Table 1: Mathematical definitions of measures (see main text for an informal discussion). All binary and undirected measures are accompanied by their weighted and directed generalizations. Generalizations which have not been previously reported (to the best of our knowledge) are marked with a star (*). The Brain Connectivity Toolbox contains Matlab functions to compute most measures in this table.

Measure Binary and undirected definitions Weighted and directed definitions

Basic concepts and measures

Basic concepts 푁 is the set of all nodes in the network, and 푛 is the number of nodes. Links 푖, 푗 are associated with connection weights 푤푖푗 . Henceforth we and notation 퐿 is the set of all links in the network, and 푙 is number of links. assume that weights are normalized, such that 0 ≤ 푤푖푗 ≤ 1 for all 푖 and 푗. 푖, 푗 is a link between nodes 푖 and 푗 푖, 푗 ∈ 푁 . w w 푙 is the sum of all weights in the network, computed as 푙 = 푖,푗 ∈푁 푤푖푗 . 푎푖푗 is the connection status between 푖 and 푗: 푎푖푗 = 1 when link 푖, 푗 exists (when 푖 and 푗 are neighbors); 푎푖푗 = 0 otherwise (푎푖푖 = 0 for all 푖). Directed links 푖, 푗 are ordered from 푖 to 푗. Consequently, in directed We compute the number of links as 푙 = 푖,푗 ∈푁 푎푖푗 (to avoid ambiguity with networks 푎푖푗 does not necessarily equal 푎푗푖 . directed links we count each undirected link twice, as 푎푖푗 and as 푎푗푖 ). w Degree: number Degree of a node 푖, Weighted degree of 푖, 푘푖 = 푗 ∈푁 푤푖푗 . of links 푘푖 = 푎푖푗 . connected to a out 푗 ∈푁 (Directed) out-degree of 푖, 푘푖 = 푗 ∈푁 푎푖푗 . node in (Directed) in-degree of 푖, 푘푖 = 푗 ∈푁 푎푗푖 . w Shortest path Shortest path length (distance), between nodes 푖 and 푗, Shortest weighted path length between 푖 and 푗, 푑 = w 푓 푤 , 푖푗 푎푢푣 ∈푔푖↔푗 푢푣 length: a basis 푑 = 푎 , where 푓 is a map (e.g. an inverse) from weight to length and 푔w is the for measuring 푖푗 푢푣 푖↔푗 shortest weighted path between 푖 and 푗. integration 푎푢푣 ∈푔푖↔푗 where 푔푖↔푗 is the shortest path (geodesic) between 푖 and 푗. Note that Shortest directed path length from 푖 to 푗, 푑→ = 푎 , where 푔 is 푑푖푗 = ∞ for all disconnected pairs 푖, 푗. 푖푗 푎푖푗 ∈푔푖→푗 푖푗 푖→푗 the directed shortest path from 푖 to 푗. Number of Number of triangles around a node 푖, (Weighted) geometric mean of triangles around 푖, triangles: a basis 1 w 1 1/3 푡 = 푎 푎 푎 . 푡푖 = 푗 ,푕∈푁 푤푖푗 푤푖푕 푤푗푕 . for measuring 푖 2 푖푗 푖푕 푗푕 2 segregation 푗 ,푕∈푁 Number of directed triangles around 푖, 1 푡→ = 푎 + 푎 푎 + 푎 푎 + 푎 . 푖 2 푗 ,푕∈푁 푖푗 푗푖 푖푕 푕푖 푗푕 푕푗

Measures of integration w Characteristic Characteristic path length of the network (e.g. Watts and Strogatz, 1998), w 1 푗 ∈푁,푗≠푖 푑푖푗 Weighted characteristic path length, 퐿 = 푖∈푁 . path length 1 1 푗 ∈푁,푗 ≠푖 푑푖푗 푛 푛−1 퐿 = 퐿푖 = , 푛 푛 푛 − 1 → 푖∈푁 푖∈푁 → 1 푗 ∈푁,푗 ≠푖 푑푖푗 Directed characteristic path length, 퐿 = 푖∈푁 . where 퐿푖 is the average distance between node 푖 and all other nodes. 푛 푛−1 −1 Global efficiency Global efficiency of the network (Latora and Marchiori, 2001), w 1 푗 ∈푁,푗 ≠푖 푑푖푗 −1 Weighted global efficiency, 퐸w = . 1 1 푗 ∈푁,푗 ≠푖 푑푖푗 푛 푖∈푁 푛−1 퐸 = 퐸 = , 푛 푖 푛 푛 − 1 −1 푖∈푁 푖∈푁 → 1 푗∈푁,푗≠푖 푑푖푗 where 퐸푖 is the efficiency of node 푖. Directed global efficiency, 퐸→ = . 푛 푖∈푁 푛−1

Measures of segregation Clustering of the network (Watts and Strogatz, 1998), Weighted clustering coefficient (Onnela et al., 2005), w coefficient 1 1 2푡푖 w 1 2푡푖 퐶 = 퐶 = , 퐶 = 푖∈푁 . See Saramaki et al. (2007) for other variants. 푖 푛 푘푖 푘푖−1 푛 푛 푘푖 푘푖 − 1 푖∈푁 푖∈푁 where 퐶푖 is the clustering coefficient of node 푖 (퐶푖 = 0 for 푘푖 < 2). Directed clustering coefficient (Fagiolo, 2007), → → 1 푡푖 퐶 = 푖∈푁 out in out in . 푛 푘푖 +푘푖 푘푖 +푘푖 −1 −2 푗 ∈푁 푎푖푗 푎푗푖 2푡w Transitivity Transitivity of the network (e.g. Newman, 2003), Weighted transitivity*, 푇w = 푖∈푁 푖 . 푘 (푘 −1) 푖∈푁 2푡푖 푖∈푁 푖 푖 푇 = . 푖∈푁 푘푖(푘푖 − 1) → → 푖∈푁 푡푖 Note that transitivity is not defined for individual nodes. Directed transitivity*, 푇 = out in out in . 푖∈푁 푘푖 +푘푖 푘푖 +푘푖 −1 −2 푗 ∈푁 푎푖푗 푎푗푖 Local efficiency Local efficiency of the network (Latora and Marchiori, 2001), −1 1/3 푤 푤 푑w (푁 ) −1 w 1 푗 ,푕∈푁,푗 ≠푖 푖푗 푖푕 푗푕 푖 1 1 푎 푎 푑 (푁 ) Weighted local efficiency*, 퐸loc = 푖∈푁 . 푗 ,푕∈푁,푗 ≠푖 푖푗 푖푕 푗푕 푖 푛 푘푖 푘푖−1 퐸loc = 퐸loc,푖 = , 푛 푛 푘푖 푘푖 − 1 푖∈푁 푖∈푁 Directed local efficiency*, where 퐸loc,푖 is the local efficiency of node 푖, and 푑푗푕 (푁푖) is the length of the −1 −1 푎 +푎 푎 +푎 푑→ (푁 ) + 푑→ (푁 ) shortest path between 푗 and 푕, that contains only neighbors of 푖. → 1 푗 ,푕∈푁,푗≠푖 푖푗 푗푖 푖푕 푕푖 푗푕 푖 푕푗 푖 퐸loc = 푖∈푁 out in out in . 2푛 푘푖 +푘푖 푘푖 +푘푖 −1 −2 푗 ∈푁 푎푖푗 푎푗푖 Modularity of the network (Newman, 2004b), Weighted modularity (Newman, 2004), w w 2 1 푘 푘 푄w = 푤 − 푖 푗 훿 . 푙w 푖,푗 ∈푁 푖푗 푙w 푚푖,푚푗 푄 = 푒푢푢 − 푒푢푣 ,

푢∈푀 푣∈푀 Directed modularity (Leicht and Newman, 2008), where the network is fully subdivided into a set of nonoverlapping modules 1 푘out푘in 푀, and 푒 is the proportion of all links that connect nodes in module 푢 with 푄→ = 푎 − 푖 푖 훿 . 푢푣 푙 푖,푗 ∈푁 푖푗 푙 푚푖,푚푗 nodes in module 푣.

An equivalent alternative formulation of the modularity (Newman, 2006) is 1 푘 푘 given by 푄 = 푎 − 푖 푗 훿 , where 푚 is the module 푙 푖,푗 ∈푁 푖푗 푙 푚푖,푚푗 푖

containing node 푖, and 훿푚푖,푚푗 = 1 if 푚푖 = 푚푗 , and 0 otherwise.

Measures of Closeness Closeness centrality of node 푖 (e.g. Freeman, 1978), w −1 푛−1 Weighted closeness centrality, 퐿푖 = w. 푗∈푁,푗≠푖 푑푖푗 centrality −1 푛 − 1 퐿푖 = . 푗 ∈푁,푗 ≠푖 푑푖푗 → −1 푛−1 Directed closeness centrality, 퐿푖 = →. 푗 ∈푁,푗 ≠푖 푑푖푗 Betweenness Betweenness centrality of node 푖 (e.g. Freeman, 1978), Betweenness centrality is computed equivalently on weighted and directed centrality 1 𝜌푕푗 (푖) networks, provided that path lengths are computed on respective weighted or 푏 = , 푖 푛 − 1 푛 − 2 𝜌 directed paths. 푕,푗 ∈푁 푕푗 푕≠푗 ,푕≠푖,푗 ≠푖 where 𝜌푕푗 is the number of shortest paths between 푕 and 푗, and 𝜌푕푗 (푖) is the number of shortest paths between 푕 and 푗 that pass through 푖. w w Within-module Within-module degree z-score of node 푖 (Guimera and Amaral, 2005), w 푘푖 푚푖 −푘 푚푖 Weighted within-module degree z-score, 푧푖 = w . degree z-score 푘 푚 − 푘 푚 𝜎푘 푚 푖 푖 푖 푖 푧푖 = 푘 푚 , 𝜎 푖 out out out 푘푖 푚푖 −푘 푚푖 where 푚푖 is the module containing node 푖, 푘푖 푚푖 is the within-module Within-module out-degree z-score, 푧푖 = out . 𝜎푘 푚 푖 degree of 푖 (the number of links between 푖 and all other nodes in 푚 ), and in in 푖 in 푘푖 푚푖 −푘 푚푖 푘 푚푖 Within-module in-degree z-score, 푧푖 = in . 푘 푚푖 and 𝜎 are the respective mean and standard deviation of the 𝜎푘 푚 푖 within-module 푚푖 . w 2 Participation Participation coefficient of node 푖 (Guimera and Amaral, 2005), w 푘푖 푚 2 Weighted participation coefficient, 푦 = 1 − coefficient 푖 푚∈푀 푘w 푘푖 푚 푖 푦푖 = 1 − , 푘푖 푚∈푀 out 2 out 푘푖 푚 where 푀 is the set of modules (see modularity), and 푘푖 푚 is the number of Out-degree participation coefficient, 푦푖 = 1 − 푚∈푀 out . 푘푖 links between 푖 and all nodes in module 푚. in 2 in 푘푖 푚 In-degree participation coefficient, 푦푖 = 1 − 푚∈푀 in . 푘푖

Network motifs 1 Anatomical and 퐽푕 is the number of occurrences of motif 푕 in all subsets of the network 푙푕 (Weighted) intensity of 푕 (Onnela et al., 2005), 퐼푕 = 푢 푖,푗 ∈퐿 푢 푤푖푗 , functional (subnetworks). 푕 is an 푛푕 node, 푙푕 link, directed connected pattern. 푕 will 푕 where the sum is over all occurrences of 푕 in the network, and 퐿 푢 is the set motifs occur as an anatomical motif in an 푛푕 node, 푙푕 link subnetwork, if links in 푕 the subnetwork match links in 푕 (Milo et al., 2002). 푕 will occur (possibly of links in the 푢th occurrence of 푕. ′ more than once) as a functional motif in an 푛푕 node, 푙 ≥ 푙푕 link 푕 Note that motifs are directed by definition. subnetwork, if at least one combination of 푙푕 links in the subnetwork matches links in 푕 (Sporns and Kötter, 2004). 퐼 − 퐼 Motif z-score z-score of motif 푕 (Milo, 2002), Intensity z-score of motif 푕 (Onnela et al., 2005), 푧퐼 = 푕 rand,푕 , 푕 𝜎퐼rand,푕 퐽푕 − 퐽rand,푕 퐼rand,푕 푧푕 = , where 퐼rand,푕 and 𝜎 are the respective mean and standard deviation for 𝜎퐽rand,푕 퐽rand,푕 the intensity of 푕 in an ensemble of random networks. where 퐽rand,푕 and 𝜎 are the respective mean and standard deviation for the number of occurrences of 푕 in an ensemble of random networks. Motif 푛푕 node motif fingerprint of the network (Sporns and Kötter, 2004), 푛푕 node motif intensity fingerprint of the network, 퐼 ′ 퐼 ′ fingerprint ′ ′ 퐹 푕 = 퐹 푕 = 퐼 ′ , ′ 푛푕 푖∈푁 푛푕 ,푖 푖∈푁 푕 ,푖 퐹푛푕 푕 = 퐹푛푕 ,푖 푕 = 퐽푕 ,푖, where 푕′ is any 푛 node motif, 퐹퐼 푕′ is the 푛 node motif intensity 푖∈푁 푖∈푁 푕 푛푕 ,푖 푕 where 푕′ is any 푛 node motif, 퐹 푕′ is the 푛 node motif fingerprint for ′ 푕 푛푕 ,푖 푕 fingerprint for node 푖, and 퐼푕′ ,푖 is the intensity of motif 푕 around node 푖. ′ node 푖, and 퐽푕′ ,푖 is the number of occurrences of motif 푕 around node 푖.

Measures of resilience w ′ Degree Cumulative degree distribution of the network (Barabasi and Albert, 1999), Cumulative weighted degree distribution, 푃 푘 = 푘′ ≥푘w 푝(푘 ), distribution ′ 푃 푘 = 푝(푘 ), out ′ ′ out ′ Cumulative out-degree distribution, 푃 푘 = 푘 ≥푘 푝(푘 ). 푘 ≥푘 in ′ where 푝(푘′ ) is the probability of a node having degree 푘′. Cumulative in-degree distribution, 푃 푘 = 푘′ ≥푘in 푝(푘 ). Average Average degree of neighbors of node 푖 (Pastor-Sattoras et al., 2001), Average weighted neighbor degree (modified from Barrat et al., 2004), w neighbor degree 푗 ∈푁 푎푖푗 푘푗 w 푗∈푁 푤푖푗 푘푗 푘 = . 푘nn,푖 = w . nn,푖 푘푖 푘푖 out in → 푗∈푁 푎푖푗 +푎푗푖 푘푖 +푘푖 Average directed neighbor degree*, 푘nn,푖 = out in . 2 푘푖 +푘푖 Assortativity coefficient of the network (Newman, 2002), Weighted assortativity coefficient (modified from Leung and Chau, 2007), 2 2 coefficient −1 w w −1 1 w w −1 −1 1 푙 푖,푗 ∈퐿 푤푖푗 푘푖 푘푗 − 푙 푖,푗 ∈퐿 푤푖푗 푘푖 +푘푗 푙 푖,푗 ∈퐿 푘푖푘푗 − 푙 푖,푗 ∈퐿 푘푖 + 푘푗 푟w = 2 . 2 1 2 2 1 2 푟 = . 푙−1 푤 푘w + 푘w − 푙−1 푤 푘w+푘w 1 1 2 푖,푗 ∈퐿2 푖푗 푖 푗 푖,푗 ∈퐿2 푖푗 푖 푗 푙−1 푘2 + 푘2 − 푙−1 푘 + 푘 푖,푗 ∈퐿 2 푖 푗 푖,푗 ∈퐿 2 푖 푗 Directed assortativity coefficient (Newman, 2002), 2 −1 out in −1 1 out in 푙 푘 푘 − 푙 푘 +푘 푟→ = 푖,푗 ∈퐿 푖 푗 푖,푗 ∈퐿2 푖 푗 . 1 2 2 1 2 푙−1 푘out + 푘in − 푙−1 푘out+푘in 푖,푗 ∈퐿2 푖 푗 푖,푗 ∈퐿2 푖 푗

Other concepts Degree Degree-distribution preserving randomization is implemented by iteratively The algorithm is equivalent for weighted and directed networks. In weighted distribution choosing four distinct nodes 푖1, 푗1, 푖2, 푗2 ∈ 푁 at random, such that links networks, weights may be switched together with links; in this case the preserving 푖1, 푗1 , (푖2, 푗2) ∈ 퐿, while links 푖1, 푗2 , 푖2, 푗1 ∉ 퐿. The links are then weighted degree distribution is not preserved, but may be subsequently network rewired such that 푖1, 푗2 , 푖2, 푗1 ∈ 퐿 and 푖1, 푗1 , (푖2, 푗2) ∉ 퐿 (Maslov and approximated on the topologically randomized graph with a heuristic weight randomization. Sneppen, 2002). “Latticization” (a lattice-like topology) results if an reshuffling scheme. additional constraint is imposed, 푖1 + 푗2 + 푖2 + 푗1 < 푖1 + 푗1 + 푖2 + 푗2 (Sporns and Kötter, 2004). 퐶w/퐶w Measure of Network small-worldness (Humphries et al., 2008), Weighted network small-worldness, 푆w = rand. 퐿w 퐿w network small- 퐶 퐶rand rand worldness. 푆 = , 퐿 퐿rand → → → 퐶 /퐶rand where 퐶 and 퐶rand are the clustering coefficients, and 퐿 and 퐿rand are the Directed network small-worldness, 푆 = → → . 퐿 퐿rand characteristic path lengths of the respective tested network and a random network. Small-world networks often have 푆 ≫ 1. In both cases, small-world networks often have 푆 ≫ 1.