Graph analysis of structural brain networks in Alzheimer’s disease: beyond small world properties

Abstract

Changes in brain connectivity in patients with early Alzheimer’s disease (AD) have been investigated using graph analysis. However, these studies were based on small data sets, explored a limited range of network parameters, and did not focus on more restricted sub-networks, where neurodegenerative processes may introduce more prominent alterations. In this study, we constructed structural brain networks out of 87 regions using data from 135 healthy elders and 100 early AD patients selected from the Open Access Series of Imaging Studies (OASIS) database. We evaluated the graph properties of these networks by investigating metrics of network efficiency, small world properties, segregation, product measures of complexity, and entropy. Because degenerative processes take place at different rates in different brain areas, analysis restricted to sub-networks may reveal changes otherwise undetected. Therefore, we first analyzed the graph properties of a network encompassing all brain areas considered together, and then repeated the analysis after dividing the brain areas into two sub-networks constructed by applying a clustering algorithm. At the level of large scale network, the analysis did not reveal differences between AD patients and controls. In contrast, the same analysis performed on the two sub-networks revealed that small worldness diminished with AD only in the sub-network containing the areas of medial temporal lobe known to be heaviest and earliest affected. The second sub-network, which did not present significant AD-induced modifications of ‘classical’ small world parameters, nonetheless showed a trend towards an increase in small world propensity, a novel metric that unbiasedly quantifies small world structure. Beyond small world properties, complexity and entropy measures indicated that the intricacy of connection patterns and structural diversity decreased in both sub-networks. These results show that neurodegenerative processes impact volumetric networks in a non-global fashion. Our findings provide new quantitative insights into topological principles of structural brain networks and their modifications during early stages of Alzheimer’s disease.

Appendix

Graph theoretical descriptive measures

The graph theoretical descriptive measures we used can be grouped under the following five large umbrellas: (A) metrics of network efficiency, (B) “classical” small world properties, (C) measures of segregation, (D) product measures of complexity and (E) entropy. Technically, measures in (A) may be considered as small world properties or even measures of complexity; however, here we wished to treat them separately because they have been conventionally treated as distinct measures.

A. Metrics of network efficiency were first defined by Latora and Marchiori (2001, 2003) and were further explored in brain network analysis by Achard and Bullmore (2007). These metrics measure the economical performance of the networks. Certain small world properties such as the average shortest path length can be calculated only for connected graphs, whereas the efficiency metrics can be calculated for disconnected graphs as well. Thus, these measures are especially useful at the low sparsity levels, where a graph could be disconnected.

• Global efficiency is technically defined as the harmonic mean of the minimum path length of each pair of nodes and hence is inversely proportional to the average minimum path length. It is a measure of efficiency of a parallel system where all the nodes in the network concurrently exchange packets of information (Achard and Bullmore 2007). Since there is strong prior evidence that brain supports massively parallel information processing, global efficiency is biologically a highly relevant measure for comparing patients and controls.

• Cost efficiency, the difference between global efficiency and sparsity level, is a measure that was defined (Achard and Bullmore 2007) to assess the efficiency in relation to “economical cost” (that is, sparsity) of the network. Since it has been shown previously that efficiency of a complex network monotonically increases as a function of its cost, it makes sense to account for the cost when comparing the efficiency of the graphs from patients and controls.

B. Measures related to small worldness. Small world networks are formally defined as networks in which the nodes are significantly more clustered, yet have approximately the same characteristic path length as random networks (in which the nodes are connected at random; (Watts and Strogatz 1998). In other words, small world networks are simultaneously highly segregated and integrated. Anatomical brain connectivity is thought to concurrently reconcile the opposing demands of functional integration and segregation and hence is considered to have the small world design (Tononi et al. 1994; Bassett and Bullmore 2006).

• Clustering coefficient measures the probability that the adjacent nodes of a given node are connected. For any node, it is calculated as the number of triangles around the node divided by the total number of possible edges among the neighboring nodes. The average across all the nodes represents the average clustering coefficient. It is a measure of the extent of ‘cliquishness’ of the network (He et al. 2008).

• Shortest path length (or characteristic path length) between any pair of nodes is the number of nodes in a geodesic (that is, a path with the minimal number of vertices connecting the two nodes). Averaged across all nodes, it is a measure of the extent of average connectivity or overall routing efficiency of the network (He et al. 2008).

• Small world propensity: Let C _obs denote the clustering coefficient for the given network; C _lat and C _rand denote the clustering coefficients for lattice and random networks constructed with the same number of nodes and the same degree distribution as the given network. Similarly, let L _obs, L _lat and L _rand denote the characteristic path lengths for the given network and the corresponding lattice and random networks. Muldoon et al. (2016) defined small world propensity as

  $$\phi = 1 - \sqrt {\frac{{\Delta_{C}^{2} + \Delta_{L }^{2} }}{2}}$$

  where $\Delta_{C} = \frac{{C_{\text{lat}} - C_{\text{obs}} }}{{C_{\text{lat}} - C_{\text{rand}} }}$ and $\Delta_{L} = \frac{{ L_{\text{obs}} - L_{\text{rand}} }}{{L_{\text{lat}} - L_{\text{rand}} }}$.

C. Measures of segregation

• Betweenness centrality: The betweenness B _i of a node i is defined as the number of shortest paths between any two nodes that run through node i (Freeman 1977). Normalized by the average betweenness of the network (that is, averaged across all nodes), it results in b _i , a global centrality measure that captures the influence of a node over information flow between other nodes in the network.

• Sigma is the ratio of the clustering coefficient and the characteristic path length and is considered a summarized measure of small worldness.

D. Product measures of complexity are based on the idea that networks with intricate topological features have a medium number of edges; that is, they are neither very sparse nor highly connected. It has been established that both minimally connected networks and fully connected networks have complexity values approximately zero (Kim and Wilhelm 2008). The two product measures described below are both products of two factors: F ₁ × F ₂. F ₁ assigns values near zero for sparsely connected networks and large values for highly connected networks. In contrast, F ₂ assigns values near zero for highly connected networks and large values for sparse networks, so that the product F ₁ × F ₂ will always have small values for both sparse and highly connected networks and large values for networks with medium number of edges.

• Medium articulation for graphs = R × I, is the product of two factors: redundancy, R and mutual information I. R is zero for the directed ring graph, but maximum for the fully connected graph; I varies in the opposite way (Kim and Wilhelm 2008).

• Graph index complexity is based on the largest eigenvalue (also known as the index, r) of the adjacency matrix of a graph. Among all graphs with n nodes, a simple path graph connecting all nodes has the smallest index, $2\cos \left( {\pi /\left( {n + 1} \right)} \right)$ and a clique (i.e., a graph where any pair of nodes are connected) has the largest index, (n − 1). Therefore, we normalize r by $c_{r} = {\raise0.7ex\hbox{${\left( {r - 2\cos \frac{\pi }{n + 1}} \right)}$} \!\mathord{\left/ {\vphantom {{\left( {r - 2\cos \frac{\pi }{n + 1}} \right)} {\left( {n - 1 - 2\cos \frac{\pi }{n + 1}} \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {n - 1 - 2\cos \frac{\pi }{n + 1}} \right)}$}},$so that c _r  = 0 for a path and =1 for a clique. Similarly, 1 − c _r  = 1 for a path and 0 for a clique. Graph index complexity is defined as the product, 4c _r (1 − c _r ) (Kim and Wilhelm 2008).

E. Entropy in thermodynamics is defined as the amount of disorder, that is, the number of specific ways in which a thermodynamic system may be arranged. In a similar vein, the entropy of a complex network quantifies its structural diversity. Shannon entropy formula for graphs was first used in coding/information processing theory to characterize how much information can be communicated using binary code words or symbols where certain pairs of codes may be not clearly distinguishable. In graph theory or information processing literature, entropy is sometimes also referred to as the information content of a graph. To be specific, let G be a graph, and let A ₁, …, A _k be a partition of either the nodes or the edges of the graph based on some criterion. Let p ₁, …, p _k be a probability scheme associated with each partition. The information content of the graph G corresponding to this partition is given by Shannon’s formula

$$G = \mathop \sum \limits_{i = 1}^{k} p_{i} \log_{2} p_{i}$$

(1)

Based on the various criteria for partitioning the invariants (nodes, edges, distances, etc.) of a graph, a host of entropy measures has been proposed in the complex networks literature. The entropy measures considered in this paper are described below.

• Topological information content (TIC): The information content of a network largely depends on the topology of the network. The TIC measure that we consider in our analysis is the one proposed by Rashevsky (1955) and redefined by Trucco (1956) in terms of the automorphism group of a graph and vertex orbits. This measure is based on the symmetry in a graph. It can be easily shown that TIC vanishes for vertex transitive graphs (graphs in which each vertex has the same number of neighbors; that is, same degree), and it attains maximum entropy for asymmetric graphs.

• Bertz index (BI): While TIC accounts for the symmetry of a graph in terms of the invariant used, it does not properly reflect the size, since TIC = 0 when all the nodes are equivalent (e.g., as in a regular graph mentioned above). Reasoning that complexity should increase with the number of nodes (or any other invariant, for that matter) when they are all equivalent, in the same way it does when they are all nonequivalent, Bertz (1981) introduced a measure that adds a n × log(n) factor to the Eq. (1), to give a definition of the entropy of a graph as a function of the number of invariants.

• Vertex degree information-equality based information index is an entropy measure calculated based on partitioning the nodes of a network based on their degrees. A graph with an approximately uniform degree distribution will have smaller values for this index compared to graphs with nonuniform degree distributions (Bonchev 1983).

• Graph vertex complexity index is the average of the vertex complexities across all the vertices of a graph. The vertex complexity of a vertex v is the entropy, calculated using Shannon’s formula, obtained from a partition of vertices of the graph based on their distances to the vertex v. The vertex complexity value of a vertex depends only on the number of times any distance occurring in the graph with respect to that vertex, but not on the magnitude of the distance (Raychaudhury et al. 1984).

• Mean information content on the edge equality is obtained by dividing the total information content of the edge distance equality by the total number of edge distances in the graph. This measure is calculated using Shannon’s formula based on partition of the edges obtained by grouping together the edges with the same edge distances (Bonchev et al. 1981).

• Mean information content on the edge magnitude is calculated by dividing the total information content of the edge distance magnitude by the edge Wiener index (defined as half the total edge distance). Mean information content on the edge equality depends only on the number of different edge distances in the graph, while as mean information content on the edge magnitude depends also on the magnitude of the edge distances (Bonchev et al. 1981).

• Off diagonal complexity: Concisely put, the off diagonal complexity of a network is the entropy of the normalized diagonal sums of the node–node link correlation matrix. It has been suggested that a complex graph has many different entries in its so-called node–node link correlation matrix (c _ij ). Off diagonal complexity measures this diversity. c _ij denotes the number of all neighbors with degree j ≥ I of all nodes with degree i. More specifically, off diagonal complexity is high for a graph where the nodes of a given degree have no preference for the degree of their neighbors. That is, summing up the elements in each diagonal one obtains about the same number for all diagonals (Kim and Wilhelm 2008).