Graphlets and motifs

Network motif analysis is concerned with the discovery of statistically significant classes of subgraphs in empirically recorded graphs. We here restrict ourselves to directed unweighted graphs, but the concepts apply similarly to undirected graphs and may be extended to weighted graphs [41, 42], time-evolving and multilayer graphs [43–46], and hypergraphs [47, 48]. As is usual in motif analysis, we consider weakly connected subgraphs [16, 25]. This ensures that the subgraph may represent a functional subunit where all nodes can participate in information processing.

Let denote the directed graph we want to analyze. For simplicity in comparing different representations of G, we consider G to be node-labeled. Thus, the nodes constitute an ordered set. The set of edges, indicates how the nodes are connected. By convention, a link indicates that i connects to j. Note that, since G is directed, the presence of does not imply the existence of . We denote by the number of edges. We only consider network data that form simple directed graphs, where does not contain repeated elements: this is the definition of a set. The model we propose, however, makes use of multigraphs where is a multi-set, which may contain repetitions.

A standard representation of a graph’s connectivity is its adjacency matrix—denoted A—with entries given by . If the graph is simple, then the adjacency matrix is boolean, i.e., A_ij = 1 if , otherwise A_ij = 0. When dealing with a multigraph, entries of the adjacency matrix take non-negative integer values, i.e., A_ij ≥ 1 if .

An induced subgraph g = (ν, ϵ) of G is the graph formed by a given subset of the nodes of G and all the edges connecting these nodes in G.

An undirected graph is called connected if there exists a path between all pairs of nodes in . A directed graph G is weakly connected if the undirected graph obtained by replacing all the directed edges in G with undirected ones is connected.

Two graphs g = (ν, ϵ) and g′ = (ν′, ϵ′) are isomorphic if there exists a permutation σ of the node indices of g′, such that the edges in the graphs perfectly overlap, i.e., (i, j) ∈ ϵ if and only if (σ(i), σ(j)) ∈ ϵ′. A graphlet, denoted by α, is an isomorphism class of weakly connected, induced subgraphs [49], i.e., the set α = {g : g ≅ g_α} of all graphs that are isomorphic to a given graph, g_α.

Finally, a motif is a graphlet that is statistically significant. Traditionally, a significant graphlet is defined as one whose number of occurrences in G is significantly higher than in random graphs generated by a null model [16]. Instead, we propose a method that selects a set of graphlets based on how well they allow to compress G. This lets us treat motif mining as a model selection problem through the MDL principle as we detail below.

Subgraph census

The first step of a motif inference procedure is to perform a subgraph census, consisting in counting the graphlet occurrences. Subgraph census is computationally hard and many methods have been developed to tackle it [50].

For graphs with a small number of nodes, e.g., hundreds of nodes, we implemented the parallelized FaSe algorithm [51], while for larger graphs, i.e., comprising a thousand nodes or more, we rely on its stochastic version, Rand-FaSe [52]. The algorithms use Wernicke’s ESU method (or Rand-ESU for large graphs) [53] for counting graphlet occurrences. It employs a trie data structure, termed g-trie [54], to store the graphlet labels in order to minimize the number of computationally costly subgraph isomorphism checks.

Since our algorithm relies on contracting individual subgraphs, we also need to store the location of each subgraph in G. Due to the large number of subgraphs, the space required to store this information may exceed working memory for larger graphs or graphlets (see discussion in S2 Text). Our most computationally challenging application—inference of motifs amongst all 3- to 5-node graphlets in the right mushroom body of the adult Drosophila connectome—requires storing 1.3 TB of data. In such cases, we write heavy textfiles of subgraph lists, one per graphlet, on the computer static memory, which are then retrieved individually from disk, at inference time (see S3 Text).

All scripts were run on the HPC cluster of the Institut Pasteur, but the less computationally challenging problem of inferring 3- to 4-node motifs can be run on a local workstation (see S2 Text).

Compression, model selection, and hypothesis testing

The massive number of possible graphlet combinations and the correlations between graphlet counts within a network make classic hypothesis testing-based approaches for motif mining ill-suited for discovering motif sets. For example, there are approximately 10 000 different five-node graphlets and exponentially more possible combinations of such graphlets, making multiplicity a critical problem for hypothesis testing. Additionally, these approaches define motif significance by comparison with a random graph null model, and the results may depend on the choice of null model [27, 29] (see “Numerical validation” in the results below). In the context of motif mining, this choice can lead to ambiguities [27, 29, 30], thus rendering the analysis unreliable.

To address these problems, we cast motif mining as a model selection problem. We wish to select as motifs the multiset of graphlets, that, together with a tractable dyadic graph model, provides the most adequate model for G. The minimum description length (MDL) principle [35] states that, within an inductive inference framework with finite data, the most faithful representation of the observed system is given by the model that leads to the highest compression of the data—that is, of minimum codelength. It relies on an equivalence between codelengths and probabilities [33] and formalizes the well-known Occam’s razor, or principle of parsimony. It is similar to Bayesian model selection and can be seen as a generalization of it [36].

To each dataset, model and parameter values, we associate a unique code, i.e., a label that identifies one representation. The code should be lossless, which means full reconstruction of the data from the compressed representation is possible [33, 35]. In practice, we are not interested in finding an actual code, but only in calculating the codelength of an optimal code [33], corresponding to our model.

Suppose we know the generative probability distribution of G, P_θ, parameterized by θ. Then, we can encode G using a code whose length is equal to the negative log-likelihood [35], (1) where log denotes the base-2 logarithm. (Note that an actual code would be between 1 to 2 bits longer than Eq (1) since real codewords are integer-valued and not continuous [35]). When the correct model and its parameters are unknown beforehand, we must encode both the model and the graph. To do this, we consider two-part codes, and, more generally, multi-part codes (see below). In a two-part code, we first encode the model and its parameters, using L(θ) bits, and then encode the data, G, conditioned on this model, using −log P_θ(G) bits. This results in a total codelength of (2)

With multi-part codes, we encode a hierarchical model following the same schema, where we first encode the model, then encode latent variables conditioned on the model, and then encode the data conditioned on the latent variables and the model.

When performing model selection, we consider a predefined set of models, , and we look for the one that, in an information-theoretic sense, best describes G. Following the MDL principle we select the parametrization θ* ∈ Θ that minimizes the description length, (3)

Note that the second term in Eq (2), L(θ), quantifies the model complexity, which measures, in bits, the volume for storing the model parameters—this is a lossless encoding. Thus, one must strike a balance between model likelihood and model complexity to minimize the description length, inherently penalizing overfitting.

While we focus on model selection, we also provide the absolute compression of the optimal model as an indicator of statistical significance. The link between compression and statistical significance is based on the no-hypercompression inequality [35]. It states that the probability that a given model, different from the true generating model, compresses the data more than the true model is exponentially small in the codelength difference. Formally, given a dataset G (e.g., a graph) drawn from the distribution P₀ and another description P_θ, then (4)

By identifying P₀ with a null model and P_θ with an alternative model, the no-hypercompression inequality thus provides an upper bound on the p-value, i.e., p ≤ 2^−K. Note, however, that the above relation is not guaranteed to be conservative for composite null models (such as the configuration models that we consider below) [36, 55].

Graph compression based on subgraph contractions

In practice, we compress the input graph by iteratively performing subgraph contractions each chosen from a set of possible graphlets, extending the approach of Bloem and de Rooij [34] which focused on a single graphlet. The model describes G by a reduced representation, a multigraph H, with N(H) < N(G) and E(H) < E(G), in which a subset of nodes are marked as supernodes, each formed by contracting a subgraph of G into a single node (Fig 1A).

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Fig 1. Graphlet-based graph compression.

(A) Reduced representation of a graph G obtained by contracting subgraphs into colored supernodes representing the subgraphs. (In this example, two different graphlets, colored in blue and green, are selected) The cost for encoding the reduced representation can be split into two parts: (i) encoding the multigraph H obtained by contracting subgraphs in G, L(H, ϕ) (See “Base codes and null models” section), and (ii) encoding which nodes in H are supernodes and their color, designating which graphlet they represent, [Eq (7)]. (B) Hierarchy of the four different dyadic graph models [56] used as base codes. Each node in the diagram represents a model. An edge between two nodes indicates that the upper model is less random than the lower. The models are: the Erdős-Rényi model P_(N,E) (cyan); the directed configuration model (orange); the reciprocal Erdős-Rényi model (pink); and the reciprocal configuration model (yellow). (C-E) Encoding the additional information necessary for lossless reconstruction of G from H, incurs a cost (Eq (8)) that is equal to the sum of three terms for each supernode, corresponding to encoding the labels of the nodes inside the graphlet, i.e., the graphlet’s orientation (C), and how the graphlet’s nodes are wired to other nodes in H (D,E). (C) Encoding the orientation of a graphlet is equivalent to specifying its automorphism class. For the graphlet shown in the example there are 3 possible distinguishable orientations, leading to a codelength of log 3. (D) Encoding the connections between a simple node and a supernode involves designating to which nodes in the graphlet the in- and out-going edges to the supernode are connected. In this example, there are possible wiring configurations for both the in- and out-going edges, leading to a wiring cost of log 36 (see Eq (9)). (E) Encoding the wiring configuration of the edges from a supernode i to another supernode j involves designating the edges from the group of nodes of supernode i to the group of nodes in j in the bipartite graph composed of the two groups (the edges from j to i are accounted for in the encoding of j). Here, there are such configurations, leading to a rewiring cost of log20 bits.

https://doi.org/10.1371/journal.pcbi.1012460.g001

We let Γ designate a predefined set of graphlets, which is the set of all graphlets we are interested in. In the following, we will generally consider all graphlets from three to five nodes—in which case |Γ| = 9579—but any predefined set of graphlets, or even a single graphlet, may be used. We define as a multiset of graphlets, corresponding to the subgraphs in G that we contracted to obtain H. We define as the set containing the unique elements of and let be the number of repetitions of α in . We finally let P_ϕ designate a dyadic random graph model, which is used to encode H. We consider four possible such base models (see Fig 1B and “Base codes and null models” below).

The full set of parameters and latent variables of our model is , and its codelength can be decomposed into four terms, (5) where (i) is the codelength for encoding the motif set; (ii) L(H, ϕ) is the codelength needed to encode the reduced multigraph H using a base code corresponding to P_ϕ; (iii) accounts for encoding which nodes of H are supernodes and to which graphlets they correspond (i.e., their colors, Fig 1A); (iv) corresponds to the information needed to reconstruct G from H (node labels, the orientations of the contracted subgraphs, and how the subgraph’s nodes are wired to their respective external neighborhoods, see Fig 1C and 1D). We detail each of the four terms in turn.

The first term in Eq (5), is given by (6) where is the maximal number of repetitions of any of the graphlets in , and is the codelength needed to encode an integer [35]. The first term in Eq (6) is the codelength needed to encode the identity of each inferred motif. Since there are |Γ| possible graphlets, this requires log |Γ| bits per motif. The second term is the cost of encoding the number |Γ|. The third term is the cost of encoding the number of times each of the motifs appears, requiring log m_max bits per motif. The fourth term is the cost of encoding m_max.

The second term in Eq (5), L(H, ϕ), depends on the base model used to encode H. We consider several models and detail their codelengths in the “Base codes and null models” section below.

The third term of Eq (5) is equal to (7) where the first part corresponds to the cost of labeling nodes of H as supernodes—equal to the logarithm of the number of ways to distribute the labels—and the second part corresponds to the labeling of the supernodes to show which graphlet they each correspond to—equal to the logarithm of the number of distinguishable ways to order .

The fourth and last term in Eq (5) is given by (8) Here, the first term is the cost of recovering the original node labeling of G from H. The second term encodes the orientation of each graphlet to recover the subgraphs found in G (Fig 1C)—for a given graphlet α (consisting of n_α nodes) there are n_α!/|Aut(α)| distinguishable orientations, where |Aut(α)| denotes the size of the automorphism group of α. The third term is the rewiring cost which accounts for encoding how edges in H involving a supernode are connected to the nodes of the corresponding graphlet. Denoting by n_s the number of nodes in the subgraph s that the supernode i_s replaces, the rewiring cost for one supernode is given by (9) where the first term is the cost for designating which of the possible wiring configurations involving the nodes inside a supernode and adjacent regular nodes corresponds to the configuration found in G (Fig 1D), and the second term is the cost of encoding the wiring configurations for edges from the nodes of the given supernode to the nodes of its adjacent supernodes (Fig 1E).

Base codes and null models

The latent graph code.

To encode the latent reduced graph H, we use two-part codes of the form L(H, ϕ) = −log P_ϕ(H) + L(ϕ) (Eq (2)), where L(ϕ) encodes the parameters of the chosen dyadic random graph model—the model’s parametric codelength—and P_ϕ(H) is a uniform probability distribution over a multigraph ensemble conditioned on the value of ϕ. Note that, while G is a simple graph, the subgraph contractions may generate multiple edges between the same nodes in H, which consequently is a multigraph. The models P_ϕ correspond to maximum entropy microcanonical graph ensembles [56–58], i.e., uniform distributions over graphs with certain structural properties ϕ(H), e.g., the node degrees, set to match exactly a given value, ϕ(H) = ϕ*. The microcanonical distribution is given by (10) where the normalizing constant Ω_ϕ = |{H: ϕ(H) = ϕ*}| is known as the microcanonical partition function. The codelength for encoding H using the model P_ϕ can be identified with the microcanonical entropy, (11) leading to a total codelength for encoding the model and the reduced graph of (12)

As base codes we consider four different paradigmatic random graph models, namely the Erdős-Rényi (ER) model, the configuration model (CM), and their reciprocal versions (RER and RCM, respectively). For the ER model, the parameters are the number of nodes and edges, while the configuration model constrains the nodes’ in- and out-degrees and their reciprocal versions additionally constrain the number of reciprocated edges. Both the degree distributions and the edge reciprocity have been found to be significantly non-random in biological networks, and they have been shown to influence the networks’ topology and function [8–11, 26, 40, 59–62]. Thus, it is natural to include these features in the base models, and the corresponding models have been widely employed as null models for hypothesis testing-based motif inference [2, 3, 16, 17, 21–23, 25].

Microcanonical models are defined by the features of a graph that they keep fixed [56] (see Eq (10)). We list them for each of the four models below and we give in Table 1 expressions for their entropy S_ϕ and their parametric codelength L(ϕ) (see Section A in S4 Text for details).

• The Erdős-Rényi model (ER) fixes the number of nodes and edges, ϕ = (N, E).
• The configuration model (CM) fixes the nodes’ in- and out-degrees (the number of incoming and outgoing edges), ϕ = (k⁺, k⁻).
• The reciprocal Erdős-Rényi model (RER) fixes the number of nodes, the number of reciprocal edges, and the number of non-reciprocated edges, ϕ = (N, E_m, E_d). Formally, for a simple graph, a (non-)reciprocated edge is conveyed by (a)symmetric entries of the adjacency matrix. That is, an edge (i, j) is reciprocal if A_ij = 1 and A_ji = 1 and non-reciprocated if A_ij = 1 and A_ji = 0. The definition for multigraphs extends this idea to integer counts by defining the reciprocal part of a multiedge as the minimum of A_ij and A_ji and the non-reciprocated part as the rest [63] (details can be found in Section A in S4 Text).
• The reciprocal configuration model (RCM) fixes the nodes’ reciprocal degrees—the number of reciprocal edges each node partakes in—as well as the non-reciprocated in- and out-degrees, .

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Table 1. Base- and null-model codelenghts.

The codelength of a model is equal to L(H, ϕ) = S_ϕ + L(ϕ) (Eq (12)), with the entropy S_ϕ and the model complexity L(ϕ) given by the appropriate expressions in the table. The entropy of multigraph models are given in the first four lines and the entropy of the simple graph models are given in the next four. The parametric complexity of the models is the same for multi- and simple graphs and are listed in the following four lines. Finally, expressions for common parametric codelengths are given in the last four lines. For multigraph codes, the asymmetric and symmetric parts of the adjency matrix are denoted by and , respectively. For reciprocal models (RER and RCM), is the number of non-reciprocated edges and is the number of reciprocated edges. For the configuration model (CM), denotes the out-degrees and the in-degrees. For the reciprocal CM (RCM), and are the non-reciprocated out- and in-degrees, and are the reciprocal degrees. (Details can be found in S4 Text).

https://doi.org/10.1371/journal.pcbi.1012460.t001

The different base models respect a partial order in terms of how random they are, i.e., how large their entropy is (Fig 1B) [56]. We stress that the model with the smallest entropy does not necessarily provide the shortest description of a graph H due to its higher model complexity (see Section A in S4 Text).

Optimization algorithm

To infer a motif set, we apply a greedy iterative algorithm that contracts the most compressing subgraph in each iteration. Since the number of n-node subgraphs grows super-exponentially in n, it is not convenient to consider all subgraphs at once. Thus, we developed a stochastic algorithm that randomly samples a mini-batch of subgraphs in each iteration and contracts the one that compresses the most among these (Fig 2). We give in Algorithms 1–4 pseudocode for its implementation and describe below each of the main steps involved.

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Fig 2. Greedy optimization algorithm.

(A) Illustration of a single step of the greedy stochastic algorithm. The putative compression ΔL(G, θ, s) that would be obtained by contracting each of the subgraphs in the minibatch is calculated, and the subgraph contraction resulting in the highest compression is selected (highlighted in blue). (B) Example of motif set inferred in the connectome of the right hemisphere of the mushroom bodies (MB right) of the Drosophila larva. (C) Evolution of the codelength during a single algorithm run. The algorithm is continued until no more subgraphs can be contracted. The representation θ* = θ_t with the shortest codelength is selected; here, after the 31st iteration (indicated by a vertical black dashed line). The horizontal orange dashed line indicates the codelength of the corresponding simple graph model without motifs (see Motif-free reference codes). (D) The algorithm is run a hundred times for each dyadic base model and the most compressing model is selected. Histograms represent the codelengths of models with motifs after each run of the greedy algorithm; colors correspond to the different base models (blue: ER model, orange: configuration model, pink: reciprocal ER model, yellow: reciprocal configuration model, see Fig 1B and Table 1); vertical dashed lines represent the codelengths of models without motifs, and the black dashed line indicates the codelength of the shortest-codelength model—here the configuration model with motifs.

https://doi.org/10.1371/journal.pcbi.1012460.g002

Algorithm 1 Greedy motif inference

Input: Graph G, graphlet set Γ, base model P_ϕ, subgraph minibatch size B

1: t ← 0

2: H₀ ← G

3:

4:

5: Θ ← {θ₀}

6:

7: while is not ∅ do

8:  t ← t + 1

9:  

10:  

11:  

12:  

13:  Θ ← Θ∪{θ_t}

14: end while

Output: argmin_θ∈Θ{L(G,θ)}

Algorithm 2 Sample subgraph batches

1: function SubgraphBatches

2:  

3:  for α ∈ Γ do

4:   

5:   while and |S_α| > 0 do

6:    

7:    if NonOverlappingSubgraph(H,s_α) then

8:     

9:    else

10:    end if

11:   end while

12:   

13:   

14:  end for

15:  return

16: end function

1: function SampleGraphletInstance

2:  return s_α, a subgraph sampled uniformly from

3: end function

1: function NonOverlappingSubgraph(H,s_α)

2:  b ← True

3:  for i ∈ s_α do

4:   if then

5:    b ← False       ▷ Delete node labels of already contracted subgraphs

6:   end if

7:  end for

8:  return b

9: end function

Algorithm 3 Find most compressing subgraph.

1: function MostCompressingSubgraph(G,)

2:      ▷see Section C in S4 Text.

3:  Let α ∈ Γ be the graphlet such that g_α ≅ s*.

4:  return α, s*

5: end function

Algorithm 4 Subgraph contraction

1: function SubgraphContraction

2:  for do

3:   A_ij(H)←0

4:  end for

5:  

6:  Let i_α be the label of a new supernode

7:  

8:  

9:  

10:  for l ∈ ∂s_α do

11:   

12:   for do

13:    

14:   end for

15:  end for

16:  return

17: end function

Datasets

Empirical datasets.

We apply our method to infer microcircuit motifs in synapse-resolution neural connectomes of different small animals obtained from serial electron microscopy (SEM) imaging (see Table 2 for descriptions and references of the datasets). All input raw and processed connectomes can be found in our GitLab project, in the data folder(gitlab.pasteur.fr/sincobe/brain-motifs/-/tree/master/data).

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Table 2. Connectome datasets.

For each connectome, we list its number of non-isolated nodes, N, its number of directed edges, E, its density ρ = E/[N(N − 1)], the features of the most compressing model for the connectome, its compressibility ΔL*, the difference in codelengths between the best models with and without motifs, ΔL_motifs, and the reference to the original publication of the dataset. The absolute compressibility ΔL* measures the number of bits that the shortest-codelength model compresses compared to a simple Erdős-Rényi model (Eq (15)). The difference in compression with and without motifs, ΔL_motifs, quantifies the significance of the inferred motif sets as the number of bits gained by the motif-based encoding compared to the optimal motif-free, dyadic model. For datasets where no motifs are found, this column is marked as “N/A”. All datasets are available at https://gitlab.pasteur.fr/sincobe/brain-motifs/-/tree/master/data.

https://doi.org/10.1371/journal.pcbi.1012460.t002

Numerical validation

To test the validity and performance of our motif inference procedure, we apply it to numerically generated networks with a known absence or presence of higher-order structure in the form of motifs (see “Artificial datasets”in Methods).

Null networks.

We first test the stringency of our inference method and compare it to classic, hypothesis testing-based approaches. We test whether they infer spurious motifs in random networks generated by the four dyadic random graph models (See “Randomized networks” in the Methods). Since these random networks do not have any non-random higher-order structure, a trustworthy inference procedure should find no, or at least very few, significant motifs.

Hypothesis testing-based approaches to motif inference consist of checking whether each graphlet is significantly over-represented with respect to a predefined null model (we detail the procedure in S1 Text). This approach is highly sensitive to the choice of null model and infers spurious motifs if the chosen null model does not correspond to the generative model (Fig 3A–3D). Nevertheless, when the chosen null model is the generative model, almost no spurious motifs are found using the approach (Fig 3A–3D). However, since there is no general protocol for the choice of null model in the frequentist approach, this sensitivity to null model choice is a major concern in practice.

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Fig 3. Performance of compression-based motif inference on numerically generated networks.

(A-D) Number of spurious motifs inferred using our compression-based method with MDL-based model selection and using hypothesis testing with four different null models in random networks generated from the same four null models: (A) the Erdős-Rényi model (ER); (B) the configuration model (CM); (C) the reciprocal ER model (RER); and (D) the reciprocal CM (RCM). The x-axis labels indicate which method was used for motif inference: our method (MDL) or classic hypothesis testing with each of the four null models as reference. The corresponding generative model is highlighted in boldface. To make hypothesis testing as conservative as possible, we applied a Bonferroni correction, which multiplies the raw p-values by |Γ| = 9576 and we set the uncorrected significance threshold to 0.01. The random networks in (A-D) are all generated by fixing the values of each null model’s parameters to those of the Drosophila larva right MB connectome (e.g., N = 198 and E = 6499 for the ER model). (E-H) Ability of our method to correctly identify a placed graphlet as a motif as a function of the number of times it is repeated, m_α. We show results for two selected 5-node graphlets: an hourglass structure (top row) and a clique (bottom row). The clique is the densest graphlet and is totally symmetric (the number of orientations, i.e., the number of non-automorphic node permutations, is equal to one). The hourglass has intermediary density, ρ_α = 2/5, and symmetry, with 60 non-automorphic orientations within a possible range of 1 to 5! = 120. The generated networks in (E-H) contain N = 300 nodes and an edge density of either ρ = E/N(N − 1) = 0.025 (E,G) or ρ = 0.1 (F,H). Each point is an average over five independently generated graphs. (E,F) The discovery rate is the estimated probability that the planted motif belongs to the inferred motif set, i.e., 〈1 − δ(m_α, 0)〉. (G,H) Average inferred number of repetitions of the planted motif, 〈m_α〉.

https://doi.org/10.1371/journal.pcbi.1012460.g003

By casting motif inference as a model selection problem, our approach allows us to select the most appropriate model, including amongst a selection of null models. In our test, our approach consistently selects the true generative model for the networks, i.e., one of the four null models, and thus does not infer any spurious motifs (Fig 3A–3D).

Neural connectomes

We apply our method to infer circuit motifs in structural connectomes and characterize the regularity of the connectivity of synapse-resolution brain networks of different species at different developmental stages (see Table 2). We consider boolean connectivity matrices that represent neural wiring as a binary, directed network where each node represents a neuron and an edge represents the presence of synaptic connections from one pre-synaptic neuron to a post-synaptic neuron. To keep in line with the usual definition of a motif, we exclude self-connections of neurons onto themselves, but they can be included if one wants to investigate such motifs.

We measure the compressibility of a connectome G as the difference in codelength between its encoding using a simple Erdős-Rényi model, i.e., encoding the edges individually, and its encoding using the most compressing model, (15)

As Fig 4 and Table 2 show, all the empirical connectomes are compressible, confirming their non-random structure (see S7 Fig for a comparison of all the models considered). Significant higher-order structures in the form of motifs are found in all the whole-CNS and whole-nervous-system connectomes studied here (Fig 4A) as well as in many connectomes of individual brain regions (Fig 4B and 4C). Besides motifs, we find significant non-random degree distributions of the nodes in all connectomes (Fig 4). This is consistent with node degrees being a salient feature of many biological networks, including neuronal networks [2]. Reciprocal connections are also a significant feature of almost all connectomes studied, in alignment with empirical observations from in vivo experiments [40, 60, 61, 72, 73] where modulation of neural activity is often implemented through recurrent patterns. Note that reciprocal connections are often considered a two-node motif. We chose to encode it as a dyadic feature of the base model since this is more efficient and allows for a higher compression, but it is entirely possible to encode them as graphlets by allowing also two-node graphlets as supernodes in the reduced graph (instead of restricting to 3–5 node graphlets as we did here).

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Fig 4. Compressibility of neural connectomes.

Compressibility (measured in number of bits per edge in the network) ΔL*/E of different connectomes as compared to encoding the edges independently using the Erdős-Rényi simple graph model (see Table 1). Two types of models are shown for the datasets: the best simple network encoding and the best motif-based encoding when this compresses more than the simple encoding. Asterisks highlight connectomes where motifs permit a higher compression than the reference models. (A) Whole-CNS and whole-animal connectomes. (B) Connectomes of three different regions of the adult Drosophila right hemibrain. Note that while the relative increase in compressibility of these connectomes obtained using motifs is relatively small, the motifs are highly significant due to the large size of these connectomes (Table 2). (C) Connectomes of different brain regions of first instar Drosophila larva. (D) Connectomes of C. elegans head ganglia at different developmental stages, from 0 hours to 50 (adult). While no higher-order motifs are found, the compressibility increases with maturation (and thus the size) of the connectome.

https://doi.org/10.1371/journal.pcbi.1012460.g004

For several smaller, regional connectomes, we do not find statistical evidence for higher-order motifs (Fig 4C and 4D), indicating the absence of significant higher-order circuit patterns (i.e., involving more than two neurons) in these connectomes. Note that network size did not have a significant effect on motif detectability in our numerical experiments above (see S3–S6 Figs), so the absence of motifs in these connectomes are likely due to their structural particularities rather than simply their smaller size. In particular, we do not find evidence for motifs in the C. elegans head ganglia (brain) connectomes at any developmental stage (Fig 4D). Note, however, that we do detect significant edge and node features (as encoded by the reciprocal configuration model), highlighting the non-random distribution of neuron connectivity and the importance of feedback connections in these connectomes. Furthermore, we do find higher-order motifs in the more complete C. elegans connectomes that also include sensory and motor neurons (Fig 4A), in line with what was found earlier using hypothesis-testing based motif mining [16, 65].

To study the structural properties of the inferred motif sets, we computed different average network measures of the motifs of each connectome (see definitions in S6 Text). The density of inferred motifs is much higher than the average density of the connectome (Fig 5A). While the density of motifs is high for all connectomes, it does vary significantly between them in a manner that is seemingly uncorrelated with the average connectome density. The motifs’ high density means that half of their node pairs or more are connected on average, which would lead to high numbers of reciprocal connections even if the motifs were wired at random. We indeed observe a high reciprocity of connections in the inferred motifs, and that this reciprocity is in large part explained by their high average density (Fig 5B), although we do observe significant variability and differences from this random baseline. The average number of cycles in the motifs is, on the other hand, in general completely explained by the motifs’ high density (Fig 5C). To probe the higher-order structure of the inferred motifs we measure their symmetry as measured by the graph polynomial root (GPR) [74]. As Fig 5D shows, the motifs are on average more symmetric than random graphlets of the same density even if the individual differences are often not significant. Thus, of the four aggregate topological features we investigated, the elevated density is the most salient feature of the motif sets. This does not exclude the existence of salient (higher-order) structural particularities of the motifs beyond their high density, only that such features are not captured well by these simple aggregate measures.

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Fig 5. Topological properties of motif sets.

Graph measures averaged over the inferred graphlet multiset, , i.e., for a network measure φ, one point corresponds to the quantity . The density (A), reciprocity (B) and number of cycles (C) and are standard properties of directed networks [75]. The graph polynomial root (D) measures the structural symmetry of the motifs [74]. Details can be found in S6 Text. Red squares indicate averages over the connectomes’ inferred motif sets. Blue squares are reference values, computed from average over randomized graphlets with their density conserved. To obtain the fixed-density references per motif set, we generate for each graphlet a collection of a hundred randomized configurations sharing the same density. The black dots of panel (A) show the connectomes’ global density.

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Even though the inferred motif sets are highly diverse, we observe that several motifs are found in a large fraction of the connectomes (Fig 6A). The same motifs also tend to be among the most frequent motifs, i.e., the ones making up the largest fraction of the inferred motif sets on average (Fig 6B). These tend to be highly dense graphlets, with the two most frequent motifs being the three and five node cliques, which are each found in roughly half of the connectomes and are also the most frequent motifs in the motif sets on average. The ten most frequently found motifs (Fig 6A) and the most repeated motifs (Fig 6B) do not perfectly overlap, though six of the ten motifs are the same between the two lists.

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Fig 6. Connectomes share common motifs.

Most frequently appearing motifs in the motif sets inferred for all connectomes. (A) Most frequently found motifs: fraction of connectomes in which each motif is found, . (B) Most repeated motifs: average graphlet concentration .

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