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Solutions Manual for Complexity Science The Study of Emergence by Henrik Jeldtoft Jensen

Complexity Science

Chapter 1 Exercise 1 For example, atoms forming molecules, molecules forming proteins etc. Exercise 2 For example, both stress how essential complex phenomenon are and that they can be classified. Anderson even argues that understanding emergence is a conceptually fundamental activity at least as important as is particle physics. Exercise 3 For example, an ontology consisting of hierarchies of processes seems to contain emergence as a defining aspect of the world. It appears that the distinction between weak and strong emergence evaporates and is replaced by an investigation of relationships between processes at different levels of the hierarchy. Project 1 a) Epistemic: Phenomena that emerge because of the way we represent the world or obtain knowledge about the world. Ontological: genuinely new entities at higher levels such as objects, properties or causes. b) Perhaps it would be interesting to discuss this in relation to optical phenomena such as the colour of an object and the lightning or gestalt images such as the duck-rabbit drawing. c) A short answer could be that we know the world epistemologically. d) Simon seems to think in terms of epistemic complexity when he says "... not in an ultimate, metaphysical sense, but in the important pragmatic sense...". His discussions of evolution and the advantage of hierarchies is ontological. The summary blends epistemic and ontological emergence: Simon mentions that the level of complexity depends on the description but the processes he mentions appear to have an ontological status. Anderson’s description of emergence through symmetry breaking is about ontology. e) Seems that if at the deepest ontological level we have processes rather than things and if we understand the world as processes the distinction between ontology and epistemology may fade away. Project 2 a) Biology for example in cells, organs, organisms,..., flocks etc. Also, society for example with individuals, families, organisations etc. b) The answer may include a discussion about hierarchies of selective pressure, the relationship between reproductive unit, the relation between "genotype" and "phenotype", and the difference between adaptive change and evolutionary change. c) Both ecosystems and society is organised hierarchically and even if ideal definitions of the building blocks of these hierarchies are at present lacking it appears to be impossible to describe the functionality and structure of, say, an ecosystem or, say, a political strucutre without some notion of building blocks consisting of collections (a species for example)of individuals sharing some relevant relations.

Complexity Science

d) First to answer this, think about how to define biological species and social classes. One would then probably conclude that we (at present) only can use these concepts at an epistemological level. Project 3 a) Carefully go through Section 1.1 b) For example, when using a bicycle pump it feels like the air inside the pump is an elastic medium. That the pressure of the gas gives rise to elasticity. c) No, Anderson will not consider pressure as an emergent property belonging to the "more is different" category. d) If we go with Wheeler, any accumulated property seems to acquire "emergent property" status. e) No, I cannot think of any.

Solutions Manual for Complexity Science The Study of Emergence by Henrik Jeldtoft Jensen

Complexity Science

Chapter 2 Exercise 1 a) Wolf packs, ant colonies, beehives, human tribes, companies, resting state neural network. Perhaps neurosis, depression. b) Punctuated equilibrium in macro-evolution. Social history is interspersed with political, cultural, and technological revolutions. Neuronal bursts (or avalanches) In psychology, attention and distraction are examples as well. Exercise 2 Biology: Fireflies, crickets etc. Physics: Coupled pendula, electronic circuits, etc. Economics: Herding behaviour, exchange rate variation, etc. Neuroscience: Epileptic seizures, circadian clock, brain waves as measured by EEG, etc. Exercise 3 Social class as given by income level is an emergent structure in society. Self-perception emerges as a consequence of our personal experience and history. Manstead demonstrates how the two are linked. Exercise 4 Simon discusses typically shared properties amongst complex systems. He focuses on hierarchical structure and presents arguments for why hierarchies naturally arise in heterogeneous many component systems. The key term is "near decomposability". Simon concentrates on epistemology, in the conclusion he writes: " How complex or simple a structure is dependent critically upon the way in which we describe it." Anderson focuses on the ontological differences between the whole and the components. He refers to symmetry breaking to describe the new reality that can emerge when many (in principle infinitely many) interacting (and possibly identical) components form coherent structures. Exercise 5 Below are some snapshots of possible considerations: a) The individual scale: 1 The family scale: 2, ..., 10 The town or city: 103 - 106 Nations: 106 - 109 b) The biological cell: 10−6m - 10−4m Plants: 10−3m - 10m (perhaps 102) Clusters of trees: 102 m Forests: 103m - 104m c) Cell body of neuron: 10−6 - 10−4 The axon 𝑚𝑁 Brain regions linear size: 10−2m Entire brain linear size 0.15m d) The answer will depend on whether we think of the size of structures in terms of money or the number of employees. One also needs to decide what to include in the structures of the financial system. For example, one could focus on banks and find data on bank-size in Italy and China, it would be of interest to look at the distribution of bank sizes. Perhaps look up the paper: Rank-size

Complexity Science distributions for banks: A cross-country analysis by R. Cequeti, C. Lupi, F, Peirtrovito, and A.F. Pozzolo. Physica A 585 126336 (2022)

Exercise 6 a) At least some degree of robustness: - the family - the tribe - the nation-state - institutions like hospitals, schools, companies, etc. b) How about: - individual plants, animals, ... - species or types - perhaps sub-networks of food webs c) Various brain regions have some functional robustness. For example, the visual cortex, the olfactory bulb, the auditory cortex etc. Perhaps if we think of the mind moods like happy, sad, anxiety, etc. can be considered "robust" emergent structures. d) For eg. financial institutions such as banks, building societies, pension funds and regulatory bodies such as central banks, and government institutions. Exercise 7 Below are some suggestions as to how the disappearance or breaking of symmetry may be related to these changes. a) For example change in the symmetry of the distribution of productivity per worker. Pre-industrial revolution: One narrow, presumably peaked formed distribution. Post-industrial revolution: Bimodal distribution with one peak at low value for non-industrial workers and another peak at higher value for industry workers. b) For example change from autocracy to democracy and change in the distribution of possible political influence. c) For example the change in the distribution of accepted moral behaviours when a change occurs from a narrow, strict (perhaps religiously motivated) set of values to a liberal tolerant "anything goes", individual-focused values. d) For example distribution of accepted female dress code. The change from dress and skirt only to a broader range including trousers and trouser suits. Exercise 8 The question is intended to inspire discussion. Project 1 Read de Haan’s paper carefully. Project 2 a) Matter of scale and aggregation. The intermittent dynamics occur at a systems level. For ecology that is the abrupt changes in taxonomic composition. For example mass extinction of species. For the economy, the intermittency is similarly seen in the type of extant companies. Suddenly nearly all cartwrights disappear for example.

Solutions Manual for Complexity Science The Study of Emergence by Henrik Jeldtoft Jensen

Complexity Science

b) The arrival of a new type which is disruptive to the stability of the current configuration c) Better understanding of dynamics may enable us to identify potential dangerous developments and act to avoid or minimise consequences. For example stress testing of banks followed by prudential action or firefighters to minimize damage by forest fires. d) No sample solution needed. Project 3 At present, a detailed answer to (a) and (b) is not known. But it is presumably related to the time averaging of spatial structures. Project 4 No sample solution needed Project 5 No sample solution needed

Complexity Science

Chapter 3 Exercise 1 No sample solution needed Exercise 2 a) For example, the cell life of red blood cells (RBCs) is approximately 4 months, white blood cells (WBCs) is about a year, and skin cells are just a few weeks. Heartbeats are about 1 second apart. Breath at rest is at a rate of 15 per minute, and digestion takes about 36 hours total. The lifespan for the entire body is about 70 years. b) Travel speed of signal along a motor neuron is approximately 430 km/hr. The average neuron fires about 0.16 times per second. The up-down cycle of the neuronal action potential is approximately 1 msec. The reaction time to stimuli is 200 msec and thoughts can vary from seconds to minutes, and to hours. Exercise 3 Some examples: Ice to water: Since ice and water can co-exist the length scale characterising the transition remains finite. Magnetic to non-magnetic: The magnetic ordering spans across the materials, i.e. the length scale equals the size of the system. Therefore, the characteristic length is infinite compared with the microscopic scale, say the separation between the atoms. When we look at the transitions in the biological or social systems (say a collapse of an ecosystem or a political change), we can check if the change penetrates across the entire system. Exercise 4 a) Despite critical fluctuations that have been suggested by Scheffer et al. to often occurring before a transition, it is difficult by a web search for "critical fluctuations as precursor" to find established examples. However, Olthof et al. suggest "critical fluctuations as an early-warning signal for sudden gains and losses in patients receiving psychotherapy for mood disorders" [Clinical Psychological Science 8(1) 2020. First published 24.09.2019] that the mechanism may be clinically relevant. b) There seems to be no indication that, for example, earthquakes are preceded by fluctuations. Nor does the appearance of a new infectious virus with the ensuing pandemic seem to be foreseeable from fluctuations. Exercise 5 No sample solution needed Project 1 No sample solution needed Project 2 a) No solution needed b) No solution needed c) Generically, the strength of the approach is a possible identification of essential mechanisms. The weakness is often that it can be difficult to go beyond the Ising approach in an incremental way.

Solutions Manual for Complexity Science The Study of Emergence by Henrik Jeldtoft Jensen

Complexity Science

Project 3 No sample solution needed. Project 4 a) Considerations may include the observation that one will need to distinguish between prey and predator links: if a species is a prey of many other species, i.e. high prey-degree it has a harder time avoiding being eaten. If a predator has a small predator-degree, meaning it only preys on a few species, it may be difficult to find food. b) Keystone species do not affect the stability of an ecosystem through tropic interactions only, as the example of the bee indicates. The bees are of crucial importance as pollinators, and hence for reproduction in the ecosystem, but are not eaten by many species. If all species could be considered to be of equal pairwise importance, i.e. all links between nodes representing species of equal weight, keystone species would presumably, e.g. be represented by hub nodes - these are nodes of the unusually high number of links. In reality, some links between species may be more important than others. In this case, a key species is probably connected to nodes (species) with important links. c) In a food-web structure a keystone species might sit at the bottom of a pyramid structured food-web with very many different predators dependent on it, i.e. with high prey-degree. Or the keystone could sit as a bottleneck between different communities in the network. It could possibly also be a top predatory, the removal of which will allow prey species populations to grow unchecked and thereby destablise the ecosystem. This could be difficult to see from the network structure alone and will only become clear when population dynamics on the network is accounted for. d) Foodwebs have very broad degree distributions. Such networks are stable against random removal of nodes (since a typical node is of low degree and in this respect of little consequence) fragile against target removal of high degree nodes. e) See last paragraph of ref. [220]. f) That a focus on pairwise interaction between species misses important aspects of higher order interactions and that non-local global effects may not be captured by the pairwise network representation. g) No sample solution needed. Project 5 No sample solution needed. Project 6 a) No sample solution needed. b) The result of the body’s temperature regulation is similar to the result of the Gaia mechanism, namely homeostasis. But the Gaia hypothesis involves evolutionary dynamics and selection pressures. The temperature regulation involves a negative feedback loop controlled centrally from the brain’s thermal regulation centre in the hypothalamus. c) See for example John S. Torday. "Homeostasis as the mechanism of evolution", Biology (Basel). 2015 Sep:4(3):575-590 For examples of sociological feedback loops see J.J. Fernandez and A.M. Jamie-Castillo. " Positive or negative policy feedback? Explaining popular attitudes towards pragmatic pension policy reforms. European Sociological Review. 29(4) (2013) 803-815. d) No sample solution needed.

Complexity Science

e) Entropic hierarchy: Sequences of systemic conditions allowing for an increasing number of configurations corresponding to an increasing entropy of the system. f) One may argue that given the intricacy of the phenomena under consideration, namely how the self-regulation of Gaia arises spontaneously, the model in [28] does very well on all three: the model extracts the most essential (P1). Though the model is simple compared to reality it tries not to over-simplify (P2) by using an individual-based stochastic model. Finally, even P3 seems to be well fulfilled given that the Gaia hypothesis may be at work in many different situations.

Chapter 4 Exercise 1 This is obviously an open-ended question intended for discussion. Presumably one would formulate a model which includes components like borrowers, investors, and bank agents. The paper by Viegas et al. "Ecosystems perspectives on Financial Networks: Diagnostic Tools". COMPLEXITY, 18(6) 3448(2013) found that an element of evolutionary dynamics is essential for crashes to occur. Exercise 2 One approach could be to start from simple and then add features to the model until the desired phenomenology appears. Another is to start from a description (model) which reproduces the phenomenology, but which appears to have irrelevant details. One may remove details to study which are essential. Exercise 3 a) Probably not much. The Hodgkin-Huxley model already describes well the dynamics of a single neuron. The biggest challenge for an understanding of the human brain appears to consist in understanding the emergent collective behaviour of the huge number of interacting neurons in the brain. The computer neuronal network will present the same problem. But of course, it will be easier to carry out an experiment on the computer representation of the brain. b) That is at the moment a very open question. Perhaps the artificial "brain" can help to understand the formation of information propagation and the formation of functional networks. However, it is highly unclear if the computer brain will possess anything similar to the human mind or consciousness. c) Not necessarily. It is possible. For example, the mind and consciousness are dependent on the entire history of experience back from the formation of the first brain cells in the foetus or other unknown dependencies. d) Maybe, if the computers were to develop through a "social environment" like a human does more or less all the way back to germination. Exercise 4 Very difficult term to handle rigorously. Even the so-called "fitness" used in evolutionary biology as genotype fitness or phenotype fitness needs to be defined for a given environment. Hence, it is not an intrinsic property of the organism. Project 1 a) Depends. Consider a primary school. The group of stakeholders will include: the teachers, parents, pupils, city council etc. b) Discussion question. No sample solution needed

Solutions Manual for Complexity Science The Study of Emergence by Henrik Jeldtoft Jensen

Complexity Science

c) It will depend on the time scale considered. For example, the union organising a group of employees could well be considered a robust structure for the duration of the existence of the firm. d) This is definitely an open question and will depend on context. Few stakeholders (e.g. individual parents or interest groups) can be considered as robust over time scales of the society (say decades and centuries). And if a certain stakeholder can be identified, say the Agriculture and Horticulture Development Board Potatoes or AHDB Potatoes (former Potato Council), they most likely keep changing their strategies and aims over the years. This will make it problematic to consider them as well-defined society building blocks in their own right. e) Perhaps on shorter time scales and for some specific subset of phenomena. A way to go about this question could be to ask for a model of, say, an university. Project 2 a) Yes. Nice, clear, and schematic model. b) Perhaps not. The paper has been useful to much further research, as indicated by 200 citations, and for example mentioned in Mahadevan et al. "Evaluating the sensitivity of functional connectivity measures to motion artifact in resting-state fMRI data". NeuroImage 241, 118408 (2021). But it has also been pointed out that the model approach by Vertes et al. needs to be supplemented by information theory. See e.g. Srivastava et al. "Models of communication and control for brain networks: distinctions, convergence, and future outlook". Network Neuroscience. 4(4), 1122-1159 (2020). c) Perhaps some kind of network of information processing and exchanging nodes. d) Yes. Probably some "aspects" as we already see rudimentary indications of it in current AI systems such as speaking and listening robots. Project 3 a) Most importantly, the waiting time distribution for ants leaving the nest follows not an exponential (i.e. a Poisson process) but behaves like a log-Poisson. The probability that ant leaves at time 𝜔 is proportional to 𝜔1 . b) N different types of agents (ants) on a periodic lattice of size 𝐿 ˆ 𝐿. Agents move according to a utility function representing the effect of interaction between ants. For a low "degree of stochasticity" (DS) moves are dominated by interaction. For high DS, ants move at random and independently. For low-DS, its log-Poisson. For high-DS, its Poisson. c) Difficult to answer since we do not know exactly how ants make decisions. But the model framework makes suggestions that in principle can be tested: dependence on density, behaviour independent of location of exit. etc. d) This is an open question, which can be addressed by trying out simplifications. e) The model predicts that the log-Poisson behaviour should very generally apply to agents moving around and could be applied to different types of ants, perhaps other insects, and maybe even humans moving in, say, a public space.

Complexity Science

Chapter 5 Exercise 1 a) See 7th paragraph of Section 3.2 b) We need ∞ ÿ

𝐴𝑠−𝜏 = 1

𝑠=1

𝐴−1 =

∞ ÿ −𝜏

𝑠 = 𝜁 (𝜏)

𝑠=1

which is Riemann’s Zeta function. To ensure 𝐴 ă 8 we need 𝜏 ą 1 c) Moments ă 𝑆 ą=

∞ ÿ

𝑠𝑃(𝑠) = 𝐴

∞

𝑠=1

=𝐴

∞ ÿ

𝑠(𝑎−𝜏)

𝑠=1

𝑆−(𝜏−𝑎) = 𝐴𝜁 (𝜏 ´ 𝑎)

𝑠=1

To ensure ă 𝑆𝑎 ąă 8 we need, 𝜏 ´ 𝑎 ą 1 or 𝑎 ă 𝜏 ´ 1. Exercise 2 a)

𝑔𝑠𝑁 (𝑠) =

∞

𝑃𝑠𝑁 (𝑞)𝑠 𝑞

𝑞=0

where

𝑃𝑠𝑁 (𝑞) = 𝑝𝑟𝑜𝑏t𝑠𝑁 =

𝑁 ÿ

𝑋𝑡 = 𝑞u

𝑡 =1

∞

ÿ 𝑛

𝑛=0

𝑡 1

= ÿ 𝑃𝑟𝑜𝑏t𝑁 = 𝑛u𝑃𝑟𝑜𝑏t ∞ ÿ

𝑋𝑡 = 𝑞u

𝑃𝑁 (𝑛)

𝑛=0

𝑞1

ÿ ...

𝑃𝑋(𝑞1)...𝑃𝑋(𝑞𝑛)

𝑞𝑛

𝑞1 ` 𝑞2 ` ... ` 𝑞𝑛 = 𝑞 ∞

𝑔𝑠𝑁 (𝑠) = = 𝑛=0

∞ ÿ

𝑞=0 𝑛=0

𝑃𝑁 (𝑛) 𝑞=0

𝑁 𝑃 (𝑛) 𝑞 ...𝑞 1 𝑛

𝑃𝑋(𝑞)...𝑃𝑋(𝑞𝑛)𝑠

∞ ÿ ÿ ÿ ( 𝑃𝑋(𝑞1)𝑠𝑞1 )...( 𝑃𝑋(𝑞𝑛)𝑠

𝑞1

𝑞

) 𝑞𝑛

𝑞𝑛

Solutions Manual for Complexity Science The Study of Emergence by Henrik Jeldtoft Jensen