We calculated network measures and motif distributions for two mammalian connectivity data sets, macaque cortex and cat cortex (see Methods
). The connectivity data sets are shown in , with brain areas arranged according to a cluster analysis based on the matching indices 
for all area pairs. The matching index quantifies the overlap in afferent and efferent connections between two areas, and previous studies have suggested that areas with low pair-wise similarity in their patterns of afferents and efferents tend to have different functional properties 
. The matching index matrix, when subjected to cluster analysis, then serves to group areas. For macaque and cat cortex, the resulting arrangement of areas resembles the major functional subdivisions (e.g. visual, sensorimotor, auditory, prefrontal) of mammalian cerebral cortex, confirming that groups of functionally related areas share connection patterns.
The connection matrices for macaque and cat cortex were of similar size and density. Both matrices contained a high fraction of reciprocal pathways (0.76 in the macaque cortex, 0.74 in the cat cortex). Vertex degrees for each matrix are shown in . In both matrices, degrees varied over a broad range without presenting evidence of a scale-free organization. For the remainder of this paper all areas with a degree that is at least one standard deviation greater than the mean are termed “high-degree areas” (). Both networks were fully connected, and the maximal distances (diameter) did not exceed four edges. Average path lengths and clustering coefficients indicated that macaque and cat cortex exhibit small-world attributes, confirming several earlier reports on similar data sets (reviewed in 
). The degree to which each network resembles a small world can be quantified by the small-world index 
, found to be σsw
1.4551 (±0.0408) for macaque cortex and σsw
1.3153 (±0.0148) for cat cortex (mean and s.d., n
1000 random networks).
Degree of areas in macaque and cat cortex.
We derived structural motif frequency spectra for motifs of size M=
3 for both connection matrices (data not shown). Confirming earlier results for similar connection patterns 
, motif spectra for macaque and cat cortex were highly correlated (r2
) and both data sets exhibited an overabundance of a single motif class, here denoted
(see ). Overabundance was assessed by computing z-scores for comparisons to n=
100 equivalent control networks (random and lattice, see Methods
). A motif was considered “significantly increased” if, relative to both random and lattice control networks, its z-scores exceeded z=
3. We note that lattice controls have near-equal proportions of reciprocal edges as compared to the actual data sets, indicating that a high proportion of reciprocal edges alone does not explain the overabundance of motif
. Motif analysis for larger motifs (M
5) identified several motif classes as significantly increased over both random and lattice controls, including various tilings of motif
into ring and star patterns (data not shown).
Statistical significance of motif participation for individual brain regions.
Individual brain regions make specific contributions to the overall motif distribution of the network. Specifically, we sought to identify regions that disproportionately contribute to motif class
in macaque and cat cortex. To pinpoint locations where aggregations of specific motifs might occur we examined all the participating individual areas in particular structural motifs. The preservation of degree sequences for random and lattice control networks allowed the identification of control vertices that corresponded to those in the real data set, which enabled us to perform statistical comparisons of motif participation on a vertex-by-vertex basis. shows significance profiles for individual brain regions in macaque and cat cortex, revealing that individual brain regions made very different contributions to the global motif frequency spectrum. In macaque and cat cortex the majority of significantly increased contributions involved motif
. In the macaque, motifs with participation significantly increased from both random and lattice networks were
(for 11 vertices) and
(for 3 vertices), while in the cat significant increases were observed in
(for 4 vertices),
(for 3 vertices),
(for 17 vertices), and
(for 4 vertices). In macaque areas with increased contributions to
participated in the majority in the dorsal stream of visual processing (MSTd, DP) or in polysensory integration (7a, 7b, STPp, 46, Ig), with the notable exception of areas VP and V4 that are believed to be components of the ventral visual processing stream. Significantly increased motif
in the cat also occurred among a subset of polysensory regions (PLLS, 20a, EPp, PFCL, Ia, CGp, RS), notably extending also to ‘higher’ motor (area 6 m) and sensory (visual: area 19; somatosensory: SII, SIV) regions.
Motif fingerprints summarize the participation of individual brain regions in specific motif classes 
. We derived motif fingerprints for all brain areas of macaque and cat cortex and then performed hierarchical cluster analysis and principal components analysis on these fingerprints to reveal clusters of brain regions with similar motif fingerprints (). We found that macaque and cat motif fingerprints formed approximately equal numbers of clusters, and that several of the average motif fingerprints of these clusters shared substantial similarity. The two main clusters for macaque and cat (labeled “c” and “f” in ) yielded highly similar average motif fingerprints. Following principal components analysis these two patterns were placed in close proximity (). All areas with significantly increased participation for motif
were found within these two clusters (with the exception of area PLLS in cat cortex). The cluster structure observed in cat and macaque cortex did not appear if cluster analysis was carried out on degree-matched random or lattice control networks.
Hierarchical cluster analysis of motif fingerprints for individual brain regions.
Area V4 participates in 136 instances (out of 721) of motif
, the largest contribution of any area in macaque cortex. In 96 of these instances area V4 is found at the central apex of this motif (, inset). We define the apex ratio as the fraction of apex locations out of all instances of motif
, yielding an apex ratio of 0.701 for area V4 (random placement would yield an apex ratio of 1/3). Apex ratios for all areas in macaque and cat cortex are shown in . In both species, all high-degree areas exhibit high apex ratios for motif
. High contributions to motif class
, combined with a high apex ratio, should be associated with low values for the clustering coefficient, as only a relatively small fraction of neighbors are connected with one another. shows that clustering coefficients are indeed found to be below the network mean for all high-degree areas.
Figure 5 Apex ratio for motif and clustering coefficients in macaque and cat cortex.
Apex ratios and clustering coefficients suggest that brain regions with significantly increased contributions to motif
form topological hubs of reciprocal edges, linking many diverse vertices. We might expect that these local waystations have high network centrality. Of the numerous available centrality measures we calculated two: betweenness centrality (
; ) and closeness centrality (
; ). Betweenness centrality captures the degree to which a given brain region participates in the set of shortest paths between any pair of vertices in the network. Closeness centrality captures the average closeness (defined as the inverse of the shortest path length) to all other vertices. In macaque cortex, areas V4, 46, 7a and 7b (previously identified as making significantly increased contributions to motif
, and having high apex ratios as well as low clustering coefficients) are among those with the highest betweenness centrality as well as closeness centrality. In cat cortex, areas CGp, EPp, Ia, and 20a share the same characteristics. Without exception, and in both species, areas with high degree have greater than average centrality.
Centrality measures in macaque and cat cortex.
All of the measures considered so far are interrelated, primarily through the most basic characteristic of each vertex, its degree. As expected, degree and clustering coefficient (r2
0.44 in macaque, r2
0.67 in cat) and degree and betweenness (r2
0.68 in macaque, r2
0.68 in cat) are moderately cross-correlated. Among motif classes, centrality is on average most strongly correlated with motif
0.55 in macaque cortex, r2
0.67 in cat cortex) while other highly connected motifs (e.g.
) reach comparable levels. and summarize our analysis for all high-degree nodes in macaque and cat cortex. On the basis of several intersecting criteria, we can identify areas V4, FEF, 46, 7a, TF, 5, and 7b as the strongest candidates for hub regions in macaque cortex, while areas CGp, 35, AES, Ia, 20a and EPp are the strongest candidates for hub regions in cat cortex.
Summary of results for hub identification and hub classification for high-degree areas in macaque cortex.
Summary of results for hub identification and hub classification for high-degree areas in cat cortex.
Once network hubs have been identified, hubs may be classified on the basis of whether their connections are distributed mostly within or mostly between network modules 
. Hubs may also be classified on the basis of their spatial embedding, e.g. the distribution of the metric lengths of their projections 
. We pursued both approaches to hub classification. We applied a spectral community detection algorithm 
to identify modules within macaque and cat cortex. We extracted optimal community structures with 2 (macaque) and 3 modules (cat). plots the participation coefficient P
, which expresses, for each area, the balance between connections that are made within and between modules. Following a previously published classification scheme 
, we denote high-degree areas with P>0.3 as connector hubs, while high-degree areas with P<0.3 are denoted as provincial hubs (,). In macaque cortex, the majority of hubs are connectors, while areas V4 and MT, as well as the less highly connected yet highly central area SII are classified as provincial hubs. In cat cortex, all high-degree areas are classified as connector hubs. The absence of clearly defined provincial hubs may point to a difference in the structural organization of network modules in the two species.
To visualize the structural embedding of a provincial and a connector hub, we plotted two submatrices of macaque visual cortex, comprising area V4 (a provincial hub) and area 46 (a connector hub) together with their immediate topological neighbors (). The V4 submatrix (, left) and a corresponding cortical surface representation (, right) indicates that virtually all of V4's neighbors are located within visual cortex, with most of V4's inter-regional connections spanning relatively short distances (17.09 mm±9.60 mm s.d.). Of its 42 connections with other areas, 23 are shorter than the network's mean connection length of 18 mm (). The graph structure of the V4 submatrix (, middle) suggests that V4 mediates information flow between two groups of areas, one belonging predominantly to the dorsal visual stream (with the exception of area VP) and the other belonging to the ventral visual stream (with the exception of area 46, a connector hub). In contrast, corresponding plots for area 46 () reveal that this area maintains a more diverse set of projections, including visual, somatosensory and motor regions. Many of the connections of area 46 were found to span long distances (33.41 mm±10.58 mm s.d.; significantly different from those of V4, p<0.0001, with 35 of its 39 connections longer than 18 mm), although we note that some short projections are likely missing because other prefrontal regions were not included in the connection matrix. Connector hubs are very highly interconnected amongst themselves, forming “hub complexes” with a connection density of 0.81 (macaque) and 0.85 (cat). For comparison, submatrices of areas with identical degrees sampled from randomized control networks have connection densities of 0.69±0.06 (n
1000, p<0.02) in macaque and 0.76±0.03 (n
1000, p<0.01) in cat cortex.
Lesions of hubs may be expected to have unusually large consequences on information flow and communication within the remaining network. Such consequences may structurally be assessed by plotting changes in the network's path length and clustering coefficient following the lesion. Our analysis shows that the effect of lesioning a single area on the network's small-world index is just as likely to be positive as negative. summarizes the impact of single area lesions on the small-world index for macaque cortex and cat cortex. In macaque cortex, lesions of connector hubs such as FEF, 46, 7a, 7b (or more generally, areas with high participation coefficient) resulted in large increases in the small-world index relative to the unlesioned network. This effect is due to an increase in cluster distance (expressed in an increase in path length) as well as an increase in their segregation from each other (expressed in an even greater increase in clustering). In contrast, lesions of provincial hubs (e.g. area V4) or more generally of areas with low participation coefficient (e.g. area SII) resulted in decreases of the small-world index. This decrease is due to a decrease in clustering accompanied by a smaller effect (an increase or a decrease) in the path length. Similar patterns were found in cat cortex, with lesions of high-degree connector hubs such as Ia and CGp resulting in a higher small-world index, while lesioning of areas with lower participation coefficient had the opposite effect. In both, macaque and cat cortex, distributions of small-world lesion effects and participation coefficients over all areas are highly and significantly correlated (see Text S1
). These results indicate that lesions of hub regions belonging to different classes may have differential effects on the small-world structure of the remaining network.
Impact of single area lesion on small-world index.