Nonlinearity + Networks: A 2020 Vision
NNonlinearity + Networks: A 2020 Vision
Mason A. Porter
Abstract
I briefly survey several fascinating topics in networks and nonlinearity.I highlight a few methods and ideas, including several of personal interest, that Ianticipate to be especially important during the next several years. These topicsinclude temporal networks (in which the entities and/or their interactions changein time), stochastic and deterministic dynamical processes on networks, adaptivenetworks (in which a dynamical process on a network is coupled to dynamics ofnetwork structure), and network structure and dynamics that include “higher-order”interactions (which involve three or more entities in a network). I draw examplesfrom a variety of scenarios, including contagion dynamics, opinion models, waves,and coupled oscillators.
Network analysis is one of the most exciting areas of applied and industrial math-ematics [121, 141, 143]. It is at the forefront of numerous and diverse applicationsthroughout the sciences, engineering, technology, and the humanities. The studyof networks combines tools from numerous areas of mathematics, including graphtheory, linear algebra, probability, statistics, optimization, statistical mechanics, sci-entific computation, and nonlinear dynamics.In this chapter, I give a short overview of popular and state-of-the-art topics innetwork science. My discussions of these topics, which I draw preferentially fromones that relate to nonlinear and complex systems, will be terse, but I will cite manyreview articles and highlight specific research papers for those who seek more details.This chapter is not a review or even a survey; instead, I give my perspective on theshort-term and medium-term future of network analysis in applied mathematics for2020 and beyond.My presentation proceeds as follows. In Section 2, I review a few basic conceptsfrom network analysis. In Section 3, I discuss the dynamics of networks in theform of time-dependent (“temporal”) networks. In Section 4, I discuss dynamicalprocesses — both stochastic and deterministic — on networks. In Section 5, I discussadaptive networks, in which there is coevolution of network structure and a dynamicalprocess on that structure. In Section 6, I discuss higher-order structures (specifically,hypergraphs and simplicial complexes) that aim to go beyond the standard networkparadigm of pairwise connections. I conclude with an outlook in Section 7.
Mason A. PorterDepartment of Mathematics, University of California, Los Angeles, Los Angeles, California 90095,USA e-mail: [email protected] a r X i v : . [ c s . S I] N ov Mason A. Porter
In its broadest form, a network consists of the connectivity patterns and connectionstrengths in a complex system of interacting entities [121]. The most traditional typeof network is a graph G = ( V , E ) (see Fig. 1a), where V is a set of “nodes” (i.e.,“vertices”) that encode entities and E ⊆ V × V is a set of “edges” (i.e., “links”or “ties”) that encode the interactions between those entities. However, recent usesof the term “network” have focused increasingly on connectivity patterns that aremore general than graphs [98]: a network’s nodes and/or edges (or their associatedweights) can change in time [70, 72] (see Section 3), nodes and edges can includeannotations [26], a network can include multiple types of edges and/or multiple typesof nodes [90,140], it can have associated dynamical processes [142] (see Sections 3,4, and 5), it can include memory [152], connections can occur between an arbitrarynumber of entities [127, 131] (see Section 6), and so on.Associated with a graph is an adjacency matrix A with entries a ij . In the simplestscenario, edges either exist or they don’t. If edges have directions, a ij = j to entity i and a ij = a ij =
1, node i is “adjacent” to node j (because we can reach i directly from j ), and the associated edge is “incident” from node j and to node i . The edge from j to i is an “out-edge” of j and an “in-edge” of i . The number of out-edges of a nodeis its “out-degree”, and the number of in-edges of a node is its “in-degree”. For anundirected network, a ij = a ji , and the number of edges that are attached to a node isthe node’s “degree”. One can assign weights to edges to represent connections withdifferent strengths (e.g., stronger friendships or larger transportation capacity) bydefining a function w : E −→ R . In many applications, the weights are nonnegative,although several applications [180] (such as in international relations) incorporatepositive, negative, and zero weights. In some applications, nodes can also have self-edges and multi-edges. The spectral properties of adjacency (and other) matricesgive important information about their associated graphs [121, 187]. For undirectednetworks, it is common to exploit the beneficent property that all eigenvalues ofsymmetric matrices are real. Traditional studies of networks consider time-independent structures, but most net-works evolve in time. For example, social networks of people and animals changebased on their interactions, roads are occasionally closed for repairs and new roadsare built, and airline routes change with the seasons and over the years. To studysuch time-dependent structures, one can analyze “temporal networks”. See [70, 72]for reviews and [73, 74] for edited collections.The key idea of a temporal network is that networks change in time, but thereare many ways to model such changes, and the time scales of interactions andother changes play a crucial role in the modeling process. There are also other onlinearity + Networks: A 2020 Vision 3(a)
A BC DE
Figure 1.
A graph consists of nodes (which I show as disks) that areconnected to each other by edges (which I show as arcs). [I drewthis network using
Tikz-network , by J¨urgen Hackl and available at https://github.com/hackl/tikz-network ), which allows one to draw networks(including multilayer networks) directly in a L A TEX file.] (b)(c) ( A , ) ( B , ) ( C , )( D , ) ( E , )( A , ) ( B , ) ( C , )( A , ) ( B , ) ( C , )( D , ) ( E , ) Figure 2.
An example of a multilayer network with three layers. We label eachlayer using di↵erent colours for its state nodes and its edges: black nodes and brownedges (three of which are unidirectional) for layer 1, purple nodes and green edgesfor layer 2, and pink nodes and grey edges for layer 3. Each state node (i.e. node-layer tuple) has a corresponding physical node and layer, so the tuple (A ,
3) denotesphysical node A on layer 3, the tuple (D ,
1) denotes physical node D on layer 1,and so on. We draw intralayer edges using solid arcs and interlayer edges usingbroken arcs; an interlayer edge is dashed (and magenta) if it connects correspondingentities and dotted (and blue) if it connects distinct ones. We include arrowheads torepresent unidirectional edges. We drew this network using Tikz-network (J¨urgenHackl, https://github.com/hackl/tikz-network), which allows one to draw multilayernetworks directly in a L A TEX file. (d)
Fig. 1
Several types of network structures: (a) a graph, (b) a temporal network, (c) a multilayernetwork, and (d) a simplicial complex. [I drew panels (a) and (c) using Tikz-network, which isby Jürgen Hackl and is available at https://github.com/hackl/tikz-network . Panel (b) isinspired by Fig. 1 of [72]. Panel (d), which is in the public domain, was drawn by Wikipedia userCflm001 and is available at https://en.wikipedia.org/wiki/Simplicial_complex .] important modeling considerations. To illustrate potential complications, supposethat an edge in a temporal network represents close physical proximity between twopeople in a short time window (e.g., with a duration of two minutes). It is relevant toconsider whether there is an underlying social network (e.g., the friendship networkof mathematics Ph.D. students at UCLA) or if the people in the network do not ingeneral have any other relationships with each other (e.g., two people who happento be visiting a particular museum on the same day). In both scenarios, edges thatrepresent close physical proximity still appear and disappear over time, but indirectconnections (i.e., between people who are on the same connected component, butwithout an edge between them) in a time window may play different roles in thespread of information. Moreover, network structure itself is often influenced by aspreading process or other dynamics, as perhaps one arranges a meeting to discussa topic (e.g., to give me comments on a draft of this chapter). See my discussion ofadaptive networks in Section 5. For convenience, most work on temporal networks employs discrete time (seeFig. 1(b)). Discrete time can arise from the natural discreteness of a setting, dis-
Mason A. Porter cretization of continuous activity over different time windows, data measurementthat occurs at discrete times, and so on.
One way to represent a discrete-time (or discretized-time) temporal network is touse the formalism of “multilayer networks” [90, 140]. One can also use multilayernetworks to study networks with multiple types of relations, networks with multiplesubsystems, and other complicated networked structures.A multilayer network M = ( V M , E M , V , L ) (see Fig. 1(c)) has a set V of nodes —these are sometimes called “physical nodes”, and each of them corresponds to anentity, such as a person — that have instantiations as “state nodes” (i.e., node-layertuples, which are elements of the set V M ) on layers in L . One layer in the set L isa combination, through the Cartesian product L × · · · × L d , of elementary layers.The number d indicates the number of types of layering; these are called “aspects”.A temporal network with one type of relationship has one type of layering, a time-independent network with multiple types of social relationships also has one type oflayering, a multirelational network that changes in time has two types of layering,and so on. The set of state nodes in M is V M ⊆ V × L × · · · × L d , and the set ofedges is E M ⊆ V M × V M . The edge (( i , α ) , ( j , β )) ∈ E M indicates that there is anedge from node j on layer β to node i on layer α (and vice versa, if M is undirected).For example, in Fig. 1(c), there is a directed intralayer edge from (A,1) to (B,1)and an undirected interlayer edge between (A,1) and (A,2). The multilayer networkin Fig. 1(c) has three layers, | V | = d = | V M | = | E M | =
20 edges. To consider weighted edges, one proceeds as inordinary graphs by defining a function w : E M −→ R . As in ordinary graphs, onecan also incorporate self-edges and multi-edges.Multilayer networks can include both intralayer edges (which have the samemeaning as in graphs) and interlayer edges. The multilayer network in Fig. 1(c)has 4 directed intralayer edges, 10 undirected intralayer edges, and 6 undirectedinterlayer edges. In most studies thus far of multilayer representations of temporalnetworks, researchers have included interlayer edges only between state nodes inconsecutive layers and only between state nodes that are associated with the sameentity (see Fig. 1(c)). However, this restriction is not always desirable (see [184] foran example), and one can envision interlayer couplings that incorporate ideas liketime horizons and interlayer edge weights that decay over time. For convenience,many researchers have used undirected interlayer edges in multilayer analyses oftemporal networks, but it is often desirable for such edges to be directed to reflectthe arrow of time [176]. The sequence of network layers, which constitute timelayers, can represent a discrete-time temporal network at different time instances ora continuous-time network in which one bins (i.e., aggregates) the network’s edgesto form a sequence of time windows with interactions in each window.Each d -aspect multilayer network with the same number of nodes in each layerhas an associated adjacency tensor A of order 2 ( d + ) . For unweighted multilayer onlinearity + Networks: A 2020 Vision 5 networks, each edge in E M is associated with a 1 entry of A , and the other entries(the “missing” edges) are 0. If a multilayer network does not have the same numberof nodes in each layer, one can add empty nodes so that it does, but the edgesthat are attached to such nodes are “forbidden”. There has been some research ontensorial properties of A [35] (and it is worthwhile to undertake further studiesof them), but the most common approach for computations is to flatten A into a“supra-adjacency matrix” A M [90, 140], which is the adjacency matrix of the graph G M that is associated with M . The entries of diagonal blocks of A M correspondto intralayer edges, and the entries of off-diagonal blocks correspond to interlayeredges. Following a long line of research in sociology [37], two important ingredients in thestudy of networks are examining (1) the importances (“centralities”) of nodes, edges,and other small network structures and the relationship of measures of importanceto dynamical processes on networks and (2) the large-scale organization of networks[121, 193].Studying central nodes in networks is useful for numerous applications, such asranking Web pages, football teams, or physicists [56]. It can also help reveal the rolesof nodes in networks, such as those that experience high traffic or help bridge differentparts of a network [121, 193]. Mesoscale features can impact network function anddynamics in important ways. Small subgraphs called “motifs” may appear frequentlyin some networks [111], perhaps indicating fundamental structures such as feedbackloops and other building blocks of global behavior [59]. Various types of larger-scale network structures, such as dense “communities” of nodes [47, 145] and core–periphery structures [33, 150], are also sometimes related to dynamical modules(e.g., a set of synchronized neurons) or functional modules (e.g., a set of proteinsthat are important for a certain regulatory process) [164]. A common way to studylarge-scale structures is inference using statistical models of random networks,such as through stochastic block models (SBMs) [134]. Much recent research hasgeneralized the study of large-scale network structure to temporal and multilayernetworks [3, 74, 90].Various types of centrality — including betweenness centrality [88, 173],Bonacich and Katz centrality [65,102], communicability [64], PageRank [151,191],and eigenvector centrality [46, 146] — have been generalized to temporal networksusing a variety of approaches. Such generalizations make it possible to examine hownode importances change over time as network structure evolves.In recent work, my collaborators and I used multilayer representations of tem-poral networks to generalize eigenvector-based centralities to temporal networks There are recent theoretical advances on examining network structure amidst rich but noisydata [120], and it is important for research on both network structure and dynamics to explicitlyconsider such scenarios. Mason A. Porter [175, 176]. One computes the eigenvector-based centralities of nodes for a time-independent network as the entries of the “dominant” eigenvector, which is asso-ciated with the largest positive eigenvalue (by the Perron–Frobenius theorem, theeigenvalue with the largest magnitude is guaranteed to be positive in these situa-tions) of a centrality matrix C ( A ) . Examples include eigenvector centrality (by using C ( A ) = A ) [17], hub and authority scores (by using C ( A ) = AA T for hubs and A T A for authorities) [91], and PageRank [56].Given a discrete-time temporal network in the form of a sequence of adjacencymatrices A ( t ) ∈ R N × N for t ∈ { , . . . , T } , where a ( t ) ij denotes a directed edge fromentity i to entity j in time layer t , we construct a “supracentrality matrix” C ( ω ) , whichcouples centrality matrices C ( A ( t ) ) of the individual time layers. We then computethe dominant eigenvector of C ( ω ) , where ω is an interlayer coupling strength. In [175,176], a key example was the ranking of doctoral programs in the mathematicalsciences (using data from the Mathematics Genealogy Project [147]), where an edgefrom one institution to another arises when someone with a Ph.D. from the firstinstitution supervises a Ph.D. student at the second institution. By calculating time-dependent centralities, we can study how the rankings of mathematical-sciencesdoctoral programs change over time and the dependence of such rankings on thevalue of ω . Larger values of ω impose more ranking consistency across time, socentrality trajectories are less volatile for larger ω [175, 176].Multilayer representations of temporal networks have been very insightful in thedetection of communities and how they split, merge, and otherwise evolve over time.Numerous methods for community detection — including inference via SBMs [135],maximization of objective functions (especially “modularity”) [117], and methodsbased on random walks and bottlenecks to their traversal of a network [38, 80]— have been generalized from graphs to multilayer networks. They have yieldedinsights in a diverse variety of applications, including brain networks [183], granularmaterials [129], political voting networks [113, 117], disease spreading [158], andecology and animal behavior [45, 139]. To assist with such applications, there areefforts to develop and analyze multilayer random-network models that incorporaterich and flexible structures [11], such as diverse types of interlayer correlations. Activity-driven (AD) models of temporal networks [136] are a popular family ofgenerative models that encode instantaneous time-dependent descriptions of net-work dynamics through a function called an “activity potential”, which encodes themechanism to generate connections and characterizes the interactions between enti- There is also much research on generalizing centralities (including eigenvector-based centralities[177]) to other types of multilayer networks, such as multiplex networks [4, 90]. Nodes that are good authorities tend to have good hubs that point to them, and nodes that aregood hubs tend to point to good authorities. A major open problem in multilayer network analysis is the measurement and/or inference ofvalues of ω (and generalizations of it in the form of coupling tensors) [140].onlinearity + Networks: A 2020 Vision 7 ties in a network. An activity potential encapsulates all of the information about thetemporal network dynamics of an AD model, making it tractable to study dynamicalprocesses (such as ones from Section 4) on networks that are generated by such amodel. It is also common to compare the properties of networks that are generatedby AD models to those of empirical temporal networks [74].In the original AD model of Perra et al. [136], one considers a network with N entities, which we encode by the nodes. We suppose that node i has an activity rate a i = η x i , which gives the probability per unit time to create new interactions withother nodes. The scaling factor η ensures that the mean number of active nodes perunit time is η (cid:104) x (cid:105) N , where (cid:104) x (cid:105) = N (cid:205) Ni = x i . We define the activity rates such that x i ∈ [ (cid:15), ] , where (cid:15) >
0, and we assign each x i from a probability distribution F ( x ) that can either take a desired functional form or be constructed from empirical data.The model uses the following generative process:• At each discrete time step (of length ∆ t ), start with a network G t that consists of N isolated nodes.• With a probability a i ∆ t that is independent of other nodes, node i is activeand generates m edges, each of which attaches to other nodes uniformly (i.e.,with the same probability for each node) and independently at random (withoutreplacement). Nodes that are not active can still receive edges from active nodes.• At the next time step t + ∆ t , we delete all edges from G t , so all interactions havea constant duration of ∆ t . We then generate new interactions from scratch. Thisis convenient, as it allows one to apply techniques from Markov chains.Because entities in time step t do not have any memory of previous time steps, F ( x ) encodes the network structure and dynamics.The AD model of Perra et al. [136] is overly simplistic, but it is amenable toanalysis and has provided a foundation for many more general AD models, includingones that incorporate memory [200]. In Section 6.4, I discuss a generalization ofAD models to simplicial complexes [137] that allows one to study instantaneousinteractions that involve three or more entities in a network. Many networked systems evolve continuously in time, but most investigations oftime-dependent networks rely on discrete or discretized time. It is important toundertake more analysis of continuous-time temporal networks.Researchers have examined continuous-time networks in a variety of scenarios.Examples include a compartmental model of biological contagions [185], a gen-eralization of Katz centrality to continuous time [65], generalizations of AD mod-els (see Section 3.1.3) to continuous time [198, 199], and rankings in competitivesports [115].In a recent paper [2], my collaborators and I formulated a notion of “tie-decaynetworks” for studying networks that evolve in continuous time. They distinguished
Mason A. Porter between interactions , which they modeled as discrete contacts, and ties , whichencode relationships and their strength as a function of time. For example, perhapsthe strength of a tie decays exponentially after the most recent interaction. Morerealistically, perhaps the decay rate depends on the weight of a tie, with strong tiesdecaying more slowly than weak ones. One can also use point-process models likeHawkes processes [99] to examine similar ideas using a node-centric perspective.Suppose that there are N interacting entities, and let B ( t ) be the N × N time-dependent, real, non-negative matrix whose entries b ij ( t ) encode the tie strengthbetween agents i and j at time t . In [2], we made the following simplifying assump-tions:1. As in [81], ties decay exponentially when there are no interactions: d b ij d t = − α b ij ,where α ≥ t = τ , the strength of the tie between them growsinstantaneously by 1.See [201] for a comparison of various choices, including those in [2] and [81], fortie evolution over time.In practice (e.g., in data-driven applications), one obtains B ( t ) by discretizingtime, so let’s suppose that there is at most one interaction during each time step oflength ∆ t . This occurs, for example, in a Poisson process. Such time discretizationis common in the simulation of stochastic dynamical systems, such as in Gillespiealgorithms [41,142,189]. Consider an N × N matrix A ( t ) in which a ij ( t ) = i interacts with node j at time t and a ij ( t ) = A ( t ) has exactly one nonzero entry during each time step when there is an interactionand no nonzero entries when there isn’t one. For an undirected network, because ofthe symmetric nature of interactions, there are exactly two nonzero entries in timesteps that include an interaction. We write B ( t + ∆ t ) = e − α ∆ t B ( t ) + A ( t + ∆ t ) . (1)Equivalently, if interactions between entities occur at times τ ( (cid:96) ) such that 0 ≤ τ ( ) <τ ( ) < . . . < τ ( T ) , then at time t ≥ τ ( T ) , we have B ( t ) = T (cid:213) k = e − α ( t − τ ( k ) ) A ( τ ( k ) ) . (2)In [2], my coauthors and I generalized PageRank [20, 56] to tie-decay networks.One nice feature of their tie-decay PageRank is that it is applicable not just todata sets, but also to data streams, as one updates the PageRank values as new dataarrives. By contrast, one problematic feature of many methods that rely on multilayerrepresentations of temporal networks is that one needs to recompute everything foran entire data set upon acquiring new data, rather than updating prior results in acomputationally efficient way. onlinearity + Networks: A 2020 Vision 9 A dynamical process can be discrete, continuous, or some mixture of the two; it canalso be either deterministic or stochastic. It can take the form of one or several coupledordinary differential equations (ODEs), partial differential equations (PDEs), maps,stochastic differential equations, and so on.A dynamical process requires a rule for updating the states of its dependentvariables with respect one or more independent variables (e.g., time), and one alsohas (one or a variety of) initial conditions and/or boundary conditions. To formalizea dynamical process on a network, one needs a rule for how to update the states ofthe nodes and/or edges.The nodes (of one or more types) of a network are connected to each other innontrivial ways by one or more types of edges. This leads to a natural question:How does nontrivial connectivity between nodes affect dynamical processes on anetwork [142]? When studying a dynamical process on a network, the networkstructure encodes which entities (i.e., nodes) of a system interact with each otherand which do not. If desired, one can ignore the network structure entirely andjust write out a dynamical system. However, keeping track of network structure isoften a very useful and insightful form of bookkeeping, which one can exploit tosystematically explore how particular structures affect the dynamics of particulardynamical processes.Prominent examples of dynamical processes on networks include coupled oscilla-tors [6,149], games [78], and the spread of diseases [89,130] and opinions [23,100].There is also a large body of research on the control of dynamical processes onnetworks [103, 116].Most studies of dynamics on networks have focused on extending familiar models— such as compartmental models of biological contagions [89] or Kuramoto phaseoscillators [149] — by coupling entities using various types of network structures,but it is also important to formulate new dynamical processes from scratch, ratherthan only studying more complicated generalizations of our favorite models. Whentrying to illuminate the effects of network structure on a dynamical process, it isoften insightful to provide a baseline comparison by examining the process on aconvenient ensemble of random networks [142].
A simple, but illustrative, dynamical process on a network is the Watts thresholdmodel (WTM) of a social contagion [100,142]. It provides a framework for illustrat-ing how network structure can affect state changes, such as the adoption of a productor a behavior, and for exploring which scenarios lead to “virality” (in the form ofstate changes of a large number of nodes in a network).The original WTM [194], a binary-state threshold model that resembles bootstrappercolation [24], has a deterministic update rule, so stochasticity can come only from other sources (see Section 4.2). In a binary state model, each node is in oneof two states; see [55] for a tabulation of well-known binary-state dynamics onnetworks. The WTM is a modification of Mark Granovetter’s threshold model forsocial influence in a fully-mixed population [62]. See [86, 186] for early work onthreshold models on networks that developed independently from investigations ofthe WTM. Threshold contagion models have been developed for many scenarios,including contagions with multiple stages [109], models with adoption latency [124],models with synergistic interactions [83], and situations with hipsters (who mayprefer to adopt a minority state) [84].In a binary-state threshold model such as the WTM, each node i has a threshold R i that one draws from some distribution. Suppose that R i is constant in time, althoughone can generalize it to be time-dependent. At any time, each node can be in oneof two states: 0 (which represents being inactive, not adopted, not infected, and soon) or 1 (active, adopted, infected, and so on). A binary-state model is a drasticoversimplification of reality, but the WTM is able to capture two crucial features ofsocial systems [125]: interdependence (an entity’s behavior depends on the behaviorof other entities) and heterogeneity (as nodes with different threshold values behavedifferently). One can assign a seed number or seed fraction of nodes to the active state,and one can choose the initially active nodes either deterministically or randomly.The states of the nodes change in time according to an update rule, which caneither be synchronous (such that it is a map) or asynchronous (e.g., as a discretizationof continuous time) [142]. In the WTM, the update rule is deterministic, so this choiceaffects only how long it takes to reach a steady state; it does not affect the steady stateitself. With a stochastic update rule, the synchronous and asynchronous versions ofostensibly the “same” model can behave in drastically different ways [43]. In theWTM on an undirected network, to update the state of a node, one compares itsfraction s i / k i of active neighbors (where s i is the number of active neighbors and k i is the degree of node i ) to the node’s threshold R i . An inactive node i becomesactive (i.e., it switches from state 0 to state 1) if s i / k i ≥ R i ; otherwise, it staysinactive. The states of nodes in the WTM are monotonic, in the sense that a nodethat becomes active remains active forever. This feature is convenient for derivingaccurate approximations for the global behavior of the WTM using branching-process approximations [55,142] or when analyzing the behavior of the WTM usingtools such as persistent homology [174]. A dynamical process on a network can take the form of a stochastic process [121,142].There are several possible sources of stochasticity: (1) choice of initial condition, (2)choice of which nodes or edges to update (when considering asynchronous updating),(3) the rule for updating nodes or edges, (4) the values of parameters in an updaterule, and (5) selection of particular networks from a random-graph ensemble (i.e.,a probability distribution on graphs). Some or all of these sources of randomness onlinearity + Networks: A 2020 Vision 11 can be present when studying dynamical processes on networks. It is desirable tocompare the sample mean of a stochastic process on a network to an ensembleaverage (i.e., to an expectation over a suitable probability distribution).Prominent examples of stochastic processes on networks include percolation[153], random walks [107], compartment models of biological contagions [89,130],bounded-confidence models with continuous-valued opinions [110], and other opin-ion and voter models [23, 100, 142, 148].
Compartmental models of biological contagions are a topic of intense interest innetwork science [89, 121, 130, 142]. A compartment represents a possible state of anode; examples include susceptible, infected, zombified, vaccinated, and recovered.An update rule determines how a node changes its state from one compartment toanother. One can formulate models with as many compartments as desired [18], butinvestigations of how network structure affects dynamics typically have employedexamples with only two or three compartments [89, 130].Researchers have studied various extensions of compartmental models, conta-gions on multilayer and temporal networks [4, 34, 90], metapopulation models onnetworks [30] for simultaneously studying network connectivity and subpopulationswith different characteristics, non-Markovian contagions on networks for exploringmemory effects [188], and explicit incorporation of individuals with essential soci-etal roles (e.g., health-care workers) [161]. As I discuss in Section 4.4, one can alsoexamine coupling between biological contagions and the spread of information (e.g.,“awareness”) [50, 192]. One can also use compartmental models to study phenom-ena, such as dissemination of ideas on social media [58] and forecasting of politicalelections [190], that are much different from the spread of diseases.One of the most prominent examples of a compartmental model is a susceptible–infected–recovered (SIR) model, which has three compartments. Susceptible nodesare healthy and can become infected, and infected nodes can eventually recover. Thesteady state of the basic SIR model on a network is related to a type of bond percola-tion [63,68,87,181]. There are many variants of SIR models and other compartmentalmodels on networks [89]. See [114] for an illustration using susceptible–infected–susceptible (SIS) models.Suppose that an infection is transmitted from an infected node to a susceptibleneighbor at a rate of λ . The probability of a transmission event on one edge betweenan infected node and a susceptible node in an infinitesimal time interval d t is λ d t .Assuming that all infection events are independent, the probability that a susceptiblenode with s infected neighbors becomes infected (i.e., for a node to transition fromthe S compartment to the I compartment, which represents both being infected andbeing infective) during d t is1 − ( − λ d t ) s → λ s d t as d t → . (3) If an infected node recovers at a constant rate of µ , the probability that it switchesfrom state I to state R in an infinitesimal time interval d t is µ d t . When there is no source of stochasticity, a dynamical process on a network is“deterministic”. A deterministic dynamical system can take the form of a systemof coupled maps, ODEs, PDEs, or something else. As with stochastic systems, thenetwork structure encodes which entities of a system interact with each other andwhich do not.There are numerous interesting deterministic dynamical systems on networks —just incorporate nontrivial connectivity between entities into your favorite deter-ministic model — although it is worth noting that some stochastic features (e.g.,choosing parameter values from a probability distribution or sampling choices ofinitial conditions) can arise in these models.
For concreteness, let’s consider the popular setting of coupled oscillators. Each nodein a network is associated with an oscillator, and we want to examine how networkstructure affects the collective behavior of the coupled oscillators.It is common to investigate various forms of synchronization (a type of coherentbehavior), such that the rhythms of the oscillators adjust to match each other (or tomatch a subset of the oscillators) because of their interactions [138]. A variety ofmethods, such as “master stability functions” [132], have been developed to studythe local stability of synchronized states and their generalizations [6, 142], such ascluster synchrony [133]. Cluster synchrony, which is related to work on “coupled-cellnetworks” [59], uses ideas from computational group theory to find synchronized setsof oscillators that are not synchronized with other sets of synchronized oscillators.Many studies have also examined other types of states, such as “chimera states”[128], in which some oscillators behave coherently but others behave incoherently.(Analogous phenomena sometimes occur in mathematics departments.)A ubiquitous example is coupled Kuramoto oscillators on a network [6, 39, 149],which is perhaps the most common setting for exploring and developing new methodsfor studying coupled oscillators. (In principle, one can then build on these insightsin studies of other oscillatory systems, such as in applications in neuroscience [7].)Coupled Kuramoto oscillators have been used for modeling numerous phenomena,including jetlag [104] and singing in frogs [126]. Indeed, a “Snowbird” (SIAM)conference on applied dynamical systems would not be complete without at leastseveral dozen talks on the Kuramoto model. In the Kuramoto model, each node i hasan associated phase θ i ( t ) ∈ [ , π ) . In the case of “diffusive” coupling between the onlinearity + Networks: A 2020 Vision 13 nodes , the dynamics of the i th node is governed by the equation (cid:219) θ i : = d θ i d t = ω i + N (cid:213) j = b ij a ij sin ( θ j − θ i ) , i ∈ { , . . . , N } , (4)where one typically draws the natural frequency ω i of node i from some distribution g ( ω ) , the scalar a ij is an adjacency-matrix entry of an unweighted network, b ij isthe coupling strength on oscillator i from oscillator j (so b ij a ij is an element of anadjacency matrix W of a weighted network), and f ij ( y ) = sin ( y ) is the couplingfunction, which depends only on the phase difference between oscillators i and j because of the diffusive nature of the coupling.Once one knows the natural frequencies ω i , the model (4) is a deterministicdynamical system, although there have been studies of coupled Kuramoto oscillatorswith additional stochastic terms [60]. Traditional studies of (4) and its generalizationsdraw the natural frequencies from some distribution (e.g., a Gaussian or a compactlysupported distribution), but some studies of so-called “explosive synchronization” (inwhich there is an abrupt phase transition from incoherent oscillators to synchronizedoscillators) have employed deterministic natural frequencies [16,39]. The propertiesof the frequency distribution g ( ω ) have a significant effect on the dynamics of (4).Important features of g ( ω ) include whether it has compact support or not, whetherit is symmetric or asymmetric, and whether it is unimodal or not [149, 170].The model (4) has been generalized in numerous ways. For example, researchershave considered a large variety of coupling functions f ij (including ones that are notdiffusive) and have incorporated an inertia term (cid:220) θ i to yield a second-order Kuramotooscillator at each node [149]. The latter generalization is important for studiesof coupled oscillators and synchronized dynamics in electric power grids [196].Another noteworthy direction is the analysis of Kuramoto model on “graphons”(see, e.g., [108]), an important type of structure that arises in a suitable limit of largenetworks. An increasingly prominent topic in network analysis is the examination of howmultilayer network structures — multiple system components, multiple types ofedges, co-occurrence and coupling of multiple dynamical processes, and so on —affect qualitative and quantitative dynamics [3, 34, 90]. For example, perhaps certaintypes of multilayer structures can induce unexpected instabilities or phase transitionsin certain types of dynamical processes?There are two categories of dynamical processes on multilayer networks: (1) asingle process can occur on a multilayer network; or (2) processes on different layers In this case, linearization yields Laplacian dynamics, which is closely related to a random walkon a network [107].4 Mason A. Porter of a multilayer network can interact with each other [34]. An important example of thefirst category is a random walk, where the relative speeds and probabilities of stepswithin layers versus steps between layers affect the qualitative nature of the dynamics.This, in turn, affects methods (such as community detection [38,80]) that are based onrandom walks, as well as anything else in which the diffusion is relevant [22,36]. Twoother examples of the first category are the spread of information on social media (forwhich there are multiple communication channels, such as Facebook and Twitter)and multimodal transportation systems [51]. For instance, a multilayer networkstructure can induce congestion even when a system without coupling between layersis decongested in each layer independently [1]. Examples of the second categoryof dynamical process are interactions between multiple strains of a disease andinteractions between the spread of disease and the spread of information [49,50,192].Many other examples have been studied [3], including coupling between oscillatordynamics on one layer and a biased random walk on another layer (as a model forneuronal oscillations coupled to blood flow) [122].Numerous interesting phenomena can occur when dynamical systems, such asspreading processes, are coupled to each other [192]. For example, the spreading ofone disease can facilitate infection by another [157], and the spread of awarenessabout a disease can inhibit spread of the disease itself (e.g., if people stay home whenthey are sick) [61]. Interacting spreading processes can also exhibit other fascinatingdynamics, such as oscillations that are induced by multilayer network structures ina biological contagion with multiple modes of transmission [79] and novel types ofphase transitions [34].A major simplification in most work thus far on dynamical processes on multi-layer networks is a tendency to focus on toy models. For example, a typical studyof coupled spreading processes may consider a standard (e.g., SIR) model on eachlayer, and it may draw the connectivity pattern of each layer from the same stan-dard random-graph model (e.g., an Erdős–Rényi model or a configuration model).However, when studying dynamics on multilayer networks, it is particular importantin future work to incorporate heterogeneity in network structure and/or dynamicalprocesses. For instance, diseases spread offline but information spreads both offlineand online, so investigations of coupled information and disease spread ought toconsider fundamentally different types of network structures for the two processes.
Network structures also affect the dynamics of PDEs on networks [8,31,57,77,112].Interesting examples include a study of a Burgers equation on graphs to investigatehow network structure affects the propagation of shocks [112] and investigationsof reaction–diffusion equations and Turing patterns on networks [8, 94]. The latterstudies exploit the rich theory of Laplacian dynamics on graphs (and concomitantideas from spectral graph theory) [107, 187] and examine the addition of nonlinearterms to Laplacians on various types of networks (including multilayer ones). onlinearity + Networks: A 2020 Vision 15
A mathematically oriented thread of research on PDEs on networks has builton ideas from so-called “quantum graphs” [57, 96] to study wave propagation onnetworks through the analysis of “metric graphs”. Metric graphs differ from the usual“combinatorial graphs”, which in other contexts are usually called simply “graphs”. In metric graphs, in addition to nodes and edges, each edge e has a positive length l e ∈ ( , ∞] . For many experimentally relevant scenarios (e.g., in models of circuitsof quantum wires [195]), there is a natural embedding into space, but metric graphsthat are not embedded in space are also appropriate for some applications.As the nomenclature suggests, one can equip a metric graph with a naturalmetric. If a sequence { e j } mj = of edges forms a path, the length of the path is (cid:205) j l j .The distance ρ ( v , v ) between two nodes, v and v , is the minimum path lengthbetween them. We place coordinates along each edge, so we can compute a distancebetween points x and x on a metric graph even when those points are not locatedat nodes. Traditionally, one assumes that the infinite ends (which one can construeas “leads” at infinity, as in scattering theory) of infinite edges have degree 1. It isalso traditional to assume that there is always a positive distance between distinctnodes and that there are no finite-length paths with infinitely many edges. See [96]for further discussion.To study waves on metric graphs, one needs to define operators, such as thenegative second derivative or more general Schrödinger operators. This exploits thefact that there are coordinates for all points on the edges — not only at the nodesthemselves, as in combinatorial graphs. When studying waves on metric graphs, itis also necessary to impose boundary conditions at the nodes [96].Many studies of wave propagation on metric graphs have considered generaliza-tions of nonlinear wave equations, such as the cubic nonlinear Schrödinger (NLS)equation [123] and a nonlinear Dirac equation [154]. The overwhelming majorityof studies in metric graphs (with both linear and nonlinear waves) have focusedon networks with a very small number of nodes, as even small networks yieldvery interesting dynamics. For example, Marzuola and Pelinovsky [106] analyzedsymmetry-breaking and symmetry-preserving bifurcations of standing waves of thecubic NLS on a dumbbell graph (with two rings attached to a central line segmentand Kirchhoff boundary conditions at the nodes). Kairzhan et al. [85] studied thespectral stability of half-soliton standing waves of the cubic NLS equation on bal-anced star graphs. Sobirov et al. [168] studied scattering and transmission at nodesof sine–Gordon solitons on networks (e.g., on a star graph and a small tree).A particularly interesting direction for future work is to study wave dynamicson large metric graphs. This will help extend investigations, as in ODEs and maps,of how network structures affect dynamics on networks to the realm of linear andnonlinear waves. One can readily formulate wave equations on large metric graphs byspecifying relevant boundary conditions and rules at each junction. For example, Jolyet al. [82] recently examined wave propagation of the standard linear wave equationon fractal trees. Because many natural real-life settings are spatially embedded (e.g., Combinatorial graphs, and more general combinatorial objects, are my main focus in this chapter.This subsection is an exception.6 Mason A. Porter wave propagation in granular materials [101,129] and traffic-flow patterns in cities),it will be particularly valuable to examine wave dynamics on (both synthetic andempirical) spatially-embedded networks [9]. Therefore, I anticipate that it will bevery insightful to undertake studies of wave dynamics on networks such as randomgeometric graphs, random neighborhood graphs, and other spatial structures. Akey question in network analysis is how different types of network structure affectdifferent types of dynamical processes [142], and the ability to take a limit as modelsynthetic networks become infinitely large (i.e., a thermodynamic limit) is crucialfor obtaining many key theoretical insights.
Dynamics of networks and dynamics on networks do not occur in isolation; instead,they are coupled to each other. Researchers have studied the coevolution of networkstructure and the states of nodes and/or edges in the context of “adaptive networks”(which are also known as “coevolving networks”) [66,159]. Whether it is sensible tostudy a dynamical process on a time-independent network, a temporal network withfrozen (or no) node or edge states, or an adaptive network depends on the relativetime scales of the dynamics of network structure and the states of nodes and/or edgesof a network. See [142] for a brief discussion.Models in the form of adaptive networks provide a promising mechanistic ap-proach to simultaneously explain both structural features (e.g., degree distributionsand temporal features (e.g., burstiness) of empirical data [5]. Incorporating adapta-tion into conventional models can produce extremely interesting and rich dynamics,such as the spontaneous development of extreme states in opinion models [160].Most studies of adaptive networks that include some analysis (i.e., that go beyondnumerical computations) have employed rather artificial adaption rules for adding,removing, and rewiring edges. This is relevant for mathematical tractability, but itis important to go beyond these limitations by considering more realistic types ofadaptation and coupling between network structure (including multilayer structures,as in [12]) and the states of nodes and edges.
When people are sick, they stay home from work or school. People also form andremove social connections (both online and offline) based on observed opinions andbehaviors. To study these ideas using adaptive networks, researchers have coupledmodels of biological and social contagions with time-dependent networks [100,142].An early example of an adaptive network of disease spreading is the susceptible–infected (SI) model in Gross et al. [67]. In this model, susceptible nodes sometimesrewire their incident edges to “protect themselves”. Suppose that we have an N -node onlinearity + Networks: A 2020 Vision 17 network with a constant number of undirected edges. Each node is either susceptible(i.e., of type S ) or infected (i.e., of type I ). At each time step, and for each edge— so-called “discordant edges” — between nodes of different types, the susceptiblenode becomes infected with probability λ . For each discordant edge, with someprobability κ , the incident susceptible node breaks the edge and rewires to someother susceptible node. This is a “rewire-to-same” mechanism, to use the languagefrom some adaptive opinion models [40, 97]. (In this model, multi-edges and self-edges are not allowed.) During each time step, infected nodes can also recover tobecome susceptible again.Gross et al. [67] studied how the rewiring probability affects the “basic repro-ductive number”, which measures how many secondary infections on average occurfor each primary infection [18, 89, 130]. This scalar quantity determines the sizeof a critical infection probability λ ∗ to maintain a stable epidemic (as determinedtraditionally using linear stability analysis of an endemic state). A high rewiringrate can significantly increase λ ∗ and thereby significantly reduce the prevalence ofa contagion. Although results like these are perhaps intuitively clear, other studiesof contagions on adaptive networks have yielded potentially actionable (and ar-guably nonintuitive) insights. For example, Scarpino et al. [161] demonstrated usingan adaptive compartmental model (along with some empirical evidence) that thespread of a disease can accelerate when individuals with essential societal roles(e.g., health-care workers) become ill and are replaced with healthy individuals. Another type of model with many interesting adaptive variants are opinion models[23, 142], especially in the form of generalizations of classical voter models [148].Voter dynamics were first considered in the 1970s by Clifford and Sudbury [29]as a model for species competition, and the dynamical process that they introducedwas dubbed “the voter model" by Holley and Liggett shortly thereafter [69]. Voterdynamics are fun and are popular to study [148], although it is questionable whetherit is ever possible to genuinely construe voter models as models of voters [44].Holme and Newman [71] undertook an early study of a rewire-to-same adaptivevoter model. Inspired by their research, Durrett et al. [40] compared the dynamicsfrom two different types of rewiring in an adaptive voter model. In each variant oftheir model, one considers an N -node network and supposes that each node is inone of two states. The network structure and the node states coevolve. Pick an edgeuniformly at random. If this edge is discordant, then with probability 1 − κ , one of itsincident nodes adopts the opinion state of the other. Otherwise, with complementaryprobability κ , a rewiring action occurs: one removes the discordant edge, and oneof the associated nodes attaches to a new node either through a rewire-to-samemechanism (choosing uniformly at random among the nodes with the same opinion There are several variants of “the” voter model, depending on choices such as whether one selectsnodes or edges at random, that have substantively different qualitative dynamics [107, 142].8 Mason A. Porter state) or through a “rewire-to-random” mechanism (choosing uniformly at randomamong all nodes). As with the adaptive SI model in [67], self-edges and multi-edgesare not allowed.The models in [40] evolve until there are no discordant edges. There are severalkey questions. Does the system reach a consensus (in which all nodes are in thesame state)? If so, how long does it take to converge to consensus? If not, how manyopinion clusters (each of which is a connected component, perhaps interpretable asan “echo chamber”, of the final network) are there at steady state? How long does ittake to reach this state? The answers and analysis are subtle; they depend on the initialnetwork topology, the initial conditions, and the specific choice of rewiring rule. Aswith other adaptive network models, researchers have developed some nonrigoroustheory (e.g., using mean-field approximations and their generalizations) on adaptivevoter models with simplistic rewiring schemes, but they have struggled to extendthese ideas to models with more realistic rewiring schemes. There are very fewmathematically rigorous results on adaptive voter models, although there do existsome, under various assumptions on initial network structure and edge density [10].Researchers have generalized adaptive voter models to consider more than twoopinion states [163] and more general types of rewiring schemes [105]. As withother adaptive networks, analyzing adaptive opinion models with increasingly di-verse types of rewiring schemes (ideally with a move towards increasing realism)is particularly important. In [97], Yacoub Kureh and I studied a variant of a votermodel with nonlinear rewiring (where the probability that a node rewires or adoptsis a function of how well it “fits in” within its neighborhood), including a “rewire-to-none” scheme to model unfriending and unfollowing in online social networks. It isalso important to study adaptive opinion models with more realistic types of opiniondynamics. A promising example is adaptive generalizations of bounded-confidencemodels (see the introduction of [110] for a brief review of bounded-confidence mod-els), which have continuous opinion states, with nodes interacting either with nodesor with other entities (such as media [21]) whose opinion is sufficiently close to theirs.A recent numerical study examined an adaptive bounded-confidence model [19]; thisis an important direction for future investigations.
It is also interesting to examine how the adaptation of oscillators — including theirintrinsic frequencies and/or the network structure that couples them to each other— affects the collective behavior (e.g., synchronization) of a network of oscillators[149]. Such ideas are useful for exploring mechanistic models of learning in thebrain (e.g., through adaptation of coupling between oscillators to produce a desiredlimit cycle [171]).One nice example is by Skardal et al. [167], who examined an adaptive modelof coupled Kuramoto oscillators as a toy model of learning. First, we write theKuramoto system as onlinearity + Networks: A 2020 Vision 19 d θ i d t = ω i + N (cid:213) j = f ij ( θ j − θ i ) , i ∈ { , . . . , N } , (5)where f ij is a 2 π -periodic function of the phase difference between oscillators i and j . One way to incorporate adaptation is to define an “order parameter” r i (which,in its traditional form, quantifies the amount of coherence of the coupled Kuramotooscillators [149]) for the i th oscillator by r i = N (cid:213) j = b ij a ij e i θ j , i ∈ { , . . . , N } and to consider the following dynamical system:d θ i d t = ω i + λ − D Im ( z i e − i θ i ) , τ d z i d t = r i − z i , T d b ij d t = α + β Re ( r i z ∗ i ) − b ij , (6)where Re ( ζ ) denotes the real part of a quantity ζ and Im ( ζ ) denotes its imaginarypart. In the model (6), λ D denotes the largest positive eigenvalue of the adjacencymatrix A , the variable z i ( t ) is a time-delayed version of r i with time parameter τ (with τ → z i → r i ), and z ∗ i denotes the complex conjugate of z i .One draws the frequencies ω i from some distribution (e.g., a Lorentz distribution,as in [167]), and we recall that b ij is the coupling strength on oscillator i fromoscillator j . The parameter T gives an adaptation time scale, and α ∈ R and β ∈ R are parameters (which one can adjust to study bifurcations). Skardal et al. [167]interpreted scenarios with β > β < β > β < Most studies of networks have focused on networks with pairwise connections, inwhich each edge (unless it is a self-edge, which connects a node to itself) connectsexactly two nodes to each other. However, many interactions — such as playinggames, coauthoring papers and other forms of collaboration, and horse races —often occur between three or more entities of a network. To examine such situations,researchers have increasingly studied “higher-order” structures in networks, as theycan exert a major influence on dynamical processes.
Perhaps the simplest way to account for higher-order structures in networks is togeneralize from graphs to “hypergraphs” [121]. Hypergraphs possess “hyperedges”that encode a connection between on arbitrary number of nodes, such as betweenall coauthors of a paper. This allows one to make important distinctions, such asbetween a k -clique (in which there are pairwise connections between each pair ofnodes in a set of k nodes) and a hyperedge that connects all k of those nodes to eachother, without the need for any pairwise connections.One way to study a hypergraph is as a “bipartite network”, in which nodesof a given type can be adjacent only to nodes of another type. For example, ascientist can be adjacent to a paper that they have written [119], and a legislatorcan be adjacent to a committee on which they sit [144]. It is important to generalizeideas from graph theory to hypergraphs, such as by developing models of randomhypergraphs [25, 26, 52]. Another way to study higher-order structures in networks is to use “simplicial com-plexes” [53, 54, 127]. A simplicial complex is a space that is built from a union ofpoints, edges, triangles, tetrahedra, and higher-dimensional polytopes (see Fig. 1d).Simplicial complexes approximate topological spaces and thereby capture some oftheir properties.A p -dimensional simplex (i.e., a p -simplex ) is a p -dimensional polytope that isthe convex hull of its p + simplicial complex K is a set ofsimplices such that (1) every face of a simplex from S is also in S and (2) the inter-section of any two simplices σ , σ ∈ S is a face of both σ and σ . An increasingsequence K ⊂ K ⊂ · · · ⊂ K l of simplicial complexes forms a filtered simplicialcomplex ; each K i is a subcomplex . As discussed in [127] and references therein, onecan examine the homology of each subcomplex. In studying the homology of a topo-logical space, one computes topological invariants that quantify features of differentdimensions [53]. One studies “persistent homology” (PH) of a filtered simplicialcomplex to quantify the topological structure of a data set (e.g., a point cloud) acrossmultiple scales of such data. The goal of such “topological data analysis” (TDA)is to measure the “shape” of data in the form of connected components, “holes” ofvarious dimensionality, and so on [127]. From the perspective of network analysis,this yields insight into types of large-scale structure that complement traditional ones(such as community structure). See [178] for a friendly, nontechnical introductionto TDA.A natural goal is to generalize ideas from network analysis to simplicial com-plexes. Important efforts include generalizing configuration models of randomgraphs [48] to random simplicial complexes [15, 32]; generalizing well-known net-work growth mechanisms, such as preferential attachment [13]; and developing onlinearity + Networks: A 2020 Vision 21 geometric notions, like curvature, for networks [156]. An important modeling issuewhen studying higher-order network data is the question of when it is more appro-priate (or convenient) to use the formalisms of hypergraphs or simplicial complexes.The computation of PH has yielded insights on a diverse set of models andapplications in network science and complex systems. Examples include granularmaterials [95, 129], functional brain networks [54, 165], quantification of “politicalislands” in voting data [42], percolation theory [169], contagion dynamics [174],swarming and collective behavior [179], chaotic flows in ODEs and PDEs [197],diurnal cycles in tropical cyclones [182], and mathematics education [28]. See theintroduction to [127] for pointers to numerous other applications.Most uses of simplicial complexes in network science and complex systemshave focused on TDA (especially the computation of PH) and its applications [127,131, 155]. In this chapter, however, I focus instead on a somewhat different (andincreasingly popular) topic: the generalization of dynamical processes on and ofnetworks to simplicial complexes to study the effects of higher-order interactionson network dynamics. Simplicial structures influence the collective behavior of thedynamics of coupled entities on networks (e.g., they can lead to novel bifurcations andphase transitions), and they provide a natural approach to analyze p -entity interactionterms, including for p ≥
3, in dynamical systems. Existing work includes researchon linear diffusion dynamics (in the form of Hodge Laplacians, such as in [162])and generalizations of a variety of other popular types of dynamical processes onnetworks. p -body interactions with p ≥ Given the ubiquitous study of coupled Kuramoto oscillators [149], a sensible startingpoint for exploring the impact of simultaneous coupling of three or more oscillatorson a system’s qualitative dynamics is to study a generalized Kuramoto model. Forexample, to include both two-entity (“two-body”) and three-entity interactions in amodel of coupled oscillators on networks, we write [172] (cid:219) x i = f i ( x i ) + (cid:213) j , k W ijk ( x i , x j , x k ) , (7)where f i describes the dynamics of oscillator i and the three-oscillator interactionterm W ijk includes two-oscillator interaction terms W ij ( x i , x j ) as a special case.An example of N coupled Kuramoto oscillators with three-term interactionsis [172] (cid:219) θ i = ω i + N (cid:213) j (cid:2) a ij sin ( θ ji + α ij ) + b ij sin ( θ ji + α ij ) (cid:3) + N (cid:213) j , k (cid:2) c ijk sin ( θ ji + α ijk ) cos ( θ ki + α ijk ) (cid:3) , (8)where we draw the coefficients a ij , b ij , c ijk , α ij , α ij , α ijk , α ijk from variousprobability distributions. Including three-body interactions leads to a large varietyof intricate dynamics, and I anticipate that incorporating the formalism of simplicialcomplexes will be very helpful for categorizing the possible dynamics.In the last few years, several other researchers have also studied Kuramoto mod-els with three-body interactions [92, 93, 166]. A recent study [166], for example,discovered a continuum of abrupt desynchronization transitions with no counterpartin abrupt synchronization transitions. There have been mathematical studies of cou-pled oscillators with interactions of three or more entities using methods such asnormal-form theory [14] and coupled-cell networks [59].An important point, as one can see in the above discussion (which does notemploy the mathematical formalism of simplicial complexes), is that one does notnecessarily need to explicitly use the language of simplicial complexes to studyinteractions between three or more entities in dynamical systems. Nevertheless, Ianticipate that explicitly incorporating the formalism of simplicial complexes willbe useful both for studying coupled oscillators on networks and for other dynamicalsystems. In upcoming studies, it will be important to determine when this formalismhelps illuminate the dynamics of multi-entity interactions in dynamical systems andwhen simpler approaches suffice. Several recent papers have generalized models of social dynamics by incorporatinghigher-order interactions [75, 76, 118, 137]. For example, perhaps somebody’s opin-ion is influenced by a group discussion of three or more people, so it is relevantto consider opinion updates that are based on higher-order interactions. Some ofthese papers use some of the terminology of simplicial complexes, but it is mostlyunclear (except perhaps for [75]) how the models in them take advantage of the asso-ciated mathematical formalism, so arguably it often may be unnecessary to use suchlanguage. Nevertheless, these models are very interesting and provide promisingavenues for further research.Petri and Barrat [137] generalized activity-driven models to simplicial complexes.Such a simplicial activity-driven (SAD) model generates time-dependent simplicialcomplexes, on which it is desirable to study dynamical processes (see Section 4),such as opinion dynamics, social contagions, and biological contagions.The simplest version of the SAD model is defined as follows. onlinearity + Networks: A 2020 Vision 23 • Each node i has an activity rate a i that we draw independently from a distribution F ( x ) .• At each discrete time step (of length ∆ t ), we start with N isolated nodes. Eachnode i is active with a probability of a i ∆ t , independently of all other nodes. Ifit is active, it creates a ( p − ) -simplex (forming, in network terms, a clique of p nodes) with p − p or draw p from some probability distribution.• At the next time step, we delete all edges, so all interactions have a constantduration. We then generate new interactions from scratch.This version of the SAD model is Markovian, and it is desirable to generalize it invarious ways (e.g., by incorporating memory or community structure).Iacopini et al. [76] recently developed a simplicial contagion model that general-izes an SI process on graphs. Consider a simplicial complex K with N nodes, andassociate each node i with a state x i ( t ) ∈ { , } at time t . If x i ( t ) =
0, node i is partof the susceptible class S ; if x i ( t ) =
1, it is part of the infected class I . The density ofinfected nodes at time t is ρ ( t ) = N (cid:205) Ni = x i ( t ) . Suppose that there are D parameters (cid:36) , . . . , (cid:36) D (with D ∈ { , . . . , N − } ), where (cid:36) d represents the probability perunit time that a susceptible node i that participates in a d -dimensional simplex σ isinfected from each of the faces of σ , under the condition that all of the other nodesof the face are infected. That is, (cid:36) is the probability per unit time that node i isinfected by an adjacent node j via the edge ( i , j ) . Similarly, (cid:36) is the probability perunit time that node i is infected via the 2-simplex ( i , j , k ) in which both j and k areinfected, and so on. The recovery dynamics, in which an infected node i becomessusceptible again, proceeds as in the SIR model that I discussed in Section 4.2. Onecan envision numerous interesting generalizations of this model (e.g., ones that areinspired by ideas that have been investigated in contagion models on graphs). The study of networks is one of the most exciting and rapidly expanding areas ofmathematics, and it touches on myriad other disciplines in both its methodology andits applications. Network analysis is increasingly prominent in numerous fields ofscholarship (both theoretical and applied), it interacts very closely with data science,and it is important for a wealth of applications.My focus in this chapter has been a forward-looking presentation of ideas innetwork analysis. My choices of which ideas to discuss reflect their connections todynamics and nonlinearity, although I have also mentioned a few other burgeoningareas of network analysis in passing. Through its exciting combination of graph the-ory, dynamical systems, statistical mechanics, probability, linear algebra, scientificcomputation, data analysis, and many other subjects — and through a comparablediversity of applications across the sciences, engineering, and the humanities — themathematics and science of networks has plenty to offer researchers for many years.
Acknowledgements
I thank Jesús Cuevas, Panos Kevrekidis, and Avadh Saxena for the invitation towrite this book chapter. I acknowledge financial support from the National ScienceFoundation (grant number 1922952) through the Algorithms for Threat Detection(ATD) program. I thank Mariano Beguerisse Díaz, Manlio De Domenico, JamesGleeson, Petter Holme, Renaud Lambiotte, and Hiroki Sayama for their helpfulcomments. I am particularly grateful to Heather Brooks, Michelle Feng, PanosKevrekidis, and Alice Schwarze for their thorough and insightful comments ondrafts of this chapter.
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