Graph Theoretical Analysis Reveals: Women's Brains are Better Connected than Men's
GGraph Theoretical Analysis Reveals: Women’s Brainsare Better Connected than Men’s
Bal´azs Szalkai a , B´alint Varga a , Vince Grolmusz a,b, ∗ a PIT Bioinformatics Group, E¨otv¨os University, H-1117 Budapest, Hungary b Uratim Ltd., H-1118 Budapest, Hungary
Abstract
Deep graph-theoretic ideas in the context with the graph of the World Wide Webled to the definition of Google’s PageRank and the subsequent rise of the most-popular search engine to date. Brain graphs, or connectomes, are being widelyexplored today. We believe that non-trivial graph theoretic concepts, similarlyas it happened in the case of the World Wide web, will lead to discoveriesenlightening the structural and also the functional details of the animal andhuman brains. When scientists examine large networks of tens or hundredsof millions of vertices, only fast algorithms can be applied because of the sizeconstraints. In the case of diffusion MRI-based structural human brain imaging,the effective vertex number of the connectomes, or brain graphs derived fromthe data is on the scale of several hundred today. That size facilitates applyingstrict mathematical graph algorithms even for some hard-to-compute (or NP-hard) quantities like vertex cover or balanced minimum cut.In the present work we have examined brain graphs, computed from the dataof the Human Connectome Project, recorded from male and female subjectsbetween ages 22 and 35. Significant differences were found between the maleand female structural brain graphs: we show that the average female connectomehas more edges, is a better expander graph, has larger minimal bisection width,and has more spanning trees than the average male connectome. Since theaverage female brain weights less than the brain of males, these properties showthat the female brain is more “well-connected” or perhaps, more “efficient” ina sense than the brain of males. It is known that the female brain has a largerwhite matter/gray matter ratio than the brain of males; this observation is inline with our findings concerning the number of edges, since the white matterconsists of myelinated axons, which, in turn, correspond to the connectionsin the brain graph. We have also found that the minimum bisection width,normalized with the edge number, is also significantly larger in the right and theleft hemispheres in females: therefore, that structural difference is independentfrom the difference in the number of edges. ∗ Corresponding author
Email addresses: [email protected] (Bal´azs Szalkai), [email protected] (B´alintVarga), [email protected] (Vince Grolmusz) a r X i v : . [ q - b i o . N C ] J a n . Introduction In the last several years hundreds of publications appeared describing oranalyzing structural or functional networks of the brain, frequently referred toas ”connectome” [1, 2]. Some of these publications analyzed data from healthyhumans [3, 4, 5, 6], and some compared the connectome of the healthy brainwith diseased one [7, 8, 9, 10, 11].So far, the analyses of the connectomes mostly used tools developed forvery large networks, such as the graph of the World Wide Web (with billions ofvertices), or protein-protein interaction networks (with tens or hundreds of thou-sands of vertices), and because of the huge size of original networks, these meth-ods used only very fast algorithms and frequently just primary degree statisticsand graph-edge counting between pre-defined regions or lobes of the brain [12].In the present work we demonstrate that deep and more intricate graph the-oretic parameters could also be computed by using, among other tools, contem-porary integer programming approaches for connectomes with several hundredvertices.With these mathematical tools we show statistically significant differences insome graph properties of the connectomes, computed from MRI imaging data ofmale and female brains. We will not try to associate behavioral patterns of malesand females with the discovered structural differences [12] (see also the debatethat article has generated: [13, 14, 15]), because we do not have behavioral dataof the subjects of the imaging study, and, additionally, we cannot describe high-level functional properties implied by those structural differences. However,we clearly demonstrate that deep graph-theoretic parameters show ”better”connections in a certain sense in female connectomes than in male ones.The study of [12] analyzed the 95-vertex graphs of 949 subjects aged be-tween 8 and 22 years, using basic statistics for the numbers of edges runningeither between or within different lobes of the brain (the parameters deducedwere called hemispheric connectivity ratio, modularity, transitivity and partic-ipation coefficients , see [12] for the definitions). It was found that males havesignificantly more intra-hemispheric edges than females, while females have sig-nificantly more inter-hemispheric edges than males.
2. Results and Discussion
We have analyzed the connectomes of 96 subjects, 52 females and 44males, each with 83, 129 and 234 node resolutions, and each graphs withfive different weight functions. We considered the connectomes as graphs withweighted edges, and performed graph-theoretic analyses with computing somepolynomial-time computable and also some NP-hard graph parameters on theindividual graphs, and then compared the results statistically for the male andthe female group. 2e have found that female connectomes have more edges, larger (normal-ized) minimum bisection widths, larger minimum-vertex covers and more span-ning trees than the male connectomes.In order to describe the parameters, which differ significantly among maleand female connectomes, we need to place them in the context of their graphtheoretical definitions.
We have found significantly higher number of edges (counted with 5 typesof weights and also without any weights) in both hemispheres and also in thewhole brain in females, in all resolutions. This finding is surprising, since weused the same parcellation and the same tractography and the same graph-construction methods for female and male brains, and because it is proven thatfemales have, on average, less-weighting brains than males [16]. For example, inthe 234-vertex resolution, the average number of (unweighted) edges in femaleconnectomes is 1826, in males 1742, with p = 0 . Suppose the nodes, or the vertices, of a graph are partitioned into two,disjoint, non-empty sets, say X and Y ; their union is the whole vertex-set ofthe graph. The X, Y cut is the set of all edges connecting vertices of X with thevertices of Y (Figure 1A). The size of the cut is the number of edges in the cut.In graph theory, the size of the minimum cut is an interesting quantity. Theminimum cut between vertices a and b is the minimum cut, taken for all X and Y , where vertex a is in X and b is in Y . This quantity gives the “bottleneck”, ina sense, between those two nodes (c.f., Menger theorems and Ford-Fulkerson’sMin-Cut-Max-Flow theorem [18, 19]). The minimum cut in a graph is definedto be the cut with the fewest edges for all non-empty sets X and Y , partitioningthe vertices.Clearly, for non-negative weights, the size of the minimum cut in a non-connected graph is 0. Very frequently, however, in connected graphs, the mini-mum cut is determined by just the smallest degree node: that node is the only3lement of set X and all the other vertices of the graph are in Y (Figure 1B).Because of this phenomenon, the minimum cut is frequently queried for the“balanced” case, when the size (i.e., the number of vertices) of X and Y needsto be equal (or, more exactly, may differ by at most one if the number of thevertices of the graph is odd), see Figure 1C. This problem is referred to as thebalanced minimum cut or the minimum bisection problem. If the minimum bi-section is small that means that there exist a partition of the vertices into twosets of equal size that are connected with only a few edges. If the minimumbisection is large then the two half-sets in every possible bisections of the graphare connected by many edges.Therefore, the balanced minimum cut of a graph is independent of the par-ticular labeling of the nodes. The number of all the balanced cuts in a graphwith n vertices is greater than 1 n + 1 2 n , that is, for n = 250, this number is very close to the number of atoms inthe visible universe [20]. Consequently, one cannot practically compute theminimum bisecton width by reviewing all the bisectons in a graph of that size.Moreover, the complexity of computing this quantity is known to be NP-hard[21] in general, but with contemporary integral programming approaches, forthe graph-sizes we are dealing with, the exact values are computable.In computer engineering, an important measure of the quality of an inter-connection network is its minimum bisection width [22]: the higher the widthis the better the network.For the whole brain graph, as it is anticipated, we have found that theminimum balanced cut is almost exactly represents the edges crossing the corpuscallosum , connecting the two cerebral hemispheres.We show that within both hemispheres, the minimum bisection size of femaleconnectomes are significantly larger than the minimum bisection size of themales. Much more importantly, we show that this remains true if we normalizewith the sum of all edge-weights : that is, this phenomenon cannot be due to thehigher number of edges or the greater edge weights in the female brain: it is anintrinsic property of the female brain graph in our data analyzed.For example, in the 234-vertex resolution, in the left hemisphere, the nor-malized balanced minimum cut in females, on the average, is 0 . . p = 0 . Expander graphs and the expander-property of graphs are one of the mostinteresting area of graph theory: they are closely related to the convergencerate and the ergodicity of Markov chains, and have applications in the design4 igure 1: Panel A: An X-Y cut. The cut-edges are colored black. Panel B: An un-balancedminimum cut. Panel C: A balanced cut. Panel D: The wheel graph. of communication- and sorting networks and methods for de-randomizing algo-rithms [23]. A graph is an ε -expander, if every – not too small and not toolarge – vertex-set S of the graph has at least ε | S | outgoing edges (see [23] forthe exact definition).Random walks on good expander graphs converge very fast to the limitdistribution: this means that good expander graphs, in a certain sense, are“intrinsically better” connected than bad expanders. It is known that largeeigengap of the walk transition matrix of the graph implies good expansionproperty [23].We have found that women’s connectomes have significantly larger eigengap,and, consequently, they are better expander graphs than the connectomes ofmen. For example, in the 83-node resolution, in the left hemisphere and in theunweighted graph, the average female connectome’s eigengap is 0 .
306 while inthe case of men it is 0 . p = 0 . A tree in graph theory is a connected, cycle-free graph. Any tree on n vertices has the same number of edges: n −
1. Trees, and tree-based structuresare common in science: phylogenetic trees, hierarchical clusters, data-storageon hard-disks, or a computational model called decision trees all apply graph-theoretic trees. A spanning tree is a minimal subgraph of a connected graph thatis still connected. Some graphs have no spanning trees at all: only connected5raphs have spanning trees. A tree has only one spanning tree: itself. Anyconnected graph on n vertices has a minimum of n − n ( n − / n -vertex wheel on Figure 1D has at least2 n − spanning trees (for n ≥ n vertices is n n − .If a graph is not connected, then it contains more than one connected com-ponents. Each connected component has at least one spanning tree, and thewhole graph has at least one spanning forest , comprising of the spanning treesof the components. The number of spanning forests is clearly the product ofthe numbers of the spanning trees of the components.For graphs in general, one can compute the number of their spanning forestsby Kirchoff’s matrix tree theorem [26, 27] using the eigenvalues of the Laplacianmatrix [27] of the graph.We show that female connectomes have significantly higher number of span-ning trees than the connectomes of males. For example, in the 129-vertex resolu-tion, in the left hemisphere, the logarithm of the number of the spanning forestsin the unweighted case are 162 .
01 in females, 158 .
88 in males with p = 0 .
3. Materials and Methods • Unweighted : Each edge has weight 1.6
FiberN : The number of fibers traced along the edge: this number is largerthan one if more than one fibers connect two cortical or sub-cortical areas,corresponding to the two endpoints of the edge. • FAMean : The arithmetic mean of the fractional anisotropies [30] of thefibers, belonging to the edge. • FiberLengthMean : The average length of the fibers, connecting the twoendpoints of the edge. • FiberNDivLength : The number of fibers belonging to the edge, dividedby their average length. This quantity is related to the simple electricalmodel of the nerve fibers: by modeling the fibers as electrical resistorswith resistances proportional to the average fiber length, this quantityis precisely the conductance between the two regions of interest. Addi-tionally,
FiberNDivLength can be observed as a reliability measure of theedge: longer fibers are less reliable than the shorter ones, due to possi-ble error accumulation in the tractography algorithm that constructs thefibers from the anisotropy data. Multiple fibers connecting the same twoROIs, corresponding to the endpoints, add to the reliability of the edge,because of the independently tractographed connections.By generalized adjacency matrix we mean a matrix of size n × n where n is the number of nodes (or vertices ) in the graph, whose rows and columnscorrespond to the nodes, and whose each element is either zero if there is noedge between the two nodes, or equals to the weight of the edge connecting thetwo nodes. By the generalized degree of a node we mean the sum of the weightsof the edges adjacent to that node. Note that the generalized degree of the node v is exactly the sum of the elements in the row (or column) of the generalizedadjacency matrix corresponding to v . By generalized Laplacian matrix we meanthe matrix D − A , where D is a diagonal matrix containing the generalizeddegrees, and A is the generalized adjacency matrix. We calculated various graph parameters for each brain graph and weightfunction. These parameters included: • Number of edges (
Sum ). The weighted version of this quantity is the sumof the weights of the edges. • Normalized largest eigenvalue (
AdjLMaxDivD ): The largest eigenvalue ofthe generalized adjacency matrix, divided by the average degree. Divid-ing by the average degree of vertices was necessary because the largesteigenvalue is bounded by the average- and maximum degrees, and thus isconsidered by some a kind of “average degree” itself [24]. This means thata denser graph may have a bigger λ max largest eigenvalue solely becauseof a larger average degree. We note that the average degree is alreadydefined by the sum of weights. 7 Eigengap of the transition matrix (
PGEigengap ): The transition matrix P G is obtained by dividing all the rows of the generalized adjacency matrixby the generalized degree of the corresponding node. When performing arandom walk on the graph, for nodes i and j , the corresponding matrixelement describes the probability of transitioning to node j , supposingthat we are at node i . The eigengap of a matrix is the difference ofthe largest and the second largest eigenvalue. It is characteristic to theexpander properties of the graph: the larger the gap, the better expanderis the graph (see [23] for the exact statements and proofs). • Hoffman’s bound (
HoffmanBound ): The expression1 + λ max | λ min | , where λ max and λ min denote the largest and smallest eigenvalues of theadjacency matrix. It is a lower bound for the chromatic number of thegraph. The chromatic number is generally higher for denser graphs, asthe addition of an edge may make a previously valid coloring invalid. • Logarithm of number of spanning forests (
LogAbsSpanningForestN ): Thenumber of the spanning trees in a connected graph can be calculatedfrom the spectrum of its Laplacian [26, 27]. Denser graphs tend to havemore spanning trees, as the addition of an edge introduces zero or morenew spanning trees. If a graph is not connected, then the number ofspanning forests is the product of the numbers of the spanning trees ofthe components. The parameter
LogAbsSpanningForestN equals to thelogarithm of the number of spanning forests in the unweighted case. Inthe case of other weight functions, if we define the weight of a tree by theproduct of the weights of its edges, then this parameter equals to the sumof the logarithms of the weights of the spanning trees in the forests. • Balanced minimum cut, divided by the number ofedges (
MinCutBalDivSum ): The task is to partition the graph into twosets whose size may differ from each other by at most 1, so that the num-ber of edges crossing the cut is minimal. This is the “balanced minimumcut” problem, or sometimes called the “minimum bisection width” prob-lem. For the whole brain graph, our expectation was that the minimumcut corresponds to the boundary of the two hemispheres, which was indeedproven when we analyzed the results. • Minimum cost spanning tree (
MinSpanningForest ), calculated withKruskal’s algorithm. • Minimum weighted vertex cover (
MinVertexCover ): Each vertex shouldhave a (possibly fractional) weight assigned such that, for each edge, thesum of the weights of its two endpoints is at least 1. This is the fractionalrelaxation of the NP-hard vertex-cover problem [31]. The minimum of thesum of all vertex-weights is computable by a linear programming approach.8
Minimum vertex cover (
MinVertexCoverBinary ): Same as above, buteach weight must be 0 or 1. In other words, a minimum size set of verticesis selected such that each edge is covered by at least one of the selected ver-tices. This NP-hard graph-parameter is computed only for the unweightedcase. The exact values are computed by an integer programming solverSCIP (http://scip.zib.de), [32, 33].The above 9 parameters were computed for all three resolutions and for theleft and the right hemispheres and also for the whole connectome, with all 5weight functions (with the following exceptions:
MinVertexCoverBinary wascomputed only for the unweighted case, and the
MinSpanningForest was notcomputed for the unweighted case).
Since each connectome was computed in multiple resolutions (in 83, 129 and234 nodes), we had three graphs for each brain. In addition, the parameterswere calculated separately for the connectome within the left and right hemi-spheres as well, not only the whole graph, since we intended to examine whetherstatistically significant differences can be attributed to the left or right hemi-spheres. Each subjects’ brain was corresponded to 9 graphs (3 resolutions, eachin the left and the right hemispheres, plus the whole cortex with sub-corticalareas) and for each graph we calculated 9 parameters, each (with the excep-tions noted above) with 5 different edge weights. This means that we assigned7 · · · · all the corresponding graph parameters differ significantly in sex groups at acombined significance level of 5%.We also highlighted (in italic) those p-values which were individually lessthan the threshold, meaning that these hypotheses can individually be rejectedat a level of 5%, but it is very likely that not all of these graph parameters aresignificantly different between the sexes.
4. Conclusions:
We have computed 83-, 129- and 234-vertex-graphs from the diffusion MRIimages of the 96 subjects of 52 females and 44 males, between the age of 22and 35. We have found, after a careful statistical analysis, significant differ-ences between some graph theoretical parameters of the male and female braingraphs. Our findings show that the female brain graphs have generally moreedges (counted with and without weights), have larger normalized minimum bi-section widths and have more spanning trees (counted with and without weights)than the connectomes of males (Table 1). Additionally, with weaker statisticalvalidity, some spectral properties and the minimum vertex cover also differ inthe connectomes of different sexes (each with p < .
5. Data availability:
Scale Property p (1st) p (2nd) p (corrected)129 Right MinCutBalDivSum FAMean 0.00807
83 All LogSpanningForestN FiberNDivLength 0.00003
234 All PGEigengap FiberNDivLength 0.00321
129 All PGEigengap FiberNDivLength 0.00792
83 Left MinCutBalDivSum FiberN 0.00403
83 Right MinCutBalDivSum FAMean 0.00496
129 Left PGEigengap FiberNDivLength 0.00223
234 All PGEigengap FiberN 0.00826
83 All Sum Unweighted 0.00025
129 Left MinCutBalDivSum FiberN 0.00001
83 All LogSpanningForestN FiberN 0.00001
83 Right Sum FAMean 0.00028
234 All Sum Unweighted 0.00063
234 Left PGEigengap FiberNDivLength 0.00013
129 All Sum Unweighted 0.00026
34 All Sum FAMean 0.00014
129 All LogSpanningForestN FiberN 0.00000
34 Right PGEigengap FAMean 0.00074 all the first 12 rows describe significantly different graph theo-retical properties between sexes. One-by-one, each row with italic thirdcolumn describe significant differences between sexes, with p=0.05. Forthe details we refer to the section “Statistical analysis”.
6. Acknowledgments
The authors declare no conflicts of interest.
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In this appendix we list the graph-theoretic parameters computed for theresolutions of 83, 129 and 234 vertex graphs. The tables contain their arith-metic means in the male and female groups, and the corresponding p-values.The values in these tables contain the values corresponded to round 1 (see the“Statistical analysis” subsection in the main text).The graph-parameters are defined in the caption of Table 1.Significant differences ( p < .
01) are denoted with an asterisk in the lastcolumn.
Scale 83, round 1
Property Female Male p-valueAll AdjLMaxDivD FAMean 1.36008 1.37750 0.06806All AdjLMaxDivD FiberLengthMean 1.44214 1.43602 0.72030All AdjLMaxDivD FiberN 2.02416 2.10529 0.05606All AdjLMaxDivD FiberNDivLength 1.84476 1.86864 0.41834All AdjLMaxDivD Unweighted 1.26760 1.26456 0.63251All HoffmanBound FAMean 4.36096 4.18564 0.00087 ∗ All HoffmanBound FiberLengthMean 3.21938 3.26552 0.33136All HoffmanBound FiberN 2.63525 2.55573 0.03144All HoffmanBound FiberNDivLength 2.51038 2.40550 0.01815All HoffmanBound Unweighted 4.55192 4.43931 0.04616All LogSpanningForestN FAMean 110.69890 101.82758 0.00012 ∗ All LogSpanningForestN FiberLengthMean 456.60084 452.95875 0.18687All LogSpanningForestN FiberN 397.53780 389.79037 0.00001 ∗ All LogSpanningForestN FiberNDivLength 148.03174 139.85355 0.00003 ∗ All LogSpanningForestN Unweighted 191.66035 187.85180 0.00113 ∗ All MinCutBalDivSum FAMean 0.00793 0.00474 0.14869All MinCutBalDivSum FiberLengthMean 0.03115 0.02889 0.47008All MinCutBalDivSum FiberN 0.02924 0.02711 0.34092All MinCutBalDivSum FiberNDivLength 0.02868 0.02644 0.38768All MinCutBalDivSum Unweighted 0.04001 0.03721 0.28887All MinSpanningForest FAMean 19.78188 18.63722 0.02232All MinSpanningForest FiberLengthMean 1096.37958 1112.97289 0.10506All MinSpanningForest FiberN 99.53846 102.93333 0.14280All MinSpanningForest FiberNDivLength 3.65548 3.66822 0.93669All MinVertexCoverBinary Unweighted 59.80769 59.00000 0.00716 ∗ All MinVertexCover FAMean 18.73144 18.10619 0.01699All MinVertexCover FiberLengthMean 2014.06431 1955.70824 0.37460All MinVertexCover FiberN 2427.21154 2315.20000 0.00244 ∗ All MinVertexCover FiberNDivLength 110.25657 103.59777 0.00036 ∗ All MinVertexCover Unweighted 40.90385 41.00000 0.32897All PGEigengap FAMean 0.05403 0.05071 0.28914All PGEigengap FiberLengthMean 0.04167 0.03891 0.43309All PGEigengap FiberN 0.03156 0.02829 0.03885All PGEigengap FiberNDivLength 0.03470 0.03062 0.01847All PGEigengap Unweighted 0.05214 0.04740 0.09708All Sum FAMean 222.01291 201.02562 0.00029 ∗ All Sum FiberLengthMean 16845.33062 15792.24352 0.06219All Sum FiberN 11261.65385 10237.13333 0.00000 ∗ All Sum FiberNDivLength 476.56342 433.37987 0.00002 ∗ All Sum Unweighted 567.07692 539.80000 0.00025 ∗ Left AdjLMaxDivD FAMean 1.33644 1.35216 0.15767Left AdjLMaxDivD FiberLengthMean 1.40515 1.38890 0.32795Left AdjLMaxDivD FiberN 1.90607 2.02087 0.00501 ∗ Left AdjLMaxDivD FiberNDivLength 1.71498 1.77482 0.07539Left AdjLMaxDivD Unweighted 1.24027 1.23523 0.43598Left HoffmanBound FAMean 4.55406 4.38621 0.01297Left HoffmanBound FiberLengthMean 3.25098 3.28435 0.51250Left HoffmanBound FiberN 2.71430 2.61098 0.00175 ∗ eft HoffmanBound FiberNDivLength 2.66652 2.59451 0.13782Left HoffmanBound Unweighted 4.73205 4.57434 0.01379Left LogSpanningForestN FAMean 53.30579 48.82905 0.00082 ∗ Left LogSpanningForestN FiberLengthMean 229.63370 227.32675 0.18765Left LogSpanningForestN FiberN 199.27958 195.25428 0.00012 ∗ Left LogSpanningForestN FiberNDivLength 73.53683 69.82889 0.00343 ∗ Left LogSpanningForestN Unweighted 95.46307 93.39767 0.01389Left MinCutBalDivSum FAMean 0.00687 0.00320 0.17151Left MinCutBalDivSum FiberLengthMean 0.23438 0.21147 0.01779Left MinCutBalDivSum FiberN 0.13337 0.12011 0.00403 ∗ Left MinCutBalDivSum FiberNDivLength 0.11057 0.09321 0.00031 ∗ Left MinCutBalDivSum Unweighted 0.24513 0.22019 0.00206 ∗ Left MinSpanningForest FAMean 9.57924 9.06313 0.04242Left MinSpanningForest FiberLengthMean 561.47024 560.36391 0.87722Left MinSpanningForest FiberN 51.23077 53.73333 0.26795Left MinSpanningForest FiberNDivLength 1.82447 1.89521 0.62729Left MinVertexCoverBinary Unweighted 30.23077 29.73333 0.09601Left MinVertexCover FAMean 9.23616 8.88642 0.01371Left MinVertexCover FiberLengthMean 1064.27185 1027.73430 0.35926Left MinVertexCover FiberN 1158.21154 1143.46667 0.55321Left MinVertexCover FiberNDivLength 54.26322 51.17634 0.02122Left MinVertexCover Unweighted 20.80769 20.83333 0.75017Left PGEigengap FAMean 0.33446 0.29469 0.00215 ∗ Left PGEigengap FiberLengthMean 0.33383 0.29287 0.01329Left PGEigengap FiberN 0.16980 0.15238 0.01654Left PGEigengap FiberNDivLength 0.14486 0.13413 0.02837Left PGEigengap Unweighted 0.30646 0.27160 0.00458 ∗ Left Sum FAMean 106.64056 96.80731 0.00056 ∗ Left Sum FiberLengthMean 8629.73791 8122.82646 0.13250Left Sum FiberN 5514.61538 5049.73333 0.00000 ∗ Left Sum FiberNDivLength 233.06402 213.49323 0.00043 ∗ Left Sum Unweighted 282.50000 269.06667 0.00378 ∗ Right AdjLMaxDivD FAMean 1.32878 1.34242 0.14511Right AdjLMaxDivD FiberLengthMean 1.39672 1.38478 0.30191Right AdjLMaxDivD FiberN 2.00803 2.09048 0.05380Right AdjLMaxDivD FiberNDivLength 1.76990 1.81343 0.09784Right AdjLMaxDivD Unweighted 1.25268 1.24720 0.29540Right HoffmanBound FAMean 4.47438 4.28666 0.00587 ∗ Right HoffmanBound FiberLengthMean 3.33823 3.39478 0.29902Right HoffmanBound FiberN 2.67311 2.57701 0.05411Right HoffmanBound FiberNDivLength 2.62635 2.48983 0.00560 ∗ Right HoffmanBound Unweighted 4.61480 4.50726 0.03806Right LogSpanningForestN FAMean 52.25642 48.14346 0.00067 ∗ Right LogSpanningForestN FiberLengthMean 218.25106 216.24411 0.16431Right LogSpanningForestN FiberN 190.62427 187.02757 0.00083 ∗ Right LogSpanningForestN FiberNDivLength 69.84080 66.17446 0.00022 ∗ Right LogSpanningForestN Unweighted 90.24090 88.51678 0.00541 ∗ Right MinCutBalDivSum FAMean 0.02476 0.00851 0.00496 ∗ Right MinCutBalDivSum FiberLengthMean 0.24577 0.22309 0.02216Right MinCutBalDivSum FiberN 0.13346 0.12050 0.00346 ∗ Right MinCutBalDivSum FiberNDivLength 0.10831 0.09357 0.00072 ∗ Right MinCutBalDivSum Unweighted 0.23713 0.22022 0.01629Right MinSpanningForest FAMean 10.30911 9.79708 0.10419Right MinSpanningForest FiberLengthMean 532.13580 547.85331 0.00491 ∗ Right MinSpanningForest FiberN 50.76923 52.53333 0.26282Right MinSpanningForest FiberNDivLength 1.94340 1.89232 0.58863Right MinVertexCoverBinary Unweighted 29.07692 28.73333 0.15457Right MinVertexCover FAMean 9.26572 9.03965 0.12382Right MinVertexCover FiberLengthMean 934.26071 897.95882 0.23661Right MinVertexCover FiberN 1169.63462 1122.93333 0.07986Right MinVertexCover FiberNDivLength 53.57144 51.50298 0.10452Right MinVertexCover Unweighted 20.11538 20.26667 0.10527Right PGEigengap FAMean 0.32454 0.28808 0.00112 ∗ Right PGEigengap FiberLengthMean 0.34029 0.29461 0.00949 ∗ Right PGEigengap FiberN 0.17666 0.15912 0.02617Right PGEigengap FiberNDivLength 0.15245 0.14034 0.01613Right PGEigengap Unweighted 0.29582 0.26081 0.00087 ∗ Right Sum FAMean 105.62164 95.26436 0.00028 ∗ ight Sum FiberLengthMean 7644.90330 7086.91000 0.02974Right Sum FiberN 5378.03846 4884.66667 0.00000 ∗ Right Sum FiberNDivLength 225.94776 206.97587 0.00018 ∗ Right Sum Unweighted 261.30769 248.26667 0.00019 ∗ Scale 129, round 1
Property Female Male p-valueAll AdjLMaxDivD FAMean 1.40519 1.42604 0.10040All AdjLMaxDivD FiberLengthMean 1.50483 1.50158 0.87806All AdjLMaxDivD FiberN 2.14552 2.22254 0.15242All AdjLMaxDivD FiberNDivLength 2.09783 2.04782 0.32031All AdjLMaxDivD Unweighted 1.30028 1.29097 0.27278All HoffmanBound FAMean 4.40157 4.29660 0.02644All HoffmanBound FiberLengthMean 3.19684 3.24689 0.32568All HoffmanBound FiberN 2.50604 2.48884 0.64956All HoffmanBound FiberNDivLength 2.34647 2.41938 0.07720All HoffmanBound Unweighted 4.62935 4.51267 0.01233All LogSpanningForestN FAMean 194.37749 181.03525 0.00019 ∗ All LogSpanningForestN FiberLengthMean 739.78985 732.55388 0.09867All LogSpanningForestN FiberN 599.76631 588.61699 0.00000 ∗ All LogSpanningForestN FiberNDivLength 210.52236 200.75240 0.00000 ∗ All LogSpanningForestN Unweighted 322.09324 316.62672 0.00218 ∗ All MinCutBalDivSum FAMean 0.00668 0.00324 0.05930All MinCutBalDivSum FiberLengthMean 0.01706 0.01607 0.56293All MinCutBalDivSum FiberN 0.02658 0.02429 0.26627All MinCutBalDivSum FiberNDivLength 0.02495 0.02258 0.30029All MinCutBalDivSum Unweighted 0.02218 0.02065 0.30082All MinSpanningForest FAMean 30.14746 28.58509 0.02073All MinSpanningForest FiberLengthMean 1642.68263 1664.23693 0.07510All MinSpanningForest FiberN 140.23077 140.93333 0.55077All MinSpanningForest FiberNDivLength 4.42401 4.43795 0.92181All MinVertexCoverBinary Unweighted 96.46154 96.26667 0.66793All MinVertexCover FAMean 29.56250 28.72424 0.02181All MinVertexCover FiberLengthMean 3230.07900 3121.21684 0.29100All MinVertexCover FiberN 2444.92308 2337.40000 0.00232 ∗ All MinVertexCover FiberNDivLength 120.18766 116.22553 0.02502All MinVertexCover Unweighted 63.88462 63.96667 0.35805All PGEigengap FAMean 0.03143 0.02928 0.25524All PGEigengap FiberLengthMean 0.02427 0.02260 0.43054All PGEigengap FiberN 0.02781 0.02453 0.01902All PGEigengap FiberNDivLength 0.02880 0.02498 0.00792 ∗ All PGEigengap Unweighted 0.03012 0.02725 0.09661All Sum FAMean 397.68878 360.50850 0.00015 ∗ All Sum FiberLengthMean 30670.09535 28478.19852 0.03582All Sum FiberN 12375.61538 11458.13333 0.00000 ∗ All Sum FiberNDivLength 548.61301 510.71378 0.00008 ∗ All Sum Unweighted 1020.80769 972.86667 0.00026 ∗ Left AdjLMaxDivD FAMean 1.37823 1.39812 0.12792Left AdjLMaxDivD FiberLengthMean 1.43638 1.42179 0.36739Left AdjLMaxDivD FiberN 1.84672 1.92762 0.12247Left AdjLMaxDivD FiberNDivLength 1.77313 1.80979 0.33521Left AdjLMaxDivD Unweighted 1.26380 1.25501 0.16858Left HoffmanBound FAMean 4.57539 4.44885 0.01512Left HoffmanBound FiberLengthMean 3.23550 3.25088 0.77158Left HoffmanBound FiberN 2.80373 2.74220 0.14090Left HoffmanBound FiberNDivLength 2.70077 2.64308 0.21782Left HoffmanBound Unweighted 4.75280 4.61941 0.00848 ∗ Left LogSpanningForestN FAMean 96.11000 89.25516 0.00106 ∗ Left LogSpanningForestN FiberLengthMean 373.09476 368.65582 0.08843Left LogSpanningForestN FiberN 300.77613 295.83044 0.00014 ∗ Left LogSpanningForestN FiberNDivLength 105.01323 100.80980 0.00288 ∗ Left LogSpanningForestN Unweighted 162.01302 158.88026 0.01336Left MinCutBalDivSum FAMean 0.00873 0.00273 0.05683Left MinCutBalDivSum FiberLengthMean 0.19822 0.17378 0.00892 ∗ Left MinCutBalDivSum FiberN 0.12848 0.10467 0.00001 ∗ Left MinCutBalDivSum FiberNDivLength 0.06926 0.05546 0.00019 ∗ eft MinCutBalDivSum Unweighted 0.19535 0.17339 0.00265 ∗ Left MinSpanningForest FAMean 14.57467 13.88500 0.06189Left MinSpanningForest FiberLengthMean 828.34729 834.54850 0.36946Left MinSpanningForest FiberN 69.30769 72.20000 0.02902Left MinSpanningForest FiberNDivLength 2.16989 2.25626 0.53695Left MinVertexCoverBinary Unweighted 48.76923 48.86667 0.69355Left MinVertexCover FAMean 14.65360 14.09857 0.01273Left MinVertexCover FiberLengthMean 1700.29684 1637.18742 0.30481Left MinVertexCover FiberN 1169.82692 1125.20000 0.06266Left MinVertexCover FiberNDivLength 58.76113 56.23736 0.06303Left MinVertexCover Unweighted 32.28846 32.30000 0.88865Left PGEigengap FAMean 0.22611 0.19656 0.00995 ∗ Left PGEigengap FiberLengthMean 0.23241 0.20065 0.02197Left PGEigengap FiberN 0.12346 0.10569 0.00382 ∗ Left PGEigengap FiberNDivLength 0.09689 0.08572 0.00223 ∗ Left PGEigengap Unweighted 0.20204 0.17516 0.01081Left Sum FAMean 197.41850 178.80563 0.00032 ∗ Left Sum FiberLengthMean 16079.40944 14931.40760 0.07487Left Sum FiberN 6071.96154 5641.93333 0.00000 ∗ Left Sum FiberNDivLength 269.09760 251.40080 0.00100 ∗ Left Sum Unweighted 519.53846 492.86667 0.00232 ∗ Right AdjLMaxDivD FAMean 1.35746 1.36837 0.36353Right AdjLMaxDivD FiberLengthMean 1.42015 1.41129 0.54264Right AdjLMaxDivD FiberN 2.05564 2.19134 0.01338Right AdjLMaxDivD FiberNDivLength 1.82146 1.86716 0.20816Right AdjLMaxDivD Unweighted 1.26684 1.25522 0.12057Right HoffmanBound FAMean 4.37886 4.29574 0.20294Right HoffmanBound FiberLengthMean 3.32686 3.36662 0.49418Right HoffmanBound FiberN 2.66511 2.56838 0.01727Right HoffmanBound FiberNDivLength 2.68679 2.59830 0.01992Right HoffmanBound Unweighted 4.60861 4.51407 0.08448Right LogSpanningForestN FAMean 93.41904 87.28295 0.00143 ∗ Right LogSpanningForestN FiberLengthMean 358.00491 354.73456 0.14280Right LogSpanningForestN FiberN 291.08563 285.72242 0.00045 ∗ Right LogSpanningForestN FiberNDivLength 100.74383 96.22891 0.00051 ∗ Right LogSpanningForestN Unweighted 154.36558 151.96595 0.01158Right MinCutBalDivSum FAMean 0.02361 0.01005 0.00807 ∗ Right MinCutBalDivSum FiberLengthMean 0.20000 0.17303 0.00768 ∗ Right MinCutBalDivSum FiberN 0.11452 0.10111 0.00563 ∗ Right MinCutBalDivSum FiberNDivLength 0.06865 0.06326 0.09375Right MinCutBalDivSum Unweighted 0.19180 0.16911 0.00492 ∗ Right MinSpanningForest FAMean 15.61479 14.88977 0.06537Right MinSpanningForest FiberLengthMean 808.14079 824.37649 0.03729Right MinSpanningForest FiberN 70.46154 68.93333 0.07096Right MinSpanningForest FiberNDivLength 2.32813 2.26810 0.46298Right MinVertexCoverBinary Unweighted 47.34615 47.00000 0.29760Right MinVertexCover FAMean 14.70648 14.40974 0.13709Right MinVertexCover FiberLengthMean 1516.99670 1461.52391 0.23679Right MinVertexCover FiberN 1175.50000 1166.36667 0.68666Right MinVertexCover FiberNDivLength 59.59421 58.78162 0.47843Right MinVertexCover Unweighted 31.61538 31.73333 0.20363Right PGEigengap FAMean 0.22838 0.19627 0.00296 ∗ Right PGEigengap FiberLengthMean 0.23840 0.19868 0.01013Right PGEigengap FiberN 0.12500 0.11049 0.00869 ∗ Right PGEigengap FiberNDivLength 0.10075 0.09371 0.03033Right PGEigengap Unweighted 0.20584 0.17429 0.00242 ∗ Right Sum FAMean 190.48228 172.48988 0.00062 ∗ Right Sum FiberLengthMean 13952.01182 13003.32443 0.04620Right Sum FiberN 5935.73077 5525.26667 0.00001 ∗ Right Sum FiberNDivLength 262.31420 246.32048 0.00180 ∗ Right Sum Unweighted 477.38462 454.86667 0.00068 ∗ Scale 234, round 1