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Modularity in Static and Dynamic Networks Modularity in static and dynamic networks Sarah F. Muldoon University at Buffalo, SUNY OHBM – Brain Graphs Workshop June 25, 2017 Outline 1. Introduction: What is modularity? 2. Determining community structure (static networks) 3. Comparing community structure 4. Multilayer networks: Constructing multitask and temporal multilayer dynamic networks 5. Dynamic community structure 6. Useful references OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Introduction: What is Modularity? OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon What is Modularity? Modularity (Community Structure) • A module (community) is a subset of vertices in a graph that have more connections to each other than to the rest of the network • Example social networks: groups of friends Modularity in the brain: • Structural networks: communities are groups of brain areas that are more highly connected to each other than the rest of the brain • Functional networks: communities are groups of brain areas with synchronous activity that is not synchronous with other brain activity OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Findings: The Brain is Modular Structural networks: cortical thickness correlations sensorimotor/spatial strategic/executive mnemonic/emotion olfactocentric auditory/language visual processing Chen et al. (2008) Cereb Cortex OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Findings: The Brain is Modular • Functional networks: resting state fMRI He et al. (2009) PLOS One OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Findings: The Brain is Hierarchical Modules are composed of modules Meunier et al. (2009) Front. Neuroinform. OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Determining Community Structure OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon How to Determine Community Structure? MANY methods (see Fortunato 2010 Phys Rep for a review) Here we focus on method of modularity maximization Define a quantity called modularity: Or more generally: Resolution Parameter Adjacency For Module Standard Standard Matrix Size Community i Null Model Resolution Parameter Null Model Community j Adjacency Matrix Newman (2006) PNAS; Bassett et al. (2013) Chaos OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Maximize Modularity Function! Examine all possible partitions of communities – choose partition that has the highest modularity Problem is NP-hard – need search algorithm: Louvain algorithm: Blondel et al. (2008) J of Stat Mech OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Importance of Multiple Optimizations The modularity energy landscape has multiple near- degeneracies • Must run optimization algorithm many times (~100) and determine a consensus partition • Consensus partitioning using Nodal Association Matrix – see Bassett et al. (2013) Chaos and Lancichinetti et al. (2012) Sci Rep Good et al. (2010) Phys Rev E OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon The Resolution Parameter Traditionally, γ=1 was used. However, changing γ changes the number of detected communities. γ<1: fewer communities γ>1: more communities Determining γ: 1. Look for stable regime 2. Run consensus over multiple parameter choices Alexander-Bloch et al. (2010) Front Syst Neurosci OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Summary: Static Community Detection 1. Construct brain network (structural or functional) 2. Compute modularity and determine community structure – determine choice of resolution parameter 3. Run community detection multiple times and determine a consensus partition. 4. INTERPRET the results! OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Comparing Community Structure OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Normalized Mutual Information How do we tell if detected community structure is alike between conditions? Compute NMI: 0 if partitions are independent 1 if the partitions are identical Many different ways to be “wrong” Can be adjusted for chance (see Vinh et al (2010)) Danon et al. (2005) J Stat Mech; Vinh et al. (2010) J of Mach Learn Res OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Dynamic Networks Cognitive processes are dynamic! • Memory • Learning • Attention • Fatigue • etc… Medaglia et al. (2015) J Cogn Neurosci How does network structure change and evolve over time? • Different tasks • As a function of learning • As people age • Progression of a disease Bassett et al. (2013) Chaos OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Introduction: Multilayer Networks OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Multilayer Networks • Each node belongs to any subset of the layers, and edges encompass pairwise connections between all possible combinations of nodes and layers 4 Nodes 4 Layers 2 Aspects (X,Y) 2 Elementary-Layer Sets {A,B}, {X,Y} Edges can connect nodes in any layers. • An aspect is a feature of the edge: for example, its definition, or the time in which it exists. Kivelä et al. (2014) J Complex Networks OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Multilayer Networks: An Example Network scientists who talked to each other or went to talks by each other across 3 different network science conferences/workshops. Kivelä et al. (2014) J Complex Networks OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Multiplex Networks • Sequential graphs Inter-layer identity links Temporal data: layers coupled to layers adjacent in time Categorical data: all-to-all layer coupling Mucha et al. (2010) Science OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Multitask Networks • How are networks extracted from the brain at rest different from networks extracted during task conditions? • How do task networks differ from one another? How are they similar? • How does the brain transition between tasks in a large task battery? Cole et al. (2014) Neuron OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Temporal Networks • Perhaps the most common type of network in neuroscience • How do connections evolve over time? • Learning • Development • Progression of disease • Resting state fluctuations Bassett et al. (2013) PLoS Comp. Bio OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Constructing Multitask and Temporal Multilayer Dynamic Networks OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Multiplex Networks • Sequential graphs Each layer is a single functional Inter-layer connectivity matrix identity links Temporal Networks 1 2 3 Multitask Networks 4 Mucha et al. (2010) Science OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Multitask Networks • Step 1: Create separate pairwise functional connectivity matrices for each task • Step 2: Couple each node to itself in every matrix (all-to- all between layer coupling) Task 1 Task 2 Task 3 Task 4 Task 1 Task 2 Task 3 Task 4 OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Temporal Networks • Step 1: Choose a windowing scheme (can overlap) • Step 2: Create pairwise functional connectivity matrices for each time window • Step 3: Align matrices in time and couple each node to itself in the following matrix Window 1 Window 2 Window 3 Window 4 Window 1 Window 2 Window 3 60 120 180 240 Window 4 Time (s) OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Dynamic Community Structure OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Multilayer Modularity We can define a multilayer modularity to estimate the community structure in multilayer networks. Resolution Parameter For Module Size For i and j Community Adjacency in same i in time Matrix community slice l Null Model Resolution Community Adjacency Parameter j in time Matrix for Module slice r Dynamics Optimize using a Louvain-like locally greedy algorithm. Blondel et al. 2008; Mucha et al. (2010) Science. • Mucha et al. (2010) Science; Bassett et al. (2013) Chaos OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Community Assignment in Layers Communities exist across layers (throughout time) and can come in and out of existence Bassett et al. (2013) Chaos OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Flexibility and Promiscuity Regional flexibility: The number of times a region changes its module allegiance normalized by possible number of changes Regional promiscuity: The fraction of communities in which a region participates Flexibility Promiscuity Region 1 0 0.25 Region 2 0.11 0.5 Region 3 0.33 1 Region 4 0.89 1 Region 5 0.11 0.5 Network (ave of regions) 0.29 0.65 Bassett et al. (2011) PNAS; Bassett et al. (2013) PLoS Comp Bio; Papadopoulos et al. (2016) Phys Rev E OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Flexibility Predicts Learning • Regional flexibility changes over the course of learning • Higher flexibility predicted higher learning Bassett et al. (2011) PNAS OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Module Allegiance Matrix We can also create a module allegiance matrix which gives the probability of two nodes being assigned to the same community Step 1: look across all partitions and make contingency tables Region 1 2 3 4 5 • 0 if assigned different communities • 1 if assigned same community 1 X 1 1 1 0 2 1 X 1 1 0 3 1 1 X 1 0 4 1 1 1 X 0 5 0 0 0 0 X Bassett et al. (2015) Nat Neuro OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Module Allegiance Matrix Step 2: Average across all contingency tables (across all variables) Region 1 2 3 4 5 Region 1 2 3 4 5 Subjects 1 X 1 1 1 0 1 X .5 .1 .1 0 2 1 X 1 1 0 Time/task 2 .5 X .3 .4 .4 3 1 1 X 1 0 3 .1 .3 X .7 .3 4 1 1 1 X 0 4 .1 .4 .7 X .6 Optimizations 5 0 0 0 0 X 5 0 .4 .3 .6 X Contingency table Module Allegiance Matrix Bassett et al. (2015) Nat Neuro OHBM 2017 – Brain Graphs Workshop – Sarah F. Muldoon Recruitment and Integration Measures interaction between larger scale groups (ex: visual and motor groupings) • Recruitment: strength of interaction within a group • Integration: strength of interaction between groups Module Allegiance Matrix Number of regions in group ki Recruitment if k1=k2 Integration if k1≠k2 Bassett et al. (2015) Nat Neuro OHBM 2017 – Brain Graphs Workshop – Sarah F.
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