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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

of closed 3-motifs (closed triad of links).

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

|Ci−Ci0|(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

|Ti−Ti0|(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 e−µ∆, 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 F−T−F−... 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)∼(se−s2/σ2Wigner surmise (delocalized behavior)

e−s/δ Poissonian statistics of events (localized behavior) (5)

where σ,δare some positive constants, and s=λi−λi+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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