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Spectral peculiarity and criticality of the human connectome

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We have performed the comparative spectral analysis of structural connectomes for various organisms using open-access data. Our analysis indicates several new peculiar features of the human connectome. We found that the spectral density of human connectome has the maximal deviation from the spectral density of the ran-domized network compared to all other organisms. For many animals except human structural peculiarities of connectomes are well reproduced in the network evolution induced by the preference of 3-cycles formation. To get the reliable fit , we discovered the crucial role of the conservation of local clusterization in human connectome evolution. We investigated for the first time the level spacing distribution in the spectrum of human connectome graph Laplacian. It turns out that the spectral statistics of human connectome corresponds exactly to the critical regime familiar in the condensed matter physics which is hybrid of Wigner-Dyson and Poisson distributions. This observation provides the strong support for the much debated statement of the brain criticality.
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Spectral peculiarity and criticality of the human connectome
K. Anokhin1,2, V. Avetisov3, A. Gorsky4,5, S. Nechaev6, N. Pospelov1, and O. Valba3,7
1Lomonosov Moscow State University, 119991, Moscow, Russia
2National Research Center ”‘Kurchatov Institute”’, 123098, Moscow, Russia
3N.N. Semenov Institute of Chemical Physics RAS, 119991 Moscow, Russia
4Institute for Information Transmission Problems RAS, 127051 Moscow, Russia
5Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia
6Interdisciplinary Scientiﬁc Center Poncelet (CNRS UMI 2615), 119002 Moscow, Russia
7Department of Applied Mathematics, National Research
University Higher School of Economics, 101000 Moscow, Russia
We have performed the comparative spectral analysis of structural connectomes
for various organisms using open-access data. Our analysis indicates several new
peculiar features of the human connectome. We found that the spectral density of
human connectome has the maximal deviation from the spectral density of the ran-
domized network compared to all other organisms. For many animals except human
structural peculiarities of connectomes are well reproduced in the network evolution
induced by the preference of 3-cycles formation. To get the reliable ﬁt , we discov-
ered the crucial role of the conservation of local clusterization in human connectome
evolution. We investigated for the ﬁrst time the level spacing distribution in the
spectrum of human connectome graph Laplacian. It turns out that the spectral
statistics of human connectome corresponds exactly to the critical regime familiar in
the condensed matter physics which is hybrid of Wigner-Dyson and Poisson distribu-
tions. This observation provides the strong support for the much debated statement
of the brain criticality.
I. INTRODUCTION
A. Purpose of the work
Understanding basic mechanisms of brain functioning in terms of the structure of un-
derlying anatomical and functional brain networks, is a challenging interdisciplinary issue
which worries researchers over the decades. Detailed presentation of a current state of the
comprehensive studies of structural and functional neural connectivity referred as connec-
tomics can be found in [1–4]. To summarize the mainstream directions of modern research,
one can highlight two questions of the primary interest:
Which properties of the connectome are of key importance for an eﬀective brain func-
tioning at the cognitive level and information processing?
What are the operational mechanisms of the structural network evolution, allowing to
arrive at a present pattern of the connectome organization?
Answers to these questions are currently sought in the studies ranging from the inves-
tigation of the complete network of connections among the 302 neurons of the nematode
arXiv:1812.06317v1 [q-bio.NC] 15 Dec 2018
2
Caenorhabditis elegans (C. elegans) [5] to analysis of complex mammalian brain networks
including the rat, cat, macaque and the human connectome [3, 6–9]. The initial analysis
of the C. elegans data has led to a conjecture that from the topological point of view, the
connectome is an example of a small world network with a high clusterization and short
path structure [10]. In some sense, such a topological network lies between the regular and
completely random (Erdos-Renyi) topological graphs [11–13]. It was suggested that high
clustering coeﬃcient determines eﬃciency of inter-module brain processes, while small av-
erage path length contributes to the embedding of large regions into a high-performance
network and thus allows to connect system processes on diﬀerent scales. It this respect
small-world model ﬁtted well to the properties of brain networks that combine high local
connectivity with global information transmission. It linked processing of neural information
on local and global levels with peculiar properties of the brain network architecture.
Such a line of reasoning was developed in [14] where the connectome is identiﬁed with
the scale-free network (see also [6]). Scale-free networks are characterized by a power-law
degree distribution according to which the majority of nodes show few links, but a small
number of hub-nodes have a large number of connections and ensure a high level of global
network connectivity [15]. Both small-world and scale-free architectures are considered to
be attractive candidates for eﬃcient ﬂow and integration of information across the network
[16–18].
However, it has been also recognized that some properties of the connectome, like the
hierarchical structure and the vertex degree distribution, cannot be explained in the frame-
works of the ”small world” paradigm. The hypothesis that brain networks exhibit scale-free
topology became popular at the turn of the millenium, however, nowadays there are many
evidences that connectomes on various anatomical scales deviate from networks with the
scale-free vertex degree distribution. The emerging viewpoint is that the connectome real-
izes a new type of a network architecture.
To unravel these speciﬁc organizational principles the comparative analysis of connec-
tomes of diﬀerent organisms [19] is of extreme importance. It provides hints for the iden-
tiﬁcation of key structural properties of the neuronal network, crucial for its integrative
functions across a variety of speciﬁc neuroanatomical organizations. Since any global brain
function is a collective eﬀect it should be treated by appropriate methods capable to catch
collective properties of underlying structural network. From this point of view, the com-
parative investigation of diﬀerent connectomes undertaken in [20] via the analysis of their
spectral properties, seems particularly promising (see also [21]). In our work we continue
and develop this comparative spectral approach.
Through the ultimate wiring of the neural network is cellular, the analysis of the most
of brain networks on this scale is currently complicated due to the lack of available experi-
mental data. The human brain is made up of 8.6×1010 neurons [22], and current imaging
techniques are far beyond resolving its microscopic connectivity. Only neuronal connectomes
of comparatively primitive organisms, such as P.paciﬁcus worm and a C.elegans nematode,
are currently available. The nematode connectome reconstruction had started in 1974 and
lasted 12 years, despite it contains only about 300 neurons and several thousands of synap-
tic connections [23]. Thus in a huge number of works the brain networks are studied at
large and middle scales. The nodes of such networks are either ”voxels” (cubic 3D areas
containing hundreds of thousands of neurons), or whole brain regions.
For our work we used data from open sources. The data on the macaque connectome is
3
limited to the only one network obtained in the Cocomac project [8]. Data on C.elegans and
macaque connectomes have been taken from the Open Connectome Project [24] database.
Data on human structural connectomes were extracted both from Open Connectome Project
and Human connectome project databases [25].
Classical methods of the theory of complex networks are now widely used in neurobiology
[1, 3, 4] . However, many commonly used metrics, such as betweenness centrality, eﬃciency,
and others focus rather on the properties of individual nodes of the network than on the
features of the network as a whole. In this paper our attention is focused on the global
properties of the connectome. For these purposes, the methods of spectral graph theory are
well suited. The main objects of interest in this case are the eigenvalues and eigenvectors of
the matrices characterizing a graph, for example, adjacency matrix or Laplace matrix. The
spectrum of such a matrix (i.e. a set of eigenvalues) is a unique identiﬁer of the network, its
peculiar ”ﬁngerprint”. Knowing the whole set of eigenvalues and eigenvectors, it is possible
to restore the original appearance of the network (with rare exceptions, which are described
below). In the process of spectral decomposition, matrix elements corresponding to diﬀerent
elements of the network are mixed in a complex way, creating a ”global” portrait of the
network as a single object. For example, in the case of a graph represented as a Laplace
matrix, its eigenvalues have a clear physical meaning: they represent the frequencies at
which the graph would resonate if it was made of springs.
Using this data we demonstrate that various structural and spectral properties of connec-
tomes of diﬀerent organisms can be designed evolutionary under speciﬁc constraints starting
from the undirected ”null network state” which is a randomized version of an initial graph.
The evolution of the network is carried out using the Metropolis algorithm. Without
the loss of generality, it can be described as follows: taking the ”null network state” as the
input, the Metropolis algorithm attempts to make random changes to the network. If these
changes occur in the ”right direction”, i.e. reduce the distance between the current and the
desired state of the network in a pre-selected metric, they are accepted with probability one.
If, to the contrary, the elementary rewiring moves the network away from the desired state
(for example, reduces the number of triangular motives, though the purpose of evolution
is to increase their number), it is accepted with some probability, exponentially decreasing
with the size of ”wrong” deviation. The chemical potential, µ, plays the role of a parameter,
governing the probability to which these ”wrong” steps in the network evolution are allowed.
The chemical potential approves its name by the function it performs: bringing analogy
from physics, µcontrols the amplitude of ”thermal ﬂuctuations” in the algorithm known
as ”simulated annealing”: in the absence of thermal ﬂuctuations, the network accepts only
”positive changes” in its evolution along the landscape and might be easily trapped in a
local energy minima. The possibility of some ”backward moves” allows the system to escape
from local traps, thus helping the network to reach the true ground state.
One can easily understand the sense of an evolutionary algorithm by considering an Erdos-
Renyi network as the system’s ’null state’. The constrained Erdos-Renyi network (CERN) of
Nnodes evolves under the condition that the vertex degree in each node is conserved during
the network’s rewiring. The spectral properties of CERNs were thoroughly investigated in
[26]. The ”driving force” of the network’s evolution is the attempt to increase the number
The condition of the vertex degree conservation in each graph’s node changes drasti-
cally the ﬁnal state of the evolving structural network. These constraints provide Nhidden
4
conservation laws for the stochastic network evolution (rewiring) making the corresponding
dynamic system quite special. In particular, it was found in [26] that the evolving CERN
undergoes the phase transition and gets defragmented into a set of Kdense communities
when the chemical potential, µ, of closed 3-motifs exceeds some critical value, µcr. The num-
ber of communities, K, depends on the density of the network at the preparation conditions
and can be approximately estimated as [1/p], where [...] designates the integer part of 1/p
and pis a probability to connect any two randomly chosen vertices of ER network. The
phase diagram of the random constrained Erdos-Renyi network (with ﬁxed vertex degree)
and enriched by closed 3-motifs (controlled by the chemical potential µ) is shown in Fig.1.
To make the ﬁgure more informative, we show network samples at three diﬀerent densities of
closed 3-motifs. The eﬀective way of constructing the phase diagram is discussed at length
below.
FIG. 1: Phase diagram of the CERN. Number of triangles in the ﬁnal network dramatically
increases at the point µ=µcrit which for large networks corresponds to 1st order phase transition.
Typical network structures are shown at three diﬀerent values of µ.
The peculiarities of the network dynamics for humans prompted us to suggest an existence
of Nadditional conservation laws consisting in conservation of local clusterization for each
network node. The local clusterization has been used in the analysis of generic exponential
graphs (see, for instance [27]), however in the context of the connectome, it has been applied
for the C.elegans only in rather restricted context. We shall use local clusterization as an
”order parameter” which can distinguish the connectome of humans from connectomes of
other animals.
In our work we tackle in details spectral properties of experimentally available adjacency
and Laplacian matrices of structural networks. The simplest characteristics of the spectrum
is its spectral density. However more reﬁned characteristics like the statistical correlators of
5
the spectral densities carry a bunch of the additional information about the network prop-
erties. The investigation of the level spacing distribution (i.e. the distribution of distances
between the neighboring eigenvalues) allows one to identify the level statistics using the
standard methods of spectral statistical analysis. The level spacing distribution provides
the key information concerning the localization properties of the signal propagation in the
connectome. To the best of our knowledge, the spectral statistics has not yet been discussed
in the context of the structural connectome.
Our statements are as follows:
The spectral density of adjacency matrices for human structural connectomes has
the maximal deviation from the spectral density of the randomized (via the Maslov-
Sneppen procedure) network in comparison with the other organisms in this study.
Spectral density of adjacency matrices of structural connectomes for all organisms
except humans can be satisfactorily reproduced by conserving the vertex degree in
every network node and reconnecting links under the control of the chemical potential
of triangles in the stochastically evolving network (i.e. restoring the initial connectome
level of clustering in a randomized network)
Spectral density of structural human connectomes can be reproduced well enough by
random rewiring of network’s links if besides the conservation of the vertex degree, we
demand also the conservation of the local connectivity (i.e. imposing extra conserva-
tion laws).
The level spacing distribution for human connectomes demonstrates very peculiar
behavior which corresponds precisely to the critical regime and is the hybrid of Wigner-
Dyson and Poisson level statistics which means that the human connectome is at
criticality. This diﬀers from the model of the cluster-enriched scale-free network.
B. Structural and dynamic properties of a connectome
Let us summarize key statistical and dynamic properties of connectomes of various or-
ganisms on the basis of open-sources data analysis. These properties we consider as the
reference point for our investigation. Firstly, connectomes typically have a large modularity.
Recall that the modularity measures the clusterization of the network and reﬂects its hier-
archical structure [28–30]. The large modularity is conﬁrmed in [31] by the high-resolution
analysis of adjacency matrices of human connectome for the N > 5×104nodes. Secondly,
structural connectomes have the ”short path” property typical for the small-world networks.
Thirdly, spreading of the signal in connectomes demonstrates the synchronization [32–34].
Important property of the connectome is the distribution of so-called ”motifs” – small
subgraphs of the whole network. For C.elegans this quantity has been studied for the
ﬁrst time in [35], where it was found that the number of open 3-motifs (connected pairs
of network links) signiﬁcantly exceeds the respective quantity for the random Erdos-Renyi
network with the same link formation probability. The situation with closed 3-motifs (fully
connected triads of links) is more subtle: First, it was suggested that there is no exceed of
the number of closed 3-motifs [35]. However later, a more detailed analysis showed that the
6
connectome still is enriched by the number of closed triads of links compared to an Erdos-
Renyi network of same density. Moreover, the connectome has a preferred typical topology
of a backbone between communicating neurons [36].
It has been shown in [35, 37] that all connectomes have an excess of open 3-motifs and
the directed open 3-motifs are highly important. In particular it was found that substitut-
ing open 3-motifs by closed 3-motifs, one can sharply decrease the synchronization of the
connectome which means that presence of open 3-motifs is crucial for the synchronization.
It was argued in [38] that the evolution of the connectome from the C.elegans to the human
shows the improvement of the ﬂow propagation in the network.
In [36] the authors have investigated the distribution of ”paths motifs” in connectomes.
The following topological classiﬁcation of paths has been adopted: L-paths between the
nodes with a mean degree, R-paths between the nodes with a mean degree and a hub, G-
paths between the hubs. It was found that the most often path in a human connectome has
a L-R-G-R-L motif.
Much information concerning the structural properties of the connectome is stored in
the spectral properties of corresponding adjacency and Laplacian matrices. The spectral
density of the Laplacian matrix of the connectome of C.elegans is the triangle-shaped ”con-
tinuum” zone accompanied by several low-energy isolated eigenvalues [20]. Such a shape of
the spectral density is far from the one for random Erdos-Renyi random graph, which has the
oval-shaped Laplacian spectrum. The low-lying eigenvalues and corresponding eigenfunc-
tions of the Laplacian matrix of the connectome carry the information about the transport
properties in the network. In particular, recently it has been found (see [39]) that the second
eigenvalue, λ2, is responsible for the diﬀusion of the signal between two hemispheres. The
third eigenvalue, λ3, seems to measure the radial diﬀusion in the connectome from the inner
to outer regions [39]. The largest eigenvalue of the connectome Laplacian does not devi-
ate much from the eigenvalue typical for a purely random network with the same averaged
characteristics, which means that the connectome typically does not develop the bipartite
structure.
II. MODELLING OF CONNECTOME EVOLUTION BY MOTIF-DRIVEN
REWIRING IN CONSTRAINED ’NULL-STATE’ NETWORK
A. Spectral density of networks and motif-driven network evolution
Here we describe the numerical procedure which manipulates by the experimental data
on structural connectomes taken from open sources. Our experiment is aimed to reveal
the principle ”conservation laws” which might govern the structural transformation of the
connectome during the biological evolution.
Before proceeding further, one important remark is appropriate. There is a common
belief supported by many numerical simulations , that the eigenvalue density (spectral den-
sity) of a graph adjacency matrix is a ”ﬁngerprint” of a corresponding network in generic
situation. Besides, there are known examples of ”iso-spectral” graphs which have diﬀerent
adjacency matrices, however their spectra coincide. Such situations are rather exceptional
and practically do not occur in randomly generated patterns (their Kolmogorov complexity
7
is very high). In our study we consider the spectral density as a graph invariant which
sets a ”metric” for graphs: if spectral densities of two graphs are similar, we say that the
adjacency matrices are similar, while as less two spectral densities resemble each other, as
more unlike the graphs are. For the quantitative comparison of spectral densities we use
(i) the ”transport metric” (see [40] for precise deﬁnition). Thus, rewiring the network, we
catch the evolution of the corresponding spectral density.
All the network spectra plots below were constructed by simple convolving of the set of
eigenvalues with a Gaussian curve and further normalization to make the area under the
distribution equal to 1.
The setting of the simulation is as follows. We take structural connectomes (the state S)
of C.elegans, macaque and human, deﬁned by adjacency matrices of corresponding networks,
and destroy the network patterns by random rewiring of links under the condition of the
vertex degree conservation at each graph node, thus getting the state Srand (the ”null state”).
To obtain null-state networks, we used the Maslov-Sneppen randomization (MS) algorithm
[41] (see Fig.2). The rewiring procedure retains the size of the network and its density, and
also strictly preserves the degree of all nodes. Degree distribution was shown to be of key
importance for the network’s structure, that is why we consider MS-randomized networks
as ”null-state” patterns. Despite the vertex degree distribution of randomized (Srand)- and
initial (S)-networks is the same, their topological, motif, spectral and other properties can
be essentially diﬀerent.
Now, starting from the state Srand, we are trying to recover back the S-state by continuing
random rewiring of links (again with the vertex degree conservation), however now – under
the inﬂuence of a ”driving force” via the Metropolis algorithm described above. The closeness
of two network states, the initial and the randomized one, is measured by the distance
between their spectra.
The standard Maslov-Sneppen Metropolis algorithm described above is transparent and
straightforward in implementation. However, performing evolution of MS-randomized net-
work to a highly clustered state, requires a lot of computational resources: it takes much
more time to pull the network evolution towards a state with a given density of closed 3-motif
state, than to push it to the ”null state”. To reduce signiﬁcantly the time of computations,
we have implemented another procedure which gives the same result. Instead of starting
from a completely randomized network state with the preserved degree distribution, we have
constructed a maximally clustered network (MCN) (again respecting the vertex degree con-
servation). The MCN is a graph that has maximal (or nearly maximal) number of closed
3-motives available for a given degree distribution. Thus, the initial network is fully reorga-
nized in the process of MCN construction. Taking MCN as a ”null state” and running the
Maslov-Sneppen Metropolis procedure controlled by the density of closed 3-motifs, we bring
the network to desired clustering level. Such a technical trick allows to perform computa-
tions for highly clustered graphs very eﬃciently. Figuratively, one can say that assembling
the network with speciﬁed density of closed 3-motifs from the randomized state is like a
”rising to the mountain”, while assembling the network from maximally clustered state is
like a ”descent from the mountain”, which is less energy consuming. Schematically we have
depicted these two algorithms in the Fig.2.
So, we have two types of ”null-state” networks: the MS-randomized network and MCN.
Selection between them depends on the choice between the clustering evolution in ”up” or
8
FIG. 2: (a) Single rewiring step of Maslov-Sneppen algorithm preserving degrees of all vertices; (b)
Example of a local network updating move which increases the number of triangles; (c) Maslov-
Sneppen randomized network, exposed to evolutionary process, which slowly increases the network
clustering until it reaches the initial level of the connectome network; (d) Maximally clustered
network, exposed to Maslov-Spappen randomization algorithm, resulting in a network with the
same clustering. Arrows indicate the directions of the evolutionary process.
”down” directions. In our work we use two types of additional constraints imposed on the
null-state network evolution:
The ”ﬁrst level”, the triangle-preserving constraint (TPC), is the condition of maxi-
mizing the number of closed 3-motifs in the evolving graph until its clustering reaches
the level of the initial (pattern) network. Using this procedure, we calculate only
overall number of triangles, while we do not pay attention to how these triangles are
distributed over the network. In general, such a ”total clustering preserving proce-
dure” allows to get a network state which is closer to the original one than the simple
MS-randomized network.
The ”second level”, the local clustering constraint (LCC), preserves local clustering
in each node of the network such, that the rewired network (besides to vertex degree
conservation), also preserves the number of closed 3-motifs in each node. The LCC
algorithm is thoroughly described below.
B. Network randomization with additional conservation laws
More reﬁned algorithms of network rewiring are demanded to preserve simultaneously
several characteristics of the original graph. We are guided by an attempt to propose the
”minimal” model which, on one hand, could capture the key properties of the structural
connectome and could distinguish humans from other organisms, and on the other hand,
is as simple as possible. We put forward the conjecture that such additional ”feature”
that should be conserved during the network randomization together with the vertex degree
conservation, is the number of triangles in which each node participates.
9
Rephrasing the said above, we randomize the network, however conserve in all nodes: i)
the vertex degree; ii) the clustering coeﬃcient. Below we propose our own way to solve this
problem numerically using a modiﬁcation of the Metropolis algorithm. Having the network
adjacency matrix, A={aij }, of a graph ((i, j)=1, ..., N ), consider two auxiliary diagonal
matrices, D?,D4, deﬁned as follows
D?={d1, d2, ..., dN}:di=
N
X
j=1
aij
D4={˜
d1,˜
d2, ..., ˜
dN}:˜
di=
N
X
j6=k
aijajk aki
(1)
The elements of the matrix D?are the degrees of nodes, while the elements of the matrix D4
are the numbers of triangles into which a speciﬁc node is involved. From the general point
of view, our algorithm resembles the simulation of some physical process that occurs when
a substance crystallizes. It is assumed that the ”crystal lattice” has already been formed,
however the transitions of ”individual atoms” from site to site are still permissible.
It is assumed that the destination state of the network is the conﬁguration in which clus-
tering of all nodes is the same as in the ”pattern state” (there can be many such destination
networks). Our algorithm takes as an input the network in which the information about
the clusterization is ”washed out”. Then we rewire the network, conserving all the degrees
of nodes. After each rewiring, we compute the ”distance” between the resulting modiﬁed
network and the preselected pattern network
F=
N
X
i=1
|CiCi0|(2)
where Nis the number of nodes of the network, Ciis the clustering coeﬃcient of the node i
in the evolving network, while Ci0is the clustering coeﬃcient in the pattern network which
the evolving network tends to reach. Another deﬁnition of the ”distance” which sets the
metric in the space of the ”networks similarity” is as follows
T=
N
X
i=1
|TiTi0|(3)
This deﬁnition is equivalent to (2), except that the clustering coeﬃcient is replaced by Ti,
the absolute number of triangles involving the vertex i.
The Metropolis-like algorithm trying to minimize the cost functions F(2) and T(3) is
set as follows. First, we chose the metrics (2) or (3). If the local random perturbation
(rewiring) of the network is such that the system tends to the desired destination state (i.e.
For Tdecrease), this random step is accepted with the probability 1. Otherwise, if For
Tare increased by ∆ 0 in the selected metric, the corresponding step is accepted with
the probability eµ, where µ > 0 is the chemical potential of Metripolis procedure. After
reaching the local energy minimum (which is calculated on the basis of the best time/distance
ratio), the algorithm updates the metric and repeats the procedure. We have mentioned that
such an algorithm known as the ”simulated annealing”, allows to reach the ground state of
the system without getting trapped at local minima in the very complex energy landscape.
10
One can say that computing the ”distance” from the evolving network to its ﬁnal desti-
nation, via the clustering coeﬃcients, F, we ”equalize” nodes with diﬀerent vertex degrees.
Namely, for all nodes the clustering coeﬃcient lies between 0 and 1. Thus, nodes with low
degrees, ”pull” the triangles from hubs during the network evolution as they are widely
spread. On the other hand, the rewiring procedure via T-metric satisﬁes the interests of
large nodes to the detriment of small ones. We have developed the optimization procedure in
which both metrics Fand Tare simultaneously used. The algorithm sequentially switches
between these metrics and adjusts the network for both F, respecting the interests of loosely
connected nodes, and T, which works well for hubs. Schematically the idea of the algorithm
is depicted in Fig.3 on the example of sphere packing. Vertical compression (associated with
the minimization of F) of the random 2D pile of spheres leads to a desired increase of the
density, however might be accompanied by the increase of a horizontal size (associated with
T). To squeeze the pile more, we shake the pile randomly and compress it in the horizontal
direction, then we switch back to the vertical compression, etc, until the densest packing
state is reached.
FIG. 3: Illustration of a repetitive sequential minimization of Fand T. Minimizing Fwe compress
the pile, however can slightly increase T. Minimizing Tafterwards, we compress the pile further,
however could slightly increase F. Repeating FTF... compression and shacking randomly
the pile, we reach the densest packing.
Arriving at the stationary state, when Fand Tcannot be decreased anymore during
reasonable time, we compare the adjacency matrices of the destination network with the
preselected pattern. This can be done by comparing the corresponding spectra of two
matrices. The distance between spectra (which, by virtue of the above comments about
the uniqueness of the spectrum, is understood as a quantitatively expressed degree of the
”dissimilarity” of the two networks) is measured in terms of the so-called ”earth mover’s
distance” metric (or Wasserstein’s metric). The earth mover’s distance (EMD) is a metric
based on the minimum cost of transforming one histogram into another. Representing two
distributions (two spectral densities) as two heaps of earth that need to be superposed by
transferring small pieces of earth, then EMD determines the least amount of work required
to accomplish this task. The calculation of EMD is based on solving the transport linear
programming problem, for which eﬀective algorithms are known. In our work we were using
an open-source Python package designed for fast EMD computation [40, 42].
11
III. RESULTS
A. Signiﬁcant non-randomness of a human connectome compared to other
organisms
We have compared spectral densities of adjacency matrices of structural connectomes
of various organisms with their Maslov-Sneppen randomized ”null states”. Our numeric
analysis presented in Fig.4 allows to conclude that the ”null state” network with the vertex
degree conservation constraint, reproduces with a good accuracy the spectrum of initial
connectome adjacency matrix of C.elegans, macaque, but not of a human.
FIG. 4: Spectra of adjacency matrices of a animal connectomes along with its MS-randomized ver-
sion (red), TPC-randomized version (green) and LCC-randomized version (blue) (a)-(c): Macaque
connecome, (d)-(f): Nematode connectome; MS = Maslov-sneppen, TPC = triangle preserving
constraint (ﬁrst evolutionary algorithm), LCC = local clustering constraint (second evolutionary
algorithm)
It means that the vertex degree conservation is a candidate for a ”suﬃcient” minimal
set of conservation laws which control the evolution of these organisms. Imposing the
”ﬁrst level” additional constraint (i.e. preserving overall number of triangles in the net-
work) we may slightly improve the spectra coincidence for nematode and macaque and
signiﬁcantly—for a human. However, the spectral distance between the human connectome
and its TPC-randomized version is still essential, indicating that preserving overall clustering
is not suﬃcient to reproduce the network properties.
To have the quantitative characteristics of the diﬀerence between spectral densities, we
have computed the earth mover’s distance (EMD), E, between spectral density of original
12
organism EMD from randomized ”null state”
human (1) EH1= 1.45 ±0.02
human (2) EH2= 1.53 ±0.02
macaque EM= 0.83 ±0.02
nematode EN= 0.72 ±0.02
TABLE I: Comparative analysis of animals by their earth mover’s distance (EMD) from MS-
randomized ”null state” network.
network and its MS-randomized analog, conserving vertex degrees in all network nodes. We
found that EMD for a human connectome, EH, is much larger than for connectomes of other
animals:
The results presented in the table signal that the human connectome is much farther from
the MS-randomized ”null state” than the neuronal networks of other animals. One could
speculate that such a diﬀerence is the consequence of the evolutionary selection acting on
the neuronal network. This issue will be discussed in more details in the Discussion.
To test whether the stated results are not the artifacts of the network size (the inves-
tigated human connectomes had about 1000 and 600 nodes respectively, while networks of
other animals had less than 300 vertices), we have performed numerical experiments on
smaller neuronal networks. For the experimental data taken from the ”Open Connectome
Project” we obtained networks of diﬀerent size (from 250 to 3000 nodes) and found that the
diﬀerence in the spectral distance does not change during such scaling. We have also carried
out numerical experiments with the data on human connectomes taken from other researches
(data available at UMCD database, graph construction algorithms do not coincide) to ex-
clude the possibility that our result is an artifact of algorithms used in the data processing
of databases. The eﬀect of increasing the spectral distance for humans with respect to other
animals is supported, what is an indirect proof of its generality.
B. Impact of local clustering on the network spectrum
We have seen in the previous Section that the evolution of a human connectome is much
more complicated compared to other considered organisms, and to restore back the spec-
trum of the human structure network pattern from its MS-randomized ”null state” requires
some extra constraints (conservation laws). The ”second level” algorithm preserving local
clustering permits to advance in reproducing the connectome structure of humans. Some
properties of exponential graphs with such set of local constraints were discussed in [27].
Among the characteristics of the network that aﬀect its spectrum, the number of triangles,
Tiinvolving some given node, i(i= 1, ..., N ) is of much importance. Rephrasing that, one
can say that the impact of the local clustering coeﬃcient associated with a given node i, is
crucial. Conserving these Nadditional quantities {T1, ..., TN}(for all network nodes), one
can signiﬁcantly improve the coincidence of the spectra of the pattern and MS-randomized
networks of human structural connectome in terms of the earth mover’s distance, ET, as it
is shown in Fig.4.
13
FIG. 5: Spectra of adjacency matrices of a human connectome along with its MS-randomized
version (red), TPC-randomized version (green) and LCC-randomized version (blue) A. Data from
Hagmann(2008) B. Data from Open Connectome project; MS = Maslov-Sneppen, TPC = triangle
preserving constraint (”ﬁrst level” evolutionary algorithm), LCC = local clustering constraint
(”second level” evolutionary algorithm).
One of the main conclusions of our work is as follows. The coincidence of spectral densities
between structural network of a human and its MS-randomized version exposed to evolu-
tionary process could not be achieved by preserving only the average clustering of the total
number of triangles. Instead, the full vector T={T1, ..., TN}for all network nodes should
be conserved. This result is stable for all organisms under investigation and for networks of
various sizes. It is assumed that such a feature can be associated with the important role of
local clustering in the structure of neuronal networks.
Let us recall that the similar set of conservation laws has been used in analysis of real
networks in [27]. The connectome of the C.elegans has been used as one example and it was
argued that tuning the single parameter which controls the local connectivity, is possible
to ﬁt well the spectral density. Our study provides further evidence of the importance of
various conservation laws in the evolution of connectomes.
C. Criticality of the human connectome
Let us begin with some deﬁnitions. We have deﬁned already A={aij }– the adjacency
matrix of an undirected network (i.e. aij =aji ). The matrix elements, aij, take binary
values: aij = 1, if the monomers iand j6=iare connected, and aij = 0 otherwise. The
14
absence of self-connections means that the diagonal elements vanish, i.e. aii = 0. At length
of the current work, we have studied spectral properties of adjacency matrices of networks,
however in many papers another characteristic, the Laplacian of the graph, is under the
investigation. The Laplacian matrix, L, of a network is, by deﬁnition,
L=dI A(4)
where dis the vector of vertex degrees of the network, and Iis the identity matrix. The
eigenvalues, λn(n= 1, ..., N ) of the Laplacian Lare all real. For regular graphs (i.e for
graphs with constant vertex degrees) the spectra of Aand Lare connected by a linear
transformation.
The spectrum the of Laplacian Lis positive and the minimal eigenvalue, λ1, is zero. From
the graph theory it is known that the multiplicity of the lowest eigenvalue, λ1= 0, equals
to the number of disconnected components in the network. This ﬁts with the identiﬁcation
of the number of separated discrete modes as of the number of clusters. Indeed, when some
isolated eigenvalue hits zero, the cluster becomes disconnected from the rest of the network.
The second eigenvalue, λ2, carries the essential topological information about the network,
known as the ”algebraic connectivity”, which measures the minimal number of links to be
cut to get the disconnected network. The value of λ2plays an important role in relaxation
and transport properties of the network, since it deﬁnes the inverse diﬀusion time, and plays
crucial role in determining synchronization of multiplex (multilayer) networks [43]. The
corresponding eigenvector (the so-called ”Fiedler vector”) sets the bijection between the
network layers.
It is known that one of the most informative characteristics, which delivers the information
about the localization properties of excitations on the network, is the so-called ”level spacing”
distribution function, P(s), where sis the normalized distance between nearest-neighboring
eigenvalues of the Laplacian matrix of the network. It is known from the classical theory
of random matrices (see, for example, [44]) that if P(s) shares the Wigner-Dyson level
statistics, the excitations are delocalized, while if P(s) is exponential, the Poisson-distributed
excitations are localized and the system behaves as an insulator:
P(s)(ses22Wigner surmise (delocalized behavior)
es/δ Poissonian statistics of events (localized behavior) (5)
where σ,δare some positive constants, and s=λiλi+1
is the normalized gaps between
nearest-neighboring eigenvalues.
However, there is the third critical regime for P(s) which occurs when some control
parameter is tuned exactly at the critical point and the system is at the point of phase
transition. In this case the function P(s) is hybrid of Wigner-Dyson and Poisson statistics
at all energies. It has small-s behavior of the former and the large-s behavior of the latter
one [45]. This hybrid statistics serves as the spectral mark of the criticality in the system.
We have used a standard procedure to construct the level spacing: selecting a certain
spectral region, ∆, we computed a set of gaps between sequential eigenvalues and averaged
them over ∆. Finally, we presented the distribution of gaps between adjacent eigenvalues
(in relative units) in coordinates (x, y), which allows for straightforward identiﬁcation of the
15
spectral statistics by the slope of the curve: for the x-axis we have log s, while for y-axis we
have the function L(s), deﬁned as follows
L(s) = log (log(1 C(s))) ; C(s) = Zs
−∞
P(s0)ds0(6)
where C(s) is the cumulative distribution function of s. The main question is whether the
level spacing distribution of human connectomes obeys the Wigner surmise, i.e. demon-
strates the level repulsion, typical for interacting chaotic systems, shares the Poisson statis-
tics, which means that the eigenvalues are uncorrelated or enjoys criticality? The results of
our computations for Laplacian matrices of human connectomes are presented in Fig.6.
FIG. 6: Level spacings of the Laplacian spectra in (log s, L(s))-coordinates: (a-b) data set of
human connectome human-1; (c-d) data set of human connectome human-2; (e-f) Barabasi-Albert
network. Slope=1 (blue dashed curve) indicates the Poissonian statistics, slope=2 (red dashed
curve) indicates the Wigner-Dyson regime.
The algorithm of computations is as follows. As one can see from Fig.5, the spectral
density of adjacency matrix, A, of human connectome consists of a continuous (central)
zone and a set of separated peaks (discrete zone, one separated eigenvalue per one cluster).
In the Laplacian L, deﬁned in (6), one can also split the spectral density into continuous
and discrete parts. In each such part we can determine the intervals ∆, within which the
level spacing belongs either to delocalized (Wigner), or to localized (Poissonian) subparts.
It turns out that two regimes in P(s) both in continuous and discrete part of the spectrum
are separated by the crossover which corresponds to transition from Wigner to Poissonian
statistics. The level spacing of two human connectomes are shown in Fig.6(a,c) for the
continuous part of the spectrum, and in Fig.6(b,d) for the discrete one. In Fig.6(e) we
have plotted for comparison the level spacing of Barabasi-Albert network Laplacian in the
discrete part of the spectrum. No crossover is seen and all eigenvalues are localized.
Thus the spectrum of Laplacian matrices of human connectomes demonstrates a bit sur-
prisingly ”hybrid” behavior for the level spacing distribution in all parts of the spectrum
16
with a clear-cut crossover from Wigner-Dyson to Poisson behavior in each part of the spec-
trum as a function of the energy resolution s. This is exactly the universal critical behavior
of P(s) discovered at the edge of the Anderson localization in [45] which means that human
connectome is at criticality. The conjecture of the human connectome criticality is con-
troversial and highly debated issue (see [46, 47] for the recent discussions). Certainly, this
conjecture is very attractive since in this regime we naturally have long-wave excitations
which exist for certain. Our result provides a strong support for the criticality conjecture
from the standard spectral analysis viewpoint. Note that from our result is also clear that
the 3d nature of the brain is essential since there are no localization/delocalization critical
behavior in one and two space dimensions.
IV. DISCUSSION
A. Main conclusions
The idea of randomizing a network with preserving degrees of nodes (the Maslov-Sneppen
algorithm) is not new, however in the literature it has been used typically outside the context
of the spectral graph theory, being applied mainly to the determination of the average path
length in the network, the global clustering coeﬃcient, etc. In our work, following the
ideas developed in [26], we use the Maslov-Sneppen randomization in combination with the
spectral analysis. This allows us to uncover some hidden structural properties of network
samples and analyze the stability of the spectrum with respect to the network perturbation.
We have proposed the procedure to identify diﬀerences in the architecture of the connectomes
of the organisms which stay on diﬀerent steps of the evolutionary staircase.
The results of our study clearly demonstrate that some fundamental properties of the
connectome cannot be explained by the behavior of typical network characteristics, such
as the vertex degree distribution, the averaging clustering coeﬃcient and the distribution
of clustering among network nodes. This is especially true for human connectomes for
which we have shown that the ”earth mover’s distance” (EMD) between the structural
network pattern (represented by the adjacency matrix) and the corresponding randomized
”null state” network, is essentially larger than respective EMDs for other animals. Such an
interpretation raises a natural question about the signiﬁcance of diﬀerences in the spectra
of macaque and human connectomes, which evolutionary are much closer to each other
than, the nematode. Yet the answer to this question is presently complicated due to the
heterogeneity and small sets of available data.
We have performed crosscheck of our results on various sets of data to avoid the artifacts of
speciﬁc algorithms of neurobiological data available from open sources. For this purpose we
have used diﬀerent human connectome data sets obtained by various experimental methods.
Also, the sizes of considered networks varied from several hundred to several thousand
nodes. However, for the macaque connectome such precautions are not yet possible, since the
CoCoMac project data is the only complete source of information for the brain connectivity
of this species.
The distribution of triangles for each vertex of the network seems to be a simplest invariant
preserving the shape of the network spectrum. Keeping only an average clustering coeﬃcient,
17
we are unable to restore the spectrum of the network from its randomized version. For better
reconstruction of the network topology it is not suﬃcient to know how many triangular motifs
it has, but it is crucial how these triangles are distributed among the nodes. The vectors
of triangular motifs, T, in the real network pattern and in its randomized versions are not
identical. Apparently, knowing Tis crucial for reconstruction a modular brain architecture
with several coupled hierarchical levels.
The importance of local clustering has been repeatedly emphasized in the analysis of
brain networks [19, 48]. It is suggested that the combination of high local connectivity
and the ”small world” property on a large scale is responsible for many features of a brain
functioning [49]. Perhaps, the local clustering should be considered as a crucial achievement
of evolutionary selection, which essentially distinguishes the connectome from its randomized
version.
We have provided analysis of the eigenvalue correlations in spectra of Laplacian matrices
of structural connectomes. Surprisingly the level statistics turns out to be critical. This
ﬁnding strongly support the widely discussed highly controversial brain criticality conjecture
formulated long time ago [50]. There are many arguments in favor and against this conjecture
and our spectral analysis yields one additional rock at the ’yes’ side of the balance.
In our study we used the undirected structural networks which certainly restricts the relia-
bility of our ﬁndings. Nevertheless, even for non-oriented structural connectome the spectral
analysis provides the new important insights. Recent studies [51] of oriented structural con-
nectome show that new interesting features emerge which deﬁnitely deserve elaborations of
new reliable mathematical models and clear physical explanations.
The issue of uniqueness of the human brain in its cognitive abilities and conscious infor-
mation processing has been addressed at various levels of analysis, including evolutionary
expansion of selective regions of the cerebral cortex, emergence of speciﬁc properties in the
human neocortical neurons, novel kinds of cellular interactions, new molecular pathways,
speciﬁcity of gene expression in neuronal and glial cortical cells [52–57]. Our data con-
tribute yet another dimension to this analysis by pointing at the peculiarity of the human
connectome global organization. Its spectral characteristics support an expanded range of
criticality known to maximize information transmission, sensitivity to external stimuli and
coordinated global behavior typical of conscious states [58–62].
B. Directions for further research
Besides the consideration of oriented networks, the challenging question deals with un-
derstanding of interaction and synchronization of various functional sub-networks in the
connectome. The ﬁrst step on this way consists in the consideration of a two-layer network
where there is a competition between strength of open in-layer 3-motifs and cross-layer pair-
wise interactions. In [63] it has been found that within such a model one can see the phase
transition between the phases with dominance of in-layer– or cross-layer–connections. De-
pending on the parameters of the model, these two phases can be separated either by the
sharp boundary or can be transformed one into another smoothly. One can speculate about
the applicability of a two-layer network for the description of functional interactions between
the hemispheres. As pointed in [39], the presence of open 3-motifs in each hemisphere is
crucial for the eﬀective informational ﬂow inside the hemisphere, while the links between
18
hemispheres are evidently important for the entire brain functioning. The existence of the
phase transition in the abstract two-layer dynamic network considered in [63], allows one to
suggest that in functional brain networks the competition between in-layer and cross-layer
interactions occurs either as a sharp 1st order phase transition which might be associated
with the brain disease, or as a smooth crossover.
A bunch of evident questions concerns the observed criticality of the connectome, many
of them have been already posed in the literature. The most immediate question concerns
the role of the long-wave excitations intrinsic for the critical regime in the brain functioning.
To make the problem more tractable one can apply in the full power the machinery used
for the analysis of the critical regime. For example, the fractal dimension and spectral
dimension can be evaluated. Most of the results concerning the criticality conjecture deal
with the avalanches in the neuron spiking phenomena ( not complete list of the references
includes [64–69]). Our results imply that the structural connectome organization supports
the critical behavior of the neural excitations.
The abstract two-layer network developing via Maslov-Sneppen rewiring algorithm, with
in-layer and cross-layer interactions, demonstrates also a kind of synchronization behavior.
Increasing the energy of in-layer motifs in one layer only, we can force the clustering in both
layers simultaneously. Such a synchronization is the consequence of joint conservation laws
in the network rewiring. It is worth noting that the multi-layer networks can demonstrate
a bunch of new critical phenomena [70, 71], such as collective phase transitions (see [72] for
the review) which are absent in single-layer networks. In the forthcoming works we plan to
discuss statistical properties of multi-layer networks in the connectome context.
Understanding the interplay between the spectral properties of the structural connectome
and the informational capacity of the brain ia also crucial for the consciousness problem
[73]. The discussions on this topic deals with the concept of the ”integrated information”
proposed in [74]. Though its initial formulations were of very limited practical applicability
the last reﬁnements of this theory [75] use the standard tools of the spectral analysis. of
random networks, which makes the evaluation of the integrated information more tractable.
Besides, it seems highly desirable to interpret the aspects of consciousness using the standard
notions of statistical and quantum physics, such as the entanglement entropy, entanglement
negativity and complexity – see, [76] for the review. It is highly likely that the free energy
extremization discussed in [77] could be of use for the description of the brain organization.
Since the clustered networks allow the natural embedding into the hyperbolic geometry
[78], it might be possible to use modern holographic approach for the evaluation of the
entanglement entropy [79] and the complexity [80, 81] via the geometry of the hyperbolic
space. Some of these ideas have been already implemented in [82, 83] in terms of random
networks of special architecture. We believe that the progress in the ﬁeld of brain studies
lies at the edge of spectral theory, statistical mechanics of complex entangled systems and
holography. Some initial discussion concerning the possible interplay between the criticality
of the connectome and the holographic approach can be found in [84].
Acknowledgments
We are grateful to A. Kamenev for the important comments. The work of V.A. was sup-
ported within frameworks of the state task for ICP RAS 0082-2014-0001 (state registration
19
AAAA-A17-117040610310-6). S.N. is grateful to RFBR grant 18-23-13013 for the support.
The work of A.G. was performed at the Institute for Information Transmission Problems
with the ﬁnancial support of the Russian Science Foundation (Grant No.14-50-00150). N.P.
and O.V. acknowledge the support of the RFBR grant 18-29-03167. O.V. thanks Basis
Foundation Fellowship for the support. A.G. thanks SCGP at Stony Brook University and
KITP at University of California, Santa Barbara, for the hospitality.
Appendix A: Spectral density of networks and motif-driven network evolution
We have mentioned already that in constrained Erdos-Renyi networks (CERNs) with
stochastic rewiring, the clustering occurs when the evolving network tends to increase the
number of closed 3-motifs (triangles), n4. The concentration of triangles is ﬁxed by the
chemical potential, µ. Imposing the condition of the vertex degree conservation in course
of network rewiring, together with the condition of maximization the number of closed 3-
motifs, one forces the network to clusterize respecting the conservation laws. The detailed
analysis of phase transitions in CERN has been carried out in [26] where it has been found
that the condition of maximization of number of closed triads forces the random network
with the conserved vertex degree to form a multi-clique ground state. The typical phase
diagram accompanied by the visualization of the network structures is shown in Fig.1 of
Section I A. At µ=µcthe network experiences the ﬁrst order phase transition and splits in
the collection of weakly connected clusters.
The structure of clusters (cliques) was carefully studied in [26] via the spectral analysis
of the matrix Aof the network. It has been shown that at µ<µc, the spectral density
has the shape typical for Erdos-Renyi graphs with moderate connection probability, p=
O(1) <1, being the Wigner semicircle with a single isolated eigenvalue apart. At µcthe
eigenvalues decouple from the main core and a collection of isolated eigenvalues forms the
second (nonperturbative) zone as it is shown in Fig.7a. The number of isolated eigenvalues
coincides with the number of clusters formed above µc. Averaging over ensemble of graphs
patterns smears the distribution of isolated eigenvalues in the second zone. Above µcthe
support of the spectral density in the ﬁrst (central) zone shrinks and the second zone becomes
dense and connected. The modes in the second zone are all localized, while the ones in the
central zone remain delocalized. The evolution of the spectral density of the entire network is
depicted in Fig.7b. The numerical results on spectral density evolution are obtained for the
ensembles of 50 Erdos-Renyi graphs of 256 vertices each and the bond formation probability
p= 0.08.
Two important properties of the spectral density of adjacency matrices of constrained
Erdos-Renyi networks have to be mentioned:
The spectral densities of each cluster (clique) and of the whole network are very diﬀer-
ent [26]. The spectrum of a clique is discrete, while the spectrum of the whole network
has a two-zonal structure with the continuous triangle-shape form of the ﬁrst (cen-
tral) zone. We have interpreted this eﬀect as the collectivization (or synchronization)
between the modes in diﬀerent clusters.
It was found in [85] that there is a memory of the spectrum in the central zone on the
initial state (on the preparation conditions), which is the signature of the non-ergodic
20
FIG. 7: (a) The spectral density of ensemble of constrained Erdos-Renyi graphs for various chemical
potentials µof closed 3-motifs; (b) The same as (a) in a three-dimensional representation. The
numerical results are obtained for the ensembles of 50 Erdos-Renyi graphs of 256 vertices and the
bond formation probability p= 0.08.
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(Current Biology 27, 714–720; March 6, 2017) In this article, we unintentionally omitted to expand on a citation of previously published results. In the caption of Figure 2, we stated that “F and p values indicate the significance of a phylogenetic ANCOVA testing for intercept differences between humans and other primates (see also Smaers and Rohlf [9], Supplemental Information…, and Table S2 for more detailed results)” (p. 716). We would like to clarify that in this statement, “see also Smaers and Rohlf” refers, specifically and exclusively, to the phylogenetic ANCOVA of primate prefrontal cortex to primary visual cortex and frontal motor areas using the Smaers dataset in [9]. These results were depicted in a subsection of our Figure 2 (the two top left regression plots) and were numerically presented in a subsection of our Table S2. Smaers and Rohlf presented these results as an empirical example when describing the least-squares solution of phylogenetic ANCOVA and did not discuss the wider biological implications of these results for primate brain evolution. The presentation of the previous results was discussed openly during the review process of this manuscript. The authors apologize for any confusion this oversight may have caused.