## Abstract

The interactions between different brain regions can be modeled as a graph, called connectome, whose nodes correspond to parcels from a predefined brain atlas. The edges of the graph encode the strength of the axonal connectivity between regions of the atlas that can be estimated via diffusion magnetic resonance imaging (MRI) tractography. Herein, we aim to provide a novel perspective on the problem of choosing a suitable atlas for structural connectivity studies by assessing how robustly an atlas captures the network topology across different subjects in a homogeneous cohort. We measure this robustness by assessing the alignability of the connectomes, namely the possibility to retrieve graph matchings that provide highly similar graphs. We introduce two novel concepts. First, the graph Jaccard index (GJI), a graph similarity measure based on the well-established Jaccard index between sets; the GJI exhibits natural mathematical properties that are not satisfied by previous approaches. Second, we devise WL-align, a new technique for aligning connectomes obtained by adapting the Weisfeiler-Leman (WL) graph-isomorphism test. We validated the GJI and WL-align on data from the Human Connectome Project database, inferring a strategy for choosing a suitable parcellation for structural connectivity studies. Code and data are publicly available.

## Author Summary

An important part of our current understanding of the structure of the human brain relies on the concept of brain network, which is obtained by looking at how different brain regions are connected with each other. In this paper we present a strategy for choosing a suitable parcellation of the brain for structural connectivity studies by making use of the concepts of network alignment and similarity. To do so, we design a novel similarity measure between weighted networks called graph Jaccard index, and a new network alignment technique called WL-align. By assessing the possibility to retrieve graph matchings that provide highly similar graphs, we show that morphology- and structure-based atlases define brain networks that are more topologically robust across a wide range of resolutions.

## INTRODUCTION

Because of the immense complexity of the brain, it is impossible to gain any insight into its global operation without simplifying assumptions. One such assumption, which has been widely used by neuroscientists, is that the brain, and in particular the cortical surface, can be divided into distinct and homogeneous areas. Of course the definition of homogeneous areas greatly depends on one’s point of view, which has led to a plethora of brain parcellations. For example, the cortical surface has been subdivided based on its cytoarchitecture (Brodmann, 1909), gyri (Desikan et al., 2006), functional organization (Schaefer et al., 2017), axonal connectivity (Gallardo, Wells, Deriche, & Wassermann, 2018), and combinations of these and other features (Glasser et al., 2016). There is also significant evidence that cortical regions vary in shape, size, number, and location across subjects and even across individual tasks, making the existence of a single canonical atlas unlikely. In addition to studying the characteristics of specific brain regions defined by a parcellation, there has been a growing interest in their relationship and interactions, an emerging field known as connectomics. In this context, the focus is shifted from understanding how information is segregated in the brain to how it is integrated. For example, through diffusion magnetic resonance imaging (MRI) tractography, structural connections between brain areas can be recovered. The result is a network whose nodes correspond to cortical regions and whose edge weights represent the strength of the structural connectivity between pairs of regions. A similar network can also be built from resting-state functional MRI yielding a functional, rather than structural, network. These brain networks, which encode the structural and functional connections of the brain, are referred to as connectomes (Hagmann, 2005; Sporns, Tonomi, & Kötter, 2005). Given functional or structural connectomes, their features can be compared across subjects and populations to link network changes to pathology or to further increase our understanding of its organization. An underlying assumption is that a correspondence exists between nodes of the network across subjects, a condition that is usually satisfied by using a group parcellation (Gallardo, Wells, et al., 2018; Parisot, Arslan, Passerat-Palmbach, Wells, & Rueckert, 2015). The drawback of this strategy is that it ignores any subject-specific changes in cortical organization and reduces the specificity of the results. An alternative approach is to construct a mapping between the nodes of the network prior to the comparison, therefore allowing the use of subject-specific atlases. To our knowledge, this approach has never been investigated in the field of network neuroscience.

The construction of a mapping between network nodes corresponds to what is known in
various fields as *network
alignment* or *graph matching* (Ayache & Faverjon, 1987; Barak, Chou, Lei, Schramm, & Sheng,
2019; Conte, Foggia, Sansone, &
Vento, 2004; Korula & Lattanzi,
2014; Singh, Xu, & Berger,
2008). More recently, the graph matching problem gained attention also in
the field of neuroscience, being used in the contexts of the analysis of connectome
heterogeneity across subjects (Rasero et al.,
2017; Takerkart, Auzias, Thirion,
& Ralaivola, 2014) and system-level matching of structural and
functional connectomes (Osmanlıoğlu
et al., 2019). Graph alignment solutions (called *alignments*) correspond to a permutation of the labels of the
nodes of a graph that maximizes its similarity to a second graph. There is no standard way to measure the
quality of its solutions (Bayati, Gleich, Saberi,
& Wang, 2013). This is also reflected in the neuroimaging
literature, where various measures of similarity between brain networks are used
(Becker et al., 2018; M. K. Chung, Lee, Solo, Davidson, & Pollak,
2017; Deslauriers-Gauthier, Zucchelli,
Frigo, & Deriche, 2020; Osmanlıoğlu et al., 2019; Rasero et al., 2017; Takerkart et al., 2014; Villareal-Haro, Ramirez-Manzanares, & Pichardo-Corpus, 2020). In
the context of connectomics, a graph alignment is a reordering of the labels of the
nodes of a brain network that maximizes its similarity with a second one while
preserving the topology. Describing the brain network through its connectivity (also
known as adjacency) matrix, permutations of the node labels correspond to identical
permutations of the rows and columns of the connectivity matrix. This problem is
distinct from the brain atlas correspondence and parcel matching problems (Gallardo, Gayraud, et al., 2018; Mars et al., 2016). The main difference is
that in those problems the permutation acts only on the rows of the connectivity
matrix as they find correspondences between *connectivity
fingerprints* that rely on external features. Conversely, graph
alignment does not rely on any external information and uses only information
contained in the topology of the graphs.

The complexity of finding the optimal alignment between two graphs using a näıve brute force strategy is exponential in the number of nodes. It is therefore intractable even for the smallest of brain networks, which typically have 50 cortical regions. Spectral methods are a popular approach to the alignment problem (Feizi et al., 2019; Hayhoe, Barreras, Hassani, & Preciado, 2019; Nassar, Veldt, Mohammadi, Grama, & Gleich, 2018), despite being subject to limitations (Wilson & Zhu, 2008). Modern machine learning paradigms exploit deep learning techniques for finding an alignment (Heimann, Shen, Safavi, & Koutra, 2018; Li et al., 2018; Liu, Cheung, Li, & Liao, 2016), however they make use of partially available information about the alignment itself (Liu et al., 2016), or lack explainability and interpretability.

We first introduce the *graph Jaccard index* (GJI), a natural
objective function for the network alignment problem. For a given alignment, the GJI
rewards correct matches while simultaneously penalizing mismatches, overcoming
limitations of previous approaches (Feizi et al.,
2019).

We then propose a new graph alignment heuristic, the *Weisfeiler-Leman
alignment* (WL-align), based on a weighted variant of the
Weisfeiler-Leman algorithm for graph isomorphism (Weisfeiler & Leman, 1968). WL-align is amenable to concrete
interpretability in terms of local network structure around each node (Figure 2) and can be integrated with other
heuristics. We compare WL-align against the fast approximate quadratic programming
for graph matching (FAQ) (Vogelstein et al.,
2015), another efficient brain-alignment heuristic that is solely based
on network structure.

## THEORY

A brain network is characterized as an edge-weighted graph *G* =
(*V*, *E*), where each of the *n* nodes represents a brain region and each weight *w*_{ij} encodes the strength of the
connection between regions *i* and *j*. The graph *G* can always be considered as complete, given that an edge
(*i*, *j*) ∉ *G* can be
associated with a null weight *w*_{ij} = 0.
The matrix that encodes in position (*i*, *j*) the
weight of the edge *w*_{ij} between nodes *i* and *j* is called adjacency matrix of *G* and is denoted as Adj(*G*). In the context of
connectomics (Hagmann, 2005; Sporns et al., 2005), the adjacency matrix is
also known as *connectivity* matrix. In this work we consider only
networks with nonnegative edge weights. For structural connectomes this does not
impose any special preprocessing, since they are usually constructed using
streamline count, length, or weights that are already nonnegative. However,
functional connectomes can contain negative entries because they are typically based
on the correlation of resting-state functional MRI signals. A practical solution,
already used in other studies (Deslauriers-Gauthier et al., 2020), is to threshold the connectomes,
therefore replacing negative entries by zeros.

### Brain Alignment

*G*

_{1}= (

*V*

_{1},

*E*

_{1}) and

*G*

_{2}= (

*V*

_{2},

*E*

_{2}) of

*n*

_{1}and

*n*

_{2}nodes respectively, it is possible to define an injective map

*m*:

*V*

_{1}→

*V*

_{2}(whose existence is granted whenever |

*V*

_{1}| ≤ |

*V*

_{2}|) that is called

*graph matching*or

*network alignment*. An edge (

*u*,

*v*) ∈

*E*

_{1}is correctly

*matched*by

*m*if (

*m*(

*u*),

*m*(

*v*)) ∈

*E*

_{2}and both edges have the same weight. Notice that a graph matching that matches all edges corresponds to an injective graph homomorphism. In the context of connectomics we will refer to

*m*also as a

*brain alignment*. A simple representation of this function is that of a matching matrix

*P*

_{m}of dimension

*n*

_{2}×

*n*

_{1}(with

*n*

_{2}≥

*n*

_{1}) defined as

*n*

_{1}=

*n*

_{2},

*P*

_{m}is a permutation matrix. If

*m*is an isomorphism between

*G*

_{1}and

*G*

_{2}, then the transformation between the adjacency matrices of the two graphs is fully characterized by the matching matrix and is given by

### Quality of Brain Alignments

*similarity between networks*used throughout this work fits well the standard concept of

*matrix similarity*in the particular case where the change of basis matrix is a permutation matrix. In the following, the similarity measures are defined for equal-sized networks, as typically encountered in connectomics. Classical metrics for this task are based on the comparison of the adjacency matrices of the two graphs by means of Pearson’s correlation coefficient,

*ℓ*

_{p}distance, or Frobenius distance (Vogelstein et al., 2015). The norm-based distances estimate the dissimilarity between two graphs

*G*

_{1}and

*G*

_{2}by computing the distance between their adjacency matrices as follows:

*t*indicates the type of norm (

*p*for

*ℓ*

^{p}norms and

*F*for Frobenius norm). Note that higher distance corresponds to lower similarity. Another similarity measure that has been widely adopted in neuroimaging and brain connectivity is correlation; among the many definitions of correlation, we consider

*cosine similarity*, since it corresponds to the cosine of the angle between the two vectors. Other distances based on geometrical (Venkatesh, Jaja, & Pessoa, 2020) and homological (M. K. Chung et al., 2017) properties of the networks have been proposed. All such measures capture some aspects of the similarity between two graphs, but none of them satisfies all the following requirements:

- ▪
arising as a natural generalization of other similarity measures for less structured data, for example, for sets of values without a network structure;

- ▪
being applicable to the algorithmic graph isomorphism and induced subgraph isomorphism problems, as fundamental special cases of the problem of measuring the similarity between two graphs;

- ▪
being simple enough so that its value can be easily interpreted;

- ▪
giving a straightforward notion of metric in the considered space.

We therefore propose a new measure obtained by generalizing the Jaccard similarity index, a similarity metric widely adopted in data mining, so that algorithmic problems such as induced subgraph isomorphism can be retrieved as special cases. Moreover, while our proposed measure assigns a clear meaning to the correspondence between two edges in two given graphs, it also depends on the global network structure.

#### Weighted graph Jaccard similarity index.

*Jaccard similarity index*was originally proposed in the context of set theory to measure the similarity between two sets

*A*and

*B*. It is computed as the ratio between the size of their intersection and the size of their union; that is,

An example of what is measured by the Jaccard index on sets is given in the
top panel of Figure 1. Notice that *J*(*A*, *B*) is defined in
the [0, 1] range and the extreme values are attained either when the
intersection of the two sets is empty (i.e., *A* ∩ *B* = ∅ ⇒ *J*(*A*, *B*) = 0) or when
the two sets are equal (i.e., *A* = *B* ⇒ *J*(*A*, *B*) = 1).
Both the sets need to be non-empty.

*x*,

*y*∈ ℝ

^{d}such that

*x*

_{i}≥ 0 and

*y*

_{i}≥ 0, their

*weighted Jaccard similarity index*can be computed as

*x*,

*y*are binary and their dimension

*d*is equal to the size of the union of the two sets.

*G*

_{1}and

*G*

_{2}with adjacency matrices Adj(

*G*

_{1}) =

*A*and Adj(

*G*

_{2}) =

*B*, the

*weighted graph Jaccard similarity index*(GJI) of

*G*

_{1}and

*G*

_{2}is

*B*as having been previously aligned to

*A*via Equation 2. Alternatively, the weighted graph Jaccard similarity index is defined as the weighted Jaccard index of the vectorizations of the graphs’ adjacency matrices. Notice that

*J*(

*G*

_{1},

*G*

_{2}) is not well defined when both

*G*

_{1}and

*G*

_{2}are empty (i.e.,

*E*

_{1}=

*E*

_{2}= ∅). Whenever Adj(

*G*

_{1}) = Adj(

*G*

_{2}), the min and the max in Equation 7 coincide and

*J*(

*G*

_{1},

*G*

_{2}) = 1. On the contrary, if

*G*

_{1}and

*G*

_{2}do not have any edge in common (i.e.,

*E*

_{1}∩

*E*

_{2}= ∅), the numerator of Equation 7 will be equal to 0 and

*J*(

*G*

_{1},

*G*

_{2}) = 0. A remarkable property of the weighted Jaccard similarity index is that it induces a metric in the space where it is defined. As a matter of fact, the function

*d*(

_{J}*x*,

*y*) = 1 −

*J*(

*x*,

*y*) ∈ [0, 1] respects the three properties of metrics:

Identity:

*d*_{J}(*x*,*y*) = 0 if and only if*x*=*y*;Symmetry:

*d*_{J}(*x*,*y*) =*d*_{J}(*y*,*x*);Triangle inequality:

*d*_{J}(*x*,*y*) ≤*d*_{J}(*x*,*z*) +*d*_{J}(*z*,*y*).

*J*, while the third follows as a particular case of what is presented in Charikar (2002, Lemma 1). An example of how the GJI acts on two graphs is given in the bottom panel of Figure 1.

We have so far formally established the notion of network alignment (Equation 1), and presented the graph Jaccard index as a principled way to measure the quality of an alignment (Equation 7). We are thus ready, in the next section, to describe our variant of the Weisfeiler-Leman heuristic and to show how to employ it to construct a network alignment.

### Weisfeiler-Leman Network Alignment

In this work we propose a brain-alignment technique that allows us to define the
graph matching *m* between two brain networks *G*_{1} and *G*_{2} with a
three-step procedure:

For each node

*u*in both graphs, define a vector*H*_{u}called*signature*.Define a

*complete bipartite graph*where on one side there are the nodes of the first graph and on the other side there are the nodes of the second graph; the Euclidean distance between two signatures becomes the weight of each edge of the bipartite graph.The graph matching is given by the solution of the

*minimum weight bipartite matching problem*, also known as*assignment problem*, on the bipartite graph previously defined.

The novelty element of this brain-alignment algorithm is given by the definition
of the node signature, which is determined with an algorithm inspired by the
Weisfeiler-Leman (WL) method for graph isomorphism testing (Weisfeiler & Leman, 1968). For
this reason, *WL-align* is the name we propose for our
brain-alignment algorithm.

#### Node signature.

*volume*of a node, which is defined as the sum of the weights of the edges incident to the node itself, namely

*V*is the set of nodes in the graph,

*v*is the node of which we compute the volume vol(

*v*), and

*w*

_{uv}is the weight of the edge connecting nodes

*u*and

*v*. The algorithm that defines the signature of node

*u*considers the subnetwork

*G*′ induced by the nodes that are reachable from

*u*in at most

*ℓ*hops. At each of these hops,

*G*′ retains only the

*k*nodes with highest contribution, weighted according to a function of the path that connects them to

*u*. In detail, such a contribution is computed via the following function

*w*(

*u*,

*v*) is just a more verbose notation for the edge weight

*w*

_{uv}. The subnetwork

*G*′ is a complete

*k*-ary tree of depth

*ℓ*that can be obtained from a breadth-first search (BFS) starting from

*u*, and has a total of

*d*= $\u2211i=0\u2113$

*k*

^{i}nodes. For this reason the parameters

*k*and

*ℓ*are respectively called

*width*and

*depth*. The entries of the signature

*H*

_{u}∈ ℝ

^{d}are then computed starting from

*u*and following the BFS by recursively estimating the contribution of each edge to the volume of the considered node via Equation 9. A formal description of the algorithm for computing the signature

*H*

_{u}is given in Algorithm 1, while a graphical intuition is illustrated in Figure 2.

**Algorithm 1**

Input: graph G; node u; width k; depth ℓ | |

Output: signature H _{u} | |

1: H_{u} ← empty list | ⊳ append right |

2: Q ← empty
queue | ⊳ FIFO data-structure; pop left (get and remove); append right |

3: Q ← append π = (u) | ⊳ π is the
zero-length, single-node path |

4: whileQ is not empty do | |

5: π =
(u, …, v_{h})
← pop path from Q | |

6: H_{u} ← append f(π) | |

7: ifh < ℓthen | ⊳ h is the length of π |

8: π_{z} ← (u, …, v_{h}, z), ∀z ∈ V | ⊳ if
(v_{h}, z) ∉ E then w(v_{h}, z) = 0 ⇒
f(π_{z})
= 0 |

9: z_{1},
…, z_{n} ← nodes s.t.
f(π_{z1})
≥ … ≥
f(π_{zn}) | ⊳ ties broken uniformly at random |

10: Q ← append the k paths π_{z1},…,π_{zk}. | |

11: end if | |

12: end while | |

13: returnH_{u} |

Input: graph G; node u; width k; depth ℓ | |

Output: signature H _{u} | |

1: H_{u} ← empty list | ⊳ append right |

2: Q ← empty
queue | ⊳ FIFO data-structure; pop left (get and remove); append right |

3: Q ← append π = (u) | ⊳ π is the
zero-length, single-node path |

4: whileQ is not empty do | |

5: π =
(u, …, v_{h})
← pop path from Q | |

6: H_{u} ← append f(π) | |

7: ifh < ℓthen | ⊳ h is the length of π |

8: π_{z} ← (u, …, v_{h}, z), ∀z ∈ V | ⊳ if
(v_{h}, z) ∉ E then w(v_{h}, z) = 0 ⇒
f(π_{z})
= 0 |

9: z_{1},
…, z_{n} ← nodes s.t.
f(π_{z1})
≥ … ≥
f(π_{zn}) | ⊳ ties broken uniformly at random |

10: Q ← append the k paths π_{z1},…,π_{zk}. | |

11: end if | |

12: end while | |

13: returnH_{u} |

#### Bipartite graph.

Once a signature is computed for each node of the two graphs, we define a
weighted complete bipartite graph *G*_{m} =
((*V*_{1} ∪ *V*_{2}), (*V*_{1} × *V*_{2})). The nodes on the left, that is, *V*_{1}, represent the nodes of the first graph,
while the nodes on the right, that is, *V*_{2},
represent the nodes of the second graph. The edge weights encode the
distance between the signatures of pairs of nodes belonging to different
graphs, that is, each edge (*u*, *v*) with *u* ∈ *V*_{1} and *v* ∈ *V*_{2} is weighted
according to a function *b* : *V*_{1} × *V*_{2} → ℝ defined as the
Euclidean distance between the signatures of the two end points *b*(*u*, *v*) =
∥*H*_{u} − *H*_{v}∥_{2}. Figure 3 shows a simple example of
the defined bipartite graph.

#### Assignment problem.

The final step towards finding the wanted matching with WL-align is the
resolution of the assignment problem corresponding to the bipartite graph *G*_{m} defined in the previous
paragraph. The matching can be found by selecting a minimum-weight graph
matching, namely a subset of edges of the bipartite graph such that every
node has degree 1 and the sum of the weights of all edges of the subset is
minimal. In formal terms, given the two sets *V*_{1} and *V*_{2} and the weighting function *b* that define *G*_{m}, the problem asks to
find a bijection *m* : *V*_{1} → *V*_{2}, that is, the matching, that
minimizes the function
∑_{u∈V1}*b*(*u*, *m*(*u*)). This assignment problem is
efficiently solved by the Hungarian algorithm (Jacobi, 1890; Kuhn,
1955).

## METHODS

We processed the data of 100 unrelated subjects from the Human Connectome Project (HCP) database and obtained the structural brain networks via dMRI-based tractography. For each of the 100 subjects we considered 23 parcellations (Desikan, Glasser, Gallardo at 11 different resolutions, Schaefer at 10 different resolutions), obtaining a total of 2,300 weighted graphs. For each parcellation, we retrieved a network alignment between each of the 5,050 pairs of subjects using WL-align, which is the novel technique introduced in this work, and the state-of-the-art competitor FAQ for a total of 232,300 alignments. The quality of the obtained alignments was then assessed using four network similarity measures.

### Data and Preprocessing

To build the structural brain networks, we considered the preprocessed data of the HCP database (U100 subject group; Glasser et al., 2013; Van Essen et al., 2012; WU-Minn Human Connectome Project Consortium, 2017). For each subject, a five-tissue-type image (Smith, Tournier, Calamante, & Connelly, 2012) was obtained using the FreeSurfer pipeline (Fischl, 2012) invoked through MRtrix3 (Tournier et al., 2019). A response function was estimated for the white matter, gray matter, and cerebrospinal fluid using a maximal spherial harmonic order of 8 for all tissues (Jeurissen, Tournier, Dhollander, Connelly, & Sijbers, 2014). The fiber orientation distribution functions (fODFs) were then computed using the multishell, multitissue constrained spherical deconvolution algorithm (Jeurissen et al., 2014). Finally, the fODFs were used as input for probabilistic anatomically constrained tractography performed with the iFOD2 algorithm (Smith et al., 2012) seeding from the gray matter–white matter interface and obtaining a total of five million streamlines per subject.

#### Parcellations.

The four parcellations considered in this work subdivide the cerebral cortex
following different characteristics of the brain. The Desikan (Desikan et al., 2006) parcellation is
based on the manual segmentation of a template of the brain cortex that
takes into account the morphological consistencies of healthy human brains.
For each subject, it was obtained directly from the HCP database
(*aparc+aseg.nii.gz*) together with the cortical surface
in fslr32k space. The Glasser parcellation (Glasser et al., 2016) follows a multimodal approach
that considers cortical architecture, function, connectivity, and
topography. Its projection onto the fslr32k space was obtained from the
BALSA repository (Van Essen et al.,
2017). The Gallardo parcellation (Gallardo, Wells, et al., 2018) is based on the
segmentation of the structural connectivity profiles associated with each
point of the cortical surface, and the Schaefer parcellation (Schaefer et al., 2017) is based on
the analysis of the coactivation patterns of the brain by means of the
analysis of resting-state functional connectivity. The Gallardo and the
Schaefer parcellations were computed with a granularity of 100, 200, 300,
400, 500, 600, 700, 800, 900, and 1,000 parcels. The Gallardo parcellation
was computed also with a granularity of 50 parcels. We extracted the 11
Gallardo atlases from the extrinsic connectivity parcellation of Gallardo et
al. (Gallardo, Wells, et al.,
2018). The used Schaefer atlas (Schaefer et al., 2017) was downloaded from the repository of the
CBIG laboratory (Yeo, 2020) for the *seven-network* parcellation (Yeo et al., 2011). The use of multiresolution
parcellations reflects the multiscale nature of the brain network and allows
us to inspect how the atlas resolution affects the similarity and the
alignment of brain networks.

#### Connectomes.

For each subject and parcellation, in-house software was used for counting the number of streamlines connecting each pair of regions. The obtained quantity was encoded as the weight of the edge connecting the two parcels in the brain network. All the edge weights were then divided by the sum of all the weights in the graph. A total of 23 connectomes of different sizes were obtained for each subject. Given the limitations of dMRI-based tractography, self-connections were excluded from the connectomes, that is, the diagonal of the adjacency matrix is set to 0. Because of the high resolution of some parcellations, some regions turned out to be isolated (i.e., not connected to any other region). In order to have a connected graph, which is a requirement of the WL-align algorithm, we artificially connected these isolated (i.e., zero-volume) nodes to the others by adding small-weighted edges connecting each of these nodes to all the other nodes in the graph. This weight was set to 1 (before normalization), which from the point of view of tractography is equivalent to the existence of one single streamline connecting the region to the others. The obtained graphs are undirected and weighted.

### Intracohort Variability

In order to assess the variability between the brain networks of the subjects in the studied cohort, for each subject we measured the similarity between the connectomes of each pair of subjects with three different similarity metrics: the weighted graph Jaccard index (Equation 7), the Frobenius norm (Equation 3) and the correlation (Equation 4).

### Network Alignments

In order to assess the ability of WL-align to retrieve the wanted alignment map,
we prepared the dataset in a way that allows us to test the quality of the
alignment against a known ground truth. In practice, for each parcellation *p*, we randomly permuted the node labels of the connectomes
of all subjects keeping track of the permutation maps. These permutation maps
allow us to compute the ground truth matching *m** between
each pair of brain networks computed with the same parcellation.

For the same set of brains, we also computed two graph matchings. The first is *m*_{WL}, which is computed with the
proposed WL-align technique. The width and depth parameters of the WL-align
algorithm were fixed to *k* = ⌊log_{2}*n*⌋, where *n* is the number of nodes in
the considered network (i.e., one hemisphere), and *ℓ* =
2. We limited the width for efficiency reasons (the size of the signature is
greater than *k*^{ℓ}, as
described in the previous section) and the depth since further increasing it
does not lead to substantial gain w.r.t. the quality of the alignments (the
deeper the nodes in the search, the smaller the contribution of the nodes to the
signature, as described in Equation
9).

The second is *m*_{FAQ}, which is
computed with the fast approximate quadratic programming for graph matching
(FAQ) algorithm (Vogelstein et al.,
2015), which is the state-of-the-art technique for network alignment.
FAQ works in three main steps: (a) arbitrarily choose a starting bi-stochastic matrix, which acts as a
relaxed permutation matrix that aligns the two networks; (b) find a local
solution to the relaxed quadratic assignment problem (rQAP), a dual version of
the graph matching problem; (c) project back onto the set of permutation
matrices. The solution found by FAQ transforms the adjacency matrix of the first
graph into one with approximately minimal Frobenius distance from the adjacency
matrix of the second graph. Notice that optimality with respect to the Frobenius
distance might not correspond to absolute optimality. We used the implementation
of FAQ available in the *graspologic* package (J. Chung et al., 2019; https://graspologic.readthedocs.io/), setting the number of
random initializations to 30.

Both WL-align and FAQ were run separately on each hemisphere of the brain, and the two resulting partial alignments were then combined into a single one. The motivation for this choice is that the correct hemisphere can always be assigned to a cortical region, and this property is independent from any influence potentially caused by the registration of the template atlas onto the subject-specific cortical mesh, while other properties, such as the location of a region, would be. Moreover, by studying single-hemisphere alignments we bypass the issue concerning the high degree of left-right similarity that characterizes the brain, which could drive the solution towards suboptimal alignments that are hardly distinguishable without external criteria such as the localization or geometry of the brain regions. Notice that this choice concerns the design of the experiment, not the setup of the graph matching algorithm, which could still be obtained using the full brain network, hence including the interhemispheric connections.

### Quality of Alignments

Given two networks *G*_{1} =
(*V*_{1}, *E*_{1}) and *G*_{2} = (*V*_{2}, *E*_{2}) defined on the same parcellation and given a
matching *m* between them, we consider the following metrics to
evaluate the quality of the matching *m*.

- ▪ Node matching ratio (NMr): The fraction of nodes that have been correctly matched by
*m*with respect to the ground truth matching*m** (known a priori), namelyThe NMr metric is defined in the [0, 1] range, and higher values correspond to better alignments.$NMrm=u\u2208V1:mu=m*uV1.$(10) - ▪ Graph Jaccard index
*J*: As defined in Equation 7, namelywhere, with an abuse of notation, we write$Jm=JmG1G2,$(11)*m*(*G*_{1}) to denote the relabeling of the nodes obtained by applying the matching*m*on the nodes of*G*_{1}. Recall that the graph Jaccard index is defined in the [0, 1] range, and higher values correspond to better alignment. - ▪
*J*ratio (*Jr*): The ratio between the graph Jaccard index*J*(*m*) obtained by*m*and the graph Jaccard index*J*(*m**) obtained by the ground truth matching*m**, namelyWhen the ground truth matching$Jrm=JmJm*.$(12)*m** is also an optimal matching, the denominator*J*(*m**) acts as a normalization factor, which takes into account how complex it is to retrieve the matching*m** in terms of Jaccard similarity; under such an assumption of ground truth optimality, the Jr metric takes values in the [0, 1] range, where higher values correspond to better alignment. - ▪ Frobenius norm (FRO): The Frobenius norm of the difference between the adjacency matrices of
*m*(*G*_{1}) and*G*_{2}, namelywhere, as was also done for$FROm=AdjmG1\u2212AdjG2F,$(13)*J*, we write*m*(*G*_{1}) to denote the relabeling of the nodes obtained by applying the matching*m*on the nodes of*G*_{1}. The FRO metric is defined in the [0, 2] range (since the adjacency matrices both have norm 1), and lower values correspond to better alignment.

*p*and for each network alignment algorithm of interest

*x*(either WL-align or FAQ), we report the average similarity metric, computed among all pairs of brains in the parcellation. For example, considering NMr as similarity metric, we compute

*p*and

*m*is the matching found by algorithm

*x*for the input pair of graphs

*G*

_{1},

*G*

_{2}. Analogously, this is done for all similarity metrics.

A further qualitative assessment of the accuracy of the alignments obtained with WL-align was performed by projecting the matching ratio of each node onto the cortical surface of a randomly picked subject, obtaining a visual indication of the localization of the regions that have been more or less frequently correctly matched. Projecting this information directly on the cortical surface provides insights into the spatial organization of the errors and of the correct matches.

### Statistical Analysis

In order to understand the differences between the alignments obtained with WL-align and FAQ, statistical analyses were performed with an alpha of 0.05 in all experiments. A separate analysis was performed for each of the four similarity metrics presented in the previous section. First, for each atlas and pair of subjects we computed an alignment with WL-align and FAQ. For each atlas, we compared the distributions of the values of the similarity metric computed on the alignments obtained with the two techniques using the nonparametric paired-samples Wilcoxon signed-rank test (Wilcoxon, 1945).

## RESULTS

### Experiments

We processed the data of 100 unrelated subjects from the HCP database, obtaining the structural brain networks as detailed in the Methods section. For each of the 100 subjects we considered 23 parcellations (Desikan, Glasser, Gallardo × 11, Schaefer × 10), obtaining 2,300 weighted graphs. For each parcellation, we retrieved a network alignment between each pair of subjects using WL-align and FAQ. The ability of WL-align to retrieve the correct brain-alignment map was quantitatively evaluated by means of four similarity measures. First, a novel measure of similarity between brain networks called graph Jaccard index was introduced in the Theory section as an adaptation of the concept of Jaccard index between sets. While behaving in a way that is similar to the commonly used correlation index defined in Equation 4, the graph Jaccard index has the property of defining a metric in the space of connectomes. This is a remarkable property in the context of modern data science, as many standard machine learning techniques can be applied only in metric spaces. The second considered similarity measure is the aforementioned correlation index defined in Equation 4, also known as cosine similarity. The third similarity measure is the Frobenius distance defined in Equation 3, which actually is a dissimilarity measure; therefore connectomes showing higher Frobenius distance are less similar and vice versa. The node matching ratio defined in Equation 10 is the last considered similarity measure.

### Comparison Between Similarity Measures

Each employed similarity metric answers a specific question. The node matching
ratio corresponds to what the expression suggests; namely, it counts how many
nodes were correctly matched and normalizes the result by the number of nodes in
the graph. The other similarity measures have less intuitive definitions. For
this reason, and in order to assess the intracohort similarity of the
connectomes, we measured how much the connectomes of the subjects in the
considered datasets are similar to each other with respect to each metric and
each parcellation. We recall that the dataset contains only healthy unrelated
subjects that do not exhibit any family structure (WU-Minn Human Connectome Project Consortium, 2017). This
allows us to compare how the within-group similarity reacts to the change in
resolution and type of the used parcellation. For each parcellation, Figure 4 shows how similar the subjects are
with respect to the graph Jaccard index, the Frobenius norm, and correlation. In
particular, the figure reports for each parcellation the average similarity
across all the pairs of subjects, which can be computed from the ground truth
matching that is granted by the fact that each network is defined on a known set
of nodes. Despite using the ground truth matching, the graphs are not expected
to exhibit perfect similarity (i.e., *J* = 1, FRO = 0, or *C* = 1), as their edge weights are subject-specific. This
specificity is what determines the intracohort variability that is taken into
account into the *J* ratio similarity metric defined in Equation 12. The most noticeable
fact is that the graph Jaccard index and the correlation show an inverted trend
with respect to the one of the Frobenius norm. A higher number of parcels give
both lower Jaccard/correlation index and lower Frobenius distance, which a
priori is counterintuitive. This phenomenon is attributable to the fact that the
Frobenius norm is incapable of capturing the relative difference between edge
weights and instead considers only the absolute difference between them. As a
matter of fact, parcellations with a higher number of parcels will create brain
networks with lower edge weights, since the same amount of connectivity (i.e.,
the same number of streamlines) is distributed among a number of edges that grow
quadratically with the number of regions. For this reason, the absolute value of
the edge weights will be lower, giving also a lower absolute difference. On the
contrary, the graph Jaccard index and the correlation, which are able to capture
the relative difference between edge weights, show lower similarity values
between brain networks obtained with a higher number of parcels compared with
brain networks obtained with a lower number of parcels. This difference suggests
that the graph Jaccard index and the correlation mitigate the influence of the
number of parcels in the estimation of the similarity between the compared brain
networks. Another observation can be done on the singular nature of the Desikan
and Glasser parcellations. When measured with the GJI and the correlation, both
of these parcellations exhibit an intracohort similarity in line with the one of
the Gallardo parcellation at the corresponding resolutions.

### Computing Brain Alignments With WL-Align

In this work, the concept of *similarity between networks* was
used as a proxy for the quality of a brain alignment, since a good graph
matching is expected to correspond to a higher similarity between the aligned
graph and the ground truth. A separate analysis was performed for each of the 23
considered parcellations. First, an alignment was computed between each pair of
subjects with the proposed technique WL-align and the state-of-the-art algorithm
FAQ, then the similarity between the aligned network and the ground truth
network was computed with the similarity measures listed in the Methods section. The node matching ratio
(NMr) tells the proportion of nodes that were correctly matched by the
alignment. This measure does not give any information about the topological
differences between the original and the aligned graph, but it gives an
important insight on how many nodes are correctly labeled, which may be of
fundamental importance in connectomic studies where the regions are associated
with a specific function of the brain. The second metric used is the Jaccard
similarity index introduced and described in this paper, while the third metric
employed is the Jaccard index ratio. The latter measures how the Jaccard index
performed with respect to the Jaccard index of the ground truth matching shown
in Figure 4, which is known a priori from
the design of the experiment. It differs from the raw Jaccard index in the sense
that it takes into account the complexity of the alignment problem, which we
showed in the previous section to be more difficult when the number of parcels
is higher. A final comparison was made using the Frobenius distance, which is
what the FAQ algorithm is designed to minimize. This makes it particularly
interesting since we expect FAQ to give Frobenius distance that is less or equal
to the one obtained with WL-align.

#### Subject-wise analysis.

In the context of this work, the simplest nontrivial alignment to be retrieved is the one between the brain network of a subject and its randomly permuted version. In this case, a good alignment algorithm is expected to always retrieve the ground truth alignment. In Figure 5 we report the average similarity between the ground truth and the obtained alignment. We notice that WL-align consistently achieves the best possible performance with respect to all the considered metrics. In particular, the naive metric of the node matching ratio always gives similarity equal to 1, meaning that WL-align correctly labels all the nodes whenever a structural brain network is aligned against a randomly permuted version of itself. These considerations are true for every parcellation. On the contrary, FAQ does not solve the self-alignment problem exactly. All the considered metrics highlight a poor performance of FAQ both in absolute terms and compared with WL-align. As a matter of fact, FAQ on average yields at most 40% of correctly matched nodes, while WL-align consistently gives 100% of correctly matched nodes. Also, different parcellations behave differently when FAQ is employed; for instance, the Desikan parcellation gives lower Frobenius similarity with respect to the other parcellations but shows higher Jaccard index and node matching ratio.

#### Full cohort analysis.

When all the subjects are aligned with the permuted version of each other,
the problem is more complicated. Even though we considered healthy subjects
whose acquisition followed the same protocol and that have been processed in
an identical way, the subject-specific differences and the intrinsic noise
of the data yield estimated structural brain networks that are in practice
different among each other, despite being substantially coherent. In order
to assess the ability of the proposed alignment technique to overcome these
differences and yield an alignment as close as possible to the ground truth,
we considered all the alignments between each pair of subjects, including
the ones between a subject and a randomly permuted version of itself. The
brain alignments obtained with WL-align are compared with the ones computed
with FAQ and presented in Figure 6,
which reports the average similarity between the obtained alignment and the
ground truth alignment among all the possible pairs of subjects. The
statistical significance of the differences between results obtained with
WL-align and FAQ is assessed using the nonparametric paired-samples Wilcoxon
signed-rank test (Wilcoxon, 1945).
For the studied cohort, statistically significant differences are observed
for each atlas and each employed similarity metric, as shown in section B of
the Supporting Information. In terms
of Frobenius norm, the alignments obtained with WL-align and FAQ are very
similar, with WL-align systematically showing slightly higher Frobenius
similarity. The performance of the Gallardo parcellation is
indistinguishable from that of the Schaefer parcellation. Also, the Glasser
parcellation is in line with the Schaefer and Gallardo parcellations when
the alignment is obtained with WL-align, while this is not true for the
Desikan parcellation. Recalling that FAQ is a technique that is inherently
based on the Frobenius norm and that WL-align is not, we can notice that
WL-align gives a brain alignment that also satisfies the optimality criteria
of FAQ, additionally to its own. A second thing that we can notice about the
Frobenius norm is that it exhibits the same phenomenon as in Figure 4, where the Frobenius similarity increases
with the number of parcels. This phenomenon appears for the same reason as
before; namely, the Frobenius norm does not capture the relative difference
between the edge weights in the compared networks. All the other employed
similarity metrics suggest that WL-align has superior performance with
respect to FAQ. While FAQ has almost identical performances when applied on
the Gallardo and the Schaefer parcellations, WL-align shows relevant and
previously unobserved differences between the performances of the two. In
particular the Gallardo parcellation allows retrieval of better alignments
with respect to the Schaefer parcellation. This may be because we are
studying structural connectivity, therefore the use of a function-based
parcellation like that of Schaefer may affect the quality of the alignment
when compared with the structural connectivity computed on a structure-based
parcellation like the one of Gallardo. Looking at the behavior of the
Desikan and the Glasser parcellation, we notice two different scenarios. The
Glasser parcellation shows Jaccard similarity slightly lower than the one of
the Gallardo parcellation but still higher than Schaefer’s,
suggesting that the multimodal nature of the atlas allows us to capture, at
least in part, the structural connectivity features that we are looking at.
This contrast is evident only when WL-align is employed. The Desikan
parcellation behaves differently. While exhibiting lower performance with
FAQ, when the WL-align is employed it emerges as a slightly superior
parcellation with respect to the NMr, the GJI, and the *J* ratio. We finally notice that atlases with more than 400 parcels all behave
very similarly; namely, they reach a plateau in terms of Jaccard index,
Jaccard index ratio, and node matching ratio. This is true both when
WL-align and when FAQ are employed. The performance in this range is lower
than the one in the 50–400 parcel range.

### Self-Matching Rate

Figure 7 illustrates the self-matching rate for each region of nine example atlases, that is, the fraction of times that regions were correctly matched when aligning different brains represented using the same atlas. It is clear that as the number of parcels is increased, the matching rate is reduced. This can be explained by the increased difficulty of the alignment problem, but also by a decrease in the signal-to-noise ratio of the connectomes driven by the reduction in parcel size. It is also interesting to note that the matching rate does not appear to be symmetric across hemispheres. For example, the right inferior parietal region of the Desikan atlas obtains relatively high matching rate of roughly 0.8, whereas the contralateral region only obtains roughly 0.4. This analysis gives important insights into the type of errors that are made by WL-align. In particular, it shows that the incorrect matchings do not have a particular structure that can be related to the geometry and morphology of the brain, be it some regional concentration of errors or some consistent symmetry with respect to the hemispheres.

## DISCUSSION AND CONCLUSIONS

Among the fundamental problems of network neuroscience at the scale of whole-brain structural connectivity, finding correspondences between brain regions and quantitatively assessing the similarity between brain networks are particularly important when it comes to considering massive heterogeneous datasets and modern data science techniques. In this work we considered these two problems in relation to the unresolved question concerning the choice of the parcellation for structural connectivity studies.

We proposed and analyzed a similarity index between brain networks, inspired by the Jaccard index between sets, that behaves in a similar way to the classical correlation index. Additionally, it enjoys the remarkable property of defining a metric in the space of connectomes, which is interesting both from the theoretical point of view and for data science applications. The proposed graph Jaccard index was shown to be less affected by the number of regions in the chosen parcellation than the Frobenius distance, which is one example of (dis)similarity index from the class of norm-based distances.

The second object introduced in this paper is WL-align, a novel algorithm that allows us to find the graph alignment between two brain networks. It relies solely on topological features of the brain network, which makes it particularly suitable for being applied also outside the domain of network neuroscience. When WL-align is used in our experiments in order to retrieve the alignment between a network and a permuted version of itself, it gives the exact solution. This does not happen when the main competitor FAQ is employed. The superior performance of WL-align is also evident when brain networks of different subjects are aligned. In this case, the WL-align algorithm was shown to retrieve brain alignments that are closer to the ground truth with respect to the alignments obtained by FAQ, and we showed that the difference between the WL-align and the FAQ alignments is statistically significant in the studied population. Notice that as it is designed, the WL-align algorithm builds on the construction of a feature vector for each node of the graph, which is then used as an edge weight in an assignment problem on a bipartite graph. This does not include any prior knowledge other than the topological similarity between the two networks to be aligned. The analysis provided in this work was intentionally constrained to the pure topological comparison of networks. Nevertheless, it would be possible to extend the feature vector defined in WL-align with any prior of geometrical, spatial, anatomical, or connectomic nature or to add any constraints in the assignment problem on the bipartite graph. Future work will be devoted to the design of these constraints and features.

The proposed WL-align algorithm can be further adapted to work with types of networks
other than the structural networks studied in this work, which are undirected and
have nonnegative edge weights. The most intuitive way to adapt WL-align is to change
the way in which the node signatures are defined, then set up the bipartite graph
and find the matching with the Hungarian algorithm in the canonical way. A first
interesting case is represented by weighted networks having both positive and
negative weights. This is the typical case of *functional* connectivity studies, where the connectivity between regions is evaluated as the
correlation (i.e., *w*_{ij} ∈
[−1, 1]) between the activation in different regions (Van Den Heuvel & Pol, 2010). As we defined it in this
work, WL-align would select the most relevant *d* nodes in an
unpredictable way because of the presence of negative-valued entries in Equation 9. A possible adaptation of it
would be to select the relevant edges performing the breadth-first search ignoring
the sign of the weights, then evaluating the corresponding entries of the WL
signature using the signed edge weights in Equation 9. Another possible adaptation would require the decomposition
of the adjacency matrix of the network as Adj(*A*) = *A*_{p} − *A*_{n}, where *A*_{p} is the positive part of the
matrix and *A*_{n} is the negative part of
the matrix. Notice that the graphs corresponding of both *A*_{p} and *A*_{n} will have nonnegative edge
weights. For each node, the WL signatures obtained from *A*_{p} and *A*_{n} can be concatenated, then
used in the canonical way. Another interesting case is represented by directed
networks, which in the context of brain imaging represent the concept of *effective* connectivity (Friston, 2011). Here, the only further adaptation that would be required
is a careful definition of the breadth-first search that gives the selection and the
order of nodes that are used for defining the WL signature. For directed
positive-weighted network, the algorithm works as it is, while for directed networks
with signed weights it would require the adaptations mentioned for the case of
functional connectivity. Finally, we discuss the adaptation of WL-align to temporal
networks. This type of graph has gained much interest in the context of brain
imaging since the concept of *dynamic functional connectome* (Preti, Bolton, & Van De Ville, 2017)
has been introduced and the consequent definition of specific tools for the
graph-theoretical analysis of these time-dependent networks (Sizemore & Bassett, 2018). In this case, at least two
options can be explored: First, one could concatenate the WL signatures of each node
obtained at each time point, then run WL-align in the canonical way. Alternatively,
it would be possible to perform the breadth-first search by taking into account the
temporal component, hence traversing the graph in both space and time.

An important instance of the graph matching problem that we did not consider in this work corresponds to when the two networks that are being aligned have different numbers of nodes. Being a generalization of a graph isomorphism test, WL-align does not appear to be trivially adaptable to this case. A possible solution would be to employ some dimensionality reduction technique (e.g., clustering via community detection) in the larger graph to reduce the number of nodes to the one of the smaller graph, then use WL-align to retrieve the wanted alignment.

Some remarkable conclusions concerning the parcellations to be used in structural
brain connectivity studies can be drawn from the ability of WL-align to find the
correct alignment between two brain networks. First, the function-based parcellation
of Schaefer is a poorer choice than the structure-based parcellation of Gallardo,
the multimodal parcellation of Glasser, and the morphological parcellation of
Desikan. This was expected from the fact that the whole study is centered on
measuring *structural* connectivity, hence the choice of a
function-based parcellation was never expected to be optimal from any point of view.
Allowing expression of this concept quantitatively is one of the merits of WL-align.
A second remarkable aspect is the performance of the Desikan atlas, which gave
better results in terms of alignability than any other parcellation of any
granularity. For this reason, whenever a study is designed using a coarse
parcellation of the cortex (in the 50–200 parcel range), one should consider
using the Desikan atlas as a first choice. Not only would it be a highly reliable
choice that has been consistently used throughout time in the community, but with
this study we showed that it would also allow definition of brain networks with more
consistent topological features, in particular those captured by WL-align. As far as
brain atlases with a higher number of parcels are concerned, we showed that
parcellations with a number of parcels in the range of more than 400 have lower
performance in terms of GJI and NMr. However, when the intersubject variability is
taken into account in the evaluation of the similarity, as for the case of the
Jaccard index ratio, we see that the performance is nearly constant for atlases with
more than 300 parcels.

The change in performance that we observe with the growing resolution of the atlas could also be due to the number of streamlines employed in the tractography pipeline, which could be adapted to the atlas used, but in practice is the same for every atlas at each resolution. On the other hand, the standardized tractography pipeline (including the identical number of streamlines in each tractogram) is what allowed us to present a comprehensive analysis and comparison of the performance across resolutions. In order to disambiguate this point, it would first be necessary to analyze how the strongest connections in the network (hence those considered by WL-align) are affected by the number of tracked streamlines. An alternative solution could be to employ a tractography filtering technique such as SIFT2 (Smith, Tournier, Calamante, & Connelly, 2015), COMMIT (Daducci, Dal Palù, Lemkaddem, & Thiran, 2014), or LiFE (Pestilli, Yeatman, Rokem, Kay, & Wandell, 2014) in order to mitigate the limited reliability of streamline count as a proxy of axonal connectivity (Jbabdi & Johansen-Berg, 2011). Given that tractography filtering techniques have nonnegligible effects on the topology of structural connectomes (Frigo et al., 2020), an independent analysis is due in order to assess how their use affects the alignability of connectomes.

Notice that in our analysis we used the defined similarity metrics to assess which
atlas yields connectomes with higher or lower robustness *in a certain
resolution range*. This means that we could not have used the similarity
argument to claim that, for instance, the Desikan atlas (68 parcels) should in
general be preferred to the Gallardo 1000 atlas. In this sense, we highlight how the
considered similarity metrics (GJI, Jr, NMr, and Fro) should not be used for
selecting the appropriate resolution at which structural connectivity studies should
be designed, but they provide a well-grounded tool for assessing which of the
available atlases at the wanted resolution is most suitable for the considered type
of study.

As highlighted throughout the paper, this work analyzes the problems of parcellation
selection and brain alignment in the context of *structural* connectivity. Any conclusion we made should not be straightforwardly generalized to
functional connectivity or effective connectivity studies, which would require a
separate analysis that was out of the scope of this work.

## ACKNOWLEDGMENTS

The authors would like to thank Dr. Guillermo Gallardo for help in computing the Gallardo parcellation and Professor Joshua T. Vogelstein for the discussion on the use of FAQ. Also, we are grateful to the OPAL infrastructure from Université Côte d’Azur and Inria Sophia Antipolis–Méditerranée “NEF” computation platform for providing resources and support. Data were provided in part by the Human Connectome Project, WU-Minn Consortium (principal investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.

## DATA AVAILABILITY

The data and code used in this work are all available at open repositories, as indicated in the text. We uploaded the code used and the obtained connectomes and alignments on the Open Science Framework. They can be found at this link: https://osf.io/depux/ (Frigo & Cruciani, 2020).

## SUPPORTING INFORMATION

Supporting information for this article is available at https://doi.org/10.1162/netn_a_00199.

## AUTHOR CONTRIBUTIONS

Matteo Frigo: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Software; Visualization; Writing – original draft; Writing – review & editing. Emilio Cruciani: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Software; Visualization; Writing – original draft; Writing – review & editing. David Coudert: Methodology; Supervision; Writing – review & editing. Rachid Deriche: Funding acquisition; Methodology; Supervision; Writing – review & editing. Emanuele Natale: Conceptualization; Formal analysis; Investigation; Methodology; Project administration; Software; Writing – original draft; Writing – review & editing. Samuel Deslauriers-Gauthier: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Project administration; Software; Visualization; Writing – original draft; Writing – review & editing.

## FUNDING INFORMATION

Rachid Deriche, Matteo Frigo, and Samuel Deslauriers-Gauthier, H2020 European Research Council (https://dx.doi.org/10.13039/100010663), Award ID: 694665.

## TECHNICAL TERMS

- Parcellation:
Subdivision of the brain into distinct regions with respect to morphological, cytoarchitectonic, anatomical, topological, or functional criteria.

- Tractography:
Method for tracking the trajectory of the axonal pathways that exploits the anisotropy of the diffusion MRI signal.

- Connectome:
The network that encodes the connections of the human brain; it refers to a comprehensive description of the brain’s structural and/or functional connections.

- Functional connectome:
Network-like description of the coherence between the activity in different brain regions; it can be obtained by studying co-activation patterns in functional MRI, EEG, or MEG.

- Structural connectome:
Network that describes the structure of the white matter connections in the brain; it can be obtained via diffusion MRI-based tractography.

- Network alignment:
Function that maps nodes of a graph onto nodes of another graph, while usually trying to preserve adjacency (end points of edges in a graph should map onto end points of edges of the other graph).

- Graph similarity:
Measure of how much two networks are close with respect to some criteria of topological nature.

- Signature:
Feature vector assigned to each node of a graph.

- Bipartite graph:
Network whose nodes can be divided in two distinct and nonoverlapping sets, such that there are no edges connecting nodes in the same set.

- Breadth-first search (BFS):
Graph traversal that, starting from a root node, explores all its neighbors before moving to the neighbor’s neighbors, and so on.

- Bi-stochastic matrix:
Matrix of positive entries where each column and row sums to 1.

## REFERENCES

## External Supplement

## Author notes

These authors contributed equally to this work.

Competing Interests: The authors have declared that no competing interests exist.

Handling Editor: Alex Fornito