Energy Scaling with Control Distance in Complex Networks
Isaac Klickstein, Ishan Kafle, Sudarshan Bartaula, Francesco Sorrentino
EEnergy Scaling with Control Distancein Complex Networks
Isaac Klickstein, Ishan Kafle, Sudarshan Bartaula, and Francesco Sorrentino
Department of Mechanical EngineeringUniversity of New MexicoEmail: [email protected]
Abstract —It has recently been shown that the expected energyrequirements of a control action applied to a complex networkscales exponentially with the number of nodes that are targeted.While the exponential scaling law provides an adequate pre-diction of the mean required energy, it has also been shownthat the spread of energy values for a particular number oftargets is large. Here, we explore more closely the effect distancebetween driver nodes and target nodes and the magnitude ofself-regulation has on the energy of the control action. We findthat the energy scaling law can be written to include informationabout the distance between driver nodes and target nodes to moreaccurately predict control energy.
I. I
NTRODUCTION
The control of complex networks is an extremely active field[1]–[5]. Numerous applications can be found from regulatingpower grids, routing traffic on the internet, marketing on socialmedia, synchronization of multi-agent systems, and manyothers. Of particular interest recently is the specific controlaction that is minimal with respect to the control energy [3],[4], [6]–[8].The nodes of the network are classified as driver nodes,target nodes, or neither. We will call those nodes which receivea control signal directly driver nodes [2] and those nodeswhose state values we prescribe at the end of the controlaction target nodes [9]. It has been shown recently that theexpected amount of energy required for a particular controlaction on a complex network decreases exponentially withrespect to the number of inputs (or driver nodes) [4] andincreases exponentially with respect to the number of outputs(or target nodes) [3]. These scaling relations describe verywell the required amount of control energy when this quantityis averaged over many possible network realizations. Thevariation seen in the energy requirement about the exponentialmean value is directly related to the topology of the graphthat describes the connections between vertices. Other scalingrelations have been obtained in [10], [11], which focus onthe case that one wants to target all the network nodes. Themain difference with the previous work is that here we exploitthe recently developed framework for optimal target control ofnetworks [3], [5] to provide new insight into how the selectionof both driver and target nodes over a network affects thecontrol energy. When trying to increase the controllability ofa complex network, it is often the addition of edges [10], [11]that provides the most improvement.In section II we briefly provide background on graph theory and optimal control of complex networks. In section III wepresent results on both canonical graphs (paths and stars) aswell as large Erdos-Renyi graphs [12]. Finally, we summarizehow the energy scales with respect to the graph propertiesinvestigated in section IV.II. B
ACKGROUND
A graph G = ( V , E ) consists of a set of n nodes V = { v i | i =1 , . . . , n } and (cid:96) edges E ⊂ V × V where ( v i , v j ) ∈ E if node v j receives information from node v i . Our work focuses onundirected graphs, that is ( v i , v j ) ∈ E implies ( v j , v i ) ∈ E . Agraph can be represented as an n by n adjancency matrix A which has elements A ij = A ji = 1 if the pair ( v i , v j ) ∈ E and A ij = A ji = 0 otherwise for i (cid:54) = j . We assume there is aself-loop at every node, that is, ( v i , v i ) ∈ E i = 1 , . . . , n , withweight − k i so the diagonal values of the adjacency matrixare A ii = − k i . Given a pair of nodes ( v i , v j ) , we define the distance between them d ij as the length of the shortest pathfrom v i to v j . As the graph G is undirected d ij = d ji .Many dynamical systems can be described by graphs wherethe edges of the graph represent transfer of power, traffic,influence, or information. We consider linear dynamics wherethe n by n state matrix A is the adjacency matrix of thegraph and the n by m control matrix B describes where in thenetwork the control inputs u k ( t ) , k = 1 , . . . , m are attached.The p by n output matrix C selects the target nodes. ˙ x ( t ) = A x ( t ) + B u ( t ) , x ( t ) = x y ( t ) = C x ( t ) , y ( t f ) = y f (1)The initial condition of the system x is known and the finalcondition for the target nodes y f is prescribed. To enforce thegraph nature of this problem, we restrict B ( C ) to have n -dimensional, independent versors as columns (rows). If entry B ij = 1 , then node i is a driver node, and if entry C ji = 1 ,then node i is a target node.In the following statements, we will assume the triplet ( A, B, C ) is output controllable [13], that is the rank of thematrix rank ( K ) = rank (cid:2) CB | CAB | . . . | CA n − N (cid:3) = p . Thecontrol input u ∗ ( t ) that is minimal with respect to energy E = (cid:82) t f t || u ( t ) || dt that drives the system in Eq. (1) fromthe initial condition x to the prescribed final condition forthe targets y f is known from optimal control theory [3], u ( t ) = B T e A T ( t f − t ) C T (cid:0) CW ( t f ) C T (cid:1) − g (2) a r X i v : . [ phy s i c s . s o c - ph ] J a n n a k s l o p e d l og ( E ) k = 3 k = 11 k = 25 k = 100 b n c b l og ( E ) d = 2 d = 3 d = 4 d = 5 d Fig. 1. Energy requirements for two types of graphs. a A path graph of length d = n − . b The energy required when the driver node is at one end of thepath and the target node is chosen to be distance d away for different values of self-regulation parameter k . As k increases, the curves become steeper, whileremaining linear. The inset shows how the slope of these curves increase with increasing regulation parameter k . c A balloon graph of length d and numberof branches b so that n = b ( d −
1) + 2 . The driver node is placed at one of the hubs and the target is chosen as the other hub. d The energy to control onehub with the other hub of the balloon graph for varying b and d . As the number of branches increase, while holding the distance between driver and targetconstant, the energy required decreases by a few orders of magnitude.
12 653 47 8 9 10 1112 13 14 15 16 kkk k kkk- δ k- δ k- δ k- δ k- δ kk k kk a l og ( E ) d = 1, δ = 0 . d = 2, δ = 0 . d = 3, δ = 0 . d = 1, δ = 0 . d = 2, δ = 0 . d = 3, δ = 0 . b Fig. 2. Analyzing the energy requirements for a star graph while increasingthe number of targets. a A star graph with branches of length . We emphasizethat we analyzed the case where each target added to the set of target nodesis at the same distance from the single driver node located at the hub. b The exponential scaling with respect to the cardinality of the target set p isexpected. On the other hand, we see that the rate of the exponential increasewhen adding target nodes at a given distance is approximately independentof the distance (see Table I). The symmetric semi-positive definite matrix W ( t ) is thecontrollability Gramian which solves the linear ODE ˙ W = AW ( t ) + W ( t ) A T + BB T . The vector g = (cid:0) y f − Ce A ( t f − t ) x (cid:1) is the difference between the prescribedfinal conditions and the zero-input evolution of the systemwhen t = t f . If the matrix A is Hurwitz, then the ODE thatdescribes the evolution of W ( t ) is globally, asymptoticallystable with fixed point W ∗ that solves the equation, O n = AW ∗ + W ∗ A T + BB T (3)Equation (3) is an algebraic Lyapunov equation which canbe solved efficiently. The characteristic energy of the controlaction can be written succinctly in terms of the asymptoticcontrollability Gramian W ∗ . E ∗ = max || g || =1 lim t f →∞ (cid:90) t f t || u ( t ) || dt = max || g || =1 g T (cid:0) CW ∗ C T (cid:1) − g = 1 µ ∗ min (4)The positive scalar µ ∗ min is the smallest eigenvalue of the ma-trix CW C T . Note that if p = 1 , then C has a single non-zeroentry located at position i and so CW ∗ C T = W ∗ ii = µ ∗ min .The case when m = 1 and p = 1 is of particular interest tous as it represents the energetic contribution when controllinga single target node from a single driver node.As we will assume that A is Hurwitz, we choose thediagonal values of the adjacency matrix k i to ensure thisproperty holds. By the Gerschgorin Disc Theorem, we choose − k i < − max { deg ( i ) } where deg ( i ) is the degree of node i so that k i > (cid:80) nj =1 ,j (cid:54) = i | A ij | .From our studies of the control energy scaling of complexnetworks, we hypothesize that the particular contribution tothe control energy any one target node provides is a functionof three main quantities; (i) the distance between the drivernode and the target node, (ii) the redundancy of paths betweenthe driver node and the target node, and (iii) the cardinality of d l og ( E ) Path GraphER: ‘ = 400ER: ‘ = 800 ER: ‘ = 1200ER: ‘ = 1600ER: ‘ = 2000 Fig. 3. The energy requirements for the one driver and one target problemin Erdos-Renyi graphs of different density, (cid:96)/n . There is at most one path oflength d = 1 between two nodes and so the energy for pairs of nodes in anyof the ER graphs of distance apart is equal. As the distance between pairsof nodes grows, the number of paths of that distance also grows and so, aswas seen in the analysis of the balloon graphs, the energy decreases at most afew orders of magnitudes. Similar results were obtained for k -regular graphsand scale-free graphs. the set of target nodes. These three aspects will be investigatedin the next section. III. R ESULTS
For the first property that determines the energy contributionof a target node, we examine the role that distance betweendriver nodes and target nodes plays. We consider a path graphof length n where the driver node is chosen to be at oneend which we define to be node . A path graph is shownin Fig. 1 a . The target node is chosen along the path so thatif node i is the target node, then the driver node and targetnode are distance ( i − apart. The energy of the controlaction is E ∗ = 1 /W ∗ ii , and log( E ∗ ) is shown in Fig. 1 b . Weperform the simulation for path graphs with increasing valuesof k along the main diagonal of the adjacency matrix. As k grows, the curves in Fig. 1 b become steeper. Interestingly, wesee there is an exponential scaling of the energy with distancebetween driver node and target node so that log( E ∗ ) ∝ f ( k ) d ,where f ( k ) represents the slope of the energy curve for a pathgraph with − k along the diagonal of the adjacency matrix. Theinset of Fig. 1 b shows how the slope of these energy curvesincreases with respect to k where we see linear dependencesuch that f ( k ) ∝ k ) . Combining these results, log( E ) ∝ k ) d (5)While the energy scaling in Eq. (5) provides a reasonableprediction for the path graph, when examining the energyrequirements for the case of one driver node and one targetnode in a large complex network, it is far less successful. Infact, given a set of pairs of driver nodes and target nodes ina complex network of all the same distance apart, the energymay vary across multiple orders of magnitude. This is due, inthe case of a complex network, to the redundancy of paths ofa given length between pairs of nodes.To investigate this phenomenon, we turn to a generalizationof the path graph which we call a balloon graph as shown in d l og ( E ) a d l og ( E ) b d l og ( E ) c d l og ( E ) d Fig. 4. The spread of energy requirements for the one driver and one targetcase in Erdos-Renyi graphs of increasing density. a m = 400 , b m = 800 , c m = 1600 , d m = 2000 . The top dashed line in each panel is the maximumvalue of log E ∗ among all pairs of nodes at distance d . The bottom dashedline in each panel is the minimum value of log E ∗ among all pairs of nodesat distance d . We see that the upper limit of the energy is bounded by a pathgraph with equal regulation parameter k along the diagonal of the adjacencymatrix. The lower limit of the energy is at most a few orders of magnitudeless as was seen in the analysis of the balloon graph. Fig. 1 c . A balloon graph consists of two hubs labeled and n with b vertex-disjoint branches that connect the hubs, each oflength d so that the whole graph consists of n = b ( d −
1) + 2 nodes. When b = 1 , the balloon graph contracts to a pathgraph of length d . The energy curves for different numbersof branches of different lengths is shown in Fig. 1 d . Wesee that as the number of branches increases from one tothree, the energy is reduced most drastically As the numberof branches continues to grow, the energy continues to reduceabout linearly. This decay provides a modification to Eq. (5) byincluding a term that describes the number of vertex-disjointpaths of length d . log( E ) ∝ k ) d − αb (6)The coefficient α is obtained by finding the slope of the linearportions of the curves in Fig. 1 d where we see α ≈ forall curves. We have seen that for the case where we includeadditonal paths of length larger than d , the decrease in energyis much less substantial, that is, the energy value is dominatedby the length of the shortest path between driver node andtarget node. On the other hand, including an additional pathof length smaller than d reduces the energy far more than seenin Fig. 1 d as we have reduced the distance between the pair ofnodes. This indicates that though all the paths between pairs ofnodes may contribute to the control energy, the shortest pathbetween them is the one whose contribution is dominant.Here we extend our study to the case of mutliple targetnodes. As the shortest distance is the dominating factor inaffecting the control energy, we choose target nodes that are allat the same distance from a given driver node. We consider the ABLE IA
PPROXIMATE SLOPES FOR THE CURVES IN F IG . 2 B d = 1 d = 2 d = 3 δ = 0 . . . . δ = 0 . . . . δ = 0 . . . . star graph in Fig. 2 a , where the driver node is in red and all thenodes at distance 2 are in blue. We can vary both the distanceof the target nodes, as well as the number p of target nodesat that given distance. Without loss of generality we label thetarget nodes , , ..., p . In order to maintain controllability, weset the self loop weights at each target node, k i = k − iδ , i = 1 , . . . , p , so that the parameter δ measures the minimumdifference between any two self loop gains. The results ofour simulations are shown in Fig. 2 b . As can be seen, theenergy increases exponentially with the number of target nodes p at a given distance d , with the rate of exponential growthdepending on the particular choice of the parameter δ , i.e., thesmaller the value δ the higher is the rate.It is interesting to see that however, the rate of exponentialgrowth of the energy is little affected by the distance of thedriver nodes d . This is shown in table I as we vary both thedistance d and the parameter δ . The observed variations of theexponential rate with the distance d are by far less significantthan those seen when the parameter δ is varied.From the results in [3], the expected value of the energyincreases exponentially with the number of target nodes,but the variation of energy values is large as can be seenby the wide standard deviation bars across multiple ordersof magnitude. We consider Erdos-Renyi (ER) graphs with n = 400 and (cid:96) edges so that the average degree (cid:96)n . For everypair of nodes ( v i , v j ) we compute the distance d ij and theenergy when v i is the only driver node and v j is the only targetnode, E ∗ ij . The mean of the energy is plotted with respect to thedistance between the two nodes with error bars representingone standard deviation are shown in Fig. 3. Also included isthe energy values from a path graph with the same value k as in all of the ER graphs. We see that the energy along thepath graph provides an upper bound to the energy betweenany two nodes in the ER graphs. As we have assumed thereare no multi-edges , there can be at most one path of length d = 1 between two nodes. As such, one would expect that theenergy requirement when the driver node and target node areadjacenct would be approximately the same as the energy ofthe path graph when d = 1 . We see this is the case in Fig. 3for all ER graphs examined when d = 1 . When d = 2 , thepossibility of multiple paths of length arises and we beginto see some energy values are less than the pair of nodes atdistance d = 2 in the path graph. As d increases from to , the number of possible paths of length d increases. Theaverage energy between two nodes of distance d increasesmore slowly than the path graph as the density of the ER graph grows. In the most sparse graph, (cid:96) = 400 , the energyrequirements between pairs of nodes is nearly equal to thepath graph.The minimum and maximum energy over any pair of nodesof distance d for the ER graphs is shown in Fig. 4. For the case (cid:96) = 400 in Fig. 4 a , so that the average degree is , we see verylittle deviation from the energy values of the path graph. Theaverage degree of the path graph is n − /n ≈ as well.For the other ER graphs, the maximum value of the energybetween a pair of nodes at distance d remains nearly equalto the path graph. The minimum value of the energy on theother hand remains within a few orders of magnitude of thepath graph energy as was predicted by the balloon graph.IV. C ONCLUSION
It has been established previously that the control energyrequired to drive a subset of nodes (the target nodes) of acomplex network scales exponentially with the cardinality ofthe set of target nodes [3]. Here, we examine in more detailthe role any individual node plays as a target node when pairedwith a driver node. Given a distance between driver node andtarget node d , we have seen the path graph of length d wherethe driver node and target node are on opposite ends providesan upper bound on the energy contribution. Additional pathsof the same length can reduce the energy by a few orders ofmagnitude. Including additional targets at the same distance d from the sole driver node exponentially increases the energybut the rate of the increase is independent of the particulardistance and instead is a function of the individual self-loopweights, characterized by the parameter δ . Finally, we examinethe energy between pairs of driver nodes and target nodes incomplex networks and we see that our predictions based onthe path graph, balloon graph and star graph hold.This work provides the framework for two design aspectsof complex networks. Given a driver node target node pair atdistance d , one can reduce the energy requirement by eitheradding an edge to reduce the distance between them d (cid:48) < d , oradding an edge to increase the number of paths of length d . Onthe other hand, adding an additional edge that accomplishesneither of these tasks will not yield any benefit. Rather thanadding edges, if one is interested in removing edges, any edgethat is neither a part of one of the shortest paths betweenthe driver node and target node will not increase the controlenergy. In the future, these relations can be used to optimizethe design of complex networks with respect to the controlenergy. A CKNOWLEDGEMENT
This work was funded by the National Science Foundationthrough NSF grant CMMI- 1400193, NSF grant CRISP-1541148 and ONR Award No. N00014-16-1-2637.R
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