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\begin{document}
\title{Notes: Complex Networks - Characteristics, Analysis and Modeling}
\author[1]{Richard Gast}%
\affil[1]{Affiliation not available}%
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\date{\today}
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\section*{Lynn \& Bassett (2019): The physics of brain network structure,
function and
control}
{\label{298276}}\par\null
\subsection*{Network models of structural connections in the
brain:}
{\label{241692}}
\begin{enumerate}
\tightlist
\item
Random networks: Can be generated by the Erd\selectlanguage{ngerman}ös-Renyi model wherein
each pair of nodes is connected independently with a fixed
probability.
\item
Community structure: Characterized by dense intra-community coupling
and sparse inter-community coupling. Can be generated via the
stochastic block model, where the coupling probabilities depend on the
community assignments of the to-be-coupled nodes~
\item
Small world structure: Characterized by a high clustering coefficient
and average path lengths that are shorter than expected in random
networks. Can be generated by the Watts-Strogatz model which is an
interpolation of a circular nearest neighbor network and a random
network.
\item
Hub structure: Characterized by high-degree hubs that form a densely
interconnected structural core. Can be identified by a heavy-tailed
degree distribution, such as observed in scale-free networks. Can be
generated by the Barabasi-Albert model that adds nodes sequentially to
a network and connects them to existing network nodes with a
probability proportional to the degree of the node.~ ~
\end{enumerate}
\par\null
\subsection*{Physical models of brain network
function:}
{\label{408692}}
Statistical mechanics established that the thermodynamic laws governing
the macroscopic behavior of gas molecules can be derived from the
microscopic dynamics of the molecules themselves. The question is,
whether similar laws exist for the collective behavior of neurons.~
\section*{Chialvo (2010) Emergent Complex Neural
Dynamics}
{\label{744199}}\par\null
\subsection*{Complex systems and
emergence}
{\label{351960}}
Almost all macroscopic phenomena, from superconductivity to gravity and
from economics to photosynthesis, are emergent phenomena of the
underlying collective dynamics of their microscopic components.
Emergence refers to the unexpected collective spatiotemporal patterns
exhibited by large complex systems. In this case, unexpected refers to
our inability to derive the emergent patterns from the equations of the
underlying microscopic elements. In general, complex systems consist of
large numbers of interacting elements that exhibit non-linear dynamics.~
Thereby, the interaction can also be indirect, through some kind of
mean-field, for example.~ The three features ((1) a large number of
individual components, (2) interaction between the components, and (3)
nonlinearity of the components) are necessary, but not sufficient
conditions for a system to exhibit emergent complex phenomena. For
example, a small number of isolated linear elements cannot produce any
unanticipated (emergent) behavior.
\par\null
\subsection*{Emergence in the brain}
{\label{238081}}
Scale-free dynamics (1/f noise) of brain activity under healthy
conditions as well as of human behavior~ support the interpretation of
brain activity as an emergent phenomenon. Importantly, mathematical
modeling of emergent behaviors should reveal conditions under which
something complex emerges from the interaction of less complex elements.
Important attempts at this have been inspired by concepts of criticality
and phase transitions from physics. Critical dynamics have for example
been reported in iron magnets, in which the mean magnetization decreases
when heated, and eventually vanishes beyond a critical temperature. Near
the critical point the system is most susceptible, showing the largest
fluctuations in magnetization, with scale-invariant temporal dynamics
and long-range correlations in the local spins (following a power-law
distribution). At the critical point, a single spin perturbation can
start an avalanche that reshapes the entire system state. Coming back to
the brain, neuromodulators could play the role of such critical, system
changing parameters (such as the temperature in the above example). An
important property of critical phenomena is that long-range
spatiotemporally correlated patterns~emerge out of short-range
interactions. Since the largest number of meta-stable states exists near
a point of transition, the brain could access the largest repertoire of
behaviors by keeping its state near an critical instability. In a very
macroscopic argument, Chialvo states that the brain has to operate near
to a critical state, since the world is in a way critical, i.e.
surprising events can happen, but with a finite probability. If the
brain would be sub-critical, it couldn't adapt and if it would be
super-critical, it couldn't learn from previous experiences.
\par\null
Beggs and Plenz (2003, 2004) reported empirical evidence for neural
critical dynamics. They described so called neural avalanches that
engage variable numbers of neurons and show power-law distributions of
avalanche lifetime and sizes with an exponent \textasciitilde{}= 3/2,
which agrees with the theoretically expected value for a critical
branching process.~ On a macroscopic scale, Kelso and colleagues (1992)
found transitions in the MEG at critical points of transitions in
behavior. Another interesting dynamic phenomenon in the brain is the
transition between different resting state networks as measured via
fMRI/MEG. Similarly to the Ising and Kuramoto model, the probability of
the length, two brain regions couple at rest seems to follow a power-law
distribution (Kitzbichler et al. 2009). Furthermore, Fraiman and
colleagues (2009) showed that~ only near the critical temperature, the
2d Ising model produced correlation networks similar to the brains
resting-state networks.~~
\par\null\par\null
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