n2 by G 1 and G 2 that each has n 1 and n 2 nodes, respectively. Nodes at each side are only connected to nodes on the other side. The total number of nodes is n = n 1 + n 2 and let us assume that n 1 ≤ n 2 . In this case, the network (6) is marginally stable if and only if τ < π 2n . The symmetric topology of K n1,n2 entails that all R ε (|y| i ) belonging to one side (either G 1 or G 2 ) are identical. We may categorize the set of complete bipartite graphs into two subclasses: graphs with n 1 = 1 and the ones with n 1 > 1. The first class contains all star graphs K 1,n−1 , where G 1 consists of only node with label 1 and G 2 contains all nodes with labels 2, . . . , n. From Table 1, it is straightforward to verify that for all nodes j ∈ {2, . . . , n} in subgraph G 2 . Similar to the wheel graph, the central node in a star graph attains the highest risk value among all nodes in the network, regardless of the time delay value. When n 1 > 1, from Table 1 and some elementary analysis, it can be shown that there exists a unique τ * = τ * (n 1 , n 2 ) such that for all i in subgraph G 1 and all j in subgraph G 2 the following inequalities hold For n 1 > 1, we conclude that, unlike graphs with star topology, time delay plays an active role in determining which of the subgraphs G 1 or G 2 contains riskier nodes. This point is illustrated with two examples in Figure 5.
Path graph. In networks for which agents are connected to a uniformly bounded number of neighbors, such as path and ring graphs, the marginal stability range for time delay does not shrink to zero as the network size 5 It can be shown that  Figure 5. Illustration of the systemic risk on K 1,4 and K 2,8 . The first example is the star graph. The central node is the riskier one. The second scheme has a group of two and eight nodes, respectively. In the second scheme, we observe that the larger group is safer for small delay and riskier for large delay.  Figure 6. Systemic risk for P 6 and P 7 . For small delays, the risk increases monotonically for as nodes lie closer to the center, where as for large delays, the risk monotonically decreases. When n is odd and τ approaches the critical delay value, the risk of the central node increases towards infinity at a significantly lower rate than the rest of the network. grows. A network with path graph topology P n is marginally stable iff τ < π 4(1+cos( π n )) . A simple calculation reveals that the network is marginally stable for all n iff τ < π 8 . From Table 1, it can be verified that for all i = 1, . . . , n.
There exists a critical time delay τ * in the marginal stability range that for all time delays τ < τ * agents residing toward the tails of the path graph are safer than the ones located closer to the center. In this case, central nodes are more prone to large fluctuations. When τ > τ * , agents closer to the center become safer and the ones that are closer to the tails become riskier. In conclusion, agents residing in the boundaries are the safest for small time delays, and in contrary, agents located in the center are the safest for large time delays. When n is odd, there is a central agent who becomes significantly safer for large time delays. Figure 6 illustrates these analytic design rules of thumb graphically.
Ring graph. The dynamical network (6) with ring graph topology R n is marginally stable iff The symmetric structure of R n necessitates that all agents to have identical risk values. In the right plot of Figure 4, the value of the systemic risk measure for one of the agents, which is given in Table 1, is drawn for different values of n. One observes that the systemic risk measure monotonically increases by n for small time delays. However, when time delay increases and approaches the marginal stability border, networks with odd number of agents appear safer than networks with even number of agents.

Simulations
We verify our theoretical findings through two numerical examples.
. The corresponding rows are the motion of the average, the deviation of nodes i = 1 and i = 5 from the average as well as the difference between nodes 2 and 3 and the difference between nodes 2 and 5. The results are presented in Figures 8 and 9. The risk of the average of motion becomes unbounded, as predicted. Finally, we notice a consistent pattern on behavior of risk among the observables c 1 , . . . , c 5 : the risk involving nodes with stronger coupling is systematically smaller than the risk that involves nodes with weaker coupling, whereas the risk on the pairwise disagreement between the strongly coupled node 2 and the loosely coupled node 5 appears to be the highest.

Example 2.
We consider a network with 100 agents and an arbitrary graph topology that satisfies Assumption 1. We will numerically calculate the value of the steady-state systemic risk in expectation (22) with respect to output matrix C = M 100 as a function of time delay in its marginal stability region. This way, an observable   For this example, we generate a network whose marginal stability range is τ < π 2λn = 0.3524. We consider risk w.r.t. quadratic utility function with cutoff threshold ε = √ 0.05. For each class, we also record average of (weighted) node degrees as a connectivity measure. The left sub-plot in Figure 10 pictures the number of agents in each class for every τ ∈ [0, 0.35). The right sub-plot in Figure 10 depicts average of (weighted) node degrees. For small time delays, the transition from safe mode to marginally safe mode starts with those nodes that have smaller than average degrees (which is about 3.65). For larger time delays, the transition from marginally safe mode to the unsafe mode starts with those nodes that have larger than average degrees. This remarkable phenomenon has been repeatedly observed for almost all networks with similar characteristics.

Example 3.
We consider a network with 100 agents and an arbitrary graph topology that satisfies Assumption 1, where the output matrix is incidence matrix of the graph, i.e., C = B 100 . The generated network is marginally stable for all time delay τ < 0.3415. The number of observable outputs is 4950 and the results of our simulations are illustrated in Figure 11. The effective graph resistance between nodes i and j, is used as a connectivity measure. For two distinct agents, the larger the effective resistance between them, the weaker these agents are connected to each other. Depending on the value of the risk measure with quadratic volatility for each individual event |y i | > δ i for i = 1, . . . , 4950, these events can be labeled as being safe (negative), marginally safe (positive bounded), and unsafe (infinity). The sub-plots in Figure 11 asserts that: for small time delays, the transition from safe mode to marginally safe mode happens between those pair of nodes that maintain a larger than average effective resistance (i.e., lower than average coupling strength), where the average effective resistance is about 0.544. For larger time delay, the transition from marginally safe mode to unsafe mode occurs for those pair of nodes that have a lower than average effective resistance (i.e., higher than average coupling strength).

Discussion
There are several points that need to be discussed. The focus of this paper is on the risk of large fluctuations in probability and in expectation. The value of these two types of risk measures capture behavior of relevant microscopic features of a network. Moreover, it turned out that their values depend on network topology through Other design objectives can be obtained by combining these three design procedures. Our particular interest is to design networks that strike a balance among connectivity, performance, and risk. This is one of our current research directions.
where f : 0, π 2 → R + is defined through Proof. Given the fundamental solution ϕ, we define the quantity: (44), the following hold:

Claim 1. For V as in
The second row at ρ = τ reads Solving for V (0), we have .
Proof of Claim 1. For (1.) we see that because ϕ is a fundamental solution. For (2.) we calculate From the symmetry of V , so long as ρ > 0. Finally, while (45) yieldṡ from property (2.) we obtainV (0) = −aV (τ ). The result occurs after equating the right hand sides of these two expressions.

B. Proofs Of Technical Results
Proof of Lemma 1. Direct calculations yield where Φ L (t) = QΦ(t)Q T in which Φ(t) = diag ϕ 1 (t), . . . , ϕ n (t) and ϕ k 's are eigensolutions of the unperturbed network. By letting C = [c 1 , . . . , c N ] T , the elements of CΣ t C T are given by According to our assumptions, it follows that ϕ 1 (t) ≡ 1 and all ϕ k (t) for k = 2, . . . , n are exponentially fast vanishing functions. Therefore, matrix CΣ t C is bounded if and only if k (c T i q 1 )(c T j q 1 ) = 0 for all i, j ∈ V. But for i = j we have (c T i q 1 ) 2 = 0 for all i. The reverse part is trivial.

Proof of Proposition 1. (i.)
If z is independent of ω ∈ Ω, then P(z > δ) = 0 for any δ arbitrarily close to z from the right. In view of ε ∈ [0, 1), the definition of R ε (z) imposes the infimum of the sum that is achieved when all δ attain the infimum value which is z.
(iii.) Let z 1 , z 2 ∈ L 2 with z 1 ≤ z 2 almost surely. Then {z 1 > δ} ⊆ {z 2 > δ} that implies P z 1 > δ ≤ P z 2 > δ . The decrease of δ on z 1 to achieve ε is smaller than the corresponding decrease on z 2 . So the risk of z 1 cannot be larger than the risk of z 2 .
If x ≤ y, then P z > δ ≤ P(y > δ while, if x ≥ y, then P z > δ ≤ P(x > δ Using the last two inequalities and property (iii.) the result follows.
(ii.) Due to the convexity and monotonicity properties of v, it is easy to verify that A is convex. Now let {z i } i=1,2 ∈ A and {δ i } i=1,2 ≤ 0 the corresponding risk values. In view of (i.), it holds that T ε (z i − δ i ) = 0, i.e. z i − δ i ∈ A and by convexity Again from (i.) we get which yields the result.
. Consequently the cost δ to bring v(z 2 − δ) closer to v(ε) is smaller than the respective cost on u(z 1 − δ).
Proof of Lemma 2. We write σ 2 t (Q) = b 2 c T Q t 0 Φ 2 (t − s) dsQ T c and with little abuse of notation our focus is on the matrix Σ t = b 2 Q t 0 Φ 2 (t − s) ds Q T . This is a diagonal form for every t and characterizes the algebraic multiplicity of the eigenvalues ζ k = ζ k (t) = t 0 ϕ 2 k (t − s) ds, i ∈ V. Fix t and i ∈ V. Note that we can write is the algebraic multiplicity of ζ k . We take cases. m k = 1: the corresponding normal (real) eigenvector is unique up to a sign. Consequently, is repeated m k times while it produces m k linearly independent normal eigenvectors that can be chosen mutually orthogonal. Then the collection of the eigenvectors of Q 1 that span the subspace of ζ k (t), say Q ζ k 1 and the corresponding collection of Q ζ k 2 of Q 2 are associated with an orthogonal matrix P such that For t and k being arbitrary, the result follows.
Proof of Theorem 1. From the discussion in §6, y t ∼ N (µ t , σ 2 t ) with µ t and σ 2 t as in (15) and (16). Then |y t | follows a folded normal distribution with mean µ |yt| and variance σ 2 |yt| defined in the statement of the theorem. For the volatility risks we distinguish between v(x) = x 2 and v(x) = e βx .

Proof of Theorem 3. Direct calculations, yield
From this and the fact the systemic risk measures are increasing functions of σ, it immediately follows that R ε or T ε are strictly increasing functions of time delay as well.
Proof of Theorem 4. We observe that R ε (|y|) is a set-valued function whose value is equivalent to the solution of the following multi-dimensional joint chance-constrained optimization problem In general, this problem is very difficult to solve even numerically, let alone find closed form solutions. To overcome computational complexity and calculate bounds, we decompose the joint probability constraint (47) into several individual constraints. Let us consider positive numbers ε 1 , . . . , ε q such that (30) holds. Suppose that δ * k is the optimal solution of the following optimization problem for k = 1, . . . , q. is an upper bound for the optimal solution of (46)-(47). Let us denote δ * = [δ * 1 , . . . , δ * q ] T . From Bonferroni inequality [12], we have that From constraint (30) and (49), it follows that This implies that δ * is a feasible solution for optimization problem (46)-(47). Thus, we conclude that By using the fact that R ε k (|y k |) = S ε k (0)S ε (0) −1 R ε (|y k |), we can rewrite (50) to conclude that the optimal solution set must satisfy the upper bound of W Rε . In order to prove the lower bound, we use an equivalent representation of (28) in the following form Suppose that δ * * is an optimal solution of (46)-(47). Then, From Fréchet inequality [33], it follows that P |y k | > δ * * k ≤ ε for all k = 1, . . . , q. Thus, δ * * k is a feasible solution of (48)-(49) and its optimal solution is upper bounded by δ * * k . This implies that the solution set R ε (|y|) must honor the lower bound of W Rε .
Proof of Theorem 5. We recall from Section 4 that in steady-state y ∼ N (0, Σ) if and only if 1 n is in the kernel of output matrix C. Thus, constraint E v(|y| − δ) ≤ v(ε) in (33) with quadratic function is equivalent to where σ 2 i is the i'th diagonal element of the covariance matrix Σ. From completing the square, we obtain The constraint is clearly feasible if and only if r > 0. The feasible set lies on the disk Taking the infimum over delta, we arrive at the subset on the sphere that satisfies which identifies with the set in the second part in the statement of the Theorem, concluding the proof.
Proof of Theorem 6. It suffices to show that the individual margins δ i , stacked as a vector (δ 1 , . . . , δ q ) T belong to the subset T ε (|y|) of R q , as in Theorem 5. We recall from (22) Proof of Theorem 7. It suffices to calculate the lower bound of the stead state variance, σ * , as in (39). The corresponding fundamental limits on risk measures are then directly derived. Recall the steady-state variance σ 2 from Theorem 2 and observe that: as the infimum of f (z), z ∈ (0, π/2) is achieved at z that satisfies z = cos(z), and i (c Q i ) 2 = c 2 for c Q 1 = 0, Q orthogonal basis.
Proof of Theorem 8. For the proof of this result we will need the following Lemma: Lemma 4. For arbitrary but fixed α, β ≥ 0, the function attains a unique minimum at (0, π/2).
We directly calculate, where p α k ,β k (x) is as in Lemma with α k = k − 1 and β k = 2(n−k) π . Now all the limits in (i.), (ii.) and (iii.) occur after combining (39) with the results of Theorem 2. For the proof for the trade-offs we work as follows: (i.) The mean value theorem yields: By virtue of (39) and (41) then For (ii.) we recall the inequality ln(1 + x) ≥ x x+1 for x > −1 and we calculate: The best bound we can now get is For (iii.) the result follows immediately from (41).
Proof of Corollary 3. (i) let δ i the i th element of T ε (|y|). From Theorem 5, it follows that δ i ∈ 2 π σ i −r, 2 π σ i + r . It is straightforward to verify that where the result follows similarly to the steps in the proof of Theorem 7. (ii) Let us denote T ε (|y|) = (δ 1 , . . . , δ q ) and recall Eq. (36). From the Jensen's inequality: . Substitute this lower bound to the first inequality of the proof and apply this bound to Eq. (36) to conclude.
Proof of Lemma 4. It can be easily shown that lim x↑ π 2 p α,β (x) = lim x↓0 p α,β (x) = +∞. This means that, there exists at least one extremum that is a minimum of p α,β (x) in (0, π/2). Now this extremum must satisfy The claim is concluded if we show that, I(x) = (βx+2α) cos(x)−βx 2 −αx, has a unique solution in any compact subset of (0, π/2). At first note that I(0) = 2α > 0 and I(π/2) = −β π 2 2 − α π 2 < 0. By the Intermediate Value Theorem, I(x) attains a root in (0, π/2). If this is not unique then I (x) = 0 should have a solution in the interval between the two distinct roots of I(x). Now, Plugging the above relation to I(x) we have Consequently, there exists x > 0 such that I(x ) > 0 and I (x) < 0 for x ∈ (x , π/2) proving that I vanishes at a unique point in (0, π/2) for every fixed pair of α and β.

APPENDIX C: Derivation of Risk Formulas
The output vector c is set as c T = m i for the deviation of the average measure. We remark that, m T i q k = q ik , i.e. the i th element of the k th eigenvector of L. Finally, we will take b = 1 for the diffusion coefficient.

The Complete Graph, K n
This is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. The eigenspace of K n is λ 1 = 0 with q 1 = 1 √ n 1, and λ k = n for k > 1 with linearly independent, unitary eigenvectors 1 √ 2 (e 1 −e k ) that are normal, but not mutually orthogonal. For this topology we therefore assume nτ < π 2 . While a Gram-Schmidt process could produce an orthonormal basis, this is not necessary here. From Theorem 1 we calculate the variancē The symmetry of K n is particularly convenient for an explicit calculation of the the variance of all elements of |ȳ| and which is of course identical for all |ȳ| i . The fluctuation risks are

Wheel graph W n+1
This graph occurs out of adding in the circulant graph of n nodes an additional central node. We denote this central node with i = 1. The next result establishes the eigen-structure of W n+1 .
Proof of Lemma 5. Let's assign n = 1 to the center node. The (n + 1) × (n + 1) laplacian matrix is The eigenvalues of W n+1 are calculated in [25]. The eigenvector for λ n+1 is directly shown to satisfy the eigenvalue equation Lq n+1 = (n + 1)q n+1 . For λ k = λ n+1 , the first row nq 1 − n+1 j=2 q k = λ k q 1 ⇔ nq 1 + q 1 = λ k q 1 ⇔ q 1 = 0 by virtue of n+1 j=1 q j = 0. Since k is arbitrary, all the eigenvectors, other than q 1 and q n+1 , have their first element to be zero. As for the rest elements of q k , these are calculated from the n × n submatrix of L, after one removes the first row and column. This sub-matrix is circulant {q k } n k=2 as in the statement of the Lemma consist a set of orthonormal eigenvectors (see also [7]).
The latter inequality holds true in view of ∂ ∂λ cos(λτ ) 1−sin(λτ ) > 0. Consequently the variance of the central node is larger than the variance of the rest ones.
The Complete Bipartite Graph, K n 1 ,n 2 .
This graph consists of two sets of nodes denoted as [n 1 ] and [n 2 ], where, without loss of generality, n 1 ≤ n 2 , and n = n 1 + n2. Each node of one set is connected to all other nodes of the other set. There are no links between nodes of a same set. The following Lemma describes the eigenspaces of K n1,n2 . Lemma 6. The eigenstructure of K n1,n2 is a. λ 1 = 0 with multiplicity one and eigenvector q 1 = 1 √ n 1 b. λ k = n 1 with multiplicity n 2 − 1 and eigenvectors, with k = 2, . . . , n 2 .
Let now c = m i for i ∈ [n] the output vector on the deviation of the i th agent from the average. We take cases.