The Princeton Companion to Applied Mathematics

800 VII. Application Areas

is not ε). Then the Lyapunov exponent λmax defined as marketing and advertising spending in the United King- dom and the United States is now spent online: so dig- x(q)max lim = λmax q→∞ q ital media marketing (“buzz” marketing), though in its infancy, requires a deeper understanding of the nature (and analogously for λmin) exists for almost all se- of social communication networks. This is an impor- quences and is independent of the initial (finite) choice tant area of complexity theory, since there is no closed of x(0). This result shows that, in principle, it is pos- theory (analogous to conservation laws for molecular sible to calculate the efficiency measures discussed above. The main difference is that, although the Lya- dynamics or chemical reactions) available at the micro- punov exponents exist, there is no nice prescription for scopic “unit” level. Instead, here we must consider irra- their calculation. The development of efficient numer- tional, inconsistent, and ever-changing people. More- ical algorithms is an open problem that the interested over, while the passage of ideas is mediated by the reader might, perhaps, wish to consider! networking behavior, the very existence of such ideas may cause communication to take place: systems can Further Reading therefore be fully coupled. In observing peer-to-peer communication in mobile Butkoviˇc,P. 2010. Max-linear Systems: Theory and Algo- phone networks, messaging, email, and online chats, rithms. London: Springer. the size of communities is a substantial challenge. Gaubert, S., and M. Plus. 1997. Methods and applications of Equally, from a conceptual modeling perspective, it (max,+) linear algebra. In STACS ’97: 14th Annual Sympo- sium on Theoretical Aspects of Computer Science. Lecture is clear that being able to simulate, anticipate, and Notes in Computer Science, volume 1200, pp. 261–82. infer behavior in real time, or on short timescales, Berlin: Springer. may be critical in designing interventions or spotting Heidergott, B., G. J. Olsder, and J. van der Woude. 2006. Max sudden aberrations. This field therefore requires and Plus at Work: Modeling and Analysis of Synchronized Sys- has inspired new ideas in both applied mathematical tems: A Course on Max-Plus Algebra and Its Applications. models and methods. Princeton, NJ: Princeton University Press. Litvinov, G. L. 2007. Maslov dequantization, idempotent and tropical mathematics: a brief introduction, Journal 2 Evolving Networks in of Mathematical Sciences 140:426–44. Continuous and Discrete Time Pachter, L., and B. Sturmfels. 2005. Algebraic Statistics for Computational Biology. Cambridge: Cambridge University Consider a population of N individuals (agents/actors) Press. connected through a dynamically evolving undirected network representing pairwise voice calls or online chats. Let A(t) denote the N × N binary adjacency VII.5 Evolving Social Networks, matrix for this network at time t, having a zero diago- Attitudes, and Beliefs—and nal. At future times, A(t) is a stochastic object defined Counterterrorism by a probability distribution over the set of all possible Peter Grindrod adjacency matrices. Each edge within this network will be assumed to evolve independently over time, though 1 Introduction it is conditionally dependent upon the current network (so any edges conditional on related current substruc- The central objects of interest here are tures may well be highly correlated over time). Rather (i) an evolving digital network of peer-to-peer commu- than model a full probability distribution for future net- nication, work evolution, conditional on its current structure, P + | (ii) the dynamics of information, ideas, and beliefs that say δt (A(t δt) A(t)), it is enough to specify its + | can propagate through that digital network, and expected value E(A(t δt) A(t)) (a matrix contain- (iii) how such networks become important in matters ing all edge probabilities, from which edges may be of national security and defense. generated independently). Their equivalence is trivial, since The applications of this theory spread far beyond these E(A(t + δt) | A(t)) = BP (B | A(t)), topics though. For example, more than a quarter of all δt B

For general queries, contact [email protected] © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher.

VII.5. Evolving Social Networks, Attitudes, and Beliefs—and Counterterrorism 801 and to others, while its column sums represent the abili- N−1,N ties of the corresponding people to receive messages Bi,j − P | = − 1 Bi,j δt (B A(t)) Wi,j (1 Wi,j ) , from others. Such performance measures are useful in i=1,j=i+1 identifying influential people within evolving networks. where W = E(A(t + δt) | A(t)). Hence we shall specify This idea has recently been extended so as to succes- our model for the stochastic network evolution via sively discount the older networks in order to produce E(A(t + δt) | A(t)) = A(t) + δtF(A(t)), (1) better inferences. → F valid as δt 0. Here the real matrix-valued function 3 Nonlinear Effects: Seen and Unseen is symmetric, it has a zero diagonal, and all elements within the right-hand side will be in [0, 1]. We write In the sociology literature the simplest form of nonlin- earity occurs when people introduce their friends to F(A(t)) =−A(t) ◦ ω(A(t)) + (C − A(t)) ◦ α(A(t)). each other. So, in (2), if two nonadjacent people are Here C denotes the adjacency matrix for the clique connected to a common friend at step k, then it is 1 − where all 2 N(N 1) edges are present (all elements are more likely that those two people will be directly con- 1s except for 0s on the diagonal), so C − A(t) denotes nected at step k+1. To model this triad closure dynamic the adjacency matrix for the graph complement of A(t); we may use ω(A˜ k) = γC, so all edges have the same ω(A(t)) and α(A(t)) are both real nonnegative sym- step-to-step death probability, γ ∈ [0, 1], and metric matrix functions containing conditional edge α(A˜ ) = δC + εA2. death rates and conditional edge birth rates, respec- k k tively; and ◦ denotes the Hadamard, or element-wise, Here δ and ε are positive and such that δ+ε(N−2)<1. 2 matrix product. The element (A )i,j counts the number of mutual con- In many cases we can usefully consider a discrete- nections that person i and person j have at step k. This { }K equation is ergodic and yet it is destined to spend most time version of the above evolution. Let Ak k=1 denote an ordered sequence of adjacency matrices (binary, of its time close to states where the density of edges symmetric with zero diagonals) representing a discrete- means that there is a balance between edge births and time evolving network with value Ak at time step tk. deaths. A mean-field approach can be applied, approxi- We shall then assume that edges evolve independently mating Ak with its expectation, which may be assumed from time step to time step with each new network con- to be of the form pkC (an erd˝os–rényirandom graph ditionally dependent on the previous one. A first-order [IV.18 §4.1] with edge density pk). In the mean-field model is given by a Markov process dynamic one obtains 2 + = 1 − + 1 − + − 2 (3) E(Ak+1 | Ak) = Ak ◦(C −ω(A˜ k))+(C −Ak)◦α(A˜ k). (2) pk 1 pk( γ) ( pk)(δ (N )εpk). If δ is small and ω< 1 ε(N − 2), then this nonlin- Here ω(A˜ k) is a real nonnegative symmetric matrix 4 function containing conditional death probabilities, ear iteration has three fixed points: two stable ones, at δ/γ + O(δ2) and 1 + ( 1 − γ/(ε(N − 2)))1/2 + O(δ), each in [0, 1], and α(A˜ k) is a real nonnegative symmet- 2 4 ric matrix function containing conditional edge birth and one unstable one in the middle. Thus the extracted probabilities, each in [0, 1]. mean-field behavior is bistable. In practice, one might As before, the edge independence assumption im- observe the edge density of such a network approach- ing one or other stable mean-field equilibrium and jig- plies that P(Ak+1 | Ak) can be reconstructed from gling around it for a very long time, without any aware- E(Ak+1 | Ak). A generalization of katz [IV.18 §3.4] centrality for ness that another type of orbit or pseudostable edge such discrete-time evolving networks can be obtained. density could exist. Direct comparisons of transient orbits from (2), incorporating triad closure, with their In particular, if 0 <μ<1/ max{ρ(Ak)}, then the communicability matrix mean-field approximations in (3) are very good over − − − short to medium timescales. Yet though we have cap- Q=(I − μA ) 1(I − μA ) 1 ···(I − μA ) 1 1 2 K tured the nonlinear effects well in (3), the stochastic provides a weighted count of all possible dynamic nature of (2) must eventually cause orbits to diverge paths between all pairs of vertices. It is nonsymmetric from the deterministic stability seen in (3). (due to time’s arrow) and its row sums represent the The phenomenon seen here explains the events of abilities of the corresponding people to send messages a new undergraduate’s first week at university are so

802 VII. Application Areas important in forming high-density connected social containing the degrees of the vertices. This system has networks among student year groups. If we do not per- an equilibrium at X = X∗, say, where the ith column of turb them with a mix of opportunities to meet, they X∗ is given by x∗ for all i = 1,...,N. may be condemned to remain close to the low-density Now consider an evolution equation for A(t),inthe (few-friends) state for a very long time. form of (1), coupled to the states X: E(A(t + δt) | A(t)) 4 Fully Coupled Systems = A(t) + δt(−A(t) ◦ (C − Φ(X(t)))γ There is a large literature within psychology that is + (C − A(t)) ◦ Φ(X(t))δ). (6) based on individuals’ attitudes and behaviors being in Here δ and γ are positive constants representing the a tensioned equilibrium between excitory (activating) maximum birth rate and the maximum death rate, processes and inhibiting processes. Typically, the state respectively; and the homophily effects are governed by of an individual is represented by a set of state vari- the pairwise similarity matrix, Φ(X(t)), such that each ables, some measuring activating elements and some term Φ(X(t)) ∈ [0, 1] is a monotonically decreasing measuring the inhibiting elements. i,j function of a suitable seminorm x (t) − x (t).We Activator–inhibitor systems have had an impact j i shall assume that Φ(X(t)) ∼ 1 for x (t)−x (t) <ε, within mathematical models where a uniformity equi- i,j j i and Φ(X(t)) = 0 otherwise, for some suitably chosen librium across a population of individual systems i,j ε>0. becomes destabilized by the very act of simple “pas- There are equilibria at X = X∗ with either A = 0or sive” coupling between them. Such Turing instabilities A = C (the full clique). To understand their stability, can sometimes seem counterintuitive. let us assume that δ and γ → 0. Then A(t) evolves very Homophily is a term that describes how associations slowly via (6). Let 0 = λ λ ··· λ be the eigen- are more likely to occur between people who have 1 2 N values of Δ. Then it can be shown that X∗ is asymptot- similar attitudes and views. Here we show how indi- ically stable only if all N matrices, df(x∗) − Dλ , are viduals’ activator–inhibitor dynamics coupled through i simultaneously stability matrices; and conversely, it is a homophilic evolving network produce systems that unstable in the ith mode of Δ if df(x∗) − Dλ has an have pseudoperiodic consensus and fractionation. i eigenvalue with positive real part. Consider a population of N identical individuals, Now one can see the possible tension between each described by a set of m state variables that are m homophily and the attitude dynamics. continuous functions of time t. Let xi(t) ∈ R denote Consider the spectrum of df(x∗) − Dλ as a function the ith individual’s attitudinal state. Let A(t) denote of λ.Ifλ is small then this is dominated by the stability the adjacency matrix for the communication network, of the uncoupled system, df(x∗).Ifλ is large, then this as it does in (1). Then consider is again a stability matrix, since D is positive-definite. N x˙ = f(x ) + D A (x − x ), i = 1,...,N. (4) The situation, dependent on some collusion between i i ij j i ∗ j=1 choices of D and df(x ), where there is a window of Here f is a given smooth field over Rm, drawn from a instability for an intermediate range of λ, is known as → class of activator–inhibitor systems, and is such that a Turing instability. Note that, as A(t) C, we have → f(x∗) = 0 for some x∗, and the Jacobian there, λi N, for i>1. So if N lies within the window of insta- df(x∗), is a stability matrix (that is, all its eigenvalues bility, we are assured that the systems can never reach a have negative real parts). D is a real diagonal nonnega- stable consensual fully connected equilibrium. Instead, tive matrix containing the maximal transmission coef- Turing instabilities can drive the breakup (weakening) ficients (diffusion rates) for the corresponding attitu- of the network into relatively well-connected subnet- dinal variables between adjacent neighbors. Let X(t) works. These in turn may restabilize the equilibrium dynamics (as the eigenvalues leave the window of insta- denote the m×N matrix with ith column given by xi(t), and let F(X)be the m×N matrix with ith column given bility), and then the whole process can begin again as homophily causes any absent edges to reappear. Thus by f(xi(t)). Then (4) may be written as we expect a pseudocyclic emergence and diminution of ˙ = − X F(X) DXΔ. (5) patterns, representing transient variations in attitudes. Here Δ(t) denotes the graph Laplacian for A(t), given In simulations, by projecting the network A(t) onto two by Δ(t) = Γ(t)−A(t), where Γ(t)is the diagonal matrix dimensions using the Frobenius matrix inner product,

VII.5. Evolving Social Networks, Attitudes, and Beliefs—and Counterterrorism 803 one may observe directly the cyclic nature of consensus So far we have discussed peer-to-peer networks in and division. general terms. But we are faced with some specific Even if the stochastic dynamics in (6) are replaced by challenges that stress the importance of social and deterministic dynamics for a weighted communication communications networks in enabling terrorist threats: adjacency matrix, one obtains a system that exhibits • aperiodic, wandering, and also sensitive dependence. In the analysis of very large communications net- such cases the orbits are chaotic: we know that they will works, in real time; oscillate, but we cannot predict whether any specific • the identification of influential individuals; individuals will become relatively inhibited or relatively • inferring how such networks should evolve in the activated within future cycles. This phenomenon even future (and thus spotting aberrant behavior); and occurs when N = 2. • recognizing that fully coupled systems may natu- These models show that, when individuals, who are rally lead to diverse views, and pattern formation. each in a dynamic equilibrium between their acti- All of these things become ever more essential. Popula- vational and inhibitory tendencies, are coupled in a tion-wide data from digital platforms requires efficient homophilic way, we should expect a relative lack of and effective applicable mathematics. global social convergence to be the norm. Radical and conservative behaviors can coexist across a population Modern adversaries may be most likely to be and are in a constant state of flux. While the macro- • organized through an actor network of transient scopic situation is predictable, the journeys for indi- affiliations appropriate for (i) time-limited oppor- viduals are not, within both deterministic and stochas- tunities and trophy or inspired goals; (ii) procure- tic versions of the model. There are some commen- ment, intelligence, reconnaissance and planning; tators in socioeconomic fields who assert that diver- and (iii) empowering individuals and encouraging gent attitudes, beliefs, and social norms require lead- both innovation and replication through competi- ers and are imposed on populations; or else they are tion; driven by partial experiences and events. But here we • employing an operational digital communication can see that the transient existence of locally clustered network that enables and empowers action while subgroups, holding diverse views, can be an emergent maximizing agility (self-adaptation and reducing behavior within fully coupled systems. This can be the the time to act) through the flow of information, normal state of affairs within societies, even without ideas, and innovations; and externalities and forcing terms. • reliant upon a third-party dissemination network Sociology studies have in the past focused on rather within the public and media space (social media, small groups of subjects under experimental condi- broadcast media, and so forth) so as to maximize tions. Digital platforms and modern applied mathe- the impact of their actions. matics will transform this situation: computation and social science can use vast data sets from very large There are thus at least three independent networks numbers of users of online platforms (Twitter, Face- operating on the side of those who would threaten book, blogs, group discussions, multiplayer online security. games) to analyze how norms, opinions, emotions, Here we have set out a framework for analyzing the and collective action emerge and spread through local form and dynamics of large evolving peer-to-peer com- interactions. munications networks. It seems likely that the challenge of modeling their behavior may lead us to develop 5 Networks on Security and Defense new models and methods in the future. “It takes a network to defeat a network” is the man- tra expressed by the most senior U.S. command in Further Reading Afghanistan and Iraq. This might equally be said of the Bohannon, J. 2009. Counterterrorism’s new tool: “metanet- threats posed by terrorists, or in post-conflict peace- work” analysis. Science July 24:409–11. keeping (theaters of asymmetric warfare), and even by Grindrod, P. 1995. Patterns and Waves, 2nd edn. Oxford: the recent summer riots and looting within U.K. cities. Oxford University Press. But what type of networks must be defeated, and what Grindrod, P., and D. J. Higham. 2013. A matrix iteration for type of network thinking will be required? dynamic network summaries. SIAM Review 55(1):118–28.

804 VII. Application Areas

Grindrod, P., D. J. Higham, and M. C. Parsons. 2012. Bista- 1.1 Hierarchical Chip Design bility through triadic closure. Internet Mathematics 8(4): 402–23. Due to its enormous complexity, the design of VLSI Grindrod, P., D. J. Higham, M. C. Parsons, and E. Estrada. chips is usually done hierarchically. A hierarchical 2011. Communicability across evolving networks. Physi- design makes it possible to distribute the design task cal Review E 83:046120. to different teams. Moreover, it can reduce the over- Grindrod, P., and M. C. Parsons. 2012. Complex dynamics in all effort, and it makes the design process more pre- a model of social norms. Technical Report, University of dictable and more manageable. Reading. For hierarchical design, a chip is subdivided into log- McCrystal, S. A. 2011. It takes a network: the new front line ical units, each of which may be subdivided into sev- of modern warfare. Foreign Policy (March/April). eral levels of smaller units. An obvious advantage of Ward, J. A., and P. Grindrod. 2012. Complex dynamics in a deterministic model of cultural dissemination. Technical hierarchical design is that components that are used Report, University of Reading. multiple times need to be designed only once. In particular, almost all chips are designed based on a library of so-called books, predesigned integrated circuits that VII.6 Chip Design realize simple logical functions such as AND or NOT or Stephan Held, Stefan Hougardy, and a simple memory element. A chip often contains many instances of the same book; these instances are often Jens Vygen called circuits. The books are composed of relatively few transistors 1 Introduction and are predesigned at an early stage. For their design one needs to work at the transistor level and hence fol- An integrated circuit or chip contains a collection of low more complicated rules. Once a book (or any hier- electronic circuits—composed of transistors—that are archical unit) is designed, the properties it has that are connected by wires to fulfill some desired functional- relevant for the design of the next higher level (e.g., ity. The first integrated circuit was built in 1958 by minimum-distance constraints, timing behavior, power Jack Kilby. It contained a single transistor. As predicted consumption) are computed and stored. Most books by Gordon Moore in 1965, the number of transistors are designed so that they have a rectangular shape and per chip doubles roughly every two years. The pro- the same height, making it easier to place them in rows cess of creating chips soon became known as very- or columns. large-scale integration (VLSI). In 2014 the most complex chips contain billions of transistors on a few square 1.2 The Chip-Design Process centimeters. In this article we concentrate on the design of dig- The first step in chip design is the specification of ital logic chips. Analog integrated circuits have many the desired functionality and the technology that will fewer transistors and more complex design rules and be used. In logic design, this functionality is made are therefore still largely designed manually. In a mem- precise using some hardware description language. ory chip, the transistors are packed in a very regular This hardware description is converted into a netlist structure, which makes their design rather easy. In con- that specifies which circuits have to be used and how trast, the design of VLSI digital logic chips is impossible they have to be connected to achieve the required without advanced mathematics. functionality. New technological challenges, exponentially increas- The physical-design step takes this netlist as input ing transistor counts, and shifting objectives like and outputs the physical location of each circuit and decreased power consumption or increased yield con- each wire on the chip. It will also change the netlist stantly create new and challenging mathematical prob- (in a logically equivalent way) in order to meet timing lems. This has made chip design one of the most inter- constraints. esting application areas for mathematics during the Before fabricating the chip (or fixing a hierarchical last forty years, and we expect this to continue to be unit for later use on the next level up), one conducts the case for at least the next two decades, although physical verification to confirm that the physical lay- technology scaling might slow down at some point. out meets all constraints and implements the desired

For general queries, contact [email protected]