Controllability of Complex Networks. Yang-Yu Liu, Jean-Jacques Slotine, Albert-Laszlo Barbasi Presented By Arindam Bhattacharya

Transcription

1 Controllability of Complex Networks Yang-Yu Liu, Jean-Jacques Slotine, Albert-Laszlo Barbasi Presented By Arindam Bhattacharya

2 Index Overview Network Controllability Controllability of real networks An analytical approach to controllability Robustness of control Conclusions

3 Index Overview Network Controllability Controllability of real networks An analytical approach to controllability Robustness of control Conclusions

4 Overview (Broad Ideas) The Ultimate proof of our understanding of natural and technological systems is reflected in our ability to control them. Q: Meaning of controllability of arbitrary complex directed networks? A:If with suitable choice of inputs we can drive a system from its initial state to a desired final state within a finite time.(philosophical) A:Identifying the set of driver nodes, that can guide the entire system s dynamics.(practical)

5 Difficulties The system s architecture. The dynamical rules that capture the interactions between the components.

6 Index Overview Network Controllability Controllability of real networks An analytical approach to controllability Robustness of control Conclusions

7 Network Controllability Real system -> Non Linear process -> (structurally similar)linear systems X(t)=(x 1 (t),,x n (t)) captures the state of a system of N nodes at a time t. The N X N matrix A describes the system s wiring diagram. (interaction strength). B is a NxM input matrix(m N) identifying nodes under outside control. The system is controlled using input vector u(t)=( u 1 (t),,u m (t)). Drive Nodes (if we find these, half the battle is won)

8 Network Controllability Real system -> Non Linear process -> (structurally similar)linear systems The system is controlled using input vector u(t)=( u 1 (t),,u m (t)). DriverNodes (if we find these, half the battle is won) To control a system we need to find the minimum number of these driver nodes, which is sufficient to control these system From now on this will be referred to as N D

9 Network Controllability Cont d Finding the minimum number of driver nodes N D.This Network can be controlled by an input vector U(u 1 (t),u 2 (t));

10 Network Controllability Cont d The system is controllable if Controllability matrix is full rank.(rank(c) = N) Kalman s controllability rank condition Controllability (practical def): finding the min number of nodes that satisfy the above equation.

11 Network Controllability Cont d Finding A in most real system is difficult. The system(a,b) Structurally Controllable if we can choose nonzero weights for A and B that satisfy (rank(c) = N). Structural controllability helps us to overcome the inherent incomplete knowledge of link weights A.

12 Network Controllability Cont d How to find the min number of nodes? Minimum number of input driver nodes needed to maintain full control of the network is determined by the maximum matching in the network. Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent edges; that is, no two edges share a common vertex.

13 Maximum Matching We gain full control of the directed network iff we directly control each unmatched node and there are directed path from the input signals to matched nodes. Solved the Problem of N D.(One of their main result)

14 Index Overview Network Controllability Controllability of real networks An analytical approach to controllability Robustness of control Conclusions

15 Controllability of real Networks n D =N D /N Explore the controllability of several real networks: gene regulatory- should be efficient from a control perspective small n D Social networks high n D, WWW not even sure what n D represents.

16 Controllability of real Networks n D =N D /N

17 Driver Nodes and hubs Node divided into (low, medium and high) according to degree K Mean degree of driver nodes as a function of mean degree <K> Driver Nodes tend to avoid hubs(high K)

18 What defines Controllability? What topological features determine network controllability?

19 Controllability of networks Randomize keeping N and L same Randomize keeping in degree and out degree same N D is determined by number of incoming and outgoing links

20 What defines Controllability? What topological features determine network controllability? A:Thus we see a system's controllability is to a great extent encoded by the underlying network distribution.

21 Index Overview Network Controllability Controllability of real networks An analytical approach to controllability Robustness of control Conclusions

22 An analytical approach to controllability The importance of degree distribution allows us to determine N D analytically for a network with an arbitrary P(Kin,Kout). Cavity method: derive a set of self consistent equations whose input is the degree distribution and solution is the average N D Over all network realizations compatible to P(k in,k out ).

23 An analytical approach to controllability Predicted Nd correlates well to Nd Real

24 An analytical approach to controllability

25 An analytical approach to controllability Driver node density as a function of K with different values of ϒ.

26 Index Overview Network Controllability Controllability of real networks An analytical approach to controllability Robustness of control Conclusions

27 Robustness of control Classify each link as Critical- increase driver nodes if link fails. Redundant-Can be removed. Ordinary-Can be removed for current state

28 Robustness of control Classify each link as Critical- increase driver nodes if link fails. Redundant-Can be removed. Ordinary-Most links are ordinary

29 Behavior of Lc

30 Behavior of Lc

31 Behavior of Lc

32 Behavior of Lc

33 Index Overview Network Controllability Controllability of real networks An analytical approach to controllability Robustness of control Conclusions

34 Conclusions Control is a central issue in most complex system, very little was known about how to control large directed networks. The key finding here is that N d is determined mainly by the degree distribution. Tools to predict N D from P(K in,k out ). EXPLORE ARBITARY NETWORK TOPOLOGIES.