12TH ICCRTS Adapting C2 to the 21st Century

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1 12TH ICCRTS Adapting C2 to the 21st Century Title of Paper: The Ipact of Entropic Drag on Coand and Control Paper ID: I-2 Track(s): Network Centric Experientation and Applications, C2 Concepts Theory and Policy, Organizational Issues Authors: David Scheidt, Michael Pekala Point of Contact: David Scheidt Nae of Organization: Johns Hopkins University Applied Physics Laboratory Coplete Address: Johns Hopkins University Applied Physics Laboratory 111 Johns Hopkins Road, Laurel, MD 2723 Telephone: (24) E-ail: Abstract: A coonly believed axio in signal detection theory is that "ore inforation is good" [1]. That is, when attepting to deterine the state of a partially observable syste the addition of correct inforation onotonically iproves the correctness of the state assessent. When diagnosing static systes the assertion that "the effect of inforation is to increase the likelihood of getting correct diagnosis, while reducing the likelihood of incorrect diagnosis" holds. However, when diagnosing dynaic systes, the iproveent in diagnosis is offset by increases in uncertainty generated by the dynaic forces within the syste being observed. A siilar effect occurs when controlling dynaic, stochastic systes. The act of exercising control requires a finite aount of tie, during which uncertainty enters into the syste reducing the efficacy of the control policy. The ipact of inforation processing delays increases in relevance as the coplexity and pace of both the syste and control apparatus increase. This paper defines a atheatical fraework describing the effect of inforation and inforation processing on the diagnosis of dynaic systes. 1

2 Introduction Regardless of the coand and control structure, the quality and speed of inforation flow between tea ebers directly ipacts the perforance of a tea. In all structures, latencies are caused by each hop in the counications chain. Counication liitations have been a topic of growing popularity since Shannon [2]. The iportance Shannon's inforation entropy has on coand and control has been studied by Perry [3], Moffett [4] and Caves [5]. However, these studies are tie insensitive and do not attept to easure and account for the dynaic forces associated with coplex, changing environents. In this paper we take a step toward filling that gap after initially laying the technical and conceptual foundations of abstract systes and inforation theory. Systes and Worlds An understanding of the inforation required to represent a syste begins with a foral definition of that syste. We refer to a syste being diagnosed or controlled as a world (W). Worlds are defined in ters of eleents, attributes, events, and state transitions, as follows. Definition: A world W = (K, E, A, Att, Val, f ), where K = {k 1, k n } is a finite set of eleents E = {e 1, e r } is a set of events A = {a 1, a p } is a finite set of attributes, each applicable to one or ore eleents. Att K A A 2... A P is an eleent-attribute relation in which ( k,( a,..., a )) Att i if and only if ki1 ki i( 1 i ) a k i are those and only j those attributes that apply to eleent k i. [Notation: A k denotes the set of k s attributes; i.e., A = a,..., a } k,( a,..., a )) ] k { ki 1 ki [Notation: a A ki j ki of values for attribute a ] ( i ki1 ki V a ki is the range j k i K, Val k i V a... V ki a 1 ki is a valuation relation for k i s attributes. Val = U Val k is the collection of i k i K valuation relations for all eleents in W. f : K Val E Val is a state transition function. Eleents represent real or siulated objects, e.g., baffles, therostats, switches, and ducts or their representations in a real or siulated HVAC world. Attributes provide descriptions of eleents, such as location and orientation. These are finite, constrained values which collectively define the scope of the world. The state of an eleent is an assignent of values to all its attributes; i.e., the state space (P k ) of eleent k is Val k. ki j P = V (1) k al k An individual state of k is represented by x k or equivalently by the list ( x,, ) k x k 1 K of values of k s attributes. Eleents change states via events. Laws that govern state transitions are apped through the function f. P W denotes the state space of W and is defined as the power set of the state spaces for all k K, i.e.; 2

3 Pw Pk Pk = P (2) Each eleent of P W is called a state of the world, or world state, soe of which are feasible (P f ) and others are infeasible (P i ). P f and P i are utually exclusive and coprehensive: P f Pi = Ø (3) Pw = Pf Pi (4) Inforation and Entropy The inforation content (I) of essage is the aount of inforation required to uniquely identify a state. At this point we only consider essages whose inforation content is ideally coded for the set of possible values that could be stored in the essage. That is, the representation schee uses the iniu nuber of bits to copletely represent the set of possible values. Inforation content relative to state space P is defined by Brillouin [6] as: cln P I = (5) Here, c is a constant associated with the size of the language used to represent a world state. In odern coputers and counications this language is binary allowing us to define inforation content as: k n I = log 2 P (6) Through Brillouin the nuber of feasible states that exist for a world after essage describing that world has been received (and processed) is I (log2 P 2 w ) P f = (7) or, alternatively, 2 I P w f = I I (8) The feasible space represents disorder, lack of knowledge, or inforation entropy (h) [7]. Shannon defines inforation entropy of a piece of inforation, a, as P 2 1 h ( a) = log2 (9) Pr( a) in which Pr(a) is the general probability of a. In a world of fixed size, the post-essage entropy of that world is: H = log 2 Pf = log2 Pw I (1) Inforation entropy is a useful construct that we will extend to collections of inforation and inforation exchanges. Inforation entropy and uncertainty are frequently used interchangeably. While both are used to easure disorder in a syste we distinguish the by the units in which they are easured. Inforation entropy, being a for of Shannon inforation, is easured in bits. Uncertainty is easured in natural units for the syste being described (e.g., eters). Equation (1) shows how new inforation can be used to reduce entropy within the representation of a world. In static worlds the inforation content of a essage that contains positive inforation (I >) decreases inforation entropy. For ideally coded essages, providing inforation about a previously unknown world (worlds in which H = log 2 P w ) the decrease in inforation entropy is equal to the size of the essage. ΔH I I ; Pw 2 = log2 Pw ; Pw < 2 I (11) If soe knowledge of the world exists prior to receipt of a essage, a portion of the essage ay be redundant, reducing the essage's inforation content. The inforation content of a essage about a world for which a priori inforation is available is expressed as: Δ H = log 2 P I p( I ) I (12) f where p(x) is a decoding function that produces a set of feasible states fro x. Uncertainty Due to Fidelity 3

4 In practical ters worlds are represented as abstractions. In odern counications, seantics for encoding attributes (and eleents, in this paper) are defined and encoded in bits. Soe real-world inforation is naturally discrete. Soe, though, is naturally continuous and can only be approxiated with binary representations. The aount of uncertainty associated with a discrete representation of continuous attributes (ε r ) is liited (i.e., bounded fro below) by the nuber of bits used to encode the data and the diensionality of the data. For exaple, when counicating attributes describing positions in Cartesian space, a essage ust be transitted digitally at soe point, liiting the fidelity by the least precise representation used in the counications path, which is the unit distance (r ˆ) of the representation. In this discussion we assue that a noralized representation schee is used and all states are equally probable. MacKay [8] showed that a noralized representation schee axiizes the inforation content per transitted bit. Since we are priarily interested in expressing the liits of counication we can assue optial, noralized encoding is used and one bit of transitted data equals one bit of un-decayed Shannon inforation. The unit distance defines the iniu uncertainty of our knowledge such that the iniu aount of error is: n ˆ 2 n ε r = r; r Z (13) where r is the unit size of representation and n is the diensionality of the space. For a world of fixed size, the size of the state space is derived fro the range of attributes within the syste and the unit distance of each attribute. Specifically, xi x Pk = (14) x X rˆ x where x and x i are the iniu and axiu values of an ordered attribute and rˆ is the unit distance of the attribute. When a attribute diensions are equivalent (e.g. Cartesian coordinates), then P k n xi x = (15) and the size of the state space of the world is P w = x i rˆ x rˆ K n (16) The inforation content of a regular world (i.e. a world in which transitions between states are Markovian) can now be shown as a function of diensions of the world and the unit distance: I w = K n( log 2 ( xi x ) log 2 rˆ ) (17) Entropic Drag So far we have liited our discussion to static worlds, in which inforation entropy is decreased by each successive essage that describes the world. On the other hand, in dynaic worlds the decrease in entropy achieved by a essage can be offset by an increase in entropy due to the world's dynaic forces. We call this entropic drag (Γ), as the increase in entropy creates a drag on an observer's ability to understand the world. Entropic drag is derived fro the rate at which the dynaic forces create unpredictable change. To define these dynaic forces recall that a world's state space P w is divided into feasible and infeasible states and that the transition between these states is Markovian, defined by the transition function ƒ, defined above. For any set of infeasible states there exists a boundary layer between it and feasible states. The rate at which infeasible states transition to feasible states is the driving force behind entropic drag. We forally define entropic drag as the log 2 of the rate at which previously infeasible world states becoe feasible: di dx Γ w ( t) = H w( I ) = log2 Pf ( x) (18) dt dt 4

5 The ipact of entropic drag on inforation is shown as a decrease in the inforation content on a world after an observation such that: t I = ( + ) Γ( ) w I I 1 t dt (19) where t is the tie required to process the essage, I is the inforation content on the world prior to processing the essage and I is the inforation content of the essage. The forces of entropic drag continuously degrade inforation on a world whether essages continue to arrive or not. Only through the continual acquisition of new inforation can a state of knowledge of a dynaic world be aintained. If a coander's objective is to ake a decision with axial inforation he ust be cognizant of the rate of entropy and the value of new inforation. An iportant feature of equation (19) is that, if the tie required to process a essage is sufficiently long, the loss of inforation associated with entropic drag will exceed the inforation content of the essage and the net result of processing the essage will be an inforation loss! Obtaining and using inforation takes soe finite aount of tie. The process of obtaining and using inforation was described by Boyd as the Observe, Orient, Decide and Act (OODA) loop. The first part of this loop consists of a sensor observing the environent and transitting a essage to an inforation sink where it is fused with additional inforation to create a consistent odel of the world. The delay (δ) between an observation (beginning) at tie t o and the incorporation of the observed data at t' is the su of the sensor processing (δ s ), counications (δ c ) and data fusion (δ f ) delays. t ' = t + δ = t + δ s + δ c + δ f (2) We assue there are no additional delays, for instance placing on hold a essage between the copletion of processing and the onset of counications. The sensor processing and counications channels are assued to process inforation at a fixed rate easured in bits per second ( β ). The data processing profile for data fusion varies by technique, with soe techniques, such as nearest-tie replaceent, operating at a fixed rate, and others, such as search-based techniques, incurring exponential increases in latency as the nuber of observations increase. For the purpose of this paper we will assue that each eleent in the observe-orient chain operates at a fixed rate in bits per second. or δ β I δ = (21a) I t = β = (21b) By applying this definition to (19) we can express inforation content in ters of essage size and bandwidth. I I ( ) ( ) w = I + I 1 Γ β t dt (22) and I Γ ( I + I ) β t dt ΔI w = I ( ) (23) Fro (18) we can see that inforation is only gained fro a single essage if the rate of inforation entropy is less than the bandwidth. I > iff β > Γ (24) Δ w Inforation entropy progresses along a frontier between the feasible state space and the infeasible state space. The frontier (S) is set of piecewise sooth surfaces (s i S) within P w that are defined as the boundaries between infeasible states x` P i that are reachable fro a feasible state x`` P f through an event in accordance to the transition function. Each piecewise sooth surface is either a eber of a set of surfaces that for a closed n-diensional surface that encapsulates positive or negative inforation, or a eber of a 5

6 sei-closed surface whose edges abut liits of P w space. S changes at a rate that is defined by the entropic drag with the boundary perpetually growing the aount of feasible space and shrinking the infeasible space. The entropic drag for a surface s i is: dh r dx Γs ( t) = H s ( t) = log si v i i 2 (25) dt dt where v r is the n-diensional vector of unpredictable change noralized to s i. The entropic drag for the entire world is the su of the entropic drag for each surface. dh Γw ( t ) = H s ( t) i (26) dt s Positive and Negative Inforation i Positive inforation (P + ) is a set of assertions that one or ore states are true. + P = { x1,x2,..,xn} (27) Positive inforation usually represents highly iprobable states, so essages containing positive inforation provide a large aount of inforation content. For exaple, a essage stating that an ant was observed at a given location at tie t is positive inforation that, through inference, ensures that that specific ant was not at any other location at t. Negative inforation (P - ) is a set of assertions that one or ore states are false; i.e., that the negation of each state holds. P = { x1, x2,.., xn} (28) Figure 1 A negative observation in P w projected onto Z 2 space For exaple, a essage stating that an observation at tie t found that an eleent did not exist at a specific location is negative inforation. Inforation partially describing a world state can be expressed either through positive inforation or negative inforation. In practice, observations frequently produce a ixture of positive and negative inforation. Positive and negative inforation both create infeasible regions in the state space and the inforation gain fro a essage is proportional to the reduction in feasible states. This can be visualized as If essage encodes negative inforation, the inforation gain fro is I = P log 2 (29) inforation describing a hole of infeasibility in the state space [Fig. 1]. 6

7 envisioned as the feasibility frontier growing outward fro a clustered set of feasible states [Fig. 3]. Inforation content decreases until eventually all states becoe feasible. As negative inforation entropies, inforation is decreasingly lost in proportion to the diensionality of the world. This can be envisioned as the feasibility frontier growing inward, reoving a diinishing set of clustered infeasible states [Fig. 4]. Figure 2 A positive observation of a unique object in P w projected onto Z 2 space The inforation gained fro positive inforation given in essage is equal to the size of the space outside of the positive inforation set: + + I = log 2 Pw log2 P (3) This can be visualized as a feasibility volue surrounded by infeasibility [Fig. 2]. Figure 4 Reducing infeasible state space after a negative observation Figure 3 Reducing infeasible state space after a positive observation Each closed surface either grows (I + ) or shrinks (I - ) due to entropic drag until it: (i) terinates in a singularity; (ii) terinates against other surfaces; or (iii) terinates against the boundaries of the state space. The inforation loss for the world is a piecewise linear surface that is the su of the inforation loss for each surface [Fig. 5]. In Fig. 5, s i is the border between the shaded feasibility state space and the unshaded infeasible state space. As inforation fro positive observations entropies, inforation is increasingly lost at a rate that increases in proportion to the diensionality of the original essages. The loss of positive inforation can be 7

8 Figure 5 Piecewise linear state space resulting fro I + and I - essages Inforation Channels Inforation channels (C(t)) are streas of inforation, with inforation flow easured in bits per second being the bandwidth of a channel (β). We use inforation channel as an abstraction representing the aggregate ability of a syste to acquire and process new inforation although inforation can be, and often is, a traditional counications channel. The axiu capacity of an inforation channel defines the rate at which a syste is capable of acquiring new inforation. In ideal conditions where C(t) = ax β, inforation on a static world increases linearly until the inforation capacity of the world is reached [Fig. 6]. When the inforation in a channel describes a dynaic world, the rate of inforation gained fro a constant inforation channel is reduced by the entropic drag on inforation within the channel and the entropic drag on previously acquired inforation. The aount of inforation accuulation on a world fro an inforation channel is: I β I = I + ( C t) Γ( t ) ( dt (31) ) As entioned briefly above, edge effects can play an iportant role in the rate of entropic drag. We highlight two basic cases here. In the first case the inforation gathering bandwidth is sufficient to exhaustively describe the world before the inforation content of the first essage has vanished in a singularity or against an edge. In this case the inforation bandwidth exceeds the axiu entropic drag and the inforation content will becoe axiized when the inforation space has been transitted. log 2 ( P w ) tax (32) β Figure 7 - Inforation accuulation with entropic drag Figure 6 Inforation accuulation fro ideal counications Once again assuing our ideal continuous channel where C(t) = ax β, the inforation accuulation for a dynaic syste with sufficient bandwidth is shown in Fig. 7. In this figure the axiu inforation is 8

9 log2( P ax = log2 Pw Γ β ) w I (33) In this case inforation onotonically increases until tie t ax after which further essages can only aintain the current aount of inforation, as the inforation gained by successive essages cannot exceed the cobination of inforation lost due to entropic drag and the inforation loss through redundancy with prior inforation. Figure 8 Inforation accuulation In the second case the bandwidth is less than the axiu entropic drag on the entire world and, as the entropic drag increases in proportion to the inforation known about the world, a stable point will be reached when the entropic drag equals the bandwidth. This point of inforational stability is the axiu feasible inforation for the syste [Fig. 8]. Note that in both cases the axiu inforation reached is less than the inforation space of the world. This gap is the iniu inforation loss associated with describing a dynaic world. Uncertainty Redux A key design decision for the construction of coand and control systes is the selection of the fidelity of inforation being counicated. Fidelity is adjustable as the designer can arbitrarily decide the unit easure for each diension in the world within sensor liitations. For exaple, a designer can choose to send positions in which the lowest bit of inforation represents a illieter, eter or kiloeter. Typically the fidelity, or bit accuracy, of a easureent is set to be slightly less than the resolving power of the sensor. This is a useful heuristic when the uncertainty associated with sensor fidelity is uch larger than the uncertainty associated with inforation entropy. However, when the speed of that which is being observed increases, or the coplexity of the world being observed increases, or the nuber of recipients of the observed inforation increases the uncertainty related to inforation can becoe an iportant design criterion for the coand and control syste. It is these cases that we are ost interested in understanding. Earlier we showed how a iniu aount of uncertainty (ε) is associated with the choice of representation. The iniu uncertainty of a syste (U) at soe tie t is the aggregation of the syste's representational uncertainty (U r ) and the feasible state space translated fro bits to real world units. U = U + rˆ (34) r P f If a syste designer wanted to design a syste to obtain the axiu aount of inforation possible, and an ideal sensor capable of observing the entire world was available, what level of fidelity should be used? At observation tie our ideal sensor would reduce the feasible state space to a single state: P ƒ = 1. The representational uncertainty is the product of the size of the inforation frontier and the inforational error: U r = S t ε r (35) where s i S t is the set of piecewise sooth surfaces that define the boundary between P f and P i at tie t. By substitution in (34), uncertainty becoes 9

10 U St ε r + r Pf (36) Both ters of the uncertainty equation (34) vary as a function of size unit vector. However, the rate at which they vary differs, with the uncertainty due to representational error becoing predoinant as the unit representation grows and the uncertainty due to entropy becoing predoinant as the unit representation shrinks. This duality allows us to identify the optial unit representation for axiizing uncertainty. = ˆ To explore this relationship let us exaine the siple case in which a C2 designer has an oniscient caera capable of copletely viewing the entire world at any level of fidelity. Fro (16), the aount of tie required to counicate the caera's observation is δ log2 Pw K xi x = = log 2 β β x X rˆ (37) x For a regular world in which all n diensions are equal, this evolves to K n δ = ( log 2 a log 2 r) (38) β in which a is a constant proportional to the size of a diension. If we assue independent entities, the feasibility space will be δ r P f = K v( t) dt (39) where v r is again the n-diensional vector of unpredictable change noralized to S t. In this case S t is a set of K unit possibilities. The perieter of this space is r S t = K v( t) dt (4) δ By applying (39) to (36) we obtain a representational uncertainty of K rˆ n r U r = v( t) dt 2 δ and by (36), the overall uncertainty is (41) K rˆ n v δv U = ( t) dt + rˆ K δ ( t) dt (42) 2 This is a soewhat essy equation that expresses inial uncertainty as a function of the unit representation (because the tie δ over which the functions are now evaluated in ters of the unit representation). This equation can be evaluated when the exact diensions of the world are known and a odel of the entropic forces is available. When evaluated it shows inial uncertainty for a given unit length. An exaple evaluation for a siple twodiensional world is shown in Fig. 9, in which the graph of uncertainty diinishes rapidly as fidelity increases (oving fro right to left in the figure) until it reaches the optial resolution at the local iniu after which uncertainty begins to increase as entropic drag takes hold. Figure 9 Uncertainty as a Function of Unit Resolution Observation Inefficiencies We have been assuing that essages are ideal, with each observation bit translating to a single bit of inforation. This is true if and only if no a priori knowledge exists. This includes infeasible states derived fro prior observations as well as probability distributions across the state space. When a priori knowledge exists, the actual inforation gained is the product of the efficiency (γ) of an observation and the 1

11 axiu capacity of the counications channel. I actual C(t) = γ (43) I I = I + β γ ( C t) Γ( t ) ) ( dt (44) Koopan showed that optially efficient search patterns can exist for regular worlds [9], yet the ajority of real-world intelligence, surveillance and reconnaissance (ISR) systes are fundaentally inefficient because, over tie, sensors will traverse an area over which soe a priori knowledge exists. The rate of observational inefficiencies over tie can be highly nonlinear as oving sensors cross feasibility frontiers. However, inefficiency rates for observing a regular environent can be approxiated by decaying the inforation achieved with each successive observation such that I & = I( 1 λ) (45) where the rate of decay ( λ ) is proportional to the probability that a state deeed infeasible by the observation was previously known as infeasible ties inforation bandwidth. λ = β P( p i p ) o i a (46) In order to explore entropic relationships within inefficient systes we notionally describe this efficiency loss as a decay function. The loss in efficiency is highly dependent upon the syste and its current use the use of a decay function to represent loss of efficiency is a useful approxiation of real-world ISR [1]. Inefficiency results in a reduction in channel capacity, C(t). Because it reduces the aount of knowledge accuulated over tie, the forces of entropic drag increase the utility of researching previously observed areas. We easure this effect by replacing the constant inforation value in equations (19) with a decay function λ(i,t) I to show the relationships between inforation variant essages and inforation entropy. I ( ( I, t) Γ( λ( I, t)) ) = t λ dt (47) When deploying a sensor otion strategy that searches the coplete range of the world the entropic drag can becoe so large that it overcoes inforation gain. This effect is shown in Fig. 1 which shows the effect of a linear entropic drag on the observation environent shown in Fig. 9. Fig. 9 shows that the axiu aount of inforation is found after an ideal nuber of observations have occurred. If the goal of a coand and control syste is to ake decisions based upon the ost coplete inforation possible, decisions would be ade iediately after these observations as the entropic drag effect causes degradation beyond this point. Figure 1 While we use a linear function in our exaple, we recognize that the inforation function ay be highly non-linear. However, non-linearity does not negate the principle shown here. Entropic Drag and Network Topologies Siulation experients were conducted to exaine the effect of entropic drag on a siple coand and control network infrastructure. These experients exained the flow and relevance of inforation throughout the network. The bulk of the siulation s requireents lie in the ters 11

12 C2 network infrastructure and inforation. A coand and control network infrastructure consists of a set of counicating entities (or nodes) whose counications topology fors an acyclic tree. The lines of counication between nodes (edges) also correspond to the chain of coand. Thus, a node s authority in the network is inversely proportional to its depth in the counications tree (i.e., the root node of the tree has the highest authority and leaf nodes have the least). An additional property of the counications/coand tree is that all nodes at the sae depth in the tree have the sae nuber of directly reporting subordinates (child nodes). This apping of rank to nuber of iediate subordinates is the property that distinguishes alternate C2 infrastructures in the siulator. Within the siulator, inforation consists of data gathered fro the environent by leaf nodes in the C2 infrastructure. Each piece of inforation is an abstract quantity that is independent of other pieces of inforation (e.g., inforation does not overlap or correspond to ultiple easureents of a known target in the environent as ight be the case in a filtering proble). Inforation can be generated by leaf nodes periodically or stochastically, depending on the siulator configuration. The value of each piece of inforation is a nuber in the interval [,1], where 1 corresponds to axiu value and corresponds to no value. Another property of inforation is that it only becoes useful to an entity after the inforation has been fused into the local world odel. Thus, inforation is subject to two priary sources of latency before it can increase the knowledge of a network entity: latency due to network counications and latency due to the local inforation fusion algorith. Siulation Structure The C2 network inforation siulation is a odular, discrete tie-step-based siulation whose priary coponents are network entities, counications links, and inforation processing algoriths. In addition, the siulation tracks a nuber of etrics for each piece of inforation. Two key etrics are inforation area (the nuber of nodes that finished fusing the inforation in the current tie step) and inforation volue (the nuber of nodes that have fused the inforation at or before the current tie step). These etrics are taken fro the literature on the perforance of real-world networks. The overall value of a piece of inforation is derived by ultiplying the inforation volue by the associated entropic drag. Since these area and volue etrics are affected by the nuber of nodes in the network, when coparing different C2 topologies we typically consider topologies with the sae nuber of leaf nodes and then restrict the results to leaf node areas and volues. The siulation has a nuber of variable input paraeters including siulation duration (tie steps), C2 topology, entropic drag (value lost/tie step), latency along a network counications link (tie steps/observation), and latency due to fusion. Unlike other siulation paraeters which are constant, fusion latency can optionally be a function of the nuber of prior fusion operations (allowing for nonlinear fusion coplexity). Siulation Outputs Fig. 11 shows a saple inforation volue plot for a single siulation run. These plots generally contain three pieces of inforation: the ideal inforation volue (red line), the inforation volue without accounting for entropic drag (blue line) and the inforation volue with entropic drag (green line). The ideal inforation volue depicts the spread of inforation throughout the network assuing no latency due to counications or fusion and hence no 12

13 entropic drag. Thus, it provides an (adittedly unrealistic) upper bound on the aount of inforation in the network. Nonetheless, it provides a strong indication that an upper bound exists. The un-decayed inforation volue depicts the network inforation volue that would result when considering latency but not entropic drag. The decayed inforation volue takes into account both the effects of latency and entropic drag, and represents the ain result of interest. One expects both the ideal and un-decayed inforation volues to onotonically increase, with the entropic drag volue trailing behind the ideal volue. The entropic drag volue shows that there exists a axiu inforation volue (tie=175) for the syste. Further, it shows that this peak occurs prior to the coplete distribution of inforation across the network across the network (tie=24). evidence. We can also see that the tie at which the inforation axiu occurs varies with respect to the entropic drag. The inforation axiu occurs at tie step 23 for the runs with lower inforation drag (lines at the top of the graph), at tie step 175 for the runs with higher entropic drag (lines at the botto) and equally across tie step 23 and 175 for the lines in the iddle. Figure 12 Inforation volue as Γ varies Future Efforts Figure 11 Inforation volue in ideal static worlds, latent static worlds and dynaic worlds A key feature of the siulation output is the ability to copare one or ore siulation runs. Fig. 12 is a plot coparing ultiple runs of the siulation where the rate of entropic drag was varied while the other siulation properties were held constant. The upper lines in the plot have progressively lower decays. In Fig. 12 one the existence of axiu inforation volue prior to full disseination of inforation across the network is again in This paper is a first step in the application of inforation theoretic entropy to coand and control. Large bodies of work in inforation anageent, particularly data fusion, networking, and the pantheon of group control strategies need to be looked at though the lens of entropic drag. Our forward looking hypothesis is that an understanding of the entropic effects of inforation will allow C2 designers and inthe-field decision akers to eploy coand and control strategies that are optiized for any given situation. The next significant step will be to extend the study of entropic drag into the latter portion of the OODA loop, that portion which involves decisions and actions; Fry [11] akes the case that the inforation entropy ipact on the "DA" portion of the 13

14 loop is the dual of the "OO" portion of the loop. This view is also reflected by Alberts & Hayes [12]. We concur that the ipact of entropic drag on control, when viewed independently of observation, is equivalent to the ipact of entropic drag on observation. However, control is dependent upon observation and the ipact of entropic drag is likely to be exacerbated in the control portion of the OODA loop. Another area of investigation involves an application of these foraliss to realworld scenarios. The state of a dynaical syste changes according to the rules specified by the transition function f. The nature of this transition function can have a doinant effect on what set of states can becoe feasible in the future. The laws that govern the transition function are a priori inforation, reducing the state space of the syste. In this paper we assued ideal coding of inforation describing the state space. In practice it is ore coon (and practical) to encode inforation generally and to use inference based upon a priori inforation to reduce the working feasibility space. The investigation of these relationships will be a key part of the scenario-specific investigation. Another iportant aspect of this investigation ust be the entropic effects of the counications infrastructure. We divide counications infrastructures into channel counications and topologies. Further work will also be required to iprove the etrics that are used to easure the dynais within an environent. Environental dynaics are driven by coplexity and the pace of environental change; however, is not well understood how coplexity and pace should be easured in real-world environents. These further advanceents should enable the pursuit of our long-range objective, naely the construction of an adaptive coand and control syste that autonoously observes the environent and changes the network topology and inforation and decision-aking strategies to optiize C2 perforance. Conclusion We have shown that the loss of inforation due to dynaic forces within an environent can have a substantial ipact upon the inforation content of one or ore essages about the environent. This effect, called entropic drag, fundaentally ipacts the effectiveness of coand and control systes. The principles outlined in this paper can be used provide a better understanding of the utility of existing coand and control systes and to iprove the design of future ones. Entropic drag ipacts C2 systes in several iportant ways. First, entropic drag enforces a fundaental liit to the aount of inforation that can be known about a dynaic syste. This liit can be used by C2 designers to identify the axiu useful fidelity of C2 seantics. Second, by expressing the relationship between inforation and tie entropic drag allows a decision aker to identify when the optial aount of inforation has been acquired. Third, entropic drag provides a fraework for understanding the flow of inforation across a networked counity, providing insight into the utility (or lack thereof) of sharing inforation with each eber of the counity as well as providing insight into the utility of alternative network topologies. Fourth, entropic drag provides tools to better understand the tradeoff between inforation and latency in control, allowing decision akers to select the optial aount of inforation to use for perforing a specific task or set of tasks. Fifth, entropic drag provides tools to understand the utility of collaboration in shared decision aking, allowing decision akers to correctly scope the degree of collaboration for optial perforance of a task. Finally, entropic drag provides a fraework for the developent of next generation, adaptive C2 infrastructures, 14

15 infrastructures that autonoously adapt inforation and decision sharing strategies and networking topology at run tie in response to an environent's changing dynaic forces. References [1] Heeger, D., Signal Detection Theory, New York University. [2] Shannon, C. E., A Matheatical Theory of Counications, The Bell Systes Technical Journal, Vol 27, pp , , July, Oct; [3] Perry, Burton et al, Measures of Effectiveness for the Inforation Age Navy: The Effects of Net-Centric Operations on Cobat Outcoes. (22) RAND Santa Monica, CA, USA. [4] Moffat, Coplexity Theory and Network Centric Warfare. DoD Coand and Control Research Progra, 23. [5] Caves, C., Entropy and Inforation: How Much Inforation is Needed to Assign a Probability?, Coplexity, Entropy and the Physics of Inforation, Zuruk, W. (ed.), Westview Press, [6] Brillouin, L., Science and Inforation Theory 2 nd Ed., Dover Publications Inc., Mineola, NY, 24. [7] Sith, Effects Based Operations, DoD Coand and Control Research Progra, 22. [8] MacKay, D., Inforation Theory, Inference and Learning Algoriths, Cabridge University Press, pg 67, 25. [9] Koopan, B., Search and Screening. Operations Evaluations Group Report No. 56, Center for Naval Analyses, Alexandria, VA, [1] Stone, L., Theory of Optial Search. Acadeic Press, New York, [11] Fry, R. An inforation-theoretic forulation of intelligent weapon systes. Technical Meo A1F(1)6-U-53, Johns Hopkins University Applied Physics laboratory, Dec., 26. [12] Alberts, D., Hayes, R., Planning Coplex Endeavors [Counity Review version], CCRP, 26, pg. 67. Acknowledgeents This research effort was supported by Johns Hopkins University Applied Physics Laboratory internal research grant X9ANXC2M. Thanks to Mike Miller, Joe Peri and David Viel for their excellent critique and advice. Biographies David H. Scheidt received a B.S. in coputer engineering fro Case Western Reserve University in He is a branch scientist at the Johns Hopkins University Applied Physics Laboratory where he chairs the laboratory's Autonoy Research Thrust. His current research concentrations are: distributed robotics, coand and control of decentralized systes, distributed intelligent control and cooperative systes. Mr. Scheidt has twice received the National Perforance Review s Haer Award. Michael J. Pekala is a Coputer Scientist for the National Security Technology Departent of the Johns Hopkins University Applied Physics Laboratory. He received B.S. and M.S. degrees in Coputer Science fro the Johns Hopkins University in 1998 and 21 respectively. Mr. Pekala joined the Laboratory in 1998, and his current focus is on the research and developent of autonoous systes. His eail address is ike.pekala@jhuapl.edu. 15