Locality & Hierarchy

Locality. A form of closeness. In space, in time, or in an abstract realm of information. It constrains with ubiquity. So much so that we are unable to see it.

Local in Space

We often forget that we act in space. We get caught up in abstract lands – “freedom”, “justice”, “power” – and the constraints of space seem inconsequential. But everything and everyone exists somewhere, and can only physically act in that somewhere.

Local in Time

Every moment is an instance in time. We can only act physically in the world in this instance. We have the weight of the past, and the potential of the future, but we cannot act in these places.

Local in Information

Einstein got this. It forms the basis of relativity. Relativity takes the axiom that light has a fixed speed, and that light is the fastest we can carry information, and builds the crazy world that results.

This video playlist illustrates the speed of light at different scales. When we come to receiving light from Mars or the Sun, we see how large the distances are, and how an “instance” in these places, may be only seen on Earth at a later point in time. We can only act locally in space and time. You cannot have a simultaneous event with actors on both Earth and Mars.

The communication of information is limited by the speed of light. Fibre optic or radio transmissions all use the electromagnetic spectrum; they all travel through space at the speed of light. Indeed, actually transmission speeds for information across the Internet is often much less than the speed of light, due to reflections, switching and other latencies. These are such that it is even extremely difficult to hold a live concert in Tokyo and London simultaneously.

Physicality

Physical actions within the world are much more limited that the observation of a burning ball of hydrogen in space. Objects occlude and obstruct at short ranges. The curvature of the Earth comes into play at longer ranges. While light is fast, physical movement is about 300 million times slower!

This is compounded if we consider physical interactions. Generally “events” don’t exist in isolation. If you move at speeds of metres per second, then any interaction with a similar entity will also be at these speeds. Physical actions, like light, are also limited by the physical environment. Your actions can typically only transport physical objects across distances of metres. You are also limited by the energy available for movement. This is finite.

Human Relationships

The Internet has up-ended our concepts of locality. You can communicate in what appears to be real-time (at least within seconds) with anyone anywhere in the world. We feel we can change human minds everywhere.

But real life is offline and exists in the physical realm. It is where we live. It is where we work. It is where we go to school or do our shopping. We often cannot choose this. This is where our actions count, regardless of where or how they were triggered.

I’ve always had a soft spot for Dunbar’s number. This suggests, given the constraints of the brain and extrapolating from primate groups, that human beings have somewhere between 100 to 200 social connections. These reflect early tribal group sizes (in both humans and primates). In modern life, this would include family, friends, work colleagues and neighbours. Although there are criticisms, it seems to capture at least a portion of the truth.

If you look at those 500+ “connections” on LinkedIn or 100s of Facebook “friends” how many do you actually interact with? How many know you as a person? How many do you interact with on a weekly basis? It seems much less – research suggests a core group of 10-20 (e.g. family and close friends), with increasingly distant sets of 10-20 people (e.g. work colleagues, neighbours, clients, and distant friends). Dunbar’s number feels right given these scales.

Now Dunbar’s number is not an abstract academic metric, it’s an indirect measurement of the limits of space and time. Even baboons can’t interact with everyone; they are limited by time and geography.

Business

On and off in business you see the craze for “flat management” structures. These ignore locality.

In you consider Dunbar’s number as reflecting the limits of the number of human beings we can care about, and work constructively with, then you can quickly see that a company with 100,000 employees cannot have a “flat management” structure. Indeed, teams appear to have a natural limit of around 10 before the spacetime complexities overwhelm efforts.

To coordinate efforts with a larger number of people, you need to compose these smaller building blocks. You can have a team of 100 people by arranging 10 teams of 10, where one person (a “manager”) has relationships with 1 person in each team. You can have a team of 1000, 10000 or 100000 people by recursively repeating this pattern. This is why “management” structures resemble a tree. “Management” becomes less about keeping workers in line than the simple business of interacting with people given the limits of locality. “Flat management” can work, but it can only work in start-ups or small businesses with a small number of people working together (10-50 realistically).

There is strong public support, both on the ground and in the boardroom, for removing layers of “management”. These layers are seen as costly and unnecessary. “All this person does is talk with those 10 people below and that one person above“. “Why can’t we have one person that talks to 100 people, and so save the cost of 9 salaries?“.

The answer is that the one person that talks to 100 people is going to struggle to form meaningful connections with those 100 people. They will not be able to communicate a shared sense of purpose or keep track of everything and everyone. The effort will be off-loaded onto impersonal metrics. Relationships become mechanised. This appears to work, at least at first. But it is fragile and leaves room for exploitation. It is also likely not good for the mental health of the 100 people or the 1 person tasked with working with them.

Ironically, if you look at all levels of business Dunbar’s number appears to hold well in the background. The eye-watering increases in executive pay and the revolving door within boardrooms is heavily influenced by relatively small social circles. Job hires are more often filled via “a mate of a mate” than explicit advertising. Nepotism is not necessarily wanton corruption, more the natural mode of human cooperation.

Knowing the constraints of a system does not necessarily mean you need to accept the system that you have. But it does make it easier to plan for real change. If you know nepotism cannot be avoided, you allow for its benign release (e.g. under the same standards as others) but construct the system so that it cannot always win without merit. If you want diversity within a community (from the frontline to the boardroom), you need to make sure that those from diverse backgrounds interact within the social circles of that community. You also need to recognise that luck and privilege pay a large part in fixing Dunbar’s social circles of 100 or so people, and even the local geography of the world you operate in. Creating opportunity for mixing on the ground, whether at work, in sports grounds, in the shops or in the pub, seems to help break open these circles. Gated communities, local segregation by income, and physical exclusivity act in opposition to this and calcify groupings.

Also if you know that your business circles are heavily influenced by your social circles, which are themselves a large part due to luck and the random movement of particles, you can invest more time and effort into ensuring that life is supported, meaningful and fulfilling within any group.

Politics

Politics in the UK, Europe and the US has led many to look afresh at the millennia-old discussions of direct vs indirect democracy. Much is made in the press of “the people” and their “will”. Direct democracy is seen as positive, direct rule by the people.

However, indirect democracy such as representative democracies appears to be an evolved solution to the problems of governing in a direct democracy. These are problems of locality. It is impossible, just concerning naive constraints of space and time for millions of people to meaningfully interact with a few in power. This is why we have smaller constituencies and local districts, themselves often defined via geography, and representatives for those regions. This makes it easier to communicate with representatives and for representatives to raise local issues that would be drowned out by any mass democracy.

One factor behind the modern ills of politics is the break down of local geography as a proxy for group ownership. This used to work when there was local diversity, e.g. in terms of class, profession, industry, education etc. However, modern countries are increasing becoming segregated by education and location (both being correlated with urban living), and group participation is increasing coordinated online, where there are no national boundaries.

The answer appears to be not to throw out the model of representative democracy in favour of online petitions or mass referendums. Instead, the hierarchy of representation needs to be strengthened, with more devolution and connection of power with local needs. There also needs to be an opportunity for issues to be raised and debated outside of local concerns, perhaps increasing the role of cross-party committees.

The Brain

How does locality affect our brains?

For me there are two major areas where it comes into play:

  1. We perceive a world of local structure.
  2. Our brains are physical objects within the world.

1) A World of Local Structure

When we look around a room, or at a countryside scene, we don’t see a bunch of white noise. We see objects that have a limited extent in the world, with interconnected portions. Things in the world are constrained by the four-dimensional world we live in. Branches have edges and extend. Mountains divide the land and the sky yet are created by physical processes that obey physical laws, including gravity. Put another way, objects don’t pop in and out of three-dimensional space at random.

This local structure in the world must be reflected in the models of the world our brain creates. We see this in lower sensory areas (M1, S1, A1 and V1) where there is a well-defined topographical mapping between an extent or scale in the physical world and input into the cortex. Neighbouring points on our retina capture neighbouring points in space and are mapped to neighbouring areas of V1. Neighbouring frequencies detected in our ears are mapped to neighbouring areas of A1. We receive input from the body in S1 in a manner that is topographically mapped to different areas of the body; sensation from your feet is received in an area that neighbours sensation from your legs (although these areas may not always be of the same size, you receive more input from the hands and lips than from the back).

2) Brain = Physical

The brain is a physical object. It is limited by the physical laws of time and space. Connections within the brain are constrained based on principles very similar to the constraints on our social lives.

Although there are many long range connections in the brain, the majority of the information flow within the cortex is local. The long range connections are often driven by genetic constraint, e.g. point-to-point mappings of cortical areas to sub-cortical structures.

Within the cortex, there appears to be a first stage of local processing within its laminar structure. Many think that computation is arranged in units referred to as cortical columns. These have between 6 to 12 vertical layers, and repeat horizontally across the thin two-dimensional sheet of the cortex. Within a cortical column, there appear to be well-defined couplings between cells. Across cortical columns, couplings are also local; neighbouring columns are often interconnected with the likelihood of connection falling off with distance.

Both neurons and people predominantly communicate with those nearby. Neurons interconnect via physio-chemical couplings: synapses. A synapse requires a portion of one neuron to be very close to a portion on another neuron. A neuron has about 1000 synapses on average (figures on the Internet tend to lie within this order of magnitude, with 7000 also being given). But the brain has eighty six billion neurons. Hence, any one neuron is only connected to around 0.000001% of the brain’s population. This is similar to our social selves: indeed 100 / 7 billion is also weirdly enough the same percentage – 0.000001%. There appear to be networking “laws” at play here that are the expression of the constraints on connection within spacetime.

Hierarchy, Intelligence & Culture

Given 1) and 2) above, how can we have intelligence?

This maybe a similar question to how can we have culture, when we only connect with a few people in our lives?

In the brain, intelligence appears to arise from a relatively large number of small-scale local transformation of information across the brain.

In culture, we have myths of genius, spontaneous invention (“Eureka”), and divine inspiration. But the reality is more prosaic. We only see the result not the effort. And visibility of the result is often driven by luck more than anything else (50% or more of the contribution). But ask any researcher on one of the cultural “greats”, whether in art, philosophy or music, and you will see someone that works for decades, that is supported by a small yet strong social network, and that is inspired by a similarly small community across the ages. We don’t see the joins at scale.

Lessons for Machine Learning

What lessons does locality provide for the burgeoning field of machine learning?

We can already see the success of convolutional neural networks, which apply a small spatial filter (typically 3 by 3 pixels) yet can generate predictions that accurately classify complex objects.

Markov also was in the right ball-park, when he suggests that complex processes may be modelled with a series of local state changes. The independence of history within a Markov chain may be considered to model local interactions are constrained by locality – distances in spacetime effect de-facto independence. Bayesian networks, causal inference and graph theory also seem to pick up on different aspects of locality.

On the other hand, it suggests that a move to larger and larger matrices of weights is not an efficient (or even possible) solution to the problem of dimensionality. We can’t just chuck everything into a vector and apply some weights and biases.

It also suggests that looking at surface correlations, without considering the structure that produces them, is not going to provide robust solutions. Instead, it is going to act more like a direct democracy, likely to become infatuated with statistical demagogues.

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