## Birds on a wire and the Ising model

30 November 2009

Statistical physics is very good at describing lots of physical systems, but one of the basic tenets underlying our technology is that statistical physics is also a good framework for describing computer network traffic. Lots of recent work by lots of people has focused on applying statistical physics to nontraditional areas: behavioral economics, link analysis (what the physicists abusively call network theory), automobile traffic, etc.

In this post I’m going to talk about a way in which one of the simplest models from statistical physics might inform group dynamics in birds (and probably even people in similar situations). As far as I know, the experiment hasn’t been done–the closest work to it seems to be on flocking (though I’ll give $.50 and a Sprite to the first person to point out a direct reference to this sort of thing). I’ve been kicking it around for years and I think that at varying scopes and levels of complexity, it might constitute anything from a really good high school science fair project to a PhD dissertation. In fact I may decide to run with this idea myself some day, and I hope that anyone else out there who wants to do the same will let me know. The basic idea is simple. But first let me show you a couple of pictures. Notice how the tree in the picture above looks? There doesn’t seem to be any wind. But I bet that either the birds flocked to the wire together or there was at least a breeze when the picture below was taken: Because the birds are on wires, they can face in essentially one of two directions. In the first picture it looks very close to a 60%-40% split, with most of the roughly 60 birds facing left. In the second picture, 14 birds are facing right and only one is facing left. Now let me show you an equation: $H = -J\sum_{\langle i j \rangle} s_i s_j - K\sum_i s_i.$ If you are a physicist you already know that this is the Hamiltonian for the spin-1/2 Ising model with an applied field, but I will explain this briefly. The Hamiltonian $H$ is really just a fancy word for energy. It is the energy of a model (notionally magnetic) system in which spins $s_i$ that occupy sites that are (typically) on a lattice (e.g., a one-dimensional lattice of equally spaced points) take the values $\pm 1$ and can be taken as caricatures of dipoles. The notation $\langle i j \rangle$ indicates that the first sum is taken over nearest neighbors in the lattice: the spins interact, but only with their neighbors, and the strength of this interaction is reflected in the exchange energy $J.$ The strength of the spins’ interaction with an applied (again notionally magnetic) field is governed by the field strength $K.$ This is the archetype of spin models in statistical physics, and it won’t serve much for me to reproduce a discussion that can be found many other places (you may like to refer to Goldenfeld’s Lectures on Phase Transitions and the Renormalization Group, which also covers the the renormalization group method that inspires the data reduction techniques used in our software). Suffice it to say that these sorts of models comprise a vast field of study and already have an enormous number of applications in lots of different areas. Now let me talk about what the pictures and the model have in common. The (local or global) average spin is called the magnetization. Ignoring an arbitrary sign, in the first picture the magnetization is roughly 0.2, and in the second it’s about 0.87. The 1D spin-1/2 Ising model is famous for exhibiting a simple phase transition in magnetization: indeed, the expected value of the magnetization for in the thermodynamic limit is shown in every introductory statistical physics course worth the name to be $\langle s \rangle = \frac{\sinh \beta K}{\sqrt{\sinh^2 \beta K + e^{-4\beta J}}}$ where $\beta \equiv 1/T$ is the inverse temperature (in natural units). As ever, a picture is worth a thousand words: For $K = 0$ and $T > 0,$ it’s easy to see that $\langle s \rangle = 0.$ But if $K \ne 0, J > 0$ and $T \downarrow 0$, then taking the subsequent limit $K \rightarrow 0^\pm$ yields a magnetization of $\pm 1.$ At zero temperature the model becomes completely magnetized–i.e., totally ordered. (Finite-temperature phase transitions in magnetization in the real world are of paramount importance for superconductivity.) And at long last, here’s the point. I am willing to bet ($.50 and a Sprite, as usual) that the arrangement of birds on wires can be well described by a simple spin model, and probably the spin-1/2 Ising model provided that the spacing between birds isn’t too wide. I expect that the same model with varying parameters works for many–or even most or all–species in some regime, which is a bet on a particularly strong kind of universality. Neglecting spacing between birds, I expect the effective exchange strength to depend on the species of bird, and the effective applied field to depend on the wind speed and angle, and possibly the sun’s relative location (and probably a transient to model the effects of arriving on the wire in a flock). I don’t have any firm suspicions on what might govern an effective temperature here, but I wouldn’t be surprised to see something that could be well described by Kawasaki or Glauber dynamics for spin flips: that is, I reckon that–as usual–it’s necessary to take timescales into account in order to unambiguously assign a formal or effective temperature (if the birds effectively stay still, then dynamics aren’t relevant and the temperature should be regarded as being already accounted for in the exchange and field parameters). I used to think about doing this kind of experiment using tagged photographs or their ilk near windsocks or something similar, but I can’t see how to get any decent results that way without more effort than a direct experiment. I think it probably ought to be done (at least initially) in a controlled environment.

Anyways, there it is. The experiment always wins, but I have a hunch how it would turn out.

UPDATE 30 Jan 2010: Somebody had another interesting idea involving birds on wires.

## Happy Thanksgiving

26 November 2009

I'm thankful for seeing truth presented with beauty.

This is a picture to help understand an Anosov flow obtained from the cat map. It’s part of research on a technique we’ve used to analyze network traffic.

## Random bits

24 November 2009

Information Dissemenation highlights a McAfee report and points out that Chinese SSBNs are really noisy.

Henry Markram publicly rips Dharmendra Modha; see also here. Sounds like it might be time for them to step into the Octagon. I met Modha at a recent DARPA industry day and had heard about Markram’s work before. Though it’s been over 15 years since I took a mathematical neurophysiology course, Markram’s Blue Brain effort seems very interesting to me.

First collisions at the LHC

## Capability of the PRC to conduct cyber warfare and computer network exploitation

23 November 2009

I just finished reading a recent report [pdf] with this title produced for the US-China Economic and Security Review Commission. Though there’s a lot of filler material, it’s pretty good. I’ll spare you the trouble of reading all 88 pages and start with what I thought were the most salient themes covered in the executive summary:

• Some evidence exists suggesting limited collaboration between individual elite hackers and the Chinese government; however
• The constant barrage of network penetrations from China (comprising most of what Mandiant calls “the advanced persistent threat“) “is difficult at best without some type of state-sponsorship”.
• The modus operandi of the penetrations “suggests the existence of a collection management infrastructure”; and
• PLA CNE aims during a military conflict would be “to delay US deployments and impact combat effectiveness of troops already in theater”.

The PLA’s “Integrated Network Electronic Warfare” doctrine is based on attacking a few carefully selected network nodes controlling C2 and logistics. The INEW doctrine was apparently validated in a 2004 OPFOR exercise when the red force (NB. the Chinese use red to denote themselves) C2 network got pwned within minutes, and it is likely that PRC leadership would authorize preemptive cyberattacks if they think it wouldn’t cross any “red lines”. This preemptive strategy is apparently favored by some in the PLA who view cyber as a “strategic deterrent comparable to nuclear weapons but posessing greater precision, leaving far fewer casualties, and possessing longer range than any weapon in the PLA arsenal“. [emphasis original]

One aspect of this thinking that I think is underappreciated is that the PRC is already deterring the US by its apparent low-level attacks. These attacks demonstrate a capability of someone in no uncertain terms and in fact may be a cornerstone of the PLA’s overall deterrence strategy. In short, if the PLA convinces US leadership that it can (at least) throw a monkey wrench in US deployments, suddenly the PRC has more leverage over Taiwan, where the PLA would need to mount a quick amphibious operation. And because it’s possible to view the Chinese Communist Party’s claim to legitimacy as deriving first of all from its vow to reunite China (i.e., retake the “renegade province” of Taiwan) one day, there is a clear path from the PLA cyber strategy to the foundations of Chinese politics.

The paper goes on to note that “much of China’s contemporary military history reflects a willingness to use force in situations where the PRC was clearly the weaker entity” and suggests that such uses of force were based on forestalling the consequences of an even greater disadvantage in the future. This putative mindset also bears on cyber, particularly through the Taiwan lens. The PLA has concluded that cyber attacks focusing on C2 and logistics would buy it time, and presumably enough time (in its calculations) to achieve its strategic aims during a conflict. This strategy requires laying a foundation, and thus the PRC is presumably penetrating networks: not just for government and industrial espionage, but also to make its central war plan credible.

In practice a lot of the exploitation would consist of throttling encrypted communications and corrupting unencrypted comms, and it is likely that the PLA is deliberately probing the boundaries of what can and cannot be detected by the US. But this generally shouldn’t be conflated with hacktivism or any civilian attacks originating from China, as there’s little reason to believe that the PLA needs or wants anything to do with this sort of thing. While it’s possible that there is some benefit to creating a noisy threat environment, executing precise cyberattacks in the INEW doctrine requires exploitation that can be undermined by hacktivism or civilian (especially amateur) attacks.

The end of the meaty part of the report talks about what’s being done and what should be done. It talks about the ineffectiveness of signature-based IDS/IPS and the promise of network behavior analysis, but also its higher overhead and false alarm rates. This is precisely the sort of thing our software is aimed at mitigating, by combining dynamical network traffic profiles and interactively configurable automated alerts with a framework for low-overhead monitoring and fast drill-down.

## Random bits

18 November 2009

Life imitates 60 Minutes

Sen. Whitehouse: “If, in fact, the NSA has technical capabilities beyond those of [internet] providers, why should you be relying on the providers in areas where the NSA actually has greater capability?” I think there are at least two ways this question could be interpreted.

The cyberwar plan

Doug Natelson blogs on the rise of graphene (covered here previously)

## DIMACS workshop on designing networks for manageability

14 November 2009

The highlight of the DIMACS workshop on designing networks for manageability for me was Nick Duffield‘s talk on characterizing IP flows at network scale. The basic idea is to use machine learning to identify the flow predicates that best reproduce packet-level classifications. By sampling flows according to a simple dynamical weighting, Duffield et al. demonstrate that this sort of flow classification is accurate (to a few percent, with the misclassifications largely due to overloading of HTTP, e.g., with media over web), scalable (i.e., faster than real-time), versatile (i.e., independent of the particular ML classifier), and stable (over space and time, with a deployment on a separate but similar network producing essentially equivalent results over several months). This work is more recent than related research we’ve cited in our whitepaper “Scalable visual traffic analysis” (on our downloads page) detailing the rationale behind our own traffic aggregation methods.

Much of the workshop (especially its first day) was more focused on current deployment and engineering issues than I would have thought for an overarching focus on “algorithmic foundations of the internet”. Both another mathematician that came with me and I expected to see some work on (or at least suggesting the use of) sparse linear algebra to deal with traffic matrices. I was surprised not to see anyone talk about some kind of agent-based configuration methods for networks–this sort of approach has been used to great effect on hosts.

But there were a number of other talks I found interesting: Aditya Akella from Wisconsin talked about an entropy characterization of “reachability sets” describing packets that can be sent between pairs of routers based on their configurations, and used this to construct a routing complexity measure for networks. Dan Rubenstein from Columbia talked about a “canonical graph” method for efficiently detecting misconfigurations for routing protocols. Iraj Saniee talked about why networks are globally hyperbolic (using a result of Gromov’s well-known work on groups), a conclusion that seems intuitively obvious to me if the existence of a global curvature (bound) is assumed. (Basically a network spreads out if it’s drawn in any reasonable way, and hyperbolic geometry amounts to expansion.)

Mung Chiang from Princeton talked about the results in “Link-State Routing with Hop-by-Hop Forwarding Can Achieve Optimal Traffic Engineering” first presented at INFOCOM 2008. He and coworkers perturb assumptions behind routing protocols to obviate the need for hard optimization problems (i.e., computation of optimal link weights to input to OSPF is NP-hard, but changing OSPF can make the corresponding optimization problem easier). From what I could tell OSPF corresponds to a “zero-temperature” protocol, whereas the improved protocol corresponds to a “finite-temperature” one.

Michael Schapira from Yale and Berkeley talked about game-theoretic and economic perspectives on routing. It is a happy “accident” that the internet is BGP stable (usually, although a notable event where a Pakistani ISP set all its hop counts to 1 some time ago created a routing “black hole”). Although ISPs are selfish, economic considerations tend to result in stability. But that’s not a guarantee. So Schapira and coworkers analyzed the situation and found that “interdomain routing with BGP is a game” in which the ASes are the players, the BGP stable states are pure Nash equilibria, and BGP is the “best response“. I mentioned to him that the “accidental” nature of this stablity is likely due to reciprocity, in that an ISP that discovers one of its neighbors engaging in predatory routing is likely to retaliate in the future. I think the use of economic and game theory is generally a good idea. An emphasis of the economics of cybercrime has developed recently, and understanding the market forces at play here and elsewhere is likely to lead to improvements in the reliability and security of networks.

## @ Rutgers

10 November 2009

I will be at the DIMACS workshop on designing networks for manageability at Rutgers for the rest of the week. Looking forward to some good talks…

## Solution of second-order matrix difference equations

9 November 2009

While browsing through my notes recently I came across this cute and not-quite trivial, but entirely elementary result. Since it doesn’t seem to be available anywhere else, I present it as blog fodder. (As usual, I will offer \$.50 and a Sprite to the first person who provides a usable reference in the event that this result is already known.)

Consider the second-order matrix difference equation

$x_{k+1} = Ax_k + \tilde Ax_{k-1}$

with $x_0, x_1$ given. Some algebra shows that (e.g.)

$x_2 = (A)x_1 + \tilde A x_0,$

$x_3 = (A^2 + \tilde A)x_1 + (A) \tilde A x_0,$

$x_4 = (A^3 + bsp_{1,1}[A, \tilde A])x_1 + (A^2 + \tilde A) \tilde A x_0,$ etc.

where the bosonic symmetrized product $bsp_{j,k}[A, \tilde A]$ for two “species” of order $(j,k)$ is defined as the sum of all distinguishable products with $j$ occurences of the first species and $k$ of the second (if $j$ or $k = 0$ then the BSP is a pure power of the nontrivial species).

For example, $bsp_{3,2}[A,\tilde A]$ equals

$A A A\tilde A \tilde A + A A\tilde A A \tilde A + A A \tilde A \tilde A A + A\tilde A A A \tilde A + A\tilde A A \tilde A A$

$+ A \tilde A \tilde A A A + \tilde A A A A \tilde A + \tilde A A A \tilde A A + \tilde A A \tilde A A A + \tilde A \tilde A A A A.$

It is not hard to verify that

$bsp_{j,k}[A, \tilde A] = A \cdot bsp_{j-1,k} + \tilde A \cdot bsp_{j,k-1}[A, \tilde A]$

and more generally for $\ell = 0,\dots \min(j,k)$ that

$bsp_{j,k}[A, \tilde A] = \sum_{m=0}^\ell bsp_{\ell-m,m}[A, \tilde A] \cdot bsp_{j-(\ell-m),k-m}[A, \tilde A].$

The same algebra that facilitated writing down the first few terms for the difference equation quickly leads to the general solution, viz.

$x_k = \left( A^{k-1} + \sum_{j=1}^{(k-1)/2 - 1} bsp_{k-2j-1,j}[A,\tilde A] + \tilde A^{(k-1)/2} \right) x_1$

$+ \left( A^{k-2} + \sum_{j=1}^{(k-1)/2-1} bsp_{k-2j-2,j}[A, \tilde A] \right) \tilde A x_0$

for $k$ odd, and

$x_k = \left( A^{k-1} + \sum_{j=1}^{k/2 - 1} bsp_{k-2j-1,j}[A,\tilde A] \right) x_1$

$+ \left( A^{k-2} + \sum_{j=1}^{k/2-2} bsp_{k-2j-2,j}[A, \tilde A] + \tilde A^{k/2-1} \right) \tilde A x_0$

for $k$ even.

I used this to understand what happens when one of the internal states of a classical Bose gas becomes “more fermionic” in the sense that fewer particles can occupy that state, but the microscopic transition probabilities are otherwise unchanged. (The underlying motivation was to apply this to get a handle on the effects of some complex configurations for an older mathematically-oriented network monitoring framework in a tractable traffic regime.) It turns out that this is completely uninteresting unless the remaining “bosonic” states are close to equally likely to occur: in this event you get a slight but also quite counter-intuitive effect on the state distributions. Pseudocolor figures of the energy functions corresponding to these distributions will illustrate:

3 totally bosonic interal states. The distribution is multigeometric, and the energy is affine.

Capping the occupancy of one internal state

A more restrictive cap

A "fermionic" internal state

Note that the color scales differ for each figure. I think the pictures indicate how surprising the behavior of this (completely classical) “poor man’s exclusion principle” is: the effect is as unexpected as it is slight.

Hopefully posting the main result will save some other folks a head-scratch or two. I’d be interested to know if it’s applied elsewhere.