Psychopathologies of Automata I: Autonomous Vehicle Systems - Computational Psychiatry: A Systems Biology Approach to the Epigenetics of Mental Disorders 1st ed.

Computational Psychiatry: A Systems Biology Approach to the Epigenetics of Mental Disorders 1st ed.

8. Psychopathologies of Automata I: Autonomous Vehicle Systems

Rodrick Wallace¹

(1)

New York State Psychiatric Institute, New York, NY, USA

Summary

We apply the perspectives of computational psychiatry to autonomous ground vehicles under intelligent traffic control in which swarms of interacting, self-driving devices are inherently unstable as a consequence of the basic irregularities of traffic interactions and the road network. It appears that such systems will experience large-scale failures analogous to the vast propagating fronts of power network blackouts, and possibly less benign, but more subtle patterns of pathology and consequent failure at various scales.

8.1 Introduction

Current marketing hype surrounding autonomous vehicles runs something like this:

Since more than 90% of highway deaths are related to driver error , automating out the driver will reduce loss of life by more than 90%.

Individual vehicles, however, are nested and enmeshed within larger milieus, creating a multi-scale, multi-level synergism determining crash and fatality rates. Individual vehicles are only one part of that system, not the system itself. Asserting that part of a thing is the whole thing is the infamous mereological fallacy, an important tool for the construction of political lies and other forms of advertising.

One is reminded of another—if different—logical fallacy:

If a woman can gestate a child in nine months, nine women should be able to do it in a month.

Given the inherently complicated nature of transport system safety , assertions regarding the effects of autonomous vehicles on traffic fatalities are entirely speculative and cannot be used as a sound basis for policy development.

Here, we ask a more fundamental question: are large-scale autonomous vehicle systems actually practical, particularly in the context of a rapidly deteriorating social and physical infrastructure? To do this, we examine an “end stage” limit in which many different kinds of vehicles communicate with each other (V2V), and with an intelligent roadway infrastructure (V2I), both embedded in a highly stochastic environment.

In effect, we adapt mathematical models from computational psychiatry to explore the dynamics of a rapid-acting, inherently unstable command, communication and control system (C³) that is cognitive in the sense that it must, in an appropriate “real time,” evaluate a large number of possible actions and choose a small subset for implementation. Such choice decreases uncertainty, in a formal manner, and reduction in uncertainty implies the existence of an information source (Wallace 2012, 2015a).

We particularly study autonomous V2V/V2I systems through the prism of the Data Rate Theorem (Nair et al. 2007), extending the argument to more general phase transition analogs, and developing statistical tools useful at different scales and levels of organization.

8.2 Central Problems

V2V/V2I autonomous systems operate along geodesics in a densely convoluted “map quotient space” that is in contrast to the much more straightforward problem of air traffic control , where locally stable vehicle paths are seen as thick braid geodesics in a simpler Euclidean quotient space (Hu et al. 2001). Such geodesics are generalizations of the streamline characteristics of hydrodynamic flow (Landau and Lifshitz 1987).

Hu et al. (2001) show that, in the context of air traffic control, finding collision-free maneuvers for multiple agents on a Euclidean plane surface $\mathcal{R}^{2}$ is the same as finding the shortest geodesic in a particular manifold with nonsmooth boundary. Given n vehicles, the Hu geodesic is calculated for the topological quotient space $\mathcal{R}^{2n}/W(r)$ , where W(r) is defined by the requirement that no vehicles are closer together than some critical Euclidean distance r. For autonomous ground vehicles, $\mathcal{R}^{2}$ must be replaced by a far more topologically complex roadmap space $\mathcal{M}^{2}$ subject to traffic jams and other “snowflake” condensation geometries in real time. Geodesics for n vehicles are then in a highly irregular quotient space $\mathcal{M}^{2n}/W(r)$ whose dynamics are subject to phase transitions in vehicle density ρ (Kerner and Klenov 2009; Kerner et al. 2015; Jin et al. 2013) that, we will show, represent cognitive groupoid symmetry breaking.

In first order, given the factoring out of most of the topological structure by the construction of geodesics in the quotient space $\mathcal{M}^{2n}/W(r)$ , the only independent system parameter is the density of vehicles per unit length, which we call ρ. Figure 8.1 shows, for streets in Rome, Japan, and Flanders, the number of vehicles per unit time as a function of, respectively, vehicles per mile, per kilometer, and percent occupancy: the “fundamental diagram” of traffic flow. There is a clear “phase transition” at about 40 vehicles/mile for the former two examples, and at about 10% occupancy for the latter.

Fig. 8.1

(a ) Vehicles per hour as a function of vehicle density per mile for a street in Rome (Blandin et al. 2011). Both streamline geodesic flow and the phase transition to “crystallized” turbulent flow at critical traffic density are evident at about 40 vehicle/mile. Some of the states may be “supercooled,” i.e., delayed “crystallization” in spite of high traffic density. “Fine structure” can be expected within both geodesic and turbulent modes. (b ) One month of data at a single point on a Japanese freeway, flow per 5 min vs. vehicles per kilometer. The critical value is about 25 vehicles∕km = 39. 1 vehicles∕mile (Sugiyama et al. 2008). (c ) 49 Mondays on a Flanders freeway. The ellipses contain 97.5% of data points for the free flow and congested regimes (Maerivoet and De Moor 2006). Breakdown begins just shy of 10% occupancy

We shall extend a simple vehicle density measure to a more complicated unsymmetric density matrix that includes multimodal vehicle indices, an inverse measure of roadway quality, and can be extended to measures of information channel congestion.

Kerner et al. (2015) explicitly apply insights from statistical physics to traffic flow, finding that in many equilibrium and dissipative metastable systems of natural science there can be a spontaneous phase transition from one metastable phase to another metastable phase of a system. Such spontaneous phase transition occurs when a nucleus for the transition appears randomly in an initial metastable phase of the system: The growth of the nucleus leads to the phase transition. The nucleus can be a fluctuation within the initial system phase whose amplitude is equal or larger than an amplitude of a critical nucleus required for spontaneous phase transition. Nuclei for such spontaneous phase transitions can be observed in empirical and experimental studies of many equilibrium and dissipative metastable systems. There can also be another source for the occurrence of a nucleus, rather than fluctuations: A nucleus can be induced by an external disturbance applied to the initial phase. In this case, the phase transition is called an induced phase transition.

A Data Rate Theorem (DRT) approach to stability and flow of autonomous vehicle/traffic control systems, via spontaneous symmetry breaking in cognitive groupoids, generalizes and extends these insights, implying a far more complex picture of control requirements for inherently unstable systems than is suggested by the Theorem itself, or by “physics” models of phase transition. That is, “higher order” instabilities can appear. Such systems can require inordinate levels of control information.

By “higher order” instabilities we mean that C³ systems may remain “stable” in the strict sense of the DRT, but can collapse into a ground state analogous to certain psychopathologies, or into even more complicated pathological dynamics. In biological circumstances, such failures can be associated with the onset of senescence (Wallace 2014, 2015b). Apparently, rapidly responding inherently unstable, C³systems can display recognizable analogs to senility under fog-of-war demands.

Using these ideas, it becomes possible to formally represent the interaction of cognitive ground state collapse in autonomous vehicle/intelligent road systems with critical transitions in traffic flow.

Defining “stability” as the ability to return, after perturbation, to the streamline geodesic trajectory of the embedding, topologically complex, road network, it is clear that individual autonomous vehicles are inherently unstable and require a constant flow of control information for safe operation, unlike aircraft that can, in fact, be made inherently stable by placing the center of pressure well behind the center of gravity. There is no such configuration possible for ground-based vehicles following sinuous road geometries in heavy, shifting, traffic.

Recall Fig. 8.1. Again, the vertical axis shows the number of vehicles per hour, the horizontal, the density of vehicles per mile. The streamline geodesic flow and deviations from it at critical vehicle density are evident. Some of the phases may be “supercooled”—fast-flowing “liquid” at higher-than-critical densities. Additional “fine structure” should be expected within both geodesic and turbulent modes.

Classic traffic flow models based on extensions of hydrodynamic perspectives involving hyperbolic partial differential equations (HPDEs) can be analogously factored using the methods of characteristic curves and Riemann invariants—streamlines (Landau and Lifshitz 1987). Along characteristic curves, HPDEs are projected down to ordinary differential equations (ODEs) that are usually far easier to solve. The ODE solution or solutions can then be projected upward as solutions to the HPDEs. Here, we will show, reduction involves expressing complex dynamics in terms of relatively simple stochastic differential equations and their stability properties. Those stability properties, marking the onset of “turbulence,” will be of central interest.

Taking a somewhat larger view, cognitive phase transitions in V2V/V2I systems, particularly ground state collapse to some equivalent of “all possible targets are enemies,” should become synergistic with more familiar traffic flow phase transitions to produce truly monumental traffic jams.

A heuristic argument is as follows: Consider a random network of roads between nodal points—intersections. If the average probability of passage falls below a critical value, the Erdos/Renyi “giant component” that connects across the full network breaks into a set of disjoint connected equivalence class subcomponents, with “bottlenecks” at which traffic jams occur marking corridors between them. Li et al. (2015), in fact, explicitly apply a percolation model to explain this effect for road congestion in a district of Beijing. The underlying road network is shown in Fig. 8.2, and in Fig. 8.3 a cross section taken during rush hour showing disjoint sections when regions with average velocity below 40% of observed maximum for the road link have been removed.

Fig. 8.2

Adapted from Li et al. (2015). The full road network near central Beijing

Fig. 8.3

Adapted from Li et al. (2015). Disconnected subcomponents of the Beijing central road network at rush hour. Sections with average vehicle velocity less than 40% of maximum observed have been removed. Disjoint pieces form equivalence classes that permit definition of a groupoid symmetry

Such equivalence classes define a groupoid, an extension of the idea of a symmetry group (Weinstein 1996). Below, we will define the cognitive groupoid to be associated with a C³ structure, here a system of autonomous vehicles linked together in a V2V “swarm intelligence” embedded in a larger vehicle to infrastructure (V2I) traffic management system. Individual vehicle spacings, speed, acceleration, lane-change, and so on are determined by this encompassing distributed cognitive machine that attempts to optimize traffic flow and safety. The associated individual groupoids are the basic transitive groupoids that build a larger composite groupoid of the cognitive system (Wallace 2012, 2015a). Thus, under declining probability of passage, related to traffic congestion and viewed as a temperature analog, this “vehicle/road” groupoid undergoes a symmetry breaking transition into a combined cognitive ground state collapse and traffic jam mode—essentially a transition from “laminar” geodesic to “turbulent” or “crystallized” flow. Autonomous vehicle systems that become senile under fog-of-war demands will likely trigger traffic jams that are far different from those associated with human-controlled vehicles. There is no reason to believe that such differences will be benign.

We begin the formal development leading to this result with a restatement of the Data Rate Theorem that characterizes the minimum rate of control information needed to ensure stability for an inherently unstable system.

8.3 Data Rate Theorem

To reiterate a central point, unlike aircraft, that can be constructed to be inherently stable in linear flight by placing the aerodynamic center of pressure sufficiently behind the mechanical center of gravity, the complex nature of road geometry and the local dynamics of vehicular traffic ensure that V2V/V2I systems will be inherently unstable, requiring constant input of control information to prevent crashes, traffic jams, and other tie-ups.

The Data Rate Theorem (Nair et al. 2007) establishes the minimum rate at which externally supplied control information must be provided for an inherently unstable system to maintain stability. Given the linear expansion near a nonequilibrium steady state, an n-dimensional vector of system parameters at time t, x_t, determines the state at time t + 1 according to the model of Fig. 8.4, so that

$\displaystyle{ x_{t+1} = \mathbf{A}x_{t} + \mathbf{B}u_{t} + W_{t} }$

(8.1)

where A, B are fixed n × n matrices, u_tis the vector of control information, and W_tis an n-dimensional vector of white noise. The Data Rate Theorem (DRT) under such conditions states that the minimum control information rate $\mathcal{H}$ is determined by the relation

$\displaystyle{ \mathcal{H}>\log[|\det(\mathbf{A}^{m})|]\equiv a_{0} }$

(8.2)

where, for m ≤ n, A^mis the subcomponent of A having eigenvalues ≥ 1. The right-hand side of Eq. (8.2) is interpreted as the rate at which the system generates “topological information.” The proof of Eq. (8.2) is not particularly straightforward (Nair et al. 2007), and the Mathematical Appendix uses the Rate Distortion Theorem (RDT) to derive a more general version of the DRT.

Fig. 8.4

A linear expansion near a nonequilibrium steady state of an inherently unstable control system, for which x_t₊₁ = A x_t+B u_t+ W _t. A, B are square matrices, x_tthe vector of system parameters at time t, u_tthe control vector at time t, and W_ta white noise vector. The Data Rate Theorem states that the minimum rate at which control information must be provided for system stability is $\mathcal{H} >\log [\vert \det [\mathbf{A}^{m}\vert ]$ , where A ^mis the subcomponent of A having eigenvalues ≥ 1

For a simple traffic flow system on a fixed highway network, the source of “topological information” is the linear vehicle density ρ. The “fundamental diagram” of traffic flow studies relates the total vehicle flow to the linear vehicle density, shown in Fig. 8.1. A similar pattern can be expected from “macroscopic fundamental diagrams” that examine multimodal travel networks (Geroliminis et al. 2014; Chiabaut 2015).

Given ρ as the critical traffic density parameter, we can extend Eq. (8.2) as

$\displaystyle{ \mathcal{H}(\rho ) > f(\rho )a_{0} }$

(8.3)

where a₀ is a road network constant and f(ρ) is a positive, monotonically increasing function. The Mathematical Appendix uses a Black–Scholes model to approximate the “cost” of $\mathcal{H}$ as a function of the “investment” ρ. The first approximation is linear, so that $\mathcal{H}\approx \kappa _{1}\rho +\kappa _{2}$ . Expanding f(ρ) to similar order,

$\displaystyle{ f(\rho ) \approx \kappa _{3}\rho +\kappa _{4} }$

(8.4)

the limit condition for stability becomes

$\displaystyle{ \mathcal{T} \equiv \frac{\kappa _{1}\rho +\kappa _{2}} {\kappa _{3}\rho +\kappa _{4}} > a_{0} }$

(8.5)

For ρ = 0, the stability condition is κ ₂∕κ₄ > a ₀. At large ρ this becomes κ₁∕κ ₃ > a₀. If κ ₂∕κ₄ ≫ κ ₁∕κ₃, the stability condition may be violated at high traffic densities, and instability becomes manifest, as at the higher ranges of Fig. 8.1. See Fig. 8.5.

Fig. 8.5

The horizontal line represents the critical limit a₀. If κ ₂∕κ₄ ≫ κ ₁∕κ₃, at some intermediate value of linear traffic density ρ, the temperature analog $\mathcal{T} \equiv (\kappa _{1}\rho +\kappa _{2})/(\kappa _{3}\rho +\kappa _{4})$ falls below that limit, traffic flow becomes “supercooled,” and traffic jams become increasingly probable

8.4 Multimodal Traffic on Bad Roads

For vehicles embedded in a larger traffic stream there are many other possible critical densities that must interact: different kinds of vehicles per linear mile, V2V/V2I communications bandwidth crowding, and an inverse index of roadway quality that one might call “potholes per mile,” and so on. There is not, then, a simple “density” index, but rather a possibly large non-symmetric density matrix $\hat{\rho }$ having interacting components with ρ_i_, j≠ ρ_j_, i.

Can there still be some scalar “ρ” under such complex circumstances so that the conditions of Fig. 8.5 apply? An n × n matrix $\hat{\rho }$ has n invariants r_i, i = 1. . n, that remain fixed when “principal component analysis” transformations are applied to data, and these can be used to construct an invariant scalar measure, using the polynomial relation

$\displaystyle{ p(\lambda ) =\det (\hat{\rho }-\lambda I) =\lambda ^{n} + r_{ 1}\lambda ^{n-1} + \cdots + r_{ n-1}\lambda + r_{n} }$

(8.6)

det is the determinant, λ is a parameter, and I the n × n identity matrix. The invariants are the coefficients of λ in p(λ), normalized so that the coefficient of λⁿis 1. Typically, the first invariant will be the matrix trace and the last ± the matrix determinant.

For an n × n matrix it then becomes possible to define a composite scalar index Γ as a monotonic increasing function of these invariants

$\displaystyle{ \varGamma = f(r_{1},\ldots,r_{n}) }$

(8.7)

The simplest example, for a 2 × 2 matrix, would be

$\displaystyle{ \varGamma = m_{1}\mathrm{Tr}[\hat{\rho }] + m_{2}\vert \det [\hat{\rho }]\vert + m_{3}\mathrm{Tr}[\hat{\rho }]\vert \det [\hat{\rho }]\vert }$

(8.8)

for positive m_i. Recall that, for n = 2, Tr $[\hat{\rho }] =\rho _{11} +\rho _{22}$ and $\det [\hat{\rho }] =\rho _{11}\rho _{22} -\rho _{12}\rho _{21}$ . In terms of the two possible eigenvalues α₁, α ₂, Tr $[\hat{\rho }] =\alpha _{1} +\alpha _{2},\det [\hat{\rho }] =\alpha _{1}\alpha _{2}$ .

Again, an n × n matrix will have n such invariants from which a scalar index Γ can be constructed.

In Eq. (8.5) defining $\mathcal{T}$ , ρ is then replaced by the composite density index Γ.

The method is a variant of the “Rate Distortion Manifold” of Glazebrook and Wallace (2009) or the “Generalized Retina” of Wallace and Wallace (2013, Sect. 10.1) in which high dimensional data flows can be projected down onto lower dimensional, shifting, tunable “tangent spaces” with minimal loss of essential information.

8.5 The Dynamics of Service Collapse

We next examine the dynamics of $\mathcal{T} (\varGamma )$ itself under stochastic circumstances. We begin by asking how a control signal u_tin Fig. 8.4 is expressed in the system response x_t₊₁. We suppose it possible to deterministically retranslate an observed sequence of system outputs Xⁱ = x₁ ⁱ, x ₂ ⁱ, … into a sequence of possible control signals $\hat{U}^{i} =\hat{ u}_{0}^{i},\hat{u}_{1}^{i},\ldots$ and to compare that sequence with the original control sequence Uⁱ = u ₀ ⁱ, u₁ ⁱ, …, with the difference between them having a particular value under some chosen distortion measure and hence having an average distortion

$\displaystyle{ \langle d\rangle =\sum _{i}p(U^{i})d(U^{i},\hat{U}^{i}) }$

(8.9)

where p(Uⁱ) is the probability of the sequence Uⁱand $d(U^{i},\hat{U}^{i})$ is the distortion between Uⁱand the sequence of control signals that has been deterministically reconstructed from the system output.

We can then apply a classic Rate Distortion argument. According to the Rate Distortion Theorem, there exists a Rate Distortion Function, R(D), that determines the minimum channel capacity necessary to keep the average distortion below some fixed limit D (Cover and Thomas 2006). Based on Feynman’s (2000) interpretation of information as a form of free energy, it becomes possible to construct a Boltzmann-like pseudoprobability density in the “temperature” $\mathcal{T}$ as

$\displaystyle{ dP(R,\mathcal{T} ) = \frac{\exp [-R/\mathcal{T} ]dR} {\int _{0}^{\infty }\exp [-R/\mathcal{T} ]dR} }$

(8.10)

since higher $\mathcal{T}$ must necessarily be associated with greater channel capacity.

The denominator can be interpreted as a statistical mechanical partition function, and it becomes possible to define a “free energy” Morse Function (Pettini 2007) $\mathcal{F}$ as

$\displaystyle{ \exp [-\mathcal{F}/\mathcal{T} ] \equiv \int _{0}^{\infty }\exp [-R/\mathcal{T} ]dR = \mathcal{T} }$

(8.11)

so that $\mathcal{F}(\mathcal{T} ) = -\mathcal{T}\log [\mathcal{T} ]$ .

Then an “entropy” can also be defined as the Legendre transform of $\mathcal{F}$ ,

$\displaystyle{ \mathcal{S}\equiv \mathcal{F}(\mathcal{T} ) -\mathcal{T} d\mathcal{F}/d\mathcal{T} = \mathcal{T} }$

(8.12)

The Onsager treatment of nonequilibrium thermodynamics (de Groot and Mazur 1984) can now be invoked, based on the gradient of $\mathcal{S}$ in $\mathcal{T}$ , so that a stochastic Onsager equation can be written as

$\displaystyle{ d\mathcal{T}_{t} = (\mu d\mathcal{S}/d\mathcal{T} )dt +\beta \mathcal{T}_{t}dW_{t} =\mu dt +\beta \mathcal{T}_{t}dW_{t} }$

(8.13)

where μ is a diffusion coefficient and β is the magnitude of the impinging white noise dW_t.

Again, applying the Ito chain rule to $\log (\mathcal{T} )$ in Eq. (8.13) (Protter 1990), Jensen’s inequality for a concave function gives the nonequilibrium steady state (nss) expectation of $\mathcal{T}$ as

$\displaystyle{ E(\mathcal{T}_{t}) \geq \frac{\mu}{\beta^{2}/2} }$

(8.14)

In the V2V/V2I context, μ is a “diffusion coefficient” representing attempts by the system to meet service demand, and β the magnitude of a traffic flow/roadway state “white noise” dW_tcontrary to those attempts.

Recall that, in the multimodal extension of the model, the condition for stability is

$\displaystyle{\mathcal{T} \approx \frac{\kappa _{1}\varGamma +\kappa _{2}} {\kappa _{3}\varGamma +\kappa _{4}} > a_{0}}$

The inference is that sufficient system noise, β, can drive $\mathcal{T}$ below critical values in Fig. 8.5, triggering a system collapse analogous to a large, propagating traffic jam. Under real-world conditions, adequate service will simultaneously raise μ and lower β. Nonetheless, Eq. (8.14) is an expectation, and there will always be some probability that $\mathcal{T} < a_{0}$ , i.e., that the condition for stability is violated. The system then becomes “supercooled” and subject to a raised likelihood of sudden, rapidly propagating, traffic jam-like condensations in the sense of Kerner et al. (2015).

8.6 Multiple Phases of Dysfunction

The DRT argument implies a raised probability of a transition between stable and unstable behavior if the temperature analog $\mathcal{T} (\varGamma )$ falls below a critical value. Kerner et al. (2015), however, argue that traffic flow can be subject to more than two phases. We can recover something similar via a “cognitive paradigm” like that used by Atlan and Cohen (1998) in their study of the immune system. They view a system as cognitive if it must compare incoming signals with a learned or inherited picture of the world, then actively choose a response from a larger set of those possible to it. V2V/V2I systems are clearly cognitive in that sense. Such choice, however, implies the existence of an information source, since it reduces uncertainty in a formal way. See Chap. 1 or Wallace (2015a,b) for details of the argument.

Given the “dual” information source associated with the inherently unstable cognitive V2V/V2I system, an equivalence class algebra can be constructed by choosing different system origin states and defining the equivalence of subsequent states at a later time by the existence of a high probability path connecting them to the same origin state. Disjoint partition by equivalence class, analogous to orbit equivalence classes in dynamical systems, defines a symmetry groupoid associated with the cognitive process. Groupoids are generalizations of group symmetries in which there is not necessarily a product defined for each possible element pair (Weinstein 1996), for example, in the disjoint union of different groups.

The equivalence classes across possible origin states define a set of information sources dual to different cognitive states available to the inherently unstable V2V/V2I system. These create a large groupoid, with each orbit corresponding to a transitive groupoid whose disjoint union is the full groupoid. Each subgroupoid is associated with its own dual information source, and larger groupoids must have richer dual information sources than smaller.

Let $X_{G_{i}}$ be the system’s dual information source associated with groupoid element G_i. Given the argument leading to Eqs. (8.5)–(8.7), we construct another Morse Function (Pettini 2007) as follows.

Let $H(X_{G_{i}}) \equiv H_{G_{i}}$ be the Shannon uncertainty of the information source associated with the groupoid element G_i. We define another pseudoprobability as

$\displaystyle{ P[H_{G_{i}}] \equiv \frac{\exp [-H_{G_{i}}/\mathcal{T} ]} {\sum _{j}\exp [-H_{G_{j}}/\mathcal{T} ]} }$

(8.15)

where the sum is over the different possible cognitive modes of the full system.

Another, more complicated, “free energy” Morse Function F can then be defined as

$\displaystyle{ \exp [-F/\mathcal{T} ] \equiv \sum _{j}\exp [-H_{G_{j}}/\mathcal{T} ] }$

(8.16)

or, more explicitly,

$\displaystyle{ F = -\mathcal{T}\log \Big[\sum _{j}\exp [-H_{G_{j}}/\mathcal{T} ]\Big] }$

(8.17)

As a consequence of the groupoid structures associated with complicated cognition, as opposed to a “simple” stable–unstable control system, we can now apply an extension of Landau’s version of phase transition (Pettini 2007). Landau saw spontaneous symmetry breaking as representing phase change in physical systems, with the higher energies available at higher temperatures being more symmetric. The shift between symmetries is highly punctuated in the temperature index, here the “temperature” analog of Eq. (8.5), in terms of the scalar construct Γ, but in the context of groupoid rather than group symmetries. Usually, for physical systems, there are only a few phases possible. Kerner et al. (2015) recognize three phases in ordinary traffic flow, but V2V/V2I systems may have relatively complex stages of dysfunction, with highly punctuated transitions between them as various density indices change and interact.

Section 5.9 above examined sufficient conditions for a pathological ground state to “lock-in” and become highly resistant to managerial intervention, that is, in this context, for a highly persistent large-scale traffic jam.

In this context, Birkoff’s (1960, p. 146) perspective on the central role of groups in fluid mechanics is of considerable interest:

[Group symmetry] underlies the entire theories of dimensional analysis and modeling. In the form of “inspectional analysis” it greatly generalizes these theories…[R]ecognition of groups…often makes possible reductions in the number of independent variables involved in partial differential equations…[E]ven after the number of independent variables is reduced to one…the resulting system of ordinary differential equations can often be integrated most easily by the use of group-theoretic considerations.

We argue here that, for “cognitive fluids” like vehicle traffic flows, groupoid generalizations of group theory become central.

Decline in the richness of control information, or in the ability of that information to influence the system as measured by the “temperature” index $\mathcal{T} (\varGamma )$ , can lead to punctuated decline in the complexity of cognitive process possible within the C³ system, driving it into a ground state collapse that may not be actual “instability” but rather a kind of dead zone in which, using the armed drone example, “all possible targets are enemies.” This condition represents a dysfunctionally simple cognitive groupoid structure roughly akin to certain individual human psychopathologies, as described in the previous chapters.

It appears that, for large-scale autonomous vehicle/intelligent infrastructure systems, the ground state dead zone involves massive, propagating tie-ups that far more resemble power network blackouts than traditional traffic jams. Again, the essential feature is the role of composite system “temperature” $\mathcal{T} (\varGamma )$ . Most of the topology of the inherently unstable vehicles/roads system will be “factored out” via the construction of geodesics in a topological quotient space, so that $\mathcal{T} (\varGamma )$ inversely indexes the rate of topological information generation for an extended DRT.

Lowering the “temperature” $\mathcal{T}$ forces the system to pass from high symmetry “free flow” to different forms of “crystalline” structure—broken symmetries representing platoons, shock fronts, traffic jams, and more complicated system-wide patterns of breakdown.

In the next section, the underlying dynamic is treated in finer detail by viewing the initial phase transition as the first order onset of a kind of “turbulence,” a transition from free flow to “flock” structures like those studied in “active matter” physics. Indeed, the traffic engineering perspective is quite precisely the inverse of mainstream active matter studies, which Ramaswamy (2010) describes as follows:

It is natural for a condensed matter physicist to regard a coherently moving flock of birds, beasts, or bacteria as an orientationally ordered phase of living matter. …[M]odels showed a nonequilibrium phase transition from a disordered state to a flock with long-range order…in the particle velocities as the noise strength was decreased or the concentration of particles was raised.

In traffic engineering, the appearance of such “long-range order” is the first stage of a traffic jam (Kerner and Klenov 2009; Kerner et al. 2015), a relation made explicit by Helbing (2001, Sect. VI) in his comprehensive review of traffic and related self-driven many-particle systems.

While flocking and schooling have obvious survival value against predation for animals in three-dimensional venues, long-range order—aggregation—among blood cells flowing along arteries is a blood clot and can be rapidly fatal.

8.7 Turbulence

The “free energy” function F in Eq. (8.17) can be used to explore dynamics within a particular system phase defined by the associated groupoid.

Given a vector of system parameters K, in standard manner it is possible to define an “entropy” from F as the Legendre transform

$\displaystyle{ S \equiv F(\mathbf{K}) -\mathbf{K} \cdot \nabla _{\mathbf{K}}F }$

(8.18)

and a nonequilibrium Onsager stochastic differential equation for dynamics in terms of the gradients in S (de Groot and Mazur 1984), which can be written in one dimension as

$\displaystyle{ dK_{t} = [\mu \partial S/\partial K_{t}]dt +\sigma K_{t}dB_{t} }$

(8.19)

where μ is a diffusion coefficient. The last term represents a macroscopic volatility—proportional to the parameter K—in which dB _tis a noise term that may not be white, i.e., the quadratic variation [B_t, B _t] may not be proportional to t. While details will depend on the particular circumstances, such systems are subject to a distressingly rich spectrum of possible instabilities (Khasminskii 2012). The full set of equations would involve properly indexed sums across the parameters making up the vector K.

A simple example. If a system following Eq. (8.19) has been initially placed in a characteristic eigenmode—e.g., the smooth part of a “fundamental diagram” flow on some traffic network—then the dynamic equation for deviation of a parameter K(t) from that mode can be written, in first order, as

$\displaystyle{ dK_{t} \approx aK_{t}dt +\sigma K_{t}dW_{t} }$

(8.20)

where dW_trepresents white noise having uniform spectrum. Then, using the Ito chain rule on log[K],

$\displaystyle{ d\log [K_{t}] \approx (a -\sigma ^{2}/2)dt +\sigma dW_{ t} }$

(8.21)

The expectation is then

$\displaystyle{ E[K_{t}] \propto \exp [(a -\sigma ^{2}/2)t] }$

(8.22)

so that, if a < σ²∕2, E[K_t] → 0. σ² then—quite counterintuitively as described in Wallace (2016a)—is a kind of control information in the sense of the Data Rate Theorem that serves to stabilize system dynamics.

For an inherently unstable traffic flow, we impose closure on the model by taking $\sigma ^{2}/2 = \mathcal{T} (\varGamma )$ so that higher “temperature” means more “noise,” an intuitive result. Then, antiparalleling the arguments of Belletti et al. (2015, Sect. 2.3), for this simple example a “traffic Froude number” (TFN) $\mathcal{F}$ that defines regimes of free and turbulent flow, can be defined as

$\displaystyle{ \mathcal{F}\equiv 1 - [a -\mathcal{T} ] }$

(8.23)

where a is then a₀ in Eq. (8.5).

When $\mathcal{F} > 1$ , the system is in “laminar” free-flow, and becomes “turbulent” when $\mathcal{F} < 1$ .

A different characterization, from this perspective, is that $\mathcal{T}$ represents a kind of viscosity index so that $\mathcal{F}$ is more akin to a Reynolds number than to a classical Froude number.

A contrast between our approach and that of Belletti et al. (2015) lies in the central object-of-interest. They invoke a hydrodynamic perspective involving the “flow” of individual vehicles in a channel that finds “instability” to be associated with unconstrained travel speed. The focus here is on the stability of geodesics in the complex topological quotient space $\mathcal{M}^{2n}/W(r)$ . This is, in a sense, the inverse of their problem.

As argued, raising Γ lowers $\mathcal{T}$ , “freezing” the system from “liquid flow” to “crystallized” broken symmetries—platoons, shock fronts, jams, and myriad other “snowflake” structures that may include large-scale system-wide “lock-in” as described in Sects. 4.3 and 5.9 above.

8.8 Reconsidering Network Flow

The analysis of traffic flow failure on a road network is, conceptually, somewhat similar to characterizing the propagation of a signal via the Markov “network dynamics” formalism of Wallace (2016a) or Gould and Wallace (1994). This is an approach that might be used to empirically identify geodesic eigenmodes of real road network systems under different conditions, as opposed to individual vehicle dynamics or flow on a single road.

Following Gould and Wallace (1994), the spread of a “signal” on a particular network of interacting sites—between and within—is described at nonequilibrium steady state in terms of an equilibrium distribution ε_i“per unit area” A_iof a Markov process, where A scales with the different “size” of each node, taken as distinguishable by a scale variable A (for example, number of entering streets or average total traffic flow) as well as by its “position” i or the associated probability-of-contact matrix (POCM). The POCM is normalized to a stochastic matrix Q having unit row sums, and the vector ε calculated as ε = ε Q

There is a vector set of dimensionless network flows $\mathcal{X}_{t}^{i}$ , i = 1, …, n at time t. These are each determined by some relation

$\displaystyle{ \mathcal{X}_{t}^{i} = g(t,\epsilon _{ i}/A_{i}) }$

(8.24)

Here, i is the index of the node of interest, $\mathcal{X}_{t}^{i}$ is the corresponding dimensionless scaled ith signal, t the time, and g an appropriate function. Again, ε _iis defined by the relation ε = ε Q for a stochastic matrix Q, calculated as the network probability-of-contact matrix between regions, normalized to unit row sums. Using Q, we have broken out the underlying network topology, a fixed between-and-within travel configuration weighted by usage that is assumed to change relatively slowly on the timescale of observation compared to the time needed to approach the nonequilibrium steady state distribution.

Since the $\mathcal{X}$ are expressed in dimensionless form, g, t, and A must be rewritten as dimensionless as well giving, for the monotonic increasing (or threshold-triggered) function F

$\displaystyle{ \mathcal{X}_{\tau }^{i} = G\left [\tau, \frac{\epsilon _{i}} {A_{i}} \times \mathcal{A}_{\tau }\right ] }$

(8.25)

where $\mathcal{A}_{\tau }$ is the value of a “characteristic area” variate that represents the spread of the perturbation signal—evolving into a traffic jam under worst-case conditions—at (dimensionless) characteristic time τ = t∕T ₀.

G may be quite complicated, including dimensionless “structural” variates for each individual geographic node i. The idea is that the characteristic “area” $\mathcal{A}_{\tau }$ grows according to a stochastic process, even though G may be a deterministic mixmaster driven by systematic local probability-of-contact or flow patterns. Then the appropriate model for $\mathcal{A}_{\tau }$ of a spreading traffic jam becomes something like Eq. (8.20), with K replaced by $\mathcal{A}$ and t by τ. Thus, for the network, the signal Y_τmust again have a “noise”/vehicle density threshold condition like Eq. (8.23) for large-scale propagation of a traffic jam across the full network—something that would look very similar to the spread of a power blackout.

Zhang (2015) uses a similar Markov method to examine taxicab GPS data for transit within and between 12 empirically identified “hot zones” in Shanghai, determining the POCM and its equilibrium distribution.

This approach is something in the spirit of a long line of work summarized by Cassidy et al. (2011) that attempts to extend the idea of a fundamental diagram for a single road to a full transport network. As they put it,

Macroscopic fundamental diagrams (MFDs)…relate the total time spent to the total distance traveled…It is proposed that these macrolevel relations should be observed if the data come from periods when all lanes on all links throughout the network are in either the congested or the uncontested regime…

Following our arguments here, such conditions might apply when $\mathcal{A}_{\tau }\rightarrow 0$ , or when it encompasses the entire network domain. Indeed, Fig. 8.3 suggests why MFDs cannot be constructed in general: Congested and free flowing sections of traffic networks will often, and perhaps usually, coexist in an essentially random manner depending on local traffic densities. Figure 8.6, adapted from Geroliminis and Sun (2011), shows the limitations of the MFD approach. It examines the flow, in vehicles/5 min intervals, vs. percent occupancy over a 3 day period for the Minnesota Twin Cities freeway network that connects St. Paul and Minneapolis. See Fig. 1 of their paper for details of the road and sensor spacing. Evidently, while the unconstrained region of occupancy permits characterization of a geodesic mode, both strong hysteresis and phase transition effects are evident after about 8% occupancy, analogous to the “nucleation” dynamics of Fig. 8.1 at high traffic density. Again, as in Fig. 8.1 “fine structure” should be expected within both geodesic and turbulent modes, depending on local parameters.

Fig. 8.6

Adapted from Geroliminis and Sun (2011). Breakdown of the macroscopic fundamental diagram for the freeway network connecting St. Paul and Minneapolis at high vehicle densities. Both nucleation and hysteresis effects are evident, showing the fine structure within the turbulent mode. As in Fig. 8.1c, breakdown begins near 8–10% occupancy

Daganzo et al. (2010) further find that MFD flow, when it can be characterized at all, will become unstable if the average network traffic density is sufficiently high. They find that, for certain network configurations, the stable congested state

…is one of complete gridlock with zero flow. It is therefore important to ensure that in real-world applications that a network’s [traffic] density never be allowed to approach this critical value.

Daqing et al. (2014) examine the dynamic spread of traffic congestion on the Beijing central road network. They characterize the failure of a road segment to be a traffic velocity less than 20 km/h and use observational data to define a spatial correlation length in terms of the Euclidean distances between failed nodes. Our equivalent might be something like $\sqrt{ \mathcal{A}_{\tau }}$ . Adapting their results, Fig. 8.7 shows the daily pattern of the correlation length of cascading traffic jams over a 9 day period. The two commuting maxima are evident, and greatest correlation lengths reach the diameter of the main part of the city. Even at rush hour, no MFD can be defined, as, according to Fig. 8.3, the network will be a dynamic patchwork of free and congested components.

Fig. 8.7

Adapted from Daqing et al. (2014). Daily cycle of traffic jam correlation length over a 9 day period in central Beijing. The maxima cover most of the central city. For rush hour, no macroscopic fundamental diagram can be defined since the region is characterized by a patchwork of free and congested parts, as shown in Fig. 8.3

A next step would be to allow ρ, or a more general Γ, to vary in space and time, i.e., to parameterize the model using the moments of various density indices.

Figure 8.8, adapted from The Rand Fire Project (1979, Fig. 6.4), provides a disturbing counterexample to these careful empirical and theoretical results on network traffic flow, one with unfortunate results. Summarizing observations carried out by the Rand Fire Project, it represents a repeated sampling of “travel time vs. distance” for the full Trenton NJ road network in 1975 under varying conditions of time-of-day, day-of-week, weather, and so on, by fire companies responding to calls for service. This was an attempt to create a Macroscopic Fundamental Diagram in the sense used above, but without any reference at all to traffic density.

Fig. 8.8

Adapted from Fig. 6.4 of The Rand Fire Project (1979). Relation between fire company travel time and response distance for the full Trenton, NJ road network, 1975. The Rand Fire Project collapsed evident large-scale traffic turbulence into a simple “square root-linear” model used to design fire service deployment policies in high fire incidence, high population density neighborhoods of many US cities, including the infamous South Bronx. The impacts were literally devastating (Wallace and Wallace 1998)

Indeed, fire service responses are a traffic flow “best case” as fire units are permitted to bypass one-way restrictions, traffic lights, and so on, and usually able to surmount even the worst weather conditions. In spite of best-case circumstances, the scatterplot evidently samples whole-network turbulent flow, not unlike that to the right of the local geodesic in Figs. 8.1 and 8.6, part of a single street and a highway network, respectively, and consistent with the assertions of Cassidy et al. (2011) that MFD relations can only be defined under very restrictive conditions, i.e., either complete free flow or full network congestion.

The Rand Fire Project, when confronted with intractable whole-network traffic turbulence, simply collapsed the data onto a “square root-linear” relation, as indicated on the figure. The computer models resulting from this gross oversimplification were used to determine fire service deployment strategies for high fire incidence, overcrowded neighborhoods in a number of US cities, with literally devastating results and consequent massive impacts on public health and public order. Wallace and Wallace (1998), produced under an Investigator Award in Health Policy Research from the Robert Wood Johnson Foundation, document the New York City case history. The Rand models are still in use by the New York City Fire Department, for political purposes outlined in that analysis.

8.9 Directed Homotopy

The symmetry breaking phase transition argument of Sect. 8.6 can be rephrased in terms of “directed homotopy” —dihomotopy—groupoids on an underlying road network, again parameterized by the “temperature” index $\mathcal{T}$ . Classical homotopy characterizes topological structures in terms of the number of ways a loop within the object can be continuously reduced to a base point (Hatcher 2001). For a sphere, all loops can be reduced. For a toroid—a donut shape—there is a hole so that two classes of loops cannot be reduced to a point. One then composes loops to create the “fundamental group” of the topological object. The construction is standard. Vehicles on a road network, however, are generally traveling from some initial point So to a final destination S1, as in Fig. 8.9, and directed paths, not loops are the “natural” objects, at least over a short time period, as in commuting.

Fig. 8.9

Two equivalence classes of deformable paths connect the origin So with the destination S1, defining a groupoid. At high $\mathcal{T}$ , both sets of paths are available for travel. At some point the synergism between crowding and road conditions creates a blockage in one or the other routes around the triangular “hole,” breaking the groupoid symmetry. The “order parameter” is the number of blockages, which becomes zero at high symmetries

Given some “hole” in the road network, there will usually be more than one way to reach S1 from So, as indicated. An equivalence class of directed paths is defined by paths that can be smoothly deformed into one another without crossing barrier zones (Fajstrup et al. 2016; Grandis 2009), as indicated in the figure. At high values of the composite index $\mathcal{T}$ , many different sets of paths will be possible, creating a large groupoid. As $\mathcal{T}$ declines, roadways and junctions become increasingly jammed, eliminating entire equivalence classes, and lowering the groupoid symmetry: phase transitions via classic symmetry breaking on a network. The “order parameter” that disappears at high $\mathcal{T}$ is then simply the number of blocked roadways.

These results extend to higher dihomotopy groupoids via introduction of cylindrical paths rather than one-dimensional lines, producing a more general version of the quotient space geodesic method of Hu et al. (2001).

8.10 Discussion and Conclusions

Ruelle (1983) raises a red flag that must apply to any traffic flow analysis:

…[A] deductive theory of developed turbulence does not exist, and a mathematical basis for the important theoretical literature on the subject is still lacking…A purely deductive analysis starting with the Navier–Stokes equation…does not appear feasible…and might be inappropriate because of the approximate nature of the…equation.

Or, as the mathematician Garrett Birkoff (1960, p. 5) puts it,

…[V]ery few of the deductions of rational hydrodynamics can be established rigorously.

Similar difficulties constrain the Black–Scholes models of financial engineering, and institutions that rely heavily on them have often gone bankrupt in the face of market turbulence (Wallace 2015c).

Turbulence in traffic flow does not represent simple drift from steady linear or even parallel travel trajectories. Traffic turbulence involves the exponential amplification of small perturbations into large-scale deviations from complicated streamline geodesics in a topologically complex map quotient space. This is the mechanism of groupoid “symmetry breaking” by which the system undergoes a phase transition from “liquid” geodesic flow to “crystalline” phases of shock fronts, platoons, and outright jams.

Under such circumstances, cognitive system initiative serves as a mechanism for returning to geodesic flows. Inhibition of cognitive initiative occurs when the composite density index Γ exceeds a critical limit, triggering complex dynamic condensation patterns and, for autonomous vehicle systems, perhaps even more disruptive behaviors.

It is, then, not enough to envision atomistic autonomous ground vehicles as having only local dynamics in an embedding traffic stream, as seems in the current American and European practice. Traffic light strategies, road quality, the usually rapid-shifting road map space, the dynamic composition of the traffic stream, bandwidth limits, and so on create the synergistic context in which single vehicles operate and which constitutes the individual “driving experience.” It is necessary to understand the dynamics of that full system, not simply the behavior of a vehicle atom within it. The properties of that system will be both overtly and subtly emergent, as will, we assert, the responses of cognitive vehicles enmeshed in context, whether controlled by humans or computers.

One inference from this analysis is that failure modes afflicting large-scale V2V/V2I systems are likely to be more akin to power blackouts than to traffic jams as we know them, and the description by Kinney et al. (2005) is of interest:

Today the North American power grid is one of the most complex and interconnected systems of our time, and about one half of all domestic generation is sold over ever-increasing distances on the wholesale market before it is delivered to customers…Unfortunately the same capabilities that allow power to be transferred over hundreds of miles also enable the propagation of local failures into grid-wide events…It is increasingly recognized that understanding the complex emergent behaviors of the power grid can only be understood from a systems perspective, taking advantage of the recent advances in complex network theory…

Dobson (2007) puts it as follows:

[P]robabalistic models of cascading failure and power system simulations suggest that there is a critical loading at which expected blackout size sharply increases and there is a power law in the distribution of blackout size…There are two attributes of the critical loading: 1. A sharp change in gradient of some quantity such as expected blackout size as one passes through the critical loading. 2. A power law region in probability distribution of blackout size at the critical loading. We use the terminology “critical” because this behavior is analogous to a critical phase transition in statistical physics.

Daqing et al. (2014), in fact, explicitly link traffic jams and power failures:

Cascading failures have become major threats to network robustness due to their potential catastrophic consequences, where local perturbations can induce global propagation of failures…[that] propagate through collective interactions among system components…. [W]e find by analyzing our collected data that jams in city traffic and faults in power grid are spatially long-range correlated with correlations decaying slowly with distance. Moreover, we find in the daily traffic, that the correlation length increases dramatically and reaches maximum, when morning or evening rush hour is approaching…

While clever V2V/V2I management strategies might keep traffic streams in supercooled high-flow mode beyond critical densities, such a state is notoriously unstable, subject to both random and deliberately caused “condensation” into large-scale frozen zones. More subtle patterns of autonomous vehicle “psychopathology” may be even less benign, as studied in detail elsewhere (Wallace 2016b).

It is difficult to escape the inference that, despite understandable marketing hype and other wishful thinking, large-scale V2V/V2I autonomous vehicle systems may simply not be practical, particularly in a context of coupled social and infrastructure deterioration.

It has been said that “The language of business is the language of dreams.” Business dreams, however, do not necessarily serve as a sound foundation for the design and implementation of public policies affecting the well-being of large populations.

Acknowledgements

The author thanks Dr. D.N. Wallace for useful discussions.

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