@normonics' Intro to Applied Complexity #ACS101 #SpringA2021 Highlights
Session 6: Networks and Connectivity
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Session 6: Networks and Connectivity
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https://twitter.com/evolvingcalm/status/1366543917160665091
Networks are the organisation structure that shape the interactions of a system.
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Intuitively complex systems have a lot of internal connectivity (complexity depends on connectivity).
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If you form a network in different ways you end up with different kinds of connectivity.
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Different connectivity - different implications.
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How many hops does something have to take to spread across the network?
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Topology = pattern of connectivity.
Intuitively in complex dynamics topology changes in concert with the activity of networks - coevolving connections with dynamics.
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Intuitively in complex dynamics topology changes in concert with the activity of networks - coevolving connections with dynamics.
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Consider the brain.
The firing patterns in your neurons are having effects on what kinds of new connections are made, what kinds stay, and what kinds are pruned.
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The firing patterns in your neurons are having effects on what kinds of new connections are made, what kinds stay, and what kinds are pruned.
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Why do networks come up so much?
Because they look at connectivity as such.
But too often we hear things like - "it's all networks, everything in the world is a network."
Of course, it's not that things are 'really' networks.
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Because they look at connectivity as such.
But too often we hear things like - "it's all networks, everything in the world is a network."
Of course, it's not that things are 'really' networks.
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Networks are a useful way to model things, that are whatever they are.
The nervous system modelling well as a network doesn't mean it's 'really' a network.
It means it's a good model but you might model it differently for different reasons like looking at different features.
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The nervous system modelling well as a network doesn't mean it's 'really' a network.
It means it's a good model but you might model it differently for different reasons like looking at different features.
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Not the end all be all modelling paradigm but it's useful.
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We accept it as a given but the topology of cellular automata is really a constraint on a type of graph.
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The ways graphs are studied in large part, especially from a structural standpoint, is by looking at various properties of the graph.
Some properties are global, some are local.
Different properties and metrics will give you different insight.
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Some properties are global, some are local.
Different properties and metrics will give you different insight.
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You're often given a graph generated by a process or data, or arbitrarily, etc. & study it by looking at different properties of that structure - there are many, many, ways you could do that.
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Undirected graph (linked symmetric relationships)
"It's hard to be friends with some not friends with you"
"It's really hard to be married with someone who's not married to you"
(You can't really)
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"It's hard to be friends with some not friends with you"
"It's really hard to be married with someone who's not married to you"
(You can't really)
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A model decision
If you are going to try apply a model to some real system - models have assumptions
- take care which assumptions you are using and why.
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If you are going to try apply a model to some real system - models have assumptions
- take care which assumptions you are using and why.
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The devil is in the details.
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Two of the ways to look at a graph are kinds of node centrality & each will tell you something different.
Degree centrality: How many connections? (good question)
Betweenness centrality: how many shortest paths cross through the node?
(some arbitrary graphs)
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Degree centrality: How many connections? (good question)
Betweenness centrality: how many shortest paths cross through the node?
(some arbitrary graphs)
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It's important to see just because we say there is a local measure of a graph it's often still being derived from global structure of the graph.
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While local metrics, these node measures are applied to a given node.
Actually, the global structure of the network is needed & has a strong impact on what that value ends up being.
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Actually, the global structure of the network is needed & has a strong impact on what that value ends up being.
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It's not trivial that things that appear to be local properties aren't really.
You can't be account for them only by local considerations.
Somehow you have to scope out - look at the whole thing & then you can say something about the local property or measure etc.
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You can't be account for them only by local considerations.
Somehow you have to scope out - look at the whole thing & then you can say something about the local property or measure etc.
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Really not a trivial thing actually - not possible to account for reductionistically - bends our mind around what's really a 'local property' vs 'non-local property'?
A node might be said to have betweenness but there's no between the node without the rest of the network.
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A node might be said to have betweenness but there's no between the node without the rest of the network.
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Small world:
As you start to add long-range connections to a network with only a moderate number (it only takes a few) the average path length collapses,
resulting in high local clustering, short path lengths.
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As you start to add long-range connections to a network with only a moderate number (it only takes a few) the average path length collapses,
resulting in high local clustering, short path lengths.
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There's a clustering coefficient so as you start to randomly rewire nodes this drops much slower than the average path length.
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In the real world air travel/long range flights are a perfect physical example of the small world.
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Yes, sitting in your local environment everything seems pretty local; you're only interacting with those close to you etc.
People on the other side of the world seem far away, but the number of hops you need to make between yourself & them are very small.
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People on the other side of the world seem far away, but the number of hops you need to make between yourself & them are very small.
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With only a moderate amount of rewiring introducing long-range connections the amount of hops to travel the network collapses to very few.
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This is very observable (another example being your social network) and has huge impact on things like disease spreading, communication networks, mimetics spreading, any kind of contagious/multiplicative process.
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When you understand how ubiquitous this is - it sensitises you to how fast things can evolve when they do spread on the space of networks relative to what naïve intuition might give.
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So what is space?
A lattice is this Euclidean idea of space - you have to travel through the intervening locations to get to other ones.
Adding long-range connections really changes the topology.
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A lattice is this Euclidean idea of space - you have to travel through the intervening locations to get to other ones.
Adding long-range connections really changes the topology.
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Obviously in a airplane you still have travel through space, so in some sense we still live in a sense objectively in a Euclidean space, but if you think about the time scales, what we are now operating over is if the topology has all these long-range connections built in.
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We're adding long-range connections to an insane degree to the point where we maybe are actually crossing into the territory where we are losing local clustering.
Especially now with everyone staying at home and only interacting on the internet etc.
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Especially now with everyone staying at home and only interacting on the internet etc.
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We need to be sober in considering how close just everybody else really is on the network now.
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Some neurons have axons that are myelinated specifically for long-range connections and others don't bother.
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Pandemics used to last a very long time, decades sometimes, as they swept across continents.
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It didn't use to be 'oh s**t last week they were locking down Wuhan and now it's in the nursing home down the street'.
Everyone is really sensitised to the idea of small world networks now.
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Everyone is really sensitised to the idea of small world networks now.
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Scale free: no characteristic scale.
Degree distribution: some node will have most of the connectivity and many will have little.
This often happens in networks through a process of preferential attachment.
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Degree distribution: some node will have most of the connectivity and many will have little.
This often happens in networks through a process of preferential attachment.
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The nodes that have more connections already are biased to get more new connections.
A kind of founder effect/power law.
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A kind of founder effect/power law.
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You see preferential attachment all the time when the probabilities of things happening are not independent.
A preference to go to things already highly connected.
The rich get richer.
The Matthew principle.
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A preference to go to things already highly connected.
The rich get richer.
The Matthew principle.
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A symmetry breaking process, however it starts, starts to aggregate over time.
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There is an important connection between this idea of dependence of random events and generations of power laws/fat tails.
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Why do we have this distribution between small villages and megacities?
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All over the place you see power laws, book sales, wealth, size of astral bodies, etc.
If you look under the hood, it's often generated by this mechanism of preferential attachment as a dynamical process over time.
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If you look under the hood, it's often generated by this mechanism of preferential attachment as a dynamical process over time.
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When you have fat tails you generally have some form of dependency.
Independence leads to thin tails
Dependence leads to patters, and these might be fat tails.
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Independence leads to thin tails
Dependence leads to patters, and these might be fat tails.
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If there is a dependency of avoidance this would cause a very thin tailed pattern.
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Adaptive networks: the dynamics on and of the network are coevolving.
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Start with a random graph.
Start updating difference equation - dynamics start evolving.
There's some transient period.
Eventually an attractor is hit that is either a fixed point or some kind of cycle (or otherwise chaotic)
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Start updating difference equation - dynamics start evolving.
There's some transient period.
Eventually an attractor is hit that is either a fixed point or some kind of cycle (or otherwise chaotic)
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On a discrete system you will always hit a cycle but it might take a very long time.
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Rewiring rule leads the network evolving itself to the critical value.
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Self-organising criticality:
When there's a critical value when the system organises itself, to get to that critical value.
A general phenomenon.
You don't tune a system to a critical point - the dynamics of the system evolves itself to that critical point.
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When there's a critical value when the system organises itself, to get to that critical value.
A general phenomenon.
You don't tune a system to a critical point - the dynamics of the system evolves itself to that critical point.
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Taxonomy of models
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